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<h1 class="ltx_title ltx_title_document">Predicting is not Understanding:
<br class="ltx_break">Recognizing and Addressing Underspecification
<br class="ltx_break">in Machine Learning</h1>
<div class="ltx_authors">
<span class="ltx_creator ltx_role_author">
<span class="ltx_personname">Damien Teney<math id="id1.1.m1.2" class="ltx_Math" alttext="{}^{1,\!3}" display="inline"><semantics id="id1.1.m1.2a"><msup id="id1.1.m1.2.2" xref="id1.1.m1.2.2.cmml"><mi id="id1.1.m1.2.2a" xref="id1.1.m1.2.2.cmml"></mi><mrow id="id1.1.m1.2.2.2.4" xref="id1.1.m1.2.2.2.3.cmml"><mn id="id1.1.m1.1.1.1.1" xref="id1.1.m1.1.1.1.1.cmml">1</mn><mpadded width="0.275em"><mo id="id1.1.m1.2.2.2.4.1" xref="id1.1.m1.2.2.2.3.cmml">,</mo></mpadded><mn id="id1.1.m1.2.2.2.2" xref="id1.1.m1.2.2.2.2.cmml">3</mn></mrow></msup><annotation-xml encoding="MathML-Content" id="id1.1.m1.2b"><apply id="id1.1.m1.2.2.cmml" xref="id1.1.m1.2.2"><list id="id1.1.m1.2.2.2.3.cmml" xref="id1.1.m1.2.2.2.4"><cn type="integer" id="id1.1.m1.1.1.1.1.cmml" xref="id1.1.m1.1.1.1.1">1</cn><cn type="integer" id="id1.1.m1.2.2.2.2.cmml" xref="id1.1.m1.2.2.2.2">3</cn></list></apply></annotation-xml><annotation encoding="application/x-tex" id="id1.1.m1.2c">{}^{1,\!3}</annotation></semantics></math>  Maxime Peyrard<sup id="id7.7.id1" class="ltx_sup">2</sup>  Ehsan Abbasnejad<sup id="id8.8.id2" class="ltx_sup">3</sup>
<br class="ltx_break"><sup id="id9.9.id3" class="ltx_sup">1</sup>Idiap Research Institute  <sup id="id10.10.id4" class="ltx_sup">2</sup>EPFL  <sup id="id11.11.id5" class="ltx_sup">3</sup>Australian Institute for Machine Learning
</span></span>
</div>

<div class="ltx_abstract">
<h6 class="ltx_title ltx_title_abstract">Abstract</h6>
<p id="id12.id1" class="ltx_p"><span id="id12.id1.1" class="ltx_text">Machine learning (ML) models are typically optimized for their accuracy on a given dataset.
However, this predictive criterion rarely captures all desirable properties of a model, in particular how well it matches a domain expert’s <em id="id12.id1.1.1" class="ltx_emph ltx_font_italic">understanding</em> of a task.
Underspecification <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib12" title="" class="ltx_ref">12</a>]</cite> refers to the existence of multiple models that are indistinguishable in their in-domain accuracy, even though they differ in other desirable properties such as out-of-distribution (OOD) performance.
Identifying these situations is critical for assessing the reliability of ML models.</span></p>
<p id="id13.id2" class="ltx_p"><span id="id13.id2.1" class="ltx_text">We formalize the concept of underspecification and propose a method to identify and partially address it.
We train multiple models with an independence constraint that forces them to implement different functions.
They discover predictive features that are otherwise ignored by standard empirical risk minimization (ERM), which we then distill into a global model with superior OOD performance.
Importantly, we constrain the models to align with the data manifold to ensure that they discover meaningful features.
We demonstrate the method on multiple datasets in computer vision (collages, WILDS-Camelyon17, GQA) and discuss general implications of underspecification.
Most notably, in-domain performance cannot serve for OOD model selection without additional assumptions.</span></p>
</div>
<section id="S1" class="ltx_section">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">1 </span>Introduction</h2>

<div id="S1.p1" class="ltx_para ltx_noindent">
<p id="S1.p1.1" class="ltx_p"><span id="S1.p1.1.1" class="ltx_text ltx_font_bold">Is data all you need?</span> 
A finite set of i.i.d. examples is almost never sufficient to learn a task.
Inductive biases have long been known to be necessary for in-domain generalization <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib49" title="" class="ltx_ref">49</a>, <a href="#bib.bib82" title="" class="ltx_ref">82</a>]</cite>.
Out-of-distribution (OOD)<span id="footnote1" class="ltx_note ltx_role_footnote"><sup class="ltx_note_mark">1</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">1</sup><span class="ltx_tag ltx_tag_note">1</span>In this paper, OOD refers to test data that has undergone a covariate shift <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib67" title="" class="ltx_ref">67</a>]</cite> compared to the training data.</span></span></span>
generalization complicates things further since one also needs to determine which predictive patterns of the training data will remain relevant at test time.
Correlations between inputs and labels that are
important for the task may be indistinguishable from spurious ones that result from dataset-specific artefacts such as selection biases.</p>
</div>
<div id="S1.p2" class="ltx_para ltx_noindent">
<p id="S1.p2.1" class="ltx_p"><span id="S1.p2.1.1" class="ltx_text ltx_font_bold">An example in image recognition.</span>  Image labels are often correlated with objects and the backgrounds they appear in (<em id="S1.p2.1.2" class="ltx_emph ltx_font_italic">e.g</em>.<span id="S1.p2.1.3" class="ltx_text"></span> cars in cities, birds in nature).
Recognizing either often suffice to predict correct labels.
However, robust OOD generalization (<em id="S1.p2.1.4" class="ltx_emph ltx_font_italic">e.g</em>.<span id="S1.p2.1.5" class="ltx_text"></span> correctly labeling images of birds in street scenes) requires to rely on shapes and to ignore the background.
When this requirement cannot be deduced from the data
(because both features leave a similar signature in the joint training distribution),
the task is said to be underspecified.
In this example, the task requires the additional knowledge that labels refer to object shapes rather than background textures <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib20" title="" class="ltx_ref">20</a>]</cite>.
Such knowledge is often task-specific.
For example, the opposite choice of prioritizing color or texture over shape would be sensible for recognizing traffic signs or segmenting medical images.</p>
</div>
<div id="S1.p3" class="ltx_para">
<span id="S1.p3.1" class="ltx_inline-block ltx_framed ltx_framed_rectangle" style="border-color: #000000;">
<span id="S1.p3.1.1" class="ltx_p"><span id="S1.p3.1.1.1" class="ltx_text ltx_font_bold">Underspecification gap</span>:
the difference between the information provided in a dataset and the information required to perform <em id="S1.p3.1.1.2" class="ltx_emph ltx_font_italic">as desired</em> on a task.</span>
</span>
<div class="ltx_pagination ltx_role_newpage"></div>
</div>
<div id="S1.p4" class="ltx_para ltx_noindent">
<p id="S1.p4.1" class="ltx_p">The qualifier “<em id="S1.p4.1.1" class="ltx_emph ltx_font_italic">as desired</em>” captures the fact that different use cases require different properties such as adversarial robustness, interpretability, fairness, or OOD generalization. The latter is the focus of this paper.
Underspecification arises because these properties do not necessarily correlate with the ERM objective <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib77" title="" class="ltx_ref">77</a>]</cite> typically used to train models.</p>
</div>
<div id="S1.p5" class="ltx_para">
<p id="S1.p5.1" class="ltx_p">This paper argues that <span id="S1.p5.1.1" class="ltx_text ltx_font_bold">identifying underspecification</span>
is important for
assessing the reliability of ML models,
their reliance on hidden assumptions,
and for identifying the information missing for OOD robustness.
We identify underspecification by <span id="S1.p5.1.2" class="ltx_text ltx_font_bold">discovering multiple understandings</span> of the data.
We learn multiple predictive models compatible with a given dataset and hypothesis class (low in-domain risk).
We force them to rely on different predictive features by encouraging orthogonality of their input gradients. We also ensure that these features remain semantically meaningful by constraining the input gradients to the data manifold.
Training multiple models stands in contrast with the standard practice of optimizing a single solution to a learning problem – which hides the existence of underspecification.
With our method, we discover predictive features otherwise ignored by standard ERM.
This alone produces candidate models with superior OOD performance.
In addition, we show how to distill selected features from multiple candidate models into one that is robust across a wider range of distribution shifts.
In all cases, a selection strategy must be provided (see Section <a href="#S5" title="5 Practical considerations ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">5</span></a>) such as an OOD validation set, domain expertise, task-specific heuristics, etc.</p>
</div>
<div id="S1.p6" class="ltx_para ltx_noindent">
<p id="S1.p6.1" class="ltx_p"><span id="S1.p6.1.1" class="ltx_text ltx_font_bold">Experiments.</span> 
We apply the method to controlled data (collages <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib66" title="" class="ltx_ref">66</a>, <a href="#bib.bib72" title="" class="ltx_ref">72</a>]</cite>) and computer vision benchmarks (WILDS-Camelyon17 <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib41" title="" class="ltx_ref">41</a>]</cite>, GQA <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib32" title="" class="ltx_ref">32</a>, <a href="#bib.bib37" title="" class="ltx_ref">37</a>]</cite>).
On visual question answering, we show that multiple models can produce similar answers while relying on different visual features (Figure <a href="#S1.F1" title="Figure 1 ‣ 1 Introduction ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">1</span></a>).</p>
</div>
<div id="S1.p7" class="ltx_para ltx_noindent">
<p id="S1.p7.1" class="ltx_p"><span id="S1.p7.1.1" class="ltx_text ltx_font_bold">Implications.</span> 
Our work complements other studies <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib12" title="" class="ltx_ref">12</a>, <a href="#bib.bib47" title="" class="ltx_ref">47</a>]</cite> in formalizing underspecification as a root cause of multiple challenges in ML including shortcut learning, distribution shifts, and even adversarial vulnerabilities (an extreme case of OOD inputs).
Our formalization of underspecification makes it obvious that ID and OOD performance are not necessarily coupled.
Therefore, without further assumptions, in-domain validation performance is not a reliable model selection strategy for OOD performance despite contradictory suggestions made in the literature <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib24" title="" class="ltx_ref">24</a>, <a href="#bib.bib48" title="" class="ltx_ref">48</a>]</cite>.
The prevalence of underspecification <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib12" title="" class="ltx_ref">12</a>]</cite> also suggests that task-specific knowledge and assumptions are often necessary to build robust ML models, since they cannot emerge from simply scaling up data and architectures.</p>
</div>
<div id="S1.p8" class="ltx_para ltx_noindent">
<p id="S1.p8.1" class="ltx_p"><span id="S1.p8.1.1" class="ltx_text ltx_font_bold">Summary of contributions.</span></p>
<ol id="S1.I1" class="ltx_enumerate">
<li id="S1.I1.i1" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_item">1.</span> 
<div id="S1.I1.i1.p1" class="ltx_para">
<p id="S1.I1.i1.p1.1" class="ltx_p">We propose a mathematical framework for quantifying and addressing underspecification.</p>
</div>
</li>
<li id="S1.I1.i2" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_item">2.</span> 
<div id="S1.I1.i2.p1" class="ltx_para">
<p id="S1.I1.i2.p1.1" class="ltx_p">We derive a method to learn a set of models compatible with a given dataset that exhibit distinct OOD behaviour. We force the models to rely on different features (independence objective) that are nonetheless semantically meaningful (on-manifold constraint).
</p>
</div>
</li>
<li id="S1.I1.i3" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_item">3.</span> 
<div id="S1.I1.i3.p1" class="ltx_para">
<p id="S1.I1.i3.p1.1" class="ltx_p">We use the method for (1) highlighting underspecification in given dataset/architecture pairs, and (2) building models with superior OOD performance. We use controlled and real-world datasets (collages <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib66" title="" class="ltx_ref">66</a>, <a href="#bib.bib72" title="" class="ltx_ref">72</a>]</cite>, WILDS-Camelyon17 <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib41" title="" class="ltx_ref">41</a>]</cite>, GQA <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib32" title="" class="ltx_ref">32</a>, <a href="#bib.bib37" title="" class="ltx_ref">37</a>]</cite>).</p>
</div>
</li>
</ol>
</div>
<figure id="S1.F1" class="ltx_figure"><img src="/html/2207.02598/assets/x1.png" id="S1.F1.g1" class="ltx_graphics ltx_centering ltx_img_landscape" width="461" height="127" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span id="S1.F1.3.1.1" class="ltx_text" style="font-size:90%;">Figure 1</span>: </span><span id="S1.F1.4.2" class="ltx_text" style="font-size:90%;">
Example of underspecification in visual question answering.
Our method trains multiple models that each discover different predictive features.
We obtain three models producing identical answers on most training and validation data, even though they rely on different visual clues (evidenced by grad-CAM visualizations over object proposals <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib64" title="" class="ltx_ref">64</a>]</cite>).
Each model reflects a <span id="S1.F1.4.2.1" class="ltx_text ltx_font_bold">different understanding of the task</span> compatible with the data (possibly incomprehensible to humans) which reveals ambiguity in its specification.
</span></figcaption>
</figure>
<div class="ltx_pagination ltx_role_newpage"></div>
</section>
<section id="S2" class="ltx_section">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">2 </span>Related work</h2>

<div id="S2.p1" class="ltx_para ltx_noindent">
<p id="S2.p1.1" class="ltx_p"><span id="S2.p1.1.1" class="ltx_text ltx_font_bold">Underspecification in ML.</span> 
An empirical study by d’Amour <em id="S2.p1.1.2" class="ltx_emph ltx_font_italic">et al</em>.<span id="S2.p1.1.3" class="ltx_text"></span> in <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib12" title="" class="ltx_ref">12</a>]</cite> showed that models trained with different random seeds have wide variations in OOD performance despite similar in-domain performance.
The effect was surprising in its prevalence but it echoed earlier observations <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib47" title="" class="ltx_ref">47</a>]</cite>.
It results from the well-known impossibility of achieving OOD generalization solely through ERM and i.i.d. data <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib4" title="" class="ltx_ref">4</a>, <a href="#bib.bib63" title="" class="ltx_ref">63</a>]</cite>.
Indeed, since this learning objective does not constrain the model’s behaviour outside the training distribution (<em id="S2.p1.1.4" class="ltx_emph ltx_font_italic">i.e</em>.<span id="S2.p1.1.5" class="ltx_text"></span> multiple hypotheses are plausible), additional assumptions/expertise is necessary <em id="S2.p1.1.6" class="ltx_emph ltx_font_italic">e.g</em>.<span id="S2.p1.1.7" class="ltx_text"></span> to tweak the network architecture to the task.
Data-driven OOD generalization is also possible but it requires heterogeneous (non-i.i.d.) data such as multiple environments <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib4" title="" class="ltx_ref">4</a>, <a href="#bib.bib54" title="" class="ltx_ref">54</a>, <a href="#bib.bib11" title="" class="ltx_ref">11</a>]</cite>, counterfactual examples <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib28" title="" class="ltx_ref">28</a>, <a href="#bib.bib72" title="" class="ltx_ref">72</a>]</cite>, or non-stationary data <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib26" title="" class="ltx_ref">26</a>, <a href="#bib.bib33" title="" class="ltx_ref">33</a>, <a href="#bib.bib56" title="" class="ltx_ref">56</a>]</cite>.
The impossibility of OOD generalization from i.i.d. data is intimately related to identifiability in causal discovery <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib6" title="" class="ltx_ref">6</a>]</cite> and non-linear ICA <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib38" title="" class="ltx_ref">38</a>]</cite>, and the impossibility of unsupervised disentanglement in representation learning <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib45" title="" class="ltx_ref">45</a>]</cite>.</p>
</div>
<section id="S2.SS0.SSS0.Px1" class="ltx_paragraph">
<h4 class="ltx_title ltx_title_paragraph">Multiple hypotheses compatible with the data.</h4>

<div id="S2.SS0.SSS0.Px1.p1" class="ltx_para">
<p id="S2.SS0.SSS0.Px1.p1.1" class="ltx_p">The <span id="S2.SS0.SSS0.Px1.p1.1.1" class="ltx_text ltx_font_bold">Rashomon effect</span> <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib9" title="" class="ltx_ref">9</a>]</cite> is almost synonymous with underspecfication but it is agnostic to OOD data.
The Rashomon set <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib16" title="" class="ltx_ref">16</a>]</cite>
is the set of models of a given class whose training loss or in-domain (ID) risk lies below a threshold <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib31" title="" class="ltx_ref">31</a>, <a href="#bib.bib62" title="" class="ltx_ref">62</a>, <a href="#bib.bib65" title="" class="ltx_ref">65</a>]</cite>.
<span id="S2.SS0.SSS0.Px1.p1.1.2" class="ltx_text ltx_font_bold">Predictive multiplicity</span> <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib8" title="" class="ltx_ref">8</a>, <a href="#bib.bib46" title="" class="ltx_ref">46</a>, <a href="#bib.bib58" title="" class="ltx_ref">58</a>]</cite> refers to the existence of multiple models compatible with the data that make conflicting predictions (on ID data again, rather than OOD).
This paper is about models with low ID risk,
yet different OOD behaviour.
We propose a method to <span id="S2.SS0.SSS0.Px1.p1.1.3" class="ltx_text ltx_font_bold">learn multiple models</span> compatible with the data.
Related approaches were described for ensembling <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib60" title="" class="ltx_ref">60</a>]</cite>, interpretability <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib61" title="" class="ltx_ref">61</a>]</cite>, and control of inductive biases <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib72" title="" class="ltx_ref">72</a>]</cite>.
The number of models required in <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib72" title="" class="ltx_ref">72</a>]</cite> was very large.
We solve this issue by constraining models to align with the data manifold.
Other works concerned with the identification of multiple solutions to a learning problem include <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib52" title="" class="ltx_ref">52</a>]</cite>, Bayesian deep learning <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib80" title="" class="ltx_ref">80</a>]</cite>, and classical feature selection <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib25" title="" class="ltx_ref">25</a>]</cite>,
none of which are suitable to large-scale models and datasets.</p>
</div>
</section>
<section id="S2.SS0.SSS0.Px2" class="ltx_paragraph">
<h4 class="ltx_title ltx_title_paragraph">Ensembles.</h4>

<div id="S2.SS0.SSS0.Px2.p1" class="ltx_para">
<p id="S2.SS0.SSS0.Px2.p1.1" class="ltx_p">Our method for combining robust features resembles traditional ensembling, which lowers variance by combining predictors with uncorrelated errors.
In comparison however, we lower prediction bias by selecting features causally related to the target.
The key in our approach is to discover predictive features that are otherwise missed by ERM.</p>
</div>
</section>
<section id="S2.SS0.SSS0.Px3" class="ltx_paragraph">
<h4 class="ltx_title ltx_title_paragraph">Diversity in feature space.</h4>

<div id="S2.SS0.SSS0.Px3.p1" class="ltx_para">
<p id="S2.SS0.SSS0.Px3.p1.1" class="ltx_p">Allen-Zhu and Li <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib2" title="" class="ltx_ref">2</a>]</cite> explained the success of deep ensembles with the use of different features by different networks simply because of different initializations.
Their argument is also based on the ubiquity of underspecification which they call “multi-view structure” of data.
Concurrently to our work, <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib85" title="" class="ltx_ref">85</a>]</cite> used this explanation to build better ensembles by encouraging diversity in <em id="S2.SS0.SSS0.Px3.p1.1.1" class="ltx_emph ltx_font_italic">feature space</em> with a Bayesian framework.
Even more recently, Heljakka <em id="S2.SS0.SSS0.Px3.p1.1.2" class="ltx_emph ltx_font_italic">et al</em>.<span id="S2.SS0.SSS0.Px3.p1.1.3" class="ltx_text"></span> <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib29" title="" class="ltx_ref">29</a>]</cite> showed how to quantify representational (<em id="S2.SS0.SSS0.Px3.p1.1.4" class="ltx_emph ltx_font_italic">i.e</em>.<span id="S2.SS0.SSS0.Px3.p1.1.5" class="ltx_text"></span> feature-wise) multiplicity.</p>
</div>
</section>
<section id="S2.SS0.SSS0.Px4" class="ltx_paragraph">
<h4 class="ltx_title ltx_title_paragraph">Diversity in prediction space.</h4>

<div id="S2.SS0.SSS0.Px4.p1" class="ltx_para">
<p id="S2.SS0.SSS0.Px4.p1.1" class="ltx_p">Concurrently to our work, <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib51" title="" class="ltx_ref">51</a>]</cite> and <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib44" title="" class="ltx_ref">44</a>]</cite>
proposed to learn models that maximally disagree in OOD predictions.
Both learn only two models.
Obviously, neither would do well on our <span id="S2.SS0.SSS0.Px4.p1.1.1" class="ltx_text ltx_font_italic">collage</span> dataset (Section <a href="#S6.SS1" title="6.1 Experiments on controlled data: collages ‣ 6 Experiments ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">6.1</span></a>) which requires discovering 4 predictive signals.
On real vision datasets, our experiments also suggest that many predictive features (<math id="S2.SS0.SSS0.Px4.p1.1.m1.1" class="ltx_Math" alttext="\gg" display="inline"><semantics id="S2.SS0.SSS0.Px4.p1.1.m1.1a"><mo id="S2.SS0.SSS0.Px4.p1.1.m1.1.1" xref="S2.SS0.SSS0.Px4.p1.1.m1.1.1.cmml">≫</mo><annotation-xml encoding="MathML-Content" id="S2.SS0.SSS0.Px4.p1.1.m1.1b"><csymbol cd="latexml" id="S2.SS0.SSS0.Px4.p1.1.m1.1.1.cmml" xref="S2.SS0.SSS0.Px4.p1.1.m1.1.1">much-greater-than</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S2.SS0.SSS0.Px4.p1.1.m1.1c">\gg</annotation></semantics></math>2) can be discovered.</p>
</div>
</section>
</section>
<section id="S3" class="ltx_section">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">3 </span>Formalizing underspecification</h2>

<div id="S3.p1" class="ltx_para">
<p id="S3.p1.11" class="ltx_p">Let us focus on binary classification tasks.
A dataset provides <span id="S3.p1.11.1" class="ltx_text ltx_font_bold">labeled examples</span> <math id="S3.p1.1.m1.1" class="ltx_Math" alttext="\mathcal{D}_{\mathrm{tr}}=\{({\boldsymbol{x}}_{i},y_{i})\}_{i}" display="inline"><semantics id="S3.p1.1.m1.1a"><mrow id="S3.p1.1.m1.1.1" xref="S3.p1.1.m1.1.1.cmml"><msub id="S3.p1.1.m1.1.1.3" xref="S3.p1.1.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p1.1.m1.1.1.3.2" xref="S3.p1.1.m1.1.1.3.2.cmml">𝒟</mi><mi id="S3.p1.1.m1.1.1.3.3" xref="S3.p1.1.m1.1.1.3.3.cmml">tr</mi></msub><mo id="S3.p1.1.m1.1.1.2" xref="S3.p1.1.m1.1.1.2.cmml">=</mo><msub id="S3.p1.1.m1.1.1.1" xref="S3.p1.1.m1.1.1.1.cmml"><mrow id="S3.p1.1.m1.1.1.1.1.1" xref="S3.p1.1.m1.1.1.1.1.2.cmml"><mo stretchy="false" id="S3.p1.1.m1.1.1.1.1.1.2" xref="S3.p1.1.m1.1.1.1.1.2.cmml">{</mo><mrow id="S3.p1.1.m1.1.1.1.1.1.1.2" xref="S3.p1.1.m1.1.1.1.1.1.1.3.cmml"><mo stretchy="false" id="S3.p1.1.m1.1.1.1.1.1.1.2.3" xref="S3.p1.1.m1.1.1.1.1.1.1.3.cmml">(</mo><msub 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xref="S3.p1.1.m1.1.1.1.1.1.1.1.1.3">𝑖</ci></apply><apply id="S3.p1.1.m1.1.1.1.1.1.1.2.2.cmml" xref="S3.p1.1.m1.1.1.1.1.1.1.2.2"><csymbol cd="ambiguous" id="S3.p1.1.m1.1.1.1.1.1.1.2.2.1.cmml" xref="S3.p1.1.m1.1.1.1.1.1.1.2.2">subscript</csymbol><ci id="S3.p1.1.m1.1.1.1.1.1.1.2.2.2.cmml" xref="S3.p1.1.m1.1.1.1.1.1.1.2.2.2">𝑦</ci><ci id="S3.p1.1.m1.1.1.1.1.1.1.2.2.3.cmml" xref="S3.p1.1.m1.1.1.1.1.1.1.2.2.3">𝑖</ci></apply></interval></set><ci id="S3.p1.1.m1.1.1.1.3.cmml" xref="S3.p1.1.m1.1.1.1.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.1.m1.1c">\mathcal{D}_{\mathrm{tr}}=\{({\boldsymbol{x}}_{i},y_{i})\}_{i}</annotation></semantics></math> with <math id="S3.p1.2.m2.1" class="ltx_Math" alttext="{\boldsymbol{x}}\in\mathbb{R}^{d_{\textrm{in}}}" display="inline"><semantics id="S3.p1.2.m2.1a"><mrow id="S3.p1.2.m2.1.1" xref="S3.p1.2.m2.1.1.cmml"><mi id="S3.p1.2.m2.1.1.2" xref="S3.p1.2.m2.1.1.2.cmml">𝒙</mi><mo id="S3.p1.2.m2.1.1.1" xref="S3.p1.2.m2.1.1.1.cmml">∈</mo><msup id="S3.p1.2.m2.1.1.3" xref="S3.p1.2.m2.1.1.3.cmml"><mi id="S3.p1.2.m2.1.1.3.2" xref="S3.p1.2.m2.1.1.3.2.cmml">ℝ</mi><msub id="S3.p1.2.m2.1.1.3.3" xref="S3.p1.2.m2.1.1.3.3.cmml"><mi id="S3.p1.2.m2.1.1.3.3.2" xref="S3.p1.2.m2.1.1.3.3.2.cmml">d</mi><mtext id="S3.p1.2.m2.1.1.3.3.3" xref="S3.p1.2.m2.1.1.3.3.3a.cmml">in</mtext></msub></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.2.m2.1b"><apply id="S3.p1.2.m2.1.1.cmml" xref="S3.p1.2.m2.1.1"><in id="S3.p1.2.m2.1.1.1.cmml" xref="S3.p1.2.m2.1.1.1"></in><ci id="S3.p1.2.m2.1.1.2.cmml" xref="S3.p1.2.m2.1.1.2">𝒙</ci><apply id="S3.p1.2.m2.1.1.3.cmml" xref="S3.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="S3.p1.2.m2.1.1.3.1.cmml" xref="S3.p1.2.m2.1.1.3">superscript</csymbol><ci id="S3.p1.2.m2.1.1.3.2.cmml" xref="S3.p1.2.m2.1.1.3.2">ℝ</ci><apply id="S3.p1.2.m2.1.1.3.3.cmml" xref="S3.p1.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="S3.p1.2.m2.1.1.3.3.1.cmml" xref="S3.p1.2.m2.1.1.3.3">subscript</csymbol><ci id="S3.p1.2.m2.1.1.3.3.2.cmml" xref="S3.p1.2.m2.1.1.3.3.2">𝑑</ci><ci id="S3.p1.2.m2.1.1.3.3.3a.cmml" xref="S3.p1.2.m2.1.1.3.3.3"><mtext mathsize="50%" id="S3.p1.2.m2.1.1.3.3.3.cmml" xref="S3.p1.2.m2.1.1.3.3.3">in</mtext></ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.2.m2.1c">{\boldsymbol{x}}\in\mathbb{R}^{d_{\textrm{in}}}</annotation></semantics></math>, <math id="S3.p1.3.m3.2" class="ltx_Math" alttext="y_{i}\in\{0,\!1\}" display="inline"><semantics id="S3.p1.3.m3.2a"><mrow id="S3.p1.3.m3.2.3" xref="S3.p1.3.m3.2.3.cmml"><msub id="S3.p1.3.m3.2.3.2" xref="S3.p1.3.m3.2.3.2.cmml"><mi id="S3.p1.3.m3.2.3.2.2" xref="S3.p1.3.m3.2.3.2.2.cmml">y</mi><mi id="S3.p1.3.m3.2.3.2.3" xref="S3.p1.3.m3.2.3.2.3.cmml">i</mi></msub><mo id="S3.p1.3.m3.2.3.1" xref="S3.p1.3.m3.2.3.1.cmml">∈</mo><mrow id="S3.p1.3.m3.2.3.3.2" xref="S3.p1.3.m3.2.3.3.1.cmml"><mo stretchy="false" id="S3.p1.3.m3.2.3.3.2.1" xref="S3.p1.3.m3.2.3.3.1.cmml">{</mo><mn id="S3.p1.3.m3.1.1" xref="S3.p1.3.m3.1.1.cmml">0</mn><mpadded width="0.275em"><mo id="S3.p1.3.m3.2.3.3.2.2" xref="S3.p1.3.m3.2.3.3.1.cmml">,</mo></mpadded><mn id="S3.p1.3.m3.2.2" xref="S3.p1.3.m3.2.2.cmml">1</mn><mo stretchy="false" id="S3.p1.3.m3.2.3.3.2.3" xref="S3.p1.3.m3.2.3.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.3.m3.2b"><apply id="S3.p1.3.m3.2.3.cmml" xref="S3.p1.3.m3.2.3"><in id="S3.p1.3.m3.2.3.1.cmml" xref="S3.p1.3.m3.2.3.1"></in><apply id="S3.p1.3.m3.2.3.2.cmml" xref="S3.p1.3.m3.2.3.2"><csymbol cd="ambiguous" id="S3.p1.3.m3.2.3.2.1.cmml" xref="S3.p1.3.m3.2.3.2">subscript</csymbol><ci id="S3.p1.3.m3.2.3.2.2.cmml" xref="S3.p1.3.m3.2.3.2.2">𝑦</ci><ci id="S3.p1.3.m3.2.3.2.3.cmml" xref="S3.p1.3.m3.2.3.2.3">𝑖</ci></apply><set id="S3.p1.3.m3.2.3.3.1.cmml" xref="S3.p1.3.m3.2.3.3.2"><cn type="integer" id="S3.p1.3.m3.1.1.cmml" xref="S3.p1.3.m3.1.1">0</cn><cn type="integer" id="S3.p1.3.m3.2.2.cmml" xref="S3.p1.3.m3.2.2">1</cn></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.3.m3.2c">y_{i}\in\{0,\!1\}</annotation></semantics></math>.
The goal of a learning algorithm is to identify a <span id="S3.p1.11.2" class="ltx_text ltx_font_bold">predictor</span>
<math id="S3.p1.4.m4.1" class="ltx_Math" alttext="f:\leavevmode\nobreak\ \mathbb{R}^{d_{\textrm{in}}}\rightarrow\mathbb{R}" display="inline"><semantics id="S3.p1.4.m4.1a"><mrow id="S3.p1.4.m4.1.1" xref="S3.p1.4.m4.1.1.cmml"><mi id="S3.p1.4.m4.1.1.2" xref="S3.p1.4.m4.1.1.2.cmml">f</mi><mo lspace="0.278em" rspace="0.778em" id="S3.p1.4.m4.1.1.1" xref="S3.p1.4.m4.1.1.1.cmml">:</mo><mrow id="S3.p1.4.m4.1.1.3" xref="S3.p1.4.m4.1.1.3.cmml"><msup id="S3.p1.4.m4.1.1.3.2" xref="S3.p1.4.m4.1.1.3.2.cmml"><mi id="S3.p1.4.m4.1.1.3.2.2" xref="S3.p1.4.m4.1.1.3.2.2.cmml">ℝ</mi><msub id="S3.p1.4.m4.1.1.3.2.3" xref="S3.p1.4.m4.1.1.3.2.3.cmml"><mi id="S3.p1.4.m4.1.1.3.2.3.2" xref="S3.p1.4.m4.1.1.3.2.3.2.cmml">d</mi><mtext id="S3.p1.4.m4.1.1.3.2.3.3" xref="S3.p1.4.m4.1.1.3.2.3.3a.cmml">in</mtext></msub></msup><mo stretchy="false" id="S3.p1.4.m4.1.1.3.1" xref="S3.p1.4.m4.1.1.3.1.cmml">→</mo><mi id="S3.p1.4.m4.1.1.3.3" xref="S3.p1.4.m4.1.1.3.3.cmml">ℝ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.4.m4.1b"><apply id="S3.p1.4.m4.1.1.cmml" xref="S3.p1.4.m4.1.1"><ci id="S3.p1.4.m4.1.1.1.cmml" xref="S3.p1.4.m4.1.1.1">:</ci><ci id="S3.p1.4.m4.1.1.2.cmml" xref="S3.p1.4.m4.1.1.2">𝑓</ci><apply id="S3.p1.4.m4.1.1.3.cmml" xref="S3.p1.4.m4.1.1.3"><ci id="S3.p1.4.m4.1.1.3.1.cmml" xref="S3.p1.4.m4.1.1.3.1">→</ci><apply id="S3.p1.4.m4.1.1.3.2.cmml" xref="S3.p1.4.m4.1.1.3.2"><csymbol cd="ambiguous" id="S3.p1.4.m4.1.1.3.2.1.cmml" xref="S3.p1.4.m4.1.1.3.2">superscript</csymbol><ci id="S3.p1.4.m4.1.1.3.2.2.cmml" xref="S3.p1.4.m4.1.1.3.2.2">ℝ</ci><apply id="S3.p1.4.m4.1.1.3.2.3.cmml" xref="S3.p1.4.m4.1.1.3.2.3"><csymbol cd="ambiguous" id="S3.p1.4.m4.1.1.3.2.3.1.cmml" xref="S3.p1.4.m4.1.1.3.2.3">subscript</csymbol><ci id="S3.p1.4.m4.1.1.3.2.3.2.cmml" xref="S3.p1.4.m4.1.1.3.2.3.2">𝑑</ci><ci id="S3.p1.4.m4.1.1.3.2.3.3a.cmml" xref="S3.p1.4.m4.1.1.3.2.3.3"><mtext mathsize="50%" id="S3.p1.4.m4.1.1.3.2.3.3.cmml" xref="S3.p1.4.m4.1.1.3.2.3.3">in</mtext></ci></apply></apply><ci id="S3.p1.4.m4.1.1.3.3.cmml" xref="S3.p1.4.m4.1.1.3.3">ℝ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.4.m4.1c">f:\leavevmode\nobreak\ \mathbb{R}^{d_{\textrm{in}}}\rightarrow\mathbb{R}</annotation></semantics></math>
to estimate labels<span id="footnote2" class="ltx_note ltx_role_footnote"><sup class="ltx_note_mark">2</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">2</sup><span class="ltx_tag ltx_tag_note">2</span>We define <math id="footnote2.m1.1" class="ltx_Math" alttext="f" display="inline"><semantics id="footnote2.m1.1b"><mi id="footnote2.m1.1.1" xref="footnote2.m1.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="footnote2.m1.1c"><ci id="footnote2.m1.1.1.cmml" xref="footnote2.m1.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="footnote2.m1.1d">f</annotation></semantics></math> to output logits. A binary prediction <math id="footnote2.m2.1" class="ltx_Math" alttext="\hat{y}" display="inline"><semantics id="footnote2.m2.1b"><mover accent="true" id="footnote2.m2.1.1" xref="footnote2.m2.1.1.cmml"><mi id="footnote2.m2.1.1.2" xref="footnote2.m2.1.1.2.cmml">y</mi><mo id="footnote2.m2.1.1.1" xref="footnote2.m2.1.1.1.cmml">^</mo></mover><annotation-xml encoding="MathML-Content" id="footnote2.m2.1c"><apply id="footnote2.m2.1.1.cmml" xref="footnote2.m2.1.1"><ci id="footnote2.m2.1.1.1.cmml" xref="footnote2.m2.1.1.1">^</ci><ci id="footnote2.m2.1.1.2.cmml" xref="footnote2.m2.1.1.2">𝑦</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote2.m2.1d">\hat{y}</annotation></semantics></math> is obtained as <math id="footnote2.m3.3" class="ltx_Math" alttext="\hat{y}=\operatorname{round}\big{(}\sigma\big{(}f({\boldsymbol{x}})\big{)}\big{)}" display="inline"><semantics id="footnote2.m3.3b"><mrow id="footnote2.m3.3.3" xref="footnote2.m3.3.3.cmml"><mover accent="true" id="footnote2.m3.3.3.3" xref="footnote2.m3.3.3.3.cmml"><mi id="footnote2.m3.3.3.3.2" xref="footnote2.m3.3.3.3.2.cmml">y</mi><mo id="footnote2.m3.3.3.3.1" xref="footnote2.m3.3.3.3.1.cmml">^</mo></mover><mo id="footnote2.m3.3.3.2" xref="footnote2.m3.3.3.2.cmml">=</mo><mrow id="footnote2.m3.3.3.1.1" xref="footnote2.m3.3.3.1.2.cmml"><mi id="footnote2.m3.2.2" xref="footnote2.m3.2.2.cmml">round</mi><mo id="footnote2.m3.3.3.1.1b" xref="footnote2.m3.3.3.1.2.cmml">⁡</mo><mrow id="footnote2.m3.3.3.1.1.1" xref="footnote2.m3.3.3.1.2.cmml"><mo maxsize="120%" minsize="120%" id="footnote2.m3.3.3.1.1.1.2" xref="footnote2.m3.3.3.1.2.cmml">(</mo><mrow id="footnote2.m3.3.3.1.1.1.1" xref="footnote2.m3.3.3.1.1.1.1.cmml"><mi id="footnote2.m3.3.3.1.1.1.1.3" xref="footnote2.m3.3.3.1.1.1.1.3.cmml">σ</mi><mo lspace="0em" rspace="0em" id="footnote2.m3.3.3.1.1.1.1.2" xref="footnote2.m3.3.3.1.1.1.1.2.cmml">​</mo><mrow id="footnote2.m3.3.3.1.1.1.1.1.1" xref="footnote2.m3.3.3.1.1.1.1.1.1.1.cmml"><mo maxsize="120%" minsize="120%" id="footnote2.m3.3.3.1.1.1.1.1.1.2" xref="footnote2.m3.3.3.1.1.1.1.1.1.1.cmml">(</mo><mrow id="footnote2.m3.3.3.1.1.1.1.1.1.1" xref="footnote2.m3.3.3.1.1.1.1.1.1.1.cmml"><mi id="footnote2.m3.3.3.1.1.1.1.1.1.1.2" xref="footnote2.m3.3.3.1.1.1.1.1.1.1.2.cmml">f</mi><mo lspace="0em" rspace="0em" id="footnote2.m3.3.3.1.1.1.1.1.1.1.1" xref="footnote2.m3.3.3.1.1.1.1.1.1.1.1.cmml">​</mo><mrow id="footnote2.m3.3.3.1.1.1.1.1.1.1.3.2" xref="footnote2.m3.3.3.1.1.1.1.1.1.1.cmml"><mo stretchy="false" id="footnote2.m3.3.3.1.1.1.1.1.1.1.3.2.1" xref="footnote2.m3.3.3.1.1.1.1.1.1.1.cmml">(</mo><mi id="footnote2.m3.1.1" xref="footnote2.m3.1.1.cmml">𝒙</mi><mo stretchy="false" id="footnote2.m3.3.3.1.1.1.1.1.1.1.3.2.2" xref="footnote2.m3.3.3.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo maxsize="120%" minsize="120%" id="footnote2.m3.3.3.1.1.1.1.1.1.3" xref="footnote2.m3.3.3.1.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo maxsize="120%" minsize="120%" id="footnote2.m3.3.3.1.1.1.3" xref="footnote2.m3.3.3.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="footnote2.m3.3c"><apply id="footnote2.m3.3.3.cmml" xref="footnote2.m3.3.3"><eq id="footnote2.m3.3.3.2.cmml" xref="footnote2.m3.3.3.2"></eq><apply id="footnote2.m3.3.3.3.cmml" xref="footnote2.m3.3.3.3"><ci id="footnote2.m3.3.3.3.1.cmml" xref="footnote2.m3.3.3.3.1">^</ci><ci id="footnote2.m3.3.3.3.2.cmml" xref="footnote2.m3.3.3.3.2">𝑦</ci></apply><apply id="footnote2.m3.3.3.1.2.cmml" xref="footnote2.m3.3.3.1.1"><ci id="footnote2.m3.2.2.cmml" xref="footnote2.m3.2.2">round</ci><apply id="footnote2.m3.3.3.1.1.1.1.cmml" xref="footnote2.m3.3.3.1.1.1.1"><times id="footnote2.m3.3.3.1.1.1.1.2.cmml" xref="footnote2.m3.3.3.1.1.1.1.2"></times><ci id="footnote2.m3.3.3.1.1.1.1.3.cmml" xref="footnote2.m3.3.3.1.1.1.1.3">𝜎</ci><apply id="footnote2.m3.3.3.1.1.1.1.1.1.1.cmml" xref="footnote2.m3.3.3.1.1.1.1.1.1"><times id="footnote2.m3.3.3.1.1.1.1.1.1.1.1.cmml" xref="footnote2.m3.3.3.1.1.1.1.1.1.1.1"></times><ci id="footnote2.m3.3.3.1.1.1.1.1.1.1.2.cmml" xref="footnote2.m3.3.3.1.1.1.1.1.1.1.2">𝑓</ci><ci id="footnote2.m3.1.1.cmml" xref="footnote2.m3.1.1">𝒙</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="footnote2.m3.3d">\hat{y}=\operatorname{round}\big{(}\sigma\big{(}f({\boldsymbol{x}})\big{)}\big{)}</annotation></semantics></math>.</span></span></span>
of examples from a test set <math id="S3.p1.5.m5.1" class="ltx_Math" alttext="\mathcal{D}_{\mathrm{test}}=\{{\boldsymbol{x}}_{i}\}_{i}" display="inline"><semantics id="S3.p1.5.m5.1a"><mrow id="S3.p1.5.m5.1.1" xref="S3.p1.5.m5.1.1.cmml"><msub id="S3.p1.5.m5.1.1.3" xref="S3.p1.5.m5.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p1.5.m5.1.1.3.2" xref="S3.p1.5.m5.1.1.3.2.cmml">𝒟</mi><mi id="S3.p1.5.m5.1.1.3.3" xref="S3.p1.5.m5.1.1.3.3.cmml">test</mi></msub><mo id="S3.p1.5.m5.1.1.2" xref="S3.p1.5.m5.1.1.2.cmml">=</mo><msub id="S3.p1.5.m5.1.1.1" xref="S3.p1.5.m5.1.1.1.cmml"><mrow id="S3.p1.5.m5.1.1.1.1.1" xref="S3.p1.5.m5.1.1.1.1.2.cmml"><mo stretchy="false" id="S3.p1.5.m5.1.1.1.1.1.2" xref="S3.p1.5.m5.1.1.1.1.2.cmml">{</mo><msub id="S3.p1.5.m5.1.1.1.1.1.1" xref="S3.p1.5.m5.1.1.1.1.1.1.cmml"><mi id="S3.p1.5.m5.1.1.1.1.1.1.2" xref="S3.p1.5.m5.1.1.1.1.1.1.2.cmml">𝒙</mi><mi id="S3.p1.5.m5.1.1.1.1.1.1.3" xref="S3.p1.5.m5.1.1.1.1.1.1.3.cmml">i</mi></msub><mo stretchy="false" id="S3.p1.5.m5.1.1.1.1.1.3" xref="S3.p1.5.m5.1.1.1.1.2.cmml">}</mo></mrow><mi id="S3.p1.5.m5.1.1.1.3" xref="S3.p1.5.m5.1.1.1.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.5.m5.1b"><apply id="S3.p1.5.m5.1.1.cmml" xref="S3.p1.5.m5.1.1"><eq id="S3.p1.5.m5.1.1.2.cmml" xref="S3.p1.5.m5.1.1.2"></eq><apply id="S3.p1.5.m5.1.1.3.cmml" xref="S3.p1.5.m5.1.1.3"><csymbol cd="ambiguous" id="S3.p1.5.m5.1.1.3.1.cmml" xref="S3.p1.5.m5.1.1.3">subscript</csymbol><ci id="S3.p1.5.m5.1.1.3.2.cmml" xref="S3.p1.5.m5.1.1.3.2">𝒟</ci><ci id="S3.p1.5.m5.1.1.3.3.cmml" xref="S3.p1.5.m5.1.1.3.3">test</ci></apply><apply id="S3.p1.5.m5.1.1.1.cmml" xref="S3.p1.5.m5.1.1.1"><csymbol cd="ambiguous" id="S3.p1.5.m5.1.1.1.2.cmml" xref="S3.p1.5.m5.1.1.1">subscript</csymbol><set id="S3.p1.5.m5.1.1.1.1.2.cmml" xref="S3.p1.5.m5.1.1.1.1.1"><apply id="S3.p1.5.m5.1.1.1.1.1.1.cmml" xref="S3.p1.5.m5.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.p1.5.m5.1.1.1.1.1.1.1.cmml" xref="S3.p1.5.m5.1.1.1.1.1.1">subscript</csymbol><ci id="S3.p1.5.m5.1.1.1.1.1.1.2.cmml" xref="S3.p1.5.m5.1.1.1.1.1.1.2">𝒙</ci><ci id="S3.p1.5.m5.1.1.1.1.1.1.3.cmml" xref="S3.p1.5.m5.1.1.1.1.1.1.3">𝑖</ci></apply></set><ci id="S3.p1.5.m5.1.1.1.3.cmml" xref="S3.p1.5.m5.1.1.1.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.5.m5.1c">\mathcal{D}_{\mathrm{test}}=\{{\boldsymbol{x}}_{i}\}_{i}</annotation></semantics></math>.
While the input data <math id="S3.p1.6.m6.1" class="ltx_Math" alttext="{\boldsymbol{x}}" display="inline"><semantics id="S3.p1.6.m6.1a"><mi id="S3.p1.6.m6.1.1" xref="S3.p1.6.m6.1.1.cmml">𝒙</mi><annotation-xml encoding="MathML-Content" id="S3.p1.6.m6.1b"><ci id="S3.p1.6.m6.1.1.cmml" xref="S3.p1.6.m6.1.1">𝒙</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.6.m6.1c">{\boldsymbol{x}}</annotation></semantics></math> is typically high-dimensional (<em id="S3.p1.11.3" class="ltx_emph ltx_font_italic">e.g</em>.<span id="S3.p1.11.4" class="ltx_text"></span> vectorized images),
natural data (<em id="S3.p1.11.5" class="ltx_emph ltx_font_italic">e.g</em>.<span id="S3.p1.11.6" class="ltx_text"></span> photographs) occupies only a fraction of the input space assumed to form a low-dimensional <span id="S3.p1.11.7" class="ltx_text ltx_font_bold">manifold</span> <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib81" title="" class="ltx_ref">81</a>]</cite> <math id="S3.p1.7.m7.1" class="ltx_Math" alttext="\mathcal{M}\subset\mathbb{R}^{d_{\textrm{in}}}" display="inline"><semantics id="S3.p1.7.m7.1a"><mrow id="S3.p1.7.m7.1.1" xref="S3.p1.7.m7.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p1.7.m7.1.1.2" xref="S3.p1.7.m7.1.1.2.cmml">ℳ</mi><mo id="S3.p1.7.m7.1.1.1" xref="S3.p1.7.m7.1.1.1.cmml">⊂</mo><msup id="S3.p1.7.m7.1.1.3" xref="S3.p1.7.m7.1.1.3.cmml"><mi id="S3.p1.7.m7.1.1.3.2" xref="S3.p1.7.m7.1.1.3.2.cmml">ℝ</mi><msub id="S3.p1.7.m7.1.1.3.3" xref="S3.p1.7.m7.1.1.3.3.cmml"><mi id="S3.p1.7.m7.1.1.3.3.2" xref="S3.p1.7.m7.1.1.3.3.2.cmml">d</mi><mtext id="S3.p1.7.m7.1.1.3.3.3" xref="S3.p1.7.m7.1.1.3.3.3a.cmml">in</mtext></msub></msup></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.7.m7.1b"><apply id="S3.p1.7.m7.1.1.cmml" xref="S3.p1.7.m7.1.1"><subset id="S3.p1.7.m7.1.1.1.cmml" xref="S3.p1.7.m7.1.1.1"></subset><ci id="S3.p1.7.m7.1.1.2.cmml" xref="S3.p1.7.m7.1.1.2">ℳ</ci><apply id="S3.p1.7.m7.1.1.3.cmml" xref="S3.p1.7.m7.1.1.3"><csymbol cd="ambiguous" id="S3.p1.7.m7.1.1.3.1.cmml" xref="S3.p1.7.m7.1.1.3">superscript</csymbol><ci id="S3.p1.7.m7.1.1.3.2.cmml" xref="S3.p1.7.m7.1.1.3.2">ℝ</ci><apply id="S3.p1.7.m7.1.1.3.3.cmml" xref="S3.p1.7.m7.1.1.3.3"><csymbol cd="ambiguous" id="S3.p1.7.m7.1.1.3.3.1.cmml" xref="S3.p1.7.m7.1.1.3.3">subscript</csymbol><ci id="S3.p1.7.m7.1.1.3.3.2.cmml" xref="S3.p1.7.m7.1.1.3.3.2">𝑑</ci><ci id="S3.p1.7.m7.1.1.3.3.3a.cmml" xref="S3.p1.7.m7.1.1.3.3.3"><mtext mathsize="50%" id="S3.p1.7.m7.1.1.3.3.3.cmml" xref="S3.p1.7.m7.1.1.3.3.3">in</mtext></ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.7.m7.1c">\mathcal{M}\subset\mathbb{R}^{d_{\textrm{in}}}</annotation></semantics></math>.
The dimensionality <math id="S3.p1.8.m8.1" class="ltx_Math" alttext="{d_{\textrm{manifold}}}" display="inline"><semantics id="S3.p1.8.m8.1a"><msub id="S3.p1.8.m8.1.1" xref="S3.p1.8.m8.1.1.cmml"><mi id="S3.p1.8.m8.1.1.2" xref="S3.p1.8.m8.1.1.2.cmml">d</mi><mtext id="S3.p1.8.m8.1.1.3" xref="S3.p1.8.m8.1.1.3a.cmml">manifold</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.p1.8.m8.1b"><apply id="S3.p1.8.m8.1.1.cmml" xref="S3.p1.8.m8.1.1"><csymbol cd="ambiguous" id="S3.p1.8.m8.1.1.1.cmml" xref="S3.p1.8.m8.1.1">subscript</csymbol><ci id="S3.p1.8.m8.1.1.2.cmml" xref="S3.p1.8.m8.1.1.2">𝑑</ci><ci id="S3.p1.8.m8.1.1.3a.cmml" xref="S3.p1.8.m8.1.1.3"><mtext mathsize="70%" id="S3.p1.8.m8.1.1.3.cmml" xref="S3.p1.8.m8.1.1.3">manifold</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.8.m8.1c">{d_{\textrm{manifold}}}</annotation></semantics></math> (<math id="S3.p1.9.m9.1" class="ltx_Math" alttext="&lt;{d_{\textrm{in}}}" display="inline"><semantics id="S3.p1.9.m9.1a"><mrow id="S3.p1.9.m9.1.1" xref="S3.p1.9.m9.1.1.cmml"><mi id="S3.p1.9.m9.1.1.2" xref="S3.p1.9.m9.1.1.2.cmml"></mi><mo id="S3.p1.9.m9.1.1.1" xref="S3.p1.9.m9.1.1.1.cmml">&lt;</mo><msub id="S3.p1.9.m9.1.1.3" xref="S3.p1.9.m9.1.1.3.cmml"><mi id="S3.p1.9.m9.1.1.3.2" xref="S3.p1.9.m9.1.1.3.2.cmml">d</mi><mtext id="S3.p1.9.m9.1.1.3.3" xref="S3.p1.9.m9.1.1.3.3a.cmml">in</mtext></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.p1.9.m9.1b"><apply id="S3.p1.9.m9.1.1.cmml" xref="S3.p1.9.m9.1.1"><lt id="S3.p1.9.m9.1.1.1.cmml" xref="S3.p1.9.m9.1.1.1"></lt><csymbol cd="latexml" id="S3.p1.9.m9.1.1.2.cmml" xref="S3.p1.9.m9.1.1.2">absent</csymbol><apply id="S3.p1.9.m9.1.1.3.cmml" xref="S3.p1.9.m9.1.1.3"><csymbol cd="ambiguous" id="S3.p1.9.m9.1.1.3.1.cmml" xref="S3.p1.9.m9.1.1.3">subscript</csymbol><ci id="S3.p1.9.m9.1.1.3.2.cmml" xref="S3.p1.9.m9.1.1.3.2">𝑑</ci><ci id="S3.p1.9.m9.1.1.3.3a.cmml" xref="S3.p1.9.m9.1.1.3.3"><mtext mathsize="70%" id="S3.p1.9.m9.1.1.3.3.cmml" xref="S3.p1.9.m9.1.1.3.3">in</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.9.m9.1c">&lt;{d_{\textrm{in}}}</annotation></semantics></math>) is known as the <span id="S3.p1.11.8" class="ltx_text ltx_font_bold">intrinsic dimensionality</span> of the data.
Training and test data are drawn from a distribution on this manifold <math id="S3.p1.10.m10.1" class="ltx_Math" alttext="\textrm{P}_{\mathrm{ID}}" display="inline"><semantics id="S3.p1.10.m10.1a"><msub id="S3.p1.10.m10.1.1" xref="S3.p1.10.m10.1.1.cmml"><mtext id="S3.p1.10.m10.1.1.2" xref="S3.p1.10.m10.1.1.2a.cmml">P</mtext><mi id="S3.p1.10.m10.1.1.3" xref="S3.p1.10.m10.1.1.3.cmml">ID</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p1.10.m10.1b"><apply id="S3.p1.10.m10.1.1.cmml" xref="S3.p1.10.m10.1.1"><csymbol cd="ambiguous" id="S3.p1.10.m10.1.1.1.cmml" xref="S3.p1.10.m10.1.1">subscript</csymbol><ci id="S3.p1.10.m10.1.1.2a.cmml" xref="S3.p1.10.m10.1.1.2"><mtext id="S3.p1.10.m10.1.1.2.cmml" xref="S3.p1.10.m10.1.1.2">P</mtext></ci><ci id="S3.p1.10.m10.1.1.3.cmml" xref="S3.p1.10.m10.1.1.3">ID</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.10.m10.1c">\textrm{P}_{\mathrm{ID}}</annotation></semantics></math> (in-domain examples) while unbiased natural data (free of dataset-specific selection biases) is drawn from a distribution <math id="S3.p1.11.m11.1" class="ltx_Math" alttext="\textrm{P}_{\mathrm{OOD}}" display="inline"><semantics id="S3.p1.11.m11.1a"><msub id="S3.p1.11.m11.1.1" xref="S3.p1.11.m11.1.1.cmml"><mtext id="S3.p1.11.m11.1.1.2" xref="S3.p1.11.m11.1.1.2a.cmml">P</mtext><mi id="S3.p1.11.m11.1.1.3" xref="S3.p1.11.m11.1.1.3.cmml">OOD</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p1.11.m11.1b"><apply id="S3.p1.11.m11.1.1.cmml" xref="S3.p1.11.m11.1.1"><csymbol cd="ambiguous" id="S3.p1.11.m11.1.1.1.cmml" xref="S3.p1.11.m11.1.1">subscript</csymbol><ci id="S3.p1.11.m11.1.1.2a.cmml" xref="S3.p1.11.m11.1.1.2"><mtext id="S3.p1.11.m11.1.1.2.cmml" xref="S3.p1.11.m11.1.1.2">P</mtext></ci><ci id="S3.p1.11.m11.1.1.3.cmml" xref="S3.p1.11.m11.1.1.3">OOD</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p1.11.m11.1c">\textrm{P}_{\mathrm{OOD}}</annotation></semantics></math> of typically broader support.</p>
</div>
<div id="S3.p2" class="ltx_para ltx_noindent">
<p id="S3.p2.9" class="ltx_p"><span id="S3.p2.9.1" class="ltx_text ltx_font_bold">Inductive biases</span>  are the properties of a learning algorithm that determine what model <math id="S3.p2.1.m1.1" class="ltx_Math" alttext="f_{\theta^{\star}}" display="inline"><semantics id="S3.p2.1.m1.1a"><msub id="S3.p2.1.m1.1.1" xref="S3.p2.1.m1.1.1.cmml"><mi id="S3.p2.1.m1.1.1.2" xref="S3.p2.1.m1.1.1.2.cmml">f</mi><msup id="S3.p2.1.m1.1.1.3" xref="S3.p2.1.m1.1.1.3.cmml"><mi id="S3.p2.1.m1.1.1.3.2" xref="S3.p2.1.m1.1.1.3.2.cmml">θ</mi><mo id="S3.p2.1.m1.1.1.3.3" xref="S3.p2.1.m1.1.1.3.3.cmml">⋆</mo></msup></msub><annotation-xml encoding="MathML-Content" id="S3.p2.1.m1.1b"><apply id="S3.p2.1.m1.1.1.cmml" xref="S3.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S3.p2.1.m1.1.1.1.cmml" xref="S3.p2.1.m1.1.1">subscript</csymbol><ci id="S3.p2.1.m1.1.1.2.cmml" xref="S3.p2.1.m1.1.1.2">𝑓</ci><apply id="S3.p2.1.m1.1.1.3.cmml" xref="S3.p2.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.p2.1.m1.1.1.3.1.cmml" xref="S3.p2.1.m1.1.1.3">superscript</csymbol><ci id="S3.p2.1.m1.1.1.3.2.cmml" xref="S3.p2.1.m1.1.1.3.2">𝜃</ci><ci id="S3.p2.1.m1.1.1.3.3.cmml" xref="S3.p2.1.m1.1.1.3.3">⋆</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.1.m1.1c">f_{\theta^{\star}}</annotation></semantics></math> is returned for a dataset <math id="S3.p2.2.m2.1" class="ltx_Math" alttext="\mathcal{D}" display="inline"><semantics id="S3.p2.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="S3.p2.2.m2.1.1" xref="S3.p2.2.m2.1.1.cmml">𝒟</mi><annotation-xml encoding="MathML-Content" id="S3.p2.2.m2.1b"><ci id="S3.p2.2.m2.1.1.cmml" xref="S3.p2.2.m2.1.1">𝒟</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.2.m2.1c">\mathcal{D}</annotation></semantics></math> from a hypothesis class
<math id="S3.p2.3.m3.2" class="ltx_Math" alttext="\mathcal{H}=\{f_{\boldsymbol{\theta}},\forall\,{\boldsymbol{\theta}}\}" display="inline"><semantics id="S3.p2.3.m3.2a"><mrow id="S3.p2.3.m3.2.2" xref="S3.p2.3.m3.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p2.3.m3.2.2.4" xref="S3.p2.3.m3.2.2.4.cmml">ℋ</mi><mo id="S3.p2.3.m3.2.2.3" xref="S3.p2.3.m3.2.2.3.cmml">=</mo><mrow id="S3.p2.3.m3.2.2.2.2" xref="S3.p2.3.m3.2.2.2.3.cmml"><mo stretchy="false" id="S3.p2.3.m3.2.2.2.2.3" xref="S3.p2.3.m3.2.2.2.3.cmml">{</mo><msub id="S3.p2.3.m3.1.1.1.1.1" xref="S3.p2.3.m3.1.1.1.1.1.cmml"><mi id="S3.p2.3.m3.1.1.1.1.1.2" xref="S3.p2.3.m3.1.1.1.1.1.2.cmml">f</mi><mi id="S3.p2.3.m3.1.1.1.1.1.3" xref="S3.p2.3.m3.1.1.1.1.1.3.cmml">𝜽</mi></msub><mo id="S3.p2.3.m3.2.2.2.2.4" xref="S3.p2.3.m3.2.2.2.3.cmml">,</mo><mrow id="S3.p2.3.m3.2.2.2.2.2" xref="S3.p2.3.m3.2.2.2.2.2.cmml"><mo rspace="0.337em" id="S3.p2.3.m3.2.2.2.2.2.1" xref="S3.p2.3.m3.2.2.2.2.2.1.cmml">∀</mo><mi id="S3.p2.3.m3.2.2.2.2.2.2" xref="S3.p2.3.m3.2.2.2.2.2.2.cmml">𝜽</mi></mrow><mo stretchy="false" id="S3.p2.3.m3.2.2.2.2.5" xref="S3.p2.3.m3.2.2.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.p2.3.m3.2b"><apply id="S3.p2.3.m3.2.2.cmml" xref="S3.p2.3.m3.2.2"><eq id="S3.p2.3.m3.2.2.3.cmml" xref="S3.p2.3.m3.2.2.3"></eq><ci id="S3.p2.3.m3.2.2.4.cmml" xref="S3.p2.3.m3.2.2.4">ℋ</ci><set id="S3.p2.3.m3.2.2.2.3.cmml" xref="S3.p2.3.m3.2.2.2.2"><apply id="S3.p2.3.m3.1.1.1.1.1.cmml" xref="S3.p2.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.p2.3.m3.1.1.1.1.1.1.cmml" xref="S3.p2.3.m3.1.1.1.1.1">subscript</csymbol><ci id="S3.p2.3.m3.1.1.1.1.1.2.cmml" xref="S3.p2.3.m3.1.1.1.1.1.2">𝑓</ci><ci id="S3.p2.3.m3.1.1.1.1.1.3.cmml" xref="S3.p2.3.m3.1.1.1.1.1.3">𝜽</ci></apply><apply id="S3.p2.3.m3.2.2.2.2.2.cmml" xref="S3.p2.3.m3.2.2.2.2.2"><csymbol cd="latexml" id="S3.p2.3.m3.2.2.2.2.2.1.cmml" xref="S3.p2.3.m3.2.2.2.2.2.1">for-all</csymbol><ci id="S3.p2.3.m3.2.2.2.2.2.2.cmml" xref="S3.p2.3.m3.2.2.2.2.2.2">𝜽</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.3.m3.2c">\mathcal{H}=\{f_{\boldsymbol{\theta}},\forall\,{\boldsymbol{\theta}}\}</annotation></semantics></math> where <math id="S3.p2.4.m4.1" class="ltx_Math" alttext="f_{\boldsymbol{\theta}}" display="inline"><semantics id="S3.p2.4.m4.1a"><msub id="S3.p2.4.m4.1.1" xref="S3.p2.4.m4.1.1.cmml"><mi id="S3.p2.4.m4.1.1.2" xref="S3.p2.4.m4.1.1.2.cmml">f</mi><mi id="S3.p2.4.m4.1.1.3" xref="S3.p2.4.m4.1.1.3.cmml">𝜽</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p2.4.m4.1b"><apply id="S3.p2.4.m4.1.1.cmml" xref="S3.p2.4.m4.1.1"><csymbol cd="ambiguous" id="S3.p2.4.m4.1.1.1.cmml" xref="S3.p2.4.m4.1.1">subscript</csymbol><ci id="S3.p2.4.m4.1.1.2.cmml" xref="S3.p2.4.m4.1.1.2">𝑓</ci><ci id="S3.p2.4.m4.1.1.3.cmml" xref="S3.p2.4.m4.1.1.3">𝜽</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.4.m4.1c">f_{\boldsymbol{\theta}}</annotation></semantics></math> is a model with free parameters <math id="S3.p2.5.m5.1" class="ltx_Math" alttext="\theta" display="inline"><semantics id="S3.p2.5.m5.1a"><mi id="S3.p2.5.m5.1.1" xref="S3.p2.5.m5.1.1.cmml">θ</mi><annotation-xml encoding="MathML-Content" id="S3.p2.5.m5.1b"><ci id="S3.p2.5.m5.1.1.cmml" xref="S3.p2.5.m5.1.1">𝜃</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.5.m5.1c">\theta</annotation></semantics></math>.
Inductive biases enable generalization from finite data <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib49" title="" class="ltx_ref">49</a>]</cite> by encoding assumptions on the relation between <math id="S3.p2.6.m6.1" class="ltx_Math" alttext="\mathcal{D}" display="inline"><semantics id="S3.p2.6.m6.1a"><mi class="ltx_font_mathcaligraphic" id="S3.p2.6.m6.1.1" xref="S3.p2.6.m6.1.1.cmml">𝒟</mi><annotation-xml encoding="MathML-Content" id="S3.p2.6.m6.1b"><ci id="S3.p2.6.m6.1.1.cmml" xref="S3.p2.6.m6.1.1">𝒟</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.6.m6.1c">\mathcal{D}</annotation></semantics></math> and <math id="S3.p2.7.m7.1" class="ltx_Math" alttext="\mathcal{D}_{\mathrm{test}}" display="inline"><semantics id="S3.p2.7.m7.1a"><msub id="S3.p2.7.m7.1.1" xref="S3.p2.7.m7.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p2.7.m7.1.1.2" xref="S3.p2.7.m7.1.1.2.cmml">𝒟</mi><mi id="S3.p2.7.m7.1.1.3" xref="S3.p2.7.m7.1.1.3.cmml">test</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p2.7.m7.1b"><apply id="S3.p2.7.m7.1.1.cmml" xref="S3.p2.7.m7.1.1"><csymbol cd="ambiguous" id="S3.p2.7.m7.1.1.1.cmml" xref="S3.p2.7.m7.1.1">subscript</csymbol><ci id="S3.p2.7.m7.1.1.2.cmml" xref="S3.p2.7.m7.1.1.2">𝒟</ci><ci id="S3.p2.7.m7.1.1.3.cmml" xref="S3.p2.7.m7.1.1.3">test</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.7.m7.1c">\mathcal{D}_{\mathrm{test}}</annotation></semantics></math>.
In particular, classical learning theory assumes that <math id="S3.p2.8.m8.1" class="ltx_Math" alttext="\mathcal{D}" display="inline"><semantics id="S3.p2.8.m8.1a"><mi class="ltx_font_mathcaligraphic" id="S3.p2.8.m8.1.1" xref="S3.p2.8.m8.1.1.cmml">𝒟</mi><annotation-xml encoding="MathML-Content" id="S3.p2.8.m8.1b"><ci id="S3.p2.8.m8.1.1.cmml" xref="S3.p2.8.m8.1.1">𝒟</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.8.m8.1c">\mathcal{D}</annotation></semantics></math> and <math id="S3.p2.9.m9.1" class="ltx_Math" alttext="\mathcal{D}_{\mathrm{test}}" display="inline"><semantics id="S3.p2.9.m9.1a"><msub id="S3.p2.9.m9.1.1" xref="S3.p2.9.m9.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.p2.9.m9.1.1.2" xref="S3.p2.9.m9.1.1.2.cmml">𝒟</mi><mi id="S3.p2.9.m9.1.1.3" xref="S3.p2.9.m9.1.1.3.cmml">test</mi></msub><annotation-xml encoding="MathML-Content" id="S3.p2.9.m9.1b"><apply id="S3.p2.9.m9.1.1.cmml" xref="S3.p2.9.m9.1.1"><csymbol cd="ambiguous" id="S3.p2.9.m9.1.1.1.cmml" xref="S3.p2.9.m9.1.1">subscript</csymbol><ci id="S3.p2.9.m9.1.1.2.cmml" xref="S3.p2.9.m9.1.1.2">𝒟</ci><ci id="S3.p2.9.m9.1.1.3.cmml" xref="S3.p2.9.m9.1.1.3">test</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p2.9.m9.1c">\mathcal{D}_{\mathrm{test}}</annotation></semantics></math> contain i.i.d. samples from the same distribution. For completeness, we summarize a standard training workflow.
</p>
</div>
<div id="S3.p3" class="ltx_para">
<span id="S3.p3.1" class="ltx_inline-block ltx_framed ltx_framed_rectangle" style="border-color: #000000;">
<span id="S3.I1" class="ltx_enumerate">
<span id="S3.I1.i1" class="ltx_item" style="list-style-type:none;"><span class="ltx_tag ltx_tag_item">1.</span> 
<span id="S3.I1.i1.p1" class="ltx_para">
<span id="S3.I1.i1.p1.1" class="ltx_p"><span id="S3.I1.i1.p1.1.1" class="ltx_text" style="font-size:90%;">Randomly split the data into training and validation sets: </span><math id="S3.I1.i1.p1.1.m1.1" class="ltx_Math" alttext="\mathcal{D}=\mathcal{D}_{\mathrm{tr}}\cup\mathcal{D}_{\mathrm{val}}" display="inline"><semantics id="S3.I1.i1.p1.1.m1.1a"><mrow id="S3.I1.i1.p1.1.m1.1.1" xref="S3.I1.i1.p1.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" mathsize="90%" id="S3.I1.i1.p1.1.m1.1.1.2" xref="S3.I1.i1.p1.1.m1.1.1.2.cmml">𝒟</mi><mo mathsize="90%" id="S3.I1.i1.p1.1.m1.1.1.1" xref="S3.I1.i1.p1.1.m1.1.1.1.cmml">=</mo><mrow id="S3.I1.i1.p1.1.m1.1.1.3" xref="S3.I1.i1.p1.1.m1.1.1.3.cmml"><msub id="S3.I1.i1.p1.1.m1.1.1.3.2" xref="S3.I1.i1.p1.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" mathsize="90%" id="S3.I1.i1.p1.1.m1.1.1.3.2.2" xref="S3.I1.i1.p1.1.m1.1.1.3.2.2.cmml">𝒟</mi><mi mathsize="90%" id="S3.I1.i1.p1.1.m1.1.1.3.2.3" xref="S3.I1.i1.p1.1.m1.1.1.3.2.3.cmml">tr</mi></msub><mo mathsize="90%" id="S3.I1.i1.p1.1.m1.1.1.3.1" xref="S3.I1.i1.p1.1.m1.1.1.3.1.cmml">∪</mo><msub id="S3.I1.i1.p1.1.m1.1.1.3.3" xref="S3.I1.i1.p1.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" mathsize="90%" id="S3.I1.i1.p1.1.m1.1.1.3.3.2" xref="S3.I1.i1.p1.1.m1.1.1.3.3.2.cmml">𝒟</mi><mi mathsize="90%" id="S3.I1.i1.p1.1.m1.1.1.3.3.3" xref="S3.I1.i1.p1.1.m1.1.1.3.3.3.cmml">val</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i1.p1.1.m1.1b"><apply id="S3.I1.i1.p1.1.m1.1.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1"><eq id="S3.I1.i1.p1.1.m1.1.1.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1.1"></eq><ci id="S3.I1.i1.p1.1.m1.1.1.2.cmml" xref="S3.I1.i1.p1.1.m1.1.1.2">𝒟</ci><apply id="S3.I1.i1.p1.1.m1.1.1.3.cmml" xref="S3.I1.i1.p1.1.m1.1.1.3"><union id="S3.I1.i1.p1.1.m1.1.1.3.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1.3.1"></union><apply id="S3.I1.i1.p1.1.m1.1.1.3.2.cmml" xref="S3.I1.i1.p1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="S3.I1.i1.p1.1.m1.1.1.3.2.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1.3.2">subscript</csymbol><ci id="S3.I1.i1.p1.1.m1.1.1.3.2.2.cmml" xref="S3.I1.i1.p1.1.m1.1.1.3.2.2">𝒟</ci><ci id="S3.I1.i1.p1.1.m1.1.1.3.2.3.cmml" xref="S3.I1.i1.p1.1.m1.1.1.3.2.3">tr</ci></apply><apply id="S3.I1.i1.p1.1.m1.1.1.3.3.cmml" xref="S3.I1.i1.p1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="S3.I1.i1.p1.1.m1.1.1.3.3.1.cmml" xref="S3.I1.i1.p1.1.m1.1.1.3.3">subscript</csymbol><ci id="S3.I1.i1.p1.1.m1.1.1.3.3.2.cmml" xref="S3.I1.i1.p1.1.m1.1.1.3.3.2">𝒟</ci><ci id="S3.I1.i1.p1.1.m1.1.1.3.3.3.cmml" xref="S3.I1.i1.p1.1.m1.1.1.3.3.3">val</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i1.p1.1.m1.1c">\mathcal{D}=\mathcal{D}_{\mathrm{tr}}\cup\mathcal{D}_{\mathrm{val}}</annotation></semantics></math><span id="S3.I1.i1.p1.1.2" class="ltx_text" style="font-size:90%;">.</span></span>
</span></span>
<span id="S3.I1.i2" class="ltx_item" style="list-style-type:none;"><span class="ltx_tag ltx_tag_item">2.</span> 
<span id="S3.I1.i2.p1" class="ltx_para">
<span id="S3.I1.i2.p1.2" class="ltx_p"><span id="S3.I1.i2.p1.2.1" class="ltx_text" style="font-size:90%;">A </span><span id="S3.I1.i2.p1.2.2" class="ltx_text ltx_font_bold" style="font-size:90%;">hypothesis class</span><span id="S3.I1.i2.p1.2.3" class="ltx_text" style="font-size:90%;"> </span><math id="S3.I1.i2.p1.1.m1.2" class="ltx_Math" alttext="\mathcal{H}=\{f_{\boldsymbol{\theta}},\forall\,{\boldsymbol{\theta}}\}" display="inline"><semantics id="S3.I1.i2.p1.1.m1.2a"><mrow id="S3.I1.i2.p1.1.m1.2.2" xref="S3.I1.i2.p1.1.m1.2.2.cmml"><mi class="ltx_font_mathcaligraphic" mathsize="90%" id="S3.I1.i2.p1.1.m1.2.2.4" xref="S3.I1.i2.p1.1.m1.2.2.4.cmml">ℋ</mi><mo mathsize="90%" id="S3.I1.i2.p1.1.m1.2.2.3" xref="S3.I1.i2.p1.1.m1.2.2.3.cmml">=</mo><mrow id="S3.I1.i2.p1.1.m1.2.2.2.2" xref="S3.I1.i2.p1.1.m1.2.2.2.3.cmml"><mo maxsize="90%" minsize="90%" id="S3.I1.i2.p1.1.m1.2.2.2.2.3" xref="S3.I1.i2.p1.1.m1.2.2.2.3.cmml">{</mo><msub id="S3.I1.i2.p1.1.m1.1.1.1.1.1" xref="S3.I1.i2.p1.1.m1.1.1.1.1.1.cmml"><mi mathsize="90%" id="S3.I1.i2.p1.1.m1.1.1.1.1.1.2" xref="S3.I1.i2.p1.1.m1.1.1.1.1.1.2.cmml">f</mi><mi mathsize="90%" id="S3.I1.i2.p1.1.m1.1.1.1.1.1.3" xref="S3.I1.i2.p1.1.m1.1.1.1.1.1.3.cmml">𝜽</mi></msub><mo mathsize="90%" id="S3.I1.i2.p1.1.m1.2.2.2.2.4" xref="S3.I1.i2.p1.1.m1.2.2.2.3.cmml">,</mo><mrow id="S3.I1.i2.p1.1.m1.2.2.2.2.2" xref="S3.I1.i2.p1.1.m1.2.2.2.2.2.cmml"><mo mathsize="90%" rspace="0.337em" id="S3.I1.i2.p1.1.m1.2.2.2.2.2.1" xref="S3.I1.i2.p1.1.m1.2.2.2.2.2.1.cmml">∀</mo><mi mathsize="90%" id="S3.I1.i2.p1.1.m1.2.2.2.2.2.2" xref="S3.I1.i2.p1.1.m1.2.2.2.2.2.2.cmml">𝜽</mi></mrow><mo maxsize="90%" minsize="90%" id="S3.I1.i2.p1.1.m1.2.2.2.2.5" xref="S3.I1.i2.p1.1.m1.2.2.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.1.m1.2b"><apply id="S3.I1.i2.p1.1.m1.2.2.cmml" xref="S3.I1.i2.p1.1.m1.2.2"><eq id="S3.I1.i2.p1.1.m1.2.2.3.cmml" xref="S3.I1.i2.p1.1.m1.2.2.3"></eq><ci id="S3.I1.i2.p1.1.m1.2.2.4.cmml" xref="S3.I1.i2.p1.1.m1.2.2.4">ℋ</ci><set id="S3.I1.i2.p1.1.m1.2.2.2.3.cmml" xref="S3.I1.i2.p1.1.m1.2.2.2.2"><apply id="S3.I1.i2.p1.1.m1.1.1.1.1.1.cmml" xref="S3.I1.i2.p1.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.I1.i2.p1.1.m1.1.1.1.1.1.1.cmml" xref="S3.I1.i2.p1.1.m1.1.1.1.1.1">subscript</csymbol><ci id="S3.I1.i2.p1.1.m1.1.1.1.1.1.2.cmml" xref="S3.I1.i2.p1.1.m1.1.1.1.1.1.2">𝑓</ci><ci id="S3.I1.i2.p1.1.m1.1.1.1.1.1.3.cmml" xref="S3.I1.i2.p1.1.m1.1.1.1.1.1.3">𝜽</ci></apply><apply id="S3.I1.i2.p1.1.m1.2.2.2.2.2.cmml" xref="S3.I1.i2.p1.1.m1.2.2.2.2.2"><csymbol cd="latexml" id="S3.I1.i2.p1.1.m1.2.2.2.2.2.1.cmml" xref="S3.I1.i2.p1.1.m1.2.2.2.2.2.1">for-all</csymbol><ci id="S3.I1.i2.p1.1.m1.2.2.2.2.2.2.cmml" xref="S3.I1.i2.p1.1.m1.2.2.2.2.2.2">𝜽</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.1.m1.2c">\mathcal{H}=\{f_{\boldsymbol{\theta}},\forall\,{\boldsymbol{\theta}}\}</annotation></semantics></math><span id="S3.I1.i2.p1.2.4" class="ltx_text" style="font-size:90%;"> is chosen </span><em id="S3.I1.i2.p1.2.5" class="ltx_emph ltx_font_italic" style="font-size:90%;">e.g</em><span id="S3.I1.i2.p1.2.6" class="ltx_text" style="font-size:90%;">.</span><span id="S3.I1.i2.p1.2.7" class="ltx_text"></span><span id="S3.I1.i2.p1.2.8" class="ltx_text" style="font-size:90%;"> by defining a neural architecture </span><math id="S3.I1.i2.p1.2.m2.1" class="ltx_Math" alttext="f" display="inline"><semantics id="S3.I1.i2.p1.2.m2.1a"><mi mathsize="90%" id="S3.I1.i2.p1.2.m2.1.1" xref="S3.I1.i2.p1.2.m2.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S3.I1.i2.p1.2.m2.1b"><ci id="S3.I1.i2.p1.2.m2.1.1.cmml" xref="S3.I1.i2.p1.2.m2.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i2.p1.2.m2.1c">f</annotation></semantics></math><span id="S3.I1.i2.p1.2.9" class="ltx_text" style="font-size:90%;">.</span></span>
</span></span>
<span id="S3.I1.i3" class="ltx_item" style="list-style-type:none;"><span class="ltx_tag ltx_tag_item">3.</span> 
<span id="S3.I1.i3.p1" class="ltx_para">
<span id="S3.I1.i3.p1.4" class="ltx_p"><span id="S3.I1.i3.p1.4.1" class="ltx_text ltx_font_bold" style="font-size:90%;">Empirical risk minimization</span><span id="S3.I1.i3.p1.4.2" class="ltx_text" style="font-size:90%;"> serves to optimize the free parameters of </span><math id="S3.I1.i3.p1.1.m1.1" class="ltx_Math" alttext="f" display="inline"><semantics id="S3.I1.i3.p1.1.m1.1a"><mi mathsize="90%" id="S3.I1.i3.p1.1.m1.1.1" xref="S3.I1.i3.p1.1.m1.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="S3.I1.i3.p1.1.m1.1b"><ci id="S3.I1.i3.p1.1.m1.1.1.cmml" xref="S3.I1.i3.p1.1.m1.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i3.p1.1.m1.1c">f</annotation></semantics></math><span id="S3.I1.i3.p1.4.3" class="ltx_text" style="font-size:90%;"> as </span><math id="S3.I1.i3.p1.2.m2.3" class="ltx_Math" alttext="{\boldsymbol{\theta}}^{\star}\leavevmode\nobreak\ =\leavevmode\nobreak\ \textrm{argmin}_{\boldsymbol{\theta}}\;\operatorname*{\mathcal{R}}\big{(}f_{\boldsymbol{\theta}},\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\mathcal{D}_{\mathrm{tr}}}\big{)}" display="inline"><semantics id="S3.I1.i3.p1.2.m2.3a"><mrow id="S3.I1.i3.p1.2.m2.3.3" xref="S3.I1.i3.p1.2.m2.3.3.cmml"><msup id="S3.I1.i3.p1.2.m2.3.3.4" xref="S3.I1.i3.p1.2.m2.3.3.4.cmml"><mi mathsize="90%" id="S3.I1.i3.p1.2.m2.3.3.4.2" xref="S3.I1.i3.p1.2.m2.3.3.4.2.cmml">𝜽</mi><mo mathsize="90%" id="S3.I1.i3.p1.2.m2.3.3.4.3" xref="S3.I1.i3.p1.2.m2.3.3.4.3.cmml">⋆</mo></msup><mo lspace="0.728em" mathsize="90%" rspace="0.728em" id="S3.I1.i3.p1.2.m2.3.3.3" xref="S3.I1.i3.p1.2.m2.3.3.3.cmml">=</mo><mrow id="S3.I1.i3.p1.2.m2.3.3.2" xref="S3.I1.i3.p1.2.m2.3.3.2.cmml"><msub id="S3.I1.i3.p1.2.m2.3.3.2.4" xref="S3.I1.i3.p1.2.m2.3.3.2.4.cmml"><mtext mathsize="90%" id="S3.I1.i3.p1.2.m2.3.3.2.4.2" xref="S3.I1.i3.p1.2.m2.3.3.2.4.2a.cmml">argmin</mtext><mi mathsize="90%" id="S3.I1.i3.p1.2.m2.3.3.2.4.3" xref="S3.I1.i3.p1.2.m2.3.3.2.4.3.cmml">𝜽</mi></msub><mo lspace="0.167em" rspace="0em" id="S3.I1.i3.p1.2.m2.3.3.2.3" xref="S3.I1.i3.p1.2.m2.3.3.2.3.cmml">​</mo><mrow id="S3.I1.i3.p1.2.m2.3.3.2.2.2" xref="S3.I1.i3.p1.2.m2.3.3.2.2.3.cmml"><mo class="ltx_font_mathcaligraphic" mathsize="90%" rspace="0em" id="S3.I1.i3.p1.2.m2.1.1" xref="S3.I1.i3.p1.2.m2.1.1.cmml">ℛ</mo><mrow id="S3.I1.i3.p1.2.m2.3.3.2.2.2.2" xref="S3.I1.i3.p1.2.m2.3.3.2.2.3.cmml"><mo maxsize="120%" minsize="120%" id="S3.I1.i3.p1.2.m2.3.3.2.2.2.2.3" xref="S3.I1.i3.p1.2.m2.3.3.2.2.3.cmml">(</mo><msub id="S3.I1.i3.p1.2.m2.2.2.1.1.1.1.1" xref="S3.I1.i3.p1.2.m2.2.2.1.1.1.1.1.cmml"><mi mathsize="90%" id="S3.I1.i3.p1.2.m2.2.2.1.1.1.1.1.2" xref="S3.I1.i3.p1.2.m2.2.2.1.1.1.1.1.2.cmml">f</mi><mi mathsize="90%" id="S3.I1.i3.p1.2.m2.2.2.1.1.1.1.1.3" xref="S3.I1.i3.p1.2.m2.2.2.1.1.1.1.1.3.cmml">𝜽</mi></msub><mo mathsize="90%" id="S3.I1.i3.p1.2.m2.3.3.2.2.2.2.4" xref="S3.I1.i3.p1.2.m2.3.3.2.2.3.cmml">,</mo><msub id="S3.I1.i3.p1.2.m2.3.3.2.2.2.2.2" xref="S3.I1.i3.p1.2.m2.3.3.2.2.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" mathcolor="#0000FF" mathsize="90%" id="S3.I1.i3.p1.2.m2.3.3.2.2.2.2.2.2" xref="S3.I1.i3.p1.2.m2.3.3.2.2.2.2.2.2.cmml">𝒟</mi><mi mathcolor="#0000FF" mathsize="90%" id="S3.I1.i3.p1.2.m2.3.3.2.2.2.2.2.3" xref="S3.I1.i3.p1.2.m2.3.3.2.2.2.2.2.3.cmml">tr</mi></msub><mo mathcolor="#0000FF" maxsize="120%" minsize="120%" id="S3.I1.i3.p1.2.m2.3.3.2.2.2.2.5" xref="S3.I1.i3.p1.2.m2.3.3.2.2.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I1.i3.p1.2.m2.3b"><apply id="S3.I1.i3.p1.2.m2.3.3.cmml" xref="S3.I1.i3.p1.2.m2.3.3"><eq id="S3.I1.i3.p1.2.m2.3.3.3.cmml" xref="S3.I1.i3.p1.2.m2.3.3.3"></eq><apply id="S3.I1.i3.p1.2.m2.3.3.4.cmml" xref="S3.I1.i3.p1.2.m2.3.3.4"><csymbol cd="ambiguous" id="S3.I1.i3.p1.2.m2.3.3.4.1.cmml" xref="S3.I1.i3.p1.2.m2.3.3.4">superscript</csymbol><ci id="S3.I1.i3.p1.2.m2.3.3.4.2.cmml" xref="S3.I1.i3.p1.2.m2.3.3.4.2">𝜽</ci><ci id="S3.I1.i3.p1.2.m2.3.3.4.3.cmml" xref="S3.I1.i3.p1.2.m2.3.3.4.3">⋆</ci></apply><apply id="S3.I1.i3.p1.2.m2.3.3.2.cmml" xref="S3.I1.i3.p1.2.m2.3.3.2"><times id="S3.I1.i3.p1.2.m2.3.3.2.3.cmml" xref="S3.I1.i3.p1.2.m2.3.3.2.3"></times><apply id="S3.I1.i3.p1.2.m2.3.3.2.4.cmml" xref="S3.I1.i3.p1.2.m2.3.3.2.4"><csymbol cd="ambiguous" id="S3.I1.i3.p1.2.m2.3.3.2.4.1.cmml" xref="S3.I1.i3.p1.2.m2.3.3.2.4">subscript</csymbol><ci id="S3.I1.i3.p1.2.m2.3.3.2.4.2a.cmml" xref="S3.I1.i3.p1.2.m2.3.3.2.4.2"><mtext mathsize="90%" id="S3.I1.i3.p1.2.m2.3.3.2.4.2.cmml" xref="S3.I1.i3.p1.2.m2.3.3.2.4.2">argmin</mtext></ci><ci id="S3.I1.i3.p1.2.m2.3.3.2.4.3.cmml" xref="S3.I1.i3.p1.2.m2.3.3.2.4.3">𝜽</ci></apply><apply id="S3.I1.i3.p1.2.m2.3.3.2.2.3.cmml" xref="S3.I1.i3.p1.2.m2.3.3.2.2.2"><ci id="S3.I1.i3.p1.2.m2.1.1.cmml" xref="S3.I1.i3.p1.2.m2.1.1">ℛ</ci><apply id="S3.I1.i3.p1.2.m2.2.2.1.1.1.1.1.cmml" xref="S3.I1.i3.p1.2.m2.2.2.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.I1.i3.p1.2.m2.2.2.1.1.1.1.1.1.cmml" xref="S3.I1.i3.p1.2.m2.2.2.1.1.1.1.1">subscript</csymbol><ci id="S3.I1.i3.p1.2.m2.2.2.1.1.1.1.1.2.cmml" xref="S3.I1.i3.p1.2.m2.2.2.1.1.1.1.1.2">𝑓</ci><ci id="S3.I1.i3.p1.2.m2.2.2.1.1.1.1.1.3.cmml" xref="S3.I1.i3.p1.2.m2.2.2.1.1.1.1.1.3">𝜽</ci></apply><apply id="S3.I1.i3.p1.2.m2.3.3.2.2.2.2.2.cmml" xref="S3.I1.i3.p1.2.m2.3.3.2.2.2.2.2"><csymbol cd="ambiguous" id="S3.I1.i3.p1.2.m2.3.3.2.2.2.2.2.1.cmml" xref="S3.I1.i3.p1.2.m2.3.3.2.2.2.2.2">subscript</csymbol><ci id="S3.I1.i3.p1.2.m2.3.3.2.2.2.2.2.2.cmml" xref="S3.I1.i3.p1.2.m2.3.3.2.2.2.2.2.2">𝒟</ci><ci id="S3.I1.i3.p1.2.m2.3.3.2.2.2.2.2.3.cmml" xref="S3.I1.i3.p1.2.m2.3.3.2.2.2.2.2.3">tr</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i3.p1.2.m2.3c">{\boldsymbol{\theta}}^{\star}\leavevmode\nobreak\ =\leavevmode\nobreak\ \textrm{argmin}_{\boldsymbol{\theta}}\;\operatorname*{\mathcal{R}}\big{(}f_{\boldsymbol{\theta}},\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\mathcal{D}_{\mathrm{tr}}}\big{)}</annotation></semantics></math><span id="S3.I1.i3.p1.4.4" class="ltx_text" style="font-size:90%;">
where the empirical risk is defined as </span><math id="S3.I1.i3.p1.3.m3.9" class="ltx_Math" alttext="\operatorname*{\mathcal{R}}(f,\mathcal{D})=\Sigma_{({\boldsymbol{x}},y)\in\mathcal{D}}\;\mathcal{L}_{\textrm{pred}}\big{(}y,\sigma\big{(}f({\boldsymbol{x}})\big{)}\big{)}\;/\;|\mathcal{D}|" display="inline"><semantics id="S3.I1.i3.p1.3.m3.9a"><mrow id="S3.I1.i3.p1.3.m3.9.9" xref="S3.I1.i3.p1.3.m3.9.9.cmml"><mrow id="S3.I1.i3.p1.3.m3.9.9.3.2" xref="S3.I1.i3.p1.3.m3.9.9.3.1.cmml"><mo class="ltx_font_mathcaligraphic" mathsize="90%" rspace="0em" id="S3.I1.i3.p1.3.m3.3.3" xref="S3.I1.i3.p1.3.m3.3.3.cmml">ℛ</mo><mrow id="S3.I1.i3.p1.3.m3.9.9.3.2.1" xref="S3.I1.i3.p1.3.m3.9.9.3.1.cmml"><mo maxsize="90%" minsize="90%" id="S3.I1.i3.p1.3.m3.9.9.3.2.1.1" xref="S3.I1.i3.p1.3.m3.9.9.3.1.cmml">(</mo><mi mathsize="90%" id="S3.I1.i3.p1.3.m3.4.4" xref="S3.I1.i3.p1.3.m3.4.4.cmml">f</mi><mo mathsize="90%" id="S3.I1.i3.p1.3.m3.9.9.3.2.1.2" xref="S3.I1.i3.p1.3.m3.9.9.3.1.cmml">,</mo><mi class="ltx_font_mathcaligraphic" mathsize="90%" id="S3.I1.i3.p1.3.m3.5.5" xref="S3.I1.i3.p1.3.m3.5.5.cmml">𝒟</mi><mo maxsize="90%" minsize="90%" id="S3.I1.i3.p1.3.m3.9.9.3.2.1.3" xref="S3.I1.i3.p1.3.m3.9.9.3.1.cmml">)</mo></mrow></mrow><mo mathsize="90%" id="S3.I1.i3.p1.3.m3.9.9.2" xref="S3.I1.i3.p1.3.m3.9.9.2.cmml">=</mo><mrow id="S3.I1.i3.p1.3.m3.9.9.1" xref="S3.I1.i3.p1.3.m3.9.9.1.cmml"><mrow id="S3.I1.i3.p1.3.m3.9.9.1.1" xref="S3.I1.i3.p1.3.m3.9.9.1.1.cmml"><msub id="S3.I1.i3.p1.3.m3.9.9.1.1.3" xref="S3.I1.i3.p1.3.m3.9.9.1.1.3.cmml"><mi mathsize="90%" mathvariant="normal" id="S3.I1.i3.p1.3.m3.9.9.1.1.3.2" xref="S3.I1.i3.p1.3.m3.9.9.1.1.3.2.cmml">Σ</mi><mrow id="S3.I1.i3.p1.3.m3.2.2.2" xref="S3.I1.i3.p1.3.m3.2.2.2.cmml"><mrow id="S3.I1.i3.p1.3.m3.2.2.2.4.2" xref="S3.I1.i3.p1.3.m3.2.2.2.4.1.cmml"><mo maxsize="90%" minsize="90%" id="S3.I1.i3.p1.3.m3.2.2.2.4.2.1" xref="S3.I1.i3.p1.3.m3.2.2.2.4.1.cmml">(</mo><mi mathsize="90%" id="S3.I1.i3.p1.3.m3.1.1.1.1" 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xref="S3.I1.i3.p1.3.m3.9.9.2"></eq><apply id="S3.I1.i3.p1.3.m3.9.9.3.1.cmml" xref="S3.I1.i3.p1.3.m3.9.9.3.2"><ci id="S3.I1.i3.p1.3.m3.3.3.cmml" xref="S3.I1.i3.p1.3.m3.3.3">ℛ</ci><ci id="S3.I1.i3.p1.3.m3.4.4.cmml" xref="S3.I1.i3.p1.3.m3.4.4">𝑓</ci><ci id="S3.I1.i3.p1.3.m3.5.5.cmml" xref="S3.I1.i3.p1.3.m3.5.5">𝒟</ci></apply><apply id="S3.I1.i3.p1.3.m3.9.9.1.cmml" xref="S3.I1.i3.p1.3.m3.9.9.1"><divide id="S3.I1.i3.p1.3.m3.9.9.1.2.cmml" xref="S3.I1.i3.p1.3.m3.9.9.1.2"></divide><apply id="S3.I1.i3.p1.3.m3.9.9.1.1.cmml" xref="S3.I1.i3.p1.3.m3.9.9.1.1"><times id="S3.I1.i3.p1.3.m3.9.9.1.1.2.cmml" xref="S3.I1.i3.p1.3.m3.9.9.1.1.2"></times><apply id="S3.I1.i3.p1.3.m3.9.9.1.1.3.cmml" xref="S3.I1.i3.p1.3.m3.9.9.1.1.3"><csymbol cd="ambiguous" id="S3.I1.i3.p1.3.m3.9.9.1.1.3.1.cmml" xref="S3.I1.i3.p1.3.m3.9.9.1.1.3">subscript</csymbol><ci id="S3.I1.i3.p1.3.m3.9.9.1.1.3.2.cmml" xref="S3.I1.i3.p1.3.m3.9.9.1.1.3.2">Σ</ci><apply id="S3.I1.i3.p1.3.m3.2.2.2.cmml" xref="S3.I1.i3.p1.3.m3.2.2.2"><in 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xref="S3.I1.i3.p1.3.m3.9.9.1.1.1.1"><ci id="S3.I1.i3.p1.3.m3.7.7.cmml" xref="S3.I1.i3.p1.3.m3.7.7">𝑦</ci><apply id="S3.I1.i3.p1.3.m3.9.9.1.1.1.1.1.cmml" xref="S3.I1.i3.p1.3.m3.9.9.1.1.1.1.1"><times id="S3.I1.i3.p1.3.m3.9.9.1.1.1.1.1.2.cmml" xref="S3.I1.i3.p1.3.m3.9.9.1.1.1.1.1.2"></times><ci id="S3.I1.i3.p1.3.m3.9.9.1.1.1.1.1.3.cmml" xref="S3.I1.i3.p1.3.m3.9.9.1.1.1.1.1.3">𝜎</ci><apply id="S3.I1.i3.p1.3.m3.9.9.1.1.1.1.1.1.1.1.cmml" xref="S3.I1.i3.p1.3.m3.9.9.1.1.1.1.1.1.1"><times id="S3.I1.i3.p1.3.m3.9.9.1.1.1.1.1.1.1.1.1.cmml" xref="S3.I1.i3.p1.3.m3.9.9.1.1.1.1.1.1.1.1.1"></times><ci id="S3.I1.i3.p1.3.m3.9.9.1.1.1.1.1.1.1.1.2.cmml" xref="S3.I1.i3.p1.3.m3.9.9.1.1.1.1.1.1.1.1.2">𝑓</ci><ci id="S3.I1.i3.p1.3.m3.6.6.cmml" xref="S3.I1.i3.p1.3.m3.6.6">𝒙</ci></apply></apply></interval></apply><apply id="S3.I1.i3.p1.3.m3.9.9.1.3.1.cmml" xref="S3.I1.i3.p1.3.m3.9.9.1.3.2"><abs id="S3.I1.i3.p1.3.m3.9.9.1.3.1.1.cmml" xref="S3.I1.i3.p1.3.m3.9.9.1.3.2.1"></abs><ci id="S3.I1.i3.p1.3.m3.8.8.cmml" xref="S3.I1.i3.p1.3.m3.8.8">𝒟</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i3.p1.3.m3.9c">\operatorname*{\mathcal{R}}(f,\mathcal{D})=\Sigma_{({\boldsymbol{x}},y)\in\mathcal{D}}\;\mathcal{L}_{\textrm{pred}}\big{(}y,\sigma\big{(}f({\boldsymbol{x}})\big{)}\big{)}\;/\;|\mathcal{D}|</annotation></semantics></math><span id="S3.I1.i3.p1.4.5" class="ltx_text" style="font-size:90%;">,</span>
<br class="ltx_break"><span id="S3.I1.i3.p1.4.6" class="ltx_text" style="font-size:90%;">and </span><math id="S3.I1.i3.p1.4.m4.1" class="ltx_Math" alttext="\mathcal{L}_{\textrm{pred}}" display="inline"><semantics id="S3.I1.i3.p1.4.m4.1a"><msub id="S3.I1.i3.p1.4.m4.1.1" xref="S3.I1.i3.p1.4.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" mathsize="90%" id="S3.I1.i3.p1.4.m4.1.1.2" xref="S3.I1.i3.p1.4.m4.1.1.2.cmml">ℒ</mi><mtext mathsize="90%" id="S3.I1.i3.p1.4.m4.1.1.3" xref="S3.I1.i3.p1.4.m4.1.1.3a.cmml">pred</mtext></msub><annotation-xml encoding="MathML-Content" id="S3.I1.i3.p1.4.m4.1b"><apply id="S3.I1.i3.p1.4.m4.1.1.cmml" xref="S3.I1.i3.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S3.I1.i3.p1.4.m4.1.1.1.cmml" xref="S3.I1.i3.p1.4.m4.1.1">subscript</csymbol><ci id="S3.I1.i3.p1.4.m4.1.1.2.cmml" xref="S3.I1.i3.p1.4.m4.1.1.2">ℒ</ci><ci id="S3.I1.i3.p1.4.m4.1.1.3a.cmml" xref="S3.I1.i3.p1.4.m4.1.1.3"><mtext mathsize="63%" id="S3.I1.i3.p1.4.m4.1.1.3.cmml" xref="S3.I1.i3.p1.4.m4.1.1.3">pred</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i3.p1.4.m4.1c">\mathcal{L}_{\textrm{pred}}</annotation></semantics></math><span id="S3.I1.i3.p1.4.7" class="ltx_text" style="font-size:90%;"> is a predictive loss such as binary cross-entropy.</span></span>
</span></span>
<span id="S3.I1.i4" class="ltx_item" style="list-style-type:none;"><span class="ltx_tag ltx_tag_item">4.</span> 
<span id="S3.I1.i4.p1" class="ltx_para">
<span id="S3.I1.i4.p1.2" class="ltx_p"><span id="S3.I1.i4.p1.2.1" class="ltx_text ltx_font_bold" style="font-size:90%;">Validation performance</span><span id="S3.I1.i4.p1.2.2" class="ltx_text" style="font-size:90%;"> serves to refine various choices (architecture, regularizers, …) by trial and error, </span><em id="S3.I1.i4.p1.2.3" class="ltx_emph ltx_font_italic" style="font-size:90%;">i.e</em><span id="S3.I1.i4.p1.2.4" class="ltx_text" style="font-size:90%;">.</span><span id="S3.I1.i4.p1.2.5" class="ltx_text"></span><span id="S3.I1.i4.p1.2.6" class="ltx_text" style="font-size:90%;"> loosely solving </span><math id="S3.I1.i4.p1.1.m1.5" class="ltx_Math" alttext="f^{\prime}_{{\boldsymbol{\theta}}^{\star^{\prime}}}\leavevmode\nobreak\ =\leavevmode\nobreak\ \textrm{argmin}_{f,\,\kern-0.63004pt.\kern-0.63004pt.\kern-0.63004pt.}\operatorname*{\mathcal{R}}(f_{{\boldsymbol{\theta}}^{\star}},\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\mathcal{D}_{\mathrm{val}}})" 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id="S3.I1.i4.p1.1.m1.5c">f^{\prime}_{{\boldsymbol{\theta}}^{\star^{\prime}}}\leavevmode\nobreak\ =\leavevmode\nobreak\ \textrm{argmin}_{f,\,\kern-0.63004pt.\kern-0.63004pt.\kern-0.63004pt.}\operatorname*{\mathcal{R}}(f_{{\boldsymbol{\theta}}^{\star}},\color[rgb]{0,0,1}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,1}{\mathcal{D}_{\mathrm{val}}})</annotation></semantics></math><span id="S3.I1.i4.p1.2.7" class="ltx_text" style="font-size:90%;">
where </span><math id="S3.I1.i4.p1.2.m2.1" class="ltx_Math" alttext="\operatorname*{\mathcal{R}}" display="inline"><semantics id="S3.I1.i4.p1.2.m2.1a"><mo class="ltx_font_mathcaligraphic" mathsize="90%" id="S3.I1.i4.p1.2.m2.1.1" xref="S3.I1.i4.p1.2.m2.1.1.cmml">ℛ</mo><annotation-xml encoding="MathML-Content" id="S3.I1.i4.p1.2.m2.1b"><ci id="S3.I1.i4.p1.2.m2.1.1.cmml" xref="S3.I1.i4.p1.2.m2.1.1">ℛ</ci></annotation-xml><annotation encoding="application/x-tex" id="S3.I1.i4.p1.2.m2.1c">\operatorname*{\mathcal{R}}</annotation></semantics></math><span id="S3.I1.i4.p1.2.8" class="ltx_text" style="font-size:90%;"> is often substituted with a task-specific metric such as the error rate.</span></span>
</span></span>
</span>
</span>
<p id="S3.p3.2" class="ltx_p">There is often a multitude of models satisfying the above procedure, not all are equally desirable because they differ in properties that the procedure does not constrain.
The degree of underspecification indicates the importance of arbitrary and stochastic factors in the outcome of the learning process.</p>
</div>
<div id="S3.p4" class="ltx_para">
<p id="S3.p4.1" class="ltx_p">This paper focuses on <span id="S3.p4.1.1" class="ltx_text ltx_font_bold">differences in OOD performance</span> among predictive models.
OOD performance is the predictive performance of a model (in terms of risk, accuracy, or another task-specific metric) on test data drawn from a distribution <math id="S3.p4.1.m1.1" class="ltx_Math" alttext="\textrm{P}_{\mathrm{OOD}}\neq\textrm{P}_{\mathrm{ID}}" display="inline"><semantics id="S3.p4.1.m1.1a"><mrow id="S3.p4.1.m1.1.1" xref="S3.p4.1.m1.1.1.cmml"><msub id="S3.p4.1.m1.1.1.2" xref="S3.p4.1.m1.1.1.2.cmml"><mtext id="S3.p4.1.m1.1.1.2.2" xref="S3.p4.1.m1.1.1.2.2a.cmml">P</mtext><mi id="S3.p4.1.m1.1.1.2.3" xref="S3.p4.1.m1.1.1.2.3.cmml">OOD</mi></msub><mo id="S3.p4.1.m1.1.1.1" xref="S3.p4.1.m1.1.1.1.cmml">≠</mo><msub id="S3.p4.1.m1.1.1.3" xref="S3.p4.1.m1.1.1.3.cmml"><mtext id="S3.p4.1.m1.1.1.3.2" xref="S3.p4.1.m1.1.1.3.2a.cmml">P</mtext><mi id="S3.p4.1.m1.1.1.3.3" xref="S3.p4.1.m1.1.1.3.3.cmml">ID</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S3.p4.1.m1.1b"><apply id="S3.p4.1.m1.1.1.cmml" xref="S3.p4.1.m1.1.1"><neq id="S3.p4.1.m1.1.1.1.cmml" xref="S3.p4.1.m1.1.1.1"></neq><apply id="S3.p4.1.m1.1.1.2.cmml" xref="S3.p4.1.m1.1.1.2"><csymbol cd="ambiguous" id="S3.p4.1.m1.1.1.2.1.cmml" xref="S3.p4.1.m1.1.1.2">subscript</csymbol><ci id="S3.p4.1.m1.1.1.2.2a.cmml" xref="S3.p4.1.m1.1.1.2.2"><mtext id="S3.p4.1.m1.1.1.2.2.cmml" xref="S3.p4.1.m1.1.1.2.2">P</mtext></ci><ci id="S3.p4.1.m1.1.1.2.3.cmml" xref="S3.p4.1.m1.1.1.2.3">OOD</ci></apply><apply id="S3.p4.1.m1.1.1.3.cmml" xref="S3.p4.1.m1.1.1.3"><csymbol cd="ambiguous" id="S3.p4.1.m1.1.1.3.1.cmml" xref="S3.p4.1.m1.1.1.3">subscript</csymbol><ci id="S3.p4.1.m1.1.1.3.2a.cmml" xref="S3.p4.1.m1.1.1.3.2"><mtext id="S3.p4.1.m1.1.1.3.2.cmml" xref="S3.p4.1.m1.1.1.3.2">P</mtext></ci><ci id="S3.p4.1.m1.1.1.3.3.cmml" xref="S3.p4.1.m1.1.1.3.3">ID</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.p4.1.m1.1c">\textrm{P}_{\mathrm{OOD}}\neq\textrm{P}_{\mathrm{ID}}</annotation></semantics></math>.
On OOD data, features that were predictive in the training data may become irrelevant or misleading, causing a drop in performance of a model that relies on them.
By definition, OOD performance is underspecified by the ERM objective, since the empirical risk is estimated on in-domain data.</p>
</div>
<div id="S3.p5" class="ltx_para">
<p id="S3.p5.1" class="ltx_p">To capture variability in OOD performance, we propose a definition of underspecification based on the number of ways to fit the data with the above procedure and produce different OOD predictions.<span id="footnote3" class="ltx_note ltx_role_footnote"><sup class="ltx_note_mark">3</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">3</sup><span class="ltx_tag ltx_tag_note">3</span>See <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib31" title="" class="ltx_ref">31</a>, <a href="#bib.bib65" title="" class="ltx_ref">65</a>]</cite> for related uses of the <em id="footnote3.1" class="ltx_emph ltx_font_italic">volume</em> of a hypothesis spaces in the definition of Rashomon sets.</span></span></span></p>
</div>
<div id="S3.p6" class="ltx_para ltx_noindent">
<span id="S3.p6.1" class="ltx_inline-block ltx_framed ltx_framed_rectangle" style="border-color: #000000;">
<span id="Thmdefinition1" class="ltx_theorem ltx_theorem_definition">
<h6 class="ltx_title ltx_runin ltx_title_theorem">
<span class="ltx_tag ltx_tag_theorem"><span id="Thmdefinition1.1.1.1" class="ltx_text ltx_font_bold">Definition 1</span></span><span id="Thmdefinition1.2.2" class="ltx_text ltx_font_bold">.</span>
</h6>
<span id="Thmdefinition1.p1" class="ltx_para">
<span id="Thmdefinition1.p1.8" class="ltx_p">The <span id="Thmdefinition1.p1.8.1" class="ltx_text ltx_font_bold">degree of underspecification</span> of a dataset <math id="Thmdefinition1.p1.1.m1.1" class="ltx_Math" alttext="\mathcal{D}=\mathcal{D}_{\mathrm{tr}}\cup\mathcal{D}_{\mathrm{val}}" display="inline"><semantics id="Thmdefinition1.p1.1.m1.1a"><mrow id="Thmdefinition1.p1.1.m1.1.1" xref="Thmdefinition1.p1.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmdefinition1.p1.1.m1.1.1.2" xref="Thmdefinition1.p1.1.m1.1.1.2.cmml">𝒟</mi><mo id="Thmdefinition1.p1.1.m1.1.1.1" xref="Thmdefinition1.p1.1.m1.1.1.1.cmml">=</mo><mrow id="Thmdefinition1.p1.1.m1.1.1.3" xref="Thmdefinition1.p1.1.m1.1.1.3.cmml"><msub id="Thmdefinition1.p1.1.m1.1.1.3.2" xref="Thmdefinition1.p1.1.m1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmdefinition1.p1.1.m1.1.1.3.2.2" xref="Thmdefinition1.p1.1.m1.1.1.3.2.2.cmml">𝒟</mi><mi id="Thmdefinition1.p1.1.m1.1.1.3.2.3" xref="Thmdefinition1.p1.1.m1.1.1.3.2.3.cmml">tr</mi></msub><mo id="Thmdefinition1.p1.1.m1.1.1.3.1" xref="Thmdefinition1.p1.1.m1.1.1.3.1.cmml">∪</mo><msub id="Thmdefinition1.p1.1.m1.1.1.3.3" xref="Thmdefinition1.p1.1.m1.1.1.3.3.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmdefinition1.p1.1.m1.1.1.3.3.2" xref="Thmdefinition1.p1.1.m1.1.1.3.3.2.cmml">𝒟</mi><mi id="Thmdefinition1.p1.1.m1.1.1.3.3.3" xref="Thmdefinition1.p1.1.m1.1.1.3.3.3.cmml">val</mi></msub></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmdefinition1.p1.1.m1.1b"><apply id="Thmdefinition1.p1.1.m1.1.1.cmml" xref="Thmdefinition1.p1.1.m1.1.1"><eq id="Thmdefinition1.p1.1.m1.1.1.1.cmml" xref="Thmdefinition1.p1.1.m1.1.1.1"></eq><ci id="Thmdefinition1.p1.1.m1.1.1.2.cmml" xref="Thmdefinition1.p1.1.m1.1.1.2">𝒟</ci><apply id="Thmdefinition1.p1.1.m1.1.1.3.cmml" xref="Thmdefinition1.p1.1.m1.1.1.3"><union id="Thmdefinition1.p1.1.m1.1.1.3.1.cmml" xref="Thmdefinition1.p1.1.m1.1.1.3.1"></union><apply id="Thmdefinition1.p1.1.m1.1.1.3.2.cmml" xref="Thmdefinition1.p1.1.m1.1.1.3.2"><csymbol cd="ambiguous" id="Thmdefinition1.p1.1.m1.1.1.3.2.1.cmml" xref="Thmdefinition1.p1.1.m1.1.1.3.2">subscript</csymbol><ci id="Thmdefinition1.p1.1.m1.1.1.3.2.2.cmml" xref="Thmdefinition1.p1.1.m1.1.1.3.2.2">𝒟</ci><ci id="Thmdefinition1.p1.1.m1.1.1.3.2.3.cmml" xref="Thmdefinition1.p1.1.m1.1.1.3.2.3">tr</ci></apply><apply id="Thmdefinition1.p1.1.m1.1.1.3.3.cmml" xref="Thmdefinition1.p1.1.m1.1.1.3.3"><csymbol cd="ambiguous" id="Thmdefinition1.p1.1.m1.1.1.3.3.1.cmml" xref="Thmdefinition1.p1.1.m1.1.1.3.3">subscript</csymbol><ci id="Thmdefinition1.p1.1.m1.1.1.3.3.2.cmml" xref="Thmdefinition1.p1.1.m1.1.1.3.3.2">𝒟</ci><ci id="Thmdefinition1.p1.1.m1.1.1.3.3.3.cmml" xref="Thmdefinition1.p1.1.m1.1.1.3.3.3">val</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition1.p1.1.m1.1c">\mathcal{D}=\mathcal{D}_{\mathrm{tr}}\cup\mathcal{D}_{\mathrm{val}}</annotation></semantics></math>, input manifold <math id="Thmdefinition1.p1.2.m2.1" class="ltx_Math" alttext="\mathcal{M}" display="inline"><semantics id="Thmdefinition1.p1.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="Thmdefinition1.p1.2.m2.1.1" xref="Thmdefinition1.p1.2.m2.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="Thmdefinition1.p1.2.m2.1b"><ci id="Thmdefinition1.p1.2.m2.1.1.cmml" xref="Thmdefinition1.p1.2.m2.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition1.p1.2.m2.1c">\mathcal{M}</annotation></semantics></math>, and hypothesis class <math id="Thmdefinition1.p1.3.m3.2" class="ltx_Math" alttext="\mathcal{H}=\{f_{\boldsymbol{\theta}},\forall\,{\boldsymbol{\theta}}\}" display="inline"><semantics id="Thmdefinition1.p1.3.m3.2a"><mrow id="Thmdefinition1.p1.3.m3.2.2" xref="Thmdefinition1.p1.3.m3.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmdefinition1.p1.3.m3.2.2.4" xref="Thmdefinition1.p1.3.m3.2.2.4.cmml">ℋ</mi><mo id="Thmdefinition1.p1.3.m3.2.2.3" xref="Thmdefinition1.p1.3.m3.2.2.3.cmml">=</mo><mrow id="Thmdefinition1.p1.3.m3.2.2.2.2" xref="Thmdefinition1.p1.3.m3.2.2.2.3.cmml"><mo stretchy="false" id="Thmdefinition1.p1.3.m3.2.2.2.2.3" xref="Thmdefinition1.p1.3.m3.2.2.2.3.cmml">{</mo><msub id="Thmdefinition1.p1.3.m3.1.1.1.1.1" xref="Thmdefinition1.p1.3.m3.1.1.1.1.1.cmml"><mi id="Thmdefinition1.p1.3.m3.1.1.1.1.1.2" xref="Thmdefinition1.p1.3.m3.1.1.1.1.1.2.cmml">f</mi><mi id="Thmdefinition1.p1.3.m3.1.1.1.1.1.3" xref="Thmdefinition1.p1.3.m3.1.1.1.1.1.3.cmml">𝜽</mi></msub><mo id="Thmdefinition1.p1.3.m3.2.2.2.2.4" xref="Thmdefinition1.p1.3.m3.2.2.2.3.cmml">,</mo><mrow id="Thmdefinition1.p1.3.m3.2.2.2.2.2" xref="Thmdefinition1.p1.3.m3.2.2.2.2.2.cmml"><mo rspace="0.337em" id="Thmdefinition1.p1.3.m3.2.2.2.2.2.1" xref="Thmdefinition1.p1.3.m3.2.2.2.2.2.1.cmml">∀</mo><mi id="Thmdefinition1.p1.3.m3.2.2.2.2.2.2" xref="Thmdefinition1.p1.3.m3.2.2.2.2.2.2.cmml">𝜽</mi></mrow><mo stretchy="false" id="Thmdefinition1.p1.3.m3.2.2.2.2.5" xref="Thmdefinition1.p1.3.m3.2.2.2.3.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmdefinition1.p1.3.m3.2b"><apply id="Thmdefinition1.p1.3.m3.2.2.cmml" xref="Thmdefinition1.p1.3.m3.2.2"><eq id="Thmdefinition1.p1.3.m3.2.2.3.cmml" xref="Thmdefinition1.p1.3.m3.2.2.3"></eq><ci id="Thmdefinition1.p1.3.m3.2.2.4.cmml" xref="Thmdefinition1.p1.3.m3.2.2.4">ℋ</ci><set id="Thmdefinition1.p1.3.m3.2.2.2.3.cmml" xref="Thmdefinition1.p1.3.m3.2.2.2.2"><apply id="Thmdefinition1.p1.3.m3.1.1.1.1.1.cmml" xref="Thmdefinition1.p1.3.m3.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmdefinition1.p1.3.m3.1.1.1.1.1.1.cmml" xref="Thmdefinition1.p1.3.m3.1.1.1.1.1">subscript</csymbol><ci id="Thmdefinition1.p1.3.m3.1.1.1.1.1.2.cmml" xref="Thmdefinition1.p1.3.m3.1.1.1.1.1.2">𝑓</ci><ci id="Thmdefinition1.p1.3.m3.1.1.1.1.1.3.cmml" xref="Thmdefinition1.p1.3.m3.1.1.1.1.1.3">𝜽</ci></apply><apply id="Thmdefinition1.p1.3.m3.2.2.2.2.2.cmml" xref="Thmdefinition1.p1.3.m3.2.2.2.2.2"><csymbol cd="latexml" id="Thmdefinition1.p1.3.m3.2.2.2.2.2.1.cmml" xref="Thmdefinition1.p1.3.m3.2.2.2.2.2.1">for-all</csymbol><ci id="Thmdefinition1.p1.3.m3.2.2.2.2.2.2.cmml" xref="Thmdefinition1.p1.3.m3.2.2.2.2.2.2">𝜽</ci></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition1.p1.3.m3.2c">\mathcal{H}=\{f_{\boldsymbol{\theta}},\forall\,{\boldsymbol{\theta}}\}</annotation></semantics></math> 
is the ratio of volumes
<math id="Thmdefinition1.p1.4.m4.2" class="ltx_Math" alttext="\mathop{\textrm{vol}}({\mathcal{H}^{\prime}})/\mathop{\textrm{vol}}(\mathcal{H})" display="inline"><semantics id="Thmdefinition1.p1.4.m4.2a"><mrow id="Thmdefinition1.p1.4.m4.2.2" xref="Thmdefinition1.p1.4.m4.2.2.cmml"><mtext id="Thmdefinition1.p1.4.m4.2.2.2" xref="Thmdefinition1.p1.4.m4.2.2.2a.cmml">vol</mtext><mrow id="Thmdefinition1.p1.4.m4.2.2.1" xref="Thmdefinition1.p1.4.m4.2.2.1.cmml"><mrow id="Thmdefinition1.p1.4.m4.2.2.1.1.1" xref="Thmdefinition1.p1.4.m4.2.2.1.1.1.1.cmml"><mo stretchy="false" id="Thmdefinition1.p1.4.m4.2.2.1.1.1.2" xref="Thmdefinition1.p1.4.m4.2.2.1.1.1.1.cmml">(</mo><msup id="Thmdefinition1.p1.4.m4.2.2.1.1.1.1" xref="Thmdefinition1.p1.4.m4.2.2.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmdefinition1.p1.4.m4.2.2.1.1.1.1.2" xref="Thmdefinition1.p1.4.m4.2.2.1.1.1.1.2.cmml">ℋ</mi><mo id="Thmdefinition1.p1.4.m4.2.2.1.1.1.1.3" xref="Thmdefinition1.p1.4.m4.2.2.1.1.1.1.3.cmml">′</mo></msup><mo stretchy="false" id="Thmdefinition1.p1.4.m4.2.2.1.1.1.3" xref="Thmdefinition1.p1.4.m4.2.2.1.1.1.1.cmml">)</mo></mrow><mo id="Thmdefinition1.p1.4.m4.2.2.1.2" xref="Thmdefinition1.p1.4.m4.2.2.1.2.cmml">/</mo><mrow id="Thmdefinition1.p1.4.m4.2.2.1.3" xref="Thmdefinition1.p1.4.m4.2.2.1.3.cmml"><mtext id="Thmdefinition1.p1.4.m4.2.2.1.3.1" xref="Thmdefinition1.p1.4.m4.2.2.1.3.1a.cmml">vol</mtext><mrow id="Thmdefinition1.p1.4.m4.2.2.1.3.2.2" xref="Thmdefinition1.p1.4.m4.2.2.1.3.cmml"><mo stretchy="false" id="Thmdefinition1.p1.4.m4.2.2.1.3.2.2.1" xref="Thmdefinition1.p1.4.m4.2.2.1.3.cmml">(</mo><mi class="ltx_font_mathcaligraphic" id="Thmdefinition1.p1.4.m4.1.1" xref="Thmdefinition1.p1.4.m4.1.1.cmml">ℋ</mi><mo stretchy="false" id="Thmdefinition1.p1.4.m4.2.2.1.3.2.2.2" xref="Thmdefinition1.p1.4.m4.2.2.1.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmdefinition1.p1.4.m4.2b"><apply id="Thmdefinition1.p1.4.m4.2.2.cmml" xref="Thmdefinition1.p1.4.m4.2.2"><ci id="Thmdefinition1.p1.4.m4.2.2.2a.cmml" xref="Thmdefinition1.p1.4.m4.2.2.2"><mtext id="Thmdefinition1.p1.4.m4.2.2.2.cmml" xref="Thmdefinition1.p1.4.m4.2.2.2">vol</mtext></ci><apply id="Thmdefinition1.p1.4.m4.2.2.1.cmml" xref="Thmdefinition1.p1.4.m4.2.2.1"><divide id="Thmdefinition1.p1.4.m4.2.2.1.2.cmml" xref="Thmdefinition1.p1.4.m4.2.2.1.2"></divide><apply id="Thmdefinition1.p1.4.m4.2.2.1.1.1.1.cmml" xref="Thmdefinition1.p1.4.m4.2.2.1.1.1"><csymbol cd="ambiguous" id="Thmdefinition1.p1.4.m4.2.2.1.1.1.1.1.cmml" xref="Thmdefinition1.p1.4.m4.2.2.1.1.1">superscript</csymbol><ci id="Thmdefinition1.p1.4.m4.2.2.1.1.1.1.2.cmml" xref="Thmdefinition1.p1.4.m4.2.2.1.1.1.1.2">ℋ</ci><ci id="Thmdefinition1.p1.4.m4.2.2.1.1.1.1.3.cmml" xref="Thmdefinition1.p1.4.m4.2.2.1.1.1.1.3">′</ci></apply><apply id="Thmdefinition1.p1.4.m4.2.2.1.3.cmml" xref="Thmdefinition1.p1.4.m4.2.2.1.3"><ci id="Thmdefinition1.p1.4.m4.2.2.1.3.1a.cmml" xref="Thmdefinition1.p1.4.m4.2.2.1.3.1"><mtext id="Thmdefinition1.p1.4.m4.2.2.1.3.1.cmml" xref="Thmdefinition1.p1.4.m4.2.2.1.3.1">vol</mtext></ci><ci id="Thmdefinition1.p1.4.m4.1.1.cmml" xref="Thmdefinition1.p1.4.m4.1.1">ℋ</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition1.p1.4.m4.2c">\mathop{\textrm{vol}}({\mathcal{H}^{\prime}})/\mathop{\textrm{vol}}(\mathcal{H})</annotation></semantics></math> of the largest subset of models
<math id="Thmdefinition1.p1.5.m5.1" class="ltx_Math" alttext="\mathcal{H}^{\prime}\subset\mathcal{H}" display="inline"><semantics id="Thmdefinition1.p1.5.m5.1a"><mrow id="Thmdefinition1.p1.5.m5.1.1" xref="Thmdefinition1.p1.5.m5.1.1.cmml"><msup id="Thmdefinition1.p1.5.m5.1.1.2" xref="Thmdefinition1.p1.5.m5.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmdefinition1.p1.5.m5.1.1.2.2" xref="Thmdefinition1.p1.5.m5.1.1.2.2.cmml">ℋ</mi><mo id="Thmdefinition1.p1.5.m5.1.1.2.3" xref="Thmdefinition1.p1.5.m5.1.1.2.3.cmml">′</mo></msup><mo id="Thmdefinition1.p1.5.m5.1.1.1" xref="Thmdefinition1.p1.5.m5.1.1.1.cmml">⊂</mo><mi class="ltx_font_mathcaligraphic" id="Thmdefinition1.p1.5.m5.1.1.3" xref="Thmdefinition1.p1.5.m5.1.1.3.cmml">ℋ</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmdefinition1.p1.5.m5.1b"><apply id="Thmdefinition1.p1.5.m5.1.1.cmml" xref="Thmdefinition1.p1.5.m5.1.1"><subset id="Thmdefinition1.p1.5.m5.1.1.1.cmml" xref="Thmdefinition1.p1.5.m5.1.1.1"></subset><apply id="Thmdefinition1.p1.5.m5.1.1.2.cmml" xref="Thmdefinition1.p1.5.m5.1.1.2"><csymbol cd="ambiguous" id="Thmdefinition1.p1.5.m5.1.1.2.1.cmml" xref="Thmdefinition1.p1.5.m5.1.1.2">superscript</csymbol><ci id="Thmdefinition1.p1.5.m5.1.1.2.2.cmml" xref="Thmdefinition1.p1.5.m5.1.1.2.2">ℋ</ci><ci id="Thmdefinition1.p1.5.m5.1.1.2.3.cmml" xref="Thmdefinition1.p1.5.m5.1.1.2.3">′</ci></apply><ci id="Thmdefinition1.p1.5.m5.1.1.3.cmml" xref="Thmdefinition1.p1.5.m5.1.1.3">ℋ</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition1.p1.5.m5.1c">\mathcal{H}^{\prime}\subset\mathcal{H}</annotation></semantics></math>
such that its elements
<math id="Thmdefinition1.p1.6.m6.1" class="ltx_Math" alttext="\{f_{{\boldsymbol{\theta}}_{m}}\}_{m}" display="inline"><semantics id="Thmdefinition1.p1.6.m6.1a"><msub id="Thmdefinition1.p1.6.m6.1.1" xref="Thmdefinition1.p1.6.m6.1.1.cmml"><mrow id="Thmdefinition1.p1.6.m6.1.1.1.1" xref="Thmdefinition1.p1.6.m6.1.1.1.2.cmml"><mo stretchy="false" id="Thmdefinition1.p1.6.m6.1.1.1.1.2" xref="Thmdefinition1.p1.6.m6.1.1.1.2.cmml">{</mo><msub id="Thmdefinition1.p1.6.m6.1.1.1.1.1" xref="Thmdefinition1.p1.6.m6.1.1.1.1.1.cmml"><mi id="Thmdefinition1.p1.6.m6.1.1.1.1.1.2" xref="Thmdefinition1.p1.6.m6.1.1.1.1.1.2.cmml">f</mi><msub id="Thmdefinition1.p1.6.m6.1.1.1.1.1.3" xref="Thmdefinition1.p1.6.m6.1.1.1.1.1.3.cmml"><mi id="Thmdefinition1.p1.6.m6.1.1.1.1.1.3.2" xref="Thmdefinition1.p1.6.m6.1.1.1.1.1.3.2.cmml">𝜽</mi><mi id="Thmdefinition1.p1.6.m6.1.1.1.1.1.3.3" xref="Thmdefinition1.p1.6.m6.1.1.1.1.1.3.3.cmml">m</mi></msub></msub><mo stretchy="false" id="Thmdefinition1.p1.6.m6.1.1.1.1.3" xref="Thmdefinition1.p1.6.m6.1.1.1.2.cmml">}</mo></mrow><mi id="Thmdefinition1.p1.6.m6.1.1.3" xref="Thmdefinition1.p1.6.m6.1.1.3.cmml">m</mi></msub><annotation-xml encoding="MathML-Content" id="Thmdefinition1.p1.6.m6.1b"><apply id="Thmdefinition1.p1.6.m6.1.1.cmml" xref="Thmdefinition1.p1.6.m6.1.1"><csymbol cd="ambiguous" id="Thmdefinition1.p1.6.m6.1.1.2.cmml" xref="Thmdefinition1.p1.6.m6.1.1">subscript</csymbol><set id="Thmdefinition1.p1.6.m6.1.1.1.2.cmml" xref="Thmdefinition1.p1.6.m6.1.1.1.1"><apply id="Thmdefinition1.p1.6.m6.1.1.1.1.1.cmml" xref="Thmdefinition1.p1.6.m6.1.1.1.1.1"><csymbol cd="ambiguous" id="Thmdefinition1.p1.6.m6.1.1.1.1.1.1.cmml" xref="Thmdefinition1.p1.6.m6.1.1.1.1.1">subscript</csymbol><ci id="Thmdefinition1.p1.6.m6.1.1.1.1.1.2.cmml" xref="Thmdefinition1.p1.6.m6.1.1.1.1.1.2">𝑓</ci><apply id="Thmdefinition1.p1.6.m6.1.1.1.1.1.3.cmml" xref="Thmdefinition1.p1.6.m6.1.1.1.1.1.3"><csymbol cd="ambiguous" id="Thmdefinition1.p1.6.m6.1.1.1.1.1.3.1.cmml" xref="Thmdefinition1.p1.6.m6.1.1.1.1.1.3">subscript</csymbol><ci id="Thmdefinition1.p1.6.m6.1.1.1.1.1.3.2.cmml" xref="Thmdefinition1.p1.6.m6.1.1.1.1.1.3.2">𝜽</ci><ci id="Thmdefinition1.p1.6.m6.1.1.1.1.1.3.3.cmml" xref="Thmdefinition1.p1.6.m6.1.1.1.1.1.3.3">𝑚</ci></apply></apply></set><ci id="Thmdefinition1.p1.6.m6.1.1.3.cmml" xref="Thmdefinition1.p1.6.m6.1.1.3">𝑚</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition1.p1.6.m6.1c">\{f_{{\boldsymbol{\theta}}_{m}}\}_{m}</annotation></semantics></math>
all have, for small constants <math id="Thmdefinition1.p1.7.m7.1" class="ltx_Math" alttext="\epsilon_{\textrm{tr}}" display="inline"><semantics id="Thmdefinition1.p1.7.m7.1a"><msub id="Thmdefinition1.p1.7.m7.1.1" xref="Thmdefinition1.p1.7.m7.1.1.cmml"><mi id="Thmdefinition1.p1.7.m7.1.1.2" xref="Thmdefinition1.p1.7.m7.1.1.2.cmml">ϵ</mi><mtext id="Thmdefinition1.p1.7.m7.1.1.3" xref="Thmdefinition1.p1.7.m7.1.1.3a.cmml">tr</mtext></msub><annotation-xml encoding="MathML-Content" id="Thmdefinition1.p1.7.m7.1b"><apply id="Thmdefinition1.p1.7.m7.1.1.cmml" xref="Thmdefinition1.p1.7.m7.1.1"><csymbol cd="ambiguous" id="Thmdefinition1.p1.7.m7.1.1.1.cmml" xref="Thmdefinition1.p1.7.m7.1.1">subscript</csymbol><ci id="Thmdefinition1.p1.7.m7.1.1.2.cmml" xref="Thmdefinition1.p1.7.m7.1.1.2">italic-ϵ</ci><ci id="Thmdefinition1.p1.7.m7.1.1.3a.cmml" xref="Thmdefinition1.p1.7.m7.1.1.3"><mtext mathsize="70%" id="Thmdefinition1.p1.7.m7.1.1.3.cmml" xref="Thmdefinition1.p1.7.m7.1.1.3">tr</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition1.p1.7.m7.1c">\epsilon_{\textrm{tr}}</annotation></semantics></math>, <math id="Thmdefinition1.p1.8.m8.1" class="ltx_Math" alttext="\epsilon_{\textrm{val}}" display="inline"><semantics id="Thmdefinition1.p1.8.m8.1a"><msub id="Thmdefinition1.p1.8.m8.1.1" xref="Thmdefinition1.p1.8.m8.1.1.cmml"><mi id="Thmdefinition1.p1.8.m8.1.1.2" xref="Thmdefinition1.p1.8.m8.1.1.2.cmml">ϵ</mi><mtext id="Thmdefinition1.p1.8.m8.1.1.3" xref="Thmdefinition1.p1.8.m8.1.1.3a.cmml">val</mtext></msub><annotation-xml encoding="MathML-Content" id="Thmdefinition1.p1.8.m8.1b"><apply id="Thmdefinition1.p1.8.m8.1.1.cmml" xref="Thmdefinition1.p1.8.m8.1.1"><csymbol cd="ambiguous" id="Thmdefinition1.p1.8.m8.1.1.1.cmml" xref="Thmdefinition1.p1.8.m8.1.1">subscript</csymbol><ci id="Thmdefinition1.p1.8.m8.1.1.2.cmml" xref="Thmdefinition1.p1.8.m8.1.1.2">italic-ϵ</ci><ci id="Thmdefinition1.p1.8.m8.1.1.3a.cmml" xref="Thmdefinition1.p1.8.m8.1.1.3"><mtext mathsize="70%" id="Thmdefinition1.p1.8.m8.1.1.3.cmml" xref="Thmdefinition1.p1.8.m8.1.1.3">val</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition1.p1.8.m8.1c">\epsilon_{\textrm{val}}</annotation></semantics></math>:</span>
<span id="S3.I2" class="ltx_itemize">
<span id="S3.I2.i1" class="ltx_item" style="list-style-type:none;"><span class="ltx_tag ltx_tag_item">•</span> 
<span id="S3.I2.i1.p1" class="ltx_para">
<span id="S3.I2.i1.p1.1" class="ltx_p">A low training risk:   <math id="S3.I2.i1.p1.1.m1.3" class="ltx_Math" alttext="\operatorname*{\mathcal{R}}(f_{{\boldsymbol{\theta}}_{m}},\mathcal{D}_{\mathrm{tr}})&lt;\epsilon_{\textrm{tr}},\leavevmode\nobreak\ \forall\,f_{\boldsymbol{\theta}}\in\mathcal{H}^{\prime}" display="inline"><semantics id="S3.I2.i1.p1.1.m1.3a"><mrow id="S3.I2.i1.p1.1.m1.3.3.2" xref="S3.I2.i1.p1.1.m1.3.3.3.cmml"><mrow id="S3.I2.i1.p1.1.m1.2.2.1.1" xref="S3.I2.i1.p1.1.m1.2.2.1.1.cmml"><mrow id="S3.I2.i1.p1.1.m1.2.2.1.1.2.2" xref="S3.I2.i1.p1.1.m1.2.2.1.1.2.3.cmml"><mo class="ltx_font_mathcaligraphic" rspace="0em" id="S3.I2.i1.p1.1.m1.1.1" xref="S3.I2.i1.p1.1.m1.1.1.cmml">ℛ</mo><mrow id="S3.I2.i1.p1.1.m1.2.2.1.1.2.2.2" xref="S3.I2.i1.p1.1.m1.2.2.1.1.2.3.cmml"><mo stretchy="false" id="S3.I2.i1.p1.1.m1.2.2.1.1.2.2.2.3" xref="S3.I2.i1.p1.1.m1.2.2.1.1.2.3.cmml">(</mo><msub id="S3.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1" xref="S3.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.cmml"><mi id="S3.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.2" xref="S3.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.2.cmml">f</mi><msub id="S3.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.3" xref="S3.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.3.cmml"><mi id="S3.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.3.2" xref="S3.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.3.2.cmml">𝜽</mi><mi id="S3.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.3.3" xref="S3.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.3.3.cmml">m</mi></msub></msub><mo id="S3.I2.i1.p1.1.m1.2.2.1.1.2.2.2.4" xref="S3.I2.i1.p1.1.m1.2.2.1.1.2.3.cmml">,</mo><msub id="S3.I2.i1.p1.1.m1.2.2.1.1.2.2.2.2" xref="S3.I2.i1.p1.1.m1.2.2.1.1.2.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.I2.i1.p1.1.m1.2.2.1.1.2.2.2.2.2" xref="S3.I2.i1.p1.1.m1.2.2.1.1.2.2.2.2.2.cmml">𝒟</mi><mi id="S3.I2.i1.p1.1.m1.2.2.1.1.2.2.2.2.3" xref="S3.I2.i1.p1.1.m1.2.2.1.1.2.2.2.2.3.cmml">tr</mi></msub><mo stretchy="false" id="S3.I2.i1.p1.1.m1.2.2.1.1.2.2.2.5" xref="S3.I2.i1.p1.1.m1.2.2.1.1.2.3.cmml">)</mo></mrow></mrow><mo id="S3.I2.i1.p1.1.m1.2.2.1.1.3" xref="S3.I2.i1.p1.1.m1.2.2.1.1.3.cmml">&lt;</mo><msub id="S3.I2.i1.p1.1.m1.2.2.1.1.4" xref="S3.I2.i1.p1.1.m1.2.2.1.1.4.cmml"><mi id="S3.I2.i1.p1.1.m1.2.2.1.1.4.2" xref="S3.I2.i1.p1.1.m1.2.2.1.1.4.2.cmml">ϵ</mi><mtext id="S3.I2.i1.p1.1.m1.2.2.1.1.4.3" xref="S3.I2.i1.p1.1.m1.2.2.1.1.4.3a.cmml">tr</mtext></msub></mrow><mo rspace="0.667em" id="S3.I2.i1.p1.1.m1.3.3.2.3" xref="S3.I2.i1.p1.1.m1.3.3.3a.cmml">,</mo><mrow id="S3.I2.i1.p1.1.m1.3.3.2.2" xref="S3.I2.i1.p1.1.m1.3.3.2.2.cmml"><mrow id="S3.I2.i1.p1.1.m1.3.3.2.2.2" xref="S3.I2.i1.p1.1.m1.3.3.2.2.2.cmml"><mo rspace="0.337em" id="S3.I2.i1.p1.1.m1.3.3.2.2.2.1" xref="S3.I2.i1.p1.1.m1.3.3.2.2.2.1.cmml">∀</mo><msub id="S3.I2.i1.p1.1.m1.3.3.2.2.2.2" xref="S3.I2.i1.p1.1.m1.3.3.2.2.2.2.cmml"><mi id="S3.I2.i1.p1.1.m1.3.3.2.2.2.2.2" xref="S3.I2.i1.p1.1.m1.3.3.2.2.2.2.2.cmml">f</mi><mi id="S3.I2.i1.p1.1.m1.3.3.2.2.2.2.3" xref="S3.I2.i1.p1.1.m1.3.3.2.2.2.2.3.cmml">𝜽</mi></msub></mrow><mo id="S3.I2.i1.p1.1.m1.3.3.2.2.1" xref="S3.I2.i1.p1.1.m1.3.3.2.2.1.cmml">∈</mo><msup id="S3.I2.i1.p1.1.m1.3.3.2.2.3" xref="S3.I2.i1.p1.1.m1.3.3.2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.I2.i1.p1.1.m1.3.3.2.2.3.2" xref="S3.I2.i1.p1.1.m1.3.3.2.2.3.2.cmml">ℋ</mi><mo id="S3.I2.i1.p1.1.m1.3.3.2.2.3.3" xref="S3.I2.i1.p1.1.m1.3.3.2.2.3.3.cmml">′</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i1.p1.1.m1.3b"><apply id="S3.I2.i1.p1.1.m1.3.3.3.cmml" xref="S3.I2.i1.p1.1.m1.3.3.2"><csymbol cd="ambiguous" id="S3.I2.i1.p1.1.m1.3.3.3a.cmml" xref="S3.I2.i1.p1.1.m1.3.3.2.3">formulae-sequence</csymbol><apply id="S3.I2.i1.p1.1.m1.2.2.1.1.cmml" xref="S3.I2.i1.p1.1.m1.2.2.1.1"><lt id="S3.I2.i1.p1.1.m1.2.2.1.1.3.cmml" xref="S3.I2.i1.p1.1.m1.2.2.1.1.3"></lt><apply id="S3.I2.i1.p1.1.m1.2.2.1.1.2.3.cmml" xref="S3.I2.i1.p1.1.m1.2.2.1.1.2.2"><ci id="S3.I2.i1.p1.1.m1.1.1.cmml" xref="S3.I2.i1.p1.1.m1.1.1">ℛ</ci><apply id="S3.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.cmml" xref="S3.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.1.cmml" xref="S3.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1">subscript</csymbol><ci id="S3.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.2.cmml" xref="S3.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.2">𝑓</ci><apply id="S3.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.3.cmml" xref="S3.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.3.1.cmml" xref="S3.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.3">subscript</csymbol><ci id="S3.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.3.2.cmml" xref="S3.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.3.2">𝜽</ci><ci id="S3.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.3.3.cmml" xref="S3.I2.i1.p1.1.m1.2.2.1.1.1.1.1.1.3.3">𝑚</ci></apply></apply><apply id="S3.I2.i1.p1.1.m1.2.2.1.1.2.2.2.2.cmml" xref="S3.I2.i1.p1.1.m1.2.2.1.1.2.2.2.2"><csymbol cd="ambiguous" id="S3.I2.i1.p1.1.m1.2.2.1.1.2.2.2.2.1.cmml" xref="S3.I2.i1.p1.1.m1.2.2.1.1.2.2.2.2">subscript</csymbol><ci id="S3.I2.i1.p1.1.m1.2.2.1.1.2.2.2.2.2.cmml" xref="S3.I2.i1.p1.1.m1.2.2.1.1.2.2.2.2.2">𝒟</ci><ci id="S3.I2.i1.p1.1.m1.2.2.1.1.2.2.2.2.3.cmml" xref="S3.I2.i1.p1.1.m1.2.2.1.1.2.2.2.2.3">tr</ci></apply></apply><apply id="S3.I2.i1.p1.1.m1.2.2.1.1.4.cmml" xref="S3.I2.i1.p1.1.m1.2.2.1.1.4"><csymbol cd="ambiguous" id="S3.I2.i1.p1.1.m1.2.2.1.1.4.1.cmml" xref="S3.I2.i1.p1.1.m1.2.2.1.1.4">subscript</csymbol><ci id="S3.I2.i1.p1.1.m1.2.2.1.1.4.2.cmml" xref="S3.I2.i1.p1.1.m1.2.2.1.1.4.2">italic-ϵ</ci><ci id="S3.I2.i1.p1.1.m1.2.2.1.1.4.3a.cmml" xref="S3.I2.i1.p1.1.m1.2.2.1.1.4.3"><mtext mathsize="70%" id="S3.I2.i1.p1.1.m1.2.2.1.1.4.3.cmml" xref="S3.I2.i1.p1.1.m1.2.2.1.1.4.3">tr</mtext></ci></apply></apply><apply id="S3.I2.i1.p1.1.m1.3.3.2.2.cmml" xref="S3.I2.i1.p1.1.m1.3.3.2.2"><in id="S3.I2.i1.p1.1.m1.3.3.2.2.1.cmml" xref="S3.I2.i1.p1.1.m1.3.3.2.2.1"></in><apply id="S3.I2.i1.p1.1.m1.3.3.2.2.2.cmml" xref="S3.I2.i1.p1.1.m1.3.3.2.2.2"><csymbol cd="latexml" id="S3.I2.i1.p1.1.m1.3.3.2.2.2.1.cmml" xref="S3.I2.i1.p1.1.m1.3.3.2.2.2.1">for-all</csymbol><apply id="S3.I2.i1.p1.1.m1.3.3.2.2.2.2.cmml" xref="S3.I2.i1.p1.1.m1.3.3.2.2.2.2"><csymbol cd="ambiguous" id="S3.I2.i1.p1.1.m1.3.3.2.2.2.2.1.cmml" xref="S3.I2.i1.p1.1.m1.3.3.2.2.2.2">subscript</csymbol><ci id="S3.I2.i1.p1.1.m1.3.3.2.2.2.2.2.cmml" xref="S3.I2.i1.p1.1.m1.3.3.2.2.2.2.2">𝑓</ci><ci id="S3.I2.i1.p1.1.m1.3.3.2.2.2.2.3.cmml" xref="S3.I2.i1.p1.1.m1.3.3.2.2.2.2.3">𝜽</ci></apply></apply><apply id="S3.I2.i1.p1.1.m1.3.3.2.2.3.cmml" xref="S3.I2.i1.p1.1.m1.3.3.2.2.3"><csymbol cd="ambiguous" id="S3.I2.i1.p1.1.m1.3.3.2.2.3.1.cmml" xref="S3.I2.i1.p1.1.m1.3.3.2.2.3">superscript</csymbol><ci id="S3.I2.i1.p1.1.m1.3.3.2.2.3.2.cmml" xref="S3.I2.i1.p1.1.m1.3.3.2.2.3.2">ℋ</ci><ci id="S3.I2.i1.p1.1.m1.3.3.2.2.3.3.cmml" xref="S3.I2.i1.p1.1.m1.3.3.2.2.3.3">′</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S3.I2.i1.p1.1.m1.3c">\operatorname*{\mathcal{R}}(f_{{\boldsymbol{\theta}}_{m}},\mathcal{D}_{\mathrm{tr}})&lt;\epsilon_{\textrm{tr}},\leavevmode\nobreak\ \forall\,f_{\boldsymbol{\theta}}\in\mathcal{H}^{\prime}</annotation></semantics></math>,</span>
</span></span>
<span id="S3.I2.i2" class="ltx_item" style="list-style-type:none;"><span class="ltx_tag ltx_tag_item">•</span> 
<span id="S3.I2.i2.p1" class="ltx_para">
<span id="S3.I2.i2.p1.1" class="ltx_p">A low validation risk:   <math id="S3.I2.i2.p1.1.m1.3" class="ltx_Math" alttext="\operatorname*{\mathcal{R}}(f_{{\boldsymbol{\theta}}_{m}},\mathcal{D}_{\mathrm{val}})&lt;\epsilon_{\textrm{val}},\leavevmode\nobreak\ \forall\,f_{\boldsymbol{\theta}}\in\mathcal{H}^{\prime}" display="inline"><semantics id="S3.I2.i2.p1.1.m1.3a"><mrow id="S3.I2.i2.p1.1.m1.3.3.2" xref="S3.I2.i2.p1.1.m1.3.3.3.cmml"><mrow id="S3.I2.i2.p1.1.m1.2.2.1.1" xref="S3.I2.i2.p1.1.m1.2.2.1.1.cmml"><mrow id="S3.I2.i2.p1.1.m1.2.2.1.1.2.2" xref="S3.I2.i2.p1.1.m1.2.2.1.1.2.3.cmml"><mo class="ltx_font_mathcaligraphic" rspace="0em" id="S3.I2.i2.p1.1.m1.1.1" xref="S3.I2.i2.p1.1.m1.1.1.cmml">ℛ</mo><mrow id="S3.I2.i2.p1.1.m1.2.2.1.1.2.2.2" xref="S3.I2.i2.p1.1.m1.2.2.1.1.2.3.cmml"><mo stretchy="false" id="S3.I2.i2.p1.1.m1.2.2.1.1.2.2.2.3" xref="S3.I2.i2.p1.1.m1.2.2.1.1.2.3.cmml">(</mo><msub id="S3.I2.i2.p1.1.m1.2.2.1.1.1.1.1.1" xref="S3.I2.i2.p1.1.m1.2.2.1.1.1.1.1.1.cmml"><mi 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xref="S3.I2.i2.p1.1.m1.2.2.1.1.3.cmml">&lt;</mo><msub id="S3.I2.i2.p1.1.m1.2.2.1.1.4" xref="S3.I2.i2.p1.1.m1.2.2.1.1.4.cmml"><mi id="S3.I2.i2.p1.1.m1.2.2.1.1.4.2" xref="S3.I2.i2.p1.1.m1.2.2.1.1.4.2.cmml">ϵ</mi><mtext id="S3.I2.i2.p1.1.m1.2.2.1.1.4.3" xref="S3.I2.i2.p1.1.m1.2.2.1.1.4.3a.cmml">val</mtext></msub></mrow><mo rspace="0.667em" id="S3.I2.i2.p1.1.m1.3.3.2.3" xref="S3.I2.i2.p1.1.m1.3.3.3a.cmml">,</mo><mrow id="S3.I2.i2.p1.1.m1.3.3.2.2" xref="S3.I2.i2.p1.1.m1.3.3.2.2.cmml"><mrow id="S3.I2.i2.p1.1.m1.3.3.2.2.2" xref="S3.I2.i2.p1.1.m1.3.3.2.2.2.cmml"><mo rspace="0.337em" id="S3.I2.i2.p1.1.m1.3.3.2.2.2.1" xref="S3.I2.i2.p1.1.m1.3.3.2.2.2.1.cmml">∀</mo><msub id="S3.I2.i2.p1.1.m1.3.3.2.2.2.2" xref="S3.I2.i2.p1.1.m1.3.3.2.2.2.2.cmml"><mi id="S3.I2.i2.p1.1.m1.3.3.2.2.2.2.2" xref="S3.I2.i2.p1.1.m1.3.3.2.2.2.2.2.cmml">f</mi><mi id="S3.I2.i2.p1.1.m1.3.3.2.2.2.2.3" xref="S3.I2.i2.p1.1.m1.3.3.2.2.2.2.3.cmml">𝜽</mi></msub></mrow><mo id="S3.I2.i2.p1.1.m1.3.3.2.2.1" xref="S3.I2.i2.p1.1.m1.3.3.2.2.1.cmml">∈</mo><msup id="S3.I2.i2.p1.1.m1.3.3.2.2.3" xref="S3.I2.i2.p1.1.m1.3.3.2.2.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S3.I2.i2.p1.1.m1.3.3.2.2.3.2" xref="S3.I2.i2.p1.1.m1.3.3.2.2.3.2.cmml">ℋ</mi><mo id="S3.I2.i2.p1.1.m1.3.3.2.2.3.3" xref="S3.I2.i2.p1.1.m1.3.3.2.2.3.3.cmml">′</mo></msup></mrow></mrow><annotation-xml encoding="MathML-Content" id="S3.I2.i2.p1.1.m1.3b"><apply id="S3.I2.i2.p1.1.m1.3.3.3.cmml" xref="S3.I2.i2.p1.1.m1.3.3.2"><csymbol cd="ambiguous" id="S3.I2.i2.p1.1.m1.3.3.3a.cmml" xref="S3.I2.i2.p1.1.m1.3.3.2.3">formulae-sequence</csymbol><apply id="S3.I2.i2.p1.1.m1.2.2.1.1.cmml" xref="S3.I2.i2.p1.1.m1.2.2.1.1"><lt id="S3.I2.i2.p1.1.m1.2.2.1.1.3.cmml" xref="S3.I2.i2.p1.1.m1.2.2.1.1.3"></lt><apply id="S3.I2.i2.p1.1.m1.2.2.1.1.2.3.cmml" xref="S3.I2.i2.p1.1.m1.2.2.1.1.2.2"><ci id="S3.I2.i2.p1.1.m1.1.1.cmml" xref="S3.I2.i2.p1.1.m1.1.1">ℛ</ci><apply id="S3.I2.i2.p1.1.m1.2.2.1.1.1.1.1.1.cmml" xref="S3.I2.i2.p1.1.m1.2.2.1.1.1.1.1.1"><csymbol cd="ambiguous" id="S3.I2.i2.p1.1.m1.2.2.1.1.1.1.1.1.1.cmml" xref="S3.I2.i2.p1.1.m1.2.2.1.1.1.1.1.1">subscript</csymbol><ci id="S3.I2.i2.p1.1.m1.2.2.1.1.1.1.1.1.2.cmml" xref="S3.I2.i2.p1.1.m1.2.2.1.1.1.1.1.1.2">𝑓</ci><apply id="S3.I2.i2.p1.1.m1.2.2.1.1.1.1.1.1.3.cmml" xref="S3.I2.i2.p1.1.m1.2.2.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="S3.I2.i2.p1.1.m1.2.2.1.1.1.1.1.1.3.1.cmml" xref="S3.I2.i2.p1.1.m1.2.2.1.1.1.1.1.1.3">subscript</csymbol><ci id="S3.I2.i2.p1.1.m1.2.2.1.1.1.1.1.1.3.2.cmml" xref="S3.I2.i2.p1.1.m1.2.2.1.1.1.1.1.1.3.2">𝜽</ci><ci id="S3.I2.i2.p1.1.m1.2.2.1.1.1.1.1.1.3.3.cmml" xref="S3.I2.i2.p1.1.m1.2.2.1.1.1.1.1.1.3.3">𝑚</ci></apply></apply><apply id="S3.I2.i2.p1.1.m1.2.2.1.1.2.2.2.2.cmml" xref="S3.I2.i2.p1.1.m1.2.2.1.1.2.2.2.2"><csymbol cd="ambiguous" id="S3.I2.i2.p1.1.m1.2.2.1.1.2.2.2.2.1.cmml" xref="S3.I2.i2.p1.1.m1.2.2.1.1.2.2.2.2">subscript</csymbol><ci id="S3.I2.i2.p1.1.m1.2.2.1.1.2.2.2.2.2.cmml" xref="S3.I2.i2.p1.1.m1.2.2.1.1.2.2.2.2.2">𝒟</ci><ci id="S3.I2.i2.p1.1.m1.2.2.1.1.2.2.2.2.3.cmml" xref="S3.I2.i2.p1.1.m1.2.2.1.1.2.2.2.2.3">val</ci></apply></apply><apply id="S3.I2.i2.p1.1.m1.2.2.1.1.4.cmml" xref="S3.I2.i2.p1.1.m1.2.2.1.1.4"><csymbol cd="ambiguous" id="S3.I2.i2.p1.1.m1.2.2.1.1.4.1.cmml" xref="S3.I2.i2.p1.1.m1.2.2.1.1.4">subscript</csymbol><ci id="S3.I2.i2.p1.1.m1.2.2.1.1.4.2.cmml" xref="S3.I2.i2.p1.1.m1.2.2.1.1.4.2">italic-ϵ</ci><ci id="S3.I2.i2.p1.1.m1.2.2.1.1.4.3a.cmml" xref="S3.I2.i2.p1.1.m1.2.2.1.1.4.3"><mtext mathsize="70%" id="S3.I2.i2.p1.1.m1.2.2.1.1.4.3.cmml" xref="S3.I2.i2.p1.1.m1.2.2.1.1.4.3">val</mtext></ci></apply></apply><apply id="S3.I2.i2.p1.1.m1.3.3.2.2.cmml" xref="S3.I2.i2.p1.1.m1.3.3.2.2"><in id="S3.I2.i2.p1.1.m1.3.3.2.2.1.cmml" xref="S3.I2.i2.p1.1.m1.3.3.2.2.1"></in><apply id="S3.I2.i2.p1.1.m1.3.3.2.2.2.cmml" xref="S3.I2.i2.p1.1.m1.3.3.2.2.2"><csymbol cd="latexml" id="S3.I2.i2.p1.1.m1.3.3.2.2.2.1.cmml" 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id="S3.I2.i2.p1.1.m1.3c">\operatorname*{\mathcal{R}}(f_{{\boldsymbol{\theta}}_{m}},\mathcal{D}_{\mathrm{val}})&lt;\epsilon_{\textrm{val}},\leavevmode\nobreak\ \forall\,f_{\boldsymbol{\theta}}\in\mathcal{H}^{\prime}</annotation></semantics></math>,</span>
</span></span>
<span id="S3.I2.i3" class="ltx_item" style="list-style-type:none;"><span class="ltx_tag ltx_tag_item">•</span> 
<span id="S3.I2.i3.p1" class="ltx_para">
<span id="S3.I2.i3.p1.1" class="ltx_p">Distinct OOD predictions:   <math id="S3.I2.i3.p1.1.m1.3" class="ltx_math_unparsed" alttext="\operatorname*{\textrm{P}}\big{(}\operatorname{round}(\sigma(f_{{\boldsymbol{\theta}}1}({\boldsymbol{x}}))\neq\operatorname{round}(\sigma(f_{{\boldsymbol{\theta}}2}({\boldsymbol{x}}))\big{)}\approx 1,\\
\leavevmode\nobreak\ \forall\leavevmode\nobreak\ f_{{\boldsymbol{\theta}}1},f_{{\boldsymbol{\theta}}2}\in\mathcal{H}^{\prime},\leavevmode\nobreak\ \leavevmode\nobreak\ f_{{\boldsymbol{\theta}}1}\neq f_{{\boldsymbol{\theta}}2},\leavevmode\nobreak\ \leavevmode\nobreak\ {\boldsymbol{x}}\!\sim\!\textrm{P}_{\mathrm{OOD}}" display="inline"><semantics id="S3.I2.i3.p1.1.m1.3a"><mrow id="S3.I2.i3.p1.1.m1.3b"><mtext id="S3.I2.i3.p1.1.m1.3.4">P</mtext><mrow id="S3.I2.i3.p1.1.m1.3.5"><mo maxsize="120%" minsize="120%" id="S3.I2.i3.p1.1.m1.3.5.1">(</mo><mi id="S3.I2.i3.p1.1.m1.3.5.2">round</mi><mrow id="S3.I2.i3.p1.1.m1.3.5.3"><mo stretchy="false" id="S3.I2.i3.p1.1.m1.3.5.3.1">(</mo><mi id="S3.I2.i3.p1.1.m1.3.5.3.2">σ</mi><mrow id="S3.I2.i3.p1.1.m1.3.5.3.3"><mo stretchy="false" id="S3.I2.i3.p1.1.m1.3.5.3.3.1">(</mo><msub id="S3.I2.i3.p1.1.m1.3.5.3.3.2"><mi id="S3.I2.i3.p1.1.m1.3.5.3.3.2.2">f</mi><mrow id="S3.I2.i3.p1.1.m1.3.5.3.3.2.3"><mi id="S3.I2.i3.p1.1.m1.3.5.3.3.2.3.2">𝜽</mi><mo lspace="0em" rspace="0em" id="S3.I2.i3.p1.1.m1.3.5.3.3.2.3.1">​</mo><mn id="S3.I2.i3.p1.1.m1.3.5.3.3.2.3.3">1</mn></mrow></msub><mrow id="S3.I2.i3.p1.1.m1.3.5.3.3.3"><mo stretchy="false" id="S3.I2.i3.p1.1.m1.3.5.3.3.3.1">(</mo><mi id="S3.I2.i3.p1.1.m1.1.1">𝒙</mi><mo stretchy="false" id="S3.I2.i3.p1.1.m1.3.5.3.3.3.2">)</mo></mrow><mo stretchy="false" id="S3.I2.i3.p1.1.m1.3.5.3.3.4">)</mo></mrow><mo id="S3.I2.i3.p1.1.m1.3.5.3.4">≠</mo><mi id="S3.I2.i3.p1.1.m1.3.3">round</mi><mrow id="S3.I2.i3.p1.1.m1.3.5.3.5"><mo stretchy="false" id="S3.I2.i3.p1.1.m1.3.5.3.5.1">(</mo><mi id="S3.I2.i3.p1.1.m1.3.5.3.5.2">σ</mi><mrow id="S3.I2.i3.p1.1.m1.3.5.3.5.3"><mo stretchy="false" id="S3.I2.i3.p1.1.m1.3.5.3.5.3.1">(</mo><msub id="S3.I2.i3.p1.1.m1.3.5.3.5.3.2"><mi id="S3.I2.i3.p1.1.m1.3.5.3.5.3.2.2">f</mi><mrow id="S3.I2.i3.p1.1.m1.3.5.3.5.3.2.3"><mi id="S3.I2.i3.p1.1.m1.3.5.3.5.3.2.3.2">𝜽</mi><mo lspace="0em" rspace="0em" id="S3.I2.i3.p1.1.m1.3.5.3.5.3.2.3.1">​</mo><mn id="S3.I2.i3.p1.1.m1.3.5.3.5.3.2.3.3">2</mn></mrow></msub><mrow id="S3.I2.i3.p1.1.m1.3.5.3.5.3.3"><mo stretchy="false" id="S3.I2.i3.p1.1.m1.3.5.3.5.3.3.1">(</mo><mi id="S3.I2.i3.p1.1.m1.2.2">𝒙</mi><mo stretchy="false" id="S3.I2.i3.p1.1.m1.3.5.3.5.3.3.2">)</mo></mrow><mo stretchy="false" id="S3.I2.i3.p1.1.m1.3.5.3.5.3.4">)</mo></mrow><mo maxsize="120%" minsize="120%" id="S3.I2.i3.p1.1.m1.3.5.3.5.4">)</mo></mrow><mo id="S3.I2.i3.p1.1.m1.3.5.3.6">≈</mo><mn id="S3.I2.i3.p1.1.m1.3.5.3.7">1</mn><mo rspace="0.667em" id="S3.I2.i3.p1.1.m1.3.5.3.8">,</mo><mo rspace="0.667em" id="S3.I2.i3.p1.1.m1.3.5.3.9">∀</mo><msub id="S3.I2.i3.p1.1.m1.3.5.3.10"><mi id="S3.I2.i3.p1.1.m1.3.5.3.10.2">f</mi><mrow id="S3.I2.i3.p1.1.m1.3.5.3.10.3"><mi id="S3.I2.i3.p1.1.m1.3.5.3.10.3.2">𝜽</mi><mo lspace="0em" rspace="0em" id="S3.I2.i3.p1.1.m1.3.5.3.10.3.1">​</mo><mn id="S3.I2.i3.p1.1.m1.3.5.3.10.3.3">1</mn></mrow></msub><mo id="S3.I2.i3.p1.1.m1.3.5.3.11">,</mo><msub id="S3.I2.i3.p1.1.m1.3.5.3.12"><mi id="S3.I2.i3.p1.1.m1.3.5.3.12.2">f</mi><mrow id="S3.I2.i3.p1.1.m1.3.5.3.12.3"><mi id="S3.I2.i3.p1.1.m1.3.5.3.12.3.2">𝜽</mi><mo lspace="0em" rspace="0em" id="S3.I2.i3.p1.1.m1.3.5.3.12.3.1">​</mo><mn id="S3.I2.i3.p1.1.m1.3.5.3.12.3.3">2</mn></mrow></msub><mo id="S3.I2.i3.p1.1.m1.3.5.3.13">∈</mo><msup id="S3.I2.i3.p1.1.m1.3.5.3.14"><mi class="ltx_font_mathcaligraphic" id="S3.I2.i3.p1.1.m1.3.5.3.14.2">ℋ</mi><mo id="S3.I2.i3.p1.1.m1.3.5.3.14.3">′</mo></msup><mo rspace="1.167em" id="S3.I2.i3.p1.1.m1.3.5.3.15">,</mo><msub id="S3.I2.i3.p1.1.m1.3.5.3.16"><mi id="S3.I2.i3.p1.1.m1.3.5.3.16.2">f</mi><mrow id="S3.I2.i3.p1.1.m1.3.5.3.16.3"><mi id="S3.I2.i3.p1.1.m1.3.5.3.16.3.2">𝜽</mi><mo lspace="0em" rspace="0em" id="S3.I2.i3.p1.1.m1.3.5.3.16.3.1">​</mo><mn id="S3.I2.i3.p1.1.m1.3.5.3.16.3.3">1</mn></mrow></msub><mo id="S3.I2.i3.p1.1.m1.3.5.3.17">≠</mo><msub id="S3.I2.i3.p1.1.m1.3.5.3.18"><mi id="S3.I2.i3.p1.1.m1.3.5.3.18.2">f</mi><mrow id="S3.I2.i3.p1.1.m1.3.5.3.18.3"><mi id="S3.I2.i3.p1.1.m1.3.5.3.18.3.2">𝜽</mi><mo lspace="0em" rspace="0em" id="S3.I2.i3.p1.1.m1.3.5.3.18.3.1">​</mo><mn id="S3.I2.i3.p1.1.m1.3.5.3.18.3.3">2</mn></mrow></msub><mo rspace="1.167em" id="S3.I2.i3.p1.1.m1.3.5.3.19">,</mo><mi id="S3.I2.i3.p1.1.m1.3.5.3.20">𝒙</mi><mo lspace="0.108em" rspace="0.108em" id="S3.I2.i3.p1.1.m1.3.5.3.21">∼</mo><msub id="S3.I2.i3.p1.1.m1.3.5.3.22"><mtext id="S3.I2.i3.p1.1.m1.3.5.3.22.2">P</mtext><mi id="S3.I2.i3.p1.1.m1.3.5.3.22.3">OOD</mi></msub></mrow></mrow></mrow><annotation encoding="application/x-tex" id="S3.I2.i3.p1.1.m1.3c">\operatorname*{\textrm{P}}\big{(}\operatorname{round}(\sigma(f_{{\boldsymbol{\theta}}1}({\boldsymbol{x}}))\neq\operatorname{round}(\sigma(f_{{\boldsymbol{\theta}}2}({\boldsymbol{x}}))\big{)}\approx 1,\\
\leavevmode\nobreak\ \forall\leavevmode\nobreak\ f_{{\boldsymbol{\theta}}1},f_{{\boldsymbol{\theta}}2}\in\mathcal{H}^{\prime},\leavevmode\nobreak\ \leavevmode\nobreak\ f_{{\boldsymbol{\theta}}1}\neq f_{{\boldsymbol{\theta}}2},\leavevmode\nobreak\ \leavevmode\nobreak\ {\boldsymbol{x}}\!\sim\!\textrm{P}_{\mathrm{OOD}}</annotation></semantics></math>.</span>
</span></span>
</span>
</span>
</span>
</span>
<p id="S3.p6.2" class="ltx_p">In the next section, we derive an algorithm to learn a set of models with these properties.</p>
</div>
</section>
<section id="S4" class="ltx_section">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">4 </span>Proposed method</h2>

<div id="S4.p1" class="ltx_para ltx_noindent">
<p id="S4.p1.1" class="ltx_p"><span id="S4.p1.1.1" class="ltx_text ltx_font_bold">Overview.</span> 
We train multiple models with the same architecture and data while enforcing them to represent different functions and use different features.
The models use different initializations, but this does not always suffice to produce significantly-different models.
We add two regularizers that enforce (1) independence of the models (mutually-orthogonal input gradients)
and (2) alignment with the data manifold such that the models learn meaningful features.</p>
</div>
<div id="S4.p2" class="ltx_para">
<p id="S4.p2.1" class="ltx_p">Since the constraints directly follow from Definition <a href="#Thmdefinition1" title="Definition 1. ‣ 3 Formalizing underspecification ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">1</span></a>, the number of models trainable to satisfy them indicates the degree of underspecification.
The only existence of multiple such models thus highlights cases of underspecification.
The models also discover some predictive features missed by standard ERM, which can be combined by distillation into a predictor with superior OOD performance.
The next sections describe how implement the two constraints as differentiable regularizers.
</p>
</div>
<section id="S4.SS1" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">4.1 </span>Independent models</h3>

<div id="S4.SS1.p1" class="ltx_para">
<p id="S4.SS1.p1.5" class="ltx_p">To optimize for distinct OOD predictions, we turn the criteria of Definition <a href="#Thmdefinition1" title="Definition 1. ‣ 3 Formalizing underspecification ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">1</span></a> into a differentiable objective.
We reuse the concept of independent models from Ross <em id="S4.SS1.p1.5.1" class="ltx_emph ltx_font_italic">et al</em>.<span id="S4.SS1.p1.5.2" class="ltx_text"></span> <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib61" title="" class="ltx_ref">61</a>, <a href="#bib.bib60" title="" class="ltx_ref">60</a>]</cite>.


<span id="S4.SS1.p1.5.3" class="ltx_inline-block ltx_framed ltx_framed_rectangle" style="border-color: #000000;">
<span id="Thmdefinition2" class="ltx_theorem ltx_theorem_definition">
<h6 class="ltx_title ltx_runin ltx_title_theorem">
<span class="ltx_tag ltx_tag_theorem"><span id="Thmdefinition2.1.1.1" class="ltx_text ltx_font_bold">Definition 2</span></span><span id="Thmdefinition2.2.2" class="ltx_text ltx_font_bold">.</span>
</h6>
<span id="Thmdefinition2.p1" class="ltx_para">
<span id="Thmdefinition2.p1.6" class="ltx_p">A pair of predictors <math id="Thmdefinition2.p1.1.m1.1" class="ltx_Math" alttext="f_{{\boldsymbol{\theta}}_{1}}" display="inline"><semantics id="Thmdefinition2.p1.1.m1.1a"><msub id="Thmdefinition2.p1.1.m1.1.1" xref="Thmdefinition2.p1.1.m1.1.1.cmml"><mi id="Thmdefinition2.p1.1.m1.1.1.2" xref="Thmdefinition2.p1.1.m1.1.1.2.cmml">f</mi><msub id="Thmdefinition2.p1.1.m1.1.1.3" xref="Thmdefinition2.p1.1.m1.1.1.3.cmml"><mi id="Thmdefinition2.p1.1.m1.1.1.3.2" xref="Thmdefinition2.p1.1.m1.1.1.3.2.cmml">𝜽</mi><mn id="Thmdefinition2.p1.1.m1.1.1.3.3" xref="Thmdefinition2.p1.1.m1.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="Thmdefinition2.p1.1.m1.1b"><apply id="Thmdefinition2.p1.1.m1.1.1.cmml" xref="Thmdefinition2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="Thmdefinition2.p1.1.m1.1.1.1.cmml" xref="Thmdefinition2.p1.1.m1.1.1">subscript</csymbol><ci id="Thmdefinition2.p1.1.m1.1.1.2.cmml" xref="Thmdefinition2.p1.1.m1.1.1.2">𝑓</ci><apply id="Thmdefinition2.p1.1.m1.1.1.3.cmml" xref="Thmdefinition2.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="Thmdefinition2.p1.1.m1.1.1.3.1.cmml" xref="Thmdefinition2.p1.1.m1.1.1.3">subscript</csymbol><ci id="Thmdefinition2.p1.1.m1.1.1.3.2.cmml" xref="Thmdefinition2.p1.1.m1.1.1.3.2">𝜽</ci><cn type="integer" id="Thmdefinition2.p1.1.m1.1.1.3.3.cmml" xref="Thmdefinition2.p1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition2.p1.1.m1.1c">f_{{\boldsymbol{\theta}}_{1}}</annotation></semantics></math>, <math id="Thmdefinition2.p1.2.m2.1" class="ltx_Math" alttext="f_{{\boldsymbol{\theta}}_{2}}" display="inline"><semantics id="Thmdefinition2.p1.2.m2.1a"><msub id="Thmdefinition2.p1.2.m2.1.1" xref="Thmdefinition2.p1.2.m2.1.1.cmml"><mi id="Thmdefinition2.p1.2.m2.1.1.2" xref="Thmdefinition2.p1.2.m2.1.1.2.cmml">f</mi><msub id="Thmdefinition2.p1.2.m2.1.1.3" xref="Thmdefinition2.p1.2.m2.1.1.3.cmml"><mi id="Thmdefinition2.p1.2.m2.1.1.3.2" xref="Thmdefinition2.p1.2.m2.1.1.3.2.cmml">𝜽</mi><mn id="Thmdefinition2.p1.2.m2.1.1.3.3" xref="Thmdefinition2.p1.2.m2.1.1.3.3.cmml">2</mn></msub></msub><annotation-xml encoding="MathML-Content" id="Thmdefinition2.p1.2.m2.1b"><apply id="Thmdefinition2.p1.2.m2.1.1.cmml" xref="Thmdefinition2.p1.2.m2.1.1"><csymbol cd="ambiguous" id="Thmdefinition2.p1.2.m2.1.1.1.cmml" xref="Thmdefinition2.p1.2.m2.1.1">subscript</csymbol><ci id="Thmdefinition2.p1.2.m2.1.1.2.cmml" xref="Thmdefinition2.p1.2.m2.1.1.2">𝑓</ci><apply id="Thmdefinition2.p1.2.m2.1.1.3.cmml" xref="Thmdefinition2.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="Thmdefinition2.p1.2.m2.1.1.3.1.cmml" xref="Thmdefinition2.p1.2.m2.1.1.3">subscript</csymbol><ci id="Thmdefinition2.p1.2.m2.1.1.3.2.cmml" xref="Thmdefinition2.p1.2.m2.1.1.3.2">𝜽</ci><cn type="integer" id="Thmdefinition2.p1.2.m2.1.1.3.3.cmml" xref="Thmdefinition2.p1.2.m2.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition2.p1.2.m2.1c">f_{{\boldsymbol{\theta}}_{2}}</annotation></semantics></math> are <span id="Thmdefinition2.p1.6.1" class="ltx_text ltx_font_bold">locally independent</span> at <math id="Thmdefinition2.p1.3.m3.1" class="ltx_Math" alttext="{\boldsymbol{x}}" display="inline"><semantics id="Thmdefinition2.p1.3.m3.1a"><mi id="Thmdefinition2.p1.3.m3.1.1" xref="Thmdefinition2.p1.3.m3.1.1.cmml">𝒙</mi><annotation-xml encoding="MathML-Content" id="Thmdefinition2.p1.3.m3.1b"><ci id="Thmdefinition2.p1.3.m3.1.1.cmml" xref="Thmdefinition2.p1.3.m3.1.1">𝒙</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition2.p1.3.m3.1c">{\boldsymbol{x}}</annotation></semantics></math> iff their predictions are statistically independent for Gaussian perturbations around <math id="Thmdefinition2.p1.4.m4.1" class="ltx_Math" alttext="{\boldsymbol{x}}" display="inline"><semantics id="Thmdefinition2.p1.4.m4.1a"><mi id="Thmdefinition2.p1.4.m4.1.1" xref="Thmdefinition2.p1.4.m4.1.1.cmml">𝒙</mi><annotation-xml encoding="MathML-Content" id="Thmdefinition2.p1.4.m4.1b"><ci id="Thmdefinition2.p1.4.m4.1.1.cmml" xref="Thmdefinition2.p1.4.m4.1.1">𝒙</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition2.p1.4.m4.1c">{\boldsymbol{x}}</annotation></semantics></math>:
<math id="Thmdefinition2.p1.5.m5.2" class="ltx_Math" alttext="f_{{\boldsymbol{\theta}}_{1}}(\widetilde{{\boldsymbol{x}}})\nolinebreak\perp\nolinebreak{f}_{{\boldsymbol{\theta}}_{2}}(\widetilde{{\boldsymbol{x}}})" display="inline"><semantics id="Thmdefinition2.p1.5.m5.2a"><mrow id="Thmdefinition2.p1.5.m5.2.3" xref="Thmdefinition2.p1.5.m5.2.3.cmml"><mrow id="Thmdefinition2.p1.5.m5.2.3.2" xref="Thmdefinition2.p1.5.m5.2.3.2.cmml"><msub id="Thmdefinition2.p1.5.m5.2.3.2.2" xref="Thmdefinition2.p1.5.m5.2.3.2.2.cmml"><mi id="Thmdefinition2.p1.5.m5.2.3.2.2.2" xref="Thmdefinition2.p1.5.m5.2.3.2.2.2.cmml">f</mi><msub id="Thmdefinition2.p1.5.m5.2.3.2.2.3" xref="Thmdefinition2.p1.5.m5.2.3.2.2.3.cmml"><mi id="Thmdefinition2.p1.5.m5.2.3.2.2.3.2" xref="Thmdefinition2.p1.5.m5.2.3.2.2.3.2.cmml">𝜽</mi><mn id="Thmdefinition2.p1.5.m5.2.3.2.2.3.3" xref="Thmdefinition2.p1.5.m5.2.3.2.2.3.3.cmml">1</mn></msub></msub><mo lspace="0em" rspace="0em" id="Thmdefinition2.p1.5.m5.2.3.2.1" xref="Thmdefinition2.p1.5.m5.2.3.2.1.cmml">​</mo><mrow id="Thmdefinition2.p1.5.m5.2.3.2.3.2" xref="Thmdefinition2.p1.5.m5.1.1.cmml"><mo stretchy="false" id="Thmdefinition2.p1.5.m5.2.3.2.3.2.1" xref="Thmdefinition2.p1.5.m5.1.1.cmml">(</mo><mover accent="true" id="Thmdefinition2.p1.5.m5.1.1" xref="Thmdefinition2.p1.5.m5.1.1.cmml"><mi id="Thmdefinition2.p1.5.m5.1.1.2" xref="Thmdefinition2.p1.5.m5.1.1.2.cmml">𝒙</mi><mo id="Thmdefinition2.p1.5.m5.1.1.1" xref="Thmdefinition2.p1.5.m5.1.1.1.cmml">~</mo></mover><mo stretchy="false" id="Thmdefinition2.p1.5.m5.2.3.2.3.2.2" xref="Thmdefinition2.p1.5.m5.1.1.cmml">)</mo></mrow></mrow><mo id="Thmdefinition2.p1.5.m5.2.3.1" xref="Thmdefinition2.p1.5.m5.2.3.1.cmml">⟂</mo><mrow id="Thmdefinition2.p1.5.m5.2.3.3" xref="Thmdefinition2.p1.5.m5.2.3.3.cmml"><msub id="Thmdefinition2.p1.5.m5.2.3.3.2" xref="Thmdefinition2.p1.5.m5.2.3.3.2.cmml"><mi id="Thmdefinition2.p1.5.m5.2.3.3.2.2" xref="Thmdefinition2.p1.5.m5.2.3.3.2.2.cmml">f</mi><msub id="Thmdefinition2.p1.5.m5.2.3.3.2.3" xref="Thmdefinition2.p1.5.m5.2.3.3.2.3.cmml"><mi id="Thmdefinition2.p1.5.m5.2.3.3.2.3.2" xref="Thmdefinition2.p1.5.m5.2.3.3.2.3.2.cmml">𝜽</mi><mn id="Thmdefinition2.p1.5.m5.2.3.3.2.3.3" xref="Thmdefinition2.p1.5.m5.2.3.3.2.3.3.cmml">2</mn></msub></msub><mo lspace="0em" rspace="0em" id="Thmdefinition2.p1.5.m5.2.3.3.1" xref="Thmdefinition2.p1.5.m5.2.3.3.1.cmml">​</mo><mrow id="Thmdefinition2.p1.5.m5.2.3.3.3.2" xref="Thmdefinition2.p1.5.m5.2.2.cmml"><mo stretchy="false" id="Thmdefinition2.p1.5.m5.2.3.3.3.2.1" xref="Thmdefinition2.p1.5.m5.2.2.cmml">(</mo><mover accent="true" id="Thmdefinition2.p1.5.m5.2.2" xref="Thmdefinition2.p1.5.m5.2.2.cmml"><mi id="Thmdefinition2.p1.5.m5.2.2.2" xref="Thmdefinition2.p1.5.m5.2.2.2.cmml">𝒙</mi><mo id="Thmdefinition2.p1.5.m5.2.2.1" xref="Thmdefinition2.p1.5.m5.2.2.1.cmml">~</mo></mover><mo stretchy="false" id="Thmdefinition2.p1.5.m5.2.3.3.3.2.2" xref="Thmdefinition2.p1.5.m5.2.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmdefinition2.p1.5.m5.2b"><apply id="Thmdefinition2.p1.5.m5.2.3.cmml" xref="Thmdefinition2.p1.5.m5.2.3"><csymbol cd="latexml" id="Thmdefinition2.p1.5.m5.2.3.1.cmml" xref="Thmdefinition2.p1.5.m5.2.3.1">perpendicular-to</csymbol><apply id="Thmdefinition2.p1.5.m5.2.3.2.cmml" xref="Thmdefinition2.p1.5.m5.2.3.2"><times id="Thmdefinition2.p1.5.m5.2.3.2.1.cmml" xref="Thmdefinition2.p1.5.m5.2.3.2.1"></times><apply id="Thmdefinition2.p1.5.m5.2.3.2.2.cmml" xref="Thmdefinition2.p1.5.m5.2.3.2.2"><csymbol cd="ambiguous" id="Thmdefinition2.p1.5.m5.2.3.2.2.1.cmml" xref="Thmdefinition2.p1.5.m5.2.3.2.2">subscript</csymbol><ci id="Thmdefinition2.p1.5.m5.2.3.2.2.2.cmml" xref="Thmdefinition2.p1.5.m5.2.3.2.2.2">𝑓</ci><apply id="Thmdefinition2.p1.5.m5.2.3.2.2.3.cmml" xref="Thmdefinition2.p1.5.m5.2.3.2.2.3"><csymbol cd="ambiguous" id="Thmdefinition2.p1.5.m5.2.3.2.2.3.1.cmml" xref="Thmdefinition2.p1.5.m5.2.3.2.2.3">subscript</csymbol><ci id="Thmdefinition2.p1.5.m5.2.3.2.2.3.2.cmml" xref="Thmdefinition2.p1.5.m5.2.3.2.2.3.2">𝜽</ci><cn type="integer" id="Thmdefinition2.p1.5.m5.2.3.2.2.3.3.cmml" xref="Thmdefinition2.p1.5.m5.2.3.2.2.3.3">1</cn></apply></apply><apply id="Thmdefinition2.p1.5.m5.1.1.cmml" xref="Thmdefinition2.p1.5.m5.2.3.2.3.2"><ci id="Thmdefinition2.p1.5.m5.1.1.1.cmml" xref="Thmdefinition2.p1.5.m5.1.1.1">~</ci><ci id="Thmdefinition2.p1.5.m5.1.1.2.cmml" xref="Thmdefinition2.p1.5.m5.1.1.2">𝒙</ci></apply></apply><apply id="Thmdefinition2.p1.5.m5.2.3.3.cmml" xref="Thmdefinition2.p1.5.m5.2.3.3"><times id="Thmdefinition2.p1.5.m5.2.3.3.1.cmml" xref="Thmdefinition2.p1.5.m5.2.3.3.1"></times><apply id="Thmdefinition2.p1.5.m5.2.3.3.2.cmml" xref="Thmdefinition2.p1.5.m5.2.3.3.2"><csymbol cd="ambiguous" id="Thmdefinition2.p1.5.m5.2.3.3.2.1.cmml" xref="Thmdefinition2.p1.5.m5.2.3.3.2">subscript</csymbol><ci id="Thmdefinition2.p1.5.m5.2.3.3.2.2.cmml" xref="Thmdefinition2.p1.5.m5.2.3.3.2.2">𝑓</ci><apply id="Thmdefinition2.p1.5.m5.2.3.3.2.3.cmml" xref="Thmdefinition2.p1.5.m5.2.3.3.2.3"><csymbol cd="ambiguous" id="Thmdefinition2.p1.5.m5.2.3.3.2.3.1.cmml" xref="Thmdefinition2.p1.5.m5.2.3.3.2.3">subscript</csymbol><ci id="Thmdefinition2.p1.5.m5.2.3.3.2.3.2.cmml" xref="Thmdefinition2.p1.5.m5.2.3.3.2.3.2">𝜽</ci><cn type="integer" id="Thmdefinition2.p1.5.m5.2.3.3.2.3.3.cmml" xref="Thmdefinition2.p1.5.m5.2.3.3.2.3.3">2</cn></apply></apply><apply id="Thmdefinition2.p1.5.m5.2.2.cmml" xref="Thmdefinition2.p1.5.m5.2.3.3.3.2"><ci id="Thmdefinition2.p1.5.m5.2.2.1.cmml" xref="Thmdefinition2.p1.5.m5.2.2.1">~</ci><ci id="Thmdefinition2.p1.5.m5.2.2.2.cmml" xref="Thmdefinition2.p1.5.m5.2.2.2">𝒙</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition2.p1.5.m5.2c">f_{{\boldsymbol{\theta}}_{1}}(\widetilde{{\boldsymbol{x}}})\nolinebreak\perp\nolinebreak{f}_{{\boldsymbol{\theta}}_{2}}(\widetilde{{\boldsymbol{x}}})</annotation></semantics></math>,  <math id="Thmdefinition2.p1.6.m6.2" class="ltx_Math" alttext="\widetilde{{\boldsymbol{x}}}\sim\mathcal{N}({\boldsymbol{x}},\sigma{\boldsymbol{I}})" display="inline"><semantics id="Thmdefinition2.p1.6.m6.2a"><mrow id="Thmdefinition2.p1.6.m6.2.2" xref="Thmdefinition2.p1.6.m6.2.2.cmml"><mover accent="true" id="Thmdefinition2.p1.6.m6.2.2.3" xref="Thmdefinition2.p1.6.m6.2.2.3.cmml"><mi id="Thmdefinition2.p1.6.m6.2.2.3.2" xref="Thmdefinition2.p1.6.m6.2.2.3.2.cmml">𝒙</mi><mo id="Thmdefinition2.p1.6.m6.2.2.3.1" xref="Thmdefinition2.p1.6.m6.2.2.3.1.cmml">~</mo></mover><mo id="Thmdefinition2.p1.6.m6.2.2.2" xref="Thmdefinition2.p1.6.m6.2.2.2.cmml">∼</mo><mrow id="Thmdefinition2.p1.6.m6.2.2.1" xref="Thmdefinition2.p1.6.m6.2.2.1.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmdefinition2.p1.6.m6.2.2.1.3" xref="Thmdefinition2.p1.6.m6.2.2.1.3.cmml">𝒩</mi><mo lspace="0em" rspace="0em" id="Thmdefinition2.p1.6.m6.2.2.1.2" xref="Thmdefinition2.p1.6.m6.2.2.1.2.cmml">​</mo><mrow id="Thmdefinition2.p1.6.m6.2.2.1.1.1" xref="Thmdefinition2.p1.6.m6.2.2.1.1.2.cmml"><mo stretchy="false" id="Thmdefinition2.p1.6.m6.2.2.1.1.1.2" xref="Thmdefinition2.p1.6.m6.2.2.1.1.2.cmml">(</mo><mi id="Thmdefinition2.p1.6.m6.1.1" xref="Thmdefinition2.p1.6.m6.1.1.cmml">𝒙</mi><mo id="Thmdefinition2.p1.6.m6.2.2.1.1.1.3" xref="Thmdefinition2.p1.6.m6.2.2.1.1.2.cmml">,</mo><mrow id="Thmdefinition2.p1.6.m6.2.2.1.1.1.1" xref="Thmdefinition2.p1.6.m6.2.2.1.1.1.1.cmml"><mi id="Thmdefinition2.p1.6.m6.2.2.1.1.1.1.2" xref="Thmdefinition2.p1.6.m6.2.2.1.1.1.1.2.cmml">σ</mi><mo lspace="0em" rspace="0em" id="Thmdefinition2.p1.6.m6.2.2.1.1.1.1.1" xref="Thmdefinition2.p1.6.m6.2.2.1.1.1.1.1.cmml">​</mo><mi id="Thmdefinition2.p1.6.m6.2.2.1.1.1.1.3" xref="Thmdefinition2.p1.6.m6.2.2.1.1.1.1.3.cmml">𝑰</mi></mrow><mo stretchy="false" id="Thmdefinition2.p1.6.m6.2.2.1.1.1.4" xref="Thmdefinition2.p1.6.m6.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmdefinition2.p1.6.m6.2b"><apply id="Thmdefinition2.p1.6.m6.2.2.cmml" xref="Thmdefinition2.p1.6.m6.2.2"><csymbol cd="latexml" id="Thmdefinition2.p1.6.m6.2.2.2.cmml" xref="Thmdefinition2.p1.6.m6.2.2.2">similar-to</csymbol><apply id="Thmdefinition2.p1.6.m6.2.2.3.cmml" xref="Thmdefinition2.p1.6.m6.2.2.3"><ci id="Thmdefinition2.p1.6.m6.2.2.3.1.cmml" xref="Thmdefinition2.p1.6.m6.2.2.3.1">~</ci><ci id="Thmdefinition2.p1.6.m6.2.2.3.2.cmml" xref="Thmdefinition2.p1.6.m6.2.2.3.2">𝒙</ci></apply><apply id="Thmdefinition2.p1.6.m6.2.2.1.cmml" xref="Thmdefinition2.p1.6.m6.2.2.1"><times id="Thmdefinition2.p1.6.m6.2.2.1.2.cmml" xref="Thmdefinition2.p1.6.m6.2.2.1.2"></times><ci id="Thmdefinition2.p1.6.m6.2.2.1.3.cmml" xref="Thmdefinition2.p1.6.m6.2.2.1.3">𝒩</ci><interval closure="open" id="Thmdefinition2.p1.6.m6.2.2.1.1.2.cmml" xref="Thmdefinition2.p1.6.m6.2.2.1.1.1"><ci id="Thmdefinition2.p1.6.m6.1.1.cmml" xref="Thmdefinition2.p1.6.m6.1.1">𝒙</ci><apply id="Thmdefinition2.p1.6.m6.2.2.1.1.1.1.cmml" xref="Thmdefinition2.p1.6.m6.2.2.1.1.1.1"><times id="Thmdefinition2.p1.6.m6.2.2.1.1.1.1.1.cmml" xref="Thmdefinition2.p1.6.m6.2.2.1.1.1.1.1"></times><ci id="Thmdefinition2.p1.6.m6.2.2.1.1.1.1.2.cmml" xref="Thmdefinition2.p1.6.m6.2.2.1.1.1.1.2">𝜎</ci><ci id="Thmdefinition2.p1.6.m6.2.2.1.1.1.1.3.cmml" xref="Thmdefinition2.p1.6.m6.2.2.1.1.1.1.3">𝑰</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition2.p1.6.m6.2c">\widetilde{{\boldsymbol{x}}}\sim\mathcal{N}({\boldsymbol{x}},\sigma{\boldsymbol{I}})</annotation></semantics></math>.</span>
</span>
</span>
<span id="Thmdefinition3" class="ltx_theorem ltx_theorem_definition">
<h6 class="ltx_title ltx_runin ltx_title_theorem">
<span class="ltx_tag ltx_tag_theorem"><span id="Thmdefinition3.1.1.1" class="ltx_text ltx_font_bold">Definition 3</span></span><span id="Thmdefinition3.2.2" class="ltx_text ltx_font_bold">.</span>
</h6>
<span id="Thmdefinition3.p1" class="ltx_para">
<span id="Thmdefinition3.p1.3" class="ltx_p">A set of predictors <math id="Thmdefinition3.p1.1.m1.3" class="ltx_Math" alttext="\{f_{{\boldsymbol{\theta}}_{1}},\kern-1.00006pt.\kern-1.00006pt.\kern-1.00006pt.,f_{{\boldsymbol{\theta}}_{M}}\}" display="inline"><semantics id="Thmdefinition3.p1.1.m1.3a"><mrow id="Thmdefinition3.p1.1.m1.3.3.2" xref="Thmdefinition3.p1.1.m1.3.3.3.cmml"><mo stretchy="false" id="Thmdefinition3.p1.1.m1.3.3.2.3" xref="Thmdefinition3.p1.1.m1.3.3.3.cmml">{</mo><msub id="Thmdefinition3.p1.1.m1.2.2.1.1" xref="Thmdefinition3.p1.1.m1.2.2.1.1.cmml"><mi id="Thmdefinition3.p1.1.m1.2.2.1.1.2" xref="Thmdefinition3.p1.1.m1.2.2.1.1.2.cmml">f</mi><msub id="Thmdefinition3.p1.1.m1.2.2.1.1.3" xref="Thmdefinition3.p1.1.m1.2.2.1.1.3.cmml"><mi id="Thmdefinition3.p1.1.m1.2.2.1.1.3.2" xref="Thmdefinition3.p1.1.m1.2.2.1.1.3.2.cmml">𝜽</mi><mn id="Thmdefinition3.p1.1.m1.2.2.1.1.3.3" xref="Thmdefinition3.p1.1.m1.2.2.1.1.3.3.cmml">1</mn></msub></msub><mo id="Thmdefinition3.p1.1.m1.3.3.2.4" xref="Thmdefinition3.p1.1.m1.3.3.3.cmml">,</mo><mi mathvariant="normal" id="Thmdefinition3.p1.1.m1.1.1" xref="Thmdefinition3.p1.1.m1.1.1.cmml">…</mi><mo id="Thmdefinition3.p1.1.m1.3.3.2.5" xref="Thmdefinition3.p1.1.m1.3.3.3.cmml">,</mo><msub id="Thmdefinition3.p1.1.m1.3.3.2.2" xref="Thmdefinition3.p1.1.m1.3.3.2.2.cmml"><mi id="Thmdefinition3.p1.1.m1.3.3.2.2.2" xref="Thmdefinition3.p1.1.m1.3.3.2.2.2.cmml">f</mi><msub id="Thmdefinition3.p1.1.m1.3.3.2.2.3" xref="Thmdefinition3.p1.1.m1.3.3.2.2.3.cmml"><mi id="Thmdefinition3.p1.1.m1.3.3.2.2.3.2" xref="Thmdefinition3.p1.1.m1.3.3.2.2.3.2.cmml">𝜽</mi><mi id="Thmdefinition3.p1.1.m1.3.3.2.2.3.3" xref="Thmdefinition3.p1.1.m1.3.3.2.2.3.3.cmml">M</mi></msub></msub><mo stretchy="false" id="Thmdefinition3.p1.1.m1.3.3.2.6" xref="Thmdefinition3.p1.1.m1.3.3.3.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="Thmdefinition3.p1.1.m1.3b"><set id="Thmdefinition3.p1.1.m1.3.3.3.cmml" xref="Thmdefinition3.p1.1.m1.3.3.2"><apply id="Thmdefinition3.p1.1.m1.2.2.1.1.cmml" xref="Thmdefinition3.p1.1.m1.2.2.1.1"><csymbol cd="ambiguous" id="Thmdefinition3.p1.1.m1.2.2.1.1.1.cmml" xref="Thmdefinition3.p1.1.m1.2.2.1.1">subscript</csymbol><ci id="Thmdefinition3.p1.1.m1.2.2.1.1.2.cmml" xref="Thmdefinition3.p1.1.m1.2.2.1.1.2">𝑓</ci><apply id="Thmdefinition3.p1.1.m1.2.2.1.1.3.cmml" xref="Thmdefinition3.p1.1.m1.2.2.1.1.3"><csymbol cd="ambiguous" id="Thmdefinition3.p1.1.m1.2.2.1.1.3.1.cmml" xref="Thmdefinition3.p1.1.m1.2.2.1.1.3">subscript</csymbol><ci id="Thmdefinition3.p1.1.m1.2.2.1.1.3.2.cmml" xref="Thmdefinition3.p1.1.m1.2.2.1.1.3.2">𝜽</ci><cn type="integer" id="Thmdefinition3.p1.1.m1.2.2.1.1.3.3.cmml" xref="Thmdefinition3.p1.1.m1.2.2.1.1.3.3">1</cn></apply></apply><ci id="Thmdefinition3.p1.1.m1.1.1.cmml" xref="Thmdefinition3.p1.1.m1.1.1">…</ci><apply id="Thmdefinition3.p1.1.m1.3.3.2.2.cmml" xref="Thmdefinition3.p1.1.m1.3.3.2.2"><csymbol cd="ambiguous" id="Thmdefinition3.p1.1.m1.3.3.2.2.1.cmml" xref="Thmdefinition3.p1.1.m1.3.3.2.2">subscript</csymbol><ci id="Thmdefinition3.p1.1.m1.3.3.2.2.2.cmml" xref="Thmdefinition3.p1.1.m1.3.3.2.2.2">𝑓</ci><apply id="Thmdefinition3.p1.1.m1.3.3.2.2.3.cmml" xref="Thmdefinition3.p1.1.m1.3.3.2.2.3"><csymbol cd="ambiguous" id="Thmdefinition3.p1.1.m1.3.3.2.2.3.1.cmml" xref="Thmdefinition3.p1.1.m1.3.3.2.2.3">subscript</csymbol><ci id="Thmdefinition3.p1.1.m1.3.3.2.2.3.2.cmml" xref="Thmdefinition3.p1.1.m1.3.3.2.2.3.2">𝜽</ci><ci id="Thmdefinition3.p1.1.m1.3.3.2.2.3.3.cmml" xref="Thmdefinition3.p1.1.m1.3.3.2.2.3.3">𝑀</ci></apply></apply></set></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition3.p1.1.m1.3c">\{f_{{\boldsymbol{\theta}}_{1}},\kern-1.00006pt.\kern-1.00006pt.\kern-1.00006pt.,f_{{\boldsymbol{\theta}}_{M}}\}</annotation></semantics></math> are <span id="Thmdefinition3.p1.3.1" class="ltx_text ltx_font_bold">globally independent</span> on a dataset <math id="Thmdefinition3.p1.2.m2.1" class="ltx_Math" alttext="\mathcal{D}" display="inline"><semantics id="Thmdefinition3.p1.2.m2.1a"><mi class="ltx_font_mathcaligraphic" id="Thmdefinition3.p1.2.m2.1.1" xref="Thmdefinition3.p1.2.m2.1.1.cmml">𝒟</mi><annotation-xml encoding="MathML-Content" id="Thmdefinition3.p1.2.m2.1b"><ci id="Thmdefinition3.p1.2.m2.1.1.cmml" xref="Thmdefinition3.p1.2.m2.1.1">𝒟</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition3.p1.2.m2.1c">\mathcal{D}</annotation></semantics></math> iff every pair of them are locally independent at every <math id="Thmdefinition3.p1.3.m3.1" class="ltx_Math" alttext="{\boldsymbol{x}}\in\mathcal{D}" display="inline"><semantics id="Thmdefinition3.p1.3.m3.1a"><mrow id="Thmdefinition3.p1.3.m3.1.1" xref="Thmdefinition3.p1.3.m3.1.1.cmml"><mi id="Thmdefinition3.p1.3.m3.1.1.2" xref="Thmdefinition3.p1.3.m3.1.1.2.cmml">𝒙</mi><mo id="Thmdefinition3.p1.3.m3.1.1.1" xref="Thmdefinition3.p1.3.m3.1.1.1.cmml">∈</mo><mi class="ltx_font_mathcaligraphic" id="Thmdefinition3.p1.3.m3.1.1.3" xref="Thmdefinition3.p1.3.m3.1.1.3.cmml">𝒟</mi></mrow><annotation-xml encoding="MathML-Content" id="Thmdefinition3.p1.3.m3.1b"><apply id="Thmdefinition3.p1.3.m3.1.1.cmml" xref="Thmdefinition3.p1.3.m3.1.1"><in id="Thmdefinition3.p1.3.m3.1.1.1.cmml" xref="Thmdefinition3.p1.3.m3.1.1.1"></in><ci id="Thmdefinition3.p1.3.m3.1.1.2.cmml" xref="Thmdefinition3.p1.3.m3.1.1.2">𝒙</ci><ci id="Thmdefinition3.p1.3.m3.1.1.3.cmml" xref="Thmdefinition3.p1.3.m3.1.1.3">𝒟</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmdefinition3.p1.3.m3.1c">{\boldsymbol{x}}\in\mathcal{D}</annotation></semantics></math>.</span>
</span>
</span>
</span>

This formalizes the notion that models can rely on different features.
In our case, we seek a set of models globally independent from one another.
We obtain a tractable objective using the relation between statistical independence and geometric orthogonality developed in <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib61" title="" class="ltx_ref">61</a>]</cite>.


<span id="S4.SS1.p1.5.4" class="ltx_inline-block ltx_framed ltx_framed_rectangle" style="border-color: #000000;">
<span id="Thmproposition1" class="ltx_theorem ltx_theorem_proposition">
<h6 class="ltx_title ltx_runin ltx_title_theorem">
<span class="ltx_tag ltx_tag_theorem"><span id="Thmproposition1.1.1.1" class="ltx_text ltx_font_bold">Proposition 1</span></span><span id="Thmproposition1.2.2" class="ltx_text ltx_font_bold">.</span>
</h6>
<span id="Thmproposition1.p1" class="ltx_para">
<span id="Thmproposition1.p1.5" class="ltx_p">A pair of predictors <math id="Thmproposition1.p1.1.m1.1" class="ltx_Math" alttext="f_{{\boldsymbol{\theta}}_{1}}" display="inline"><semantics id="Thmproposition1.p1.1.m1.1a"><msub id="Thmproposition1.p1.1.m1.1.1" xref="Thmproposition1.p1.1.m1.1.1.cmml"><mi id="Thmproposition1.p1.1.m1.1.1.2" xref="Thmproposition1.p1.1.m1.1.1.2.cmml">f</mi><msub id="Thmproposition1.p1.1.m1.1.1.3" xref="Thmproposition1.p1.1.m1.1.1.3.cmml"><mi id="Thmproposition1.p1.1.m1.1.1.3.2" xref="Thmproposition1.p1.1.m1.1.1.3.2.cmml">𝜽</mi><mn id="Thmproposition1.p1.1.m1.1.1.3.3" xref="Thmproposition1.p1.1.m1.1.1.3.3.cmml">1</mn></msub></msub><annotation-xml encoding="MathML-Content" id="Thmproposition1.p1.1.m1.1b"><apply id="Thmproposition1.p1.1.m1.1.1.cmml" xref="Thmproposition1.p1.1.m1.1.1"><csymbol cd="ambiguous" id="Thmproposition1.p1.1.m1.1.1.1.cmml" xref="Thmproposition1.p1.1.m1.1.1">subscript</csymbol><ci id="Thmproposition1.p1.1.m1.1.1.2.cmml" xref="Thmproposition1.p1.1.m1.1.1.2">𝑓</ci><apply id="Thmproposition1.p1.1.m1.1.1.3.cmml" xref="Thmproposition1.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="Thmproposition1.p1.1.m1.1.1.3.1.cmml" xref="Thmproposition1.p1.1.m1.1.1.3">subscript</csymbol><ci id="Thmproposition1.p1.1.m1.1.1.3.2.cmml" xref="Thmproposition1.p1.1.m1.1.1.3.2">𝜽</ci><cn type="integer" id="Thmproposition1.p1.1.m1.1.1.3.3.cmml" xref="Thmproposition1.p1.1.m1.1.1.3.3">1</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition1.p1.1.m1.1c">f_{{\boldsymbol{\theta}}_{1}}</annotation></semantics></math>, <math id="Thmproposition1.p1.2.m2.1" class="ltx_Math" alttext="f_{{\boldsymbol{\theta}}_{2}}" display="inline"><semantics id="Thmproposition1.p1.2.m2.1a"><msub id="Thmproposition1.p1.2.m2.1.1" xref="Thmproposition1.p1.2.m2.1.1.cmml"><mi id="Thmproposition1.p1.2.m2.1.1.2" xref="Thmproposition1.p1.2.m2.1.1.2.cmml">f</mi><msub id="Thmproposition1.p1.2.m2.1.1.3" xref="Thmproposition1.p1.2.m2.1.1.3.cmml"><mi id="Thmproposition1.p1.2.m2.1.1.3.2" xref="Thmproposition1.p1.2.m2.1.1.3.2.cmml">𝜽</mi><mn id="Thmproposition1.p1.2.m2.1.1.3.3" xref="Thmproposition1.p1.2.m2.1.1.3.3.cmml">2</mn></msub></msub><annotation-xml encoding="MathML-Content" id="Thmproposition1.p1.2.m2.1b"><apply id="Thmproposition1.p1.2.m2.1.1.cmml" xref="Thmproposition1.p1.2.m2.1.1"><csymbol cd="ambiguous" id="Thmproposition1.p1.2.m2.1.1.1.cmml" xref="Thmproposition1.p1.2.m2.1.1">subscript</csymbol><ci id="Thmproposition1.p1.2.m2.1.1.2.cmml" xref="Thmproposition1.p1.2.m2.1.1.2">𝑓</ci><apply id="Thmproposition1.p1.2.m2.1.1.3.cmml" xref="Thmproposition1.p1.2.m2.1.1.3"><csymbol cd="ambiguous" id="Thmproposition1.p1.2.m2.1.1.3.1.cmml" xref="Thmproposition1.p1.2.m2.1.1.3">subscript</csymbol><ci id="Thmproposition1.p1.2.m2.1.1.3.2.cmml" xref="Thmproposition1.p1.2.m2.1.1.3.2">𝜽</ci><cn type="integer" id="Thmproposition1.p1.2.m2.1.1.3.3.cmml" xref="Thmproposition1.p1.2.m2.1.1.3.3">2</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition1.p1.2.m2.1c">f_{{\boldsymbol{\theta}}_{2}}</annotation></semantics></math> are locally independent at <math id="Thmproposition1.p1.3.m3.1" class="ltx_Math" alttext="{\boldsymbol{x}}" display="inline"><semantics id="Thmproposition1.p1.3.m3.1a"><mi id="Thmproposition1.p1.3.m3.1.1" xref="Thmproposition1.p1.3.m3.1.1.cmml">𝒙</mi><annotation-xml encoding="MathML-Content" id="Thmproposition1.p1.3.m3.1b"><ci id="Thmproposition1.p1.3.m3.1.1.cmml" xref="Thmproposition1.p1.3.m3.1.1">𝒙</ci></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition1.p1.3.m3.1c">{\boldsymbol{x}}</annotation></semantics></math> (Definition <a href="#Thmdefinition2" title="Definition 2. ‣ 4.1 Independent models ‣ 4 Proposed method ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">2</span></a>) iff the mutual information
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id="Thmproposition1.p1.4.m4.5.5.2.2.2.2.2.3.3.cmml" xref="Thmproposition1.p1.4.m4.5.5.2.2.2.2.2.3.3">2</cn></apply></apply><apply id="Thmproposition1.p1.4.m4.2.2.cmml" xref="Thmproposition1.p1.4.m4.5.5.2.2.2.2.3.2"><ci id="Thmproposition1.p1.4.m4.2.2.1.cmml" xref="Thmproposition1.p1.4.m4.2.2.1">~</ci><ci id="Thmproposition1.p1.4.m4.2.2.2.cmml" xref="Thmproposition1.p1.4.m4.2.2.2">𝒙</ci></apply></apply></apply><cn type="integer" id="Thmproposition1.p1.4.m4.5.5.4.cmml" xref="Thmproposition1.p1.4.m4.5.5.4">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition1.p1.4.m4.5c">\operatorname*{\textrm{MI}}(f_{{\boldsymbol{\theta}}_{1}}(\widetilde{{\boldsymbol{x}}}),\,f_{{\boldsymbol{\theta}}_{2}}(\widetilde{{\boldsymbol{x}}}))=0</annotation></semantics></math> with <math id="Thmproposition1.p1.5.m5.2" class="ltx_Math" alttext="\widetilde{{\boldsymbol{x}}}\leavevmode\nobreak\ \sim\leavevmode\nobreak\ \mathcal{N}({\boldsymbol{x}},\sigma{\boldsymbol{I}})" display="inline"><semantics id="Thmproposition1.p1.5.m5.2a"><mrow id="Thmproposition1.p1.5.m5.2.2" xref="Thmproposition1.p1.5.m5.2.2.cmml"><mover accent="true" id="Thmproposition1.p1.5.m5.2.2.3" xref="Thmproposition1.p1.5.m5.2.2.3.cmml"><mi id="Thmproposition1.p1.5.m5.2.2.3.2" xref="Thmproposition1.p1.5.m5.2.2.3.2.cmml">𝒙</mi><mo id="Thmproposition1.p1.5.m5.2.2.3.1" xref="Thmproposition1.p1.5.m5.2.2.3.1.cmml">~</mo></mover><mo lspace="0.778em" rspace="0.778em" id="Thmproposition1.p1.5.m5.2.2.2" xref="Thmproposition1.p1.5.m5.2.2.2.cmml">∼</mo><mrow id="Thmproposition1.p1.5.m5.2.2.1" xref="Thmproposition1.p1.5.m5.2.2.1.cmml"><mi class="ltx_font_mathcaligraphic" id="Thmproposition1.p1.5.m5.2.2.1.3" xref="Thmproposition1.p1.5.m5.2.2.1.3.cmml">𝒩</mi><mo lspace="0em" rspace="0em" id="Thmproposition1.p1.5.m5.2.2.1.2" xref="Thmproposition1.p1.5.m5.2.2.1.2.cmml">​</mo><mrow id="Thmproposition1.p1.5.m5.2.2.1.1.1" xref="Thmproposition1.p1.5.m5.2.2.1.1.2.cmml"><mo stretchy="false" id="Thmproposition1.p1.5.m5.2.2.1.1.1.2" xref="Thmproposition1.p1.5.m5.2.2.1.1.2.cmml">(</mo><mi id="Thmproposition1.p1.5.m5.1.1" xref="Thmproposition1.p1.5.m5.1.1.cmml">𝒙</mi><mo id="Thmproposition1.p1.5.m5.2.2.1.1.1.3" xref="Thmproposition1.p1.5.m5.2.2.1.1.2.cmml">,</mo><mrow id="Thmproposition1.p1.5.m5.2.2.1.1.1.1" xref="Thmproposition1.p1.5.m5.2.2.1.1.1.1.cmml"><mi id="Thmproposition1.p1.5.m5.2.2.1.1.1.1.2" xref="Thmproposition1.p1.5.m5.2.2.1.1.1.1.2.cmml">σ</mi><mo lspace="0em" rspace="0em" id="Thmproposition1.p1.5.m5.2.2.1.1.1.1.1" xref="Thmproposition1.p1.5.m5.2.2.1.1.1.1.1.cmml">​</mo><mi id="Thmproposition1.p1.5.m5.2.2.1.1.1.1.3" xref="Thmproposition1.p1.5.m5.2.2.1.1.1.1.3.cmml">𝑰</mi></mrow><mo stretchy="false" id="Thmproposition1.p1.5.m5.2.2.1.1.1.4" xref="Thmproposition1.p1.5.m5.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="Thmproposition1.p1.5.m5.2b"><apply id="Thmproposition1.p1.5.m5.2.2.cmml" xref="Thmproposition1.p1.5.m5.2.2"><csymbol cd="latexml" id="Thmproposition1.p1.5.m5.2.2.2.cmml" xref="Thmproposition1.p1.5.m5.2.2.2">similar-to</csymbol><apply id="Thmproposition1.p1.5.m5.2.2.3.cmml" xref="Thmproposition1.p1.5.m5.2.2.3"><ci id="Thmproposition1.p1.5.m5.2.2.3.1.cmml" xref="Thmproposition1.p1.5.m5.2.2.3.1">~</ci><ci id="Thmproposition1.p1.5.m5.2.2.3.2.cmml" xref="Thmproposition1.p1.5.m5.2.2.3.2">𝒙</ci></apply><apply id="Thmproposition1.p1.5.m5.2.2.1.cmml" xref="Thmproposition1.p1.5.m5.2.2.1"><times id="Thmproposition1.p1.5.m5.2.2.1.2.cmml" xref="Thmproposition1.p1.5.m5.2.2.1.2"></times><ci id="Thmproposition1.p1.5.m5.2.2.1.3.cmml" xref="Thmproposition1.p1.5.m5.2.2.1.3">𝒩</ci><interval closure="open" id="Thmproposition1.p1.5.m5.2.2.1.1.2.cmml" xref="Thmproposition1.p1.5.m5.2.2.1.1.1"><ci id="Thmproposition1.p1.5.m5.1.1.cmml" xref="Thmproposition1.p1.5.m5.1.1">𝒙</ci><apply id="Thmproposition1.p1.5.m5.2.2.1.1.1.1.cmml" xref="Thmproposition1.p1.5.m5.2.2.1.1.1.1"><times id="Thmproposition1.p1.5.m5.2.2.1.1.1.1.1.cmml" xref="Thmproposition1.p1.5.m5.2.2.1.1.1.1.1"></times><ci id="Thmproposition1.p1.5.m5.2.2.1.1.1.1.2.cmml" xref="Thmproposition1.p1.5.m5.2.2.1.1.1.1.2">𝜎</ci><ci id="Thmproposition1.p1.5.m5.2.2.1.1.1.1.3.cmml" xref="Thmproposition1.p1.5.m5.2.2.1.1.1.1.3">𝑰</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="Thmproposition1.p1.5.m5.2c">\widetilde{{\boldsymbol{x}}}\leavevmode\nobreak\ \sim\leavevmode\nobreak\ \mathcal{N}({\boldsymbol{x}},\sigma{\boldsymbol{I}})</annotation></semantics></math>.</span>
</span>
</span>
</span>

For infinitesimally small perturbations (<math id="S4.SS1.p1.1.m1.1" class="ltx_Math" alttext="\sigma\rightarrow 0" display="inline"><semantics id="S4.SS1.p1.1.m1.1a"><mrow id="S4.SS1.p1.1.m1.1.1" xref="S4.SS1.p1.1.m1.1.1.cmml"><mi id="S4.SS1.p1.1.m1.1.1.2" xref="S4.SS1.p1.1.m1.1.1.2.cmml">σ</mi><mo stretchy="false" id="S4.SS1.p1.1.m1.1.1.1" xref="S4.SS1.p1.1.m1.1.1.1.cmml">→</mo><mn id="S4.SS1.p1.1.m1.1.1.3" xref="S4.SS1.p1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.1.m1.1b"><apply id="S4.SS1.p1.1.m1.1.1.cmml" xref="S4.SS1.p1.1.m1.1.1"><ci id="S4.SS1.p1.1.m1.1.1.1.cmml" xref="S4.SS1.p1.1.m1.1.1.1">→</ci><ci id="S4.SS1.p1.1.m1.1.1.2.cmml" xref="S4.SS1.p1.1.m1.1.1.2">𝜎</ci><cn type="integer" id="S4.SS1.p1.1.m1.1.1.3.cmml" xref="S4.SS1.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.1.m1.1c">\sigma\rightarrow 0</annotation></semantics></math>), the samples <math id="S4.SS1.p1.2.m2.1" class="ltx_Math" alttext="\widetilde{{\boldsymbol{x}}}" display="inline"><semantics id="S4.SS1.p1.2.m2.1a"><mover accent="true" id="S4.SS1.p1.2.m2.1.1" xref="S4.SS1.p1.2.m2.1.1.cmml"><mi id="S4.SS1.p1.2.m2.1.1.2" xref="S4.SS1.p1.2.m2.1.1.2.cmml">𝒙</mi><mo id="S4.SS1.p1.2.m2.1.1.1" xref="S4.SS1.p1.2.m2.1.1.1.cmml">~</mo></mover><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.2.m2.1b"><apply id="S4.SS1.p1.2.m2.1.1.cmml" xref="S4.SS1.p1.2.m2.1.1"><ci id="S4.SS1.p1.2.m2.1.1.1.cmml" xref="S4.SS1.p1.2.m2.1.1.1">~</ci><ci id="S4.SS1.p1.2.m2.1.1.2.cmml" xref="S4.SS1.p1.2.m2.1.1.2">𝒙</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.2.m2.1c">\widetilde{{\boldsymbol{x}}}</annotation></semantics></math> can be approximated through linearization by the input gradients <math id="S4.SS1.p1.3.m3.1" class="ltx_Math" alttext="\nabla_{\boldsymbol{x}}f" display="inline"><semantics id="S4.SS1.p1.3.m3.1a"><mrow id="S4.SS1.p1.3.m3.1.1" xref="S4.SS1.p1.3.m3.1.1.cmml"><msub id="S4.SS1.p1.3.m3.1.1.1" xref="S4.SS1.p1.3.m3.1.1.1.cmml"><mo id="S4.SS1.p1.3.m3.1.1.1.2" xref="S4.SS1.p1.3.m3.1.1.1.2.cmml">∇</mo><mi id="S4.SS1.p1.3.m3.1.1.1.3" xref="S4.SS1.p1.3.m3.1.1.1.3.cmml">𝒙</mi></msub><mi id="S4.SS1.p1.3.m3.1.1.2" xref="S4.SS1.p1.3.m3.1.1.2.cmml">f</mi></mrow><annotation-xml encoding="MathML-Content" id="S4.SS1.p1.3.m3.1b"><apply id="S4.SS1.p1.3.m3.1.1.cmml" xref="S4.SS1.p1.3.m3.1.1"><apply id="S4.SS1.p1.3.m3.1.1.1.cmml" xref="S4.SS1.p1.3.m3.1.1.1"><csymbol cd="ambiguous" id="S4.SS1.p1.3.m3.1.1.1.1.cmml" xref="S4.SS1.p1.3.m3.1.1.1">subscript</csymbol><ci id="S4.SS1.p1.3.m3.1.1.1.2.cmml" xref="S4.SS1.p1.3.m3.1.1.1.2">∇</ci><ci id="S4.SS1.p1.3.m3.1.1.1.3.cmml" xref="S4.SS1.p1.3.m3.1.1.1.3">𝒙</ci></apply><ci id="S4.SS1.p1.3.m3.1.1.2.cmml" xref="S4.SS1.p1.3.m3.1.1.2">𝑓</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.3.m3.1c">\nabla_{\boldsymbol{x}}f</annotation></semantics></math>.
These are 1D Gaussian random variables whose correlation is given by
<math id="S4.SS1.p1.4.m4.2" class="ltx_math_unparsed" alttext="\cos\big{(}\nabla_{\boldsymbol{x}}f_{{\boldsymbol{\theta}}_{1}}(x),\nabla_{\boldsymbol{x}}f_{{\boldsymbol{\theta}}_{2}}x)\big{)}" display="inline"><semantics id="S4.SS1.p1.4.m4.2a"><mrow id="S4.SS1.p1.4.m4.2b"><mi id="S4.SS1.p1.4.m4.2.2">cos</mi><mrow id="S4.SS1.p1.4.m4.2.3"><mo maxsize="120%" minsize="120%" id="S4.SS1.p1.4.m4.2.3.1">(</mo><msub id="S4.SS1.p1.4.m4.2.3.2"><mo rspace="0.167em" id="S4.SS1.p1.4.m4.2.3.2.2">∇</mo><mi id="S4.SS1.p1.4.m4.2.3.2.3">𝒙</mi></msub><msub id="S4.SS1.p1.4.m4.2.3.3"><mi id="S4.SS1.p1.4.m4.2.3.3.2">f</mi><msub id="S4.SS1.p1.4.m4.2.3.3.3"><mi id="S4.SS1.p1.4.m4.2.3.3.3.2">𝜽</mi><mn id="S4.SS1.p1.4.m4.2.3.3.3.3">1</mn></msub></msub><mrow id="S4.SS1.p1.4.m4.2.3.4"><mo stretchy="false" id="S4.SS1.p1.4.m4.2.3.4.1">(</mo><mi id="S4.SS1.p1.4.m4.1.1">x</mi><mo stretchy="false" id="S4.SS1.p1.4.m4.2.3.4.2">)</mo></mrow><mo id="S4.SS1.p1.4.m4.2.3.5">,</mo><msub id="S4.SS1.p1.4.m4.2.3.6"><mo rspace="0.167em" id="S4.SS1.p1.4.m4.2.3.6.2">∇</mo><mi id="S4.SS1.p1.4.m4.2.3.6.3">𝒙</mi></msub><msub id="S4.SS1.p1.4.m4.2.3.7"><mi id="S4.SS1.p1.4.m4.2.3.7.2">f</mi><msub id="S4.SS1.p1.4.m4.2.3.7.3"><mi id="S4.SS1.p1.4.m4.2.3.7.3.2">𝜽</mi><mn id="S4.SS1.p1.4.m4.2.3.7.3.3">2</mn></msub></msub><mi id="S4.SS1.p1.4.m4.2.3.8">x</mi><mo stretchy="false" id="S4.SS1.p1.4.m4.2.3.9">)</mo></mrow><mo maxsize="120%" minsize="120%" id="S4.SS1.p1.4.m4.2.4">)</mo></mrow><annotation encoding="application/x-tex" id="S4.SS1.p1.4.m4.2c">\cos\big{(}\nabla_{\boldsymbol{x}}f_{{\boldsymbol{\theta}}_{1}}(x),\nabla_{\boldsymbol{x}}f_{{\boldsymbol{\theta}}_{2}}x)\big{)}</annotation></semantics></math>
and their mutual information <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib23" title="" class="ltx_ref">23</a>]</cite>:
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id="S4.SS1.p1.5.m5.4.4.1.1.1.1.1.3.3.3.2.2.2.3.2.cmml" xref="S4.SS1.p1.5.m5.4.4.1.1.1.1.1.3.3.3.2.2.2.3.2">𝜽</ci><cn type="integer" id="S4.SS1.p1.5.m5.4.4.1.1.1.1.1.3.3.3.2.2.2.3.3.cmml" xref="S4.SS1.p1.5.m5.4.4.1.1.1.1.1.3.3.3.2.2.2.3.3">2</cn></apply></apply></apply><ci id="S4.SS1.p1.5.m5.2.2.cmml" xref="S4.SS1.p1.5.m5.2.2">𝑥</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS1.p1.5.m5.4c">-\frac{1}{2}\ln\big{(}1-\cos^{2}\big{(}\nabla_{\boldsymbol{x}}f_{{\boldsymbol{\theta}}_{1}}(x),\nabla_{\boldsymbol{x}}f_{{\boldsymbol{\theta}}_{2}}(x)\big{)}\big{)}</annotation></semantics></math>.
Therefore, the statistical independence between the models’ outputs as their inputs are perturbed by small Gaussian variations can be enforced by making their input gradients orthogonal.
Our <span id="S4.SS1.p1.5.5" class="ltx_text ltx_font_bold">local independence loss</span> for a pair of models is:</p>
<table id="A8.EGx1" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="S4.E1"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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\cos^{2}\big{(}\nabla_{\boldsymbol{x}}f_{{\boldsymbol{\theta}}_{m_{1}}}({\boldsymbol{x}}),\leavevmode\nobreak\ \nabla_{\boldsymbol{x}}f_{{\boldsymbol{\theta}}_{m_{2}}}({\boldsymbol{x}})\big{)}</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(1)</span></td>
</tr></tbody>
</table>
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xref="S4.SS1.p1.6.m1.1.1.cmml">𝒗</mi><mo id="S4.SS1.p1.6.m1.3.3.1.1.2.2" xref="S4.SS1.p1.6.m1.3.3.1.2.cmml">,</mo><mi id="S4.SS1.p1.6.m1.2.2" xref="S4.SS1.p1.6.m1.2.2.cmml">𝒘</mi><mo stretchy="false" id="S4.SS1.p1.6.m1.3.3.1.1.2.3" xref="S4.SS1.p1.6.m1.3.3.1.2.cmml">)</mo></mrow></mrow><mo id="S4.SS1.p1.6.m1.6.6.5" xref="S4.SS1.p1.6.m1.6.6.5.cmml">=</mo><mrow id="S4.SS1.p1.6.m1.6.6.4" xref="S4.SS1.p1.6.m1.6.6.4.cmml"><mrow id="S4.SS1.p1.6.m1.5.5.3.2" xref="S4.SS1.p1.6.m1.5.5.3.2.cmml"><msup id="S4.SS1.p1.6.m1.4.4.2.1.1" xref="S4.SS1.p1.6.m1.4.4.2.1.1.cmml"><mrow id="S4.SS1.p1.6.m1.4.4.2.1.1.1.1" xref="S4.SS1.p1.6.m1.4.4.2.1.1.1.1.1.cmml"><mo stretchy="false" id="S4.SS1.p1.6.m1.4.4.2.1.1.1.1.2" xref="S4.SS1.p1.6.m1.4.4.2.1.1.1.1.1.cmml">(</mo><mrow id="S4.SS1.p1.6.m1.4.4.2.1.1.1.1.1" xref="S4.SS1.p1.6.m1.4.4.2.1.1.1.1.1.cmml"><msup id="S4.SS1.p1.6.m1.4.4.2.1.1.1.1.1.2" xref="S4.SS1.p1.6.m1.4.4.2.1.1.1.1.1.2.cmml"><mi id="S4.SS1.p1.6.m1.4.4.2.1.1.1.1.1.2.2" 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encoding="application/x-tex" id="S4.SS1.p1.6.m1.6c">\cos^{2}({\boldsymbol{v}},{\boldsymbol{w}})=({\boldsymbol{v}}^{\intercal}{\boldsymbol{w}})^{2}\,/\,({\boldsymbol{v}}^{\intercal}{\boldsymbol{v}})({\boldsymbol{w}}^{\intercal}{\boldsymbol{w}})</annotation></semantics></math>.
To enforce <em id="S4.SS1.p1.6.1" class="ltx_emph ltx_font_italic">global</em> independence, this loss will be applied to all training points and pairs of models in the overall objective of Eq. (<a href="#S4.Ex1" title="4.2 On-manifold constraint ‣ 4 Proposed method ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">4.2</span></a>).</p>
</div>
</section>
<section id="S4.SS2" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">4.2 </span>On-manifold constraint</h3>

<figure id="S4.F2" class="ltx_figure"><img src="/html/2207.02598/assets/x2.png" id="S4.F2.g1" class="ltx_graphics ltx_img_landscape" width="438" height="223" alt="Refer to caption">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span id="S4.F2.2.1.1" class="ltx_text" style="font-size:90%;">Figure 2</span>: </span><span id="S4.F2.3.2" class="ltx_text" style="font-size:90%;">Method overview
<br class="ltx_break">during training.</span></figcaption>
</figure>
<div id="S4.SS2.p1" class="ltx_para">
<p id="S4.SS2.p1.1" class="ltx_p">The independence constraint (<a href="#S4.E1" title="In 4.1 Independent models ‣ 4 Proposed method ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">1</span></a>) makes models’ input gradients orthogonal to one another.
The number of models satisfying it grows exponentially with the input dimension (<math id="S4.SS2.p1.1.m1.1" class="ltx_Math" alttext="{d_{\textrm{in}}}" display="inline"><semantics id="S4.SS2.p1.1.m1.1a"><msub id="S4.SS2.p1.1.m1.1.1" xref="S4.SS2.p1.1.m1.1.1.cmml"><mi id="S4.SS2.p1.1.m1.1.1.2" xref="S4.SS2.p1.1.m1.1.1.2.cmml">d</mi><mtext id="S4.SS2.p1.1.m1.1.1.3" xref="S4.SS2.p1.1.m1.1.1.3a.cmml">in</mtext></msub><annotation-xml encoding="MathML-Content" id="S4.SS2.p1.1.m1.1b"><apply id="S4.SS2.p1.1.m1.1.1.cmml" xref="S4.SS2.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p1.1.m1.1.1.1.cmml" xref="S4.SS2.p1.1.m1.1.1">subscript</csymbol><ci id="S4.SS2.p1.1.m1.1.1.2.cmml" xref="S4.SS2.p1.1.m1.1.1.2">𝑑</ci><ci id="S4.SS2.p1.1.m1.1.1.3a.cmml" xref="S4.SS2.p1.1.m1.1.1.3"><mtext mathsize="70%" id="S4.SS2.p1.1.m1.1.1.3.cmml" xref="S4.SS2.p1.1.m1.1.1.3">in</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p1.1.m1.1c">{d_{\textrm{in}}}</annotation></semantics></math>) but many are practically irrelevant because the natural data manifold usually occupies much fewer dimensions.
Intuitively, when the constraint affects a model’s gradients in dimensions pointing outward the manifold, it does not affect its predictions on natural data.
Consequently, the independence constraint could be satisfied by models that produce identical predictions on every natural input (thus defeating its purpose) because their decision boundaries are identical when projected on the manifold.
The issue stems from the <em id="S4.SS2.p1.1.1" class="ltx_emph ltx_font_italic">isotropic</em> perturbations in Eq. (<a href="#Thmproposition1" title="Proposition 1. ‣ 4.1 Independent models ‣ 4 Proposed method ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">1</span></a>). Only perturbations <em id="S4.SS2.p1.1.2" class="ltx_emph ltx_font_italic">on</em> the manifold are meaningful.</p>
</div>
<div id="S4.SS2.p2" class="ltx_para">
<p id="S4.SS2.p2.1" class="ltx_p">One straightforward solution would be to enforce independence after projecting the data on a learned approximation of the manifold.
This approach, proposed in <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib60" title="" class="ltx_ref">60</a>, <a href="#bib.bib61" title="" class="ltx_ref">61</a>]</cite> failed in our early experiments because of the difficulty of optimizing the independence objective under such a strict on-manifold constraint.
Instead, we implement a soft constraint as a regularizer that proved easy to train and resilient to imperfect models of the manifold.</p>
</div>
<div id="S4.SS2.p3" class="ltx_para ltx_noindent">
<p id="S4.SS2.p3.5" class="ltx_p">To learn the data manifold <math id="S4.SS2.p3.1.m1.1" class="ltx_Math" alttext="\mathcal{M}" display="inline"><semantics id="S4.SS2.p3.1.m1.1a"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p3.1.m1.1.1" xref="S4.SS2.p3.1.m1.1.1.cmml">ℳ</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.1.m1.1b"><ci id="S4.SS2.p3.1.m1.1.1.cmml" xref="S4.SS2.p3.1.m1.1.1">ℳ</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.1.m1.1c">\mathcal{M}</annotation></semantics></math>, we need unlabeled examples, ideally containing the type of OOD data expected at test time <em id="S4.SS2.p3.5.1" class="ltx_emph ltx_font_italic">e.g</em>.<span id="S4.SS2.p3.5.2" class="ltx_text"></span> a broad collection of natural images: <math id="S4.SS2.p3.2.m2.1" class="ltx_Math" alttext="\mathcal{D}_{\mathrm{OOD}}=\{{\boldsymbol{x}}_{i}\}\sim\textrm{P}_{\mathrm{OOD}}" display="inline"><semantics id="S4.SS2.p3.2.m2.1a"><mrow id="S4.SS2.p3.2.m2.1.1" xref="S4.SS2.p3.2.m2.1.1.cmml"><msub id="S4.SS2.p3.2.m2.1.1.3" xref="S4.SS2.p3.2.m2.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p3.2.m2.1.1.3.2" xref="S4.SS2.p3.2.m2.1.1.3.2.cmml">𝒟</mi><mi id="S4.SS2.p3.2.m2.1.1.3.3" xref="S4.SS2.p3.2.m2.1.1.3.3.cmml">OOD</mi></msub><mo id="S4.SS2.p3.2.m2.1.1.4" xref="S4.SS2.p3.2.m2.1.1.4.cmml">=</mo><mrow id="S4.SS2.p3.2.m2.1.1.1.1" xref="S4.SS2.p3.2.m2.1.1.1.2.cmml"><mo stretchy="false" id="S4.SS2.p3.2.m2.1.1.1.1.2" xref="S4.SS2.p3.2.m2.1.1.1.2.cmml">{</mo><msub id="S4.SS2.p3.2.m2.1.1.1.1.1" xref="S4.SS2.p3.2.m2.1.1.1.1.1.cmml"><mi id="S4.SS2.p3.2.m2.1.1.1.1.1.2" xref="S4.SS2.p3.2.m2.1.1.1.1.1.2.cmml">𝒙</mi><mi id="S4.SS2.p3.2.m2.1.1.1.1.1.3" xref="S4.SS2.p3.2.m2.1.1.1.1.1.3.cmml">i</mi></msub><mo stretchy="false" id="S4.SS2.p3.2.m2.1.1.1.1.3" xref="S4.SS2.p3.2.m2.1.1.1.2.cmml">}</mo></mrow><mo id="S4.SS2.p3.2.m2.1.1.5" xref="S4.SS2.p3.2.m2.1.1.5.cmml">∼</mo><msub id="S4.SS2.p3.2.m2.1.1.6" xref="S4.SS2.p3.2.m2.1.1.6.cmml"><mtext id="S4.SS2.p3.2.m2.1.1.6.2" xref="S4.SS2.p3.2.m2.1.1.6.2a.cmml">P</mtext><mi id="S4.SS2.p3.2.m2.1.1.6.3" xref="S4.SS2.p3.2.m2.1.1.6.3.cmml">OOD</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.2.m2.1b"><apply id="S4.SS2.p3.2.m2.1.1.cmml" xref="S4.SS2.p3.2.m2.1.1"><and id="S4.SS2.p3.2.m2.1.1a.cmml" xref="S4.SS2.p3.2.m2.1.1"></and><apply id="S4.SS2.p3.2.m2.1.1b.cmml" xref="S4.SS2.p3.2.m2.1.1"><eq id="S4.SS2.p3.2.m2.1.1.4.cmml" xref="S4.SS2.p3.2.m2.1.1.4"></eq><apply id="S4.SS2.p3.2.m2.1.1.3.cmml" xref="S4.SS2.p3.2.m2.1.1.3"><csymbol cd="ambiguous" id="S4.SS2.p3.2.m2.1.1.3.1.cmml" xref="S4.SS2.p3.2.m2.1.1.3">subscript</csymbol><ci id="S4.SS2.p3.2.m2.1.1.3.2.cmml" xref="S4.SS2.p3.2.m2.1.1.3.2">𝒟</ci><ci id="S4.SS2.p3.2.m2.1.1.3.3.cmml" xref="S4.SS2.p3.2.m2.1.1.3.3">OOD</ci></apply><set id="S4.SS2.p3.2.m2.1.1.1.2.cmml" xref="S4.SS2.p3.2.m2.1.1.1.1"><apply id="S4.SS2.p3.2.m2.1.1.1.1.1.cmml" xref="S4.SS2.p3.2.m2.1.1.1.1.1"><csymbol cd="ambiguous" id="S4.SS2.p3.2.m2.1.1.1.1.1.1.cmml" xref="S4.SS2.p3.2.m2.1.1.1.1.1">subscript</csymbol><ci id="S4.SS2.p3.2.m2.1.1.1.1.1.2.cmml" xref="S4.SS2.p3.2.m2.1.1.1.1.1.2">𝒙</ci><ci id="S4.SS2.p3.2.m2.1.1.1.1.1.3.cmml" xref="S4.SS2.p3.2.m2.1.1.1.1.1.3">𝑖</ci></apply></set></apply><apply id="S4.SS2.p3.2.m2.1.1c.cmml" xref="S4.SS2.p3.2.m2.1.1"><csymbol cd="latexml" id="S4.SS2.p3.2.m2.1.1.5.cmml" xref="S4.SS2.p3.2.m2.1.1.5">similar-to</csymbol><share href="#S4.SS2.p3.2.m2.1.1.1.cmml" id="S4.SS2.p3.2.m2.1.1d.cmml" xref="S4.SS2.p3.2.m2.1.1"></share><apply id="S4.SS2.p3.2.m2.1.1.6.cmml" xref="S4.SS2.p3.2.m2.1.1.6"><csymbol cd="ambiguous" id="S4.SS2.p3.2.m2.1.1.6.1.cmml" xref="S4.SS2.p3.2.m2.1.1.6">subscript</csymbol><ci id="S4.SS2.p3.2.m2.1.1.6.2a.cmml" xref="S4.SS2.p3.2.m2.1.1.6.2"><mtext id="S4.SS2.p3.2.m2.1.1.6.2.cmml" xref="S4.SS2.p3.2.m2.1.1.6.2">P</mtext></ci><ci id="S4.SS2.p3.2.m2.1.1.6.3.cmml" xref="S4.SS2.p3.2.m2.1.1.6.3">OOD</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.2.m2.1c">\mathcal{D}_{\mathrm{OOD}}=\{{\boldsymbol{x}}_{i}\}\sim\textrm{P}_{\mathrm{OOD}}</annotation></semantics></math>.
We use this data off-line to prepare a function <math id="S4.SS2.p3.3.m3.2" class="ltx_Math" alttext="\mathrm{proj}_{\mathcal{M}}({\boldsymbol{x}},{\boldsymbol{v}})" display="inline"><semantics id="S4.SS2.p3.3.m3.2a"><mrow id="S4.SS2.p3.3.m3.2.3" xref="S4.SS2.p3.3.m3.2.3.cmml"><msub id="S4.SS2.p3.3.m3.2.3.2" xref="S4.SS2.p3.3.m3.2.3.2.cmml"><mi id="S4.SS2.p3.3.m3.2.3.2.2" xref="S4.SS2.p3.3.m3.2.3.2.2.cmml">proj</mi><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p3.3.m3.2.3.2.3" xref="S4.SS2.p3.3.m3.2.3.2.3.cmml">ℳ</mi></msub><mo lspace="0em" rspace="0em" id="S4.SS2.p3.3.m3.2.3.1" xref="S4.SS2.p3.3.m3.2.3.1.cmml">​</mo><mrow id="S4.SS2.p3.3.m3.2.3.3.2" xref="S4.SS2.p3.3.m3.2.3.3.1.cmml"><mo stretchy="false" id="S4.SS2.p3.3.m3.2.3.3.2.1" xref="S4.SS2.p3.3.m3.2.3.3.1.cmml">(</mo><mi id="S4.SS2.p3.3.m3.1.1" xref="S4.SS2.p3.3.m3.1.1.cmml">𝒙</mi><mo id="S4.SS2.p3.3.m3.2.3.3.2.2" xref="S4.SS2.p3.3.m3.2.3.3.1.cmml">,</mo><mi id="S4.SS2.p3.3.m3.2.2" xref="S4.SS2.p3.3.m3.2.2.cmml">𝒗</mi><mo stretchy="false" id="S4.SS2.p3.3.m3.2.3.3.2.3" xref="S4.SS2.p3.3.m3.2.3.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.3.m3.2b"><apply id="S4.SS2.p3.3.m3.2.3.cmml" xref="S4.SS2.p3.3.m3.2.3"><times id="S4.SS2.p3.3.m3.2.3.1.cmml" xref="S4.SS2.p3.3.m3.2.3.1"></times><apply id="S4.SS2.p3.3.m3.2.3.2.cmml" xref="S4.SS2.p3.3.m3.2.3.2"><csymbol cd="ambiguous" id="S4.SS2.p3.3.m3.2.3.2.1.cmml" xref="S4.SS2.p3.3.m3.2.3.2">subscript</csymbol><ci id="S4.SS2.p3.3.m3.2.3.2.2.cmml" xref="S4.SS2.p3.3.m3.2.3.2.2">proj</ci><ci id="S4.SS2.p3.3.m3.2.3.2.3.cmml" xref="S4.SS2.p3.3.m3.2.3.2.3">ℳ</ci></apply><interval closure="open" id="S4.SS2.p3.3.m3.2.3.3.1.cmml" xref="S4.SS2.p3.3.m3.2.3.3.2"><ci id="S4.SS2.p3.3.m3.1.1.cmml" xref="S4.SS2.p3.3.m3.1.1">𝒙</ci><ci id="S4.SS2.p3.3.m3.2.2.cmml" xref="S4.SS2.p3.3.m3.2.2">𝒗</ci></interval></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.3.m3.2c">\mathrm{proj}_{\mathcal{M}}({\boldsymbol{x}},{\boldsymbol{v}})</annotation></semantics></math> that projects an arbitrary vector <math id="S4.SS2.p3.4.m4.1" class="ltx_Math" alttext="{\boldsymbol{v}}" display="inline"><semantics id="S4.SS2.p3.4.m4.1a"><mi id="S4.SS2.p3.4.m4.1.1" xref="S4.SS2.p3.4.m4.1.1.cmml">𝒗</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.4.m4.1b"><ci id="S4.SS2.p3.4.m4.1.1.cmml" xref="S4.SS2.p3.4.m4.1.1">𝒗</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.4.m4.1c">{\boldsymbol{v}}</annotation></semantics></math> at <math id="S4.SS2.p3.5.m5.1" class="ltx_Math" alttext="{\boldsymbol{x}}" display="inline"><semantics id="S4.SS2.p3.5.m5.1a"><mi id="S4.SS2.p3.5.m5.1.1" xref="S4.SS2.p3.5.m5.1.1.cmml">𝒙</mi><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.5.m5.1b"><ci id="S4.SS2.p3.5.m5.1.1.cmml" xref="S4.SS2.p3.5.m5.1.1">𝒙</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.5.m5.1c">{\boldsymbol{x}}</annotation></semantics></math> in the input space onto the manifold (Figure <a href="#S4.F3" title="Figure 3 ‣ 4.2 On-manifold constraint ‣ 4 Proposed method ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">3</span></a>).
During training, we penalize each model with the distance between its input gradients and their projection on the manifold.
The <span id="S4.SS2.p3.5.3" class="ltx_text ltx_font_bold">on-manifold loss</span> is defined as</p>
<table id="A8.EGx2" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="S4.E2"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="S4.E2.m1.5" class="ltx_Math" alttext="\displaystyle\mathcal{L}_{\textrm{manifold}}\big{(}\nabla f({\boldsymbol{x}})\big{)}=\big{|}\big{|}\mathrm{proj}_{\mathcal{M}}\big{(}{\boldsymbol{x}},\nabla_{\boldsymbol{x}}f({\boldsymbol{x}})\big{)}-\nabla_{\boldsymbol{x}}f({\boldsymbol{x}})\big{|}\big{|}_{2}^{2}." display="inline"><semantics id="S4.E2.m1.5a"><mrow id="S4.E2.m1.5.5.1" xref="S4.E2.m1.5.5.1.1.cmml"><mrow id="S4.E2.m1.5.5.1.1" xref="S4.E2.m1.5.5.1.1.cmml"><mrow id="S4.E2.m1.5.5.1.1.1" xref="S4.E2.m1.5.5.1.1.1.cmml"><msub id="S4.E2.m1.5.5.1.1.1.3" xref="S4.E2.m1.5.5.1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.E2.m1.5.5.1.1.1.3.2" xref="S4.E2.m1.5.5.1.1.1.3.2.cmml">ℒ</mi><mtext id="S4.E2.m1.5.5.1.1.1.3.3" xref="S4.E2.m1.5.5.1.1.1.3.3a.cmml">manifold</mtext></msub><mo lspace="0em" rspace="0em" id="S4.E2.m1.5.5.1.1.1.2" xref="S4.E2.m1.5.5.1.1.1.2.cmml">​</mo><mrow id="S4.E2.m1.5.5.1.1.1.1.1" xref="S4.E2.m1.5.5.1.1.1.1.1.1.cmml"><mo maxsize="120%" minsize="120%" id="S4.E2.m1.5.5.1.1.1.1.1.2" xref="S4.E2.m1.5.5.1.1.1.1.1.1.cmml">(</mo><mrow id="S4.E2.m1.5.5.1.1.1.1.1.1" xref="S4.E2.m1.5.5.1.1.1.1.1.1.cmml"><mrow id="S4.E2.m1.5.5.1.1.1.1.1.1.2" xref="S4.E2.m1.5.5.1.1.1.1.1.1.2.cmml"><mo rspace="0.167em" id="S4.E2.m1.5.5.1.1.1.1.1.1.2.1" xref="S4.E2.m1.5.5.1.1.1.1.1.1.2.1.cmml">∇</mo><mi id="S4.E2.m1.5.5.1.1.1.1.1.1.2.2" xref="S4.E2.m1.5.5.1.1.1.1.1.1.2.2.cmml">f</mi></mrow><mo lspace="0em" rspace="0em" id="S4.E2.m1.5.5.1.1.1.1.1.1.1" xref="S4.E2.m1.5.5.1.1.1.1.1.1.1.cmml">​</mo><mrow id="S4.E2.m1.5.5.1.1.1.1.1.1.3.2" xref="S4.E2.m1.5.5.1.1.1.1.1.1.cmml"><mo stretchy="false" id="S4.E2.m1.5.5.1.1.1.1.1.1.3.2.1" xref="S4.E2.m1.5.5.1.1.1.1.1.1.cmml">(</mo><mi id="S4.E2.m1.1.1" xref="S4.E2.m1.1.1.cmml">𝒙</mi><mo stretchy="false" id="S4.E2.m1.5.5.1.1.1.1.1.1.3.2.2" xref="S4.E2.m1.5.5.1.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo maxsize="120%" minsize="120%" 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encoding="application/x-tex" id="S4.E2.m1.5c">\displaystyle\mathcal{L}_{\textrm{manifold}}\big{(}\nabla f({\boldsymbol{x}})\big{)}=\big{|}\big{|}\mathrm{proj}_{\mathcal{M}}\big{(}{\boldsymbol{x}},\nabla_{\boldsymbol{x}}f({\boldsymbol{x}})\big{)}-\nabla_{\boldsymbol{x}}f({\boldsymbol{x}})\big{|}\big{|}_{2}^{2}.</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(2)</span></td>
</tr></tbody>
</table>
<p id="S4.SS2.p3.6" class="ltx_p">We describe several implementations of <math id="S4.SS2.p3.6.m1.1" class="ltx_Math" alttext="\mathrm{proj}_{\mathcal{M}}(\cdot)" display="inline"><semantics id="S4.SS2.p3.6.m1.1a"><mrow id="S4.SS2.p3.6.m1.1.2" xref="S4.SS2.p3.6.m1.1.2.cmml"><msub id="S4.SS2.p3.6.m1.1.2.2" xref="S4.SS2.p3.6.m1.1.2.2.cmml"><mi id="S4.SS2.p3.6.m1.1.2.2.2" xref="S4.SS2.p3.6.m1.1.2.2.2.cmml">proj</mi><mi class="ltx_font_mathcaligraphic" id="S4.SS2.p3.6.m1.1.2.2.3" xref="S4.SS2.p3.6.m1.1.2.2.3.cmml">ℳ</mi></msub><mo lspace="0em" rspace="0em" id="S4.SS2.p3.6.m1.1.2.1" xref="S4.SS2.p3.6.m1.1.2.1.cmml">​</mo><mrow id="S4.SS2.p3.6.m1.1.2.3.2" xref="S4.SS2.p3.6.m1.1.2.cmml"><mo stretchy="false" id="S4.SS2.p3.6.m1.1.2.3.2.1" xref="S4.SS2.p3.6.m1.1.2.cmml">(</mo><mo lspace="0em" rspace="0em" id="S4.SS2.p3.6.m1.1.1" xref="S4.SS2.p3.6.m1.1.1.cmml">⋅</mo><mo stretchy="false" id="S4.SS2.p3.6.m1.1.2.3.2.2" xref="S4.SS2.p3.6.m1.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.SS2.p3.6.m1.1b"><apply id="S4.SS2.p3.6.m1.1.2.cmml" xref="S4.SS2.p3.6.m1.1.2"><times id="S4.SS2.p3.6.m1.1.2.1.cmml" xref="S4.SS2.p3.6.m1.1.2.1"></times><apply id="S4.SS2.p3.6.m1.1.2.2.cmml" xref="S4.SS2.p3.6.m1.1.2.2"><csymbol cd="ambiguous" id="S4.SS2.p3.6.m1.1.2.2.1.cmml" xref="S4.SS2.p3.6.m1.1.2.2">subscript</csymbol><ci id="S4.SS2.p3.6.m1.1.2.2.2.cmml" xref="S4.SS2.p3.6.m1.1.2.2.2">proj</ci><ci id="S4.SS2.p3.6.m1.1.2.2.3.cmml" xref="S4.SS2.p3.6.m1.1.2.2.3">ℳ</ci></apply><ci id="S4.SS2.p3.6.m1.1.1.cmml" xref="S4.SS2.p3.6.m1.1.1">⋅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS2.p3.6.m1.1c">\mathrm{proj}_{\mathcal{M}}(\cdot)</annotation></semantics></math> in Appendix <a href="#A2" title="Appendix B Projections on the manifold ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">B</span></a> using a variational auto-encoder (VAE) or a simple PCA.
In summary, the on-manifold loss encourages a model to be sensitive to variations in the input that are likely to be encountered in natural test data.
It typically has no effect on in-domain performance (Figure <a href="#S6.F6.sf3" title="In Figure 6 ‣ 6.1 Experiments on controlled data: collages ‣ 6 Experiments ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">6(c)</span></a>) since it only removes a model’s sensitivity to “unnatural” inputs such as variations of isolated pixels, which are unlikely to appear in natural images.</p>
</div>
<div id="S4.SS2.p4" class="ltx_para">
<p id="S4.SS2.p4.1" class="ltx_p">The overall learning objective combines the predictive, independence, and on-manifold losses:</p>
<table id="A8.EGx3" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="S4.Ex1"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="S4.Ex1.m1.2" class="ltx_Math" alttext="\displaystyle\mathcal{L}(\mathcal{D}_{\mathrm{tr}},{\boldsymbol{\theta}}_{1}\,\kern-1.00006pt.\kern-1.00006pt.\kern-1.00006pt.\,{\boldsymbol{\theta}}_{M})={\Sigma}_{{\boldsymbol{x}}\in\mathcal{D}_{\mathrm{tr}}}" display="inline"><semantics id="S4.Ex1.m1.2a"><mrow id="S4.Ex1.m1.2.2" xref="S4.Ex1.m1.2.2.cmml"><mrow id="S4.Ex1.m1.2.2.2" xref="S4.Ex1.m1.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex1.m1.2.2.2.4" xref="S4.Ex1.m1.2.2.2.4.cmml">ℒ</mi><mo lspace="0em" rspace="0em" id="S4.Ex1.m1.2.2.2.3" xref="S4.Ex1.m1.2.2.2.3.cmml">​</mo><mrow id="S4.Ex1.m1.2.2.2.2.2" xref="S4.Ex1.m1.2.2.2.2.3.cmml"><mo stretchy="false" id="S4.Ex1.m1.2.2.2.2.2.3" xref="S4.Ex1.m1.2.2.2.2.3.cmml">(</mo><msub id="S4.Ex1.m1.1.1.1.1.1.1" xref="S4.Ex1.m1.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.Ex1.m1.1.1.1.1.1.1.2" xref="S4.Ex1.m1.1.1.1.1.1.1.2.cmml">𝒟</mi><mi 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xref="S4.Ex1.m1.2.2.4.3.3"><csymbol cd="ambiguous" id="S4.Ex1.m1.2.2.4.3.3.1.cmml" xref="S4.Ex1.m1.2.2.4.3.3">subscript</csymbol><ci id="S4.Ex1.m1.2.2.4.3.3.2.cmml" xref="S4.Ex1.m1.2.2.4.3.3.2">𝒟</ci><ci id="S4.Ex1.m1.2.2.4.3.3.3.cmml" xref="S4.Ex1.m1.2.2.4.3.3.3">tr</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex1.m1.2c">\displaystyle\mathcal{L}(\mathcal{D}_{\mathrm{tr}},{\boldsymbol{\theta}}_{1}\,\kern-1.00006pt.\kern-1.00006pt.\kern-1.00006pt.\,{\boldsymbol{\theta}}_{M})={\Sigma}_{{\boldsymbol{x}}\in\mathcal{D}_{\mathrm{tr}}}</annotation></semantics></math></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="S4.Ex1.m2.2" class="ltx_math_unparsed" alttext="\displaystyle\Big{[}\leavevmode\nobreak\ \leavevmode\nobreak\ ({\scriptstyle 1}/{\scriptstyle M})\,\leavevmode\nobreak\ \leavevmode\nobreak\ {\Sigma}_{m=1}^{M}\mathcal{L}_{\textrm{pred}}\big{(}y,\,\sigma(f_{{\boldsymbol{\theta}}_{m}}({\boldsymbol{x}}))\big{)}" display="inline"><semantics id="S4.Ex1.m2.2a"><mrow id="S4.Ex1.m2.2b"><mo maxsize="160%" minsize="160%" rspace="1.000em" id="S4.Ex1.m2.2.3">[</mo><mrow id="S4.Ex1.m2.2.4"><mo stretchy="false" id="S4.Ex1.m2.2.4.1">(</mo><mn mathsize="70%" id="S4.Ex1.m2.2.4.2">1</mn><mo id="S4.Ex1.m2.2.4.3">/</mo><mi mathsize="70%" id="S4.Ex1.m2.2.4.4">M</mi><mo stretchy="false" id="S4.Ex1.m2.2.4.5">)</mo></mrow><mspace width="1.16666666666667em" id="S4.Ex1.m2.2.5"></mspace><msubsup id="S4.Ex1.m2.2.6"><mi mathvariant="normal" id="S4.Ex1.m2.2.6.2.2">Σ</mi><mrow id="S4.Ex1.m2.2.6.2.3"><mi id="S4.Ex1.m2.2.6.2.3.2">m</mi><mo id="S4.Ex1.m2.2.6.2.3.1">=</mo><mn id="S4.Ex1.m2.2.6.2.3.3">1</mn></mrow><mi id="S4.Ex1.m2.2.6.3">M</mi></msubsup><msub id="S4.Ex1.m2.2.7"><mi class="ltx_font_mathcaligraphic" id="S4.Ex1.m2.2.7.2">ℒ</mi><mtext id="S4.Ex1.m2.2.7.3">pred</mtext></msub><mrow id="S4.Ex1.m2.2.8"><mo maxsize="120%" minsize="120%" id="S4.Ex1.m2.2.8.1">(</mo><mi id="S4.Ex1.m2.2.2">y</mi><mo rspace="0.337em" id="S4.Ex1.m2.2.8.2">,</mo><mi id="S4.Ex1.m2.2.8.3">σ</mi><mrow id="S4.Ex1.m2.2.8.4"><mo stretchy="false" id="S4.Ex1.m2.2.8.4.1">(</mo><msub id="S4.Ex1.m2.2.8.4.2"><mi id="S4.Ex1.m2.2.8.4.2.2">f</mi><msub id="S4.Ex1.m2.2.8.4.2.3"><mi id="S4.Ex1.m2.2.8.4.2.3.2">𝜽</mi><mi id="S4.Ex1.m2.2.8.4.2.3.3">m</mi></msub></msub><mrow id="S4.Ex1.m2.2.8.4.3"><mo stretchy="false" id="S4.Ex1.m2.2.8.4.3.1">(</mo><mi id="S4.Ex1.m2.1.1">𝒙</mi><mo stretchy="false" id="S4.Ex1.m2.2.8.4.3.2">)</mo></mrow><mo stretchy="false" id="S4.Ex1.m2.2.8.4.4">)</mo></mrow><mo maxsize="120%" minsize="120%" id="S4.Ex1.m2.2.8.5">)</mo></mrow></mrow><annotation encoding="application/x-tex" id="S4.Ex1.m2.2c">\displaystyle\Big{[}\leavevmode\nobreak\ \leavevmode\nobreak\ ({\scriptstyle 1}/{\scriptstyle M})\,\leavevmode\nobreak\ \leavevmode\nobreak\ {\Sigma}_{m=1}^{M}\mathcal{L}_{\textrm{pred}}\big{(}y,\,\sigma(f_{{\boldsymbol{\theta}}_{m}}({\boldsymbol{x}}))\big{)}</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="S4.Ex2"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_eqn_cell"></td>
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xref="S4.Ex2.m1.5.5.3.3.2.2.2.2.3.3">subscript</csymbol><ci id="S4.Ex2.m1.5.5.3.3.2.2.2.2.3.3.2.cmml" xref="S4.Ex2.m1.5.5.3.3.2.2.2.2.3.3.2">𝑚</ci><cn type="integer" id="S4.Ex2.m1.5.5.3.3.2.2.2.2.3.3.3.cmml" xref="S4.Ex2.m1.5.5.3.3.2.2.2.2.3.3.3">2</cn></apply></apply></apply></apply><ci id="S4.Ex2.m1.2.2.cmml" xref="S4.Ex2.m1.2.2">𝒙</ci></apply></interval></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.Ex2.m1.5c">\displaystyle+({\scriptstyle 1}/{\scriptstyle M^{2}})\leavevmode\nobreak\ {\Sigma}_{m_{1}=1}^{M}{\Sigma}_{m_{2}=1}^{M}{\lambda_{\textrm{indep}}}\leavevmode\nobreak\ \,\mathcal{L}_{\textrm{indep}}\big{(}\nabla_{\boldsymbol{x}}f_{{\boldsymbol{\theta}}_{m_{1}}}({\boldsymbol{x}}),\nabla_{\boldsymbol{x}}f_{{\boldsymbol{\theta}}_{m_{2}}}({\boldsymbol{x}})\big{)}</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
</tr></tbody>
<tbody id="S4.E3"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_eqn_cell"></td>
<td class="ltx_td ltx_align_left ltx_eqn_cell"><math id="S4.E3.m1.1" class="ltx_math_unparsed" alttext="\displaystyle+({\scriptstyle 1}/{\scriptstyle M})\,\leavevmode\nobreak\ \leavevmode\nobreak\ {\Sigma}_{m=1}^{M}{\lambda_{\textrm{manifold}}}\leavevmode\nobreak\ \,\mathcal{L}_{\textrm{manifold}}\big{(}\nabla_{\boldsymbol{x}}f_{{\boldsymbol{\theta}}_{m}}({\boldsymbol{x}})\big{)}\leavevmode\nobreak\ \Big{]}." display="inline"><semantics id="S4.E3.m1.1a"><mrow id="S4.E3.m1.1b"><mo id="S4.E3.m1.1.2">+</mo><mrow id="S4.E3.m1.1.3"><mo stretchy="false" id="S4.E3.m1.1.3.1">(</mo><mn mathsize="70%" id="S4.E3.m1.1.3.2">1</mn><mo id="S4.E3.m1.1.3.3">/</mo><mi mathsize="70%" id="S4.E3.m1.1.3.4">M</mi><mo stretchy="false" id="S4.E3.m1.1.3.5">)</mo></mrow><mspace width="1.16666666666667em" id="S4.E3.m1.1.4"></mspace><msubsup id="S4.E3.m1.1.5"><mi mathvariant="normal" id="S4.E3.m1.1.5.2.2">Σ</mi><mrow id="S4.E3.m1.1.5.2.3"><mi id="S4.E3.m1.1.5.2.3.2">m</mi><mo id="S4.E3.m1.1.5.2.3.1">=</mo><mn id="S4.E3.m1.1.5.2.3.3">1</mn></mrow><mi id="S4.E3.m1.1.5.3">M</mi></msubsup><msub id="S4.E3.m1.1.6"><mi id="S4.E3.m1.1.6.2">λ</mi><mtext id="S4.E3.m1.1.6.3">manifold</mtext></msub><msub id="S4.E3.m1.1.7"><mi class="ltx_font_mathcaligraphic" id="S4.E3.m1.1.7.2">ℒ</mi><mtext id="S4.E3.m1.1.7.3">manifold</mtext></msub><mrow id="S4.E3.m1.1.8"><mo maxsize="120%" minsize="120%" id="S4.E3.m1.1.8.1">(</mo><msub id="S4.E3.m1.1.8.2"><mo rspace="0.167em" id="S4.E3.m1.1.8.2.2">∇</mo><mi id="S4.E3.m1.1.8.2.3">𝒙</mi></msub><msub id="S4.E3.m1.1.8.3"><mi id="S4.E3.m1.1.8.3.2">f</mi><msub id="S4.E3.m1.1.8.3.3"><mi id="S4.E3.m1.1.8.3.3.2">𝜽</mi><mi id="S4.E3.m1.1.8.3.3.3">m</mi></msub></msub><mrow id="S4.E3.m1.1.8.4"><mo stretchy="false" id="S4.E3.m1.1.8.4.1">(</mo><mi id="S4.E3.m1.1.1">𝒙</mi><mo stretchy="false" id="S4.E3.m1.1.8.4.2">)</mo></mrow><mo maxsize="120%" minsize="120%" rspace="0.500em" id="S4.E3.m1.1.8.5">)</mo></mrow><mo maxsize="160%" minsize="160%" id="S4.E3.m1.1.9">]</mo><mo lspace="0em" id="S4.E3.m1.1.10">.</mo></mrow><annotation encoding="application/x-tex" id="S4.E3.m1.1c">\displaystyle+({\scriptstyle 1}/{\scriptstyle M})\,\leavevmode\nobreak\ \leavevmode\nobreak\ {\Sigma}_{m=1}^{M}{\lambda_{\textrm{manifold}}}\leavevmode\nobreak\ \,\mathcal{L}_{\textrm{manifold}}\big{(}\nabla_{\boldsymbol{x}}f_{{\boldsymbol{\theta}}_{m}}({\boldsymbol{x}})\big{)}\leavevmode\nobreak\ \Big{]}.</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(3)</span></td>
</tr></tbody>
</table>
</div>
<figure id="S4.F3" class="ltx_figure"><span id="S4.F3.pic1" class="ltx_picture ltx_centering" style="width:138.8pt;height:99.0pt;">\begin{overpic}[width=138.76157pt]{figManifold.pdf}
\put(60.0,39.6){$\blacksquare$}
\put(68.0,35.0){$\blacksquare$}
\put(66.0,29.6){$\blacksquare$}
\put(36.0,22.6){$\bullet$}
\put(31.5,39.2){$\bullet$}
\put(38.8,31.35){$\bullet$} \put(60.5,20.5){$\bigvarstar$}
\put(74.0,64.0){\footnotesize Input space $\mathbb{R}^{d_{\textrm{in}}}$}
\put(85.0,46.0){\footnotesize Data manifold $\mathcal{M}$}
\put(-94.0,64.0){\scriptsize Two possible decision boundaries ({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}red}/{\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}blue}),}
\put(-94.0,58.65){\scriptsize input gradients at one training point,}
\put(-94.0,53.0){\scriptsize and their projection on the manifold.}
\put(56.0,4.7){{\scriptsize$\bullet$} {\tiny$\blacksquare$}\scriptsize\leavevmode\nobreak\ \leavevmode\nobreak\ Training points of two classes.}
\put(55.0,-1.0){\scriptsize\leavevmode\nobreak\ \leavevmode\nobreak\ \,$\bigvarstar$\leavevmode\nobreak\ \leavevmode\nobreak\ \, Test point, OOD, underspecified.}
\end{overpic}</span>
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span id="S4.F3.8.2.1" class="ltx_text" style="font-size:90%;">Figure 3</span>: </span><span id="S4.F3.2.1" class="ltx_text" style="font-size:90%;">
Effect of the proposed method in input space.
Data such as natural images is assumed to lie on a low-dimensional manifold.
The training set covers a subset of this manifold (gray ellipse). OOD test data (<math id="S4.F3.2.1.m1.1" class="ltx_Math" alttext="\bigvarstar" display="inline"><semantics id="S4.F3.2.1.m1.1b"><mo id="S4.F3.2.1.m1.1.1" xref="S4.F3.2.1.m1.1.1.cmml">🟌</mo><annotation-xml encoding="MathML-Content" id="S4.F3.2.1.m1.1c"><ci id="S4.F3.2.1.m1.1.1.cmml" xref="S4.F3.2.1.m1.1.1">🟌</ci></annotation-xml><annotation encoding="application/x-tex" id="S4.F3.2.1.m1.1d">\bigvarstar</annotation></semantics></math>) lies outside this subset.
In this example, our method discovers two models (<span id="S4.F3.2.1.1" class="ltx_text" style="color:#FF0000;">red</span> and <span id="S4.F3.2.1.2" class="ltx_text" style="color:#0000BF;">blue</span> decision boundaries) whose input gradients are orthogonal (shown at one training point, in colors matching the boundary).
Even though a third model (<span id="S4.F3.2.1.3" class="ltx_text" style="color:#00BF00;">green</span> vector) could satisfy the orthogonality constraint, its input gradient would point outside the manifold.
This would violate the <em id="S4.F3.2.1.4" class="ltx_emph ltx_font_italic">on-manifold</em> constraint, which requires gradients to closely match their projection on the manifold.
</span></figcaption>
</figure>
</section>
<section id="S4.SS3" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">4.3 </span>Fine-tuning</h3>

<div id="S4.SS3.p1" class="ltx_para">
<p id="S4.SS3.p1.2" class="ltx_p">After training a set of models with (<a href="#S4.Ex1" title="4.2 On-manifold constraint ‣ 4 Proposed method ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">4.2</span></a>), we propose to relax the independence and on-manifold constraints (<math id="S4.SS3.p1.1.m1.1" class="ltx_Math" alttext="{\lambda_{\textrm{indep}}}\!\leftarrow\!0" display="inline"><semantics id="S4.SS3.p1.1.m1.1a"><mrow id="S4.SS3.p1.1.m1.1.1" xref="S4.SS3.p1.1.m1.1.1.cmml"><msub id="S4.SS3.p1.1.m1.1.1.2" xref="S4.SS3.p1.1.m1.1.1.2.cmml"><mi id="S4.SS3.p1.1.m1.1.1.2.2" xref="S4.SS3.p1.1.m1.1.1.2.2.cmml">λ</mi><mtext id="S4.SS3.p1.1.m1.1.1.2.3" xref="S4.SS3.p1.1.m1.1.1.2.3a.cmml">indep</mtext></msub><mo lspace="0.108em" rspace="0.108em" stretchy="false" id="S4.SS3.p1.1.m1.1.1.1" xref="S4.SS3.p1.1.m1.1.1.1.cmml">←</mo><mn id="S4.SS3.p1.1.m1.1.1.3" xref="S4.SS3.p1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.1.m1.1b"><apply id="S4.SS3.p1.1.m1.1.1.cmml" xref="S4.SS3.p1.1.m1.1.1"><ci id="S4.SS3.p1.1.m1.1.1.1.cmml" xref="S4.SS3.p1.1.m1.1.1.1">←</ci><apply id="S4.SS3.p1.1.m1.1.1.2.cmml" xref="S4.SS3.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p1.1.m1.1.1.2.1.cmml" xref="S4.SS3.p1.1.m1.1.1.2">subscript</csymbol><ci id="S4.SS3.p1.1.m1.1.1.2.2.cmml" xref="S4.SS3.p1.1.m1.1.1.2.2">𝜆</ci><ci id="S4.SS3.p1.1.m1.1.1.2.3a.cmml" xref="S4.SS3.p1.1.m1.1.1.2.3"><mtext mathsize="70%" id="S4.SS3.p1.1.m1.1.1.2.3.cmml" xref="S4.SS3.p1.1.m1.1.1.2.3">indep</mtext></ci></apply><cn type="integer" id="S4.SS3.p1.1.m1.1.1.3.cmml" xref="S4.SS3.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.1.m1.1c">{\lambda_{\textrm{indep}}}\!\leftarrow\!0</annotation></semantics></math>, <math id="S4.SS3.p1.2.m2.1" class="ltx_Math" alttext="{\lambda_{\textrm{manifold}}}\!\leftarrow\!0" display="inline"><semantics id="S4.SS3.p1.2.m2.1a"><mrow id="S4.SS3.p1.2.m2.1.1" xref="S4.SS3.p1.2.m2.1.1.cmml"><msub id="S4.SS3.p1.2.m2.1.1.2" xref="S4.SS3.p1.2.m2.1.1.2.cmml"><mi id="S4.SS3.p1.2.m2.1.1.2.2" xref="S4.SS3.p1.2.m2.1.1.2.2.cmml">λ</mi><mtext id="S4.SS3.p1.2.m2.1.1.2.3" xref="S4.SS3.p1.2.m2.1.1.2.3a.cmml">manifold</mtext></msub><mo lspace="0.108em" rspace="0.108em" stretchy="false" id="S4.SS3.p1.2.m2.1.1.1" xref="S4.SS3.p1.2.m2.1.1.1.cmml">←</mo><mn id="S4.SS3.p1.2.m2.1.1.3" xref="S4.SS3.p1.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.2.m2.1b"><apply id="S4.SS3.p1.2.m2.1.1.cmml" xref="S4.SS3.p1.2.m2.1.1"><ci id="S4.SS3.p1.2.m2.1.1.1.cmml" xref="S4.SS3.p1.2.m2.1.1.1">←</ci><apply id="S4.SS3.p1.2.m2.1.1.2.cmml" xref="S4.SS3.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p1.2.m2.1.1.2.1.cmml" xref="S4.SS3.p1.2.m2.1.1.2">subscript</csymbol><ci id="S4.SS3.p1.2.m2.1.1.2.2.cmml" xref="S4.SS3.p1.2.m2.1.1.2.2">𝜆</ci><ci id="S4.SS3.p1.2.m2.1.1.2.3a.cmml" xref="S4.SS3.p1.2.m2.1.1.2.3"><mtext mathsize="70%" id="S4.SS3.p1.2.m2.1.1.2.3.cmml" xref="S4.SS3.p1.2.m2.1.1.2.3">manifold</mtext></ci></apply><cn type="integer" id="S4.SS3.p1.2.m2.1.1.3.cmml" xref="S4.SS3.p1.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.2.m2.1c">{\lambda_{\textrm{manifold}}}\!\leftarrow\!0</annotation></semantics></math>) then fine-tune the models.
This eases the optimization and typically allows the models to reach a higher predictive accuracy.
Concretely, we apply binary masks on the data such that each model is fine-tuned only on the elements most relevant to itself:<span id="footnote4" class="ltx_note ltx_role_footnote"><sup class="ltx_note_mark">4</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">4</sup><span class="ltx_tag ltx_tag_note">4</span>In our implementation, masked elements are not replaced with zeros, but rater with random values from other instances in the current mini-batch.</span></span></span></p>
<table id="A8.EGx4" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="S4.E4"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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xref="S4.E4.m1.1.1.1.1.1.2.2.2.cmml">y</mi><mi id="S4.E4.m1.1.1.1.1.1.2.2.3" xref="S4.E4.m1.1.1.1.1.1.2.2.3.cmml">i</mi></msub><mo rspace="0.278em" stretchy="false" id="S4.E4.m1.1.1.1.1.1.2.5" xref="S4.E4.m1.1.1.1.1.1.3.cmml">)</mo></mrow><mo rspace="0.278em" id="S4.E4.m1.2.2.2.2.4" xref="S4.E4.m1.2.2.2.3.1.cmml">:</mo><mrow id="S4.E4.m1.2.2.2.2.2" xref="S4.E4.m1.2.2.2.2.2.cmml"><mrow id="S4.E4.m1.2.2.2.2.2.2.2" xref="S4.E4.m1.2.2.2.2.2.2.3.cmml"><mo stretchy="false" id="S4.E4.m1.2.2.2.2.2.2.2.3" xref="S4.E4.m1.2.2.2.2.2.2.3.cmml">(</mo><msub id="S4.E4.m1.2.2.2.2.2.1.1.1" xref="S4.E4.m1.2.2.2.2.2.1.1.1.cmml"><mi id="S4.E4.m1.2.2.2.2.2.1.1.1.2" xref="S4.E4.m1.2.2.2.2.2.1.1.1.2.cmml">𝒙</mi><mi id="S4.E4.m1.2.2.2.2.2.1.1.1.3" xref="S4.E4.m1.2.2.2.2.2.1.1.1.3.cmml">i</mi></msub><mo id="S4.E4.m1.2.2.2.2.2.2.2.4" xref="S4.E4.m1.2.2.2.2.2.2.3.cmml">,</mo><msub id="S4.E4.m1.2.2.2.2.2.2.2.2" xref="S4.E4.m1.2.2.2.2.2.2.2.2.cmml"><mi id="S4.E4.m1.2.2.2.2.2.2.2.2.2" xref="S4.E4.m1.2.2.2.2.2.2.2.2.2.cmml">y</mi><mi id="S4.E4.m1.2.2.2.2.2.2.2.2.3" xref="S4.E4.m1.2.2.2.2.2.2.2.2.3.cmml">i</mi></msub><mo stretchy="false" id="S4.E4.m1.2.2.2.2.2.2.2.5" xref="S4.E4.m1.2.2.2.2.2.2.3.cmml">)</mo></mrow><mo id="S4.E4.m1.2.2.2.2.2.3" xref="S4.E4.m1.2.2.2.2.2.3.cmml">∈</mo><msub id="S4.E4.m1.2.2.2.2.2.4" xref="S4.E4.m1.2.2.2.2.2.4.cmml"><mi class="ltx_font_mathcaligraphic" id="S4.E4.m1.2.2.2.2.2.4.2" xref="S4.E4.m1.2.2.2.2.2.4.2.cmml">𝒟</mi><mi id="S4.E4.m1.2.2.2.2.2.4.3" xref="S4.E4.m1.2.2.2.2.2.4.3.cmml">tr</mi></msub></mrow><mo stretchy="false" id="S4.E4.m1.2.2.2.2.5" xref="S4.E4.m1.2.2.2.3.1.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="S4.E4.m1.2b"><apply id="S4.E4.m1.2.2.cmml" xref="S4.E4.m1.2.2"><eq id="S4.E4.m1.2.2.3.cmml" xref="S4.E4.m1.2.2.3"></eq><apply id="S4.E4.m1.2.2.4.cmml" xref="S4.E4.m1.2.2.4"><csymbol cd="ambiguous" id="S4.E4.m1.2.2.4.1.cmml" xref="S4.E4.m1.2.2.4">superscript</csymbol><apply id="S4.E4.m1.2.2.4.2.cmml" xref="S4.E4.m1.2.2.4"><csymbol cd="ambiguous" id="S4.E4.m1.2.2.4.2.1.cmml" xref="S4.E4.m1.2.2.4">subscript</csymbol><ci id="S4.E4.m1.2.2.4.2.2.cmml" xref="S4.E4.m1.2.2.4.2.2">𝒟</ci><ci id="S4.E4.m1.2.2.4.2.3.cmml" xref="S4.E4.m1.2.2.4.2.3">tr</ci></apply><ci id="S4.E4.m1.2.2.4.3.cmml" xref="S4.E4.m1.2.2.4.3">𝑚</ci></apply><apply id="S4.E4.m1.2.2.2.3.cmml" xref="S4.E4.m1.2.2.2.2"><csymbol cd="latexml" id="S4.E4.m1.2.2.2.3.1.cmml" xref="S4.E4.m1.2.2.2.2.3">conditional-set</csymbol><interval closure="open" id="S4.E4.m1.1.1.1.1.1.3.cmml" xref="S4.E4.m1.1.1.1.1.1.2"><apply id="S4.E4.m1.1.1.1.1.1.1.1.cmml" xref="S4.E4.m1.1.1.1.1.1.1.1"><csymbol cd="latexml" id="S4.E4.m1.1.1.1.1.1.1.1.1.cmml" xref="S4.E4.m1.1.1.1.1.1.1.1.1">direct-product</csymbol><apply id="S4.E4.m1.1.1.1.1.1.1.1.2.cmml" xref="S4.E4.m1.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="S4.E4.m1.1.1.1.1.1.1.1.2.1.cmml" xref="S4.E4.m1.1.1.1.1.1.1.1.2">subscript</csymbol><ci 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id="S4.E4.m1.2.2.2.2.2.2.2.2.2.cmml" xref="S4.E4.m1.2.2.2.2.2.2.2.2.2">𝑦</ci><ci id="S4.E4.m1.2.2.2.2.2.2.2.2.3.cmml" xref="S4.E4.m1.2.2.2.2.2.2.2.2.3">𝑖</ci></apply></interval><apply id="S4.E4.m1.2.2.2.2.2.4.cmml" xref="S4.E4.m1.2.2.2.2.2.4"><csymbol cd="ambiguous" id="S4.E4.m1.2.2.2.2.2.4.1.cmml" xref="S4.E4.m1.2.2.2.2.2.4">subscript</csymbol><ci id="S4.E4.m1.2.2.2.2.2.4.2.cmml" xref="S4.E4.m1.2.2.2.2.2.4.2">𝒟</ci><ci id="S4.E4.m1.2.2.2.2.2.4.3.cmml" xref="S4.E4.m1.2.2.2.2.2.4.3">tr</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.E4.m1.2c">\displaystyle\mathcal{D}_{\mathrm{tr}}^{m}=\{({\boldsymbol{x}}_{i}\odot{\boldsymbol{\mathrm{mask}}}^{m}_{i},y_{i}):({\boldsymbol{x}}_{i},y_{i})\in\mathcal{D}_{\mathrm{tr}}\}</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(4)</span></td>
</tr></tbody>
</table>
<p id="S4.SS3.p1.3" class="ltx_p">with <math id="S4.SS3.p1.3.m1.2" class="ltx_Math" alttext="{\boldsymbol{\mathrm{mask}}}^{m}_{i}\in\{0,1\}^{d_{\textrm{in}}}" display="inline"><semantics id="S4.SS3.p1.3.m1.2a"><mrow id="S4.SS3.p1.3.m1.2.3" xref="S4.SS3.p1.3.m1.2.3.cmml"><msubsup id="S4.SS3.p1.3.m1.2.3.2" xref="S4.SS3.p1.3.m1.2.3.2.cmml"><mi id="S4.SS3.p1.3.m1.2.3.2.2.2" xref="S4.SS3.p1.3.m1.2.3.2.2.2.cmml">𝐦𝐚𝐬𝐤</mi><mi id="S4.SS3.p1.3.m1.2.3.2.3" xref="S4.SS3.p1.3.m1.2.3.2.3.cmml">i</mi><mi id="S4.SS3.p1.3.m1.2.3.2.2.3" xref="S4.SS3.p1.3.m1.2.3.2.2.3.cmml">m</mi></msubsup><mo id="S4.SS3.p1.3.m1.2.3.1" xref="S4.SS3.p1.3.m1.2.3.1.cmml">∈</mo><msup id="S4.SS3.p1.3.m1.2.3.3" xref="S4.SS3.p1.3.m1.2.3.3.cmml"><mrow id="S4.SS3.p1.3.m1.2.3.3.2.2" xref="S4.SS3.p1.3.m1.2.3.3.2.1.cmml"><mo stretchy="false" id="S4.SS3.p1.3.m1.2.3.3.2.2.1" xref="S4.SS3.p1.3.m1.2.3.3.2.1.cmml">{</mo><mn id="S4.SS3.p1.3.m1.1.1" xref="S4.SS3.p1.3.m1.1.1.cmml">0</mn><mo id="S4.SS3.p1.3.m1.2.3.3.2.2.2" xref="S4.SS3.p1.3.m1.2.3.3.2.1.cmml">,</mo><mn id="S4.SS3.p1.3.m1.2.2" xref="S4.SS3.p1.3.m1.2.2.cmml">1</mn><mo stretchy="false" id="S4.SS3.p1.3.m1.2.3.3.2.2.3" xref="S4.SS3.p1.3.m1.2.3.3.2.1.cmml">}</mo></mrow><msub id="S4.SS3.p1.3.m1.2.3.3.3" xref="S4.SS3.p1.3.m1.2.3.3.3.cmml"><mi id="S4.SS3.p1.3.m1.2.3.3.3.2" xref="S4.SS3.p1.3.m1.2.3.3.3.2.cmml">d</mi><mtext id="S4.SS3.p1.3.m1.2.3.3.3.3" xref="S4.SS3.p1.3.m1.2.3.3.3.3a.cmml">in</mtext></msub></msup></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.3.m1.2b"><apply id="S4.SS3.p1.3.m1.2.3.cmml" xref="S4.SS3.p1.3.m1.2.3"><in id="S4.SS3.p1.3.m1.2.3.1.cmml" xref="S4.SS3.p1.3.m1.2.3.1"></in><apply id="S4.SS3.p1.3.m1.2.3.2.cmml" xref="S4.SS3.p1.3.m1.2.3.2"><csymbol cd="ambiguous" id="S4.SS3.p1.3.m1.2.3.2.1.cmml" xref="S4.SS3.p1.3.m1.2.3.2">subscript</csymbol><apply id="S4.SS3.p1.3.m1.2.3.2.2.cmml" xref="S4.SS3.p1.3.m1.2.3.2"><csymbol cd="ambiguous" id="S4.SS3.p1.3.m1.2.3.2.2.1.cmml" xref="S4.SS3.p1.3.m1.2.3.2">superscript</csymbol><ci id="S4.SS3.p1.3.m1.2.3.2.2.2.cmml" xref="S4.SS3.p1.3.m1.2.3.2.2.2">𝐦𝐚𝐬𝐤</ci><ci id="S4.SS3.p1.3.m1.2.3.2.2.3.cmml" xref="S4.SS3.p1.3.m1.2.3.2.2.3">𝑚</ci></apply><ci id="S4.SS3.p1.3.m1.2.3.2.3.cmml" xref="S4.SS3.p1.3.m1.2.3.2.3">𝑖</ci></apply><apply id="S4.SS3.p1.3.m1.2.3.3.cmml" xref="S4.SS3.p1.3.m1.2.3.3"><csymbol cd="ambiguous" id="S4.SS3.p1.3.m1.2.3.3.1.cmml" xref="S4.SS3.p1.3.m1.2.3.3">superscript</csymbol><set id="S4.SS3.p1.3.m1.2.3.3.2.1.cmml" xref="S4.SS3.p1.3.m1.2.3.3.2.2"><cn type="integer" id="S4.SS3.p1.3.m1.1.1.cmml" xref="S4.SS3.p1.3.m1.1.1">0</cn><cn type="integer" id="S4.SS3.p1.3.m1.2.2.cmml" xref="S4.SS3.p1.3.m1.2.2">1</cn></set><apply id="S4.SS3.p1.3.m1.2.3.3.3.cmml" xref="S4.SS3.p1.3.m1.2.3.3.3"><csymbol cd="ambiguous" id="S4.SS3.p1.3.m1.2.3.3.3.1.cmml" xref="S4.SS3.p1.3.m1.2.3.3.3">subscript</csymbol><ci id="S4.SS3.p1.3.m1.2.3.3.3.2.cmml" xref="S4.SS3.p1.3.m1.2.3.3.3.2">𝑑</ci><ci id="S4.SS3.p1.3.m1.2.3.3.3.3a.cmml" xref="S4.SS3.p1.3.m1.2.3.3.3.3"><mtext mathsize="50%" id="S4.SS3.p1.3.m1.2.3.3.3.3.cmml" xref="S4.SS3.p1.3.m1.2.3.3.3.3">in</mtext></ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.3.m1.2c">{\boldsymbol{\mathrm{mask}}}^{m}_{i}\in\{0,1\}^{d_{\textrm{in}}}</annotation></semantics></math>.
The masks are computed before starting the fine-tuning to highlight the data most relevant to each model.
Each element (i.e. pixel or channel) is unmasked only for the model with the largest corresponding gradient magnitude:</p>
<table id="A8.EGx5" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="S4.E5"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
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encoding="application/x-tex" id="S4.E5.m1.3c">\displaystyle{\boldsymbol{\mathrm{mask}}}^{m}_{i}=\operatorname*{\mathbbm{1}}\big{(}m=\mathop{\textrm{argmax}}_{1\leq m\leq M}\nabla f_{{\boldsymbol{\theta}}_{m}}({\boldsymbol{x}}_{i})\big{)}\leavevmode\nobreak\ \leavevmode\nobreak\ \,\forall\,({\boldsymbol{x}}_{i},\cdot)\in\mathcal{D}_{\mathrm{tr}}.</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(5)</span></td>
</tr></tbody>
</table>
<p id="S4.SS3.p1.5" class="ltx_p">We fine-tune each model on its own masked version of the data.<span id="footnote5" class="ltx_note ltx_role_footnote"><sup class="ltx_note_mark">5</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">5</sup><span class="ltx_tag ltx_tag_note">5</span>We obtain very similar results between fine-tuning and retraining models from scratch on the masked data.</span></span></span>
This ensures that the models remain distinct despite disabling the regularizers (<math id="S4.SS3.p1.4.m1.1" class="ltx_Math" alttext="{\lambda_{\textrm{indep}}}\!\leftarrow\!0" display="inline"><semantics id="S4.SS3.p1.4.m1.1a"><mrow id="S4.SS3.p1.4.m1.1.1" xref="S4.SS3.p1.4.m1.1.1.cmml"><msub id="S4.SS3.p1.4.m1.1.1.2" xref="S4.SS3.p1.4.m1.1.1.2.cmml"><mi id="S4.SS3.p1.4.m1.1.1.2.2" xref="S4.SS3.p1.4.m1.1.1.2.2.cmml">λ</mi><mtext id="S4.SS3.p1.4.m1.1.1.2.3" xref="S4.SS3.p1.4.m1.1.1.2.3a.cmml">indep</mtext></msub><mo lspace="0.108em" rspace="0.108em" stretchy="false" id="S4.SS3.p1.4.m1.1.1.1" xref="S4.SS3.p1.4.m1.1.1.1.cmml">←</mo><mn id="S4.SS3.p1.4.m1.1.1.3" xref="S4.SS3.p1.4.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.4.m1.1b"><apply id="S4.SS3.p1.4.m1.1.1.cmml" xref="S4.SS3.p1.4.m1.1.1"><ci id="S4.SS3.p1.4.m1.1.1.1.cmml" xref="S4.SS3.p1.4.m1.1.1.1">←</ci><apply id="S4.SS3.p1.4.m1.1.1.2.cmml" xref="S4.SS3.p1.4.m1.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p1.4.m1.1.1.2.1.cmml" xref="S4.SS3.p1.4.m1.1.1.2">subscript</csymbol><ci id="S4.SS3.p1.4.m1.1.1.2.2.cmml" xref="S4.SS3.p1.4.m1.1.1.2.2">𝜆</ci><ci id="S4.SS3.p1.4.m1.1.1.2.3a.cmml" xref="S4.SS3.p1.4.m1.1.1.2.3"><mtext mathsize="70%" id="S4.SS3.p1.4.m1.1.1.2.3.cmml" xref="S4.SS3.p1.4.m1.1.1.2.3">indep</mtext></ci></apply><cn type="integer" id="S4.SS3.p1.4.m1.1.1.3.cmml" xref="S4.SS3.p1.4.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.4.m1.1c">{\lambda_{\textrm{indep}}}\!\leftarrow\!0</annotation></semantics></math>, <math id="S4.SS3.p1.5.m2.1" class="ltx_Math" alttext="{\lambda_{\textrm{manifold}}}\!\leftarrow\!0" display="inline"><semantics id="S4.SS3.p1.5.m2.1a"><mrow id="S4.SS3.p1.5.m2.1.1" xref="S4.SS3.p1.5.m2.1.1.cmml"><msub id="S4.SS3.p1.5.m2.1.1.2" xref="S4.SS3.p1.5.m2.1.1.2.cmml"><mi id="S4.SS3.p1.5.m2.1.1.2.2" xref="S4.SS3.p1.5.m2.1.1.2.2.cmml">λ</mi><mtext id="S4.SS3.p1.5.m2.1.1.2.3" xref="S4.SS3.p1.5.m2.1.1.2.3a.cmml">manifold</mtext></msub><mo lspace="0.108em" rspace="0.108em" stretchy="false" id="S4.SS3.p1.5.m2.1.1.1" xref="S4.SS3.p1.5.m2.1.1.1.cmml">←</mo><mn id="S4.SS3.p1.5.m2.1.1.3" xref="S4.SS3.p1.5.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S4.SS3.p1.5.m2.1b"><apply id="S4.SS3.p1.5.m2.1.1.cmml" xref="S4.SS3.p1.5.m2.1.1"><ci id="S4.SS3.p1.5.m2.1.1.1.cmml" xref="S4.SS3.p1.5.m2.1.1.1">←</ci><apply id="S4.SS3.p1.5.m2.1.1.2.cmml" xref="S4.SS3.p1.5.m2.1.1.2"><csymbol cd="ambiguous" id="S4.SS3.p1.5.m2.1.1.2.1.cmml" xref="S4.SS3.p1.5.m2.1.1.2">subscript</csymbol><ci id="S4.SS3.p1.5.m2.1.1.2.2.cmml" xref="S4.SS3.p1.5.m2.1.1.2.2">𝜆</ci><ci id="S4.SS3.p1.5.m2.1.1.2.3a.cmml" xref="S4.SS3.p1.5.m2.1.1.2.3"><mtext mathsize="70%" id="S4.SS3.p1.5.m2.1.1.2.3.cmml" xref="S4.SS3.p1.5.m2.1.1.2.3">manifold</mtext></ci></apply><cn type="integer" id="S4.SS3.p1.5.m2.1.1.3.cmml" xref="S4.SS3.p1.5.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S4.SS3.p1.5.m2.1c">{\lambda_{\textrm{manifold}}}\!\leftarrow\!0</annotation></semantics></math>).
See Algorithm <a href="#algorithm1" title="In 4.3 Fine-tuning ‣ 4 Proposed method ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">1</span></a> for a summary.</p>
</div>
<figure id="algorithm1" class="ltx_float ltx_algorithm">
<div id="algorithm1.19" class="ltx_listing ltx_lst_numbers_left ltx_listing">
<div id="algorithm1.4.4" class="ltx_listingline">
<span id="algorithm1.4.4.4" class="ltx_text" style="font-size:80%;">
<span id="algorithm1.4.4.4.1" class="ltx_text ltx_font_bold">Inputs:</span> Labeled examples <math id="algorithm1.1.1.1.m1.1" class="ltx_Math" alttext="\mathcal{D}_{\mathrm{tr}}" display="inline"><semantics id="algorithm1.1.1.1.m1.1a"><msub id="algorithm1.1.1.1.m1.1.1" xref="algorithm1.1.1.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="algorithm1.1.1.1.m1.1.1.2" xref="algorithm1.1.1.1.m1.1.1.2.cmml">𝒟</mi><mi id="algorithm1.1.1.1.m1.1.1.3" xref="algorithm1.1.1.1.m1.1.1.3.cmml">tr</mi></msub><annotation-xml encoding="MathML-Content" id="algorithm1.1.1.1.m1.1b"><apply id="algorithm1.1.1.1.m1.1.1.cmml" xref="algorithm1.1.1.1.m1.1.1"><csymbol cd="ambiguous" id="algorithm1.1.1.1.m1.1.1.1.cmml" xref="algorithm1.1.1.1.m1.1.1">subscript</csymbol><ci id="algorithm1.1.1.1.m1.1.1.2.cmml" xref="algorithm1.1.1.1.m1.1.1.2">𝒟</ci><ci id="algorithm1.1.1.1.m1.1.1.3.cmml" xref="algorithm1.1.1.1.m1.1.1.3">tr</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.1.1.1.m1.1c">\mathcal{D}_{\mathrm{tr}}</annotation></semantics></math>.
Unlabeled examples <math id="algorithm1.2.2.2.m2.1" class="ltx_Math" alttext="\mathcal{D}_{\mathrm{OOD}}" display="inline"><semantics id="algorithm1.2.2.2.m2.1a"><msub id="algorithm1.2.2.2.m2.1.1" xref="algorithm1.2.2.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="algorithm1.2.2.2.m2.1.1.2" xref="algorithm1.2.2.2.m2.1.1.2.cmml">𝒟</mi><mi id="algorithm1.2.2.2.m2.1.1.3" xref="algorithm1.2.2.2.m2.1.1.3.cmml">OOD</mi></msub><annotation-xml encoding="MathML-Content" id="algorithm1.2.2.2.m2.1b"><apply id="algorithm1.2.2.2.m2.1.1.cmml" xref="algorithm1.2.2.2.m2.1.1"><csymbol cd="ambiguous" id="algorithm1.2.2.2.m2.1.1.1.cmml" xref="algorithm1.2.2.2.m2.1.1">subscript</csymbol><ci id="algorithm1.2.2.2.m2.1.1.2.cmml" xref="algorithm1.2.2.2.m2.1.1.2">𝒟</ci><ci id="algorithm1.2.2.2.m2.1.1.3.cmml" xref="algorithm1.2.2.2.m2.1.1.3">OOD</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.2.2.2.m2.1c">\mathcal{D}_{\mathrm{OOD}}</annotation></semantics></math> (typically <math id="algorithm1.3.3.3.m3.1" class="ltx_Math" alttext="\mathcal{D}_{\mathrm{tr}}\subset\mathcal{D}_{\mathrm{OOD}}" display="inline"><semantics id="algorithm1.3.3.3.m3.1a"><mrow id="algorithm1.3.3.3.m3.1.1" xref="algorithm1.3.3.3.m3.1.1.cmml"><msub id="algorithm1.3.3.3.m3.1.1.2" xref="algorithm1.3.3.3.m3.1.1.2.cmml"><mi class="ltx_font_mathcaligraphic" id="algorithm1.3.3.3.m3.1.1.2.2" xref="algorithm1.3.3.3.m3.1.1.2.2.cmml">𝒟</mi><mi id="algorithm1.3.3.3.m3.1.1.2.3" xref="algorithm1.3.3.3.m3.1.1.2.3.cmml">tr</mi></msub><mo id="algorithm1.3.3.3.m3.1.1.1" xref="algorithm1.3.3.3.m3.1.1.1.cmml">⊂</mo><msub id="algorithm1.3.3.3.m3.1.1.3" xref="algorithm1.3.3.3.m3.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="algorithm1.3.3.3.m3.1.1.3.2" xref="algorithm1.3.3.3.m3.1.1.3.2.cmml">𝒟</mi><mi id="algorithm1.3.3.3.m3.1.1.3.3" xref="algorithm1.3.3.3.m3.1.1.3.3.cmml">OOD</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="algorithm1.3.3.3.m3.1b"><apply id="algorithm1.3.3.3.m3.1.1.cmml" xref="algorithm1.3.3.3.m3.1.1"><subset id="algorithm1.3.3.3.m3.1.1.1.cmml" xref="algorithm1.3.3.3.m3.1.1.1"></subset><apply id="algorithm1.3.3.3.m3.1.1.2.cmml" xref="algorithm1.3.3.3.m3.1.1.2"><csymbol cd="ambiguous" id="algorithm1.3.3.3.m3.1.1.2.1.cmml" xref="algorithm1.3.3.3.m3.1.1.2">subscript</csymbol><ci id="algorithm1.3.3.3.m3.1.1.2.2.cmml" xref="algorithm1.3.3.3.m3.1.1.2.2">𝒟</ci><ci id="algorithm1.3.3.3.m3.1.1.2.3.cmml" xref="algorithm1.3.3.3.m3.1.1.2.3">tr</ci></apply><apply id="algorithm1.3.3.3.m3.1.1.3.cmml" xref="algorithm1.3.3.3.m3.1.1.3"><csymbol cd="ambiguous" id="algorithm1.3.3.3.m3.1.1.3.1.cmml" xref="algorithm1.3.3.3.m3.1.1.3">subscript</csymbol><ci id="algorithm1.3.3.3.m3.1.1.3.2.cmml" xref="algorithm1.3.3.3.m3.1.1.3.2">𝒟</ci><ci id="algorithm1.3.3.3.m3.1.1.3.3.cmml" xref="algorithm1.3.3.3.m3.1.1.3.3">OOD</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.3.3.3.m3.1c">\mathcal{D}_{\mathrm{tr}}\subset\mathcal{D}_{\mathrm{OOD}}</annotation></semantics></math>). Architecture <math id="algorithm1.4.4.4.m4.1" class="ltx_Math" alttext="f" display="inline"><semantics id="algorithm1.4.4.4.m4.1a"><mi id="algorithm1.4.4.4.m4.1.1" xref="algorithm1.4.4.4.m4.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="algorithm1.4.4.4.m4.1b"><ci id="algorithm1.4.4.4.m4.1.1.cmml" xref="algorithm1.4.4.4.m4.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.4.4.4.m4.1c">f</annotation></semantics></math>.</span>
</div>
<div id="algorithm1.5.5" class="ltx_listingline">
<span id="algorithm1.5.5.1" class="ltx_text" style="font-size:80%;">
</span><span id="algorithm1.5.5.2" class="ltx_text ltx_font_bold" style="font-size:80%;">Result:</span><span id="algorithm1.5.5.3" class="ltx_text" style="font-size:80%;">
Set of independent models </span><math id="algorithm1.5.5.m1.1" class="ltx_Math" alttext="\{f_{{\boldsymbol{\theta}}_{1}}\,\kern-0.80005pt.\kern-0.80005pt.\kern-0.80005pt.\,f_{{\boldsymbol{\theta}}_{M}}\}" display="inline"><semantics id="algorithm1.5.5.m1.1a"><mrow id="algorithm1.5.5.m1.1.1.1" xref="algorithm1.5.5.m1.1.1.2.cmml"><mo maxsize="80%" minsize="80%" id="algorithm1.5.5.m1.1.1.1.2" xref="algorithm1.5.5.m1.1.1.2.cmml">{</mo><mrow id="algorithm1.5.5.m1.1.1.1.1" xref="algorithm1.5.5.m1.1.1.1.1.cmml"><msub id="algorithm1.5.5.m1.1.1.1.1.2" xref="algorithm1.5.5.m1.1.1.1.1.2.cmml"><mi mathsize="80%" id="algorithm1.5.5.m1.1.1.1.1.2.2" xref="algorithm1.5.5.m1.1.1.1.1.2.2.cmml">f</mi><msub id="algorithm1.5.5.m1.1.1.1.1.2.3" xref="algorithm1.5.5.m1.1.1.1.1.2.3.cmml"><mi mathsize="80%" id="algorithm1.5.5.m1.1.1.1.1.2.3.2" xref="algorithm1.5.5.m1.1.1.1.1.2.3.2.cmml">𝜽</mi><mn mathsize="80%" id="algorithm1.5.5.m1.1.1.1.1.2.3.3" xref="algorithm1.5.5.m1.1.1.1.1.2.3.3.cmml">1</mn></msub></msub><mo lspace="0em" rspace="0em" id="algorithm1.5.5.m1.1.1.1.1.1" xref="algorithm1.5.5.m1.1.1.1.1.1.cmml">​</mo><mi mathsize="80%" mathvariant="normal" id="algorithm1.5.5.m1.1.1.1.1.3" xref="algorithm1.5.5.m1.1.1.1.1.3.cmml">…</mi><mo lspace="0.170em" rspace="0em" id="algorithm1.5.5.m1.1.1.1.1.1a" xref="algorithm1.5.5.m1.1.1.1.1.1.cmml">​</mo><msub id="algorithm1.5.5.m1.1.1.1.1.4" xref="algorithm1.5.5.m1.1.1.1.1.4.cmml"><mi mathsize="80%" id="algorithm1.5.5.m1.1.1.1.1.4.2" xref="algorithm1.5.5.m1.1.1.1.1.4.2.cmml">f</mi><msub id="algorithm1.5.5.m1.1.1.1.1.4.3" xref="algorithm1.5.5.m1.1.1.1.1.4.3.cmml"><mi mathsize="80%" id="algorithm1.5.5.m1.1.1.1.1.4.3.2" xref="algorithm1.5.5.m1.1.1.1.1.4.3.2.cmml">𝜽</mi><mi mathsize="80%" id="algorithm1.5.5.m1.1.1.1.1.4.3.3" xref="algorithm1.5.5.m1.1.1.1.1.4.3.3.cmml">M</mi></msub></msub></mrow><mo maxsize="80%" minsize="80%" id="algorithm1.5.5.m1.1.1.1.3" xref="algorithm1.5.5.m1.1.1.2.cmml">}</mo></mrow><annotation-xml encoding="MathML-Content" id="algorithm1.5.5.m1.1b"><set id="algorithm1.5.5.m1.1.1.2.cmml" xref="algorithm1.5.5.m1.1.1.1"><apply id="algorithm1.5.5.m1.1.1.1.1.cmml" xref="algorithm1.5.5.m1.1.1.1.1"><times id="algorithm1.5.5.m1.1.1.1.1.1.cmml" xref="algorithm1.5.5.m1.1.1.1.1.1"></times><apply id="algorithm1.5.5.m1.1.1.1.1.2.cmml" xref="algorithm1.5.5.m1.1.1.1.1.2"><csymbol cd="ambiguous" id="algorithm1.5.5.m1.1.1.1.1.2.1.cmml" xref="algorithm1.5.5.m1.1.1.1.1.2">subscript</csymbol><ci id="algorithm1.5.5.m1.1.1.1.1.2.2.cmml" xref="algorithm1.5.5.m1.1.1.1.1.2.2">𝑓</ci><apply id="algorithm1.5.5.m1.1.1.1.1.2.3.cmml" xref="algorithm1.5.5.m1.1.1.1.1.2.3"><csymbol cd="ambiguous" id="algorithm1.5.5.m1.1.1.1.1.2.3.1.cmml" xref="algorithm1.5.5.m1.1.1.1.1.2.3">subscript</csymbol><ci id="algorithm1.5.5.m1.1.1.1.1.2.3.2.cmml" xref="algorithm1.5.5.m1.1.1.1.1.2.3.2">𝜽</ci><cn type="integer" id="algorithm1.5.5.m1.1.1.1.1.2.3.3.cmml" xref="algorithm1.5.5.m1.1.1.1.1.2.3.3">1</cn></apply></apply><ci id="algorithm1.5.5.m1.1.1.1.1.3.cmml" xref="algorithm1.5.5.m1.1.1.1.1.3">…</ci><apply id="algorithm1.5.5.m1.1.1.1.1.4.cmml" xref="algorithm1.5.5.m1.1.1.1.1.4"><csymbol cd="ambiguous" id="algorithm1.5.5.m1.1.1.1.1.4.1.cmml" xref="algorithm1.5.5.m1.1.1.1.1.4">subscript</csymbol><ci id="algorithm1.5.5.m1.1.1.1.1.4.2.cmml" xref="algorithm1.5.5.m1.1.1.1.1.4.2">𝑓</ci><apply id="algorithm1.5.5.m1.1.1.1.1.4.3.cmml" xref="algorithm1.5.5.m1.1.1.1.1.4.3"><csymbol cd="ambiguous" id="algorithm1.5.5.m1.1.1.1.1.4.3.1.cmml" xref="algorithm1.5.5.m1.1.1.1.1.4.3">subscript</csymbol><ci id="algorithm1.5.5.m1.1.1.1.1.4.3.2.cmml" xref="algorithm1.5.5.m1.1.1.1.1.4.3.2">𝜽</ci><ci id="algorithm1.5.5.m1.1.1.1.1.4.3.3.cmml" xref="algorithm1.5.5.m1.1.1.1.1.4.3.3">𝑀</ci></apply></apply></apply></set></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.5.5.m1.1c">\{f_{{\boldsymbol{\theta}}_{1}}\,\kern-0.80005pt.\kern-0.80005pt.\kern-0.80005pt.\,f_{{\boldsymbol{\theta}}_{M}}\}</annotation></semantics></math><span id="algorithm1.5.5.4" class="ltx_text" style="font-size:80%;">.</span>
</div>
<div id="algorithm1.19.20" class="ltx_listingline">
<span id="algorithm1.19.20.1" class="ltx_text" style="font-size:80%;">
</span><span id="algorithm1.19.20.2" class="ltx_text ltx_font_bold" style="font-size:80%;">Method:</span>
</div>
<div id="algorithm1.8.8" class="ltx_listingline">
<span id="algorithm1.8.8.1" class="ltx_text" style="font-size:80%;">
With </span><math id="algorithm1.6.6.m1.1" class="ltx_Math" alttext="\mathcal{D}_{\mathrm{OOD}}" display="inline"><semantics id="algorithm1.6.6.m1.1a"><msub id="algorithm1.6.6.m1.1.1" xref="algorithm1.6.6.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" mathsize="80%" id="algorithm1.6.6.m1.1.1.2" xref="algorithm1.6.6.m1.1.1.2.cmml">𝒟</mi><mi mathsize="80%" id="algorithm1.6.6.m1.1.1.3" xref="algorithm1.6.6.m1.1.1.3.cmml">OOD</mi></msub><annotation-xml encoding="MathML-Content" id="algorithm1.6.6.m1.1b"><apply id="algorithm1.6.6.m1.1.1.cmml" xref="algorithm1.6.6.m1.1.1"><csymbol cd="ambiguous" id="algorithm1.6.6.m1.1.1.1.cmml" xref="algorithm1.6.6.m1.1.1">subscript</csymbol><ci id="algorithm1.6.6.m1.1.1.2.cmml" xref="algorithm1.6.6.m1.1.1.2">𝒟</ci><ci id="algorithm1.6.6.m1.1.1.3.cmml" xref="algorithm1.6.6.m1.1.1.3">OOD</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.6.6.m1.1c">\mathcal{D}_{\mathrm{OOD}}</annotation></semantics></math><span id="algorithm1.8.8.2" class="ltx_text" style="font-size:80%;">, estimate dimensionality </span><math id="algorithm1.7.7.m2.1" class="ltx_Math" alttext="{d_{\textrm{manifold}}}" display="inline"><semantics id="algorithm1.7.7.m2.1a"><msub id="algorithm1.7.7.m2.1.1" xref="algorithm1.7.7.m2.1.1.cmml"><mi mathsize="80%" id="algorithm1.7.7.m2.1.1.2" xref="algorithm1.7.7.m2.1.1.2.cmml">d</mi><mtext mathsize="80%" id="algorithm1.7.7.m2.1.1.3" xref="algorithm1.7.7.m2.1.1.3a.cmml">manifold</mtext></msub><annotation-xml encoding="MathML-Content" id="algorithm1.7.7.m2.1b"><apply id="algorithm1.7.7.m2.1.1.cmml" xref="algorithm1.7.7.m2.1.1"><csymbol cd="ambiguous" id="algorithm1.7.7.m2.1.1.1.cmml" xref="algorithm1.7.7.m2.1.1">subscript</csymbol><ci id="algorithm1.7.7.m2.1.1.2.cmml" xref="algorithm1.7.7.m2.1.1.2">𝑑</ci><ci id="algorithm1.7.7.m2.1.1.3a.cmml" xref="algorithm1.7.7.m2.1.1.3"><mtext mathsize="56%" id="algorithm1.7.7.m2.1.1.3.cmml" xref="algorithm1.7.7.m2.1.1.3">manifold</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.7.7.m2.1c">{d_{\textrm{manifold}}}</annotation></semantics></math><span id="algorithm1.8.8.3" class="ltx_text" style="font-size:80%;"> </span><cite class="ltx_cite ltx_citemacro_cite"><span id="algorithm1.8.8.4.1" class="ltx_text" style="font-size:80%;">[</span><a href="#bib.bib57" title="" class="ltx_ref">57</a><span id="algorithm1.8.8.5.2" class="ltx_text" style="font-size:80%;">]</span></cite><span id="algorithm1.8.8.6" class="ltx_text" style="font-size:80%;">  and set the number of models </span><math id="algorithm1.8.8.m3.1" class="ltx_Math" alttext="M\leftarrow{d_{\textrm{manifold}}}" display="inline"><semantics id="algorithm1.8.8.m3.1a"><mrow id="algorithm1.8.8.m3.1.1" xref="algorithm1.8.8.m3.1.1.cmml"><mi mathsize="80%" id="algorithm1.8.8.m3.1.1.2" xref="algorithm1.8.8.m3.1.1.2.cmml">M</mi><mo mathsize="80%" stretchy="false" id="algorithm1.8.8.m3.1.1.1" xref="algorithm1.8.8.m3.1.1.1.cmml">←</mo><msub id="algorithm1.8.8.m3.1.1.3" xref="algorithm1.8.8.m3.1.1.3.cmml"><mi mathsize="80%" id="algorithm1.8.8.m3.1.1.3.2" xref="algorithm1.8.8.m3.1.1.3.2.cmml">d</mi><mtext mathsize="80%" id="algorithm1.8.8.m3.1.1.3.3" xref="algorithm1.8.8.m3.1.1.3.3a.cmml">manifold</mtext></msub></mrow><annotation-xml encoding="MathML-Content" id="algorithm1.8.8.m3.1b"><apply id="algorithm1.8.8.m3.1.1.cmml" xref="algorithm1.8.8.m3.1.1"><ci id="algorithm1.8.8.m3.1.1.1.cmml" xref="algorithm1.8.8.m3.1.1.1">←</ci><ci id="algorithm1.8.8.m3.1.1.2.cmml" xref="algorithm1.8.8.m3.1.1.2">𝑀</ci><apply id="algorithm1.8.8.m3.1.1.3.cmml" xref="algorithm1.8.8.m3.1.1.3"><csymbol cd="ambiguous" id="algorithm1.8.8.m3.1.1.3.1.cmml" xref="algorithm1.8.8.m3.1.1.3">subscript</csymbol><ci id="algorithm1.8.8.m3.1.1.3.2.cmml" xref="algorithm1.8.8.m3.1.1.3.2">𝑑</ci><ci id="algorithm1.8.8.m3.1.1.3.3a.cmml" xref="algorithm1.8.8.m3.1.1.3.3"><mtext mathsize="56%" id="algorithm1.8.8.m3.1.1.3.3.cmml" xref="algorithm1.8.8.m3.1.1.3.3">manifold</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.8.8.m3.1c">M\leftarrow{d_{\textrm{manifold}}}</annotation></semantics></math><span id="algorithm1.8.8.7" class="ltx_text" style="font-size:80%;">.</span>
</div>
<div id="algorithm1.10.10" class="ltx_listingline">
<span id="algorithm1.10.10.1" class="ltx_text" style="font-size:80%;">
With </span><math id="algorithm1.9.9.m1.1" class="ltx_Math" alttext="\mathcal{D}_{\mathrm{OOD}}" display="inline"><semantics id="algorithm1.9.9.m1.1a"><msub id="algorithm1.9.9.m1.1.1" xref="algorithm1.9.9.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" mathsize="80%" id="algorithm1.9.9.m1.1.1.2" xref="algorithm1.9.9.m1.1.1.2.cmml">𝒟</mi><mi mathsize="80%" id="algorithm1.9.9.m1.1.1.3" xref="algorithm1.9.9.m1.1.1.3.cmml">OOD</mi></msub><annotation-xml encoding="MathML-Content" id="algorithm1.9.9.m1.1b"><apply id="algorithm1.9.9.m1.1.1.cmml" xref="algorithm1.9.9.m1.1.1"><csymbol cd="ambiguous" id="algorithm1.9.9.m1.1.1.1.cmml" xref="algorithm1.9.9.m1.1.1">subscript</csymbol><ci id="algorithm1.9.9.m1.1.1.2.cmml" xref="algorithm1.9.9.m1.1.1.2">𝒟</ci><ci id="algorithm1.9.9.m1.1.1.3.cmml" xref="algorithm1.9.9.m1.1.1.3">OOD</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.9.9.m1.1c">\mathcal{D}_{\mathrm{OOD}}</annotation></semantics></math><span id="algorithm1.10.10.2" class="ltx_text" style="font-size:80%;">, prepare function </span><math id="algorithm1.10.10.m2.1" class="ltx_Math" alttext="\mathrm{proj}(\cdot)" display="inline"><semantics id="algorithm1.10.10.m2.1a"><mrow id="algorithm1.10.10.m2.1.2" xref="algorithm1.10.10.m2.1.2.cmml"><mi mathsize="80%" id="algorithm1.10.10.m2.1.2.2" xref="algorithm1.10.10.m2.1.2.2.cmml">proj</mi><mo lspace="0em" rspace="0em" id="algorithm1.10.10.m2.1.2.1" xref="algorithm1.10.10.m2.1.2.1.cmml">​</mo><mrow id="algorithm1.10.10.m2.1.2.3.2" xref="algorithm1.10.10.m2.1.2.cmml"><mo maxsize="80%" minsize="80%" id="algorithm1.10.10.m2.1.2.3.2.1" xref="algorithm1.10.10.m2.1.2.cmml">(</mo><mo lspace="0em" mathsize="80%" rspace="0em" id="algorithm1.10.10.m2.1.1" xref="algorithm1.10.10.m2.1.1.cmml">⋅</mo><mo maxsize="80%" minsize="80%" id="algorithm1.10.10.m2.1.2.3.2.2" xref="algorithm1.10.10.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm1.10.10.m2.1b"><apply id="algorithm1.10.10.m2.1.2.cmml" xref="algorithm1.10.10.m2.1.2"><times id="algorithm1.10.10.m2.1.2.1.cmml" xref="algorithm1.10.10.m2.1.2.1"></times><ci id="algorithm1.10.10.m2.1.2.2.cmml" xref="algorithm1.10.10.m2.1.2.2">proj</ci><ci id="algorithm1.10.10.m2.1.1.cmml" xref="algorithm1.10.10.m2.1.1">⋅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.10.10.m2.1c">\mathrm{proj}(\cdot)</annotation></semantics></math><span id="algorithm1.10.10.3" class="ltx_text" style="font-size:80%;">  by PCA decomp. or by training a VAE.</span>
</div>
<div id="algorithm1.14.14" class="ltx_listingline">
<span id="algorithm1.14.14.1" class="ltx_text" style="font-size:80%;"> With </span><math id="algorithm1.11.11.m1.1" class="ltx_Math" alttext="\mathcal{D}_{\mathrm{tr}}" display="inline"><semantics id="algorithm1.11.11.m1.1a"><msub id="algorithm1.11.11.m1.1.1" xref="algorithm1.11.11.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" mathsize="80%" id="algorithm1.11.11.m1.1.1.2" xref="algorithm1.11.11.m1.1.1.2.cmml">𝒟</mi><mi mathsize="80%" id="algorithm1.11.11.m1.1.1.3" xref="algorithm1.11.11.m1.1.1.3.cmml">tr</mi></msub><annotation-xml encoding="MathML-Content" id="algorithm1.11.11.m1.1b"><apply id="algorithm1.11.11.m1.1.1.cmml" xref="algorithm1.11.11.m1.1.1"><csymbol cd="ambiguous" id="algorithm1.11.11.m1.1.1.1.cmml" xref="algorithm1.11.11.m1.1.1">subscript</csymbol><ci id="algorithm1.11.11.m1.1.1.2.cmml" xref="algorithm1.11.11.m1.1.1.2">𝒟</ci><ci id="algorithm1.11.11.m1.1.1.3.cmml" xref="algorithm1.11.11.m1.1.1.3">tr</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.11.11.m1.1c">\mathcal{D}_{\mathrm{tr}}</annotation></semantics></math><span id="algorithm1.14.14.2" class="ltx_text" style="font-size:80%;">, train </span><math id="algorithm1.12.12.m2.1" class="ltx_Math" alttext="M" display="inline"><semantics id="algorithm1.12.12.m2.1a"><mi mathsize="80%" id="algorithm1.12.12.m2.1.1" xref="algorithm1.12.12.m2.1.1.cmml">M</mi><annotation-xml encoding="MathML-Content" id="algorithm1.12.12.m2.1b"><ci id="algorithm1.12.12.m2.1.1.cmml" xref="algorithm1.12.12.m2.1.1">𝑀</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.12.12.m2.1c">M</annotation></semantics></math><span id="algorithm1.14.14.3" class="ltx_text" style="font-size:80%;"> instances of </span><math id="algorithm1.13.13.m3.1" class="ltx_Math" alttext="f" display="inline"><semantics id="algorithm1.13.13.m3.1a"><mi mathsize="80%" id="algorithm1.13.13.m3.1.1" xref="algorithm1.13.13.m3.1.1.cmml">f</mi><annotation-xml encoding="MathML-Content" id="algorithm1.13.13.m3.1b"><ci id="algorithm1.13.13.m3.1.1.cmml" xref="algorithm1.13.13.m3.1.1">𝑓</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.13.13.m3.1c">f</annotation></semantics></math><span id="algorithm1.14.14.4" class="ltx_text" style="font-size:80%;"> in parallel (Eq. </span><a href="#S4.Ex1" title="4.2 On-manifold constraint ‣ 4 Proposed method ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref" style="font-size:80%;"><span class="ltx_text ltx_ref_tag">4.2</span></a><span id="algorithm1.14.14.5" class="ltx_text" style="font-size:80%;">): </span><math id="algorithm1.14.14.m4.1" class="ltx_Math" alttext="\{{\boldsymbol{\theta}}_{1}\,\kern-0.80005pt.\kern-0.80005pt.\kern-0.80005pt.\,{\boldsymbol{\theta}}_{M}\}\leftarrow\mathop{\textrm{argmin}}\mathcal{L}(\mathcal{D}_{\mathrm{tr}},{\boldsymbol{\theta}}_{1}\,\kern-0.80005pt.\kern-0.80005pt.\kern-0.80005pt.\,{\boldsymbol{\theta}}_{M})." display="inline"><semantics id="algorithm1.14.14.m4.1a"><mrow id="algorithm1.14.14.m4.1.1.1" xref="algorithm1.14.14.m4.1.1.1.1.cmml"><mrow id="algorithm1.14.14.m4.1.1.1.1" xref="algorithm1.14.14.m4.1.1.1.1.cmml"><mrow id="algorithm1.14.14.m4.1.1.1.1.1.1" xref="algorithm1.14.14.m4.1.1.1.1.1.2.cmml"><mo maxsize="80%" minsize="80%" id="algorithm1.14.14.m4.1.1.1.1.1.1.2" xref="algorithm1.14.14.m4.1.1.1.1.1.2.cmml">{</mo><mrow id="algorithm1.14.14.m4.1.1.1.1.1.1.1" xref="algorithm1.14.14.m4.1.1.1.1.1.1.1.cmml"><msub id="algorithm1.14.14.m4.1.1.1.1.1.1.1.2" xref="algorithm1.14.14.m4.1.1.1.1.1.1.1.2.cmml"><mi mathsize="80%" id="algorithm1.14.14.m4.1.1.1.1.1.1.1.2.2" xref="algorithm1.14.14.m4.1.1.1.1.1.1.1.2.2.cmml">𝜽</mi><mn mathsize="80%" id="algorithm1.14.14.m4.1.1.1.1.1.1.1.2.3" xref="algorithm1.14.14.m4.1.1.1.1.1.1.1.2.3.cmml">1</mn></msub><mo lspace="0em" rspace="0em" id="algorithm1.14.14.m4.1.1.1.1.1.1.1.1" xref="algorithm1.14.14.m4.1.1.1.1.1.1.1.1.cmml">​</mo><mi mathsize="80%" mathvariant="normal" id="algorithm1.14.14.m4.1.1.1.1.1.1.1.3" xref="algorithm1.14.14.m4.1.1.1.1.1.1.1.3.cmml">…</mi><mo lspace="0.170em" rspace="0em" id="algorithm1.14.14.m4.1.1.1.1.1.1.1.1a" xref="algorithm1.14.14.m4.1.1.1.1.1.1.1.1.cmml">​</mo><msub id="algorithm1.14.14.m4.1.1.1.1.1.1.1.4" xref="algorithm1.14.14.m4.1.1.1.1.1.1.1.4.cmml"><mi mathsize="80%" id="algorithm1.14.14.m4.1.1.1.1.1.1.1.4.2" xref="algorithm1.14.14.m4.1.1.1.1.1.1.1.4.2.cmml">𝜽</mi><mi mathsize="80%" id="algorithm1.14.14.m4.1.1.1.1.1.1.1.4.3" xref="algorithm1.14.14.m4.1.1.1.1.1.1.1.4.3.cmml">M</mi></msub></mrow><mo maxsize="80%" minsize="80%" id="algorithm1.14.14.m4.1.1.1.1.1.1.3" xref="algorithm1.14.14.m4.1.1.1.1.1.2.cmml">}</mo></mrow><mo mathsize="80%" stretchy="false" id="algorithm1.14.14.m4.1.1.1.1.4" xref="algorithm1.14.14.m4.1.1.1.1.4.cmml">←</mo><mrow id="algorithm1.14.14.m4.1.1.1.1.3" xref="algorithm1.14.14.m4.1.1.1.1.3.cmml"><mtext mathsize="80%" id="algorithm1.14.14.m4.1.1.1.1.3.3" xref="algorithm1.14.14.m4.1.1.1.1.3.3a.cmml">argmin</mtext><mrow id="algorithm1.14.14.m4.1.1.1.1.3.2" xref="algorithm1.14.14.m4.1.1.1.1.3.2.cmml"><mi class="ltx_font_mathcaligraphic" mathsize="80%" id="algorithm1.14.14.m4.1.1.1.1.3.2.4" xref="algorithm1.14.14.m4.1.1.1.1.3.2.4.cmml">ℒ</mi><mo lspace="0em" rspace="0em" id="algorithm1.14.14.m4.1.1.1.1.3.2.3" xref="algorithm1.14.14.m4.1.1.1.1.3.2.3.cmml">​</mo><mrow id="algorithm1.14.14.m4.1.1.1.1.3.2.2.2" xref="algorithm1.14.14.m4.1.1.1.1.3.2.2.3.cmml"><mo maxsize="80%" minsize="80%" id="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.3" xref="algorithm1.14.14.m4.1.1.1.1.3.2.2.3.cmml">(</mo><msub id="algorithm1.14.14.m4.1.1.1.1.2.1.1.1.1" xref="algorithm1.14.14.m4.1.1.1.1.2.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" mathsize="80%" id="algorithm1.14.14.m4.1.1.1.1.2.1.1.1.1.2" xref="algorithm1.14.14.m4.1.1.1.1.2.1.1.1.1.2.cmml">𝒟</mi><mi mathsize="80%" id="algorithm1.14.14.m4.1.1.1.1.2.1.1.1.1.3" xref="algorithm1.14.14.m4.1.1.1.1.2.1.1.1.1.3.cmml">tr</mi></msub><mo mathsize="80%" id="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.4" xref="algorithm1.14.14.m4.1.1.1.1.3.2.2.3.cmml">,</mo><mrow id="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2" xref="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.cmml"><msub id="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.2" xref="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.2.cmml"><mi mathsize="80%" id="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.2.2" xref="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.2.2.cmml">𝜽</mi><mn mathsize="80%" id="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.2.3" xref="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.2.3.cmml">1</mn></msub><mo lspace="0em" rspace="0em" id="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.1" xref="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.1.cmml">​</mo><mi mathsize="80%" mathvariant="normal" id="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.3" xref="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.3.cmml">…</mi><mo lspace="0.170em" rspace="0em" id="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.1a" xref="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.1.cmml">​</mo><msub id="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.4" xref="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.4.cmml"><mi mathsize="80%" id="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.4.2" xref="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.4.2.cmml">𝜽</mi><mi mathsize="80%" id="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.4.3" xref="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.4.3.cmml">M</mi></msub></mrow><mo maxsize="80%" minsize="80%" id="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.5" xref="algorithm1.14.14.m4.1.1.1.1.3.2.2.3.cmml">)</mo></mrow></mrow></mrow></mrow><mo lspace="0em" mathsize="80%" id="algorithm1.14.14.m4.1.1.1.2" xref="algorithm1.14.14.m4.1.1.1.1.cmml">.</mo></mrow><annotation-xml encoding="MathML-Content" id="algorithm1.14.14.m4.1b"><apply id="algorithm1.14.14.m4.1.1.1.1.cmml" xref="algorithm1.14.14.m4.1.1.1"><ci id="algorithm1.14.14.m4.1.1.1.1.4.cmml" xref="algorithm1.14.14.m4.1.1.1.1.4">←</ci><set id="algorithm1.14.14.m4.1.1.1.1.1.2.cmml" xref="algorithm1.14.14.m4.1.1.1.1.1.1"><apply id="algorithm1.14.14.m4.1.1.1.1.1.1.1.cmml" xref="algorithm1.14.14.m4.1.1.1.1.1.1.1"><times id="algorithm1.14.14.m4.1.1.1.1.1.1.1.1.cmml" xref="algorithm1.14.14.m4.1.1.1.1.1.1.1.1"></times><apply id="algorithm1.14.14.m4.1.1.1.1.1.1.1.2.cmml" xref="algorithm1.14.14.m4.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="algorithm1.14.14.m4.1.1.1.1.1.1.1.2.1.cmml" xref="algorithm1.14.14.m4.1.1.1.1.1.1.1.2">subscript</csymbol><ci id="algorithm1.14.14.m4.1.1.1.1.1.1.1.2.2.cmml" xref="algorithm1.14.14.m4.1.1.1.1.1.1.1.2.2">𝜽</ci><cn type="integer" id="algorithm1.14.14.m4.1.1.1.1.1.1.1.2.3.cmml" xref="algorithm1.14.14.m4.1.1.1.1.1.1.1.2.3">1</cn></apply><ci id="algorithm1.14.14.m4.1.1.1.1.1.1.1.3.cmml" xref="algorithm1.14.14.m4.1.1.1.1.1.1.1.3">…</ci><apply id="algorithm1.14.14.m4.1.1.1.1.1.1.1.4.cmml" xref="algorithm1.14.14.m4.1.1.1.1.1.1.1.4"><csymbol cd="ambiguous" id="algorithm1.14.14.m4.1.1.1.1.1.1.1.4.1.cmml" xref="algorithm1.14.14.m4.1.1.1.1.1.1.1.4">subscript</csymbol><ci id="algorithm1.14.14.m4.1.1.1.1.1.1.1.4.2.cmml" xref="algorithm1.14.14.m4.1.1.1.1.1.1.1.4.2">𝜽</ci><ci id="algorithm1.14.14.m4.1.1.1.1.1.1.1.4.3.cmml" xref="algorithm1.14.14.m4.1.1.1.1.1.1.1.4.3">𝑀</ci></apply></apply></set><apply id="algorithm1.14.14.m4.1.1.1.1.3.cmml" xref="algorithm1.14.14.m4.1.1.1.1.3"><ci id="algorithm1.14.14.m4.1.1.1.1.3.3a.cmml" xref="algorithm1.14.14.m4.1.1.1.1.3.3"><mtext mathsize="80%" id="algorithm1.14.14.m4.1.1.1.1.3.3.cmml" xref="algorithm1.14.14.m4.1.1.1.1.3.3">argmin</mtext></ci><apply id="algorithm1.14.14.m4.1.1.1.1.3.2.cmml" xref="algorithm1.14.14.m4.1.1.1.1.3.2"><times id="algorithm1.14.14.m4.1.1.1.1.3.2.3.cmml" xref="algorithm1.14.14.m4.1.1.1.1.3.2.3"></times><ci id="algorithm1.14.14.m4.1.1.1.1.3.2.4.cmml" xref="algorithm1.14.14.m4.1.1.1.1.3.2.4">ℒ</ci><interval closure="open" id="algorithm1.14.14.m4.1.1.1.1.3.2.2.3.cmml" xref="algorithm1.14.14.m4.1.1.1.1.3.2.2.2"><apply id="algorithm1.14.14.m4.1.1.1.1.2.1.1.1.1.cmml" xref="algorithm1.14.14.m4.1.1.1.1.2.1.1.1.1"><csymbol cd="ambiguous" id="algorithm1.14.14.m4.1.1.1.1.2.1.1.1.1.1.cmml" xref="algorithm1.14.14.m4.1.1.1.1.2.1.1.1.1">subscript</csymbol><ci id="algorithm1.14.14.m4.1.1.1.1.2.1.1.1.1.2.cmml" xref="algorithm1.14.14.m4.1.1.1.1.2.1.1.1.1.2">𝒟</ci><ci id="algorithm1.14.14.m4.1.1.1.1.2.1.1.1.1.3.cmml" xref="algorithm1.14.14.m4.1.1.1.1.2.1.1.1.1.3">tr</ci></apply><apply id="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.cmml" xref="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2"><times id="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.1.cmml" xref="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.1"></times><apply id="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.2.cmml" xref="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.2"><csymbol cd="ambiguous" id="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.2.1.cmml" xref="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.2">subscript</csymbol><ci id="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.2.2.cmml" xref="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.2.2">𝜽</ci><cn type="integer" id="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.2.3.cmml" xref="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.2.3">1</cn></apply><ci id="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.3.cmml" xref="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.3">…</ci><apply id="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.4.cmml" xref="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.4"><csymbol cd="ambiguous" id="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.4.1.cmml" xref="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.4">subscript</csymbol><ci id="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.4.2.cmml" xref="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.4.2">𝜽</ci><ci id="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.4.3.cmml" xref="algorithm1.14.14.m4.1.1.1.1.3.2.2.2.2.4.3">𝑀</ci></apply></apply></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.14.14.m4.1c">\{{\boldsymbol{\theta}}_{1}\,\kern-0.80005pt.\kern-0.80005pt.\kern-0.80005pt.\,{\boldsymbol{\theta}}_{M}\}\leftarrow\mathop{\textrm{argmin}}\mathcal{L}(\mathcal{D}_{\mathrm{tr}},{\boldsymbol{\theta}}_{1}\,\kern-0.80005pt.\kern-0.80005pt.\kern-0.80005pt.\,{\boldsymbol{\theta}}_{M}).</annotation></semantics></math>
</div>
<div id="algorithm1.15.15" class="ltx_listingline">
<span id="algorithm1.15.15.1" class="ltx_text" style="font-size:80%;">
Determine masks on input data (Eq. </span><a href="#S4.E5" title="In 4.3 Fine-tuning ‣ 4 Proposed method ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref" style="font-size:80%;"><span class="ltx_text ltx_ref_tag">5</span></a><span id="algorithm1.15.15.2" class="ltx_text" style="font-size:80%;">): </span><math id="algorithm1.15.15.m1.3" class="ltx_Math" alttext="\{{\boldsymbol{\mathrm{mask}}}^{m}_{i}\}_{i,m}" display="inline"><semantics id="algorithm1.15.15.m1.3a"><msub id="algorithm1.15.15.m1.3.3" xref="algorithm1.15.15.m1.3.3.cmml"><mrow id="algorithm1.15.15.m1.3.3.1.1" xref="algorithm1.15.15.m1.3.3.1.2.cmml"><mo maxsize="80%" minsize="80%" id="algorithm1.15.15.m1.3.3.1.1.2" xref="algorithm1.15.15.m1.3.3.1.2.cmml">{</mo><msubsup id="algorithm1.15.15.m1.3.3.1.1.1" xref="algorithm1.15.15.m1.3.3.1.1.1.cmml"><mi mathsize="80%" id="algorithm1.15.15.m1.3.3.1.1.1.2.2" xref="algorithm1.15.15.m1.3.3.1.1.1.2.2.cmml">𝐦𝐚𝐬𝐤</mi><mi mathsize="80%" id="algorithm1.15.15.m1.3.3.1.1.1.3" xref="algorithm1.15.15.m1.3.3.1.1.1.3.cmml">i</mi><mi mathsize="80%" id="algorithm1.15.15.m1.3.3.1.1.1.2.3" xref="algorithm1.15.15.m1.3.3.1.1.1.2.3.cmml">m</mi></msubsup><mo maxsize="80%" minsize="80%" id="algorithm1.15.15.m1.3.3.1.1.3" xref="algorithm1.15.15.m1.3.3.1.2.cmml">}</mo></mrow><mrow id="algorithm1.15.15.m1.2.2.2.4" xref="algorithm1.15.15.m1.2.2.2.3.cmml"><mi mathsize="80%" id="algorithm1.15.15.m1.1.1.1.1" xref="algorithm1.15.15.m1.1.1.1.1.cmml">i</mi><mo mathsize="80%" id="algorithm1.15.15.m1.2.2.2.4.1" xref="algorithm1.15.15.m1.2.2.2.3.cmml">,</mo><mi mathsize="80%" id="algorithm1.15.15.m1.2.2.2.2" xref="algorithm1.15.15.m1.2.2.2.2.cmml">m</mi></mrow></msub><annotation-xml encoding="MathML-Content" id="algorithm1.15.15.m1.3b"><apply id="algorithm1.15.15.m1.3.3.cmml" xref="algorithm1.15.15.m1.3.3"><csymbol cd="ambiguous" id="algorithm1.15.15.m1.3.3.2.cmml" xref="algorithm1.15.15.m1.3.3">subscript</csymbol><set id="algorithm1.15.15.m1.3.3.1.2.cmml" xref="algorithm1.15.15.m1.3.3.1.1"><apply id="algorithm1.15.15.m1.3.3.1.1.1.cmml" xref="algorithm1.15.15.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="algorithm1.15.15.m1.3.3.1.1.1.1.cmml" xref="algorithm1.15.15.m1.3.3.1.1.1">subscript</csymbol><apply id="algorithm1.15.15.m1.3.3.1.1.1.2.cmml" xref="algorithm1.15.15.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="algorithm1.15.15.m1.3.3.1.1.1.2.1.cmml" xref="algorithm1.15.15.m1.3.3.1.1.1">superscript</csymbol><ci id="algorithm1.15.15.m1.3.3.1.1.1.2.2.cmml" xref="algorithm1.15.15.m1.3.3.1.1.1.2.2">𝐦𝐚𝐬𝐤</ci><ci id="algorithm1.15.15.m1.3.3.1.1.1.2.3.cmml" xref="algorithm1.15.15.m1.3.3.1.1.1.2.3">𝑚</ci></apply><ci id="algorithm1.15.15.m1.3.3.1.1.1.3.cmml" xref="algorithm1.15.15.m1.3.3.1.1.1.3">𝑖</ci></apply></set><list id="algorithm1.15.15.m1.2.2.2.3.cmml" xref="algorithm1.15.15.m1.2.2.2.4"><ci id="algorithm1.15.15.m1.1.1.1.1.cmml" xref="algorithm1.15.15.m1.1.1.1.1">𝑖</ci><ci id="algorithm1.15.15.m1.2.2.2.2.cmml" xref="algorithm1.15.15.m1.2.2.2.2">𝑚</ci></list></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.15.15.m1.3c">\{{\boldsymbol{\mathrm{mask}}}^{m}_{i}\}_{i,m}</annotation></semantics></math>
</div>
<div id="algorithm1.16.16" class="ltx_listingline">
<span id="algorithm1.16.16.2" class="ltx_text" style="font-size:80%;">
</span><span id="algorithm1.16.16.3" class="ltx_text ltx_font_bold" style="font-size:80%;">foreach</span><span id="algorithm1.16.16.4" class="ltx_text" style="font-size:80%;"> </span><em id="algorithm1.16.16.1" class="ltx_emph ltx_font_italic" style="font-size:80%;"><math id="algorithm1.16.16.1.m1.1" class="ltx_Math" alttext="m" display="inline"><semantics id="algorithm1.16.16.1.m1.1a"><mi id="algorithm1.16.16.1.m1.1.1" xref="algorithm1.16.16.1.m1.1.1.cmml">m</mi><annotation-xml encoding="MathML-Content" id="algorithm1.16.16.1.m1.1b"><ci id="algorithm1.16.16.1.m1.1.1.cmml" xref="algorithm1.16.16.1.m1.1.1">𝑚</ci></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.16.16.1.m1.1c">m</annotation></semantics></math></em><span id="algorithm1.16.16.5" class="ltx_text" style="font-size:80%;"> </span><span id="algorithm1.16.16.6" class="ltx_text ltx_font_bold" style="font-size:80%;">do</span><span id="algorithm1.16.16.7" class="ltx_text" style="font-size:80%;">   </span><span id="algorithm1.16.16.8" class="ltx_text" style="font-size:80%;color:#8C8C8C;">// Optional fine-tuning on masked data</span>
</div>
<div id="algorithm1.17.17" class="ltx_listingline">
<span id="algorithm1.17.17.1" class="ltx_text" style="font-size:80%;">  </span><span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span><span id="algorithm1.17.17.2" class="ltx_text" style="font-size:80%;">   </span><span id="algorithm1.17.17.3" class="ltx_text" style="font-size:80%;">
</span><math id="algorithm1.17.17.m1.1" class="ltx_Math" alttext="\mathcal{D}_{\mathrm{tr}}^{m}\leftarrow\{({\boldsymbol{x}}_{i}\odot{\boldsymbol{\mathrm{mask}}}^{m}_{i},y_{i})\}_{i}" display="inline"><semantics id="algorithm1.17.17.m1.1a"><mrow id="algorithm1.17.17.m1.1.1" xref="algorithm1.17.17.m1.1.1.cmml"><msubsup id="algorithm1.17.17.m1.1.1.3" xref="algorithm1.17.17.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" mathsize="80%" id="algorithm1.17.17.m1.1.1.3.2.2" xref="algorithm1.17.17.m1.1.1.3.2.2.cmml">𝒟</mi><mi mathsize="80%" id="algorithm1.17.17.m1.1.1.3.2.3" xref="algorithm1.17.17.m1.1.1.3.2.3.cmml">tr</mi><mi mathsize="80%" id="algorithm1.17.17.m1.1.1.3.3" xref="algorithm1.17.17.m1.1.1.3.3.cmml">m</mi></msubsup><mo mathsize="80%" stretchy="false" id="algorithm1.17.17.m1.1.1.2" xref="algorithm1.17.17.m1.1.1.2.cmml">←</mo><msub id="algorithm1.17.17.m1.1.1.1" xref="algorithm1.17.17.m1.1.1.1.cmml"><mrow id="algorithm1.17.17.m1.1.1.1.1.1" xref="algorithm1.17.17.m1.1.1.1.1.2.cmml"><mo maxsize="80%" minsize="80%" id="algorithm1.17.17.m1.1.1.1.1.1.2" xref="algorithm1.17.17.m1.1.1.1.1.2.cmml">{</mo><mrow id="algorithm1.17.17.m1.1.1.1.1.1.1.2" xref="algorithm1.17.17.m1.1.1.1.1.1.1.3.cmml"><mo maxsize="80%" minsize="80%" id="algorithm1.17.17.m1.1.1.1.1.1.1.2.3" xref="algorithm1.17.17.m1.1.1.1.1.1.1.3.cmml">(</mo><mrow id="algorithm1.17.17.m1.1.1.1.1.1.1.1.1" xref="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.cmml"><msub id="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.2" xref="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.2.cmml"><mi mathsize="80%" id="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.2.2" xref="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.2.2.cmml">𝒙</mi><mi mathsize="80%" id="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.2.3" xref="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.2.3.cmml">i</mi></msub><mo lspace="0.222em" mathsize="80%" rspace="0.222em" id="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.1" xref="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.1.cmml">⊙</mo><msubsup id="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.3" xref="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.3.cmml"><mi mathsize="80%" id="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.3.2.2" xref="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.3.2.2.cmml">𝐦𝐚𝐬𝐤</mi><mi mathsize="80%" id="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.3.3" xref="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.3.3.cmml">i</mi><mi mathsize="80%" id="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.3.2.3" xref="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.3.2.3.cmml">m</mi></msubsup></mrow><mo mathsize="80%" id="algorithm1.17.17.m1.1.1.1.1.1.1.2.4" xref="algorithm1.17.17.m1.1.1.1.1.1.1.3.cmml">,</mo><msub id="algorithm1.17.17.m1.1.1.1.1.1.1.2.2" xref="algorithm1.17.17.m1.1.1.1.1.1.1.2.2.cmml"><mi mathsize="80%" id="algorithm1.17.17.m1.1.1.1.1.1.1.2.2.2" xref="algorithm1.17.17.m1.1.1.1.1.1.1.2.2.2.cmml">y</mi><mi mathsize="80%" id="algorithm1.17.17.m1.1.1.1.1.1.1.2.2.3" xref="algorithm1.17.17.m1.1.1.1.1.1.1.2.2.3.cmml">i</mi></msub><mo maxsize="80%" minsize="80%" id="algorithm1.17.17.m1.1.1.1.1.1.1.2.5" xref="algorithm1.17.17.m1.1.1.1.1.1.1.3.cmml">)</mo></mrow><mo maxsize="80%" minsize="80%" id="algorithm1.17.17.m1.1.1.1.1.1.3" xref="algorithm1.17.17.m1.1.1.1.1.2.cmml">}</mo></mrow><mi mathsize="80%" id="algorithm1.17.17.m1.1.1.1.3" xref="algorithm1.17.17.m1.1.1.1.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="algorithm1.17.17.m1.1b"><apply id="algorithm1.17.17.m1.1.1.cmml" xref="algorithm1.17.17.m1.1.1"><ci id="algorithm1.17.17.m1.1.1.2.cmml" xref="algorithm1.17.17.m1.1.1.2">←</ci><apply id="algorithm1.17.17.m1.1.1.3.cmml" xref="algorithm1.17.17.m1.1.1.3"><csymbol cd="ambiguous" id="algorithm1.17.17.m1.1.1.3.1.cmml" xref="algorithm1.17.17.m1.1.1.3">superscript</csymbol><apply id="algorithm1.17.17.m1.1.1.3.2.cmml" xref="algorithm1.17.17.m1.1.1.3"><csymbol cd="ambiguous" id="algorithm1.17.17.m1.1.1.3.2.1.cmml" xref="algorithm1.17.17.m1.1.1.3">subscript</csymbol><ci id="algorithm1.17.17.m1.1.1.3.2.2.cmml" xref="algorithm1.17.17.m1.1.1.3.2.2">𝒟</ci><ci id="algorithm1.17.17.m1.1.1.3.2.3.cmml" xref="algorithm1.17.17.m1.1.1.3.2.3">tr</ci></apply><ci id="algorithm1.17.17.m1.1.1.3.3.cmml" xref="algorithm1.17.17.m1.1.1.3.3">𝑚</ci></apply><apply id="algorithm1.17.17.m1.1.1.1.cmml" xref="algorithm1.17.17.m1.1.1.1"><csymbol cd="ambiguous" id="algorithm1.17.17.m1.1.1.1.2.cmml" xref="algorithm1.17.17.m1.1.1.1">subscript</csymbol><set id="algorithm1.17.17.m1.1.1.1.1.2.cmml" xref="algorithm1.17.17.m1.1.1.1.1.1"><interval closure="open" id="algorithm1.17.17.m1.1.1.1.1.1.1.3.cmml" xref="algorithm1.17.17.m1.1.1.1.1.1.1.2"><apply id="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.cmml" xref="algorithm1.17.17.m1.1.1.1.1.1.1.1.1"><csymbol cd="latexml" id="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.1.cmml" xref="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.1">direct-product</csymbol><apply id="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.2.cmml" xref="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.2.1.cmml" xref="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.2">subscript</csymbol><ci id="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.2.2.cmml" xref="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.2.2">𝒙</ci><ci id="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.2.3.cmml" xref="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.2.3">𝑖</ci></apply><apply id="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.3.cmml" xref="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.3.1.cmml" xref="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.3">subscript</csymbol><apply id="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.3.2.cmml" xref="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.3.2.1.cmml" xref="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.3">superscript</csymbol><ci id="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.3.2.2.cmml" xref="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.3.2.2">𝐦𝐚𝐬𝐤</ci><ci id="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.3.2.3.cmml" xref="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.3.2.3">𝑚</ci></apply><ci id="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.3.3.cmml" xref="algorithm1.17.17.m1.1.1.1.1.1.1.1.1.3.3">𝑖</ci></apply></apply><apply id="algorithm1.17.17.m1.1.1.1.1.1.1.2.2.cmml" xref="algorithm1.17.17.m1.1.1.1.1.1.1.2.2"><csymbol cd="ambiguous" id="algorithm1.17.17.m1.1.1.1.1.1.1.2.2.1.cmml" xref="algorithm1.17.17.m1.1.1.1.1.1.1.2.2">subscript</csymbol><ci id="algorithm1.17.17.m1.1.1.1.1.1.1.2.2.2.cmml" xref="algorithm1.17.17.m1.1.1.1.1.1.1.2.2.2">𝑦</ci><ci id="algorithm1.17.17.m1.1.1.1.1.1.1.2.2.3.cmml" xref="algorithm1.17.17.m1.1.1.1.1.1.1.2.2.3">𝑖</ci></apply></interval></set><ci id="algorithm1.17.17.m1.1.1.1.3.cmml" xref="algorithm1.17.17.m1.1.1.1.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.17.17.m1.1c">\mathcal{D}_{\mathrm{tr}}^{m}\leftarrow\{({\boldsymbol{x}}_{i}\odot{\boldsymbol{\mathrm{mask}}}^{m}_{i},y_{i})\}_{i}</annotation></semantics></math><span id="algorithm1.17.17.4" class="ltx_text" style="font-size:80%;">  </span><span id="algorithm1.17.17.5" class="ltx_text" style="font-size:80%;color:#8C8C8C;">// Prepare masked data</span>
</div>
<div id="algorithm1.18.18" class="ltx_listingline">
<span id="algorithm1.18.18.1" class="ltx_text" style="font-size:80%;">  </span><span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span><span id="algorithm1.18.18.2" class="ltx_text" style="font-size:80%;">   </span><span id="algorithm1.18.18.3" class="ltx_text" style="font-size:80%;">
</span><math id="algorithm1.18.18.m1.2" class="ltx_Math" alttext="{\lambda_{\textrm{indep}}}\leftarrow 0,\leavevmode\nobreak\ {\lambda_{\textrm{manifold}}}\leftarrow 0" display="inline"><semantics id="algorithm1.18.18.m1.2a"><mrow id="algorithm1.18.18.m1.2.2.2" xref="algorithm1.18.18.m1.2.2.3.cmml"><mrow id="algorithm1.18.18.m1.1.1.1.1" xref="algorithm1.18.18.m1.1.1.1.1.cmml"><msub id="algorithm1.18.18.m1.1.1.1.1.2" xref="algorithm1.18.18.m1.1.1.1.1.2.cmml"><mi mathsize="80%" id="algorithm1.18.18.m1.1.1.1.1.2.2" xref="algorithm1.18.18.m1.1.1.1.1.2.2.cmml">λ</mi><mtext mathsize="80%" id="algorithm1.18.18.m1.1.1.1.1.2.3" xref="algorithm1.18.18.m1.1.1.1.1.2.3a.cmml">indep</mtext></msub><mo mathsize="80%" stretchy="false" id="algorithm1.18.18.m1.1.1.1.1.1" xref="algorithm1.18.18.m1.1.1.1.1.1.cmml">←</mo><mn mathsize="80%" id="algorithm1.18.18.m1.1.1.1.1.3" xref="algorithm1.18.18.m1.1.1.1.1.3.cmml">0</mn></mrow><mo mathsize="80%" rspace="0.567em" id="algorithm1.18.18.m1.2.2.2.3" xref="algorithm1.18.18.m1.2.2.3a.cmml">,</mo><mrow id="algorithm1.18.18.m1.2.2.2.2" xref="algorithm1.18.18.m1.2.2.2.2.cmml"><msub id="algorithm1.18.18.m1.2.2.2.2.2" xref="algorithm1.18.18.m1.2.2.2.2.2.cmml"><mi mathsize="80%" id="algorithm1.18.18.m1.2.2.2.2.2.2" xref="algorithm1.18.18.m1.2.2.2.2.2.2.cmml">λ</mi><mtext mathsize="80%" id="algorithm1.18.18.m1.2.2.2.2.2.3" xref="algorithm1.18.18.m1.2.2.2.2.2.3a.cmml">manifold</mtext></msub><mo mathsize="80%" stretchy="false" id="algorithm1.18.18.m1.2.2.2.2.1" xref="algorithm1.18.18.m1.2.2.2.2.1.cmml">←</mo><mn mathsize="80%" id="algorithm1.18.18.m1.2.2.2.2.3" xref="algorithm1.18.18.m1.2.2.2.2.3.cmml">0</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm1.18.18.m1.2b"><apply id="algorithm1.18.18.m1.2.2.3.cmml" xref="algorithm1.18.18.m1.2.2.2"><csymbol cd="ambiguous" id="algorithm1.18.18.m1.2.2.3a.cmml" xref="algorithm1.18.18.m1.2.2.2.3">formulae-sequence</csymbol><apply id="algorithm1.18.18.m1.1.1.1.1.cmml" xref="algorithm1.18.18.m1.1.1.1.1"><ci id="algorithm1.18.18.m1.1.1.1.1.1.cmml" xref="algorithm1.18.18.m1.1.1.1.1.1">←</ci><apply id="algorithm1.18.18.m1.1.1.1.1.2.cmml" xref="algorithm1.18.18.m1.1.1.1.1.2"><csymbol cd="ambiguous" id="algorithm1.18.18.m1.1.1.1.1.2.1.cmml" xref="algorithm1.18.18.m1.1.1.1.1.2">subscript</csymbol><ci id="algorithm1.18.18.m1.1.1.1.1.2.2.cmml" xref="algorithm1.18.18.m1.1.1.1.1.2.2">𝜆</ci><ci id="algorithm1.18.18.m1.1.1.1.1.2.3a.cmml" xref="algorithm1.18.18.m1.1.1.1.1.2.3"><mtext mathsize="56%" id="algorithm1.18.18.m1.1.1.1.1.2.3.cmml" xref="algorithm1.18.18.m1.1.1.1.1.2.3">indep</mtext></ci></apply><cn type="integer" id="algorithm1.18.18.m1.1.1.1.1.3.cmml" xref="algorithm1.18.18.m1.1.1.1.1.3">0</cn></apply><apply id="algorithm1.18.18.m1.2.2.2.2.cmml" xref="algorithm1.18.18.m1.2.2.2.2"><ci id="algorithm1.18.18.m1.2.2.2.2.1.cmml" xref="algorithm1.18.18.m1.2.2.2.2.1">←</ci><apply id="algorithm1.18.18.m1.2.2.2.2.2.cmml" xref="algorithm1.18.18.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="algorithm1.18.18.m1.2.2.2.2.2.1.cmml" xref="algorithm1.18.18.m1.2.2.2.2.2">subscript</csymbol><ci id="algorithm1.18.18.m1.2.2.2.2.2.2.cmml" xref="algorithm1.18.18.m1.2.2.2.2.2.2">𝜆</ci><ci id="algorithm1.18.18.m1.2.2.2.2.2.3a.cmml" xref="algorithm1.18.18.m1.2.2.2.2.2.3"><mtext mathsize="56%" id="algorithm1.18.18.m1.2.2.2.2.2.3.cmml" xref="algorithm1.18.18.m1.2.2.2.2.2.3">manifold</mtext></ci></apply><cn type="integer" id="algorithm1.18.18.m1.2.2.2.2.3.cmml" xref="algorithm1.18.18.m1.2.2.2.2.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.18.18.m1.2c">{\lambda_{\textrm{indep}}}\leftarrow 0,\leavevmode\nobreak\ {\lambda_{\textrm{manifold}}}\leftarrow 0</annotation></semantics></math><span id="algorithm1.18.18.4" class="ltx_text" style="font-size:80%;">  </span><span id="algorithm1.18.18.5" class="ltx_text" style="font-size:80%;color:#8C8C8C;">// Use only predictive loss</span>
</div>
<div id="algorithm1.19.19" class="ltx_listingline">
<span id="algorithm1.19.19.1" class="ltx_text" style="font-size:80%;">  </span><span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span><span id="algorithm1.19.19.2" class="ltx_text" style="font-size:80%;">   </span><span id="algorithm1.19.19.3" class="ltx_text" style="font-size:80%;">
</span><math id="algorithm1.19.19.m1.2" class="ltx_Math" alttext="{\boldsymbol{\theta}}_{m}\leftarrow\mathop{\textrm{argmin}}\mathcal{L}(\mathcal{D}_{\mathrm{tr}}^{m},{\boldsymbol{\theta}}_{m})" display="inline"><semantics id="algorithm1.19.19.m1.2a"><mrow id="algorithm1.19.19.m1.2.2" xref="algorithm1.19.19.m1.2.2.cmml"><msub id="algorithm1.19.19.m1.2.2.4" xref="algorithm1.19.19.m1.2.2.4.cmml"><mi mathsize="80%" id="algorithm1.19.19.m1.2.2.4.2" xref="algorithm1.19.19.m1.2.2.4.2.cmml">𝜽</mi><mi mathsize="80%" id="algorithm1.19.19.m1.2.2.4.3" xref="algorithm1.19.19.m1.2.2.4.3.cmml">m</mi></msub><mo mathsize="80%" stretchy="false" id="algorithm1.19.19.m1.2.2.3" xref="algorithm1.19.19.m1.2.2.3.cmml">←</mo><mrow id="algorithm1.19.19.m1.2.2.2" xref="algorithm1.19.19.m1.2.2.2.cmml"><mtext mathsize="80%" id="algorithm1.19.19.m1.2.2.2.3" xref="algorithm1.19.19.m1.2.2.2.3a.cmml">argmin</mtext><mrow id="algorithm1.19.19.m1.2.2.2.2" xref="algorithm1.19.19.m1.2.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" mathsize="80%" id="algorithm1.19.19.m1.2.2.2.2.4" xref="algorithm1.19.19.m1.2.2.2.2.4.cmml">ℒ</mi><mo lspace="0em" rspace="0em" id="algorithm1.19.19.m1.2.2.2.2.3" xref="algorithm1.19.19.m1.2.2.2.2.3.cmml">​</mo><mrow id="algorithm1.19.19.m1.2.2.2.2.2.2" xref="algorithm1.19.19.m1.2.2.2.2.2.3.cmml"><mo maxsize="80%" minsize="80%" id="algorithm1.19.19.m1.2.2.2.2.2.2.3" xref="algorithm1.19.19.m1.2.2.2.2.2.3.cmml">(</mo><msubsup id="algorithm1.19.19.m1.1.1.1.1.1.1.1" xref="algorithm1.19.19.m1.1.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" mathsize="80%" id="algorithm1.19.19.m1.1.1.1.1.1.1.1.2.2" xref="algorithm1.19.19.m1.1.1.1.1.1.1.1.2.2.cmml">𝒟</mi><mi mathsize="80%" id="algorithm1.19.19.m1.1.1.1.1.1.1.1.2.3" xref="algorithm1.19.19.m1.1.1.1.1.1.1.1.2.3.cmml">tr</mi><mi mathsize="80%" id="algorithm1.19.19.m1.1.1.1.1.1.1.1.3" xref="algorithm1.19.19.m1.1.1.1.1.1.1.1.3.cmml">m</mi></msubsup><mo mathsize="80%" id="algorithm1.19.19.m1.2.2.2.2.2.2.4" xref="algorithm1.19.19.m1.2.2.2.2.2.3.cmml">,</mo><msub id="algorithm1.19.19.m1.2.2.2.2.2.2.2" xref="algorithm1.19.19.m1.2.2.2.2.2.2.2.cmml"><mi mathsize="80%" id="algorithm1.19.19.m1.2.2.2.2.2.2.2.2" xref="algorithm1.19.19.m1.2.2.2.2.2.2.2.2.cmml">𝜽</mi><mi mathsize="80%" id="algorithm1.19.19.m1.2.2.2.2.2.2.2.3" xref="algorithm1.19.19.m1.2.2.2.2.2.2.2.3.cmml">m</mi></msub><mo maxsize="80%" minsize="80%" id="algorithm1.19.19.m1.2.2.2.2.2.2.5" xref="algorithm1.19.19.m1.2.2.2.2.2.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm1.19.19.m1.2b"><apply id="algorithm1.19.19.m1.2.2.cmml" xref="algorithm1.19.19.m1.2.2"><ci id="algorithm1.19.19.m1.2.2.3.cmml" xref="algorithm1.19.19.m1.2.2.3">←</ci><apply id="algorithm1.19.19.m1.2.2.4.cmml" xref="algorithm1.19.19.m1.2.2.4"><csymbol cd="ambiguous" id="algorithm1.19.19.m1.2.2.4.1.cmml" xref="algorithm1.19.19.m1.2.2.4">subscript</csymbol><ci id="algorithm1.19.19.m1.2.2.4.2.cmml" xref="algorithm1.19.19.m1.2.2.4.2">𝜽</ci><ci id="algorithm1.19.19.m1.2.2.4.3.cmml" xref="algorithm1.19.19.m1.2.2.4.3">𝑚</ci></apply><apply id="algorithm1.19.19.m1.2.2.2.cmml" xref="algorithm1.19.19.m1.2.2.2"><ci id="algorithm1.19.19.m1.2.2.2.3a.cmml" xref="algorithm1.19.19.m1.2.2.2.3"><mtext mathsize="80%" id="algorithm1.19.19.m1.2.2.2.3.cmml" xref="algorithm1.19.19.m1.2.2.2.3">argmin</mtext></ci><apply id="algorithm1.19.19.m1.2.2.2.2.cmml" xref="algorithm1.19.19.m1.2.2.2.2"><times id="algorithm1.19.19.m1.2.2.2.2.3.cmml" xref="algorithm1.19.19.m1.2.2.2.2.3"></times><ci id="algorithm1.19.19.m1.2.2.2.2.4.cmml" xref="algorithm1.19.19.m1.2.2.2.2.4">ℒ</ci><interval closure="open" id="algorithm1.19.19.m1.2.2.2.2.2.3.cmml" xref="algorithm1.19.19.m1.2.2.2.2.2.2"><apply id="algorithm1.19.19.m1.1.1.1.1.1.1.1.cmml" xref="algorithm1.19.19.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="algorithm1.19.19.m1.1.1.1.1.1.1.1.1.cmml" xref="algorithm1.19.19.m1.1.1.1.1.1.1.1">superscript</csymbol><apply id="algorithm1.19.19.m1.1.1.1.1.1.1.1.2.cmml" xref="algorithm1.19.19.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="algorithm1.19.19.m1.1.1.1.1.1.1.1.2.1.cmml" xref="algorithm1.19.19.m1.1.1.1.1.1.1.1">subscript</csymbol><ci id="algorithm1.19.19.m1.1.1.1.1.1.1.1.2.2.cmml" xref="algorithm1.19.19.m1.1.1.1.1.1.1.1.2.2">𝒟</ci><ci id="algorithm1.19.19.m1.1.1.1.1.1.1.1.2.3.cmml" xref="algorithm1.19.19.m1.1.1.1.1.1.1.1.2.3">tr</ci></apply><ci id="algorithm1.19.19.m1.1.1.1.1.1.1.1.3.cmml" xref="algorithm1.19.19.m1.1.1.1.1.1.1.1.3">𝑚</ci></apply><apply id="algorithm1.19.19.m1.2.2.2.2.2.2.2.cmml" xref="algorithm1.19.19.m1.2.2.2.2.2.2.2"><csymbol cd="ambiguous" id="algorithm1.19.19.m1.2.2.2.2.2.2.2.1.cmml" xref="algorithm1.19.19.m1.2.2.2.2.2.2.2">subscript</csymbol><ci id="algorithm1.19.19.m1.2.2.2.2.2.2.2.2.cmml" xref="algorithm1.19.19.m1.2.2.2.2.2.2.2.2">𝜽</ci><ci id="algorithm1.19.19.m1.2.2.2.2.2.2.2.3.cmml" xref="algorithm1.19.19.m1.2.2.2.2.2.2.2.3">𝑚</ci></apply></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm1.19.19.m1.2c">{\boldsymbol{\theta}}_{m}\leftarrow\mathop{\textrm{argmin}}\mathcal{L}(\mathcal{D}_{\mathrm{tr}}^{m},{\boldsymbol{\theta}}_{m})</annotation></semantics></math><span id="algorithm1.19.19.4" class="ltx_text" style="font-size:80%;">  </span><span id="algorithm1.19.19.5" class="ltx_text" style="font-size:80%;color:#8C8C8C;">// Fine-tune</span>
</div>
<div id="algorithm1.19.21" class="ltx_listingline">
<span id="algorithm1.19.21.1" class="ltx_text" style="font-size:80%;">  </span><span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span><span id="algorithm1.19.21.2" class="ltx_text" style="font-size:80%;">   </span><span id="algorithm1.19.21.3" class="ltx_text" style="font-size:80%;">
</span>
</div>
<div id="algorithm1.19.22" class="ltx_listingline">
</div>
</div>
<figcaption class="ltx_caption" style="font-size:80%;"><span class="ltx_tag ltx_tag_float"><span id="algorithm1.23.1.1" class="ltx_text ltx_font_bold">Algorithm 1</span> </span>Training and fine-tuning models.</figcaption>
</figure>
</section>
<section id="S4.SS4" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">4.4 </span>Distilling multiple models into one</h3>

<div id="S4.SS4.p1" class="ltx_para">
<p id="S4.SS4.p1.1" class="ltx_p">Finally, after training/fine-tuning a set of models, we propose to combine the best of them into a global one that uses all of the most relevant features.<span id="footnote6" class="ltx_note ltx_role_footnote"><sup class="ltx_note_mark">6</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">6</sup><span class="ltx_tag ltx_tag_note">6</span>Our combination of models is not equivalent to an ensemble: we train a global model to use/ignore specific, chosen features. In comparison, a traditional ensemble combines models in prediction space, relying on their uncorrelated errors to lower the prediction variance. See also Appendix <a href="#A1" title="Appendix A Analogy with disentanglement in representation learning ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">A</span></a> for an the analogy of our method with disentanglement in predictor space.</span></span></span>
We train this global model from scratch, without regularizers, on masked data as described above, using masks from <em id="S4.SS4.p1.1.1" class="ltx_emph ltx_font_italic">multiple</em> selected models combined with a logical OR.
In our experiments, we combine the two models with the highest accuracies on an OOD validation set.
We repeat this pairwise combination as long the accuracy of the global model increases, usually for 2–3 iterations (as formalized in Algorithm <a href="#algorithm2" title="In Appendix C Distilling multiple models into one ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">2</span></a> in the Appendix).</p>
</div>
</section>
</section>
<section id="S5" class="ltx_section">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">5 </span>Practical considerations</h2>

<div id="S5.p1" class="ltx_para ltx_noindent">
<p id="S5.p1.4" class="ltx_p"><span id="S5.p1.4.1" class="ltx_text ltx_font_bold">Number of models to train.</span> 

We initially estimate the dimensionality of the manifold (<math id="S5.p1.1.m1.1" class="ltx_Math" alttext="{d_{\textrm{manifold}}}" display="inline"><semantics id="S5.p1.1.m1.1a"><msub id="S5.p1.1.m1.1.1" xref="S5.p1.1.m1.1.1.cmml"><mi id="S5.p1.1.m1.1.1.2" xref="S5.p1.1.m1.1.1.2.cmml">d</mi><mtext id="S5.p1.1.m1.1.1.3" xref="S5.p1.1.m1.1.1.3a.cmml">manifold</mtext></msub><annotation-xml encoding="MathML-Content" id="S5.p1.1.m1.1b"><apply id="S5.p1.1.m1.1.1.cmml" xref="S5.p1.1.m1.1.1"><csymbol cd="ambiguous" id="S5.p1.1.m1.1.1.1.cmml" xref="S5.p1.1.m1.1.1">subscript</csymbol><ci id="S5.p1.1.m1.1.1.2.cmml" xref="S5.p1.1.m1.1.1.2">𝑑</ci><ci id="S5.p1.1.m1.1.1.3a.cmml" xref="S5.p1.1.m1.1.1.3"><mtext mathsize="70%" id="S5.p1.1.m1.1.1.3.cmml" xref="S5.p1.1.m1.1.1.3">manifold</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.1.m1.1c">{d_{\textrm{manifold}}}</annotation></semantics></math>) from <math id="S5.p1.2.m2.1" class="ltx_Math" alttext="\mathcal{D}_{\mathrm{OOD}}" display="inline"><semantics id="S5.p1.2.m2.1a"><msub id="S5.p1.2.m2.1.1" xref="S5.p1.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.p1.2.m2.1.1.2" xref="S5.p1.2.m2.1.1.2.cmml">𝒟</mi><mi id="S5.p1.2.m2.1.1.3" xref="S5.p1.2.m2.1.1.3.cmml">OOD</mi></msub><annotation-xml encoding="MathML-Content" id="S5.p1.2.m2.1b"><apply id="S5.p1.2.m2.1.1.cmml" xref="S5.p1.2.m2.1.1"><csymbol cd="ambiguous" id="S5.p1.2.m2.1.1.1.cmml" xref="S5.p1.2.m2.1.1">subscript</csymbol><ci id="S5.p1.2.m2.1.1.2.cmml" xref="S5.p1.2.m2.1.1.2">𝒟</ci><ci id="S5.p1.2.m2.1.1.3.cmml" xref="S5.p1.2.m2.1.1.3">OOD</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.2.m2.1c">\mathcal{D}_{\mathrm{OOD}}</annotation></semantics></math>.
This is the upper bound on the number of predictors aligned with the manifold that can be mutually independent.
The dimensionality is estimated with a simple method <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib57" title="" class="ltx_ref">57</a>]</cite> based on distances between nearest neighbours.
Only the worst case (highest degree of underspecification per Definition <a href="#Thmdefinition1" title="Definition 1. ‣ 3 Formalizing underspecification ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">1</span></a>) should allow training a number of models <math id="S5.p1.3.m3.1" class="ltx_Math" alttext="M={d_{\textrm{manifold}}}" display="inline"><semantics id="S5.p1.3.m3.1a"><mrow id="S5.p1.3.m3.1.1" xref="S5.p1.3.m3.1.1.cmml"><mi id="S5.p1.3.m3.1.1.2" xref="S5.p1.3.m3.1.1.2.cmml">M</mi><mo id="S5.p1.3.m3.1.1.1" xref="S5.p1.3.m3.1.1.1.cmml">=</mo><msub id="S5.p1.3.m3.1.1.3" xref="S5.p1.3.m3.1.1.3.cmml"><mi id="S5.p1.3.m3.1.1.3.2" xref="S5.p1.3.m3.1.1.3.2.cmml">d</mi><mtext id="S5.p1.3.m3.1.1.3.3" xref="S5.p1.3.m3.1.1.3.3a.cmml">manifold</mtext></msub></mrow><annotation-xml encoding="MathML-Content" id="S5.p1.3.m3.1b"><apply id="S5.p1.3.m3.1.1.cmml" xref="S5.p1.3.m3.1.1"><eq id="S5.p1.3.m3.1.1.1.cmml" xref="S5.p1.3.m3.1.1.1"></eq><ci id="S5.p1.3.m3.1.1.2.cmml" xref="S5.p1.3.m3.1.1.2">𝑀</ci><apply id="S5.p1.3.m3.1.1.3.cmml" xref="S5.p1.3.m3.1.1.3"><csymbol cd="ambiguous" id="S5.p1.3.m3.1.1.3.1.cmml" xref="S5.p1.3.m3.1.1.3">subscript</csymbol><ci id="S5.p1.3.m3.1.1.3.2.cmml" xref="S5.p1.3.m3.1.1.3.2">𝑑</ci><ci id="S5.p1.3.m3.1.1.3.3a.cmml" xref="S5.p1.3.m3.1.1.3.3"><mtext mathsize="70%" id="S5.p1.3.m3.1.1.3.3.cmml" xref="S5.p1.3.m3.1.1.3.3">manifold</mtext></ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.3.m3.1c">M={d_{\textrm{manifold}}}</annotation></semantics></math> while satisfying the independence and on-manifold constraints.
In our experiments, underspecification is usually less severe.
We observed that the value of the predictive loss (<math id="S5.p1.4.m4.1" class="ltx_Math" alttext="\mathcal{L}_{\textrm{pred}}" display="inline"><semantics id="S5.p1.4.m4.1a"><msub id="S5.p1.4.m4.1.1" xref="S5.p1.4.m4.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="S5.p1.4.m4.1.1.2" xref="S5.p1.4.m4.1.1.2.cmml">ℒ</mi><mtext id="S5.p1.4.m4.1.1.3" xref="S5.p1.4.m4.1.1.3a.cmml">pred</mtext></msub><annotation-xml encoding="MathML-Content" id="S5.p1.4.m4.1b"><apply id="S5.p1.4.m4.1.1.cmml" xref="S5.p1.4.m4.1.1"><csymbol cd="ambiguous" id="S5.p1.4.m4.1.1.1.cmml" xref="S5.p1.4.m4.1.1">subscript</csymbol><ci id="S5.p1.4.m4.1.1.2.cmml" xref="S5.p1.4.m4.1.1.2">ℒ</ci><ci id="S5.p1.4.m4.1.1.3a.cmml" xref="S5.p1.4.m4.1.1.3"><mtext mathsize="70%" id="S5.p1.4.m4.1.1.3.cmml" xref="S5.p1.4.m4.1.1.3">pred</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S5.p1.4.m4.1c">\mathcal{L}_{\textrm{pred}}</annotation></semantics></math>) on training and validation data would remain high for some models, while it would converge normally for others.
Therefore, monitoring the loss per model could potentially serve for evaluating the degree of underspecification of a given setting (dataset and architecture) by noting the number of models that converge.
A thorough investigation and validation of this technique is an important topic left for future work.</p>
</div>
<div id="S5.p2" class="ltx_para ltx_noindent">
<p id="S5.p2.1" class="ltx_p"><span id="S5.p2.1.1" class="ltx_text ltx_font_bold">Is the data manifold necessary in the definition of underspecification ?</span> 
Our definition without the reference to the data manifold would be pointless because virtually all conceivable models would be severely underspecified.
The manifold (<em id="S5.p2.1.2" class="ltx_emph ltx_font_italic">e.g</em>.<span id="S5.p2.1.3" class="ltx_text"></span> natural photographs) only fills a fraction of the ambient input space (<em id="S5.p2.1.4" class="ltx_emph ltx_font_italic">e.g</em>.<span id="S5.p2.1.5" class="ltx_text"></span> all possible RGB pixel grids).
Our definition captures underspecification with respect to this smaller range of inputs that are expected at test time.</p>
</div>
<div id="S5.p3" class="ltx_para ltx_noindent">
<p id="S5.p3.1" class="ltx_p"><span id="S5.p3.1.1" class="ltx_text ltx_font_bold">How to select models to combine into a global predictor?</span> 

Our approach is agnostic to the model selection strategy.
Many options are possible to identify models that rely on “robust” features.
The simplest is an evaluation on an OOD validation set.
Alternatively, interpretability methods allow domain experts to inspect models and the features they rely on.
The selection may also be semi-automated with task-specific heuristics <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib15" title="" class="ltx_ref">15</a>, <a href="#bib.bib19" title="" class="ltx_ref">19</a>, <a href="#bib.bib35" title="" class="ltx_ref">35</a>, <a href="#bib.bib79" title="" class="ltx_ref">79</a>]</cite> or additional annotations <em id="S5.p3.1.2" class="ltx_emph ltx_font_italic">e.g</em>.<span id="S5.p3.1.3" class="ltx_text"></span> human attention and rationales <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib64" title="" class="ltx_ref">64</a>, <a href="#bib.bib69" title="" class="ltx_ref">69</a>]</cite>.</p>
</div>
<div id="S5.p4" class="ltx_para ltx_noindent">
<p id="S5.p4.1" class="ltx_p"><span id="S5.p4.1.1" class="ltx_text ltx_font_bold">Computational cost.</span> 
With parallelization, our method scales sublinearly in computing time and linearly in memory w.r.t. the number of models.
Models share mini-batches.
And since they use the same architecture, most computations can be parallelized with <span id="S5.p4.1.2" class="ltx_text ltx_font_bold">grouped convolutions</span> (one group per model).
The method is also applicable on a frozen feature extractor (Section <a href="#S6.SS2" title="6.2 Experiments on real data: WILDS-Camelyon17 ‣ 6 Experiments ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">6.2</span></a> and Appendix <a href="#A6" title="Appendix F Additional results: collages ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">F</span></a>).
However, fine-tuning a shared extractor with the independence loss is not possible. The model could simply dispatch redundant features on multiple channels and satisfy the independence constraint without the desired increase in diversity.
Compared to <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib72" title="" class="ltx_ref">72</a>]</cite>, we significantly reduce the number of models needed to discover useful features.</p>
</div>
</section>
<section id="S6" class="ltx_section">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">6 </span>Experiments</h2>

<div id="S6.p1" class="ltx_para">
<p id="S6.p1.1" class="ltx_p">We first present experiments that validate the method on controlled data with multiple known predictive features (collages, Section <a href="#S6.SS1" title="6.1 Experiments on controlled data: collages ‣ 6 Experiments ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">6.1</span></a>).
We then demonstrate applications to existing datasets: WILDS-Camelyon17, <a href="#S6.SS2" title="6.2 Experiments on real data: WILDS-Camelyon17 ‣ 6 Experiments ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">6.2</span></a> and GQA, Section <a href="#S6.SS3" title="6.3 Experiments on visual question answering: GQA dataset ‣ 6 Experiments ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">6.3</span></a>.</p>
</div>
<section id="S6.SS1" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">6.1 </span>Experiments on controlled data: collages</h3>

<div id="S6.SS1.p1" class="ltx_para">
<p id="S6.SS1.p1.1" class="ltx_p">This diagnostic dataset contains images with binary labels that are constructed to contain multiple predictive features <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib66" title="" class="ltx_ref">66</a>, <a href="#bib.bib72" title="" class="ltx_ref">72</a>]</cite>.
Each image contains four tiles representing one of two classes respectively from MNIST (0/1), CIFAR-10 (automobile/truck), Fashion-MNIST (pullover/coat), and SVHN (0/1).</p>
<ul id="S6.I1" class="ltx_itemize">
<li id="S6.I1.i1" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_item">•</span> 
<div id="S6.I1.i1.p1" class="ltx_para">
<p id="S6.I1.i1.p1.1" class="ltx_p">At <span id="S6.I1.i1.p1.1.1" class="ltx_text ltx_font_bold">training time</span>, the labels are perfectly correlated with the four tiles (0/1 respectively for the first/second possible class in each tile).
There are (at least) four equally-valid ways of understanding the task (<em id="S6.I1.i1.p1.1.2" class="ltx_emph ltx_font_italic">i.e</em>.<span id="S6.I1.i1.p1.1.3" class="ltx_text"></span> relying on any of the four tiles).</p>
</div>
</li>
<li id="S6.I1.i2" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_item">•</span> 
<div id="S6.I1.i2.p1" class="ltx_para">
<p id="S6.I1.i2.p1.1" class="ltx_p">At <span id="S6.I1.i2.p1.1.1" class="ltx_text ltx_font_bold">test time</span>, we evaluate a model on four test sets that represent different OOD conditions. In each, only one tile is correlated with the correct label while others tiles are randomized.
By examining the performance on the four test sets, we can identify which tile(s) the model relies on
</p>
</div>
</li>
</ul>
<span id="S6.SS1.p1.2" class="ltx_ERROR undefined">\floatsetup</span>
<p id="S6.SS1.p1.3" class="ltx_p">[figure]capposition=beside,capbesideposition=center,right,floatwidth=0.54capbesidewidth=0.42</p>
</div>
<figure id="S6.F4" class="ltx_figure">
<table id="S6.F4.4" class="ltx_tabular ltx_centering ltx_align_middle">
<tr id="S6.F4.4.4" class="ltx_tr">
<td id="S6.F4.4.4.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding-left:0.0pt;padding-right:0.0pt;">
<table id="S6.F4.4.4.5.1" class="ltx_tabular ltx_align_middle">
<tr id="S6.F4.4.4.5.1.1" class="ltx_tr">
<td id="S6.F4.4.4.5.1.1.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.F4.4.4.5.1.1.1.1" class="ltx_text ltx_font_bold" style="font-size:70%;">Class 0</span></td>
</tr>
<tr id="S6.F4.4.4.5.1.2" class="ltx_tr">
<td id="S6.F4.4.4.5.1.2.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.F4.4.4.5.1.2.1.1" class="ltx_text" style="font-size:70%;">Zero, pullover</span></td>
</tr>
<tr id="S6.F4.4.4.5.1.3" class="ltx_tr">
<td id="S6.F4.4.4.5.1.3.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.F4.4.4.5.1.3.1.1" class="ltx_text" style="font-size:70%;">automobile, zero.</span></td>
</tr>
</table>
</td>
<td id="S6.F4.1.1.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><img src="/html/2207.02598/assets/n4class0-4.png" id="S6.F4.1.1.1.g1" class="ltx_graphics ltx_img_square" width="84" height="84" alt="Refer to caption"></td>
<td id="S6.F4.2.2.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><img src="/html/2207.02598/assets/n4class0-3.png" id="S6.F4.2.2.2.g1" class="ltx_graphics ltx_img_square" width="84" height="84" alt="Refer to caption"></td>
<td id="S6.F4.4.4.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding-left:0.0pt;padding-right:0.0pt;">
<table id="S6.F4.4.4.6.1" class="ltx_tabular ltx_align_middle">
<tr id="S6.F4.4.4.6.1.1" class="ltx_tr">
<td id="S6.F4.4.4.6.1.1.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.F4.4.4.6.1.1.1.1" class="ltx_text ltx_font_bold" style="font-size:70%;">Class 1</span></td>
</tr>
<tr id="S6.F4.4.4.6.1.2" class="ltx_tr">
<td id="S6.F4.4.4.6.1.2.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.F4.4.4.6.1.2.1.1" class="ltx_text" style="font-size:70%;">One, coat</span></td>
</tr>
<tr id="S6.F4.4.4.6.1.3" class="ltx_tr">
<td id="S6.F4.4.4.6.1.3.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.F4.4.4.6.1.3.1.1" class="ltx_text" style="font-size:70%;">truck, one.</span></td>
</tr>
</table>
</td>
<td id="S6.F4.3.3.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><img src="/html/2207.02598/assets/n4class1-2.png" id="S6.F4.3.3.3.g1" class="ltx_graphics ltx_img_square" width="84" height="84" alt="Refer to caption"></td>
<td id="S6.F4.4.4.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><img src="/html/2207.02598/assets/n4class1-1.png" id="S6.F4.4.4.4.g1" class="ltx_graphics ltx_img_square" width="84" height="84" alt="Refer to caption"></td>
</tr>
</table>
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span id="S6.F4.6.1.1" class="ltx_text" style="font-size:90%;">Figure 4</span>: </span><span id="S6.F4.7.2" class="ltx_text" style="font-size:90%;">Examples of collages <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib72" title="" class="ltx_ref">72</a>]</cite>. Training labels are correlated with all four tiles. Each tile contains features of different complexity.</span></figcaption>
</figure>
<div id="S6.SS1.p2" class="ltx_para ltx_noindent">
<p id="S6.SS1.p2.2" class="ltx_p"><span id="S6.SS1.p2.2.1" class="ltx_text ltx_font_bold">Task difficulty.</span> 
This dataset is surprisingly challenging because the tiles vary greatly in learning difficulty (<em id="S6.SS1.p2.2.2" class="ltx_emph ltx_font_italic">e.g</em>.<span id="S6.SS1.p2.2.3" class="ltx_text"></span> MNIST 0s/1s are very distinct while Fashion-MNIST pullovers/coats look extremely similar).
It would be reasonable to learn a model that relies on all four tiles. However, an ERM-trained baseline surprisingly uses only a few MNIST pixels (achieving <math id="S6.SS1.p2.1.m1.1" class="ltx_Math" alttext="\sim" display="inline"><semantics id="S6.SS1.p2.1.m1.1a"><mo id="S6.SS1.p2.1.m1.1.1" xref="S6.SS1.p2.1.m1.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.1.m1.1b"><csymbol cd="latexml" id="S6.SS1.p2.1.m1.1.1.cmml" xref="S6.SS1.p2.1.m1.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.1.m1.1c">\sim</annotation></semantics></math>99% accuracy on the MNIST test set and <math id="S6.SS1.p2.2.m2.1" class="ltx_Math" alttext="\sim" display="inline"><semantics id="S6.SS1.p2.2.m2.1a"><mo id="S6.SS1.p2.2.m2.1.1" xref="S6.SS1.p2.2.m2.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S6.SS1.p2.2.m2.1b"><csymbol cd="latexml" id="S6.SS1.p2.2.m2.1.1.cmml" xref="S6.SS1.p2.2.m2.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p2.2.m2.1c">\sim</annotation></semantics></math>50% on the others), as shown in previous work on the simplicity bias of neural networks <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib72" title="" class="ltx_ref">72</a>]</cite>.</p>
</div>
<div id="S6.SS1.p3" class="ltx_para">
<p id="S6.SS1.p3.1" class="ltx_p">We follow <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib72" title="" class="ltx_ref">72</a>]</cite> and use our method to learn multiple models compatible with the data. We then report the accuracy of the best model on each test set, <em id="S6.SS1.p3.1.1" class="ltx_emph ltx_font_italic">i.e</em>.<span id="S6.SS1.p3.1.2" class="ltx_text"></span> the best accuracy assuming perfect model selection.
This avoids confounding the performance of the learning algorithm and with that of the selection strategy.</p>
</div>
<div id="S6.SS1.p4" class="ltx_para ltx_noindent">
<p id="S6.SS1.p4.2" class="ltx_p"><span id="S6.SS1.p4.2.1" class="ltx_text ltx_font_bold">Applying the proposed method.</span> 
We follow Algorithm <a href="#algorithm1" title="In 4.3 Fine-tuning ‣ 4 Proposed method ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">1</span></a>.
We prepare unlabeled data to defines the data manifold as the union of the training and test sets, thus covering all combinations of contents of the four tiles.
With this data, we estimate the dimensionality of the manifold with <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib57" title="" class="ltx_ref">57</a>]</cite> as about 23.8 (<math id="S6.SS1.p4.1.m1.1" class="ltx_Math" alttext="\sigma=0.16" display="inline"><semantics id="S6.SS1.p4.1.m1.1a"><mrow id="S6.SS1.p4.1.m1.1.1" xref="S6.SS1.p4.1.m1.1.1.cmml"><mi id="S6.SS1.p4.1.m1.1.1.2" xref="S6.SS1.p4.1.m1.1.1.2.cmml">σ</mi><mo id="S6.SS1.p4.1.m1.1.1.1" xref="S6.SS1.p4.1.m1.1.1.1.cmml">=</mo><mn id="S6.SS1.p4.1.m1.1.1.3" xref="S6.SS1.p4.1.m1.1.1.3.cmml">0.16</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.SS1.p4.1.m1.1b"><apply id="S6.SS1.p4.1.m1.1.1.cmml" xref="S6.SS1.p4.1.m1.1.1"><eq id="S6.SS1.p4.1.m1.1.1.1.cmml" xref="S6.SS1.p4.1.m1.1.1.1"></eq><ci id="S6.SS1.p4.1.m1.1.1.2.cmml" xref="S6.SS1.p4.1.m1.1.1.2">𝜎</ci><cn type="float" id="S6.SS1.p4.1.m1.1.1.3.cmml" xref="S6.SS1.p4.1.m1.1.1.3">0.16</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p4.1.m1.1c">\sigma=0.16</annotation></semantics></math> over 10 runs).
We prepare two generative models of the manifold: a PCA with 24 components (capturing <math id="S6.SS1.p4.2.m2.1" class="ltx_Math" alttext="\sim" display="inline"><semantics id="S6.SS1.p4.2.m2.1a"><mo id="S6.SS1.p4.2.m2.1.1" xref="S6.SS1.p4.2.m2.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S6.SS1.p4.2.m2.1b"><csymbol cd="latexml" id="S6.SS1.p4.2.m2.1.1.cmml" xref="S6.SS1.p4.2.m2.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p4.2.m2.1c">\sim</annotation></semantics></math>85% of the variance)
and a VAE with 24 latent dimensions (details in Appendix <a href="#A4" title="Appendix D Experimental details: collages ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">D</span></a>).
We define a simple architecture (2-layer MLPs) and train multiple instances in parallel with the proposed objective.
The only hyperparameters are the number of models and weights of independence/on-manifold constraints.
We plot a range of values in the appendix (Figure <a href="#A6.F11" title="Figure 11 ‣ Appendix F Additional results: collages ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">11</span></a>).</p>
</div>
<div id="S6.SS1.p5" class="ltx_para ltx_noindent">
<p id="S6.SS1.p5.1" class="ltx_p"><span id="S6.SS1.p5.1.1" class="ltx_text ltx_font_bold">Results.</span> 
Our method learns a set of models that focus on different parts of the images. Remarkably, learning as few as 4 models is sufficient to obtain models with high accuracy on all of the four test sets.
Let us examine several ablations.</p>
<ul id="S6.I2" class="ltx_itemize">
<li id="S6.I2.i1" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_item">•</span> 
<div id="S6.I2.i1.p1" class="ltx_para">
<p id="S6.I2.i1.p1.1" class="ltx_p">The baseline (<math id="S6.I2.i1.p1.1.m1.1" class="ltx_Math" alttext="{\lambda_{\textrm{indep}}}={\lambda_{\textrm{manifold}}}=0" display="inline"><semantics id="S6.I2.i1.p1.1.m1.1a"><mrow id="S6.I2.i1.p1.1.m1.1.1" xref="S6.I2.i1.p1.1.m1.1.1.cmml"><msub id="S6.I2.i1.p1.1.m1.1.1.2" xref="S6.I2.i1.p1.1.m1.1.1.2.cmml"><mi id="S6.I2.i1.p1.1.m1.1.1.2.2" xref="S6.I2.i1.p1.1.m1.1.1.2.2.cmml">λ</mi><mtext id="S6.I2.i1.p1.1.m1.1.1.2.3" xref="S6.I2.i1.p1.1.m1.1.1.2.3a.cmml">indep</mtext></msub><mo id="S6.I2.i1.p1.1.m1.1.1.3" xref="S6.I2.i1.p1.1.m1.1.1.3.cmml">=</mo><msub id="S6.I2.i1.p1.1.m1.1.1.4" xref="S6.I2.i1.p1.1.m1.1.1.4.cmml"><mi id="S6.I2.i1.p1.1.m1.1.1.4.2" xref="S6.I2.i1.p1.1.m1.1.1.4.2.cmml">λ</mi><mtext id="S6.I2.i1.p1.1.m1.1.1.4.3" xref="S6.I2.i1.p1.1.m1.1.1.4.3a.cmml">manifold</mtext></msub><mo id="S6.I2.i1.p1.1.m1.1.1.5" xref="S6.I2.i1.p1.1.m1.1.1.5.cmml">=</mo><mn id="S6.I2.i1.p1.1.m1.1.1.6" xref="S6.I2.i1.p1.1.m1.1.1.6.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.I2.i1.p1.1.m1.1b"><apply id="S6.I2.i1.p1.1.m1.1.1.cmml" xref="S6.I2.i1.p1.1.m1.1.1"><and id="S6.I2.i1.p1.1.m1.1.1a.cmml" xref="S6.I2.i1.p1.1.m1.1.1"></and><apply id="S6.I2.i1.p1.1.m1.1.1b.cmml" xref="S6.I2.i1.p1.1.m1.1.1"><eq id="S6.I2.i1.p1.1.m1.1.1.3.cmml" xref="S6.I2.i1.p1.1.m1.1.1.3"></eq><apply id="S6.I2.i1.p1.1.m1.1.1.2.cmml" xref="S6.I2.i1.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S6.I2.i1.p1.1.m1.1.1.2.1.cmml" xref="S6.I2.i1.p1.1.m1.1.1.2">subscript</csymbol><ci id="S6.I2.i1.p1.1.m1.1.1.2.2.cmml" xref="S6.I2.i1.p1.1.m1.1.1.2.2">𝜆</ci><ci id="S6.I2.i1.p1.1.m1.1.1.2.3a.cmml" xref="S6.I2.i1.p1.1.m1.1.1.2.3"><mtext mathsize="70%" id="S6.I2.i1.p1.1.m1.1.1.2.3.cmml" xref="S6.I2.i1.p1.1.m1.1.1.2.3">indep</mtext></ci></apply><apply id="S6.I2.i1.p1.1.m1.1.1.4.cmml" xref="S6.I2.i1.p1.1.m1.1.1.4"><csymbol cd="ambiguous" id="S6.I2.i1.p1.1.m1.1.1.4.1.cmml" xref="S6.I2.i1.p1.1.m1.1.1.4">subscript</csymbol><ci id="S6.I2.i1.p1.1.m1.1.1.4.2.cmml" xref="S6.I2.i1.p1.1.m1.1.1.4.2">𝜆</ci><ci id="S6.I2.i1.p1.1.m1.1.1.4.3a.cmml" xref="S6.I2.i1.p1.1.m1.1.1.4.3"><mtext mathsize="70%" id="S6.I2.i1.p1.1.m1.1.1.4.3.cmml" xref="S6.I2.i1.p1.1.m1.1.1.4.3">manifold</mtext></ci></apply></apply><apply id="S6.I2.i1.p1.1.m1.1.1c.cmml" xref="S6.I2.i1.p1.1.m1.1.1"><eq id="S6.I2.i1.p1.1.m1.1.1.5.cmml" xref="S6.I2.i1.p1.1.m1.1.1.5"></eq><share href="#S6.I2.i1.p1.1.m1.1.1.4.cmml" id="S6.I2.i1.p1.1.m1.1.1d.cmml" xref="S6.I2.i1.p1.1.m1.1.1"></share><cn type="integer" id="S6.I2.i1.p1.1.m1.1.1.6.cmml" xref="S6.I2.i1.p1.1.m1.1.1.6">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I2.i1.p1.1.m1.1c">{\lambda_{\textrm{indep}}}={\lambda_{\textrm{manifold}}}=0</annotation></semantics></math>) only learns about MNIST.</p>
</div>
</li>
<li id="S6.I2.i2" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_item">•</span> 
<div id="S6.I2.i2.p1" class="ltx_para">
<p id="S6.I2.i2.p1.3" class="ltx_p">The independence constraint (<math id="S6.I2.i2.p1.1.m1.1" class="ltx_Math" alttext="{\lambda_{\textrm{indep}}}&gt;" display="inline"><semantics id="S6.I2.i2.p1.1.m1.1a"><mrow id="S6.I2.i2.p1.1.m1.1.1" xref="S6.I2.i2.p1.1.m1.1.1.cmml"><msub id="S6.I2.i2.p1.1.m1.1.1.2" xref="S6.I2.i2.p1.1.m1.1.1.2.cmml"><mi id="S6.I2.i2.p1.1.m1.1.1.2.2" xref="S6.I2.i2.p1.1.m1.1.1.2.2.cmml">λ</mi><mtext id="S6.I2.i2.p1.1.m1.1.1.2.3" xref="S6.I2.i2.p1.1.m1.1.1.2.3a.cmml">indep</mtext></msub><mo id="S6.I2.i2.p1.1.m1.1.1.1" xref="S6.I2.i2.p1.1.m1.1.1.1.cmml">&gt;</mo><mi id="S6.I2.i2.p1.1.m1.1.1.3" xref="S6.I2.i2.p1.1.m1.1.1.3.cmml"></mi></mrow><annotation-xml encoding="MathML-Content" id="S6.I2.i2.p1.1.m1.1b"><apply id="S6.I2.i2.p1.1.m1.1.1.cmml" xref="S6.I2.i2.p1.1.m1.1.1"><gt id="S6.I2.i2.p1.1.m1.1.1.1.cmml" xref="S6.I2.i2.p1.1.m1.1.1.1"></gt><apply id="S6.I2.i2.p1.1.m1.1.1.2.cmml" xref="S6.I2.i2.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S6.I2.i2.p1.1.m1.1.1.2.1.cmml" xref="S6.I2.i2.p1.1.m1.1.1.2">subscript</csymbol><ci id="S6.I2.i2.p1.1.m1.1.1.2.2.cmml" xref="S6.I2.i2.p1.1.m1.1.1.2.2">𝜆</ci><ci id="S6.I2.i2.p1.1.m1.1.1.2.3a.cmml" xref="S6.I2.i2.p1.1.m1.1.1.2.3"><mtext mathsize="70%" id="S6.I2.i2.p1.1.m1.1.1.2.3.cmml" xref="S6.I2.i2.p1.1.m1.1.1.2.3">indep</mtext></ci></apply><csymbol cd="latexml" id="S6.I2.i2.p1.1.m1.1.1.3.cmml" xref="S6.I2.i2.p1.1.m1.1.1.3">absent</csymbol></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I2.i2.p1.1.m1.1c">{\lambda_{\textrm{indep}}}&gt;</annotation></semantics></math>, <math id="S6.I2.i2.p1.2.m2.1" class="ltx_Math" alttext="{\lambda_{\textrm{manifold}}}=0" display="inline"><semantics id="S6.I2.i2.p1.2.m2.1a"><mrow id="S6.I2.i2.p1.2.m2.1.1" xref="S6.I2.i2.p1.2.m2.1.1.cmml"><msub id="S6.I2.i2.p1.2.m2.1.1.2" xref="S6.I2.i2.p1.2.m2.1.1.2.cmml"><mi id="S6.I2.i2.p1.2.m2.1.1.2.2" xref="S6.I2.i2.p1.2.m2.1.1.2.2.cmml">λ</mi><mtext id="S6.I2.i2.p1.2.m2.1.1.2.3" xref="S6.I2.i2.p1.2.m2.1.1.2.3a.cmml">manifold</mtext></msub><mo id="S6.I2.i2.p1.2.m2.1.1.1" xref="S6.I2.i2.p1.2.m2.1.1.1.cmml">=</mo><mn id="S6.I2.i2.p1.2.m2.1.1.3" xref="S6.I2.i2.p1.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.I2.i2.p1.2.m2.1b"><apply id="S6.I2.i2.p1.2.m2.1.1.cmml" xref="S6.I2.i2.p1.2.m2.1.1"><eq id="S6.I2.i2.p1.2.m2.1.1.1.cmml" xref="S6.I2.i2.p1.2.m2.1.1.1"></eq><apply id="S6.I2.i2.p1.2.m2.1.1.2.cmml" xref="S6.I2.i2.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S6.I2.i2.p1.2.m2.1.1.2.1.cmml" xref="S6.I2.i2.p1.2.m2.1.1.2">subscript</csymbol><ci id="S6.I2.i2.p1.2.m2.1.1.2.2.cmml" xref="S6.I2.i2.p1.2.m2.1.1.2.2">𝜆</ci><ci id="S6.I2.i2.p1.2.m2.1.1.2.3a.cmml" xref="S6.I2.i2.p1.2.m2.1.1.2.3"><mtext mathsize="70%" id="S6.I2.i2.p1.2.m2.1.1.2.3.cmml" xref="S6.I2.i2.p1.2.m2.1.1.2.3">manifold</mtext></ci></apply><cn type="integer" id="S6.I2.i2.p1.2.m2.1.1.3.cmml" xref="S6.I2.i2.p1.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I2.i2.p1.2.m2.1c">{\lambda_{\textrm{manifold}}}=0</annotation></semantics></math>) is crucial for learning distinct models. On its own, it requires training a very large number of models (<math id="S6.I2.i2.p1.3.m3.1" class="ltx_Math" alttext="\gg" display="inline"><semantics id="S6.I2.i2.p1.3.m3.1a"><mo id="S6.I2.i2.p1.3.m3.1.1" xref="S6.I2.i2.p1.3.m3.1.1.cmml">≫</mo><annotation-xml encoding="MathML-Content" id="S6.I2.i2.p1.3.m3.1b"><csymbol cd="latexml" id="S6.I2.i2.p1.3.m3.1.1.cmml" xref="S6.I2.i2.p1.3.m3.1.1">much-greater-than</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.I2.i2.p1.3.m3.1c">\gg</annotation></semantics></math>32) before picking up features outside the MNIST tiles.
Visualizations of input gradients (Figure <a href="#S6.F6" title="Figure 6 ‣ 6.1 Experiments on controlled data: collages ‣ 6 Experiments ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">6</span></a>) reveal that these models each rely only on a single or a few pixels.
These trivial solutions to the independence constraint, akin to adversarial examples, are avoided with the on-manifold constraint.</p>
</div>
</li>
<li id="S6.I2.i3" class="ltx_item" style="list-style-type:none;">
<span class="ltx_tag ltx_tag_item">•</span> 
<div id="S6.I2.i3.p1" class="ltx_para">
<p id="S6.I2.i3.p1.2" class="ltx_p">In the full method (<math id="S6.I2.i3.p1.1.m1.1" class="ltx_Math" alttext="{\lambda_{\textrm{indep}}}&gt;0" display="inline"><semantics id="S6.I2.i3.p1.1.m1.1a"><mrow id="S6.I2.i3.p1.1.m1.1.1" xref="S6.I2.i3.p1.1.m1.1.1.cmml"><msub id="S6.I2.i3.p1.1.m1.1.1.2" xref="S6.I2.i3.p1.1.m1.1.1.2.cmml"><mi id="S6.I2.i3.p1.1.m1.1.1.2.2" xref="S6.I2.i3.p1.1.m1.1.1.2.2.cmml">λ</mi><mtext id="S6.I2.i3.p1.1.m1.1.1.2.3" xref="S6.I2.i3.p1.1.m1.1.1.2.3a.cmml">indep</mtext></msub><mo id="S6.I2.i3.p1.1.m1.1.1.1" xref="S6.I2.i3.p1.1.m1.1.1.1.cmml">&gt;</mo><mn id="S6.I2.i3.p1.1.m1.1.1.3" xref="S6.I2.i3.p1.1.m1.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.I2.i3.p1.1.m1.1b"><apply id="S6.I2.i3.p1.1.m1.1.1.cmml" xref="S6.I2.i3.p1.1.m1.1.1"><gt id="S6.I2.i3.p1.1.m1.1.1.1.cmml" xref="S6.I2.i3.p1.1.m1.1.1.1"></gt><apply id="S6.I2.i3.p1.1.m1.1.1.2.cmml" xref="S6.I2.i3.p1.1.m1.1.1.2"><csymbol cd="ambiguous" id="S6.I2.i3.p1.1.m1.1.1.2.1.cmml" xref="S6.I2.i3.p1.1.m1.1.1.2">subscript</csymbol><ci id="S6.I2.i3.p1.1.m1.1.1.2.2.cmml" xref="S6.I2.i3.p1.1.m1.1.1.2.2">𝜆</ci><ci id="S6.I2.i3.p1.1.m1.1.1.2.3a.cmml" xref="S6.I2.i3.p1.1.m1.1.1.2.3"><mtext mathsize="70%" id="S6.I2.i3.p1.1.m1.1.1.2.3.cmml" xref="S6.I2.i3.p1.1.m1.1.1.2.3">indep</mtext></ci></apply><cn type="integer" id="S6.I2.i3.p1.1.m1.1.1.3.cmml" xref="S6.I2.i3.p1.1.m1.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I2.i3.p1.1.m1.1c">{\lambda_{\textrm{indep}}}&gt;0</annotation></semantics></math>, <math id="S6.I2.i3.p1.2.m2.1" class="ltx_Math" alttext="{\lambda_{\textrm{manifold}}}&gt;0" display="inline"><semantics id="S6.I2.i3.p1.2.m2.1a"><mrow id="S6.I2.i3.p1.2.m2.1.1" xref="S6.I2.i3.p1.2.m2.1.1.cmml"><msub id="S6.I2.i3.p1.2.m2.1.1.2" xref="S6.I2.i3.p1.2.m2.1.1.2.cmml"><mi id="S6.I2.i3.p1.2.m2.1.1.2.2" xref="S6.I2.i3.p1.2.m2.1.1.2.2.cmml">λ</mi><mtext id="S6.I2.i3.p1.2.m2.1.1.2.3" xref="S6.I2.i3.p1.2.m2.1.1.2.3a.cmml">manifold</mtext></msub><mo id="S6.I2.i3.p1.2.m2.1.1.1" xref="S6.I2.i3.p1.2.m2.1.1.1.cmml">&gt;</mo><mn id="S6.I2.i3.p1.2.m2.1.1.3" xref="S6.I2.i3.p1.2.m2.1.1.3.cmml">0</mn></mrow><annotation-xml encoding="MathML-Content" id="S6.I2.i3.p1.2.m2.1b"><apply id="S6.I2.i3.p1.2.m2.1.1.cmml" xref="S6.I2.i3.p1.2.m2.1.1"><gt id="S6.I2.i3.p1.2.m2.1.1.1.cmml" xref="S6.I2.i3.p1.2.m2.1.1.1"></gt><apply id="S6.I2.i3.p1.2.m2.1.1.2.cmml" xref="S6.I2.i3.p1.2.m2.1.1.2"><csymbol cd="ambiguous" id="S6.I2.i3.p1.2.m2.1.1.2.1.cmml" xref="S6.I2.i3.p1.2.m2.1.1.2">subscript</csymbol><ci id="S6.I2.i3.p1.2.m2.1.1.2.2.cmml" xref="S6.I2.i3.p1.2.m2.1.1.2.2">𝜆</ci><ci id="S6.I2.i3.p1.2.m2.1.1.2.3a.cmml" xref="S6.I2.i3.p1.2.m2.1.1.2.3"><mtext mathsize="70%" id="S6.I2.i3.p1.2.m2.1.1.2.3.cmml" xref="S6.I2.i3.p1.2.m2.1.1.2.3">manifold</mtext></ci></apply><cn type="integer" id="S6.I2.i3.p1.2.m2.1.1.3.cmml" xref="S6.I2.i3.p1.2.m2.1.1.3">0</cn></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.I2.i3.p1.2.m2.1c">{\lambda_{\textrm{manifold}}}&gt;0</annotation></semantics></math>) the models discover distinct features that align with the semantic contents of images. The effect of the on-manifold constraint on input gradients is striking (Figure <a href="#S6.F6" title="Figure 6 ‣ 6.1 Experiments on controlled data: collages ‣ 6 Experiments ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">6</span></a>).
It forces models to be sensitive to natural variations of the data – rather than unlikely single-pixel patterns.
Remarkably, <span id="S6.I2.i3.p1.2.1" class="ltx_text ltx_font_bold">image regions emerge as meaningful features without inductive bias for spatial locality</span> (<em id="S6.I2.i3.p1.2.2" class="ltx_emph ltx_font_italic">e.g</em>.<span id="S6.I2.i3.p1.2.3" class="ltx_text"></span> no convolutions).</p>
</div>
</li>
</ul>
<span id="S6.SS1.p5.2" class="ltx_ERROR undefined">\floatsetup</span>
<p id="S6.SS1.p5.3" class="ltx_p">[figure]capposition=beside,capbesideposition=center,right,floatwidth=0.35capbesidewidth=0.61</p>
</div>
<figure id="S6.F5" class="ltx_figure"><img src="/html/2207.02598/assets/x3.png" id="S6.F5.g1" class="ltx_graphics ltx_img_landscape" width="461" height="148" alt="Refer to caption">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_figure"><span id="S6.F5.3.1.1" class="ltx_text" style="font-size:90%;">Figure 5</span>: </span><span id="S6.F5.4.2" class="ltx_text" style="font-size:90%;">Collages dataset: accuracy on the four test sets (columns) of models with best accuracy on each set (rows).
Diagonal patterns indicate that <span id="S6.F5.4.2.1" class="ltx_text ltx_font_bold">models specialize and learn different, non-overlapping features</span>.
The baseline only learns features relevant to MNIST.
</span></figcaption>
</figure>
<div id="S6.SS1.p6" class="ltx_para ltx_noindent">
<p id="S6.SS1.p6.2" class="ltx_p"><span id="S6.SS1.p6.2.1" class="ltx_text ltx_font_bold">Hyperparameters.</span> 
A number of models between 4 and 24 give excellent results. As expected, the larger this number, the more granular the features these models learn (Figure <a href="#S6.F6" title="Figure 6 ‣ 6.1 Experiments on controlled data: collages ‣ 6 Experiments ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">6</span></a>).
The effect breaks down for <math id="S6.SS1.p6.1.m1.1" class="ltx_Math" alttext="&gt;" display="inline"><semantics id="S6.SS1.p6.1.m1.1a"><mo id="S6.SS1.p6.1.m1.1.1" xref="S6.SS1.p6.1.m1.1.1.cmml">&gt;</mo><annotation-xml encoding="MathML-Content" id="S6.SS1.p6.1.m1.1b"><gt id="S6.SS1.p6.1.m1.1.1.cmml" xref="S6.SS1.p6.1.m1.1.1"></gt></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p6.1.m1.1c">&gt;</annotation></semantics></math>24 models, matching theoretical expectations since the dimensionality of the manifold was estimated at <math id="S6.SS1.p6.2.m2.1" class="ltx_Math" alttext="\sim" display="inline"><semantics id="S6.SS1.p6.2.m2.1a"><mo id="S6.SS1.p6.2.m2.1.1" xref="S6.SS1.p6.2.m2.1.1.cmml">∼</mo><annotation-xml encoding="MathML-Content" id="S6.SS1.p6.2.m2.1b"><csymbol cd="latexml" id="S6.SS1.p6.2.m2.1.1.cmml" xref="S6.SS1.p6.2.m2.1.1">similar-to</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.SS1.p6.2.m2.1c">\sim</annotation></semantics></math>24.
The method is stable over a range of regularizer weights.
Additional comparisons in Table <a href="#S6.T1" title="Table 1 ‣ 6.1 Experiments on controlled data: collages ‣ 6 Experiments ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">1</span></a> show that a VAE is better than PCA to represent the manifold.
This agrees with the general expectation that natural images form a non-linear manifold in pixel space.
We also found overall results to be robust to variations in architecture and hyperparameters of the VAE.</p>
</div>
<div id="S6.SS1.p7" class="ltx_para">
<span id="S6.SS1.p7.1" class="ltx_ERROR undefined">\floatsetup</span>
<p id="S6.SS1.p7.2" class="ltx_p">[figure]capposition=below,floatwidth=</p>
</div>
<figure id="S6.F6" class="ltx_figure">
<div class="ltx_flex_figure">
<div class="ltx_flex_cell ltx_flex_size_4">
<figure id="S6.F6.sf1" class="ltx_figure ltx_figure_panel ltx_align_center">
<table id="S6.F6.sf1.3" class="ltx_tabular ltx_centering ltx_align_middle">
<tr id="S6.F6.sf1.3.4" class="ltx_tr">
<td id="S6.F6.sf1.3.4.1" class="ltx_td ltx_nopad_l ltx_nopad_r" style="padding:0.25pt 0.0pt;"></td>
<td id="S6.F6.sf1.3.4.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.25pt 0.0pt;"><span id="S6.F6.sf1.3.4.2.1" class="ltx_text ltx_font_bold" style="font-size:50%;">Baseline</span></td>
</tr>
<tr id="S6.F6.sf1.1.1" class="ltx_tr">
<td id="S6.F6.sf1.1.1.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.25pt 0.0pt;"><span id="S6.F6.sf1.1.1.2.1" class="ltx_text" style="font-size:50%;">4 Models</span></td>
<td id="S6.F6.sf1.1.1.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.25pt 0.0pt;">
<table id="S6.F6.sf1.1.1.1.1.1" class="ltx_tabular ltx_minipage ltx_align_middle" style="width:433.6pt;">
<tr id="S6.F6.sf1.1.1.1.1.1.2" class="ltx_tr">
<td id="S6.F6.sf1.1.1.1.1.1.2.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.25pt 0.0pt;"><span id="S6.F6.sf1.1.1.1.1.1.2.1.1" class="ltx_text" style="font-size:50%;"></span></td>
</tr>
<tr id="S6.F6.sf1.1.1.1.1.1.1" class="ltx_tr">
<td id="S6.F6.sf1.1.1.1.1.1.1.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.25pt 0.0pt;">
<img src="/html/2207.02598/assets/grad-241003s0n4--b256acc1mdo0n4e8r241003wt0.00-1.00-0.00p0-mlp1sigmoid0-s1-e50figGd.png" id="S6.F6.sf1.1.1.1.1.1.1.1.g1" class="ltx_graphics ltx_img_landscape" width="97" height="23" alt="Refer to caption"><span id="S6.F6.sf1.1.1.1.1.1.1.1.1" class="ltx_text" style="font-size:50%;"></span>
</td>
</tr>
</table>
</td>
</tr>
<tr id="S6.F6.sf1.2.2" class="ltx_tr">
<td id="S6.F6.sf1.2.2.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.25pt 0.0pt;"><span id="S6.F6.sf1.2.2.2.1" class="ltx_text" style="font-size:50%;">8 Models</span></td>
<td id="S6.F6.sf1.2.2.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.25pt 0.0pt;">
<table id="S6.F6.sf1.2.2.1.1.1" class="ltx_tabular ltx_minipage ltx_align_middle" style="width:433.6pt;">
<tr id="S6.F6.sf1.2.2.1.1.1.2" class="ltx_tr">
<td id="S6.F6.sf1.2.2.1.1.1.2.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.25pt 0.0pt;"><span id="S6.F6.sf1.2.2.1.1.1.2.1.1" class="ltx_text" style="font-size:50%;"></span></td>
</tr>
<tr id="S6.F6.sf1.2.2.1.1.1.1" class="ltx_tr">
<td id="S6.F6.sf1.2.2.1.1.1.1.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.25pt 0.0pt;">
<img src="/html/2207.02598/assets/grad-241003s0n8--b256acc1mdo0n8e8r241003wt0.00-1.00-0.00p0-mlp1sigmoid0-s1-e50figGd.png" id="S6.F6.sf1.2.2.1.1.1.1.1.g1" class="ltx_graphics ltx_img_landscape" width="97" height="12" alt="Refer to caption"><span id="S6.F6.sf1.2.2.1.1.1.1.1.1" class="ltx_text" style="font-size:50%;"></span>
</td>
</tr>
</table>
</td>
</tr>
<tr id="S6.F6.sf1.3.3" class="ltx_tr">
<td id="S6.F6.sf1.3.3.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.25pt 0.0pt;"><span id="S6.F6.sf1.3.3.2.1" class="ltx_text" style="font-size:50%;">12 Models</span></td>
<td id="S6.F6.sf1.3.3.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.25pt 0.0pt;">
<table id="S6.F6.sf1.3.3.1.1.1" class="ltx_tabular ltx_minipage ltx_align_middle" style="width:433.6pt;">
<tr id="S6.F6.sf1.3.3.1.1.1.2" class="ltx_tr">
<td id="S6.F6.sf1.3.3.1.1.1.2.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.25pt 0.0pt;"><span id="S6.F6.sf1.3.3.1.1.1.2.1.1" class="ltx_text" style="font-size:50%;"></span></td>
</tr>
<tr id="S6.F6.sf1.3.3.1.1.1.1" class="ltx_tr">
<td id="S6.F6.sf1.3.3.1.1.1.1.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.25pt 0.0pt;">
<img src="/html/2207.02598/assets/grad-241003s0n16--b256acc1mdo0n16e8r241003wt0.00-1.00-0.00p0-mlp1sigmoid0-s1-e50figGd.png" id="S6.F6.sf1.3.3.1.1.1.1.1.g1" class="ltx_graphics ltx_img_landscape" width="97" height="8" alt="Refer to caption"><span id="S6.F6.sf1.3.3.1.1.1.1.1.1" class="ltx_text" style="font-size:50%;"></span>
</td>
</tr>
</table>
</td>
</tr>
</table>
<figcaption class="ltx_caption ltx_centering" style="font-size:50%;"><span class="ltx_tag ltx_tag_figure"><span id="S6.F6.sf1.7.1.1" class="ltx_text" style="font-size:180%;">(a)</span> </span><span id="S6.F6.sf1.8.2" class="ltx_text" style="font-size:140%;">With standard training, all models rely on a small, identical region of the image, despite the fact that predictive features are present all over.</span></figcaption>
</figure>
</div>
<div class="ltx_flex_cell ltx_flex_size_4">
<figure id="S6.F6.sf2" class="ltx_figure ltx_figure_panel ltx_align_center">
<table id="S6.F6.sf2.3" class="ltx_tabular ltx_centering ltx_align_middle">
<tr id="S6.F6.sf2.3.4" class="ltx_tr">
<td id="S6.F6.sf2.3.4.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.25pt 0.0pt;"><span id="S6.F6.sf2.3.4.1.1" class="ltx_text ltx_font_bold" style="font-size:50%;">+ Independence</span></td>
</tr>
<tr id="S6.F6.sf2.1.1" class="ltx_tr">
<td id="S6.F6.sf2.1.1.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.25pt 0.0pt;">
<table id="S6.F6.sf2.1.1.1.1.1" class="ltx_tabular ltx_minipage ltx_align_middle" style="width:433.6pt;">
<tr id="S6.F6.sf2.1.1.1.1.1.2" class="ltx_tr">
<td id="S6.F6.sf2.1.1.1.1.1.2.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.25pt 0.0pt;"><span id="S6.F6.sf2.1.1.1.1.1.2.1.1" class="ltx_text" style="font-size:50%;"></span></td>
</tr>
<tr id="S6.F6.sf2.1.1.1.1.1.1" class="ltx_tr">
<td id="S6.F6.sf2.1.1.1.1.1.1.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.25pt 0.0pt;">
<img src="/html/2207.02598/assets/grad-241003s0n4--b256acc1mdo0n4e8r241003wt0.00-1.00-0.70p0-mlp1sigmoid0-s1-e50figGd.png" id="S6.F6.sf2.1.1.1.1.1.1.1.g1" class="ltx_graphics ltx_img_landscape" width="97" height="23" alt="Refer to caption"><span id="S6.F6.sf2.1.1.1.1.1.1.1.1" class="ltx_text" style="font-size:50%;"></span>
</td>
</tr>
</table>
</td>
</tr>
<tr id="S6.F6.sf2.2.2" class="ltx_tr">
<td id="S6.F6.sf2.2.2.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.25pt 0.0pt;">
<table id="S6.F6.sf2.2.2.1.1.1" class="ltx_tabular ltx_minipage ltx_align_middle" style="width:433.6pt;">
<tr id="S6.F6.sf2.2.2.1.1.1.2" class="ltx_tr">
<td id="S6.F6.sf2.2.2.1.1.1.2.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.25pt 0.0pt;"><span id="S6.F6.sf2.2.2.1.1.1.2.1.1" class="ltx_text" style="font-size:50%;"></span></td>
</tr>
<tr id="S6.F6.sf2.2.2.1.1.1.1" class="ltx_tr">
<td id="S6.F6.sf2.2.2.1.1.1.1.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.25pt 0.0pt;">
<img src="/html/2207.02598/assets/grad-241003s0n8--b256acc1mdo0n8e8r241003wt0.00-1.00-0.65p0-mlp1sigmoid0-s1-e50figGd.png" id="S6.F6.sf2.2.2.1.1.1.1.1.g1" class="ltx_graphics ltx_img_landscape" width="97" height="12" alt="Refer to caption"><span id="S6.F6.sf2.2.2.1.1.1.1.1.1" class="ltx_text" style="font-size:50%;"></span>
</td>
</tr>
</table>
</td>
</tr>
<tr id="S6.F6.sf2.3.3" class="ltx_tr">
<td id="S6.F6.sf2.3.3.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.25pt 0.0pt;">
<table id="S6.F6.sf2.3.3.1.1.1" class="ltx_tabular ltx_minipage ltx_align_middle" style="width:433.6pt;">
<tr id="S6.F6.sf2.3.3.1.1.1.2" class="ltx_tr">
<td id="S6.F6.sf2.3.3.1.1.1.2.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.25pt 0.0pt;"><span id="S6.F6.sf2.3.3.1.1.1.2.1.1" class="ltx_text" style="font-size:50%;"></span></td>
</tr>
<tr id="S6.F6.sf2.3.3.1.1.1.1" class="ltx_tr">
<td id="S6.F6.sf2.3.3.1.1.1.1.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.25pt 0.0pt;">
<img src="/html/2207.02598/assets/grad-241003s0n16--b256acc1mdo0n16e8r241003wt0.00-1.00-0.55p0-mlp1sigmoid0-s1-e50figGd.png" id="S6.F6.sf2.3.3.1.1.1.1.1.g1" class="ltx_graphics ltx_img_landscape" width="97" height="8" alt="Refer to caption"><span id="S6.F6.sf2.3.3.1.1.1.1.1.1" class="ltx_text" style="font-size:50%;"></span>
</td>
</tr>
</table>
</td>
</tr>
</table>
<figcaption class="ltx_caption ltx_centering" style="font-size:50%;"><span class="ltx_tag ltx_tag_figure"><span id="S6.F6.sf2.7.1.1" class="ltx_text" style="font-size:180%;">(b)</span> </span><span id="S6.F6.sf2.8.2" class="ltx_text" style="font-size:140%;">Independence produces distinct gradients, but many models are needed to discover new features and they are sensitive to isolated pixels.</span></figcaption>
</figure>
</div>
<div class="ltx_flex_cell ltx_flex_size_4">
<figure id="S6.F6.sf3" class="ltx_figure ltx_figure_panel ltx_align_center">
<table id="S6.F6.sf3.3" class="ltx_tabular ltx_centering ltx_align_middle">
<tr id="S6.F6.sf3.3.4" class="ltx_tr">
<td id="S6.F6.sf3.3.4.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.25pt 0.0pt;"><span id="S6.F6.sf3.3.4.1.1" class="ltx_text ltx_font_bold" style="font-size:50%;">+ On-manifold</span></td>
</tr>
<tr id="S6.F6.sf3.1.1" class="ltx_tr">
<td id="S6.F6.sf3.1.1.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.25pt 0.0pt;">
<table id="S6.F6.sf3.1.1.1.1.1" class="ltx_tabular ltx_minipage ltx_align_middle" style="width:433.6pt;">
<tr id="S6.F6.sf3.1.1.1.1.1.2" class="ltx_tr">
<td id="S6.F6.sf3.1.1.1.1.1.2.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.25pt 0.0pt;"><span id="S6.F6.sf3.1.1.1.1.1.2.1.1" class="ltx_text" style="font-size:50%;"></span></td>
</tr>
<tr id="S6.F6.sf3.1.1.1.1.1.1" class="ltx_tr">
<td id="S6.F6.sf3.1.1.1.1.1.1.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.25pt 0.0pt;">
<img src="/html/2207.02598/assets/grad-5341003s0n4vae514--b256acc1mdo0n4e8r5341003wt0.00-1.00-1.00-0.00p0-mlp1sigmoid0-s1-e50figGd.png" id="S6.F6.sf3.1.1.1.1.1.1.1.g1" class="ltx_graphics ltx_img_landscape" width="97" height="23" alt="Refer to caption"><span id="S6.F6.sf3.1.1.1.1.1.1.1.1" class="ltx_text" style="font-size:50%;"></span>
</td>
</tr>
</table>
</td>
</tr>
<tr id="S6.F6.sf3.2.2" class="ltx_tr">
<td id="S6.F6.sf3.2.2.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.25pt 0.0pt;">
<table id="S6.F6.sf3.2.2.1.1.1" class="ltx_tabular ltx_minipage ltx_align_middle" style="width:433.6pt;">
<tr id="S6.F6.sf3.2.2.1.1.1.2" class="ltx_tr">
<td id="S6.F6.sf3.2.2.1.1.1.2.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.25pt 0.0pt;"><span id="S6.F6.sf3.2.2.1.1.1.2.1.1" class="ltx_text" style="font-size:50%;"></span></td>
</tr>
<tr id="S6.F6.sf3.2.2.1.1.1.1" class="ltx_tr">
<td id="S6.F6.sf3.2.2.1.1.1.1.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.25pt 0.0pt;">
<img src="/html/2207.02598/assets/grad-5341003s0n8vae514--b256acc1mdo0n8e8r5341003wt0.00-1.00-1.00-0.00p0-mlp1sigmoid0-s1-e50figGd.png" id="S6.F6.sf3.2.2.1.1.1.1.1.g1" class="ltx_graphics ltx_img_landscape" width="97" height="12" alt="Refer to caption"><span id="S6.F6.sf3.2.2.1.1.1.1.1.1" class="ltx_text" style="font-size:50%;"></span>
</td>
</tr>
</table>
</td>
</tr>
<tr id="S6.F6.sf3.3.3" class="ltx_tr">
<td id="S6.F6.sf3.3.3.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.25pt 0.0pt;">
<table id="S6.F6.sf3.3.3.1.1.1" class="ltx_tabular ltx_minipage ltx_align_middle" style="width:433.6pt;">
<tr id="S6.F6.sf3.3.3.1.1.1.2" class="ltx_tr">
<td id="S6.F6.sf3.3.3.1.1.1.2.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.25pt 0.0pt;"><span id="S6.F6.sf3.3.3.1.1.1.2.1.1" class="ltx_text" style="font-size:50%;"></span></td>
</tr>
<tr id="S6.F6.sf3.3.3.1.1.1.1" class="ltx_tr">
<td id="S6.F6.sf3.3.3.1.1.1.1.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.25pt 0.0pt;">
<img src="/html/2207.02598/assets/grad-5341003s0n12vae514--b256acc1mdo0n12e8r5341003wt0.00-1.00-1.00-0.00p0-mlp1sigmoid0-s1-e50figGd.png" id="S6.F6.sf3.3.3.1.1.1.1.1.g1" class="ltx_graphics ltx_img_landscape" width="97" height="8" alt="Refer to caption"><span id="S6.F6.sf3.3.3.1.1.1.1.1.1" class="ltx_text" style="font-size:50%;"></span>
</td>
</tr>
</table>
</td>
</tr>
</table>
<figcaption class="ltx_caption ltx_centering" style="font-size:50%;"><span class="ltx_tag ltx_tag_figure"><span id="S6.F6.sf3.7.1.1" class="ltx_text" style="font-size:180%;">(c)</span> </span><span id="S6.F6.sf3.8.2" class="ltx_text" style="font-size:140%;">The on-manifold constraint forces gradients to align with natural variations of the data. Accuracy is virtually identical to the baseline.</span></figcaption>
</figure>
</div>
<div class="ltx_flex_cell ltx_flex_size_4">
<figure id="S6.F6.sf4" class="ltx_figure ltx_figure_panel ltx_align_center">
<table id="S6.F6.sf4.3" class="ltx_tabular ltx_centering ltx_align_middle">
<tr id="S6.F6.sf4.3.4" class="ltx_tr">
<td id="S6.F6.sf4.3.4.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.25pt 0.0pt;"><span id="S6.F6.sf4.3.4.1.1" class="ltx_text ltx_font_bold" style="font-size:50%;">+ Indep. + on-manifold</span></td>
</tr>
<tr id="S6.F6.sf4.1.1" class="ltx_tr">
<td id="S6.F6.sf4.1.1.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.25pt 0.0pt;">
<table id="S6.F6.sf4.1.1.1.1.1" class="ltx_tabular ltx_minipage ltx_align_middle" style="width:433.6pt;">
<tr id="S6.F6.sf4.1.1.1.1.1.2" class="ltx_tr">
<td id="S6.F6.sf4.1.1.1.1.1.2.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.25pt 0.0pt;"><span id="S6.F6.sf4.1.1.1.1.1.2.1.1" class="ltx_text" style="font-size:50%;"></span></td>
</tr>
<tr id="S6.F6.sf4.1.1.1.1.1.1" class="ltx_tr">
<td id="S6.F6.sf4.1.1.1.1.1.1.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.25pt 0.0pt;">
<img src="/html/2207.02598/assets/grad-5341003s0n4vae514--b256acc1mdo0n4e8r5341003wt0.00-1.00-1.00-8.00p0-mlp1sigmoid0-s1-e50figGd.png" id="S6.F6.sf4.1.1.1.1.1.1.1.g1" class="ltx_graphics ltx_img_landscape" width="97" height="23" alt="Refer to caption"><span id="S6.F6.sf4.1.1.1.1.1.1.1.1" class="ltx_text" style="font-size:50%;"></span>
</td>
</tr>
</table>
</td>
</tr>
<tr id="S6.F6.sf4.2.2" class="ltx_tr">
<td id="S6.F6.sf4.2.2.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.25pt 0.0pt;">
<table id="S6.F6.sf4.2.2.1.1.1" class="ltx_tabular ltx_minipage ltx_align_middle" style="width:433.6pt;">
<tr id="S6.F6.sf4.2.2.1.1.1.2" class="ltx_tr">
<td id="S6.F6.sf4.2.2.1.1.1.2.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.25pt 0.0pt;"><span id="S6.F6.sf4.2.2.1.1.1.2.1.1" class="ltx_text" style="font-size:50%;"></span></td>
</tr>
<tr id="S6.F6.sf4.2.2.1.1.1.1" class="ltx_tr">
<td id="S6.F6.sf4.2.2.1.1.1.1.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.25pt 0.0pt;">
<img src="/html/2207.02598/assets/grad-5341003s0n8vae514--b256acc1mdo0n8e8r5341003wt0.00-1.00-1.00-0.60p0-mlp1sigmoid0normMaxThreshold0.01-continue-randFill-s1-e50figGd.png" id="S6.F6.sf4.2.2.1.1.1.1.1.g1" class="ltx_graphics ltx_img_landscape" width="97" height="12" alt="Refer to caption"><span id="S6.F6.sf4.2.2.1.1.1.1.1.1" class="ltx_text" style="font-size:50%;"></span>
</td>
</tr>
</table>
</td>
</tr>
<tr id="S6.F6.sf4.3.3" class="ltx_tr">
<td id="S6.F6.sf4.3.3.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.25pt 0.0pt;">
<table id="S6.F6.sf4.3.3.1.1.1" class="ltx_tabular ltx_minipage ltx_align_middle" style="width:433.6pt;">
<tr id="S6.F6.sf4.3.3.1.1.1.2" class="ltx_tr">
<td id="S6.F6.sf4.3.3.1.1.1.2.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.25pt 0.0pt;"><span id="S6.F6.sf4.3.3.1.1.1.2.1.1" class="ltx_text" style="font-size:50%;"></span></td>
</tr>
<tr id="S6.F6.sf4.3.3.1.1.1.1" class="ltx_tr">
<td id="S6.F6.sf4.3.3.1.1.1.1.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.25pt 0.0pt;">
<img src="/html/2207.02598/assets/grad-5341003s0n12vae514--b256acc1mdo0n12e8r5341003wt0.00-1.00-1.00-0.60p0-mlp1sigmoid0-s1-e50figGd.png" id="S6.F6.sf4.3.3.1.1.1.1.1.g1" class="ltx_graphics ltx_img_landscape" width="97" height="8" alt="Refer to caption"><span id="S6.F6.sf4.3.3.1.1.1.1.1.1" class="ltx_text" style="font-size:50%;"></span>
</td>
</tr>
</table>
</td>
</tr>
</table>
<figcaption class="ltx_caption ltx_centering" style="font-size:50%;"><span class="ltx_tag ltx_tag_figure"><span id="S6.F6.sf4.7.1.1" class="ltx_text" style="font-size:180%;">(d)</span> </span><span id="S6.F6.sf4.8.2" class="ltx_text" style="font-size:140%;">With both constraints, <span id="S6.F6.sf4.8.2.1" class="ltx_text ltx_font_bold">we learn semantically relevant features in all image regions</span> with as few as 4 models.</span></figcaption>
</figure>
</div>
</div>
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span id="S6.F6.3.1.1" class="ltx_text" style="font-size:90%;">Figure 6</span>: </span><span id="S6.F6.4.2" class="ltx_text" style="font-size:90%;">Input gradients for a random test image from the collages dataset.
It is remarkable that, with the proposed method (d) <span id="S6.F6.4.2.1" class="ltx_text ltx_font_bold">image regions emerge as meaningful features without any inductive bias for spatial locality</span> such as convolutions (models in these experiments are fully-connected MLPs).
</span></figcaption>
</figure>
<div id="S6.SS1.p8" class="ltx_para ltx_noindent">
<p id="S6.SS1.p8.1" class="ltx_p"><span id="S6.SS1.p8.1.1" class="ltx_text ltx_font_bold">Fine-tuning.</span> 
We report the accuracy of models fine-tuned on masked inputs as proposed in Section <a href="#S4.SS3" title="4.3 Fine-tuning ‣ 4 Proposed method ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">4.3</span></a>.
This optional step relaxes the independence constraint to maximize each model’s predictive performance.
The accuracy jumps significantly and almost reaches the upper-bound on each test set (Table <a href="#S6.T1" title="Table 1 ‣ 6.1 Experiments on controlled data: collages ‣ 6 Experiments ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">1</span></a>).
We experimented with relaxing both the independence and on-manifold constraints. Disabling the former has a significant effect. But the latter has no significant effect on accuracy on its own as expected and discussed in Section <a href="#S4.SS2" title="4.2 On-manifold constraint ‣ 4 Proposed method ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">4.2</span></a>.</p>
</div>
<div id="S6.SS1.p9" class="ltx_para ltx_noindent">
<p id="S6.SS1.p9.1" class="ltx_p"><span id="S6.SS1.p9.1.1" class="ltx_text ltx_font_bold">Distilling multiple models into one.</span> 
We report the performance of combinations of features described in Section <a href="#S4.SS4" title="4.4 Distilling multiple models into one ‣ 4 Proposed method ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">4.4</span></a>.
This procedure is most effective after training a large number of models (24 here).
This is unsurprising since models then discover finer-grained features.
Each combination selects features relevant to only one specific tile to achieve near-maximal accuracy on the test set of that tile.
Simple traditional ensembling of models completely failed in our experiments.</p>
</div>
<div id="S6.SS1.p10" class="ltx_para ltx_noindent">
<p id="S6.SS1.p10.1" class="ltx_p"><span id="S6.SS1.p10.1.1" class="ltx_text ltx_font_bold">Comparison with existing methods.</span> 
No other method reported in Table <a href="#S6.T1" title="Table 1 ‣ 6.1 Experiments on controlled data: collages ‣ 6 Experiments ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">1</span></a> performed well on this dataset.
The method of Teney <em id="S6.SS1.p10.1.2" class="ltx_emph ltx_font_italic">et al</em>.<span id="S6.SS1.p10.1.3" class="ltx_text"></span> <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib72" title="" class="ltx_ref">72</a>]</cite> is technically the most similar to ours, but it requires training a much larger number of models and still achieves much lower accuracy.
While all experiments of this section used a model taking raw pixels as input, we repeated the whole evaluation using a shared, frozen ResNet to extract features in Appendix <a href="#A6" title="Appendix F Additional results: collages ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">F</span></a>.
This implementation is computationally appealing for larger-scale applications, and gave essentially similar findings with higher overall accuracy thanks to the deeper architecture.</p>
</div>
<div id="S6.SS1.p11" class="ltx_para">
<span id="S6.SS1.p11.1" class="ltx_ERROR undefined">\floatsetup</span>
<p id="S6.SS1.p11.2" class="ltx_p">[table]capposition=beside,capbesideposition=center,right,floatwidth=0.55capbesidewidth=0.4</p>
</div>
<figure id="S6.T1" class="ltx_table">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table"><span id="S6.T1.12.1.1" class="ltx_text" style="font-size:90%;">Table 1</span>: </span><span id="S6.T1.13.2" class="ltx_text" style="font-size:90%;">
Accuracy on <span id="S6.T1.13.2.1" class="ltx_text ltx_font_italic">collages</span> of existing and proposed methods (8 models per method unless specified).
The 4 test sets simulate different OOD conditions: only one tile in each set is correlated with the labels.
<span id="S6.T1.13.2.2" class="ltx_text ltx_font_bold">Standard training only learns a fraction of predictive features</span>.
Existing methods cannot do better than chance except on MNIST, or they require training a large number of models.
Ours learns a variety of features and give near-optimal predictions on every test set (last row).
</span></figcaption>
<table id="S6.T1.8" class="ltx_tabular ltx_centering ltx_align_middle">
<tr id="S6.T1.8.9" class="ltx_tr">
<td id="S6.T1.8.9.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify ltx_border_tt" style="padding-left:0.0pt;padding-right:0.0pt;">
<span id="S6.T1.8.9.1.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T1.8.9.1.1.1" class="ltx_p"><span id="S6.T1.8.9.1.1.1.1" class="ltx_text ltx_font_bold" style="font-size:50%;">Collages</span><span id="S6.T1.8.9.1.1.1.2" class="ltx_text" style="font-size:50%;"> dataset (accuracy in %)</span></span>
</span>
</td>
<td id="S6.T1.8.9.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_tt" style="padding-left:0.0pt;padding-right:0.0pt;" colspan="4"><span id="S6.T1.8.9.2.1" class="ltx_text" style="font-size:50%;">Best model on</span></td>
<td id="S6.T1.8.9.3" class="ltx_td ltx_border_tt" style="padding-left:0.0pt;padding-right:0.0pt;"></td>
</tr>
<tr id="S6.T1.8.10" class="ltx_tr">
<td id="S6.T1.8.10.1" class="ltx_td ltx_nopad_l ltx_nopad_r" style="padding-left:0.0pt;padding-right:0.0pt;"></td>
<td id="S6.T1.8.10.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;">
<div id="S6.T1.8.10.2.1" class="ltx_inline-block ltx_transformed_outer" style="width:3.4pt;height:16.5pt;vertical-align:-0.0pt;"><span class="ltx_transformed_inner" style="width:16.5pt;transform:translate(-6.56pt,-6.56pt) rotate(-90deg) ;">
<p id="S6.T1.8.10.2.1.1" class="ltx_p"><span id="S6.T1.8.10.2.1.1.1" class="ltx_text" style="font-size:50%;">MNIST</span></p>
</span></div>
</td>
<td id="S6.T1.8.10.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;">
<div id="S6.T1.8.10.3.1" class="ltx_inline-block ltx_transformed_outer" style="width:3.4pt;height:19pt;vertical-align:-0.0pt;"><span class="ltx_transformed_inner" style="width:19.0pt;transform:translate(-7.81pt,-7.81pt) rotate(-90deg) ;">
<p id="S6.T1.8.10.3.1.1" class="ltx_p"><span id="S6.T1.8.10.3.1.1.1" class="ltx_text" style="font-size:50%;">SVHN</span></p>
</span></div>
</td>
<td id="S6.T1.8.10.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;">
<div id="S6.T1.8.10.4.1" class="ltx_inline-block ltx_transformed_outer" style="width:3.5pt;height:19pt;vertical-align:-0.0pt;"><span class="ltx_transformed_inner" style="width:19.0pt;transform:translate(-7.76pt,-7.76pt) rotate(-90deg) ;">
<p id="S6.T1.8.10.4.1.1" class="ltx_p"><span id="S6.T1.8.10.4.1.1.1" class="ltx_text" style="font-size:50%;">Fashion</span><span id="S6.T1.8.10.4.1.1.2" class="ltx_text ltx_phantom" style="font-size:50%;"><span style="visibility:hidden">e</span></span></p>
</span></div>
</td>
<td id="S6.T1.8.10.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;">
<div id="S6.T1.8.10.5.1" class="ltx_inline-block ltx_transformed_outer" style="width:3.4pt;height:24.4pt;vertical-align:-0.0pt;"><span class="ltx_transformed_inner" style="width:24.4pt;transform:translate(-10.51pt,-10.51pt) rotate(-90deg) ;">
<p id="S6.T1.8.10.5.1.1" class="ltx_p"><span id="S6.T1.8.10.5.1.1.1" class="ltx_text" style="font-size:50%;">CIFAR-10</span><span id="S6.T1.8.10.5.1.1.2" class="ltx_text ltx_phantom" style="font-size:50%;"><span style="visibility:hidden">e</span></span></p>
</span></div>
</td>
<td id="S6.T1.8.10.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;">
<div id="S6.T1.8.10.6.1" class="ltx_inline-block ltx_transformed_outer" style="width:4.4pt;height:17.7pt;vertical-align:-0.0pt;"><span class="ltx_transformed_inner" style="width:17.7pt;transform:translate(-6.63pt,-6.15pt) rotate(-90deg) ;">
<p id="S6.T1.8.10.6.1.1" class="ltx_p"><span id="S6.T1.8.10.6.1.1.1" class="ltx_text" style="font-size:50%;">Average</span></p>
</span></div>
</td>
</tr>
<tr id="S6.T1.8.11" class="ltx_tr">
<td id="S6.T1.8.11.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;">
<span id="S6.T1.8.11.1.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T1.8.11.1.1.1" class="ltx_p"><span id="S6.T1.8.11.1.1.1.1" class="ltx_text" style="font-size:50%;">Upper bound (training on test-domain data)</span></span>
</span>
</td>
<td id="S6.T1.8.11.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.11.2.1" class="ltx_text" style="font-size:50%;">99.9</span></td>
<td id="S6.T1.8.11.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.11.3.1" class="ltx_text" style="font-size:50%;">92.4</span></td>
<td id="S6.T1.8.11.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.11.4.1" class="ltx_text" style="font-size:50%;">80.8</span></td>
<td id="S6.T1.8.11.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.11.5.1" class="ltx_text" style="font-size:50%;">68.6</span></td>
<td id="S6.T1.8.11.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.11.6.1" class="ltx_text" style="font-size:50%;">85.5</span></td>
</tr>
<tr id="S6.T1.8.12" class="ltx_tr">
<td id="S6.T1.8.12.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;">
<span id="S6.T1.8.12.1.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T1.8.12.1.1.1" class="ltx_p"><span id="S6.T1.8.12.1.1.1.1" class="ltx_text" style="font-size:50%;">ERM Baseline</span></span>
</span>
</td>
<td id="S6.T1.8.12.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.12.2.1" class="ltx_text" style="font-size:50%;">99.8</span></td>
<td id="S6.T1.8.12.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.12.3.1" class="ltx_text" style="font-size:50%;">50.0</span></td>
<td id="S6.T1.8.12.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.12.4.1" class="ltx_text" style="font-size:50%;">50.0</span></td>
<td id="S6.T1.8.12.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.12.5.1" class="ltx_text" style="font-size:50%;">50.0</span></td>
<td id="S6.T1.8.12.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.12.6.1" class="ltx_text" style="font-size:50%;">62.5</span></td>
</tr>
<tr id="S6.T1.8.13" class="ltx_tr">
<td id="S6.T1.8.13.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding-left:0.0pt;padding-right:0.0pt;">
<span id="S6.T1.8.13.1.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T1.8.13.1.1.1" class="ltx_p"><span id="S6.T1.8.13.1.1.1.1" class="ltx_text" style="font-size:50%;">Spectral decoupling </span><cite class="ltx_cite ltx_citemacro_cite"><span id="S6.T1.8.13.1.1.1.2.1" class="ltx_text" style="font-size:50%;">[</span><a href="#bib.bib55" title="" class="ltx_ref">55</a><span id="S6.T1.8.13.1.1.1.3.2" class="ltx_text" style="font-size:50%;">]</span></cite></span>
</span>
</td>
<td id="S6.T1.8.13.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.13.2.1" class="ltx_text" style="font-size:50%;">99.9</span></td>
<td id="S6.T1.8.13.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.13.3.1" class="ltx_text" style="font-size:50%;">49.8</span></td>
<td id="S6.T1.8.13.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.13.4.1" class="ltx_text" style="font-size:50%;">50.6</span></td>
<td id="S6.T1.8.13.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.13.5.1" class="ltx_text" style="font-size:50%;">49.9</span></td>
<td id="S6.T1.8.13.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.13.6.1" class="ltx_text" style="font-size:50%;">62.5</span></td>
</tr>
<tr id="S6.T1.8.14" class="ltx_tr">
<td id="S6.T1.8.14.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding-left:0.0pt;padding-right:0.0pt;">
<span id="S6.T1.8.14.1.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T1.8.14.1.1.1" class="ltx_p"><span id="S6.T1.8.14.1.1.1.1" class="ltx_text" style="font-size:50%;">With penalty on L1 norm of gradients</span></span>
</span>
</td>
<td id="S6.T1.8.14.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.14.2.1" class="ltx_text" style="font-size:50%;">98.5</span></td>
<td id="S6.T1.8.14.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.14.3.1" class="ltx_text" style="font-size:50%;">49.6</span></td>
<td id="S6.T1.8.14.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.14.4.1" class="ltx_text" style="font-size:50%;">50.5</span></td>
<td id="S6.T1.8.14.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.14.5.1" class="ltx_text" style="font-size:50%;">50.0</span></td>
<td id="S6.T1.8.14.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.14.6.1" class="ltx_text" style="font-size:50%;">62.1</span></td>
</tr>
<tr id="S6.T1.8.15" class="ltx_tr">
<td id="S6.T1.8.15.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding-left:0.0pt;padding-right:0.0pt;">
<span id="S6.T1.8.15.1.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T1.8.15.1.1.1" class="ltx_p"><span id="S6.T1.8.15.1.1.1.1" class="ltx_text" style="font-size:50%;">With penalty on L2 norm of gradients </span><cite class="ltx_cite ltx_citemacro_cite"><span id="S6.T1.8.15.1.1.1.2.1" class="ltx_text" style="font-size:50%;">[</span><a href="#bib.bib30" title="" class="ltx_ref">30</a><span id="S6.T1.8.15.1.1.1.3.2" class="ltx_text" style="font-size:50%;">]</span></cite></span>
</span>
</td>
<td id="S6.T1.8.15.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.15.2.1" class="ltx_text" style="font-size:50%;">96.6</span></td>
<td id="S6.T1.8.15.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.15.3.1" class="ltx_text" style="font-size:50%;">52.1</span></td>
<td id="S6.T1.8.15.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.15.4.1" class="ltx_text" style="font-size:50%;">52.3</span></td>
<td id="S6.T1.8.15.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.15.5.1" class="ltx_text" style="font-size:50%;">54.3</span></td>
<td id="S6.T1.8.15.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.15.6.1" class="ltx_text" style="font-size:50%;">63.8</span></td>
</tr>
<tr id="S6.T1.8.16" class="ltx_tr">
<td id="S6.T1.8.16.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding-left:0.0pt;padding-right:0.0pt;">
<span id="S6.T1.8.16.1.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T1.8.16.1.1.1" class="ltx_p"><span id="S6.T1.8.16.1.1.1.1" class="ltx_text" style="font-size:50%;">Input dropout (best ratio: 0.9)</span></span>
</span>
</td>
<td id="S6.T1.8.16.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.16.2.1" class="ltx_text" style="font-size:50%;">97.4</span></td>
<td id="S6.T1.8.16.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.16.3.1" class="ltx_text" style="font-size:50%;">50.7</span></td>
<td id="S6.T1.8.16.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.16.4.1" class="ltx_text" style="font-size:50%;">56.1</span></td>
<td id="S6.T1.8.16.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.16.5.1" class="ltx_text" style="font-size:50%;">52.1</span></td>
<td id="S6.T1.8.16.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.16.6.1" class="ltx_text" style="font-size:50%;">64.1</span></td>
</tr>
<tr id="S6.T1.8.17" class="ltx_tr">
<td id="S6.T1.8.17.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding-left:0.0pt;padding-right:0.0pt;">
<span id="S6.T1.8.17.1.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T1.8.17.1.1.1" class="ltx_p"><span id="S6.T1.8.17.1.1.1.1" class="ltx_text" style="font-size:50%;">Independence loss (cosine similarity) </span><cite class="ltx_cite ltx_citemacro_cite"><span id="S6.T1.8.17.1.1.1.2.1" class="ltx_text" style="font-size:50%;">[</span><a href="#bib.bib60" title="" class="ltx_ref">60</a><span id="S6.T1.8.17.1.1.1.3.2" class="ltx_text" style="font-size:50%;">]</span></cite></span>
</span>
</td>
<td id="S6.T1.8.17.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.17.2.1" class="ltx_text" style="font-size:50%;">99.7</span></td>
<td id="S6.T1.8.17.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.17.3.1" class="ltx_text" style="font-size:50%;">50.4</span></td>
<td id="S6.T1.8.17.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.17.4.1" class="ltx_text" style="font-size:50%;">51.5</span></td>
<td id="S6.T1.8.17.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.17.5.1" class="ltx_text" style="font-size:50%;">50.2</span></td>
<td id="S6.T1.8.17.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.17.6.1" class="ltx_text" style="font-size:50%;">63.0</span></td>
</tr>
<tr id="S6.T1.8.18" class="ltx_tr">
<td id="S6.T1.8.18.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding-left:0.0pt;padding-right:0.0pt;">
<span id="S6.T1.8.18.1.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T1.8.18.1.1.1" class="ltx_p"><span id="S6.T1.8.18.1.1.1.1" class="ltx_text" style="font-size:50%;">Independence loss (dot product) </span><cite class="ltx_cite ltx_citemacro_cite"><span id="S6.T1.8.18.1.1.1.2.1" class="ltx_text" style="font-size:50%;">[</span><a href="#bib.bib72" title="" class="ltx_ref">72</a><span id="S6.T1.8.18.1.1.1.3.2" class="ltx_text" style="font-size:50%;">]</span></cite></span>
</span>
</td>
<td id="S6.T1.8.18.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.18.2.1" class="ltx_text" style="font-size:50%;">99.5</span></td>
<td id="S6.T1.8.18.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.18.3.1" class="ltx_text" style="font-size:50%;">53.5</span></td>
<td id="S6.T1.8.18.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.18.4.1" class="ltx_text" style="font-size:50%;">53.3</span></td>
<td id="S6.T1.8.18.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.18.5.1" class="ltx_text" style="font-size:50%;">50.5</span></td>
<td id="S6.T1.8.18.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.18.6.1" class="ltx_text" style="font-size:50%;">64.2</span></td>
</tr>
<tr id="S6.T1.8.19" class="ltx_tr">
<td id="S6.T1.8.19.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;">
<span id="S6.T1.8.19.1.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T1.8.19.1.1.1" class="ltx_p"><span id="S6.T1.8.19.1.1.1.1" class="ltx_text" style="font-size:50%;">With many more models</span></span>
</span>
</td>
<td id="S6.T1.8.19.2" class="ltx_td ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;"></td>
<td id="S6.T1.8.19.3" class="ltx_td ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;"></td>
<td id="S6.T1.8.19.4" class="ltx_td ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;"></td>
<td id="S6.T1.8.19.5" class="ltx_td ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;"></td>
<td id="S6.T1.8.19.6" class="ltx_td ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;"></td>
</tr>
<tr id="S6.T1.8.20" class="ltx_tr">
<td id="S6.T1.8.20.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding-left:0.0pt;padding-right:0.0pt;">
<span id="S6.T1.8.20.1.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T1.8.20.1.1.1" class="ltx_p"><span id="S6.T1.8.20.1.1.1.1" class="ltx_text" style="font-size:50%;">Independence loss (cosine similarity), </span><span id="S6.T1.8.20.1.1.1.2" class="ltx_text ltx_framed ltx_framed_underline" style="font-size:50%;">1024</span><span id="S6.T1.8.20.1.1.1.3" class="ltx_text" style="font-size:50%;"> models</span></span>
</span>
</td>
<td id="S6.T1.8.20.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.20.2.1" class="ltx_text" style="font-size:50%;">99.5</span></td>
<td id="S6.T1.8.20.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.20.3.1" class="ltx_text" style="font-size:50%;">58.1</span></td>
<td id="S6.T1.8.20.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.20.4.1" class="ltx_text" style="font-size:50%;">66.8</span></td>
<td id="S6.T1.8.20.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.20.5.1" class="ltx_text" style="font-size:50%;">63.0</span></td>
<td id="S6.T1.8.20.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.20.6.1" class="ltx_text" style="font-size:50%;">71.9</span></td>
</tr>
<tr id="S6.T1.8.21" class="ltx_tr">
<td id="S6.T1.8.21.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding-left:0.0pt;padding-right:0.0pt;">
<span id="S6.T1.8.21.1.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T1.8.21.1.1.1" class="ltx_p"><span id="S6.T1.8.21.1.1.1.1" class="ltx_text" style="font-size:50%;">Independence loss (dot product), </span><span id="S6.T1.8.21.1.1.1.2" class="ltx_text ltx_framed ltx_framed_underline" style="font-size:50%;">128</span><span id="S6.T1.8.21.1.1.1.3" class="ltx_text" style="font-size:50%;"> models</span></span>
</span>
</td>
<td id="S6.T1.8.21.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.21.2.1" class="ltx_text" style="font-size:50%;">98.7</span></td>
<td id="S6.T1.8.21.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.21.3.1" class="ltx_text" style="font-size:50%;">84.9</span></td>
<td id="S6.T1.8.21.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.21.4.1" class="ltx_text" style="font-size:50%;">71.6</span></td>
<td id="S6.T1.8.21.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.21.5.1" class="ltx_text" style="font-size:50%;">61.5</span></td>
<td id="S6.T1.8.21.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.21.6.1" class="ltx_text" style="font-size:50%;">79.2</span></td>
</tr>
<tr id="S6.T1.8.22" class="ltx_tr">
<td id="S6.T1.8.22.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;">
<span id="S6.T1.8.22.1.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T1.8.22.1.1.1" class="ltx_p"><span id="S6.T1.8.22.1.1.1.1" class="ltx_text" style="font-size:50%;">Proposed method (only 8 models)</span></span>
</span>
</td>
<td id="S6.T1.8.22.2" class="ltx_td ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;"></td>
<td id="S6.T1.8.22.3" class="ltx_td ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;"></td>
<td id="S6.T1.8.22.4" class="ltx_td ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;"></td>
<td id="S6.T1.8.22.5" class="ltx_td ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;"></td>
<td id="S6.T1.8.22.6" class="ltx_td ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;"></td>
</tr>
<tr id="S6.T1.8.23" class="ltx_tr">
<td id="S6.T1.8.23.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding-left:0.0pt;padding-right:0.0pt;">
<span id="S6.T1.8.23.1.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T1.8.23.1.1.1" class="ltx_p"><span id="S6.T1.8.23.1.1.1.1" class="ltx_text" style="font-size:50%;">Independence + on-manifold constraints, PCA</span></span>
</span>
</td>
<td id="S6.T1.8.23.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.23.2.1" class="ltx_text" style="font-size:50%;">97.3</span></td>
<td id="S6.T1.8.23.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.23.3.1" class="ltx_text" style="font-size:50%;">69.8</span></td>
<td id="S6.T1.8.23.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.23.4.1" class="ltx_text" style="font-size:50%;">62.2</span></td>
<td id="S6.T1.8.23.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.23.5.1" class="ltx_text" style="font-size:50%;">60.0</span></td>
<td id="S6.T1.8.23.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.23.6.1" class="ltx_text" style="font-size:50%;">72.3</span></td>
</tr>
<tr id="S6.T1.1.1" class="ltx_tr">
<td id="S6.T1.1.1.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding-left:0.0pt;padding-right:0.0pt;">
<span id="S6.T1.1.1.1.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T1.1.1.1.1.1" class="ltx_p"><span id="S6.T1.1.1.1.1.1.1" class="ltx_text" style="font-size:50%;">Independence + on-manifold constraints, VAE (</span><sup id="S6.T1.1.1.1.1.1.2" class="ltx_sup"><span id="S6.T1.1.1.1.1.1.2.1" class="ltx_text" style="font-size:50%;">∗</span></sup><span id="S6.T1.1.1.1.1.1.3" class="ltx_text" style="font-size:50%;">)</span></span>
</span>
</td>
<td id="S6.T1.1.1.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.1.1.2.1" class="ltx_text" style="font-size:50%;">96.5</span></td>
<td id="S6.T1.1.1.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.1.1.3.1" class="ltx_text" style="font-size:50%;">85.1</span></td>
<td id="S6.T1.1.1.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.1.1.4.1" class="ltx_text" style="font-size:50%;">61.1</span></td>
<td id="S6.T1.1.1.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.1.1.5.1" class="ltx_text" style="font-size:50%;">62.1</span></td>
<td id="S6.T1.1.1.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.1.1.6.1" class="ltx_text" style="font-size:50%;">76.2</span></td>
</tr>
<tr id="S6.T1.2.2" class="ltx_tr">
<td id="S6.T1.2.2.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding-left:0.0pt;padding-right:0.0pt;">
<span id="S6.T1.2.2.1.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T1.2.2.1.1.1" class="ltx_p"><span id="S6.T1.2.2.1.1.1.1" class="ltx_text" style="font-size:50%;">(</span><sup id="S6.T1.2.2.1.1.1.2" class="ltx_sup"><span id="S6.T1.2.2.1.1.1.2.1" class="ltx_text" style="font-size:50%;">∗</span></sup><span id="S6.T1.2.2.1.1.1.3" class="ltx_text" style="font-size:50%;">)  +  FT  (fine-tuning)</span></span>
</span>
</td>
<td id="S6.T1.2.2.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.2.2.2.1" class="ltx_text" style="font-size:50%;">99.7</span></td>
<td id="S6.T1.2.2.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.2.2.3.1" class="ltx_text" style="font-size:50%;">90.9</span></td>
<td id="S6.T1.2.2.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.2.2.4.1" class="ltx_text" style="font-size:50%;">81.4</span></td>
<td id="S6.T1.2.2.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.2.2.5.1" class="ltx_text" style="font-size:50%;">67.4</span></td>
<td id="S6.T1.2.2.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.2.2.6.1" class="ltx_text" style="font-size:50%;">84.8</span></td>
</tr>
<tr id="S6.T1.4.4" class="ltx_tr">
<td id="S6.T1.4.4.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding-left:0.0pt;padding-right:0.0pt;">
<span id="S6.T1.4.4.2.2" class="ltx_inline-block ltx_align_top">
<span id="S6.T1.4.4.2.2.2" class="ltx_p"><span id="S6.T1.4.4.2.2.2.1" class="ltx_text" style="font-size:50%;">(</span><sup id="S6.T1.4.4.2.2.2.2" class="ltx_sup"><span id="S6.T1.4.4.2.2.2.2.1" class="ltx_text" style="font-size:50%;">∗</span></sup><span id="S6.T1.4.4.2.2.2.3" class="ltx_text" style="font-size:50%;">)  +  FT  +  pairwise combinations (1</span><math id="S6.T1.4.4.2.2.2.m2.1" class="ltx_Math" alttext="\times" display="inline"><semantics id="S6.T1.4.4.2.2.2.m2.1a"><mo mathsize="50%" id="S6.T1.4.4.2.2.2.m2.1.1" xref="S6.T1.4.4.2.2.2.m2.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="S6.T1.4.4.2.2.2.m2.1b"><times id="S6.T1.4.4.2.2.2.m2.1.1.cmml" xref="S6.T1.4.4.2.2.2.m2.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="S6.T1.4.4.2.2.2.m2.1c">\times</annotation></semantics></math><span id="S6.T1.4.4.2.2.2.4" class="ltx_text" style="font-size:50%;">)</span></span>
</span>
</td>
<td id="S6.T1.4.4.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.4.4.3.1" class="ltx_text" style="font-size:50%;">99.9</span></td>
<td id="S6.T1.4.4.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.4.4.4.1" class="ltx_text" style="font-size:50%;">92.2</span></td>
<td id="S6.T1.4.4.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.4.4.5.1" class="ltx_text" style="font-size:50%;">79.3</span></td>
<td id="S6.T1.4.4.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.4.4.6.1" class="ltx_text" style="font-size:50%;">66.3</span></td>
<td id="S6.T1.4.4.7" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.4.4.7.1" class="ltx_text" style="font-size:50%;">84.4</span></td>
</tr>
<tr id="S6.T1.6.6" class="ltx_tr">
<td id="S6.T1.6.6.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding-left:0.0pt;padding-right:0.0pt;">
<span id="S6.T1.6.6.2.2" class="ltx_inline-block ltx_align_top">
<span id="S6.T1.6.6.2.2.2" class="ltx_p"><span id="S6.T1.6.6.2.2.2.1" class="ltx_text" style="font-size:50%;">(</span><sup id="S6.T1.6.6.2.2.2.2" class="ltx_sup"><span id="S6.T1.6.6.2.2.2.2.1" class="ltx_text" style="font-size:50%;">∗</span></sup><span id="S6.T1.6.6.2.2.2.3" class="ltx_text" style="font-size:50%;">)  +  FT  +  pairwise combinations (2</span><math id="S6.T1.6.6.2.2.2.m2.1" class="ltx_Math" alttext="\times" display="inline"><semantics id="S6.T1.6.6.2.2.2.m2.1a"><mo mathsize="50%" id="S6.T1.6.6.2.2.2.m2.1.1" xref="S6.T1.6.6.2.2.2.m2.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="S6.T1.6.6.2.2.2.m2.1b"><times id="S6.T1.6.6.2.2.2.m2.1.1.cmml" xref="S6.T1.6.6.2.2.2.m2.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="S6.T1.6.6.2.2.2.m2.1c">\times</annotation></semantics></math><span id="S6.T1.6.6.2.2.2.4" class="ltx_text" style="font-size:50%;">)</span></span>
</span>
</td>
<td id="S6.T1.6.6.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.6.6.3.1" class="ltx_text" style="font-size:50%;">99.9</span></td>
<td id="S6.T1.6.6.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.6.6.4.1" class="ltx_text" style="font-size:50%;">92.5</span></td>
<td id="S6.T1.6.6.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.6.6.5.1" class="ltx_text" style="font-size:50%;">80.2</span></td>
<td id="S6.T1.6.6.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.6.6.6.1" class="ltx_text" style="font-size:50%;">67.5</span></td>
<td id="S6.T1.6.6.7" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.6.6.7.1" class="ltx_text" style="font-size:50%;">85.0</span></td>
</tr>
<tr id="S6.T1.8.8" class="ltx_tr">
<td id="S6.T1.8.8.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify ltx_border_bb" style="padding-left:0.0pt;padding-right:0.0pt;">
<span id="S6.T1.8.8.2.2" class="ltx_inline-block ltx_align_top">
<span id="S6.T1.8.8.2.2.2" class="ltx_p"><span id="S6.T1.8.8.2.2.2.2" class="ltx_text ltx_font_bold" style="font-size:50%;">(<sup id="S6.T1.8.8.2.2.2.2.1" class="ltx_sup"><span id="S6.T1.8.8.2.2.2.2.1.1" class="ltx_text ltx_font_medium">∗</span></sup>)  +  FT  +  pairwise combinations (3<math id="S6.T1.8.8.2.2.2.2.m2.1" class="ltx_Math" alttext="\times" display="inline"><semantics id="S6.T1.8.8.2.2.2.2.m2.1a"><mo id="S6.T1.8.8.2.2.2.2.m2.1.1" xref="S6.T1.8.8.2.2.2.2.m2.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="S6.T1.8.8.2.2.2.2.m2.1b"><times id="S6.T1.8.8.2.2.2.2.m2.1.1.cmml" xref="S6.T1.8.8.2.2.2.2.m2.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="S6.T1.8.8.2.2.2.2.m2.1c">\times</annotation></semantics></math>)</span></span>
</span>
</td>
<td id="S6.T1.8.8.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_bb" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.8.3.1" class="ltx_text ltx_font_bold" style="font-size:50%;">99.9</span></td>
<td id="S6.T1.8.8.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_bb" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.8.4.1" class="ltx_text ltx_font_bold" style="font-size:50%;">92.3</span></td>
<td id="S6.T1.8.8.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_bb" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.8.5.1" class="ltx_text ltx_font_bold" style="font-size:50%;">80.8</span></td>
<td id="S6.T1.8.8.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_bb" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.8.6.1" class="ltx_text ltx_font_bold" style="font-size:50%;">68.5</span></td>
<td id="S6.T1.8.8.7" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_bb" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T1.8.8.7.1" class="ltx_text ltx_font_bold" style="font-size:50%;">85.4</span></td>
</tr>
</table>
</figure>
</section>
<section id="S6.SS2" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">6.2 </span>Experiments on real data: WILDS-Camelyon17</h3>

<div id="S6.SS2.p1" class="ltx_para ltx_noindent">
<p id="S6.SS2.p1.1" class="ltx_p"><span id="S6.SS2.p1.1.1" class="ltx_text ltx_font_bold">Dataset.</span> 
The WILDS-Camelyon17 benchmark <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib40" title="" class="ltx_ref">40</a>]</cite> provides histopathology images to classify as “<span id="S6.SS2.p1.1.2" class="ltx_text ltx_font_italic">tumor</span>” or “<span id="S6.SS2.p1.1.3" class="ltx_text ltx_font_italic">normal</span>”.
The images come from different sets of hospitals in the training, validation (val-OOD), and test splits (test-OOD).
The challenge is to learn a model that generalizes from the training hospitals to those of the test set.
The original authors of <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib40" title="" class="ltx_ref">40</a>]</cite> trained a Densenet-121 model from scratch on this data with 10 random seeds. They showed that the performance on val-OOD and test-OOD varies wildly across seeds, demonstrating that the task is severely underspecified with only the standard training images (the dataset provides additional hospital labels that could enable generalization; neither ERM nor our method uses them).</p>
</div>
<div id="S6.SS2.p2" class="ltx_para ltx_noindent">
<p id="S6.SS2.p2.1" class="ltx_p"><span id="S6.SS2.p2.1.1" class="ltx_text ltx_font_bold">Implementation of our method.</span> 
We use frozen features (last-layer activations) from one of the pretrained models from <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib40" title="" class="ltx_ref">40</a>]</cite> as input. We will show that we can recover even more variability in performance than the complete models trained on different random seeds, even while keeping the model frozen (<em id="S6.SS2.p2.1.2" class="ltx_emph ltx_font_italic">i.e</em>.<span id="S6.SS2.p2.1.3" class="ltx_text"></span> retraining only a classifier).
We first determined that the best ERM-trained classifier on frozen features is a simple linear one, rather than an MLP.
Our method simplifies in two ways with a linear classifier.
First, input gradients are equal to the classifier weights, and the proposed regularizers do not require second-order derivatives anymore during back-propagation.
Second, we found empirically that the soft on-manifold regularizer can be replaced with a hard constraint: we explicit project the input gradients onto the manifold and apply the independence regularizer on these projections, as proposed in <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib60" title="" class="ltx_ref">60</a>]</cite>.
As noted in Section <a href="#S4.SS2" title="4.2 On-manifold constraint ‣ 4 Proposed method ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">4.2</span></a>, this option completely failed in our early experiments with MLPs, but it seems viable with linear classifiers. This further simplifies the implementation and removes the hyperparameter <math id="S6.SS2.p2.1.m1.1" class="ltx_Math" alttext="{\lambda_{\textrm{manifold}}}" display="inline"><semantics id="S6.SS2.p2.1.m1.1a"><msub id="S6.SS2.p2.1.m1.1.1" xref="S6.SS2.p2.1.m1.1.1.cmml"><mi id="S6.SS2.p2.1.m1.1.1.2" xref="S6.SS2.p2.1.m1.1.1.2.cmml">λ</mi><mtext id="S6.SS2.p2.1.m1.1.1.3" xref="S6.SS2.p2.1.m1.1.1.3a.cmml">manifold</mtext></msub><annotation-xml encoding="MathML-Content" id="S6.SS2.p2.1.m1.1b"><apply id="S6.SS2.p2.1.m1.1.1.cmml" xref="S6.SS2.p2.1.m1.1.1"><csymbol cd="ambiguous" id="S6.SS2.p2.1.m1.1.1.1.cmml" xref="S6.SS2.p2.1.m1.1.1">subscript</csymbol><ci id="S6.SS2.p2.1.m1.1.1.2.cmml" xref="S6.SS2.p2.1.m1.1.1.2">𝜆</ci><ci id="S6.SS2.p2.1.m1.1.1.3a.cmml" xref="S6.SS2.p2.1.m1.1.1.3"><mtext mathsize="70%" id="S6.SS2.p2.1.m1.1.1.3.cmml" xref="S6.SS2.p2.1.m1.1.1.3">manifold</mtext></ci></apply></annotation-xml><annotation encoding="application/x-tex" id="S6.SS2.p2.1.m1.1c">{\lambda_{\textrm{manifold}}}</annotation></semantics></math>.</p>
</div>
<div id="S6.SS2.p3" class="ltx_para ltx_noindent">
<p id="S6.SS2.p3.1" class="ltx_p"><span id="S6.SS2.p3.1.1" class="ltx_text ltx_font_bold">Results.</span> 
We plot in Figure <a href="#S6.F7" title="Figure 7 ‣ 6.2 Experiments on real data: WILDS-Camelyon17 ‣ 6 Experiments ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">7</span></a> the spread of accuracies of models trained with different methods (using features from the first pretrained model from <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib40" title="" class="ltx_ref">40</a>]</cite>, see Appendix <a href="#A7" title="Appendix G Additional results: WILDS-Camelyon17 ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">G</span></a> for similar results with the others).
The <span id="S6.SS2.p3.1.2" class="ltx_text ltx_font_bold">ERM baseline</span> simply recovers the accuracy of the original complete Densenet, with essentially no variation across random seeds.
With our <span id="S6.SS2.p3.1.3" class="ltx_text ltx_font_bold">independence constraint</span>, the spread of accuracies significantly widens, both below and above the baseline.
In Figure <a href="#S6.F8" title="Figure 8 ‣ 6.2 Experiments on real data: WILDS-Camelyon17 ‣ 6 Experiments ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">8</span></a>, we see that the models span various trade-offs in accuracy on val-OOD and on test-OOD, neither of which is correlated with the accuracy on in-domain data (val-ID), thus showing evidence of underspecification.
Back to Figure <a href="#S6.F7" title="Figure 7 ‣ 6.2 Experiments on real data: WILDS-Camelyon17 ‣ 6 Experiments ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">7</span></a>, with our additional <span id="S6.SS2.p3.1.4" class="ltx_text ltx_font_bold">on-manifold constraint</span>, the best models reach higher accuracies. This also tops out when training a handful of models (about 10–14, near the intrinsic dimensionality of the data estimated at 12 with <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib57" title="" class="ltx_ref">57</a>]</cite>).
Keeping in mind that we use frozen features, these results show that the ERM-pretrained model extracts features useful for OOD performance but that are ignored by the pretrained classifier.
Similar findings were recently reported in <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib39" title="" class="ltx_ref">39</a>, <a href="#bib.bib59" title="" class="ltx_ref">59</a>]</cite>.
Our method recovers these features and produces alternative classifiers with a variety of trade-offs in performance across various OOD conditions.</p>
</div>
<div id="S6.SS2.p4" class="ltx_para ltx_noindent">
<p id="S6.SS2.p4.1" class="ltx_p"><span id="S6.SS2.p4.1.1" class="ltx_text ltx_font_bold">Ablations.</span> 
In Table <a href="#S6.T2" title="Table 2 ‣ 6.2 Experiments on real data: WILDS-Camelyon17 ‣ 6 Experiments ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">2</span></a>, we compare additional ablations of our method, using a fixed number of 10 models.
The <span id="S6.SS2.p4.1.2" class="ltx_text ltx_font_bold">essential components are the independence and on-manifold constraints</span>.
The fine-tuning and distillation steps contribute to a marginal improvement.
We report similar <em id="S6.SS2.p4.1.3" class="ltx_emph ltx_font_italic">relative</em> improvements in Appendix <a href="#A7" title="Appendix G Additional results: WILDS-Camelyon17 ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">G</span></a> with other pretrained models, but the absolute performance is very much <span id="S6.SS2.p4.1.4" class="ltx_text ltx_font_bold">dependent on a “good” pretrained model</span>.</p>
</div>
<div id="S6.SS2.p5" class="ltx_para ltx_noindent">
<p id="S6.SS2.p5.1" class="ltx_p"><span id="S6.SS2.p5.1.1" class="ltx_text ltx_font_bold">Model selection.</span> 
Our method brings similar relative improvements on either val-OOD or test-OOD but typically with different models for each (see Figure <a href="#S6.F7" title="Figure 7 ‣ 6.2 Experiments on real data: WILDS-Camelyon17 ‣ 6 Experiments ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">7</span></a>) despite both being OOD relative to the training data.
<span id="S6.SS2.p5.1.2" class="ltx_text ltx_font_bold">Model selection absent labelled target-domain data therefore remains an issue</span> on this dataset.
Fortunately, only little such data may be sufficient.
We repeated a few experiments while holding out 1% of test-OOD (less than 1,000 instances) and we observed a 99.87% correlation coefficient between the accuracy on test-OOD and this held-out data.
While all our results assume perfect “oracle” model selection, it seems reasonable that real applications could provide a small amount of labelled test data to achieve similar results.</p>
</div>
<figure id="S6.F7" class="ltx_figure"><img src="/html/2207.02598/assets/x4.png" id="S6.F7.g1" class="ltx_graphics ltx_centering ltx_img_landscape" width="461" height="116" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span id="S6.F7.9.1.1" class="ltx_text" style="font-size:90%;">Figure 7</span>: </span><span id="S6.F7.10.2" class="ltx_text" style="font-size:90%;">
Spread of accuracies on WILDS-Camelyon17 of models trained with different ablations of our method.
The upper/lower bounds of the <span id="S6.F7.10.2.1" class="ltx_text ltx_font_bold">shaded areas</span> show the highest/lowest accuracy of any model from one run, averaged over 6 seeds.
The <span id="S6.F7.10.2.2" class="ltx_text ltx_font_bold">violins</span> show the distribution of accuracies over all seeds (hence some values outside the shaded areas).
<span id="S6.F7.10.2.3" class="ltx_text ltx_font_bold">Take-aways.</span> The independence constraint (<span id="S6.F7.10.2.4" class="ltx_text ltx_font_bold">gray</span>) produces a wide variety of models compared to the baseline (<span id="S6.F7.10.2.5" class="ltx_text ltx_font_bold">red</span>).
However, the highest accuracy in each run grows slowly with the number of models.
With the on-manifold constraint (<span id="S6.F7.10.2.6" class="ltx_text ltx_font_bold">blue</span>), the improvement is clearly larger and only requires a handful of models. The fine-tuning and distillation (<span id="S6.F7.10.2.7" class="ltx_text ltx_font_bold">green</span>) bring an additional marginal improvement.
</span></figcaption>
</figure>
<figure id="S6.F8" class="ltx_figure">
<div class="ltx_flex_figure">
<div class="ltx_flex_cell ltx_flex_size_2"><img src="/html/2207.02598/assets/x5.png" id="S6.F8.g1" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_square" width="138" height="138" alt="Refer to caption"></div>
<div class="ltx_flex_cell ltx_flex_size_2"><img src="/html/2207.02598/assets/x6.png" id="S6.F8.g2" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_square" width="138" height="138" alt="Refer to caption"></div>
</div>
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span id="S6.F8.3.1.1" class="ltx_text" style="font-size:90%;">Figure 8</span>: </span><span id="S6.F8.4.2" class="ltx_text" style="font-size:90%;">On WILDS-Camelyon17, each model (same colors as Figure <a href="#S6.F7" title="Figure 7 ‣ 6.2 Experiments on real data: WILDS-Camelyon17 ‣ 6 Experiments ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">7</span></a>) achieves a different trade-off in performance on different splits, as seen with test-OOD <span id="S6.F8.4.2.1" class="ltx_text ltx_font_italic">vs.</span> val-ID/val-OOD (left/right, respectively).
Each dot represents one model at one training epoch. We plot every epoch since each model moves on these charts as training progresses.
The proposed method (green) produces models with much better performance on all splits (toward the upper right) than the baseline (red).
</span></figcaption>
</figure>
<div id="S6.SS2.p6" class="ltx_para">
<span id="S6.SS2.p6.1" class="ltx_ERROR undefined">\floatsetup</span>
<p id="S6.SS2.p6.2" class="ltx_p">[table]capposition=beside,capbesideposition=center,right,floatwidth=0.58capbesidewidth=0.40</p>
</div>
<figure id="S6.T2" class="ltx_table">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table"><span id="S6.T2.23.1.1" class="ltx_text" style="font-size:90%;">Table 2</span>: </span><span id="S6.T2.24.2" class="ltx_text" style="font-size:90%;">Accuracy on WILDS-Camelyon17 (average over 6 random seeds) while training 12 models with various ablations.
Each component of our method contributes to improving the accuracy of the best model from each run. The data also appears simple enough that a PCA approximates the manifold well enough. This allows implementing the on-manifold constraint as a hard projection rather than a soft regularizer.</span></figcaption>
<table id="S6.T2.21" class="ltx_tabular ltx_centering ltx_align_middle">
<tr id="S6.T2.21.22" class="ltx_tr">
<td id="S6.T2.21.22.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify ltx_border_tt" style="padding:0.25pt 0.0pt;">
<span id="S6.T2.21.22.1.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T2.21.22.1.1.1" class="ltx_p"><span id="S6.T2.21.22.1.1.1.1" class="ltx_text ltx_font_bold" style="font-size:90%;">WILDS-Camelyon17</span></span>
</span>
</td>
<td id="S6.T2.21.22.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_tt" style="padding:0.25pt 0.0pt;" colspan="3"><span id="S6.T2.21.22.2.1" class="ltx_text" style="font-size:90%;">Best accuracy (%) on</span></td>
</tr>
<tr id="S6.T2.21.23" class="ltx_tr">
<td id="S6.T2.21.23.1" class="ltx_td ltx_nopad_l ltx_nopad_r" style="padding:0.25pt 0.0pt;"></td>
<td id="S6.T2.21.23.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.25pt 0.0pt;"><span id="S6.T2.21.23.2.1" class="ltx_text" style="font-size:90%;">val-OOD</span></td>
<td id="S6.T2.21.23.3" class="ltx_td ltx_nopad_l ltx_nopad_r" style="padding:0.25pt 0.0pt;"></td>
<td id="S6.T2.21.23.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.25pt 0.0pt;"><span id="S6.T2.21.23.4.1" class="ltx_text" style="font-size:90%;">test-OOD</span></td>
</tr>
<tr id="S6.T2.1.1" class="ltx_tr">
<td id="S6.T2.1.1.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify ltx_border_t" style="padding:0.25pt 0.0pt;">
<span id="S6.T2.1.1.2.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T2.1.1.2.1.1" class="ltx_p"><span id="S6.T2.1.1.2.1.1.1" class="ltx_text" style="font-size:90%;">Pseudo-Label </span><cite class="ltx_cite ltx_citemacro_cite"><span id="S6.T2.1.1.2.1.1.2.1" class="ltx_text" style="font-size:90%;">[</span><a href="#bib.bib43" title="" class="ltx_ref">43</a><span id="S6.T2.1.1.2.1.1.3.2" class="ltx_text" style="font-size:90%;">]</span></cite></span>
</span>
</td>
<td id="S6.T2.1.1.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding:0.25pt 0.0pt;"><span id="S6.T2.1.1.3.1" class="ltx_text" style="font-size:90%;">–</span></td>
<td id="S6.T2.1.1.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_border_t" style="padding:0.25pt 0.0pt;"></td>
<td id="S6.T2.1.1.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding:0.25pt 0.0pt;">
<span id="S6.T2.1.1.1.1" class="ltx_text" style="font-size:90%;">67.7 </span><math id="S6.T2.1.1.1.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S6.T2.1.1.1.m1.1a"><mo mathsize="50%" id="S6.T2.1.1.1.m1.1.1" xref="S6.T2.1.1.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S6.T2.1.1.1.m1.1b"><csymbol cd="latexml" id="S6.T2.1.1.1.m1.1.1.cmml" xref="S6.T2.1.1.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.T2.1.1.1.m1.1c">\pm</annotation></semantics></math><span id="S6.T2.1.1.1.2" class="ltx_text" style="font-size:50%;">8.2</span>
</td>
</tr>
<tr id="S6.T2.2.2" class="ltx_tr">
<td id="S6.T2.2.2.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.25pt 0.0pt;">
<span id="S6.T2.2.2.2.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T2.2.2.2.1.1" class="ltx_p"><span id="S6.T2.2.2.2.1.1.1" class="ltx_text" style="font-size:90%;">DANN </span><cite class="ltx_cite ltx_citemacro_cite"><span id="S6.T2.2.2.2.1.1.2.1" class="ltx_text" style="font-size:90%;">[</span><a href="#bib.bib17" title="" class="ltx_ref">17</a><span id="S6.T2.2.2.2.1.1.3.2" class="ltx_text" style="font-size:90%;">]</span></cite></span>
</span>
</td>
<td id="S6.T2.2.2.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.25pt 0.0pt;"><span id="S6.T2.2.2.3.1" class="ltx_text" style="font-size:90%;">–</span></td>
<td id="S6.T2.2.2.4" class="ltx_td ltx_nopad_l ltx_nopad_r" style="padding:0.25pt 0.0pt;"></td>
<td id="S6.T2.2.2.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.25pt 0.0pt;">
<span id="S6.T2.2.2.1.1" class="ltx_text" style="font-size:90%;">68.4 </span><math id="S6.T2.2.2.1.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S6.T2.2.2.1.m1.1a"><mo mathsize="50%" id="S6.T2.2.2.1.m1.1.1" xref="S6.T2.2.2.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S6.T2.2.2.1.m1.1b"><csymbol cd="latexml" id="S6.T2.2.2.1.m1.1.1.cmml" xref="S6.T2.2.2.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.T2.2.2.1.m1.1c">\pm</annotation></semantics></math><span id="S6.T2.2.2.1.2" class="ltx_text" style="font-size:50%;">9.2</span>
</td>
</tr>
<tr id="S6.T2.3.3" class="ltx_tr">
<td id="S6.T2.3.3.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.25pt 0.0pt;">
<span id="S6.T2.3.3.2.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T2.3.3.2.1.1" class="ltx_p"><span id="S6.T2.3.3.2.1.1.1" class="ltx_text" style="font-size:90%;">FixMatch </span><cite class="ltx_cite ltx_citemacro_cite"><span id="S6.T2.3.3.2.1.1.2.1" class="ltx_text" style="font-size:90%;">[</span><a href="#bib.bib68" title="" class="ltx_ref">68</a><span id="S6.T2.3.3.2.1.1.3.2" class="ltx_text" style="font-size:90%;">]</span></cite></span>
</span>
</td>
<td id="S6.T2.3.3.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.25pt 0.0pt;"><span id="S6.T2.3.3.3.1" class="ltx_text" style="font-size:90%;">–</span></td>
<td id="S6.T2.3.3.4" class="ltx_td ltx_nopad_l ltx_nopad_r" style="padding:0.25pt 0.0pt;"></td>
<td id="S6.T2.3.3.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.25pt 0.0pt;">
<span id="S6.T2.3.3.1.1" class="ltx_text" style="font-size:90%;">71.0 </span><math id="S6.T2.3.3.1.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S6.T2.3.3.1.m1.1a"><mo mathsize="50%" id="S6.T2.3.3.1.m1.1.1" xref="S6.T2.3.3.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S6.T2.3.3.1.m1.1b"><csymbol cd="latexml" id="S6.T2.3.3.1.m1.1.1.cmml" xref="S6.T2.3.3.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.T2.3.3.1.m1.1c">\pm</annotation></semantics></math><span id="S6.T2.3.3.1.2" class="ltx_text" style="font-size:50%;">4.9</span>
</td>
</tr>
<tr id="S6.T2.4.4" class="ltx_tr">
<td id="S6.T2.4.4.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.25pt 0.0pt;">
<span id="S6.T2.4.4.2.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T2.4.4.2.1.1" class="ltx_p"><span id="S6.T2.4.4.2.1.1.1" class="ltx_text" style="font-size:90%;">CORAL </span><cite class="ltx_cite ltx_citemacro_cite"><span id="S6.T2.4.4.2.1.1.2.1" class="ltx_text" style="font-size:90%;">[</span><a href="#bib.bib70" title="" class="ltx_ref">70</a><span id="S6.T2.4.4.2.1.1.3.2" class="ltx_text" style="font-size:90%;">]</span></cite></span>
</span>
</td>
<td id="S6.T2.4.4.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.25pt 0.0pt;"><span id="S6.T2.4.4.3.1" class="ltx_text" style="font-size:90%;">–</span></td>
<td id="S6.T2.4.4.4" class="ltx_td ltx_nopad_l ltx_nopad_r" style="padding:0.25pt 0.0pt;"></td>
<td id="S6.T2.4.4.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.25pt 0.0pt;">
<span id="S6.T2.4.4.1.1" class="ltx_text" style="font-size:90%;">77.9 </span><math id="S6.T2.4.4.1.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S6.T2.4.4.1.m1.1a"><mo mathsize="50%" id="S6.T2.4.4.1.m1.1.1" xref="S6.T2.4.4.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S6.T2.4.4.1.m1.1b"><csymbol cd="latexml" id="S6.T2.4.4.1.m1.1.1.cmml" xref="S6.T2.4.4.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.T2.4.4.1.m1.1c">\pm</annotation></semantics></math><span id="S6.T2.4.4.1.2" class="ltx_text" style="font-size:50%;">6.6</span>
</td>
</tr>
<tr id="S6.T2.5.5" class="ltx_tr">
<td id="S6.T2.5.5.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.25pt 0.0pt;">
<span id="S6.T2.5.5.2.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T2.5.5.2.1.1" class="ltx_p"><span id="S6.T2.5.5.2.1.1.1" class="ltx_text" style="font-size:90%;">NoisyStudent </span><cite class="ltx_cite ltx_citemacro_cite"><span id="S6.T2.5.5.2.1.1.2.1" class="ltx_text" style="font-size:90%;">[</span><a href="#bib.bib84" title="" class="ltx_ref">84</a><span id="S6.T2.5.5.2.1.1.3.2" class="ltx_text" style="font-size:90%;">]</span></cite></span>
</span>
</td>
<td id="S6.T2.5.5.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.25pt 0.0pt;"><span id="S6.T2.5.5.3.1" class="ltx_text" style="font-size:90%;">–</span></td>
<td id="S6.T2.5.5.4" class="ltx_td ltx_nopad_l ltx_nopad_r" style="padding:0.25pt 0.0pt;"></td>
<td id="S6.T2.5.5.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.25pt 0.0pt;">
<span id="S6.T2.5.5.1.1" class="ltx_text ltx_font_bold ltx_framed ltx_framed_underline" style="font-size:90%;">86.7</span><span id="S6.T2.5.5.1.2" class="ltx_text" style="font-size:90%;"> </span><math id="S6.T2.5.5.1.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S6.T2.5.5.1.m1.1a"><mo mathsize="50%" id="S6.T2.5.5.1.m1.1.1" xref="S6.T2.5.5.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S6.T2.5.5.1.m1.1b"><csymbol cd="latexml" id="S6.T2.5.5.1.m1.1.1.cmml" xref="S6.T2.5.5.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.T2.5.5.1.m1.1c">\pm</annotation></semantics></math><span id="S6.T2.5.5.1.3" class="ltx_text" style="font-size:50%;">1.7</span>
</td>
</tr>
<tr id="S6.T2.7.7" class="ltx_tr">
<td id="S6.T2.7.7.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify ltx_border_t" style="padding:0.25pt 0.0pt;">
<span id="S6.T2.7.7.3.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T2.7.7.3.1.1" class="ltx_p"><span id="S6.T2.7.7.3.1.1.1" class="ltx_text" style="font-size:90%;">ERM Baseline</span></span>
</span>
</td>
<td id="S6.T2.6.6.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding:0.25pt 0.0pt;">
<span id="S6.T2.6.6.1.1" class="ltx_text" style="font-size:90%;">84.9 </span><math id="S6.T2.6.6.1.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S6.T2.6.6.1.m1.1a"><mo mathsize="50%" id="S6.T2.6.6.1.m1.1.1" xref="S6.T2.6.6.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S6.T2.6.6.1.m1.1b"><csymbol cd="latexml" id="S6.T2.6.6.1.m1.1.1.cmml" xref="S6.T2.6.6.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.T2.6.6.1.m1.1c">\pm</annotation></semantics></math><span id="S6.T2.6.6.1.2" class="ltx_text" style="font-size:50%;">0.1</span>
</td>
<td id="S6.T2.7.7.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_border_t" style="padding:0.25pt 0.0pt;"></td>
<td id="S6.T2.7.7.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding:0.25pt 0.0pt;">
<span id="S6.T2.7.7.2.1" class="ltx_text" style="font-size:90%;">68.4 </span><math id="S6.T2.7.7.2.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S6.T2.7.7.2.m1.1a"><mo mathsize="50%" id="S6.T2.7.7.2.m1.1.1" xref="S6.T2.7.7.2.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S6.T2.7.7.2.m1.1b"><csymbol cd="latexml" id="S6.T2.7.7.2.m1.1.1.cmml" xref="S6.T2.7.7.2.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.T2.7.7.2.m1.1c">\pm</annotation></semantics></math><span id="S6.T2.7.7.2.2" class="ltx_text" style="font-size:50%;">0.1</span>
</td>
</tr>
<tr id="S6.T2.9.9" class="ltx_tr">
<td id="S6.T2.9.9.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.25pt 0.0pt;">
<span id="S6.T2.9.9.3.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T2.9.9.3.1.1" class="ltx_p"><span id="S6.T2.9.9.3.1.1.1" class="ltx_text" style="font-size:90%;">+ Independence constraint</span></span>
</span>
</td>
<td id="S6.T2.8.8.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.25pt 0.0pt;">
<span id="S6.T2.8.8.1.1" class="ltx_text" style="font-size:90%;">85.3 </span><math id="S6.T2.8.8.1.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S6.T2.8.8.1.m1.1a"><mo mathsize="50%" id="S6.T2.8.8.1.m1.1.1" xref="S6.T2.8.8.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S6.T2.8.8.1.m1.1b"><csymbol cd="latexml" id="S6.T2.8.8.1.m1.1.1.cmml" xref="S6.T2.8.8.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.T2.8.8.1.m1.1c">\pm</annotation></semantics></math><span id="S6.T2.8.8.1.2" class="ltx_text" style="font-size:50%;">0.5</span>
</td>
<td id="S6.T2.9.9.4" class="ltx_td ltx_nopad_l ltx_nopad_r" style="padding:0.25pt 0.0pt;"></td>
<td id="S6.T2.9.9.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.25pt 0.0pt;">
<span id="S6.T2.9.9.2.1" class="ltx_text" style="font-size:90%;">74.6 </span><math id="S6.T2.9.9.2.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S6.T2.9.9.2.m1.1a"><mo mathsize="50%" id="S6.T2.9.9.2.m1.1.1" xref="S6.T2.9.9.2.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S6.T2.9.9.2.m1.1b"><csymbol cd="latexml" id="S6.T2.9.9.2.m1.1.1.cmml" xref="S6.T2.9.9.2.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.T2.9.9.2.m1.1c">\pm</annotation></semantics></math><span id="S6.T2.9.9.2.2" class="ltx_text" style="font-size:50%;">0.9</span>
</td>
</tr>
<tr id="S6.T2.11.11" class="ltx_tr">
<td id="S6.T2.11.11.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.25pt 0.0pt;">
<span id="S6.T2.11.11.3.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T2.11.11.3.1.1" class="ltx_p"><span id="S6.T2.11.11.3.1.1.1" class="ltx_text" style="font-size:90%;">+ On-manifold soft regularizer, VAE</span></span>
</span>
</td>
<td id="S6.T2.10.10.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.25pt 0.0pt;">
<span id="S6.T2.10.10.1.1" class="ltx_text" style="font-size:90%;">85.4 </span><math id="S6.T2.10.10.1.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S6.T2.10.10.1.m1.1a"><mo mathsize="50%" id="S6.T2.10.10.1.m1.1.1" xref="S6.T2.10.10.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S6.T2.10.10.1.m1.1b"><csymbol cd="latexml" id="S6.T2.10.10.1.m1.1.1.cmml" xref="S6.T2.10.10.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.T2.10.10.1.m1.1c">\pm</annotation></semantics></math><span id="S6.T2.10.10.1.2" class="ltx_text" style="font-size:50%;">0.4</span>
</td>
<td id="S6.T2.11.11.4" class="ltx_td ltx_nopad_l ltx_nopad_r" style="padding:0.25pt 0.0pt;"></td>
<td id="S6.T2.11.11.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.25pt 0.0pt;">
<span id="S6.T2.11.11.2.1" class="ltx_text" style="font-size:90%;">80.3 </span><math id="S6.T2.11.11.2.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S6.T2.11.11.2.m1.1a"><mo mathsize="50%" id="S6.T2.11.11.2.m1.1.1" xref="S6.T2.11.11.2.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S6.T2.11.11.2.m1.1b"><csymbol cd="latexml" id="S6.T2.11.11.2.m1.1.1.cmml" xref="S6.T2.11.11.2.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.T2.11.11.2.m1.1c">\pm</annotation></semantics></math><span id="S6.T2.11.11.2.2" class="ltx_text" style="font-size:50%;">1.7</span>
</td>
</tr>
<tr id="S6.T2.13.13" class="ltx_tr">
<td id="S6.T2.13.13.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.25pt 0.0pt;">
<span id="S6.T2.13.13.3.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T2.13.13.3.1.1" class="ltx_p"><span id="S6.T2.13.13.3.1.1.1" class="ltx_text" style="font-size:90%;">+ On-manifold hard projection, VAE</span></span>
</span>
</td>
<td id="S6.T2.12.12.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.25pt 0.0pt;">
<span id="S6.T2.12.12.1.1" class="ltx_text" style="font-size:90%;">88.2 </span><math id="S6.T2.12.12.1.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S6.T2.12.12.1.m1.1a"><mo mathsize="50%" id="S6.T2.12.12.1.m1.1.1" xref="S6.T2.12.12.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S6.T2.12.12.1.m1.1b"><csymbol cd="latexml" id="S6.T2.12.12.1.m1.1.1.cmml" xref="S6.T2.12.12.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.T2.12.12.1.m1.1c">\pm</annotation></semantics></math><span id="S6.T2.12.12.1.2" class="ltx_text" style="font-size:50%;">2.1</span>
</td>
<td id="S6.T2.13.13.4" class="ltx_td ltx_nopad_l ltx_nopad_r" style="padding:0.25pt 0.0pt;"></td>
<td id="S6.T2.13.13.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.25pt 0.0pt;">
<span id="S6.T2.13.13.2.1" class="ltx_text" style="font-size:90%;">76.3 </span><math id="S6.T2.13.13.2.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S6.T2.13.13.2.m1.1a"><mo mathsize="50%" id="S6.T2.13.13.2.m1.1.1" xref="S6.T2.13.13.2.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S6.T2.13.13.2.m1.1b"><csymbol cd="latexml" id="S6.T2.13.13.2.m1.1.1.cmml" xref="S6.T2.13.13.2.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.T2.13.13.2.m1.1c">\pm</annotation></semantics></math><span id="S6.T2.13.13.2.2" class="ltx_text" style="font-size:50%;">2.8</span>
</td>
</tr>
<tr id="S6.T2.15.15" class="ltx_tr">
<td id="S6.T2.15.15.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.25pt 0.0pt;">
<span id="S6.T2.15.15.3.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T2.15.15.3.1.1" class="ltx_p"><span id="S6.T2.15.15.3.1.1.1" class="ltx_text" style="font-size:90%;">+ On-manifold soft regularizer, PCA</span></span>
</span>
</td>
<td id="S6.T2.14.14.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.25pt 0.0pt;">
<span id="S6.T2.14.14.1.1" class="ltx_text" style="font-size:90%;">87.8 </span><math id="S6.T2.14.14.1.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S6.T2.14.14.1.m1.1a"><mo mathsize="50%" id="S6.T2.14.14.1.m1.1.1" xref="S6.T2.14.14.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S6.T2.14.14.1.m1.1b"><csymbol cd="latexml" id="S6.T2.14.14.1.m1.1.1.cmml" xref="S6.T2.14.14.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.T2.14.14.1.m1.1c">\pm</annotation></semantics></math><span id="S6.T2.14.14.1.2" class="ltx_text" style="font-size:50%;">0.3</span>
</td>
<td id="S6.T2.15.15.4" class="ltx_td ltx_nopad_l ltx_nopad_r" style="padding:0.25pt 0.0pt;"></td>
<td id="S6.T2.15.15.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.25pt 0.0pt;">
<span id="S6.T2.15.15.2.1" class="ltx_text" style="font-size:90%;">79.0 </span><math id="S6.T2.15.15.2.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S6.T2.15.15.2.m1.1a"><mo mathsize="50%" id="S6.T2.15.15.2.m1.1.1" xref="S6.T2.15.15.2.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S6.T2.15.15.2.m1.1b"><csymbol cd="latexml" id="S6.T2.15.15.2.m1.1.1.cmml" xref="S6.T2.15.15.2.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.T2.15.15.2.m1.1c">\pm</annotation></semantics></math><span id="S6.T2.15.15.2.2" class="ltx_text" style="font-size:50%;">2.9</span>
</td>
</tr>
<tr id="S6.T2.18.18" class="ltx_tr">
<td id="S6.T2.16.16.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.25pt 0.0pt;">
<span id="S6.T2.16.16.1.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T2.16.16.1.1.1" class="ltx_p"><span id="S6.T2.16.16.1.1.1.1" class="ltx_text" style="font-size:90%;">+ On-manifold hard projection, PCA</span><sup id="S6.T2.16.16.1.1.1.2" class="ltx_sup"><span id="S6.T2.16.16.1.1.1.2.1" class="ltx_text" style="font-size:90%;">∗</span></sup></span>
</span>
</td>
<td id="S6.T2.17.17.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.25pt 0.0pt;">
<span id="S6.T2.17.17.2.1" class="ltx_text ltx_font_bold" style="font-size:90%;">88.4</span><span id="S6.T2.17.17.2.2" class="ltx_text" style="font-size:90%;"> </span><math id="S6.T2.17.17.2.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S6.T2.17.17.2.m1.1a"><mo mathsize="50%" id="S6.T2.17.17.2.m1.1.1" xref="S6.T2.17.17.2.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S6.T2.17.17.2.m1.1b"><csymbol cd="latexml" id="S6.T2.17.17.2.m1.1.1.cmml" xref="S6.T2.17.17.2.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.T2.17.17.2.m1.1c">\pm</annotation></semantics></math><span id="S6.T2.17.17.2.3" class="ltx_text" style="font-size:50%;">0.7</span>
</td>
<td id="S6.T2.18.18.4" class="ltx_td ltx_nopad_l ltx_nopad_r" style="padding:0.25pt 0.0pt;"></td>
<td id="S6.T2.18.18.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.25pt 0.0pt;">
<span id="S6.T2.18.18.3.1" class="ltx_text" style="font-size:90%;">81.6 </span><math id="S6.T2.18.18.3.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S6.T2.18.18.3.m1.1a"><mo mathsize="50%" id="S6.T2.18.18.3.m1.1.1" xref="S6.T2.18.18.3.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S6.T2.18.18.3.m1.1b"><csymbol cd="latexml" id="S6.T2.18.18.3.m1.1.1.cmml" xref="S6.T2.18.18.3.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.T2.18.18.3.m1.1c">\pm</annotation></semantics></math><span id="S6.T2.18.18.3.2" class="ltx_text" style="font-size:50%;">1.4</span>
</td>
</tr>
<tr id="S6.T2.21.21" class="ltx_tr">
<td id="S6.T2.19.19.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify ltx_border_bb" style="padding:0.25pt 0.0pt;">
<span id="S6.T2.19.19.1.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T2.19.19.1.1.1" class="ltx_p"><span id="S6.T2.19.19.1.1.1.1" class="ltx_text" style="font-size:90%;">(</span><sup id="S6.T2.19.19.1.1.1.2" class="ltx_sup"><span id="S6.T2.19.19.1.1.1.2.1" class="ltx_text" style="font-size:90%;">∗</span></sup><span id="S6.T2.19.19.1.1.1.3" class="ltx_text" style="font-size:90%;">) + Fine-tuning &amp; distillation</span></span>
</span>
</td>
<td id="S6.T2.20.20.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_bb" style="padding:0.25pt 0.0pt;">
<span id="S6.T2.20.20.2.1" class="ltx_text ltx_font_bold" style="font-size:90%;">88.4</span><span id="S6.T2.20.20.2.2" class="ltx_text" style="font-size:90%;"> </span><math id="S6.T2.20.20.2.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S6.T2.20.20.2.m1.1a"><mo mathsize="50%" id="S6.T2.20.20.2.m1.1.1" xref="S6.T2.20.20.2.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S6.T2.20.20.2.m1.1b"><csymbol cd="latexml" id="S6.T2.20.20.2.m1.1.1.cmml" xref="S6.T2.20.20.2.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.T2.20.20.2.m1.1c">\pm</annotation></semantics></math><span id="S6.T2.20.20.2.3" class="ltx_text" style="font-size:50%;">0.7</span>
</td>
<td id="S6.T2.21.21.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_border_bb" style="padding:0.25pt 0.0pt;"></td>
<td id="S6.T2.21.21.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_bb" style="padding:0.25pt 0.0pt;">
<span id="S6.T2.21.21.3.1" class="ltx_text ltx_font_bold" style="font-size:90%;">82.5</span><span id="S6.T2.21.21.3.2" class="ltx_text" style="font-size:90%;"> </span><math id="S6.T2.21.21.3.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="S6.T2.21.21.3.m1.1a"><mo mathsize="50%" id="S6.T2.21.21.3.m1.1.1" xref="S6.T2.21.21.3.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="S6.T2.21.21.3.m1.1b"><csymbol cd="latexml" id="S6.T2.21.21.3.m1.1.1.cmml" xref="S6.T2.21.21.3.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="S6.T2.21.21.3.m1.1c">\pm</annotation></semantics></math><span id="S6.T2.21.21.3.3" class="ltx_text" style="font-size:50%;">2.4</span>
</td>
</tr>
</table>
</figure>
<div class="ltx_pagination ltx_role_newpage"></div>
</section>
<section id="S6.SS3" class="ltx_subsection">
<h3 class="ltx_title ltx_title_subsection">
<span class="ltx_tag ltx_tag_subsection">6.3 </span>Experiments on visual question answering: GQA dataset</h3>

<div id="S6.SS3.p1" class="ltx_para">
<p id="S6.SS3.p1.1" class="ltx_p">We now demonstrate the applicability of our method on a more complex task.
We choose visual question answering (VQA) because it is notorious for dataset biases <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib74" title="" class="ltx_ref">74</a>]</cite> that cause shortcut learning <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib13" title="" class="ltx_ref">13</a>]</cite>.
We train a simple architecture (details in Appendix <a href="#A5" title="Appendix E Experimental details: GQA ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">E</span></a>) on the GQA dataset <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib32" title="" class="ltx_ref">32</a>]</cite> and report accuracy on the GQA and GQA-OOD <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib37" title="" class="ltx_ref">37</a>]</cite> validation sets.
We focus on binary questions to lighten the computational expense.
As input the model, we use frozen 2048-dimensional image features (globally-pooled bottom-up features <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib3" title="" class="ltx_ref">3</a>]</cite>).
Our method is applied in the space of these features.
In Table <a href="#S6.T3" title="Table 3 ‣ 6.3 Experiments on visual question answering: GQA dataset ‣ 6 Experiments ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">3</span></a>, we observe that our method produces models with slightly better accuracy on all test sets.
The independence regularizer alone is not sufficient, most likely because it is unable to discover meaningful features in the large 2048-dimensional space.
The on-manifold constraint does solve this difficulty.
Interestingly, the individual models are better as well as a simple ensemble of all of these models.
This suggests a benefit from the greater variety of features learned collectively by the models.
We show in Figure <a href="#S1.F1" title="Figure 1 ‣ 1 Introduction ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">1</span></a> and in Appendix <a href="#A8" title="Appendix H Additional results: VQA ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">H</span></a> that the models typically focus each on different regions of images (with grad-CAM-weighted visualizations as in <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib64" title="" class="ltx_ref">64</a>]</cite>).
It is important to note that our method is applied across channels of image features, and that the spatial diversity emerges naturally.
We quantitatively verify this increase in diversity in Table <a href="#S6.T3" title="Table 3 ‣ 6.3 Experiments on visual question answering: GQA dataset ‣ 6 Experiments ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">3</span></a> with the average Spearman rank correlation of grad-CAM scores across models. <span id="S6.SS3.p1.1.1" class="ltx_ERROR undefined">\floatsetup</span>[table]capposition=beside,capbesideposition=center,right,floatwidth=0.61capbesidewidth=0.36</p>
</div>
<figure id="S6.T3" class="ltx_table">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table"><span id="S6.T3.2.1.1" class="ltx_text" style="font-size:90%;">Table 3</span>: </span><span id="S6.T3.3.2" class="ltx_text" style="font-size:90%;">Application to visual question answering. Models trained with the proposed method achieve higher accuracy.
They also show higher diversity in the image regions they rely on (last column, lower correlation of grad-CAM scores).</span></figcaption>
<table id="S6.T3.4" class="ltx_tabular ltx_centering ltx_align_middle">
<tr id="S6.T3.4.1" class="ltx_tr">
<td id="S6.T3.4.1.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify ltx_border_tt" style="padding-left:0.0pt;padding-right:0.0pt;">
<span id="S6.T3.4.1.1.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T3.4.1.1.1.1" class="ltx_p"><span id="S6.T3.4.1.1.1.1.1" class="ltx_text ltx_font_bold" style="font-size:50%;">GQA yes/no</span></span>
</span>
</td>
<td id="S6.T3.4.1.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_tt" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.1.2.1" class="ltx_text" style="font-size:50%;">N. of</span></td>
<td id="S6.T3.4.1.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_tt" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.1.3.1" class="ltx_text" style="font-size:50%;">GQA Val.</span></td>
<td id="S6.T3.4.1.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_tt" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.1.4.1" class="ltx_text" style="font-size:50%;">GQA Val.</span></td>
<td id="S6.T3.4.1.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_tt" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.1.5.1" class="ltx_text" style="font-size:50%;">GQA-OOD</span></td>
<td id="S6.T3.4.1.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_tt" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.1.6.1" class="ltx_text" style="font-size:50%;">GQA-OOD</span></td>
<td id="S6.T3.4.1.7" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_tt" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.1.7.1" class="ltx_text" style="font-size:50%;">Grad-CAM</span></td>
</tr>
<tr id="S6.T3.4.2" class="ltx_tr">
<td id="S6.T3.4.2.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding-left:0.0pt;padding-right:0.0pt;">
<span id="S6.T3.4.2.1.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T3.4.2.1.1.1" class="ltx_p"><span id="S6.T3.4.2.1.1.1.1" class="ltx_text" style="font-size:50%;">(accuracy in %)</span></span>
</span>
</td>
<td id="S6.T3.4.2.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.2.2.1" class="ltx_text" style="font-size:50%;">models</span></td>
<td id="S6.T3.4.2.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.2.3.1" class="ltx_text" style="font-size:70%;">(best)</span></td>
<td id="S6.T3.4.2.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.2.4.1" class="ltx_text" style="font-size:70%;">(ensemble)</span></td>
<td id="S6.T3.4.2.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;">
<span id="S6.T3.4.2.5.1" class="ltx_text" style="font-size:50%;">Val-head </span><span id="S6.T3.4.2.5.2" class="ltx_text" style="font-size:70%;">(best)</span>
</td>
<td id="S6.T3.4.2.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;">
<span id="S6.T3.4.2.6.1" class="ltx_text" style="font-size:50%;">Val-tail </span><span id="S6.T3.4.2.6.2" class="ltx_text" style="font-size:70%;">(best)</span>
</td>
<td id="S6.T3.4.2.7" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.2.7.1" class="ltx_text" style="font-size:50%;">rank corr.</span></td>
</tr>
<tr id="S6.T3.4.3" class="ltx_tr">
<td id="S6.T3.4.3.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;">
<span id="S6.T3.4.3.1.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T3.4.3.1.1.1" class="ltx_p"><span id="S6.T3.4.3.1.1.1.1" class="ltx_text" style="font-size:50%;">Baseline</span></span>
</span>
</td>
<td id="S6.T3.4.3.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.3.2.1" class="ltx_text" style="font-size:50%;">3</span></td>
<td id="S6.T3.4.3.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.3.3.1" class="ltx_text" style="font-size:50%;">67.9</span></td>
<td id="S6.T3.4.3.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.3.4.1" class="ltx_text" style="font-size:50%;">69.3</span></td>
<td id="S6.T3.4.3.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.3.5.1" class="ltx_text" style="font-size:50%;">70.4</span></td>
<td id="S6.T3.4.3.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.3.6.1" class="ltx_text" style="font-size:50%;">66.5</span></td>
<td id="S6.T3.4.3.7" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.3.7.1" class="ltx_text" style="font-size:50%;">0.68</span></td>
</tr>
<tr id="S6.T3.4.4" class="ltx_tr">
<td id="S6.T3.4.4.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding-left:0.0pt;padding-right:0.0pt;">
<span id="S6.T3.4.4.1.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T3.4.4.1.1.1" class="ltx_p"><span id="S6.T3.4.4.1.1.1.1" class="ltx_text" style="font-size:50%;">Baseline</span></span>
</span>
</td>
<td id="S6.T3.4.4.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.4.2.1" class="ltx_text" style="font-size:50%;">16</span></td>
<td id="S6.T3.4.4.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.4.3.1" class="ltx_text" style="font-size:50%;">68.6</span></td>
<td id="S6.T3.4.4.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.4.4.1" class="ltx_text" style="font-size:50%;">68.9</span></td>
<td id="S6.T3.4.4.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.4.5.1" class="ltx_text" style="font-size:50%;">71.2</span></td>
<td id="S6.T3.4.4.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.4.6.1" class="ltx_text" style="font-size:50%;">67.3</span></td>
<td id="S6.T3.4.4.7" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.4.7.1" class="ltx_text" style="font-size:50%;">–</span></td>
</tr>
<tr id="S6.T3.4.5" class="ltx_tr">
<td id="S6.T3.4.5.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;">
<span id="S6.T3.4.5.1.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T3.4.5.1.1.1" class="ltx_p"><span id="S6.T3.4.5.1.1.1.1" class="ltx_text" style="font-size:50%;">+ Independence</span></span>
</span>
</td>
<td id="S6.T3.4.5.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.5.2.1" class="ltx_text" style="font-size:50%;">3</span></td>
<td id="S6.T3.4.5.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.5.3.1" class="ltx_text" style="font-size:50%;">67.6</span></td>
<td id="S6.T3.4.5.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.5.4.1" class="ltx_text" style="font-size:50%;">69.2</span></td>
<td id="S6.T3.4.5.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.5.5.1" class="ltx_text" style="font-size:50%;">70.5</span></td>
<td id="S6.T3.4.5.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.5.6.1" class="ltx_text" style="font-size:50%;">67.1</span></td>
<td id="S6.T3.4.5.7" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.5.7.1" class="ltx_text" style="font-size:50%;">0.57</span></td>
</tr>
<tr id="S6.T3.4.6" class="ltx_tr">
<td id="S6.T3.4.6.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding-left:0.0pt;padding-right:0.0pt;">
<span id="S6.T3.4.6.1.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T3.4.6.1.1.1" class="ltx_p"><span id="S6.T3.4.6.1.1.1.1" class="ltx_text" style="font-size:50%;">+ Independence</span></span>
</span>
</td>
<td id="S6.T3.4.6.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.6.2.1" class="ltx_text" style="font-size:50%;">4</span></td>
<td id="S6.T3.4.6.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.6.3.1" class="ltx_text" style="font-size:50%;">67.6</span></td>
<td id="S6.T3.4.6.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.6.4.1" class="ltx_text" style="font-size:50%;">69.4</span></td>
<td id="S6.T3.4.6.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.6.5.1" class="ltx_text" style="font-size:50%;">70.1</span></td>
<td id="S6.T3.4.6.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.6.6.1" class="ltx_text" style="font-size:50%;">66.2</span></td>
<td id="S6.T3.4.6.7" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.6.7.1" class="ltx_text" style="font-size:50%;">–</span></td>
</tr>
<tr id="S6.T3.4.7" class="ltx_tr">
<td id="S6.T3.4.7.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding-left:0.0pt;padding-right:0.0pt;">
<span id="S6.T3.4.7.1.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T3.4.7.1.1.1" class="ltx_p"><span id="S6.T3.4.7.1.1.1.1" class="ltx_text" style="font-size:50%;">+ Independence</span></span>
</span>
</td>
<td id="S6.T3.4.7.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.7.2.1" class="ltx_text" style="font-size:50%;">8</span></td>
<td id="S6.T3.4.7.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.7.3.1" class="ltx_text" style="font-size:50%;">67.8</span></td>
<td id="S6.T3.4.7.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.7.4.1" class="ltx_text" style="font-size:50%;">70.0</span></td>
<td id="S6.T3.4.7.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.7.5.1" class="ltx_text" style="font-size:50%;">70.7</span></td>
<td id="S6.T3.4.7.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.7.6.1" class="ltx_text" style="font-size:50%;">66.7</span></td>
<td id="S6.T3.4.7.7" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.7.7.1" class="ltx_text" style="font-size:50%;">–</span></td>
</tr>
<tr id="S6.T3.4.8" class="ltx_tr">
<td id="S6.T3.4.8.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding-left:0.0pt;padding-right:0.0pt;">
<span id="S6.T3.4.8.1.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T3.4.8.1.1.1" class="ltx_p"><span id="S6.T3.4.8.1.1.1.1" class="ltx_text" style="font-size:50%;">+ Independence</span></span>
</span>
</td>
<td id="S6.T3.4.8.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.8.2.1" class="ltx_text" style="font-size:50%;">12</span></td>
<td id="S6.T3.4.8.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.8.3.1" class="ltx_text" style="font-size:50%;">68.1</span></td>
<td id="S6.T3.4.8.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.8.4.1" class="ltx_text" style="font-size:50%;">70.3</span></td>
<td id="S6.T3.4.8.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.8.5.1" class="ltx_text" style="font-size:50%;">70.6</span></td>
<td id="S6.T3.4.8.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.8.6.1" class="ltx_text" style="font-size:50%;">69.8</span></td>
<td id="S6.T3.4.8.7" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.8.7.1" class="ltx_text" style="font-size:50%;">–</span></td>
</tr>
<tr id="S6.T3.4.9" class="ltx_tr">
<td id="S6.T3.4.9.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding-left:0.0pt;padding-right:0.0pt;">
<span id="S6.T3.4.9.1.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T3.4.9.1.1.1" class="ltx_p"><span id="S6.T3.4.9.1.1.1.1" class="ltx_text" style="font-size:50%;">+ Independence</span></span>
</span>
</td>
<td id="S6.T3.4.9.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.9.2.1" class="ltx_text" style="font-size:50%;">16</span></td>
<td id="S6.T3.4.9.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.9.3.1" class="ltx_text" style="font-size:50%;">68.1</span></td>
<td id="S6.T3.4.9.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.9.4.1" class="ltx_text ltx_font_bold" style="font-size:50%;">70.5</span></td>
<td id="S6.T3.4.9.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.9.5.1" class="ltx_text" style="font-size:50%;">71.1</span></td>
<td id="S6.T3.4.9.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.9.6.1" class="ltx_text" style="font-size:50%;">69.6</span></td>
<td id="S6.T3.4.9.7" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.9.7.1" class="ltx_text" style="font-size:50%;">–</span></td>
</tr>
<tr id="S6.T3.4.10" class="ltx_tr">
<td id="S6.T3.4.10.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;">
<span id="S6.T3.4.10.1.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T3.4.10.1.1.1" class="ltx_p"><span id="S6.T3.4.10.1.1.1.1" class="ltx_text" style="font-size:50%;">+ Ind. + on-manifold</span></span>
</span>
</td>
<td id="S6.T3.4.10.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.10.2.1" class="ltx_text" style="font-size:50%;">3</span></td>
<td id="S6.T3.4.10.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.10.3.1" class="ltx_text" style="font-size:50%;">68.7</span></td>
<td id="S6.T3.4.10.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.10.4.1" class="ltx_text" style="font-size:50%;">69.7</span></td>
<td id="S6.T3.4.10.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.10.5.1" class="ltx_text" style="font-size:50%;">71.9</span></td>
<td id="S6.T3.4.10.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.10.6.1" class="ltx_text ltx_font_bold" style="font-size:50%;">72.5</span></td>
<td id="S6.T3.4.10.7" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.10.7.1" class="ltx_text" style="font-size:50%;">0.59</span></td>
</tr>
<tr id="S6.T3.4.11" class="ltx_tr">
<td id="S6.T3.4.11.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding-left:0.0pt;padding-right:0.0pt;">
<span id="S6.T3.4.11.1.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T3.4.11.1.1.1" class="ltx_p"><span id="S6.T3.4.11.1.1.1.1" class="ltx_text" style="font-size:50%;">+ Ind. + on-manifold</span></span>
</span>
</td>
<td id="S6.T3.4.11.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.11.2.1" class="ltx_text" style="font-size:50%;">4</span></td>
<td id="S6.T3.4.11.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.11.3.1" class="ltx_text" style="font-size:50%;">69.2</span></td>
<td id="S6.T3.4.11.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.11.4.1" class="ltx_text" style="font-size:50%;">70.4</span></td>
<td id="S6.T3.4.11.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.11.5.1" class="ltx_text ltx_font_bold" style="font-size:50%;">72.9</span></td>
<td id="S6.T3.4.11.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.11.6.1" class="ltx_text" style="font-size:50%;">70.5</span></td>
<td id="S6.T3.4.11.7" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.11.7.1" class="ltx_text" style="font-size:50%;">–</span></td>
</tr>
<tr id="S6.T3.4.12" class="ltx_tr">
<td id="S6.T3.4.12.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding-left:0.0pt;padding-right:0.0pt;">
<span id="S6.T3.4.12.1.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T3.4.12.1.1.1" class="ltx_p"><span id="S6.T3.4.12.1.1.1.1" class="ltx_text" style="font-size:50%;">+ Ind. + on-manifold</span></span>
</span>
</td>
<td id="S6.T3.4.12.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.12.2.1" class="ltx_text" style="font-size:50%;">8</span></td>
<td id="S6.T3.4.12.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.12.3.1" class="ltx_text" style="font-size:50%;">68.8</span></td>
<td id="S6.T3.4.12.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.12.4.1" class="ltx_text" style="font-size:50%;">70.0</span></td>
<td id="S6.T3.4.12.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.12.5.1" class="ltx_text" style="font-size:50%;">71.5</span></td>
<td id="S6.T3.4.12.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.12.6.1" class="ltx_text" style="font-size:50%;">69.1</span></td>
<td id="S6.T3.4.12.7" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.12.7.1" class="ltx_text" style="font-size:50%;">–</span></td>
</tr>
<tr id="S6.T3.4.13" class="ltx_tr">
<td id="S6.T3.4.13.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding-left:0.0pt;padding-right:0.0pt;">
<span id="S6.T3.4.13.1.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T3.4.13.1.1.1" class="ltx_p"><span id="S6.T3.4.13.1.1.1.1" class="ltx_text" style="font-size:50%;">+ Ind. + on-manifold</span></span>
</span>
</td>
<td id="S6.T3.4.13.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.13.2.1" class="ltx_text" style="font-size:50%;">12</span></td>
<td id="S6.T3.4.13.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.13.3.1" class="ltx_text" style="font-size:50%;">69.0</span></td>
<td id="S6.T3.4.13.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.13.4.1" class="ltx_text" style="font-size:50%;">70.2</span></td>
<td id="S6.T3.4.13.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.13.5.1" class="ltx_text" style="font-size:50%;">72.6</span></td>
<td id="S6.T3.4.13.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.13.6.1" class="ltx_text" style="font-size:50%;">69.4</span></td>
<td id="S6.T3.4.13.7" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.13.7.1" class="ltx_text" style="font-size:50%;">–</span></td>
</tr>
<tr id="S6.T3.4.14" class="ltx_tr">
<td id="S6.T3.4.14.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify ltx_border_bb" style="padding-left:0.0pt;padding-right:0.0pt;">
<span id="S6.T3.4.14.1.1" class="ltx_inline-block ltx_align_top">
<span id="S6.T3.4.14.1.1.1" class="ltx_p"><span id="S6.T3.4.14.1.1.1.1" class="ltx_text" style="font-size:50%;">+ Ind. + on-manifold</span></span>
</span>
</td>
<td id="S6.T3.4.14.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_bb" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.14.2.1" class="ltx_text" style="font-size:50%;">16</span></td>
<td id="S6.T3.4.14.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_bb" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.14.3.1" class="ltx_text ltx_font_bold" style="font-size:50%;">69.3</span></td>
<td id="S6.T3.4.14.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_bb" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.14.4.1" class="ltx_text" style="font-size:50%;">70.3</span></td>
<td id="S6.T3.4.14.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_bb" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.14.5.1" class="ltx_text" style="font-size:50%;">72.4</span></td>
<td id="S6.T3.4.14.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_bb" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.14.6.1" class="ltx_text" style="font-size:50%;">71.5</span></td>
<td id="S6.T3.4.14.7" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_bb" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="S6.T3.4.14.7.1" class="ltx_text" style="font-size:50%;">–</span></td>
</tr>
</table>
</figure>
<figure id="S6.F9" class="ltx_figure">
<div class="ltx_flex_figure">
<div class="ltx_flex_cell ltx_flex_size_1"><img src="/html/2207.02598/assets/x7.png" id="S6.F9.g1" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_landscape" width="230" height="69" alt="Refer to caption"></div>
<div class="ltx_flex_break"></div>
<div class="ltx_flex_cell ltx_flex_size_1"><img src="/html/2207.02598/assets/x8.png" id="S6.F9.g2" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_landscape" width="230" height="88" alt="Refer to caption"></div>
<div class="ltx_flex_break"></div>
<div class="ltx_flex_cell ltx_flex_size_1"><img src="/html/2207.02598/assets/x9.png" id="S6.F9.g3" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_landscape" width="230" height="87" alt="Refer to caption"></div>
<div class="ltx_flex_break"></div>
</div>
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span id="S6.F9.3.1.1" class="ltx_text" style="font-size:90%;">Figure 9</span>: </span><span id="S6.F9.4.2" class="ltx_text" style="font-size:90%;">
Examples from the GQA validation set. We show the input question, ground truth answer, and predicted answer from 3 models trained with our method. The input images are weighted with grad-CAM scores over object detections.
The models often predict the same answer while focusing on different regions.
This diversity in spatial locations emerges naturally: our independence constraint is applied across the <em id="S6.F9.4.2.1" class="ltx_emph ltx_font_italic">channels</em> of globally-pooled visual features.
</span></figcaption>
</figure>
<div class="ltx_pagination ltx_role_newpage"></div>
</section>
</section>
<section id="S7" class="ltx_section">
<h2 class="ltx_title ltx_title_section">
<span class="ltx_tag ltx_tag_section">7 </span>Discussion</h2>

<div id="S7.p1" class="ltx_para">
<p id="S7.p1.1" class="ltx_p">We presented a method that highlights cases of underspecification by training multiple models with similar in-domain performance yet different OOD behaviour.
This method offers a partial solution to building robust models since it discovers features that are otherwise missed by standard ERM due to shortcut learning or other implicit inductive biases <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib50" title="" class="ltx_ref">50</a>, <a href="#bib.bib66" title="" class="ltx_ref">66</a>]</cite>.</p>
</div>
<div id="S7.p2" class="ltx_para ltx_noindent">
<p id="S7.p2.1" class="ltx_p"><span id="S7.p2.1.1" class="ltx_text ltx_font_bold">What do we gain from identifying cases of underspecification?</span> 
The level of underspecification (indicated by the number of models that can be trained with the proposed constraints) shows how far from unique a solution to a learning problem is.
Diagnosing underspecification is not a pass-or-fail test: all but the simplest tasks and models are underspecified to some extent.
Measuring underspecification should help determining the level of trust attributed to an ML model.
Our constructive approach has the added advantage of exposing the range of predictive features present in the data.</p>
</div>
<div id="S7.p3" class="ltx_para ltx_noindent">
<p id="S7.p3.1" class="ltx_p"><span id="S7.p3.1.1" class="ltx_text ltx_font_bold">Importance to both engineering and science.</span> 
There is a continuing source of research questions in the apparent mismatch between empirical practices in ML and some hard limitations of learning methods.
The concept of underspecification has the potential to unify phenomena including shortcut learning, distribution shifts, and adversarial robustness.
These are important for ML as an engineering discipline (improving reliability and applicability of ML methods) as well as a scientific endeavour (understanding the structure of real-world data and how/why existing methods work).</p>
</div>
<div id="S7.p4" class="ltx_para ltx_noindent">
<p id="S7.p4.1" class="ltx_p"><span id="S7.p4.1.1" class="ltx_text ltx_font_bold">A first implication</span>  of underspecification is that ERM is insufficient to guarantee OOD generalization.
Identified cases of underspecification point at the need for additional task-specific information in the design of reliable learning methods.
If such information cannot be integrated, learned models are at risk of unexpected behaviour when deployed on OOD data, because the depend on stochastic or arbitrary factors (<em id="S7.p4.1.2" class="ltx_emph ltx_font_italic">e.g</em>.<span id="S7.p4.1.3" class="ltx_text"></span> reliance on texture <em id="S7.p4.1.4" class="ltx_emph ltx_font_italic">vs</em>.<span id="S7.p4.1.5" class="ltx_text"></span> shape in image classification <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib20" title="" class="ltx_ref">20</a>]</cite>).</p>
</div>
<div id="S7.p5" class="ltx_para ltx_noindent">
<p id="S7.p5.1" class="ltx_p"><span id="S7.p5.1.1" class="ltx_text ltx_font_bold">A second implication</span>  is that ID and OOD performance are not necessarily coupled.
Without further assumptions, in-domain validation is not a reliable model selection strategy for OOD performance despite contradictory suggestions made in the literature <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib24" title="" class="ltx_ref">24</a>, <a href="#bib.bib48" title="" class="ltx_ref">48</a>]</cite>.
This strategy might still be useful as a heuristic owing to some inherent structure in real-world data, but its limits of applicability are yet to be understood.</p>
</div>
<div id="S7.p6" class="ltx_para ltx_noindent">
<p id="S7.p6.1" class="ltx_p"><span id="S7.p6.1.1" class="ltx_text ltx_font_bold">A third implication</span>  is that high OOD performance of a model is no guarantee for its reliability. High apparent performance might happen by accident in an underspecified setting. In such cases, the model behaviour depends on hidden assumptions and it could still fail unexpectedly.
Identifying underspecification remains important to identify these hidden assumptions, which is
particularly important for high-stakes applications such as medical imaging <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib5" title="" class="ltx_ref">5</a>, <a href="#bib.bib21" title="" class="ltx_ref">21</a>]</cite>.</p>
</div>
<div id="S7.p7" class="ltx_para">
<p id="S7.p7.1" class="ltx_p">The proposed analysis also corroborates existing explanations for techniques that successfully improve generalization, such as data augmentation and contrastive learning.
Both were indeed shown to depend on the injection of additional knowledge, respectively in the design of the augmentations <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib10" title="" class="ltx_ref">10</a>, <a href="#bib.bib34" title="" class="ltx_ref">34</a>, <a href="#bib.bib42" title="" class="ltx_ref">42</a>]</cite> and pair selection strategy <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib86" title="" class="ltx_ref">86</a>]</cite>.
And this extra knowledge is often task-specific <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib83" title="" class="ltx_ref">83</a>]</cite>. For example, augmenting images with rotations may help in identifying flowers but not traffic signs.
Injecting task-specific knowledge is sometimes vilified in a “data-driven” culture.
This study suggests that we would rather benefit from highlighting this practice and making assumptions more explicit, thus helping one to identify the limits of applicability of various methods.</p>
</div>
<div id="S7.p8" class="ltx_para ltx_noindent">
<p id="S7.p8.1" class="ltx_p"><span id="S7.p8.1.1" class="ltx_text ltx_font_bold">Conclusion.</span> 
This paper made theoretical and methodological steps on the study of underspecification. It complements an observational study <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib12" title="" class="ltx_ref">12</a>]</cite> with a method to diagnose and address the problem.</p>
</div>
<div id="S7.p9" class="ltx_para ltx_noindent">
<p id="S7.p9.1" class="ltx_p"><span id="S7.p9.1.1" class="ltx_text ltx_font_bold">Limitations.</span> 
The proposed method for building models with better generalization is only a partial solution since it requires an external model selection procedure.
New methods for model selection <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib15" title="" class="ltx_ref">15</a>, <a href="#bib.bib19" title="" class="ltx_ref">19</a>, <a href="#bib.bib35" title="" class="ltx_ref">35</a>, <a href="#bib.bib79" title="" class="ltx_ref">79</a>]</cite>, robust evaluation <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib18" title="" class="ltx_ref">18</a>, <a href="#bib.bib36" title="" class="ltx_ref">36</a>]</cite>, and explainability <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib22" title="" class="ltx_ref">22</a>, <a href="#bib.bib75" title="" class="ltx_ref">75</a>]</cite> are all suitable to implement this selection.
Interactive approaches <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib14" title="" class="ltx_ref">14</a>]</cite> are another option that injects expert knowledge.
Another possible extension is to apply the method to the end-to-end training of larger models.
Finally, this work focused on i.i.d. training data.
We hope to extend the analysis to forms of data known to be valuable for OOD generalization such as multiple environments <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib4" title="" class="ltx_ref">4</a>, <a href="#bib.bib54" title="" class="ltx_ref">54</a>, <a href="#bib.bib11" title="" class="ltx_ref">11</a>]</cite>, counterfactual examples <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib36" title="" class="ltx_ref">36</a>, <a href="#bib.bib71" title="" class="ltx_ref">71</a>]</cite>, and non-stationary data <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib1" title="" class="ltx_ref">1</a>, <a href="#bib.bib26" title="" class="ltx_ref">26</a>, <a href="#bib.bib56" title="" class="ltx_ref">56</a>, <a href="#bib.bib78" title="" class="ltx_ref">78</a>]</cite>.
The analysis of multi-environment training, as used in domain generalization, may elucidate why these methods are often ineffective in practice <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib24" title="" class="ltx_ref">24</a>]</cite>.</p>
</div>
</section>
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</section>
<div class="ltx_pagination ltx_role_newpage"></div>
<section id="Ax1" class="ltx_appendix">
<h2 class="ltx_title ltx_title_appendix">Appendices</h2>

<div id="Ax1.p1" class="ltx_para">
<span id="Ax1.p1.1" class="ltx_ERROR undefined">\floatsetup</span>
<p id="Ax1.p1.2" class="ltx_p">[figure]capposition=below,floatwidth=
<span id="Ax1.p1.2.1" class="ltx_ERROR undefined">\floatsetup</span>[table]capposition=below,floatwidth=</p>
</div>
</section>
<section id="A1" class="ltx_appendix">
<h2 class="ltx_title ltx_title_appendix">
<span class="ltx_tag ltx_tag_appendix">Appendix A </span>Analogy with disentanglement in representation learning</h2>

<div id="A1.p1" class="ltx_para">
<p id="A1.p1.1" class="ltx_p">There are parallels between our discovery of simple predictive features and the topic of disentanglement in representation learning.
Disentanglement aims to identify all independent factors of variation in the data-generating process <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib7" title="" class="ltx_ref">7</a>]</cite> <em id="A1.p1.1.1" class="ltx_emph ltx_font_italic">i.e</em>.<span id="A1.p1.1.2" class="ltx_text"></span> in a <span id="A1.p1.1.3" class="ltx_text ltx_font_bold">task-independent</span> manner.
In comparison, we are interested only in features that are correlated with <span id="A1.p1.1.4" class="ltx_text ltx_font_bold">task-specific</span> labels in a given dataset.
Many disentanglement methods are based on the independence of the factors (even though it is usually insufficient <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib76" title="" class="ltx_ref">76</a>]</cite>) while we rely on the independence of <em id="A1.p1.1.5" class="ltx_emph ltx_font_italic">predictions</em> with each feature.
Our method can therefore be seen as a form of disentanglement in the space of predictors.
This parallel also has implications for identifiability.
Disentanglement of the correct generative factors was shown to be impossible without additional assumptions or supervision <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib45" title="" class="ltx_ref">45</a>]</cite>.
In our setting, the identification of features that are causally-related to the label (as opposed to spuriously-correlated) also requires additional information <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib6" title="" class="ltx_ref">6</a>, <a href="#bib.bib63" title="" class="ltx_ref">63</a>]</cite>.
Our formulation relegates this need to an external “model selection strategy” that allows flexibility in its actual realization.</p>
</div>
</section>
<section id="A2" class="ltx_appendix">
<h2 class="ltx_title ltx_title_appendix">
<span class="ltx_tag ltx_tag_appendix">Appendix B </span>Projections on the manifold</h2>

<div id="A2.p1" class="ltx_para">
<p id="A2.p1.2" class="ltx_p">The on-manifold loss defined in Eq. (<a href="#S4.E2" title="In 4.2 On-manifold constraint ‣ 4 Proposed method ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">2</span></a>) requires projecting a model’s input gradients on the data manifold.
The manifold is characterized by a provided set of unlabeled data <math id="A2.p1.1.m1.1" class="ltx_Math" alttext="\mathcal{D}_{\mathrm{OOD}}=\{{\boldsymbol{x}}_{i}\}\sim\textrm{P}_{\mathrm{OOD}}" display="inline"><semantics id="A2.p1.1.m1.1a"><mrow id="A2.p1.1.m1.1.1" xref="A2.p1.1.m1.1.1.cmml"><msub id="A2.p1.1.m1.1.1.3" xref="A2.p1.1.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" id="A2.p1.1.m1.1.1.3.2" xref="A2.p1.1.m1.1.1.3.2.cmml">𝒟</mi><mi id="A2.p1.1.m1.1.1.3.3" xref="A2.p1.1.m1.1.1.3.3.cmml">OOD</mi></msub><mo id="A2.p1.1.m1.1.1.4" xref="A2.p1.1.m1.1.1.4.cmml">=</mo><mrow id="A2.p1.1.m1.1.1.1.1" xref="A2.p1.1.m1.1.1.1.2.cmml"><mo stretchy="false" id="A2.p1.1.m1.1.1.1.1.2" xref="A2.p1.1.m1.1.1.1.2.cmml">{</mo><msub id="A2.p1.1.m1.1.1.1.1.1" xref="A2.p1.1.m1.1.1.1.1.1.cmml"><mi id="A2.p1.1.m1.1.1.1.1.1.2" xref="A2.p1.1.m1.1.1.1.1.1.2.cmml">𝒙</mi><mi id="A2.p1.1.m1.1.1.1.1.1.3" xref="A2.p1.1.m1.1.1.1.1.1.3.cmml">i</mi></msub><mo stretchy="false" id="A2.p1.1.m1.1.1.1.1.3" xref="A2.p1.1.m1.1.1.1.2.cmml">}</mo></mrow><mo id="A2.p1.1.m1.1.1.5" xref="A2.p1.1.m1.1.1.5.cmml">∼</mo><msub id="A2.p1.1.m1.1.1.6" xref="A2.p1.1.m1.1.1.6.cmml"><mtext id="A2.p1.1.m1.1.1.6.2" xref="A2.p1.1.m1.1.1.6.2a.cmml">P</mtext><mi id="A2.p1.1.m1.1.1.6.3" xref="A2.p1.1.m1.1.1.6.3.cmml">OOD</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="A2.p1.1.m1.1b"><apply id="A2.p1.1.m1.1.1.cmml" xref="A2.p1.1.m1.1.1"><and id="A2.p1.1.m1.1.1a.cmml" xref="A2.p1.1.m1.1.1"></and><apply id="A2.p1.1.m1.1.1b.cmml" xref="A2.p1.1.m1.1.1"><eq id="A2.p1.1.m1.1.1.4.cmml" xref="A2.p1.1.m1.1.1.4"></eq><apply id="A2.p1.1.m1.1.1.3.cmml" xref="A2.p1.1.m1.1.1.3"><csymbol cd="ambiguous" id="A2.p1.1.m1.1.1.3.1.cmml" xref="A2.p1.1.m1.1.1.3">subscript</csymbol><ci id="A2.p1.1.m1.1.1.3.2.cmml" xref="A2.p1.1.m1.1.1.3.2">𝒟</ci><ci id="A2.p1.1.m1.1.1.3.3.cmml" xref="A2.p1.1.m1.1.1.3.3">OOD</ci></apply><set id="A2.p1.1.m1.1.1.1.2.cmml" xref="A2.p1.1.m1.1.1.1.1"><apply id="A2.p1.1.m1.1.1.1.1.1.cmml" xref="A2.p1.1.m1.1.1.1.1.1"><csymbol cd="ambiguous" id="A2.p1.1.m1.1.1.1.1.1.1.cmml" xref="A2.p1.1.m1.1.1.1.1.1">subscript</csymbol><ci id="A2.p1.1.m1.1.1.1.1.1.2.cmml" xref="A2.p1.1.m1.1.1.1.1.1.2">𝒙</ci><ci id="A2.p1.1.m1.1.1.1.1.1.3.cmml" xref="A2.p1.1.m1.1.1.1.1.1.3">𝑖</ci></apply></set></apply><apply id="A2.p1.1.m1.1.1c.cmml" xref="A2.p1.1.m1.1.1"><csymbol cd="latexml" id="A2.p1.1.m1.1.1.5.cmml" xref="A2.p1.1.m1.1.1.5">similar-to</csymbol><share href="#A2.p1.1.m1.1.1.1.cmml" id="A2.p1.1.m1.1.1d.cmml" xref="A2.p1.1.m1.1.1"></share><apply id="A2.p1.1.m1.1.1.6.cmml" xref="A2.p1.1.m1.1.1.6"><csymbol cd="ambiguous" id="A2.p1.1.m1.1.1.6.1.cmml" xref="A2.p1.1.m1.1.1.6">subscript</csymbol><ci id="A2.p1.1.m1.1.1.6.2a.cmml" xref="A2.p1.1.m1.1.1.6.2"><mtext id="A2.p1.1.m1.1.1.6.2.cmml" xref="A2.p1.1.m1.1.1.6.2">P</mtext></ci><ci id="A2.p1.1.m1.1.1.6.3.cmml" xref="A2.p1.1.m1.1.1.6.3">OOD</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p1.1.m1.1c">\mathcal{D}_{\mathrm{OOD}}=\{{\boldsymbol{x}}_{i}\}\sim\textrm{P}_{\mathrm{OOD}}</annotation></semantics></math>.
We propose two implementations of the projection function <math id="A2.p1.2.m2.1" class="ltx_Math" alttext="\mathrm{proj}(\cdot)" display="inline"><semantics id="A2.p1.2.m2.1a"><mrow id="A2.p1.2.m2.1.2" xref="A2.p1.2.m2.1.2.cmml"><mi id="A2.p1.2.m2.1.2.2" xref="A2.p1.2.m2.1.2.2.cmml">proj</mi><mo lspace="0em" rspace="0em" id="A2.p1.2.m2.1.2.1" xref="A2.p1.2.m2.1.2.1.cmml">​</mo><mrow id="A2.p1.2.m2.1.2.3.2" xref="A2.p1.2.m2.1.2.cmml"><mo stretchy="false" id="A2.p1.2.m2.1.2.3.2.1" xref="A2.p1.2.m2.1.2.cmml">(</mo><mo lspace="0em" rspace="0em" id="A2.p1.2.m2.1.1" xref="A2.p1.2.m2.1.1.cmml">⋅</mo><mo stretchy="false" id="A2.p1.2.m2.1.2.3.2.2" xref="A2.p1.2.m2.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.p1.2.m2.1b"><apply id="A2.p1.2.m2.1.2.cmml" xref="A2.p1.2.m2.1.2"><times id="A2.p1.2.m2.1.2.1.cmml" xref="A2.p1.2.m2.1.2.1"></times><ci id="A2.p1.2.m2.1.2.2.cmml" xref="A2.p1.2.m2.1.2.2">proj</ci><ci id="A2.p1.2.m2.1.1.cmml" xref="A2.p1.2.m2.1.1">⋅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p1.2.m2.1c">\mathrm{proj}(\cdot)</annotation></semantics></math>, using either a simple principal component analysis (PCA) model of the manifold or a variational auto-encoder (VAE).
A PCA is fast to evaluate but can only model a linear manifold, which is likely overly simplistic for most real datasets. A VAE allows learning a non-linear manifold.
Better implementations are probably possible by taking advantage of recent developments in OOD detection and generative models such as GANs, EBMs, and diffusion models.</p>
</div>
<div id="A2.p2" class="ltx_para">
<p id="A2.p2.4" class="ltx_p">In the case of a PCA model, the projection is a straightforward linear projection on the top components of a PCA basis of <math id="A2.p2.1.m1.1" class="ltx_Math" alttext="\mathcal{D}_{\mathrm{OOD}}" display="inline"><semantics id="A2.p2.1.m1.1a"><msub id="A2.p2.1.m1.1.1" xref="A2.p2.1.m1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A2.p2.1.m1.1.1.2" xref="A2.p2.1.m1.1.1.2.cmml">𝒟</mi><mi id="A2.p2.1.m1.1.1.3" xref="A2.p2.1.m1.1.1.3.cmml">OOD</mi></msub><annotation-xml encoding="MathML-Content" id="A2.p2.1.m1.1b"><apply id="A2.p2.1.m1.1.1.cmml" xref="A2.p2.1.m1.1.1"><csymbol cd="ambiguous" id="A2.p2.1.m1.1.1.1.cmml" xref="A2.p2.1.m1.1.1">subscript</csymbol><ci id="A2.p2.1.m1.1.1.2.cmml" xref="A2.p2.1.m1.1.1.2">𝒟</ci><ci id="A2.p2.1.m1.1.1.3.cmml" xref="A2.p2.1.m1.1.1.3">OOD</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p2.1.m1.1c">\mathcal{D}_{\mathrm{OOD}}</annotation></semantics></math>.
In the case of a VAE, we train an auto-encoder on the data <math id="A2.p2.2.m2.1" class="ltx_Math" alttext="\mathcal{D}_{\mathrm{OOD}}" display="inline"><semantics id="A2.p2.2.m2.1a"><msub id="A2.p2.2.m2.1.1" xref="A2.p2.2.m2.1.1.cmml"><mi class="ltx_font_mathcaligraphic" id="A2.p2.2.m2.1.1.2" xref="A2.p2.2.m2.1.1.2.cmml">𝒟</mi><mi id="A2.p2.2.m2.1.1.3" xref="A2.p2.2.m2.1.1.3.cmml">OOD</mi></msub><annotation-xml encoding="MathML-Content" id="A2.p2.2.m2.1b"><apply id="A2.p2.2.m2.1.1.cmml" xref="A2.p2.2.m2.1.1"><csymbol cd="ambiguous" id="A2.p2.2.m2.1.1.1.cmml" xref="A2.p2.2.m2.1.1">subscript</csymbol><ci id="A2.p2.2.m2.1.1.2.cmml" xref="A2.p2.2.m2.1.1.2">𝒟</ci><ci id="A2.p2.2.m2.1.1.3.cmml" xref="A2.p2.2.m2.1.1.3">OOD</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p2.2.m2.1c">\mathcal{D}_{\mathrm{OOD}}</annotation></semantics></math>. We interpret it as a generative model of the manifold, assuming that inputs to the auto-encoder will be mapped to a nearby projection on the manifold.
We denote the auto-encoder as a function
<math id="A2.p2.3.m3.1" class="ltx_Math" alttext="\mathrm{proj}_{\boldsymbol{\psi}}:\leavevmode\nobreak\ \mathbb{R}^{d_{\textrm{in}}}\rightarrow\mathcal{M}" display="inline"><semantics id="A2.p2.3.m3.1a"><mrow id="A2.p2.3.m3.1.1" xref="A2.p2.3.m3.1.1.cmml"><msub id="A2.p2.3.m3.1.1.2" xref="A2.p2.3.m3.1.1.2.cmml"><mi id="A2.p2.3.m3.1.1.2.2" xref="A2.p2.3.m3.1.1.2.2.cmml">proj</mi><mi id="A2.p2.3.m3.1.1.2.3" xref="A2.p2.3.m3.1.1.2.3.cmml">𝝍</mi></msub><mo lspace="0.278em" rspace="0.778em" id="A2.p2.3.m3.1.1.1" xref="A2.p2.3.m3.1.1.1.cmml">:</mo><mrow id="A2.p2.3.m3.1.1.3" xref="A2.p2.3.m3.1.1.3.cmml"><msup id="A2.p2.3.m3.1.1.3.2" xref="A2.p2.3.m3.1.1.3.2.cmml"><mi id="A2.p2.3.m3.1.1.3.2.2" xref="A2.p2.3.m3.1.1.3.2.2.cmml">ℝ</mi><msub id="A2.p2.3.m3.1.1.3.2.3" xref="A2.p2.3.m3.1.1.3.2.3.cmml"><mi id="A2.p2.3.m3.1.1.3.2.3.2" xref="A2.p2.3.m3.1.1.3.2.3.2.cmml">d</mi><mtext id="A2.p2.3.m3.1.1.3.2.3.3" xref="A2.p2.3.m3.1.1.3.2.3.3a.cmml">in</mtext></msub></msup><mo stretchy="false" id="A2.p2.3.m3.1.1.3.1" xref="A2.p2.3.m3.1.1.3.1.cmml">→</mo><mi class="ltx_font_mathcaligraphic" id="A2.p2.3.m3.1.1.3.3" xref="A2.p2.3.m3.1.1.3.3.cmml">ℳ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.p2.3.m3.1b"><apply id="A2.p2.3.m3.1.1.cmml" xref="A2.p2.3.m3.1.1"><ci id="A2.p2.3.m3.1.1.1.cmml" xref="A2.p2.3.m3.1.1.1">:</ci><apply id="A2.p2.3.m3.1.1.2.cmml" xref="A2.p2.3.m3.1.1.2"><csymbol cd="ambiguous" id="A2.p2.3.m3.1.1.2.1.cmml" xref="A2.p2.3.m3.1.1.2">subscript</csymbol><ci id="A2.p2.3.m3.1.1.2.2.cmml" xref="A2.p2.3.m3.1.1.2.2">proj</ci><ci id="A2.p2.3.m3.1.1.2.3.cmml" xref="A2.p2.3.m3.1.1.2.3">𝝍</ci></apply><apply id="A2.p2.3.m3.1.1.3.cmml" xref="A2.p2.3.m3.1.1.3"><ci id="A2.p2.3.m3.1.1.3.1.cmml" xref="A2.p2.3.m3.1.1.3.1">→</ci><apply id="A2.p2.3.m3.1.1.3.2.cmml" xref="A2.p2.3.m3.1.1.3.2"><csymbol cd="ambiguous" id="A2.p2.3.m3.1.1.3.2.1.cmml" xref="A2.p2.3.m3.1.1.3.2">superscript</csymbol><ci id="A2.p2.3.m3.1.1.3.2.2.cmml" xref="A2.p2.3.m3.1.1.3.2.2">ℝ</ci><apply id="A2.p2.3.m3.1.1.3.2.3.cmml" xref="A2.p2.3.m3.1.1.3.2.3"><csymbol cd="ambiguous" id="A2.p2.3.m3.1.1.3.2.3.1.cmml" xref="A2.p2.3.m3.1.1.3.2.3">subscript</csymbol><ci id="A2.p2.3.m3.1.1.3.2.3.2.cmml" xref="A2.p2.3.m3.1.1.3.2.3.2">𝑑</ci><ci id="A2.p2.3.m3.1.1.3.2.3.3a.cmml" xref="A2.p2.3.m3.1.1.3.2.3.3"><mtext mathsize="50%" id="A2.p2.3.m3.1.1.3.2.3.3.cmml" xref="A2.p2.3.m3.1.1.3.2.3.3">in</mtext></ci></apply></apply><ci id="A2.p2.3.m3.1.1.3.3.cmml" xref="A2.p2.3.m3.1.1.3.3">ℳ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p2.3.m3.1c">\mathrm{proj}_{\boldsymbol{\psi}}:\leavevmode\nobreak\ \mathbb{R}^{d_{\textrm{in}}}\rightarrow\mathcal{M}</annotation></semantics></math>
of parameters <math id="A2.p2.4.m4.1" class="ltx_Math" alttext="{\boldsymbol{\theta}}" display="inline"><semantics id="A2.p2.4.m4.1a"><mi id="A2.p2.4.m4.1.1" xref="A2.p2.4.m4.1.1.cmml">𝜽</mi><annotation-xml encoding="MathML-Content" id="A2.p2.4.m4.1b"><ci id="A2.p2.4.m4.1.1.cmml" xref="A2.p2.4.m4.1.1">𝜽</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.p2.4.m4.1c">{\boldsymbol{\theta}}</annotation></semantics></math> such that</p>
<table id="A8.EGx6" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A2.E6"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="A2.E6.m1.2" class="ltx_Math" alttext="\displaystyle\mathrm{proj}_{\boldsymbol{\psi}}({\boldsymbol{x}})\approx{\boldsymbol{x}}\quad\forall\leavevmode\nobreak\ {\boldsymbol{x}}\sim\textrm{P}_{\mathrm{OOD}}." display="inline"><semantics id="A2.E6.m1.2a"><mrow id="A2.E6.m1.2.2.1"><mrow id="A2.E6.m1.2.2.1.1.2" xref="A2.E6.m1.2.2.1.1.3.cmml"><mrow id="A2.E6.m1.2.2.1.1.1.1" xref="A2.E6.m1.2.2.1.1.1.1.cmml"><mrow id="A2.E6.m1.2.2.1.1.1.1.2" xref="A2.E6.m1.2.2.1.1.1.1.2.cmml"><msub id="A2.E6.m1.2.2.1.1.1.1.2.2" xref="A2.E6.m1.2.2.1.1.1.1.2.2.cmml"><mi id="A2.E6.m1.2.2.1.1.1.1.2.2.2" xref="A2.E6.m1.2.2.1.1.1.1.2.2.2.cmml">proj</mi><mi id="A2.E6.m1.2.2.1.1.1.1.2.2.3" xref="A2.E6.m1.2.2.1.1.1.1.2.2.3.cmml">𝝍</mi></msub><mo lspace="0em" rspace="0em" id="A2.E6.m1.2.2.1.1.1.1.2.1" xref="A2.E6.m1.2.2.1.1.1.1.2.1.cmml">​</mo><mrow id="A2.E6.m1.2.2.1.1.1.1.2.3.2" xref="A2.E6.m1.2.2.1.1.1.1.2.cmml"><mo stretchy="false" id="A2.E6.m1.2.2.1.1.1.1.2.3.2.1" xref="A2.E6.m1.2.2.1.1.1.1.2.cmml">(</mo><mi id="A2.E6.m1.1.1" xref="A2.E6.m1.1.1.cmml">𝒙</mi><mo stretchy="false" id="A2.E6.m1.2.2.1.1.1.1.2.3.2.2" xref="A2.E6.m1.2.2.1.1.1.1.2.cmml">)</mo></mrow></mrow><mo id="A2.E6.m1.2.2.1.1.1.1.1" xref="A2.E6.m1.2.2.1.1.1.1.1.cmml">≈</mo><mi id="A2.E6.m1.2.2.1.1.1.1.3" xref="A2.E6.m1.2.2.1.1.1.1.3.cmml">𝒙</mi></mrow><mspace width="1.167em" id="A2.E6.m1.2.2.1.1.2.3" xref="A2.E6.m1.2.2.1.1.3a.cmml"></mspace><mrow id="A2.E6.m1.2.2.1.1.2.2" xref="A2.E6.m1.2.2.1.1.2.2.cmml"><mrow id="A2.E6.m1.2.2.1.1.2.2.2" xref="A2.E6.m1.2.2.1.1.2.2.2.cmml"><mo rspace="0.667em" id="A2.E6.m1.2.2.1.1.2.2.2.1" xref="A2.E6.m1.2.2.1.1.2.2.2.1.cmml">∀</mo><mi id="A2.E6.m1.2.2.1.1.2.2.2.2" xref="A2.E6.m1.2.2.1.1.2.2.2.2.cmml">𝒙</mi></mrow><mo id="A2.E6.m1.2.2.1.1.2.2.1" xref="A2.E6.m1.2.2.1.1.2.2.1.cmml">∼</mo><msub id="A2.E6.m1.2.2.1.1.2.2.3" xref="A2.E6.m1.2.2.1.1.2.2.3.cmml"><mtext id="A2.E6.m1.2.2.1.1.2.2.3.2" xref="A2.E6.m1.2.2.1.1.2.2.3.2a.cmml">P</mtext><mi id="A2.E6.m1.2.2.1.1.2.2.3.3" xref="A2.E6.m1.2.2.1.1.2.2.3.3.cmml">OOD</mi></msub></mrow></mrow><mo lspace="0em" id="A2.E6.m1.2.2.1.2">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.E6.m1.2b"><apply id="A2.E6.m1.2.2.1.1.3.cmml" xref="A2.E6.m1.2.2.1.1.2"><csymbol cd="ambiguous" id="A2.E6.m1.2.2.1.1.3a.cmml" xref="A2.E6.m1.2.2.1.1.2.3">formulae-sequence</csymbol><apply id="A2.E6.m1.2.2.1.1.1.1.cmml" xref="A2.E6.m1.2.2.1.1.1.1"><approx id="A2.E6.m1.2.2.1.1.1.1.1.cmml" xref="A2.E6.m1.2.2.1.1.1.1.1"></approx><apply id="A2.E6.m1.2.2.1.1.1.1.2.cmml" xref="A2.E6.m1.2.2.1.1.1.1.2"><times id="A2.E6.m1.2.2.1.1.1.1.2.1.cmml" xref="A2.E6.m1.2.2.1.1.1.1.2.1"></times><apply id="A2.E6.m1.2.2.1.1.1.1.2.2.cmml" xref="A2.E6.m1.2.2.1.1.1.1.2.2"><csymbol cd="ambiguous" id="A2.E6.m1.2.2.1.1.1.1.2.2.1.cmml" xref="A2.E6.m1.2.2.1.1.1.1.2.2">subscript</csymbol><ci id="A2.E6.m1.2.2.1.1.1.1.2.2.2.cmml" xref="A2.E6.m1.2.2.1.1.1.1.2.2.2">proj</ci><ci id="A2.E6.m1.2.2.1.1.1.1.2.2.3.cmml" xref="A2.E6.m1.2.2.1.1.1.1.2.2.3">𝝍</ci></apply><ci id="A2.E6.m1.1.1.cmml" xref="A2.E6.m1.1.1">𝒙</ci></apply><ci id="A2.E6.m1.2.2.1.1.1.1.3.cmml" xref="A2.E6.m1.2.2.1.1.1.1.3">𝒙</ci></apply><apply id="A2.E6.m1.2.2.1.1.2.2.cmml" xref="A2.E6.m1.2.2.1.1.2.2"><csymbol cd="latexml" id="A2.E6.m1.2.2.1.1.2.2.1.cmml" xref="A2.E6.m1.2.2.1.1.2.2.1">similar-to</csymbol><apply id="A2.E6.m1.2.2.1.1.2.2.2.cmml" xref="A2.E6.m1.2.2.1.1.2.2.2"><csymbol cd="latexml" id="A2.E6.m1.2.2.1.1.2.2.2.1.cmml" xref="A2.E6.m1.2.2.1.1.2.2.2.1">for-all</csymbol><ci id="A2.E6.m1.2.2.1.1.2.2.2.2.cmml" xref="A2.E6.m1.2.2.1.1.2.2.2.2">𝒙</ci></apply><apply id="A2.E6.m1.2.2.1.1.2.2.3.cmml" xref="A2.E6.m1.2.2.1.1.2.2.3"><csymbol cd="ambiguous" id="A2.E6.m1.2.2.1.1.2.2.3.1.cmml" xref="A2.E6.m1.2.2.1.1.2.2.3">subscript</csymbol><ci id="A2.E6.m1.2.2.1.1.2.2.3.2a.cmml" xref="A2.E6.m1.2.2.1.1.2.2.3.2"><mtext id="A2.E6.m1.2.2.1.1.2.2.3.2.cmml" xref="A2.E6.m1.2.2.1.1.2.2.3.2">P</mtext></ci><ci id="A2.E6.m1.2.2.1.1.2.2.3.3.cmml" xref="A2.E6.m1.2.2.1.1.2.2.3.3">OOD</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E6.m1.2c">\displaystyle\mathrm{proj}_{\boldsymbol{\psi}}({\boldsymbol{x}})\approx{\boldsymbol{x}}\quad\forall\leavevmode\nobreak\ {\boldsymbol{x}}\sim\textrm{P}_{\mathrm{OOD}}.</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(6)</span></td>
</tr></tbody>
</table>
<p id="A2.p2.6" class="ltx_p">Therefore, the reconstruction error <math id="A2.p2.5.m1.2" class="ltx_Math" alttext="||{\boldsymbol{x}}-\mathrm{proj}_{\boldsymbol{\psi}}({\boldsymbol{x}})||" display="inline"><semantics id="A2.p2.5.m1.2a"><mrow id="A2.p2.5.m1.2.2.1" xref="A2.p2.5.m1.2.2.2.cmml"><mo stretchy="false" id="A2.p2.5.m1.2.2.1.2" xref="A2.p2.5.m1.2.2.2.1.cmml">‖</mo><mrow id="A2.p2.5.m1.2.2.1.1" xref="A2.p2.5.m1.2.2.1.1.cmml"><mi id="A2.p2.5.m1.2.2.1.1.2" xref="A2.p2.5.m1.2.2.1.1.2.cmml">𝒙</mi><mo id="A2.p2.5.m1.2.2.1.1.1" xref="A2.p2.5.m1.2.2.1.1.1.cmml">−</mo><mrow id="A2.p2.5.m1.2.2.1.1.3" xref="A2.p2.5.m1.2.2.1.1.3.cmml"><msub id="A2.p2.5.m1.2.2.1.1.3.2" xref="A2.p2.5.m1.2.2.1.1.3.2.cmml"><mi id="A2.p2.5.m1.2.2.1.1.3.2.2" xref="A2.p2.5.m1.2.2.1.1.3.2.2.cmml">proj</mi><mi id="A2.p2.5.m1.2.2.1.1.3.2.3" xref="A2.p2.5.m1.2.2.1.1.3.2.3.cmml">𝝍</mi></msub><mo lspace="0em" rspace="0em" id="A2.p2.5.m1.2.2.1.1.3.1" xref="A2.p2.5.m1.2.2.1.1.3.1.cmml">​</mo><mrow id="A2.p2.5.m1.2.2.1.1.3.3.2" xref="A2.p2.5.m1.2.2.1.1.3.cmml"><mo stretchy="false" id="A2.p2.5.m1.2.2.1.1.3.3.2.1" xref="A2.p2.5.m1.2.2.1.1.3.cmml">(</mo><mi id="A2.p2.5.m1.1.1" xref="A2.p2.5.m1.1.1.cmml">𝒙</mi><mo stretchy="false" id="A2.p2.5.m1.2.2.1.1.3.3.2.2" xref="A2.p2.5.m1.2.2.1.1.3.cmml">)</mo></mrow></mrow></mrow><mo stretchy="false" id="A2.p2.5.m1.2.2.1.3" xref="A2.p2.5.m1.2.2.2.1.cmml">‖</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.p2.5.m1.2b"><apply id="A2.p2.5.m1.2.2.2.cmml" xref="A2.p2.5.m1.2.2.1"><csymbol cd="latexml" id="A2.p2.5.m1.2.2.2.1.cmml" xref="A2.p2.5.m1.2.2.1.2">norm</csymbol><apply id="A2.p2.5.m1.2.2.1.1.cmml" xref="A2.p2.5.m1.2.2.1.1"><minus id="A2.p2.5.m1.2.2.1.1.1.cmml" xref="A2.p2.5.m1.2.2.1.1.1"></minus><ci id="A2.p2.5.m1.2.2.1.1.2.cmml" xref="A2.p2.5.m1.2.2.1.1.2">𝒙</ci><apply id="A2.p2.5.m1.2.2.1.1.3.cmml" xref="A2.p2.5.m1.2.2.1.1.3"><times id="A2.p2.5.m1.2.2.1.1.3.1.cmml" xref="A2.p2.5.m1.2.2.1.1.3.1"></times><apply id="A2.p2.5.m1.2.2.1.1.3.2.cmml" xref="A2.p2.5.m1.2.2.1.1.3.2"><csymbol cd="ambiguous" id="A2.p2.5.m1.2.2.1.1.3.2.1.cmml" xref="A2.p2.5.m1.2.2.1.1.3.2">subscript</csymbol><ci id="A2.p2.5.m1.2.2.1.1.3.2.2.cmml" xref="A2.p2.5.m1.2.2.1.1.3.2.2">proj</ci><ci id="A2.p2.5.m1.2.2.1.1.3.2.3.cmml" xref="A2.p2.5.m1.2.2.1.1.3.2.3">𝝍</ci></apply><ci id="A2.p2.5.m1.1.1.cmml" xref="A2.p2.5.m1.1.1">𝒙</ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p2.5.m1.2c">||{\boldsymbol{x}}-\mathrm{proj}_{\boldsymbol{\psi}}({\boldsymbol{x}})||</annotation></semantics></math> is minimal for points <math id="A2.p2.6.m2.1" class="ltx_Math" alttext="{\boldsymbol{x}}" display="inline"><semantics id="A2.p2.6.m2.1a"><mi id="A2.p2.6.m2.1.1" xref="A2.p2.6.m2.1.1.cmml">𝒙</mi><annotation-xml encoding="MathML-Content" id="A2.p2.6.m2.1b"><ci id="A2.p2.6.m2.1.1.cmml" xref="A2.p2.6.m2.1.1">𝒙</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.p2.6.m2.1c">{\boldsymbol{x}}</annotation></semantics></math> on to the manifold.</p>
</div>
<div id="A2.p3" class="ltx_para">
<p id="A2.p3.5" class="ltx_p">The function <math id="A2.p3.1.m1.1" class="ltx_Math" alttext="\mathrm{proj}" display="inline"><semantics id="A2.p3.1.m1.1a"><mi id="A2.p3.1.m1.1.1" xref="A2.p3.1.m1.1.1.cmml">proj</mi><annotation-xml encoding="MathML-Content" id="A2.p3.1.m1.1b"><ci id="A2.p3.1.m1.1.1.cmml" xref="A2.p3.1.m1.1.1">proj</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.p3.1.m1.1c">\mathrm{proj}</annotation></semantics></math> is trained to reconstruct points, but it is also readily capable of projecting a vector <math id="A2.p3.2.m2.1" class="ltx_Math" alttext="{\boldsymbol{v}}\in\mathbb{R}^{d_{\textrm{in}}}" display="inline"><semantics id="A2.p3.2.m2.1a"><mrow id="A2.p3.2.m2.1.1" xref="A2.p3.2.m2.1.1.cmml"><mi id="A2.p3.2.m2.1.1.2" xref="A2.p3.2.m2.1.1.2.cmml">𝒗</mi><mo id="A2.p3.2.m2.1.1.1" xref="A2.p3.2.m2.1.1.1.cmml">∈</mo><msup id="A2.p3.2.m2.1.1.3" xref="A2.p3.2.m2.1.1.3.cmml"><mi id="A2.p3.2.m2.1.1.3.2" xref="A2.p3.2.m2.1.1.3.2.cmml">ℝ</mi><msub id="A2.p3.2.m2.1.1.3.3" xref="A2.p3.2.m2.1.1.3.3.cmml"><mi id="A2.p3.2.m2.1.1.3.3.2" xref="A2.p3.2.m2.1.1.3.3.2.cmml">d</mi><mtext id="A2.p3.2.m2.1.1.3.3.3" xref="A2.p3.2.m2.1.1.3.3.3a.cmml">in</mtext></msub></msup></mrow><annotation-xml encoding="MathML-Content" id="A2.p3.2.m2.1b"><apply id="A2.p3.2.m2.1.1.cmml" xref="A2.p3.2.m2.1.1"><in id="A2.p3.2.m2.1.1.1.cmml" xref="A2.p3.2.m2.1.1.1"></in><ci id="A2.p3.2.m2.1.1.2.cmml" xref="A2.p3.2.m2.1.1.2">𝒗</ci><apply id="A2.p3.2.m2.1.1.3.cmml" xref="A2.p3.2.m2.1.1.3"><csymbol cd="ambiguous" id="A2.p3.2.m2.1.1.3.1.cmml" xref="A2.p3.2.m2.1.1.3">superscript</csymbol><ci id="A2.p3.2.m2.1.1.3.2.cmml" xref="A2.p3.2.m2.1.1.3.2">ℝ</ci><apply id="A2.p3.2.m2.1.1.3.3.cmml" xref="A2.p3.2.m2.1.1.3.3"><csymbol cd="ambiguous" id="A2.p3.2.m2.1.1.3.3.1.cmml" xref="A2.p3.2.m2.1.1.3.3">subscript</csymbol><ci id="A2.p3.2.m2.1.1.3.3.2.cmml" xref="A2.p3.2.m2.1.1.3.3.2">𝑑</ci><ci id="A2.p3.2.m2.1.1.3.3.3a.cmml" xref="A2.p3.2.m2.1.1.3.3.3"><mtext mathsize="50%" id="A2.p3.2.m2.1.1.3.3.3.cmml" xref="A2.p3.2.m2.1.1.3.3.3">in</mtext></ci></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p3.2.m2.1c">{\boldsymbol{v}}\in\mathbb{R}^{d_{\textrm{in}}}</annotation></semantics></math> originating at <math id="A2.p3.3.m3.1" class="ltx_Math" alttext="{\boldsymbol{x}}" display="inline"><semantics id="A2.p3.3.m3.1a"><mi id="A2.p3.3.m3.1.1" xref="A2.p3.3.m3.1.1.cmml">𝒙</mi><annotation-xml encoding="MathML-Content" id="A2.p3.3.m3.1b"><ci id="A2.p3.3.m3.1.1.cmml" xref="A2.p3.3.m3.1.1">𝒙</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.p3.3.m3.1c">{\boldsymbol{x}}</annotation></semantics></math>.
In our case, we will use it to project the input gradient (<math id="A2.p3.4.m4.1" class="ltx_Math" alttext="{\boldsymbol{v}}=\nabla_{\boldsymbol{x}}f{\boldsymbol{\theta}}({\boldsymbol{x}})" display="inline"><semantics id="A2.p3.4.m4.1a"><mrow id="A2.p3.4.m4.1.2" xref="A2.p3.4.m4.1.2.cmml"><mi id="A2.p3.4.m4.1.2.2" xref="A2.p3.4.m4.1.2.2.cmml">𝒗</mi><mo id="A2.p3.4.m4.1.2.1" xref="A2.p3.4.m4.1.2.1.cmml">=</mo><mrow id="A2.p3.4.m4.1.2.3" xref="A2.p3.4.m4.1.2.3.cmml"><mrow id="A2.p3.4.m4.1.2.3.2" xref="A2.p3.4.m4.1.2.3.2.cmml"><msub id="A2.p3.4.m4.1.2.3.2.1" xref="A2.p3.4.m4.1.2.3.2.1.cmml"><mo rspace="0.167em" id="A2.p3.4.m4.1.2.3.2.1.2" xref="A2.p3.4.m4.1.2.3.2.1.2.cmml">∇</mo><mi id="A2.p3.4.m4.1.2.3.2.1.3" xref="A2.p3.4.m4.1.2.3.2.1.3.cmml">𝒙</mi></msub><mrow id="A2.p3.4.m4.1.2.3.2.2" xref="A2.p3.4.m4.1.2.3.2.2.cmml"><mi id="A2.p3.4.m4.1.2.3.2.2.2" xref="A2.p3.4.m4.1.2.3.2.2.2.cmml">f</mi><mo lspace="0em" rspace="0em" id="A2.p3.4.m4.1.2.3.2.2.1" xref="A2.p3.4.m4.1.2.3.2.2.1.cmml">​</mo><mi id="A2.p3.4.m4.1.2.3.2.2.3" xref="A2.p3.4.m4.1.2.3.2.2.3.cmml">𝜽</mi></mrow></mrow><mo lspace="0em" rspace="0em" id="A2.p3.4.m4.1.2.3.1" xref="A2.p3.4.m4.1.2.3.1.cmml">​</mo><mrow id="A2.p3.4.m4.1.2.3.3.2" xref="A2.p3.4.m4.1.2.3.cmml"><mo stretchy="false" id="A2.p3.4.m4.1.2.3.3.2.1" xref="A2.p3.4.m4.1.2.3.cmml">(</mo><mi id="A2.p3.4.m4.1.1" xref="A2.p3.4.m4.1.1.cmml">𝒙</mi><mo stretchy="false" id="A2.p3.4.m4.1.2.3.3.2.2" xref="A2.p3.4.m4.1.2.3.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.p3.4.m4.1b"><apply id="A2.p3.4.m4.1.2.cmml" xref="A2.p3.4.m4.1.2"><eq id="A2.p3.4.m4.1.2.1.cmml" xref="A2.p3.4.m4.1.2.1"></eq><ci id="A2.p3.4.m4.1.2.2.cmml" xref="A2.p3.4.m4.1.2.2">𝒗</ci><apply id="A2.p3.4.m4.1.2.3.cmml" xref="A2.p3.4.m4.1.2.3"><times id="A2.p3.4.m4.1.2.3.1.cmml" xref="A2.p3.4.m4.1.2.3.1"></times><apply id="A2.p3.4.m4.1.2.3.2.cmml" xref="A2.p3.4.m4.1.2.3.2"><apply id="A2.p3.4.m4.1.2.3.2.1.cmml" xref="A2.p3.4.m4.1.2.3.2.1"><csymbol cd="ambiguous" id="A2.p3.4.m4.1.2.3.2.1.1.cmml" xref="A2.p3.4.m4.1.2.3.2.1">subscript</csymbol><ci id="A2.p3.4.m4.1.2.3.2.1.2.cmml" xref="A2.p3.4.m4.1.2.3.2.1.2">∇</ci><ci id="A2.p3.4.m4.1.2.3.2.1.3.cmml" xref="A2.p3.4.m4.1.2.3.2.1.3">𝒙</ci></apply><apply id="A2.p3.4.m4.1.2.3.2.2.cmml" xref="A2.p3.4.m4.1.2.3.2.2"><times id="A2.p3.4.m4.1.2.3.2.2.1.cmml" xref="A2.p3.4.m4.1.2.3.2.2.1"></times><ci id="A2.p3.4.m4.1.2.3.2.2.2.cmml" xref="A2.p3.4.m4.1.2.3.2.2.2">𝑓</ci><ci id="A2.p3.4.m4.1.2.3.2.2.3.cmml" xref="A2.p3.4.m4.1.2.3.2.2.3">𝜽</ci></apply></apply><ci id="A2.p3.4.m4.1.1.cmml" xref="A2.p3.4.m4.1.1">𝒙</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p3.4.m4.1c">{\boldsymbol{v}}=\nabla_{\boldsymbol{x}}f{\boldsymbol{\theta}}({\boldsymbol{x}})</annotation></semantics></math>).
We overload the function as <math id="A2.p3.5.m5.1" class="ltx_Math" alttext="\mathrm{proj}_{\boldsymbol{\psi}}:\mathbb{R}^{d_{\textrm{in}}}\times\mathbb{R}^{d_{\textrm{in}}}\,\rightarrow\,\mathcal{M}" display="inline"><semantics id="A2.p3.5.m5.1a"><mrow id="A2.p3.5.m5.1.1" xref="A2.p3.5.m5.1.1.cmml"><msub id="A2.p3.5.m5.1.1.2" xref="A2.p3.5.m5.1.1.2.cmml"><mi id="A2.p3.5.m5.1.1.2.2" xref="A2.p3.5.m5.1.1.2.2.cmml">proj</mi><mi id="A2.p3.5.m5.1.1.2.3" xref="A2.p3.5.m5.1.1.2.3.cmml">𝝍</mi></msub><mo lspace="0.278em" rspace="0.278em" id="A2.p3.5.m5.1.1.1" xref="A2.p3.5.m5.1.1.1.cmml">:</mo><mrow id="A2.p3.5.m5.1.1.3" xref="A2.p3.5.m5.1.1.3.cmml"><mrow id="A2.p3.5.m5.1.1.3.2" xref="A2.p3.5.m5.1.1.3.2.cmml"><msup id="A2.p3.5.m5.1.1.3.2.2" xref="A2.p3.5.m5.1.1.3.2.2.cmml"><mi id="A2.p3.5.m5.1.1.3.2.2.2" xref="A2.p3.5.m5.1.1.3.2.2.2.cmml">ℝ</mi><msub id="A2.p3.5.m5.1.1.3.2.2.3" xref="A2.p3.5.m5.1.1.3.2.2.3.cmml"><mi id="A2.p3.5.m5.1.1.3.2.2.3.2" xref="A2.p3.5.m5.1.1.3.2.2.3.2.cmml">d</mi><mtext id="A2.p3.5.m5.1.1.3.2.2.3.3" xref="A2.p3.5.m5.1.1.3.2.2.3.3a.cmml">in</mtext></msub></msup><mo lspace="0.222em" rspace="0.222em" id="A2.p3.5.m5.1.1.3.2.1" xref="A2.p3.5.m5.1.1.3.2.1.cmml">×</mo><msup id="A2.p3.5.m5.1.1.3.2.3" xref="A2.p3.5.m5.1.1.3.2.3.cmml"><mi id="A2.p3.5.m5.1.1.3.2.3.2" xref="A2.p3.5.m5.1.1.3.2.3.2.cmml">ℝ</mi><msub id="A2.p3.5.m5.1.1.3.2.3.3" xref="A2.p3.5.m5.1.1.3.2.3.3.cmml"><mi id="A2.p3.5.m5.1.1.3.2.3.3.2" xref="A2.p3.5.m5.1.1.3.2.3.3.2.cmml">d</mi><mtext id="A2.p3.5.m5.1.1.3.2.3.3.3" xref="A2.p3.5.m5.1.1.3.2.3.3.3a.cmml">in</mtext></msub></msup></mrow><mo rspace="0.448em" stretchy="false" id="A2.p3.5.m5.1.1.3.1" xref="A2.p3.5.m5.1.1.3.1.cmml">→</mo><mi class="ltx_font_mathcaligraphic" id="A2.p3.5.m5.1.1.3.3" xref="A2.p3.5.m5.1.1.3.3.cmml">ℳ</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.p3.5.m5.1b"><apply id="A2.p3.5.m5.1.1.cmml" xref="A2.p3.5.m5.1.1"><ci id="A2.p3.5.m5.1.1.1.cmml" xref="A2.p3.5.m5.1.1.1">:</ci><apply id="A2.p3.5.m5.1.1.2.cmml" xref="A2.p3.5.m5.1.1.2"><csymbol cd="ambiguous" id="A2.p3.5.m5.1.1.2.1.cmml" xref="A2.p3.5.m5.1.1.2">subscript</csymbol><ci id="A2.p3.5.m5.1.1.2.2.cmml" xref="A2.p3.5.m5.1.1.2.2">proj</ci><ci id="A2.p3.5.m5.1.1.2.3.cmml" xref="A2.p3.5.m5.1.1.2.3">𝝍</ci></apply><apply id="A2.p3.5.m5.1.1.3.cmml" xref="A2.p3.5.m5.1.1.3"><ci id="A2.p3.5.m5.1.1.3.1.cmml" xref="A2.p3.5.m5.1.1.3.1">→</ci><apply id="A2.p3.5.m5.1.1.3.2.cmml" xref="A2.p3.5.m5.1.1.3.2"><times id="A2.p3.5.m5.1.1.3.2.1.cmml" xref="A2.p3.5.m5.1.1.3.2.1"></times><apply id="A2.p3.5.m5.1.1.3.2.2.cmml" xref="A2.p3.5.m5.1.1.3.2.2"><csymbol cd="ambiguous" id="A2.p3.5.m5.1.1.3.2.2.1.cmml" xref="A2.p3.5.m5.1.1.3.2.2">superscript</csymbol><ci id="A2.p3.5.m5.1.1.3.2.2.2.cmml" xref="A2.p3.5.m5.1.1.3.2.2.2">ℝ</ci><apply id="A2.p3.5.m5.1.1.3.2.2.3.cmml" xref="A2.p3.5.m5.1.1.3.2.2.3"><csymbol cd="ambiguous" id="A2.p3.5.m5.1.1.3.2.2.3.1.cmml" xref="A2.p3.5.m5.1.1.3.2.2.3">subscript</csymbol><ci id="A2.p3.5.m5.1.1.3.2.2.3.2.cmml" xref="A2.p3.5.m5.1.1.3.2.2.3.2">𝑑</ci><ci id="A2.p3.5.m5.1.1.3.2.2.3.3a.cmml" xref="A2.p3.5.m5.1.1.3.2.2.3.3"><mtext mathsize="50%" id="A2.p3.5.m5.1.1.3.2.2.3.3.cmml" xref="A2.p3.5.m5.1.1.3.2.2.3.3">in</mtext></ci></apply></apply><apply id="A2.p3.5.m5.1.1.3.2.3.cmml" xref="A2.p3.5.m5.1.1.3.2.3"><csymbol cd="ambiguous" id="A2.p3.5.m5.1.1.3.2.3.1.cmml" xref="A2.p3.5.m5.1.1.3.2.3">superscript</csymbol><ci id="A2.p3.5.m5.1.1.3.2.3.2.cmml" xref="A2.p3.5.m5.1.1.3.2.3.2">ℝ</ci><apply id="A2.p3.5.m5.1.1.3.2.3.3.cmml" xref="A2.p3.5.m5.1.1.3.2.3.3"><csymbol cd="ambiguous" id="A2.p3.5.m5.1.1.3.2.3.3.1.cmml" xref="A2.p3.5.m5.1.1.3.2.3.3">subscript</csymbol><ci id="A2.p3.5.m5.1.1.3.2.3.3.2.cmml" xref="A2.p3.5.m5.1.1.3.2.3.3.2">𝑑</ci><ci id="A2.p3.5.m5.1.1.3.2.3.3.3a.cmml" xref="A2.p3.5.m5.1.1.3.2.3.3.3"><mtext mathsize="50%" id="A2.p3.5.m5.1.1.3.2.3.3.3.cmml" xref="A2.p3.5.m5.1.1.3.2.3.3.3">in</mtext></ci></apply></apply></apply><ci id="A2.p3.5.m5.1.1.3.3.cmml" xref="A2.p3.5.m5.1.1.3.3">ℳ</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p3.5.m5.1c">\mathrm{proj}_{\boldsymbol{\psi}}:\mathbb{R}^{d_{\textrm{in}}}\times\mathbb{R}^{d_{\textrm{in}}}\,\rightarrow\,\mathcal{M}</annotation></semantics></math> with</p>
<table id="A8.EGx7" class="ltx_equationgroup ltx_eqn_align ltx_eqn_table">

<tbody id="A2.E7"><tr class="ltx_equation ltx_eqn_row ltx_align_baseline">
<td class="ltx_eqn_cell ltx_eqn_center_padleft"></td>
<td class="ltx_td ltx_align_right ltx_eqn_cell"><math id="A2.E7.m1.3" class="ltx_Math" alttext="\displaystyle\mathrm{proj}_{\boldsymbol{\psi}}({\boldsymbol{x}},{\boldsymbol{v}})\approx{\boldsymbol{v}}\quad\forall\leavevmode\nobreak\ {\boldsymbol{x}}\sim\textrm{P}_{\mathrm{OOD}},\leavevmode\nobreak\ \leavevmode\nobreak\ ({\boldsymbol{x}}+{\boldsymbol{v}})\sim\textrm{P}_{\mathrm{OOD}}." display="inline"><semantics id="A2.E7.m1.3a"><mrow id="A2.E7.m1.3.3.1"><mrow id="A2.E7.m1.3.3.1.1.2" xref="A2.E7.m1.3.3.1.1.3.cmml"><mrow id="A2.E7.m1.3.3.1.1.1.1" xref="A2.E7.m1.3.3.1.1.1.1.cmml"><mrow id="A2.E7.m1.3.3.1.1.1.1.2" xref="A2.E7.m1.3.3.1.1.1.1.2.cmml"><msub id="A2.E7.m1.3.3.1.1.1.1.2.2" xref="A2.E7.m1.3.3.1.1.1.1.2.2.cmml"><mi id="A2.E7.m1.3.3.1.1.1.1.2.2.2" xref="A2.E7.m1.3.3.1.1.1.1.2.2.2.cmml">proj</mi><mi id="A2.E7.m1.3.3.1.1.1.1.2.2.3" xref="A2.E7.m1.3.3.1.1.1.1.2.2.3.cmml">𝝍</mi></msub><mo lspace="0em" rspace="0em" id="A2.E7.m1.3.3.1.1.1.1.2.1" xref="A2.E7.m1.3.3.1.1.1.1.2.1.cmml">​</mo><mrow id="A2.E7.m1.3.3.1.1.1.1.2.3.2" xref="A2.E7.m1.3.3.1.1.1.1.2.3.1.cmml"><mo stretchy="false" id="A2.E7.m1.3.3.1.1.1.1.2.3.2.1" xref="A2.E7.m1.3.3.1.1.1.1.2.3.1.cmml">(</mo><mi id="A2.E7.m1.1.1" xref="A2.E7.m1.1.1.cmml">𝒙</mi><mo id="A2.E7.m1.3.3.1.1.1.1.2.3.2.2" xref="A2.E7.m1.3.3.1.1.1.1.2.3.1.cmml">,</mo><mi id="A2.E7.m1.2.2" xref="A2.E7.m1.2.2.cmml">𝒗</mi><mo stretchy="false" id="A2.E7.m1.3.3.1.1.1.1.2.3.2.3" xref="A2.E7.m1.3.3.1.1.1.1.2.3.1.cmml">)</mo></mrow></mrow><mo id="A2.E7.m1.3.3.1.1.1.1.1" xref="A2.E7.m1.3.3.1.1.1.1.1.cmml">≈</mo><mi id="A2.E7.m1.3.3.1.1.1.1.3" xref="A2.E7.m1.3.3.1.1.1.1.3.cmml">𝒗</mi></mrow><mspace width="1.167em" id="A2.E7.m1.3.3.1.1.2.3" xref="A2.E7.m1.3.3.1.1.3a.cmml"></mspace><mrow id="A2.E7.m1.3.3.1.1.2.2.2" xref="A2.E7.m1.3.3.1.1.2.2.3.cmml"><mrow id="A2.E7.m1.3.3.1.1.2.2.1.1" xref="A2.E7.m1.3.3.1.1.2.2.1.1.cmml"><mrow id="A2.E7.m1.3.3.1.1.2.2.1.1.2" xref="A2.E7.m1.3.3.1.1.2.2.1.1.2.cmml"><mo rspace="0.667em" id="A2.E7.m1.3.3.1.1.2.2.1.1.2.1" xref="A2.E7.m1.3.3.1.1.2.2.1.1.2.1.cmml">∀</mo><mi id="A2.E7.m1.3.3.1.1.2.2.1.1.2.2" xref="A2.E7.m1.3.3.1.1.2.2.1.1.2.2.cmml">𝒙</mi></mrow><mo id="A2.E7.m1.3.3.1.1.2.2.1.1.1" xref="A2.E7.m1.3.3.1.1.2.2.1.1.1.cmml">∼</mo><msub id="A2.E7.m1.3.3.1.1.2.2.1.1.3" xref="A2.E7.m1.3.3.1.1.2.2.1.1.3.cmml"><mtext id="A2.E7.m1.3.3.1.1.2.2.1.1.3.2" xref="A2.E7.m1.3.3.1.1.2.2.1.1.3.2a.cmml">P</mtext><mi id="A2.E7.m1.3.3.1.1.2.2.1.1.3.3" xref="A2.E7.m1.3.3.1.1.2.2.1.1.3.3.cmml">OOD</mi></msub></mrow><mo rspace="1.167em" id="A2.E7.m1.3.3.1.1.2.2.2.3" xref="A2.E7.m1.3.3.1.1.2.2.3a.cmml">,</mo><mrow id="A2.E7.m1.3.3.1.1.2.2.2.2" xref="A2.E7.m1.3.3.1.1.2.2.2.2.cmml"><mrow id="A2.E7.m1.3.3.1.1.2.2.2.2.1.1" xref="A2.E7.m1.3.3.1.1.2.2.2.2.1.1.1.cmml"><mo stretchy="false" id="A2.E7.m1.3.3.1.1.2.2.2.2.1.1.2" xref="A2.E7.m1.3.3.1.1.2.2.2.2.1.1.1.cmml">(</mo><mrow id="A2.E7.m1.3.3.1.1.2.2.2.2.1.1.1" xref="A2.E7.m1.3.3.1.1.2.2.2.2.1.1.1.cmml"><mi id="A2.E7.m1.3.3.1.1.2.2.2.2.1.1.1.2" xref="A2.E7.m1.3.3.1.1.2.2.2.2.1.1.1.2.cmml">𝒙</mi><mo id="A2.E7.m1.3.3.1.1.2.2.2.2.1.1.1.1" xref="A2.E7.m1.3.3.1.1.2.2.2.2.1.1.1.1.cmml">+</mo><mi id="A2.E7.m1.3.3.1.1.2.2.2.2.1.1.1.3" xref="A2.E7.m1.3.3.1.1.2.2.2.2.1.1.1.3.cmml">𝒗</mi></mrow><mo stretchy="false" id="A2.E7.m1.3.3.1.1.2.2.2.2.1.1.3" xref="A2.E7.m1.3.3.1.1.2.2.2.2.1.1.1.cmml">)</mo></mrow><mo id="A2.E7.m1.3.3.1.1.2.2.2.2.2" xref="A2.E7.m1.3.3.1.1.2.2.2.2.2.cmml">∼</mo><msub id="A2.E7.m1.3.3.1.1.2.2.2.2.3" xref="A2.E7.m1.3.3.1.1.2.2.2.2.3.cmml"><mtext id="A2.E7.m1.3.3.1.1.2.2.2.2.3.2" xref="A2.E7.m1.3.3.1.1.2.2.2.2.3.2a.cmml">P</mtext><mi id="A2.E7.m1.3.3.1.1.2.2.2.2.3.3" xref="A2.E7.m1.3.3.1.1.2.2.2.2.3.3.cmml">OOD</mi></msub></mrow></mrow></mrow><mo lspace="0em" id="A2.E7.m1.3.3.1.2">.</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.E7.m1.3b"><apply id="A2.E7.m1.3.3.1.1.3.cmml" xref="A2.E7.m1.3.3.1.1.2"><csymbol cd="ambiguous" id="A2.E7.m1.3.3.1.1.3a.cmml" xref="A2.E7.m1.3.3.1.1.2.3">formulae-sequence</csymbol><apply id="A2.E7.m1.3.3.1.1.1.1.cmml" xref="A2.E7.m1.3.3.1.1.1.1"><approx id="A2.E7.m1.3.3.1.1.1.1.1.cmml" xref="A2.E7.m1.3.3.1.1.1.1.1"></approx><apply id="A2.E7.m1.3.3.1.1.1.1.2.cmml" xref="A2.E7.m1.3.3.1.1.1.1.2"><times id="A2.E7.m1.3.3.1.1.1.1.2.1.cmml" xref="A2.E7.m1.3.3.1.1.1.1.2.1"></times><apply id="A2.E7.m1.3.3.1.1.1.1.2.2.cmml" xref="A2.E7.m1.3.3.1.1.1.1.2.2"><csymbol cd="ambiguous" id="A2.E7.m1.3.3.1.1.1.1.2.2.1.cmml" xref="A2.E7.m1.3.3.1.1.1.1.2.2">subscript</csymbol><ci id="A2.E7.m1.3.3.1.1.1.1.2.2.2.cmml" xref="A2.E7.m1.3.3.1.1.1.1.2.2.2">proj</ci><ci id="A2.E7.m1.3.3.1.1.1.1.2.2.3.cmml" xref="A2.E7.m1.3.3.1.1.1.1.2.2.3">𝝍</ci></apply><interval closure="open" id="A2.E7.m1.3.3.1.1.1.1.2.3.1.cmml" xref="A2.E7.m1.3.3.1.1.1.1.2.3.2"><ci id="A2.E7.m1.1.1.cmml" xref="A2.E7.m1.1.1">𝒙</ci><ci id="A2.E7.m1.2.2.cmml" xref="A2.E7.m1.2.2">𝒗</ci></interval></apply><ci id="A2.E7.m1.3.3.1.1.1.1.3.cmml" xref="A2.E7.m1.3.3.1.1.1.1.3">𝒗</ci></apply><apply id="A2.E7.m1.3.3.1.1.2.2.3.cmml" xref="A2.E7.m1.3.3.1.1.2.2.2"><csymbol cd="ambiguous" id="A2.E7.m1.3.3.1.1.2.2.3a.cmml" xref="A2.E7.m1.3.3.1.1.2.2.2.3">formulae-sequence</csymbol><apply id="A2.E7.m1.3.3.1.1.2.2.1.1.cmml" xref="A2.E7.m1.3.3.1.1.2.2.1.1"><csymbol cd="latexml" id="A2.E7.m1.3.3.1.1.2.2.1.1.1.cmml" xref="A2.E7.m1.3.3.1.1.2.2.1.1.1">similar-to</csymbol><apply id="A2.E7.m1.3.3.1.1.2.2.1.1.2.cmml" xref="A2.E7.m1.3.3.1.1.2.2.1.1.2"><csymbol cd="latexml" id="A2.E7.m1.3.3.1.1.2.2.1.1.2.1.cmml" xref="A2.E7.m1.3.3.1.1.2.2.1.1.2.1">for-all</csymbol><ci id="A2.E7.m1.3.3.1.1.2.2.1.1.2.2.cmml" xref="A2.E7.m1.3.3.1.1.2.2.1.1.2.2">𝒙</ci></apply><apply id="A2.E7.m1.3.3.1.1.2.2.1.1.3.cmml" xref="A2.E7.m1.3.3.1.1.2.2.1.1.3"><csymbol cd="ambiguous" id="A2.E7.m1.3.3.1.1.2.2.1.1.3.1.cmml" xref="A2.E7.m1.3.3.1.1.2.2.1.1.3">subscript</csymbol><ci id="A2.E7.m1.3.3.1.1.2.2.1.1.3.2a.cmml" xref="A2.E7.m1.3.3.1.1.2.2.1.1.3.2"><mtext id="A2.E7.m1.3.3.1.1.2.2.1.1.3.2.cmml" xref="A2.E7.m1.3.3.1.1.2.2.1.1.3.2">P</mtext></ci><ci id="A2.E7.m1.3.3.1.1.2.2.1.1.3.3.cmml" xref="A2.E7.m1.3.3.1.1.2.2.1.1.3.3">OOD</ci></apply></apply><apply id="A2.E7.m1.3.3.1.1.2.2.2.2.cmml" xref="A2.E7.m1.3.3.1.1.2.2.2.2"><csymbol cd="latexml" id="A2.E7.m1.3.3.1.1.2.2.2.2.2.cmml" xref="A2.E7.m1.3.3.1.1.2.2.2.2.2">similar-to</csymbol><apply id="A2.E7.m1.3.3.1.1.2.2.2.2.1.1.1.cmml" xref="A2.E7.m1.3.3.1.1.2.2.2.2.1.1"><plus id="A2.E7.m1.3.3.1.1.2.2.2.2.1.1.1.1.cmml" xref="A2.E7.m1.3.3.1.1.2.2.2.2.1.1.1.1"></plus><ci id="A2.E7.m1.3.3.1.1.2.2.2.2.1.1.1.2.cmml" xref="A2.E7.m1.3.3.1.1.2.2.2.2.1.1.1.2">𝒙</ci><ci id="A2.E7.m1.3.3.1.1.2.2.2.2.1.1.1.3.cmml" xref="A2.E7.m1.3.3.1.1.2.2.2.2.1.1.1.3">𝒗</ci></apply><apply id="A2.E7.m1.3.3.1.1.2.2.2.2.3.cmml" xref="A2.E7.m1.3.3.1.1.2.2.2.2.3"><csymbol cd="ambiguous" id="A2.E7.m1.3.3.1.1.2.2.2.2.3.1.cmml" xref="A2.E7.m1.3.3.1.1.2.2.2.2.3">subscript</csymbol><ci id="A2.E7.m1.3.3.1.1.2.2.2.2.3.2a.cmml" xref="A2.E7.m1.3.3.1.1.2.2.2.2.3.2"><mtext id="A2.E7.m1.3.3.1.1.2.2.2.2.3.2.cmml" xref="A2.E7.m1.3.3.1.1.2.2.2.2.3.2">P</mtext></ci><ci id="A2.E7.m1.3.3.1.1.2.2.2.2.3.3.cmml" xref="A2.E7.m1.3.3.1.1.2.2.2.2.3.3">OOD</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.E7.m1.3c">\displaystyle\mathrm{proj}_{\boldsymbol{\psi}}({\boldsymbol{x}},{\boldsymbol{v}})\approx{\boldsymbol{v}}\quad\forall\leavevmode\nobreak\ {\boldsymbol{x}}\sim\textrm{P}_{\mathrm{OOD}},\leavevmode\nobreak\ \leavevmode\nobreak\ ({\boldsymbol{x}}+{\boldsymbol{v}})\sim\textrm{P}_{\mathrm{OOD}}.</annotation></semantics></math></td>
<td class="ltx_eqn_cell ltx_eqn_center_padright"></td>
<td rowspan="1" class="ltx_eqn_cell ltx_eqn_eqno ltx_align_middle ltx_align_right"><span class="ltx_tag ltx_tag_equation ltx_align_right">(7)</span></td>
</tr></tbody>
</table>
<p id="A2.p3.9" class="ltx_p">Here again, the reconstruction error <math id="A2.p3.6.m1.3" class="ltx_Math" alttext="||{\boldsymbol{v}}-\mathrm{proj}_{\boldsymbol{\psi}}({\boldsymbol{x}},{\boldsymbol{v}})||" display="inline"><semantics id="A2.p3.6.m1.3a"><mrow id="A2.p3.6.m1.3.3.1" xref="A2.p3.6.m1.3.3.2.cmml"><mo stretchy="false" id="A2.p3.6.m1.3.3.1.2" xref="A2.p3.6.m1.3.3.2.1.cmml">‖</mo><mrow id="A2.p3.6.m1.3.3.1.1" xref="A2.p3.6.m1.3.3.1.1.cmml"><mi id="A2.p3.6.m1.3.3.1.1.2" xref="A2.p3.6.m1.3.3.1.1.2.cmml">𝒗</mi><mo id="A2.p3.6.m1.3.3.1.1.1" xref="A2.p3.6.m1.3.3.1.1.1.cmml">−</mo><mrow id="A2.p3.6.m1.3.3.1.1.3" xref="A2.p3.6.m1.3.3.1.1.3.cmml"><msub id="A2.p3.6.m1.3.3.1.1.3.2" xref="A2.p3.6.m1.3.3.1.1.3.2.cmml"><mi id="A2.p3.6.m1.3.3.1.1.3.2.2" xref="A2.p3.6.m1.3.3.1.1.3.2.2.cmml">proj</mi><mi id="A2.p3.6.m1.3.3.1.1.3.2.3" xref="A2.p3.6.m1.3.3.1.1.3.2.3.cmml">𝝍</mi></msub><mo lspace="0em" rspace="0em" id="A2.p3.6.m1.3.3.1.1.3.1" xref="A2.p3.6.m1.3.3.1.1.3.1.cmml">​</mo><mrow id="A2.p3.6.m1.3.3.1.1.3.3.2" xref="A2.p3.6.m1.3.3.1.1.3.3.1.cmml"><mo stretchy="false" id="A2.p3.6.m1.3.3.1.1.3.3.2.1" xref="A2.p3.6.m1.3.3.1.1.3.3.1.cmml">(</mo><mi id="A2.p3.6.m1.1.1" xref="A2.p3.6.m1.1.1.cmml">𝒙</mi><mo id="A2.p3.6.m1.3.3.1.1.3.3.2.2" xref="A2.p3.6.m1.3.3.1.1.3.3.1.cmml">,</mo><mi id="A2.p3.6.m1.2.2" xref="A2.p3.6.m1.2.2.cmml">𝒗</mi><mo stretchy="false" id="A2.p3.6.m1.3.3.1.1.3.3.2.3" xref="A2.p3.6.m1.3.3.1.1.3.3.1.cmml">)</mo></mrow></mrow></mrow><mo stretchy="false" id="A2.p3.6.m1.3.3.1.3" xref="A2.p3.6.m1.3.3.2.1.cmml">‖</mo></mrow><annotation-xml encoding="MathML-Content" id="A2.p3.6.m1.3b"><apply id="A2.p3.6.m1.3.3.2.cmml" xref="A2.p3.6.m1.3.3.1"><csymbol cd="latexml" id="A2.p3.6.m1.3.3.2.1.cmml" xref="A2.p3.6.m1.3.3.1.2">norm</csymbol><apply id="A2.p3.6.m1.3.3.1.1.cmml" xref="A2.p3.6.m1.3.3.1.1"><minus id="A2.p3.6.m1.3.3.1.1.1.cmml" xref="A2.p3.6.m1.3.3.1.1.1"></minus><ci id="A2.p3.6.m1.3.3.1.1.2.cmml" xref="A2.p3.6.m1.3.3.1.1.2">𝒗</ci><apply id="A2.p3.6.m1.3.3.1.1.3.cmml" xref="A2.p3.6.m1.3.3.1.1.3"><times id="A2.p3.6.m1.3.3.1.1.3.1.cmml" xref="A2.p3.6.m1.3.3.1.1.3.1"></times><apply id="A2.p3.6.m1.3.3.1.1.3.2.cmml" xref="A2.p3.6.m1.3.3.1.1.3.2"><csymbol cd="ambiguous" id="A2.p3.6.m1.3.3.1.1.3.2.1.cmml" xref="A2.p3.6.m1.3.3.1.1.3.2">subscript</csymbol><ci id="A2.p3.6.m1.3.3.1.1.3.2.2.cmml" xref="A2.p3.6.m1.3.3.1.1.3.2.2">proj</ci><ci id="A2.p3.6.m1.3.3.1.1.3.2.3.cmml" xref="A2.p3.6.m1.3.3.1.1.3.2.3">𝝍</ci></apply><interval closure="open" id="A2.p3.6.m1.3.3.1.1.3.3.1.cmml" xref="A2.p3.6.m1.3.3.1.1.3.3.2"><ci id="A2.p3.6.m1.1.1.cmml" xref="A2.p3.6.m1.1.1">𝒙</ci><ci id="A2.p3.6.m1.2.2.cmml" xref="A2.p3.6.m1.2.2">𝒗</ci></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p3.6.m1.3c">||{\boldsymbol{v}}-\mathrm{proj}_{\boldsymbol{\psi}}({\boldsymbol{x}},{\boldsymbol{v}})||</annotation></semantics></math> is minimal for a point <math id="A2.p3.7.m2.1" class="ltx_Math" alttext="{\boldsymbol{x}}" display="inline"><semantics id="A2.p3.7.m2.1a"><mi id="A2.p3.7.m2.1.1" xref="A2.p3.7.m2.1.1.cmml">𝒙</mi><annotation-xml encoding="MathML-Content" id="A2.p3.7.m2.1b"><ci id="A2.p3.7.m2.1.1.cmml" xref="A2.p3.7.m2.1.1">𝒙</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.p3.7.m2.1c">{\boldsymbol{x}}</annotation></semantics></math> on the manifold and vector <math id="A2.p3.8.m3.1" class="ltx_Math" alttext="{\boldsymbol{v}}" display="inline"><semantics id="A2.p3.8.m3.1a"><mi id="A2.p3.8.m3.1.1" xref="A2.p3.8.m3.1.1.cmml">𝒗</mi><annotation-xml encoding="MathML-Content" id="A2.p3.8.m3.1b"><ci id="A2.p3.8.m3.1.1.cmml" xref="A2.p3.8.m3.1.1">𝒗</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.p3.8.m3.1c">{\boldsymbol{v}}</annotation></semantics></math> <span id="A2.p3.9.1" class="ltx_text ltx_font_bold">aligned with the manifold</span>.
We can therefore apply <math id="A2.p3.9.m4.1" class="ltx_Math" alttext="\mathrm{proj}(\cdot)" display="inline"><semantics id="A2.p3.9.m4.1a"><mrow id="A2.p3.9.m4.1.2" xref="A2.p3.9.m4.1.2.cmml"><mi id="A2.p3.9.m4.1.2.2" xref="A2.p3.9.m4.1.2.2.cmml">proj</mi><mo lspace="0em" rspace="0em" id="A2.p3.9.m4.1.2.1" xref="A2.p3.9.m4.1.2.1.cmml">​</mo><mrow id="A2.p3.9.m4.1.2.3.2" xref="A2.p3.9.m4.1.2.cmml"><mo stretchy="false" id="A2.p3.9.m4.1.2.3.2.1" xref="A2.p3.9.m4.1.2.cmml">(</mo><mo lspace="0em" rspace="0em" id="A2.p3.9.m4.1.1" xref="A2.p3.9.m4.1.1.cmml">⋅</mo><mo stretchy="false" id="A2.p3.9.m4.1.2.3.2.2" xref="A2.p3.9.m4.1.2.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.p3.9.m4.1b"><apply id="A2.p3.9.m4.1.2.cmml" xref="A2.p3.9.m4.1.2"><times id="A2.p3.9.m4.1.2.1.cmml" xref="A2.p3.9.m4.1.2.1"></times><ci id="A2.p3.9.m4.1.2.2.cmml" xref="A2.p3.9.m4.1.2.2">proj</ci><ci id="A2.p3.9.m4.1.1.cmml" xref="A2.p3.9.m4.1.1">⋅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.p3.9.m4.1c">\mathrm{proj}(\cdot)</annotation></semantics></math> to our input gradients, measure the distance with their projection on the manifold, and use this distance as our on-manifold loss in Eq. (<a href="#S4.E2" title="In 4.2 On-manifold constraint ‣ 4 Proposed method ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">2</span></a>).</p>
</div>
<div id="A2.p4" class="ltx_para ltx_noindent">
<p id="A2.p4.2" class="ltx_p"><span id="A2.p4.2.1" class="ltx_text ltx_font_bold">Reconstructing vectors with an auto-encoder.</span> 

This section describes how to take a standard auto-encoder, typically trained to reconstruct a point <math id="A2.p4.1.m1.1" class="ltx_Math" alttext="{\boldsymbol{x}}" display="inline"><semantics id="A2.p4.1.m1.1a"><mi id="A2.p4.1.m1.1.1" xref="A2.p4.1.m1.1.1.cmml">𝒙</mi><annotation-xml encoding="MathML-Content" id="A2.p4.1.m1.1b"><ci id="A2.p4.1.m1.1.1.cmml" xref="A2.p4.1.m1.1.1">𝒙</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.p4.1.m1.1c">{\boldsymbol{x}}</annotation></semantics></math> of its input space, and use it to reconstruct a vector <math id="A2.p4.2.m2.1" class="ltx_Math" alttext="{\boldsymbol{v}}" display="inline"><semantics id="A2.p4.2.m2.1a"><mi id="A2.p4.2.m2.1.1" xref="A2.p4.2.m2.1.1.cmml">𝒗</mi><annotation-xml encoding="MathML-Content" id="A2.p4.2.m2.1b"><ci id="A2.p4.2.m2.1.1.cmml" xref="A2.p4.2.m2.1.1">𝒗</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.p4.2.m2.1c">{\boldsymbol{v}}</annotation></semantics></math> in this space (input gradients in our case).
The auto-encoders used this work are compositions of linear layers, ReLU activations, and a sigmoid output activation.
Table <a href="#A2.T4" title="Table 4 ‣ Appendix B Projections on the manifold ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">4</span></a> provides the equivalent operations performed during forward propagation in any such layer for reconstructing an input or gradient.</p>
</div>
<figure id="A2.T4" class="ltx_table">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table"><span id="A2.T4.10.1.1" class="ltx_text" style="font-size:90%;">Table 4</span>: </span><span id="A2.T4.11.2" class="ltx_text" style="font-size:90%;">Operations performed at each layer of a VAE during forward propagation for reconstructing points (left column) and vectors/gradients (right column).</span></figcaption>
<table id="A2.T4.8" class="ltx_tabular ltx_centering ltx_align_middle">
<tr id="A2.T4.2.2" class="ltx_tr">
<td id="A2.T4.1.1.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left ltx_border_tt" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="A2.T4.1.1.1.1" class="ltx_text ltx_font_bold" style="font-size:90%;">Operation on point <math id="A2.T4.1.1.1.1.m1.1" class="ltx_Math" alttext="{\boldsymbol{x}}" display="inline"><semantics id="A2.T4.1.1.1.1.m1.1a"><mi id="A2.T4.1.1.1.1.m1.1.1" xref="A2.T4.1.1.1.1.m1.1.1.cmml">x</mi><annotation-xml encoding="MathML-Content" id="A2.T4.1.1.1.1.m1.1b"><ci id="A2.T4.1.1.1.1.m1.1.1.cmml" xref="A2.T4.1.1.1.1.m1.1.1">𝑥</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.T4.1.1.1.1.m1.1c">{\boldsymbol{x}}</annotation></semantics></math></span></td>
<td id="A2.T4.2.2.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left ltx_border_tt" style="padding-left:0.0pt;padding-right:0.0pt;"><span id="A2.T4.2.2.2.1" class="ltx_text ltx_font_bold" style="font-size:90%;">Operation on vector <math id="A2.T4.2.2.2.1.m1.1" class="ltx_Math" alttext="{\boldsymbol{v}}" display="inline"><semantics id="A2.T4.2.2.2.1.m1.1a"><mi id="A2.T4.2.2.2.1.m1.1.1" xref="A2.T4.2.2.2.1.m1.1.1.cmml">v</mi><annotation-xml encoding="MathML-Content" id="A2.T4.2.2.2.1.m1.1b"><ci id="A2.T4.2.2.2.1.m1.1.1.cmml" xref="A2.T4.2.2.2.1.m1.1.1">𝑣</ci></annotation-xml><annotation encoding="application/x-tex" id="A2.T4.2.2.2.1.m1.1c">{\boldsymbol{v}}</annotation></semantics></math></span></td>
</tr>
<tr id="A2.T4.8.9" class="ltx_tr">
<td id="A2.T4.8.9.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;" colspan="2"><span id="A2.T4.8.9.1.1" class="ltx_text" style="font-size:90%;">Linear layer</span></td>
</tr>
<tr id="A2.T4.4.4" class="ltx_tr">
<td id="A2.T4.3.3.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding-left:0.0pt;padding-right:0.0pt;"><math id="A2.T4.3.3.1.m1.1" class="ltx_Math" alttext="{\boldsymbol{x}}\,\leftarrow\,{\boldsymbol{W}}{\boldsymbol{x}}\,+\,\boldsymbol{b}" display="inline"><semantics id="A2.T4.3.3.1.m1.1a"><mrow id="A2.T4.3.3.1.m1.1.1" xref="A2.T4.3.3.1.m1.1.1.cmml"><mi mathsize="90%" id="A2.T4.3.3.1.m1.1.1.2" xref="A2.T4.3.3.1.m1.1.1.2.cmml">𝒙</mi><mo lspace="0.448em" mathsize="90%" rspace="0.448em" stretchy="false" id="A2.T4.3.3.1.m1.1.1.1" xref="A2.T4.3.3.1.m1.1.1.1.cmml">←</mo><mrow id="A2.T4.3.3.1.m1.1.1.3" xref="A2.T4.3.3.1.m1.1.1.3.cmml"><mrow id="A2.T4.3.3.1.m1.1.1.3.2" xref="A2.T4.3.3.1.m1.1.1.3.2.cmml"><mi mathsize="90%" id="A2.T4.3.3.1.m1.1.1.3.2.2" xref="A2.T4.3.3.1.m1.1.1.3.2.2.cmml">𝑾</mi><mo lspace="0em" rspace="0em" id="A2.T4.3.3.1.m1.1.1.3.2.1" xref="A2.T4.3.3.1.m1.1.1.3.2.1.cmml">​</mo><mi mathsize="90%" id="A2.T4.3.3.1.m1.1.1.3.2.3" xref="A2.T4.3.3.1.m1.1.1.3.2.3.cmml">𝒙</mi></mrow><mo lspace="0.392em" mathsize="90%" rspace="0.392em" id="A2.T4.3.3.1.m1.1.1.3.1" xref="A2.T4.3.3.1.m1.1.1.3.1.cmml">+</mo><mi mathsize="90%" id="A2.T4.3.3.1.m1.1.1.3.3" xref="A2.T4.3.3.1.m1.1.1.3.3.cmml">𝒃</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.T4.3.3.1.m1.1b"><apply id="A2.T4.3.3.1.m1.1.1.cmml" xref="A2.T4.3.3.1.m1.1.1"><ci id="A2.T4.3.3.1.m1.1.1.1.cmml" xref="A2.T4.3.3.1.m1.1.1.1">←</ci><ci id="A2.T4.3.3.1.m1.1.1.2.cmml" xref="A2.T4.3.3.1.m1.1.1.2">𝒙</ci><apply id="A2.T4.3.3.1.m1.1.1.3.cmml" xref="A2.T4.3.3.1.m1.1.1.3"><plus id="A2.T4.3.3.1.m1.1.1.3.1.cmml" xref="A2.T4.3.3.1.m1.1.1.3.1"></plus><apply id="A2.T4.3.3.1.m1.1.1.3.2.cmml" xref="A2.T4.3.3.1.m1.1.1.3.2"><times id="A2.T4.3.3.1.m1.1.1.3.2.1.cmml" xref="A2.T4.3.3.1.m1.1.1.3.2.1"></times><ci id="A2.T4.3.3.1.m1.1.1.3.2.2.cmml" xref="A2.T4.3.3.1.m1.1.1.3.2.2">𝑾</ci><ci id="A2.T4.3.3.1.m1.1.1.3.2.3.cmml" xref="A2.T4.3.3.1.m1.1.1.3.2.3">𝒙</ci></apply><ci id="A2.T4.3.3.1.m1.1.1.3.3.cmml" xref="A2.T4.3.3.1.m1.1.1.3.3">𝒃</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.T4.3.3.1.m1.1c">{\boldsymbol{x}}\,\leftarrow\,{\boldsymbol{W}}{\boldsymbol{x}}\,+\,\boldsymbol{b}</annotation></semantics></math></td>
<td id="A2.T4.4.4.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding-left:0.0pt;padding-right:0.0pt;"><math id="A2.T4.4.4.2.m1.1" class="ltx_Math" alttext="{\boldsymbol{v}}\,\leftarrow\,{\boldsymbol{W}}{\boldsymbol{v}}" display="inline"><semantics id="A2.T4.4.4.2.m1.1a"><mrow id="A2.T4.4.4.2.m1.1.1" xref="A2.T4.4.4.2.m1.1.1.cmml"><mi mathsize="90%" id="A2.T4.4.4.2.m1.1.1.2" xref="A2.T4.4.4.2.m1.1.1.2.cmml">𝒗</mi><mo lspace="0.448em" mathsize="90%" rspace="0.448em" stretchy="false" id="A2.T4.4.4.2.m1.1.1.1" xref="A2.T4.4.4.2.m1.1.1.1.cmml">←</mo><mrow id="A2.T4.4.4.2.m1.1.1.3" xref="A2.T4.4.4.2.m1.1.1.3.cmml"><mi mathsize="90%" id="A2.T4.4.4.2.m1.1.1.3.2" xref="A2.T4.4.4.2.m1.1.1.3.2.cmml">𝑾</mi><mo lspace="0em" rspace="0em" id="A2.T4.4.4.2.m1.1.1.3.1" xref="A2.T4.4.4.2.m1.1.1.3.1.cmml">​</mo><mi mathsize="90%" id="A2.T4.4.4.2.m1.1.1.3.3" xref="A2.T4.4.4.2.m1.1.1.3.3.cmml">𝒗</mi></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.T4.4.4.2.m1.1b"><apply id="A2.T4.4.4.2.m1.1.1.cmml" xref="A2.T4.4.4.2.m1.1.1"><ci id="A2.T4.4.4.2.m1.1.1.1.cmml" xref="A2.T4.4.4.2.m1.1.1.1">←</ci><ci id="A2.T4.4.4.2.m1.1.1.2.cmml" xref="A2.T4.4.4.2.m1.1.1.2">𝒗</ci><apply id="A2.T4.4.4.2.m1.1.1.3.cmml" xref="A2.T4.4.4.2.m1.1.1.3"><times id="A2.T4.4.4.2.m1.1.1.3.1.cmml" xref="A2.T4.4.4.2.m1.1.1.3.1"></times><ci id="A2.T4.4.4.2.m1.1.1.3.2.cmml" xref="A2.T4.4.4.2.m1.1.1.3.2">𝑾</ci><ci id="A2.T4.4.4.2.m1.1.1.3.3.cmml" xref="A2.T4.4.4.2.m1.1.1.3.3">𝒗</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.T4.4.4.2.m1.1c">{\boldsymbol{v}}\,\leftarrow\,{\boldsymbol{W}}{\boldsymbol{v}}</annotation></semantics></math></td>
</tr>
<tr id="A2.T4.8.10" class="ltx_tr">
<td id="A2.T4.8.10.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;" colspan="2"><span id="A2.T4.8.10.1.1" class="ltx_text" style="font-size:90%;">ReLU activation</span></td>
</tr>
<tr id="A2.T4.6.6" class="ltx_tr">
<td id="A2.T4.5.5.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding-left:0.0pt;padding-right:0.0pt;"><math id="A2.T4.5.5.1.m1.2" class="ltx_Math" alttext="{\boldsymbol{x}}\,\leftarrow\,{\boldsymbol{x}}\,\odot\,\operatorname*{\mathbbm{1}}({\boldsymbol{x}}&gt;0)" display="inline"><semantics id="A2.T4.5.5.1.m1.2a"><mrow id="A2.T4.5.5.1.m1.2.2" xref="A2.T4.5.5.1.m1.2.2.cmml"><mi mathsize="90%" id="A2.T4.5.5.1.m1.2.2.3" xref="A2.T4.5.5.1.m1.2.2.3.cmml">𝒙</mi><mo lspace="0.448em" mathsize="90%" rspace="0.448em" stretchy="false" id="A2.T4.5.5.1.m1.2.2.2" xref="A2.T4.5.5.1.m1.2.2.2.cmml">←</mo><mrow id="A2.T4.5.5.1.m1.2.2.1" xref="A2.T4.5.5.1.m1.2.2.1.cmml"><mi mathsize="90%" id="A2.T4.5.5.1.m1.2.2.1.3" xref="A2.T4.5.5.1.m1.2.2.1.3.cmml">𝒙</mi><mo lspace="0.392em" mathsize="90%" rspace="0.114em" id="A2.T4.5.5.1.m1.2.2.1.2" xref="A2.T4.5.5.1.m1.2.2.1.2.cmml">⊙</mo><mrow id="A2.T4.5.5.1.m1.2.2.1.1.1" xref="A2.T4.5.5.1.m1.2.2.1.1.2.cmml"><mo mathsize="90%" rspace="0em" id="A2.T4.5.5.1.m1.1.1" xref="A2.T4.5.5.1.m1.1.1.cmml">𝟙</mo><mrow id="A2.T4.5.5.1.m1.2.2.1.1.1.1" xref="A2.T4.5.5.1.m1.2.2.1.1.2.cmml"><mo maxsize="90%" minsize="90%" id="A2.T4.5.5.1.m1.2.2.1.1.1.1.2" xref="A2.T4.5.5.1.m1.2.2.1.1.2.cmml">(</mo><mrow id="A2.T4.5.5.1.m1.2.2.1.1.1.1.1" xref="A2.T4.5.5.1.m1.2.2.1.1.1.1.1.cmml"><mi mathsize="90%" id="A2.T4.5.5.1.m1.2.2.1.1.1.1.1.2" xref="A2.T4.5.5.1.m1.2.2.1.1.1.1.1.2.cmml">𝒙</mi><mo mathsize="90%" id="A2.T4.5.5.1.m1.2.2.1.1.1.1.1.1" xref="A2.T4.5.5.1.m1.2.2.1.1.1.1.1.1.cmml">&gt;</mo><mn mathsize="90%" id="A2.T4.5.5.1.m1.2.2.1.1.1.1.1.3" xref="A2.T4.5.5.1.m1.2.2.1.1.1.1.1.3.cmml">0</mn></mrow><mo maxsize="90%" minsize="90%" id="A2.T4.5.5.1.m1.2.2.1.1.1.1.3" xref="A2.T4.5.5.1.m1.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.T4.5.5.1.m1.2b"><apply id="A2.T4.5.5.1.m1.2.2.cmml" xref="A2.T4.5.5.1.m1.2.2"><ci id="A2.T4.5.5.1.m1.2.2.2.cmml" xref="A2.T4.5.5.1.m1.2.2.2">←</ci><ci id="A2.T4.5.5.1.m1.2.2.3.cmml" xref="A2.T4.5.5.1.m1.2.2.3">𝒙</ci><apply id="A2.T4.5.5.1.m1.2.2.1.cmml" xref="A2.T4.5.5.1.m1.2.2.1"><csymbol cd="latexml" id="A2.T4.5.5.1.m1.2.2.1.2.cmml" xref="A2.T4.5.5.1.m1.2.2.1.2">direct-product</csymbol><ci id="A2.T4.5.5.1.m1.2.2.1.3.cmml" xref="A2.T4.5.5.1.m1.2.2.1.3">𝒙</ci><apply id="A2.T4.5.5.1.m1.2.2.1.1.2.cmml" xref="A2.T4.5.5.1.m1.2.2.1.1.1"><csymbol cd="latexml" id="A2.T4.5.5.1.m1.1.1.cmml" xref="A2.T4.5.5.1.m1.1.1">1</csymbol><apply id="A2.T4.5.5.1.m1.2.2.1.1.1.1.1.cmml" xref="A2.T4.5.5.1.m1.2.2.1.1.1.1.1"><gt id="A2.T4.5.5.1.m1.2.2.1.1.1.1.1.1.cmml" xref="A2.T4.5.5.1.m1.2.2.1.1.1.1.1.1"></gt><ci id="A2.T4.5.5.1.m1.2.2.1.1.1.1.1.2.cmml" xref="A2.T4.5.5.1.m1.2.2.1.1.1.1.1.2">𝒙</ci><cn type="integer" id="A2.T4.5.5.1.m1.2.2.1.1.1.1.1.3.cmml" xref="A2.T4.5.5.1.m1.2.2.1.1.1.1.1.3">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.T4.5.5.1.m1.2c">{\boldsymbol{x}}\,\leftarrow\,{\boldsymbol{x}}\,\odot\,\operatorname*{\mathbbm{1}}({\boldsymbol{x}}&gt;0)</annotation></semantics></math></td>
<td id="A2.T4.6.6.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding-left:0.0pt;padding-right:0.0pt;"><math id="A2.T4.6.6.2.m1.2" class="ltx_Math" alttext="{\boldsymbol{v}}\,\leftarrow\,{\boldsymbol{v}}\,\odot\,\operatorname*{\mathbbm{1}}({\boldsymbol{x}}&gt;0)" display="inline"><semantics id="A2.T4.6.6.2.m1.2a"><mrow id="A2.T4.6.6.2.m1.2.2" xref="A2.T4.6.6.2.m1.2.2.cmml"><mi mathsize="90%" id="A2.T4.6.6.2.m1.2.2.3" xref="A2.T4.6.6.2.m1.2.2.3.cmml">𝒗</mi><mo lspace="0.448em" mathsize="90%" rspace="0.448em" stretchy="false" id="A2.T4.6.6.2.m1.2.2.2" xref="A2.T4.6.6.2.m1.2.2.2.cmml">←</mo><mrow id="A2.T4.6.6.2.m1.2.2.1" xref="A2.T4.6.6.2.m1.2.2.1.cmml"><mi mathsize="90%" id="A2.T4.6.6.2.m1.2.2.1.3" xref="A2.T4.6.6.2.m1.2.2.1.3.cmml">𝒗</mi><mo lspace="0.392em" mathsize="90%" rspace="0.114em" id="A2.T4.6.6.2.m1.2.2.1.2" xref="A2.T4.6.6.2.m1.2.2.1.2.cmml">⊙</mo><mrow id="A2.T4.6.6.2.m1.2.2.1.1.1" xref="A2.T4.6.6.2.m1.2.2.1.1.2.cmml"><mo mathsize="90%" rspace="0em" id="A2.T4.6.6.2.m1.1.1" xref="A2.T4.6.6.2.m1.1.1.cmml">𝟙</mo><mrow id="A2.T4.6.6.2.m1.2.2.1.1.1.1" xref="A2.T4.6.6.2.m1.2.2.1.1.2.cmml"><mo maxsize="90%" minsize="90%" id="A2.T4.6.6.2.m1.2.2.1.1.1.1.2" xref="A2.T4.6.6.2.m1.2.2.1.1.2.cmml">(</mo><mrow id="A2.T4.6.6.2.m1.2.2.1.1.1.1.1" xref="A2.T4.6.6.2.m1.2.2.1.1.1.1.1.cmml"><mi mathsize="90%" id="A2.T4.6.6.2.m1.2.2.1.1.1.1.1.2" xref="A2.T4.6.6.2.m1.2.2.1.1.1.1.1.2.cmml">𝒙</mi><mo mathsize="90%" id="A2.T4.6.6.2.m1.2.2.1.1.1.1.1.1" xref="A2.T4.6.6.2.m1.2.2.1.1.1.1.1.1.cmml">&gt;</mo><mn mathsize="90%" id="A2.T4.6.6.2.m1.2.2.1.1.1.1.1.3" xref="A2.T4.6.6.2.m1.2.2.1.1.1.1.1.3.cmml">0</mn></mrow><mo maxsize="90%" minsize="90%" id="A2.T4.6.6.2.m1.2.2.1.1.1.1.3" xref="A2.T4.6.6.2.m1.2.2.1.1.2.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.T4.6.6.2.m1.2b"><apply id="A2.T4.6.6.2.m1.2.2.cmml" xref="A2.T4.6.6.2.m1.2.2"><ci id="A2.T4.6.6.2.m1.2.2.2.cmml" xref="A2.T4.6.6.2.m1.2.2.2">←</ci><ci id="A2.T4.6.6.2.m1.2.2.3.cmml" xref="A2.T4.6.6.2.m1.2.2.3">𝒗</ci><apply id="A2.T4.6.6.2.m1.2.2.1.cmml" xref="A2.T4.6.6.2.m1.2.2.1"><csymbol cd="latexml" id="A2.T4.6.6.2.m1.2.2.1.2.cmml" xref="A2.T4.6.6.2.m1.2.2.1.2">direct-product</csymbol><ci id="A2.T4.6.6.2.m1.2.2.1.3.cmml" xref="A2.T4.6.6.2.m1.2.2.1.3">𝒗</ci><apply id="A2.T4.6.6.2.m1.2.2.1.1.2.cmml" xref="A2.T4.6.6.2.m1.2.2.1.1.1"><csymbol cd="latexml" id="A2.T4.6.6.2.m1.1.1.cmml" xref="A2.T4.6.6.2.m1.1.1">1</csymbol><apply id="A2.T4.6.6.2.m1.2.2.1.1.1.1.1.cmml" xref="A2.T4.6.6.2.m1.2.2.1.1.1.1.1"><gt id="A2.T4.6.6.2.m1.2.2.1.1.1.1.1.1.cmml" xref="A2.T4.6.6.2.m1.2.2.1.1.1.1.1.1"></gt><ci id="A2.T4.6.6.2.m1.2.2.1.1.1.1.1.2.cmml" xref="A2.T4.6.6.2.m1.2.2.1.1.1.1.1.2">𝒙</ci><cn type="integer" id="A2.T4.6.6.2.m1.2.2.1.1.1.1.1.3.cmml" xref="A2.T4.6.6.2.m1.2.2.1.1.1.1.1.3">0</cn></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.T4.6.6.2.m1.2c">{\boldsymbol{v}}\,\leftarrow\,{\boldsymbol{v}}\,\odot\,\operatorname*{\mathbbm{1}}({\boldsymbol{x}}&gt;0)</annotation></semantics></math></td>
</tr>
<tr id="A2.T4.8.11" class="ltx_tr">
<td id="A2.T4.8.11.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left ltx_border_t" style="padding-left:0.0pt;padding-right:0.0pt;" colspan="2"><span id="A2.T4.8.11.1.1" class="ltx_text" style="font-size:90%;">Sigmoid activation</span></td>
</tr>
<tr id="A2.T4.8.8" class="ltx_tr">
<td id="A2.T4.7.7.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left ltx_border_bb" style="padding-left:0.0pt;padding-right:0.0pt;"><math id="A2.T4.7.7.1.m1.3" class="ltx_Math" alttext="x\,\leftarrow\,\operatorname*{\sigma}(x)=1/(1+e^{-x})" display="inline"><semantics id="A2.T4.7.7.1.m1.3a"><mrow id="A2.T4.7.7.1.m1.3.3" xref="A2.T4.7.7.1.m1.3.3.cmml"><mi mathsize="90%" id="A2.T4.7.7.1.m1.3.3.3" xref="A2.T4.7.7.1.m1.3.3.3.cmml">x</mi><mo lspace="0.448em" mathsize="90%" rspace="0.170em" stretchy="false" id="A2.T4.7.7.1.m1.3.3.4" xref="A2.T4.7.7.1.m1.3.3.4.cmml">←</mo><mrow id="A2.T4.7.7.1.m1.3.3.5.2" xref="A2.T4.7.7.1.m1.3.3.5.1.cmml"><mo mathsize="90%" rspace="0em" id="A2.T4.7.7.1.m1.1.1" xref="A2.T4.7.7.1.m1.1.1.cmml">𝜎</mo><mrow id="A2.T4.7.7.1.m1.3.3.5.2.1" xref="A2.T4.7.7.1.m1.3.3.5.1.cmml"><mo maxsize="90%" minsize="90%" id="A2.T4.7.7.1.m1.3.3.5.2.1.1" xref="A2.T4.7.7.1.m1.3.3.5.1.cmml">(</mo><mi mathsize="90%" id="A2.T4.7.7.1.m1.2.2" xref="A2.T4.7.7.1.m1.2.2.cmml">x</mi><mo maxsize="90%" minsize="90%" id="A2.T4.7.7.1.m1.3.3.5.2.1.2" xref="A2.T4.7.7.1.m1.3.3.5.1.cmml">)</mo></mrow></mrow><mo mathsize="90%" id="A2.T4.7.7.1.m1.3.3.6" xref="A2.T4.7.7.1.m1.3.3.6.cmml">=</mo><mrow id="A2.T4.7.7.1.m1.3.3.1" xref="A2.T4.7.7.1.m1.3.3.1.cmml"><mn mathsize="90%" id="A2.T4.7.7.1.m1.3.3.1.3" xref="A2.T4.7.7.1.m1.3.3.1.3.cmml">1</mn><mo maxsize="90%" minsize="90%" stretchy="true" symmetric="true" id="A2.T4.7.7.1.m1.3.3.1.2" xref="A2.T4.7.7.1.m1.3.3.1.2.cmml">/</mo><mrow id="A2.T4.7.7.1.m1.3.3.1.1.1" xref="A2.T4.7.7.1.m1.3.3.1.1.1.1.cmml"><mo maxsize="90%" minsize="90%" id="A2.T4.7.7.1.m1.3.3.1.1.1.2" xref="A2.T4.7.7.1.m1.3.3.1.1.1.1.cmml">(</mo><mrow id="A2.T4.7.7.1.m1.3.3.1.1.1.1" xref="A2.T4.7.7.1.m1.3.3.1.1.1.1.cmml"><mn mathsize="90%" id="A2.T4.7.7.1.m1.3.3.1.1.1.1.2" xref="A2.T4.7.7.1.m1.3.3.1.1.1.1.2.cmml">1</mn><mo mathsize="90%" id="A2.T4.7.7.1.m1.3.3.1.1.1.1.1" xref="A2.T4.7.7.1.m1.3.3.1.1.1.1.1.cmml">+</mo><msup id="A2.T4.7.7.1.m1.3.3.1.1.1.1.3" xref="A2.T4.7.7.1.m1.3.3.1.1.1.1.3.cmml"><mi mathsize="90%" id="A2.T4.7.7.1.m1.3.3.1.1.1.1.3.2" xref="A2.T4.7.7.1.m1.3.3.1.1.1.1.3.2.cmml">e</mi><mrow id="A2.T4.7.7.1.m1.3.3.1.1.1.1.3.3" xref="A2.T4.7.7.1.m1.3.3.1.1.1.1.3.3.cmml"><mo mathsize="90%" id="A2.T4.7.7.1.m1.3.3.1.1.1.1.3.3a" xref="A2.T4.7.7.1.m1.3.3.1.1.1.1.3.3.cmml">−</mo><mi mathsize="90%" id="A2.T4.7.7.1.m1.3.3.1.1.1.1.3.3.2" xref="A2.T4.7.7.1.m1.3.3.1.1.1.1.3.3.2.cmml">x</mi></mrow></msup></mrow><mo maxsize="90%" minsize="90%" id="A2.T4.7.7.1.m1.3.3.1.1.1.3" xref="A2.T4.7.7.1.m1.3.3.1.1.1.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.T4.7.7.1.m1.3b"><apply id="A2.T4.7.7.1.m1.3.3.cmml" xref="A2.T4.7.7.1.m1.3.3"><and id="A2.T4.7.7.1.m1.3.3a.cmml" xref="A2.T4.7.7.1.m1.3.3"></and><apply id="A2.T4.7.7.1.m1.3.3b.cmml" xref="A2.T4.7.7.1.m1.3.3"><ci id="A2.T4.7.7.1.m1.3.3.4.cmml" xref="A2.T4.7.7.1.m1.3.3.4">←</ci><ci id="A2.T4.7.7.1.m1.3.3.3.cmml" xref="A2.T4.7.7.1.m1.3.3.3">𝑥</ci><apply id="A2.T4.7.7.1.m1.3.3.5.1.cmml" xref="A2.T4.7.7.1.m1.3.3.5.2"><ci id="A2.T4.7.7.1.m1.1.1.cmml" xref="A2.T4.7.7.1.m1.1.1">𝜎</ci><ci id="A2.T4.7.7.1.m1.2.2.cmml" xref="A2.T4.7.7.1.m1.2.2">𝑥</ci></apply></apply><apply id="A2.T4.7.7.1.m1.3.3c.cmml" xref="A2.T4.7.7.1.m1.3.3"><eq id="A2.T4.7.7.1.m1.3.3.6.cmml" xref="A2.T4.7.7.1.m1.3.3.6"></eq><share href="#A2.T4.7.7.1.m1.3.3.5.cmml" id="A2.T4.7.7.1.m1.3.3d.cmml" xref="A2.T4.7.7.1.m1.3.3"></share><apply id="A2.T4.7.7.1.m1.3.3.1.cmml" xref="A2.T4.7.7.1.m1.3.3.1"><divide id="A2.T4.7.7.1.m1.3.3.1.2.cmml" xref="A2.T4.7.7.1.m1.3.3.1.2"></divide><cn type="integer" id="A2.T4.7.7.1.m1.3.3.1.3.cmml" xref="A2.T4.7.7.1.m1.3.3.1.3">1</cn><apply id="A2.T4.7.7.1.m1.3.3.1.1.1.1.cmml" xref="A2.T4.7.7.1.m1.3.3.1.1.1"><plus id="A2.T4.7.7.1.m1.3.3.1.1.1.1.1.cmml" xref="A2.T4.7.7.1.m1.3.3.1.1.1.1.1"></plus><cn type="integer" id="A2.T4.7.7.1.m1.3.3.1.1.1.1.2.cmml" xref="A2.T4.7.7.1.m1.3.3.1.1.1.1.2">1</cn><apply id="A2.T4.7.7.1.m1.3.3.1.1.1.1.3.cmml" xref="A2.T4.7.7.1.m1.3.3.1.1.1.1.3"><csymbol cd="ambiguous" id="A2.T4.7.7.1.m1.3.3.1.1.1.1.3.1.cmml" xref="A2.T4.7.7.1.m1.3.3.1.1.1.1.3">superscript</csymbol><ci id="A2.T4.7.7.1.m1.3.3.1.1.1.1.3.2.cmml" xref="A2.T4.7.7.1.m1.3.3.1.1.1.1.3.2">𝑒</ci><apply id="A2.T4.7.7.1.m1.3.3.1.1.1.1.3.3.cmml" xref="A2.T4.7.7.1.m1.3.3.1.1.1.1.3.3"><minus id="A2.T4.7.7.1.m1.3.3.1.1.1.1.3.3.1.cmml" xref="A2.T4.7.7.1.m1.3.3.1.1.1.1.3.3"></minus><ci id="A2.T4.7.7.1.m1.3.3.1.1.1.1.3.3.2.cmml" xref="A2.T4.7.7.1.m1.3.3.1.1.1.1.3.3.2">𝑥</ci></apply></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.T4.7.7.1.m1.3c">x\,\leftarrow\,\operatorname*{\sigma}(x)=1/(1+e^{-x})</annotation></semantics></math></td>
<td id="A2.T4.8.8.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left ltx_border_bb" style="padding-left:0.0pt;padding-right:0.0pt;"><math id="A2.T4.8.8.2.m1.6" class="ltx_Math" alttext="{\boldsymbol{v}}\,\leftarrow\,{\boldsymbol{v}}\;\operatorname*{\sigma}(x)\;\big{(}1-\operatorname*{\sigma}({\boldsymbol{x}})\big{)}" display="inline"><semantics id="A2.T4.8.8.2.m1.6a"><mrow id="A2.T4.8.8.2.m1.6.6" xref="A2.T4.8.8.2.m1.6.6.cmml"><mi mathsize="90%" id="A2.T4.8.8.2.m1.6.6.4" xref="A2.T4.8.8.2.m1.6.6.4.cmml">𝒗</mi><mo lspace="0.448em" mathsize="90%" rspace="0.448em" stretchy="false" id="A2.T4.8.8.2.m1.6.6.3" xref="A2.T4.8.8.2.m1.6.6.3.cmml">←</mo><mrow id="A2.T4.8.8.2.m1.6.6.2" xref="A2.T4.8.8.2.m1.6.6.2.cmml"><mi mathsize="90%" id="A2.T4.8.8.2.m1.6.6.2.4" xref="A2.T4.8.8.2.m1.6.6.2.4.cmml">𝒗</mi><mo lspace="0.447em" rspace="0em" id="A2.T4.8.8.2.m1.6.6.2.3" xref="A2.T4.8.8.2.m1.6.6.2.3.cmml">​</mo><mrow id="A2.T4.8.8.2.m1.6.6.2.2.2" xref="A2.T4.8.8.2.m1.6.6.2.2.3.cmml"><mrow id="A2.T4.8.8.2.m1.5.5.1.1.1.1.2" xref="A2.T4.8.8.2.m1.5.5.1.1.1.1.1.cmml"><mo mathsize="90%" rspace="0em" id="A2.T4.8.8.2.m1.1.1" xref="A2.T4.8.8.2.m1.1.1.cmml">𝜎</mo><mrow id="A2.T4.8.8.2.m1.5.5.1.1.1.1.2.1" xref="A2.T4.8.8.2.m1.5.5.1.1.1.1.1.cmml"><mo maxsize="90%" minsize="90%" id="A2.T4.8.8.2.m1.5.5.1.1.1.1.2.1.1" xref="A2.T4.8.8.2.m1.5.5.1.1.1.1.1.cmml">(</mo><mi mathsize="90%" id="A2.T4.8.8.2.m1.2.2" xref="A2.T4.8.8.2.m1.2.2.cmml">x</mi><mo maxsize="90%" minsize="90%" id="A2.T4.8.8.2.m1.5.5.1.1.1.1.2.1.2" xref="A2.T4.8.8.2.m1.5.5.1.1.1.1.1.cmml">)</mo></mrow></mrow><mo lspace="0.280em" id="A2.T4.8.8.2.m1.6.6.2.2.2a" xref="A2.T4.8.8.2.m1.6.6.2.2.3.cmml">⁡</mo><mrow id="A2.T4.8.8.2.m1.6.6.2.2.2.2" xref="A2.T4.8.8.2.m1.6.6.2.2.3.cmml"><mo maxsize="120%" minsize="120%" id="A2.T4.8.8.2.m1.6.6.2.2.2.2.2" xref="A2.T4.8.8.2.m1.6.6.2.2.3.cmml">(</mo><mrow id="A2.T4.8.8.2.m1.6.6.2.2.2.2.1" xref="A2.T4.8.8.2.m1.6.6.2.2.2.2.1.cmml"><mn mathsize="90%" id="A2.T4.8.8.2.m1.6.6.2.2.2.2.1.2" xref="A2.T4.8.8.2.m1.6.6.2.2.2.2.1.2.cmml">1</mn><mo mathsize="90%" id="A2.T4.8.8.2.m1.6.6.2.2.2.2.1.1" xref="A2.T4.8.8.2.m1.6.6.2.2.2.2.1.1.cmml">−</mo><mrow id="A2.T4.8.8.2.m1.6.6.2.2.2.2.1.3.2" xref="A2.T4.8.8.2.m1.6.6.2.2.2.2.1.3.1.cmml"><mo lspace="0em" mathsize="90%" rspace="0em" id="A2.T4.8.8.2.m1.3.3" xref="A2.T4.8.8.2.m1.3.3.cmml">𝜎</mo><mrow id="A2.T4.8.8.2.m1.6.6.2.2.2.2.1.3.2.1" xref="A2.T4.8.8.2.m1.6.6.2.2.2.2.1.3.1.cmml"><mo maxsize="90%" minsize="90%" id="A2.T4.8.8.2.m1.6.6.2.2.2.2.1.3.2.1.1" xref="A2.T4.8.8.2.m1.6.6.2.2.2.2.1.3.1.cmml">(</mo><mi mathsize="90%" id="A2.T4.8.8.2.m1.4.4" xref="A2.T4.8.8.2.m1.4.4.cmml">𝒙</mi><mo maxsize="90%" minsize="90%" id="A2.T4.8.8.2.m1.6.6.2.2.2.2.1.3.2.1.2" xref="A2.T4.8.8.2.m1.6.6.2.2.2.2.1.3.1.cmml">)</mo></mrow></mrow></mrow><mo maxsize="120%" minsize="120%" id="A2.T4.8.8.2.m1.6.6.2.2.2.2.3" xref="A2.T4.8.8.2.m1.6.6.2.2.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="A2.T4.8.8.2.m1.6b"><apply id="A2.T4.8.8.2.m1.6.6.cmml" xref="A2.T4.8.8.2.m1.6.6"><ci id="A2.T4.8.8.2.m1.6.6.3.cmml" xref="A2.T4.8.8.2.m1.6.6.3">←</ci><ci id="A2.T4.8.8.2.m1.6.6.4.cmml" xref="A2.T4.8.8.2.m1.6.6.4">𝒗</ci><apply id="A2.T4.8.8.2.m1.6.6.2.cmml" xref="A2.T4.8.8.2.m1.6.6.2"><times id="A2.T4.8.8.2.m1.6.6.2.3.cmml" xref="A2.T4.8.8.2.m1.6.6.2.3"></times><ci id="A2.T4.8.8.2.m1.6.6.2.4.cmml" xref="A2.T4.8.8.2.m1.6.6.2.4">𝒗</ci><apply id="A2.T4.8.8.2.m1.6.6.2.2.3.cmml" xref="A2.T4.8.8.2.m1.6.6.2.2.2"><apply id="A2.T4.8.8.2.m1.5.5.1.1.1.1.1.cmml" xref="A2.T4.8.8.2.m1.5.5.1.1.1.1.2"><ci id="A2.T4.8.8.2.m1.1.1.cmml" xref="A2.T4.8.8.2.m1.1.1">𝜎</ci><ci id="A2.T4.8.8.2.m1.2.2.cmml" xref="A2.T4.8.8.2.m1.2.2">𝑥</ci></apply><apply id="A2.T4.8.8.2.m1.6.6.2.2.2.2.1.cmml" xref="A2.T4.8.8.2.m1.6.6.2.2.2.2.1"><minus id="A2.T4.8.8.2.m1.6.6.2.2.2.2.1.1.cmml" xref="A2.T4.8.8.2.m1.6.6.2.2.2.2.1.1"></minus><cn type="integer" id="A2.T4.8.8.2.m1.6.6.2.2.2.2.1.2.cmml" xref="A2.T4.8.8.2.m1.6.6.2.2.2.2.1.2">1</cn><apply id="A2.T4.8.8.2.m1.6.6.2.2.2.2.1.3.1.cmml" xref="A2.T4.8.8.2.m1.6.6.2.2.2.2.1.3.2"><ci id="A2.T4.8.8.2.m1.3.3.cmml" xref="A2.T4.8.8.2.m1.3.3">𝜎</ci><ci id="A2.T4.8.8.2.m1.4.4.cmml" xref="A2.T4.8.8.2.m1.4.4">𝒙</ci></apply></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="A2.T4.8.8.2.m1.6c">{\boldsymbol{v}}\,\leftarrow\,{\boldsymbol{v}}\;\operatorname*{\sigma}(x)\;\big{(}1-\operatorname*{\sigma}({\boldsymbol{x}})\big{)}</annotation></semantics></math></td>
</tr>
</table>
</figure>
</section>
<section id="A3" class="ltx_appendix">
<h2 class="ltx_title ltx_title_appendix">
<span class="ltx_tag ltx_tag_appendix">Appendix C </span>Distilling multiple models into one</h2>

<div id="A3.p1" class="ltx_para">
<p id="A3.p1.1" class="ltx_p">After training a set of models, we propose to combine the best of them into one a global predictor with superior OOD performance.
We train this predictor with the same fine-tuning as described in Section <a href="#S4.SS3" title="4.3 Fine-tuning ‣ 4 Proposed method ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">4.3</span></a>, using masks of selected models aggregated with a logical OR.
We outline in Algorithm <a href="#algorithm2" title="In Appendix C Distilling multiple models into one ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">2</span></a> a simple procedure to iteratively combine models pairwise and greedily.</p>
</div>
<figure id="algorithm2" class="ltx_float ltx_algorithm">
<div id="algorithm2.11" class="ltx_listing ltx_lst_numbers_left ltx_listing">
<div id="algorithm2.11.12" class="ltx_listingline">
<span id="algorithm2.11.12.1" class="ltx_text" style="font-size:90%;">
<span id="algorithm2.11.12.1.1" class="ltx_text"><span id="algorithm2.11.12.1.1.1" class="ltx_text ltx_font_bold">Inputs:</span> </span></span>
</div>
<div id="algorithm2.1.1" class="ltx_listingline">
<span id="algorithm2.1.1.1" class="ltx_text" style="font-size:90%;">
</span><math id="algorithm2.1.1.m1.1" class="ltx_Math" alttext="{\boldsymbol{S}}=\{f_{{\boldsymbol{\theta}}_{1}}\,\kern-0.90005pt.\kern-0.90005pt.\kern-0.90005pt.\,f_{{\boldsymbol{\theta}}_{M}}\}" display="inline"><semantics id="algorithm2.1.1.m1.1a"><mrow id="algorithm2.1.1.m1.1.1" xref="algorithm2.1.1.m1.1.1.cmml"><mi mathsize="90%" id="algorithm2.1.1.m1.1.1.3" xref="algorithm2.1.1.m1.1.1.3.cmml">𝑺</mi><mo mathsize="90%" id="algorithm2.1.1.m1.1.1.2" xref="algorithm2.1.1.m1.1.1.2.cmml">=</mo><mrow id="algorithm2.1.1.m1.1.1.1.1" xref="algorithm2.1.1.m1.1.1.1.2.cmml"><mo maxsize="90%" minsize="90%" id="algorithm2.1.1.m1.1.1.1.1.2" xref="algorithm2.1.1.m1.1.1.1.2.cmml">{</mo><mrow id="algorithm2.1.1.m1.1.1.1.1.1" xref="algorithm2.1.1.m1.1.1.1.1.1.cmml"><msub id="algorithm2.1.1.m1.1.1.1.1.1.2" xref="algorithm2.1.1.m1.1.1.1.1.1.2.cmml"><mi mathsize="90%" id="algorithm2.1.1.m1.1.1.1.1.1.2.2" xref="algorithm2.1.1.m1.1.1.1.1.1.2.2.cmml">f</mi><msub id="algorithm2.1.1.m1.1.1.1.1.1.2.3" xref="algorithm2.1.1.m1.1.1.1.1.1.2.3.cmml"><mi mathsize="90%" id="algorithm2.1.1.m1.1.1.1.1.1.2.3.2" xref="algorithm2.1.1.m1.1.1.1.1.1.2.3.2.cmml">𝜽</mi><mn mathsize="90%" id="algorithm2.1.1.m1.1.1.1.1.1.2.3.3" xref="algorithm2.1.1.m1.1.1.1.1.1.2.3.3.cmml">1</mn></msub></msub><mo lspace="0em" rspace="0em" id="algorithm2.1.1.m1.1.1.1.1.1.1" xref="algorithm2.1.1.m1.1.1.1.1.1.1.cmml">​</mo><mi mathsize="90%" mathvariant="normal" id="algorithm2.1.1.m1.1.1.1.1.1.3" xref="algorithm2.1.1.m1.1.1.1.1.1.3.cmml">…</mi><mo lspace="0.170em" rspace="0em" id="algorithm2.1.1.m1.1.1.1.1.1.1a" xref="algorithm2.1.1.m1.1.1.1.1.1.1.cmml">​</mo><msub id="algorithm2.1.1.m1.1.1.1.1.1.4" xref="algorithm2.1.1.m1.1.1.1.1.1.4.cmml"><mi mathsize="90%" id="algorithm2.1.1.m1.1.1.1.1.1.4.2" xref="algorithm2.1.1.m1.1.1.1.1.1.4.2.cmml">f</mi><msub id="algorithm2.1.1.m1.1.1.1.1.1.4.3" xref="algorithm2.1.1.m1.1.1.1.1.1.4.3.cmml"><mi mathsize="90%" id="algorithm2.1.1.m1.1.1.1.1.1.4.3.2" xref="algorithm2.1.1.m1.1.1.1.1.1.4.3.2.cmml">𝜽</mi><mi mathsize="90%" id="algorithm2.1.1.m1.1.1.1.1.1.4.3.3" xref="algorithm2.1.1.m1.1.1.1.1.1.4.3.3.cmml">M</mi></msub></msub></mrow><mo maxsize="90%" minsize="90%" id="algorithm2.1.1.m1.1.1.1.1.3" xref="algorithm2.1.1.m1.1.1.1.2.cmml">}</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm2.1.1.m1.1b"><apply id="algorithm2.1.1.m1.1.1.cmml" xref="algorithm2.1.1.m1.1.1"><eq id="algorithm2.1.1.m1.1.1.2.cmml" xref="algorithm2.1.1.m1.1.1.2"></eq><ci id="algorithm2.1.1.m1.1.1.3.cmml" xref="algorithm2.1.1.m1.1.1.3">𝑺</ci><set id="algorithm2.1.1.m1.1.1.1.2.cmml" xref="algorithm2.1.1.m1.1.1.1.1"><apply id="algorithm2.1.1.m1.1.1.1.1.1.cmml" xref="algorithm2.1.1.m1.1.1.1.1.1"><times id="algorithm2.1.1.m1.1.1.1.1.1.1.cmml" xref="algorithm2.1.1.m1.1.1.1.1.1.1"></times><apply id="algorithm2.1.1.m1.1.1.1.1.1.2.cmml" xref="algorithm2.1.1.m1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="algorithm2.1.1.m1.1.1.1.1.1.2.1.cmml" xref="algorithm2.1.1.m1.1.1.1.1.1.2">subscript</csymbol><ci id="algorithm2.1.1.m1.1.1.1.1.1.2.2.cmml" xref="algorithm2.1.1.m1.1.1.1.1.1.2.2">𝑓</ci><apply id="algorithm2.1.1.m1.1.1.1.1.1.2.3.cmml" xref="algorithm2.1.1.m1.1.1.1.1.1.2.3"><csymbol cd="ambiguous" id="algorithm2.1.1.m1.1.1.1.1.1.2.3.1.cmml" xref="algorithm2.1.1.m1.1.1.1.1.1.2.3">subscript</csymbol><ci id="algorithm2.1.1.m1.1.1.1.1.1.2.3.2.cmml" xref="algorithm2.1.1.m1.1.1.1.1.1.2.3.2">𝜽</ci><cn type="integer" id="algorithm2.1.1.m1.1.1.1.1.1.2.3.3.cmml" xref="algorithm2.1.1.m1.1.1.1.1.1.2.3.3">1</cn></apply></apply><ci id="algorithm2.1.1.m1.1.1.1.1.1.3.cmml" xref="algorithm2.1.1.m1.1.1.1.1.1.3">…</ci><apply id="algorithm2.1.1.m1.1.1.1.1.1.4.cmml" xref="algorithm2.1.1.m1.1.1.1.1.1.4"><csymbol cd="ambiguous" id="algorithm2.1.1.m1.1.1.1.1.1.4.1.cmml" xref="algorithm2.1.1.m1.1.1.1.1.1.4">subscript</csymbol><ci id="algorithm2.1.1.m1.1.1.1.1.1.4.2.cmml" xref="algorithm2.1.1.m1.1.1.1.1.1.4.2">𝑓</ci><apply id="algorithm2.1.1.m1.1.1.1.1.1.4.3.cmml" xref="algorithm2.1.1.m1.1.1.1.1.1.4.3"><csymbol cd="ambiguous" id="algorithm2.1.1.m1.1.1.1.1.1.4.3.1.cmml" xref="algorithm2.1.1.m1.1.1.1.1.1.4.3">subscript</csymbol><ci id="algorithm2.1.1.m1.1.1.1.1.1.4.3.2.cmml" xref="algorithm2.1.1.m1.1.1.1.1.1.4.3.2">𝜽</ci><ci id="algorithm2.1.1.m1.1.1.1.1.1.4.3.3.cmml" xref="algorithm2.1.1.m1.1.1.1.1.1.4.3.3">𝑀</ci></apply></apply></apply></set></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm2.1.1.m1.1c">{\boldsymbol{S}}=\{f_{{\boldsymbol{\theta}}_{1}}\,\kern-0.90005pt.\kern-0.90005pt.\kern-0.90005pt.\,f_{{\boldsymbol{\theta}}_{M}}\}</annotation></semantics></math><span id="algorithm2.1.1.2" class="ltx_text" style="font-size:90%;">: Set of independent models.</span>
</div>
<div id="algorithm2.2.2" class="ltx_listingline">
<span id="algorithm2.2.2.1" class="ltx_text" style="font-size:90%;">
</span><math id="algorithm2.2.2.m1.2" class="ltx_Math" alttext="\operatorname{getNBestModels}(\cdot)" display="inline"><semantics id="algorithm2.2.2.m1.2a"><mrow id="algorithm2.2.2.m1.2.3.2" xref="algorithm2.2.2.m1.2.3.1.cmml"><mi mathsize="90%" id="algorithm2.2.2.m1.1.1" xref="algorithm2.2.2.m1.1.1.cmml">getNBestModels</mi><mo id="algorithm2.2.2.m1.2.3.2a" xref="algorithm2.2.2.m1.2.3.1.cmml">⁡</mo><mrow id="algorithm2.2.2.m1.2.3.2.1" xref="algorithm2.2.2.m1.2.3.1.cmml"><mo maxsize="90%" minsize="90%" id="algorithm2.2.2.m1.2.3.2.1.1" xref="algorithm2.2.2.m1.2.3.1.cmml">(</mo><mo lspace="0em" mathsize="90%" rspace="0em" id="algorithm2.2.2.m1.2.2" xref="algorithm2.2.2.m1.2.2.cmml">⋅</mo><mo maxsize="90%" minsize="90%" id="algorithm2.2.2.m1.2.3.2.1.2" xref="algorithm2.2.2.m1.2.3.1.cmml">)</mo></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm2.2.2.m1.2b"><apply id="algorithm2.2.2.m1.2.3.1.cmml" xref="algorithm2.2.2.m1.2.3.2"><ci id="algorithm2.2.2.m1.1.1.cmml" xref="algorithm2.2.2.m1.1.1">getNBestModels</ci><ci id="algorithm2.2.2.m1.2.2.cmml" xref="algorithm2.2.2.m1.2.2">⋅</ci></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm2.2.2.m1.2c">\operatorname{getNBestModels}(\cdot)</annotation></semantics></math><span id="algorithm2.2.2.2" class="ltx_text" style="font-size:90%;">: Model selection strategy, </span><em id="algorithm2.2.2.3" class="ltx_emph ltx_font_italic" style="font-size:90%;">e.g</em><span id="algorithm2.2.2.4" class="ltx_text" style="font-size:90%;">.</span><span id="algorithm2.2.2.5" class="ltx_text"></span><span id="algorithm2.2.2.6" class="ltx_text" style="font-size:90%;"> evaluation on OOD validation set.
</span>
</div>
<div id="algorithm2.11.13" class="ltx_listingline">
<span id="algorithm2.11.13.1" class="ltx_text" style="font-size:90%;">
</span><span id="algorithm2.11.13.2" class="ltx_text" style="font-size:90%;"><span id="algorithm2.11.13.2.1" class="ltx_text ltx_font_bold">Result:</span> </span>
</div>
<div id="algorithm2.3.3" class="ltx_listingline">
<span id="algorithm2.3.3.1" class="ltx_text" style="font-size:90%;">
</span><math id="algorithm2.3.3.m1.1" class="ltx_Math" alttext="f_{{\boldsymbol{\theta}}_{\star}}" display="inline"><semantics id="algorithm2.3.3.m1.1a"><msub id="algorithm2.3.3.m1.1.1" xref="algorithm2.3.3.m1.1.1.cmml"><mi mathsize="90%" id="algorithm2.3.3.m1.1.1.2" xref="algorithm2.3.3.m1.1.1.2.cmml">f</mi><msub id="algorithm2.3.3.m1.1.1.3" xref="algorithm2.3.3.m1.1.1.3.cmml"><mi mathsize="90%" id="algorithm2.3.3.m1.1.1.3.2" xref="algorithm2.3.3.m1.1.1.3.2.cmml">𝜽</mi><mo mathsize="90%" id="algorithm2.3.3.m1.1.1.3.3" xref="algorithm2.3.3.m1.1.1.3.3.cmml">⋆</mo></msub></msub><annotation-xml encoding="MathML-Content" id="algorithm2.3.3.m1.1b"><apply id="algorithm2.3.3.m1.1.1.cmml" xref="algorithm2.3.3.m1.1.1"><csymbol cd="ambiguous" id="algorithm2.3.3.m1.1.1.1.cmml" xref="algorithm2.3.3.m1.1.1">subscript</csymbol><ci id="algorithm2.3.3.m1.1.1.2.cmml" xref="algorithm2.3.3.m1.1.1.2">𝑓</ci><apply id="algorithm2.3.3.m1.1.1.3.cmml" xref="algorithm2.3.3.m1.1.1.3"><csymbol cd="ambiguous" id="algorithm2.3.3.m1.1.1.3.1.cmml" xref="algorithm2.3.3.m1.1.1.3">subscript</csymbol><ci id="algorithm2.3.3.m1.1.1.3.2.cmml" xref="algorithm2.3.3.m1.1.1.3.2">𝜽</ci><ci id="algorithm2.3.3.m1.1.1.3.3.cmml" xref="algorithm2.3.3.m1.1.1.3.3">⋆</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm2.3.3.m1.1c">f_{{\boldsymbol{\theta}}_{\star}}</annotation></semantics></math><span id="algorithm2.3.3.2" class="ltx_text" style="font-size:90%;">: Best combined model according to given strategy.
</span>
</div>
<div id="algorithm2.11.14" class="ltx_listingline">
<span id="algorithm2.11.14.1" class="ltx_text" style="font-size:90%;">
</span><span id="algorithm2.11.14.2" class="ltx_text" style="font-size:90%;"><span id="algorithm2.11.14.2.1" class="ltx_text ltx_font_bold">Method:</span> </span>
</div>
<div id="algorithm2.4.4" class="ltx_listingline">
<span id="algorithm2.4.4.1" class="ltx_text" style="font-size:90%;">
</span><math id="algorithm2.4.4.m1.2" class="ltx_Math" alttext="{\lambda_{\textrm{indep}}}\leftarrow 0,\leavevmode\nobreak\ {\lambda_{\textrm{manifold}}}\leftarrow 0" display="inline"><semantics id="algorithm2.4.4.m1.2a"><mrow id="algorithm2.4.4.m1.2.2.2" xref="algorithm2.4.4.m1.2.2.3.cmml"><mrow id="algorithm2.4.4.m1.1.1.1.1" xref="algorithm2.4.4.m1.1.1.1.1.cmml"><msub id="algorithm2.4.4.m1.1.1.1.1.2" xref="algorithm2.4.4.m1.1.1.1.1.2.cmml"><mi mathsize="90%" id="algorithm2.4.4.m1.1.1.1.1.2.2" xref="algorithm2.4.4.m1.1.1.1.1.2.2.cmml">λ</mi><mtext mathsize="90%" id="algorithm2.4.4.m1.1.1.1.1.2.3" xref="algorithm2.4.4.m1.1.1.1.1.2.3a.cmml">indep</mtext></msub><mo mathsize="90%" stretchy="false" id="algorithm2.4.4.m1.1.1.1.1.1" xref="algorithm2.4.4.m1.1.1.1.1.1.cmml">←</mo><mn mathsize="90%" id="algorithm2.4.4.m1.1.1.1.1.3" xref="algorithm2.4.4.m1.1.1.1.1.3.cmml">0</mn></mrow><mo mathsize="90%" rspace="0.617em" id="algorithm2.4.4.m1.2.2.2.3" xref="algorithm2.4.4.m1.2.2.3a.cmml">,</mo><mrow id="algorithm2.4.4.m1.2.2.2.2" xref="algorithm2.4.4.m1.2.2.2.2.cmml"><msub id="algorithm2.4.4.m1.2.2.2.2.2" xref="algorithm2.4.4.m1.2.2.2.2.2.cmml"><mi mathsize="90%" id="algorithm2.4.4.m1.2.2.2.2.2.2" xref="algorithm2.4.4.m1.2.2.2.2.2.2.cmml">λ</mi><mtext mathsize="90%" id="algorithm2.4.4.m1.2.2.2.2.2.3" xref="algorithm2.4.4.m1.2.2.2.2.2.3a.cmml">manifold</mtext></msub><mo mathsize="90%" stretchy="false" id="algorithm2.4.4.m1.2.2.2.2.1" xref="algorithm2.4.4.m1.2.2.2.2.1.cmml">←</mo><mn mathsize="90%" id="algorithm2.4.4.m1.2.2.2.2.3" xref="algorithm2.4.4.m1.2.2.2.2.3.cmml">0</mn></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm2.4.4.m1.2b"><apply id="algorithm2.4.4.m1.2.2.3.cmml" xref="algorithm2.4.4.m1.2.2.2"><csymbol cd="ambiguous" id="algorithm2.4.4.m1.2.2.3a.cmml" xref="algorithm2.4.4.m1.2.2.2.3">formulae-sequence</csymbol><apply id="algorithm2.4.4.m1.1.1.1.1.cmml" xref="algorithm2.4.4.m1.1.1.1.1"><ci id="algorithm2.4.4.m1.1.1.1.1.1.cmml" xref="algorithm2.4.4.m1.1.1.1.1.1">←</ci><apply id="algorithm2.4.4.m1.1.1.1.1.2.cmml" xref="algorithm2.4.4.m1.1.1.1.1.2"><csymbol cd="ambiguous" id="algorithm2.4.4.m1.1.1.1.1.2.1.cmml" xref="algorithm2.4.4.m1.1.1.1.1.2">subscript</csymbol><ci id="algorithm2.4.4.m1.1.1.1.1.2.2.cmml" xref="algorithm2.4.4.m1.1.1.1.1.2.2">𝜆</ci><ci id="algorithm2.4.4.m1.1.1.1.1.2.3a.cmml" xref="algorithm2.4.4.m1.1.1.1.1.2.3"><mtext mathsize="63%" id="algorithm2.4.4.m1.1.1.1.1.2.3.cmml" xref="algorithm2.4.4.m1.1.1.1.1.2.3">indep</mtext></ci></apply><cn type="integer" id="algorithm2.4.4.m1.1.1.1.1.3.cmml" xref="algorithm2.4.4.m1.1.1.1.1.3">0</cn></apply><apply id="algorithm2.4.4.m1.2.2.2.2.cmml" xref="algorithm2.4.4.m1.2.2.2.2"><ci id="algorithm2.4.4.m1.2.2.2.2.1.cmml" xref="algorithm2.4.4.m1.2.2.2.2.1">←</ci><apply id="algorithm2.4.4.m1.2.2.2.2.2.cmml" xref="algorithm2.4.4.m1.2.2.2.2.2"><csymbol cd="ambiguous" id="algorithm2.4.4.m1.2.2.2.2.2.1.cmml" xref="algorithm2.4.4.m1.2.2.2.2.2">subscript</csymbol><ci id="algorithm2.4.4.m1.2.2.2.2.2.2.cmml" xref="algorithm2.4.4.m1.2.2.2.2.2.2">𝜆</ci><ci id="algorithm2.4.4.m1.2.2.2.2.2.3a.cmml" xref="algorithm2.4.4.m1.2.2.2.2.2.3"><mtext mathsize="63%" id="algorithm2.4.4.m1.2.2.2.2.2.3.cmml" xref="algorithm2.4.4.m1.2.2.2.2.2.3">manifold</mtext></ci></apply><cn type="integer" id="algorithm2.4.4.m1.2.2.2.2.3.cmml" xref="algorithm2.4.4.m1.2.2.2.2.3">0</cn></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm2.4.4.m1.2c">{\lambda_{\textrm{indep}}}\leftarrow 0,\leavevmode\nobreak\ {\lambda_{\textrm{manifold}}}\leftarrow 0</annotation></semantics></math><span id="algorithm2.4.4.2" class="ltx_text" style="font-size:90%;">  </span><span id="algorithm2.4.4.3" class="ltx_text" style="font-size:90%;color:#8C8C8C;">// <span id="algorithm2.4.4.3.1" class="ltx_text" style="font-size:89%;">Use only predictive loss</span></span>
</div>
<div id="algorithm2.11.15" class="ltx_listingline">
<span id="algorithm2.11.15.1" class="ltx_text" style="font-size:90%;">
</span><span id="algorithm2.11.15.2" class="ltx_text ltx_font_bold" style="font-size:90%;">do</span>
</div>
<div id="algorithm2.5.5" class="ltx_listingline">
<span id="algorithm2.5.5.1" class="ltx_text" style="font-size:90%;">  </span><span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span><span id="algorithm2.5.5.2" class="ltx_text" style="font-size:90%;">   </span><span id="algorithm2.5.5.3" class="ltx_text" style="font-size:90%;">
</span><math id="algorithm2.5.5.m1.4" class="ltx_Math" alttext="\{{\boldsymbol{\theta}}_{k},{\boldsymbol{\theta}}_{l}\}\leftarrow\operatorname{get2BestModels}({\boldsymbol{S}})" display="inline"><semantics id="algorithm2.5.5.m1.4a"><mrow id="algorithm2.5.5.m1.4.4" xref="algorithm2.5.5.m1.4.4.cmml"><mrow id="algorithm2.5.5.m1.4.4.2.2" xref="algorithm2.5.5.m1.4.4.2.3.cmml"><mo maxsize="90%" minsize="90%" id="algorithm2.5.5.m1.4.4.2.2.3" xref="algorithm2.5.5.m1.4.4.2.3.cmml">{</mo><msub id="algorithm2.5.5.m1.3.3.1.1.1" xref="algorithm2.5.5.m1.3.3.1.1.1.cmml"><mi mathsize="90%" id="algorithm2.5.5.m1.3.3.1.1.1.2" xref="algorithm2.5.5.m1.3.3.1.1.1.2.cmml">𝜽</mi><mi mathsize="90%" id="algorithm2.5.5.m1.3.3.1.1.1.3" xref="algorithm2.5.5.m1.3.3.1.1.1.3.cmml">k</mi></msub><mo mathsize="90%" id="algorithm2.5.5.m1.4.4.2.2.4" xref="algorithm2.5.5.m1.4.4.2.3.cmml">,</mo><msub id="algorithm2.5.5.m1.4.4.2.2.2" xref="algorithm2.5.5.m1.4.4.2.2.2.cmml"><mi mathsize="90%" id="algorithm2.5.5.m1.4.4.2.2.2.2" xref="algorithm2.5.5.m1.4.4.2.2.2.2.cmml">𝜽</mi><mi mathsize="90%" id="algorithm2.5.5.m1.4.4.2.2.2.3" xref="algorithm2.5.5.m1.4.4.2.2.2.3.cmml">l</mi></msub><mo maxsize="90%" minsize="90%" id="algorithm2.5.5.m1.4.4.2.2.5" xref="algorithm2.5.5.m1.4.4.2.3.cmml">}</mo></mrow><mo mathsize="90%" stretchy="false" id="algorithm2.5.5.m1.4.4.3" xref="algorithm2.5.5.m1.4.4.3.cmml">←</mo><mrow id="algorithm2.5.5.m1.4.4.4.2" xref="algorithm2.5.5.m1.4.4.4.1.cmml"><mi mathsize="90%" id="algorithm2.5.5.m1.1.1" xref="algorithm2.5.5.m1.1.1.cmml">get2BestModels</mi><mo id="algorithm2.5.5.m1.4.4.4.2a" xref="algorithm2.5.5.m1.4.4.4.1.cmml">⁡</mo><mrow id="algorithm2.5.5.m1.4.4.4.2.1" xref="algorithm2.5.5.m1.4.4.4.1.cmml"><mo maxsize="90%" minsize="90%" id="algorithm2.5.5.m1.4.4.4.2.1.1" xref="algorithm2.5.5.m1.4.4.4.1.cmml">(</mo><mi mathsize="90%" id="algorithm2.5.5.m1.2.2" xref="algorithm2.5.5.m1.2.2.cmml">𝑺</mi><mo maxsize="90%" minsize="90%" id="algorithm2.5.5.m1.4.4.4.2.1.2" xref="algorithm2.5.5.m1.4.4.4.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm2.5.5.m1.4b"><apply id="algorithm2.5.5.m1.4.4.cmml" xref="algorithm2.5.5.m1.4.4"><ci id="algorithm2.5.5.m1.4.4.3.cmml" xref="algorithm2.5.5.m1.4.4.3">←</ci><set id="algorithm2.5.5.m1.4.4.2.3.cmml" xref="algorithm2.5.5.m1.4.4.2.2"><apply id="algorithm2.5.5.m1.3.3.1.1.1.cmml" xref="algorithm2.5.5.m1.3.3.1.1.1"><csymbol cd="ambiguous" id="algorithm2.5.5.m1.3.3.1.1.1.1.cmml" xref="algorithm2.5.5.m1.3.3.1.1.1">subscript</csymbol><ci id="algorithm2.5.5.m1.3.3.1.1.1.2.cmml" xref="algorithm2.5.5.m1.3.3.1.1.1.2">𝜽</ci><ci id="algorithm2.5.5.m1.3.3.1.1.1.3.cmml" xref="algorithm2.5.5.m1.3.3.1.1.1.3">𝑘</ci></apply><apply id="algorithm2.5.5.m1.4.4.2.2.2.cmml" xref="algorithm2.5.5.m1.4.4.2.2.2"><csymbol cd="ambiguous" id="algorithm2.5.5.m1.4.4.2.2.2.1.cmml" xref="algorithm2.5.5.m1.4.4.2.2.2">subscript</csymbol><ci id="algorithm2.5.5.m1.4.4.2.2.2.2.cmml" xref="algorithm2.5.5.m1.4.4.2.2.2.2">𝜽</ci><ci id="algorithm2.5.5.m1.4.4.2.2.2.3.cmml" xref="algorithm2.5.5.m1.4.4.2.2.2.3">𝑙</ci></apply></set><apply id="algorithm2.5.5.m1.4.4.4.1.cmml" xref="algorithm2.5.5.m1.4.4.4.2"><ci id="algorithm2.5.5.m1.1.1.cmml" xref="algorithm2.5.5.m1.1.1">get2BestModels</ci><ci id="algorithm2.5.5.m1.2.2.cmml" xref="algorithm2.5.5.m1.2.2">𝑺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm2.5.5.m1.4c">\{{\boldsymbol{\theta}}_{k},{\boldsymbol{\theta}}_{l}\}\leftarrow\operatorname{get2BestModels}({\boldsymbol{S}})</annotation></semantics></math>
</div>
<div id="algorithm2.6.6" class="ltx_listingline">
<span id="algorithm2.6.6.1" class="ltx_text" style="font-size:90%;">  </span><span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span><span id="algorithm2.6.6.2" class="ltx_text" style="font-size:90%;">   </span><span id="algorithm2.6.6.3" class="ltx_text" style="font-size:90%;"> </span><math id="algorithm2.6.6.m1.2" class="ltx_Math" alttext="{\boldsymbol{\mathrm{mask}}}^{\star}_{i}\leftarrow{\boldsymbol{\mathrm{mask}}}^{k}_{i}\,\lor\,{\boldsymbol{\mathrm{mask}}}^{l}_{i}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \forall i" display="inline"><semantics id="algorithm2.6.6.m1.2a"><mrow id="algorithm2.6.6.m1.2.2" xref="algorithm2.6.6.m1.2.2.cmml"><msubsup id="algorithm2.6.6.m1.2.2.4" xref="algorithm2.6.6.m1.2.2.4.cmml"><mi mathsize="90%" id="algorithm2.6.6.m1.2.2.4.2.2" xref="algorithm2.6.6.m1.2.2.4.2.2.cmml">𝐦𝐚𝐬𝐤</mi><mi mathsize="90%" id="algorithm2.6.6.m1.2.2.4.3" xref="algorithm2.6.6.m1.2.2.4.3.cmml">i</mi><mo mathsize="90%" id="algorithm2.6.6.m1.2.2.4.2.3" xref="algorithm2.6.6.m1.2.2.4.2.3.cmml">⋆</mo></msubsup><mo mathsize="90%" stretchy="false" id="algorithm2.6.6.m1.2.2.3" xref="algorithm2.6.6.m1.2.2.3.cmml">←</mo><mrow id="algorithm2.6.6.m1.2.2.2.2" xref="algorithm2.6.6.m1.2.2.2.3.cmml"><mrow id="algorithm2.6.6.m1.1.1.1.1.1" xref="algorithm2.6.6.m1.1.1.1.1.1.cmml"><msubsup id="algorithm2.6.6.m1.1.1.1.1.1.2" xref="algorithm2.6.6.m1.1.1.1.1.1.2.cmml"><mi mathsize="90%" id="algorithm2.6.6.m1.1.1.1.1.1.2.2.2" xref="algorithm2.6.6.m1.1.1.1.1.1.2.2.2.cmml">𝐦𝐚𝐬𝐤</mi><mi mathsize="90%" id="algorithm2.6.6.m1.1.1.1.1.1.2.3" xref="algorithm2.6.6.m1.1.1.1.1.1.2.3.cmml">i</mi><mi mathsize="90%" id="algorithm2.6.6.m1.1.1.1.1.1.2.2.3" xref="algorithm2.6.6.m1.1.1.1.1.1.2.2.3.cmml">k</mi></msubsup><mo mathsize="90%" rspace="0.392em" id="algorithm2.6.6.m1.1.1.1.1.1.1" xref="algorithm2.6.6.m1.1.1.1.1.1.1.cmml">∨</mo><msubsup id="algorithm2.6.6.m1.1.1.1.1.1.3" xref="algorithm2.6.6.m1.1.1.1.1.1.3.cmml"><mi mathsize="90%" id="algorithm2.6.6.m1.1.1.1.1.1.3.2.2" xref="algorithm2.6.6.m1.1.1.1.1.1.3.2.2.cmml">𝐦𝐚𝐬𝐤</mi><mi mathsize="90%" id="algorithm2.6.6.m1.1.1.1.1.1.3.3" xref="algorithm2.6.6.m1.1.1.1.1.1.3.3.cmml">i</mi><mi mathsize="90%" id="algorithm2.6.6.m1.1.1.1.1.1.3.2.3" xref="algorithm2.6.6.m1.1.1.1.1.1.3.2.3.cmml">l</mi></msubsup></mrow><mspace width="1.517em" id="algorithm2.6.6.m1.2.2.2.2.3" xref="algorithm2.6.6.m1.2.2.2.3.cmml"></mspace><mrow id="algorithm2.6.6.m1.2.2.2.2.2" xref="algorithm2.6.6.m1.2.2.2.2.2.cmml"><mo mathsize="90%" rspace="0.167em" id="algorithm2.6.6.m1.2.2.2.2.2.1" xref="algorithm2.6.6.m1.2.2.2.2.2.1.cmml">∀</mo><mi mathsize="90%" id="algorithm2.6.6.m1.2.2.2.2.2.2" xref="algorithm2.6.6.m1.2.2.2.2.2.2.cmml">i</mi></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm2.6.6.m1.2b"><apply id="algorithm2.6.6.m1.2.2.cmml" xref="algorithm2.6.6.m1.2.2"><ci id="algorithm2.6.6.m1.2.2.3.cmml" xref="algorithm2.6.6.m1.2.2.3">←</ci><apply id="algorithm2.6.6.m1.2.2.4.cmml" xref="algorithm2.6.6.m1.2.2.4"><csymbol cd="ambiguous" id="algorithm2.6.6.m1.2.2.4.1.cmml" xref="algorithm2.6.6.m1.2.2.4">subscript</csymbol><apply id="algorithm2.6.6.m1.2.2.4.2.cmml" xref="algorithm2.6.6.m1.2.2.4"><csymbol cd="ambiguous" id="algorithm2.6.6.m1.2.2.4.2.1.cmml" xref="algorithm2.6.6.m1.2.2.4">superscript</csymbol><ci id="algorithm2.6.6.m1.2.2.4.2.2.cmml" xref="algorithm2.6.6.m1.2.2.4.2.2">𝐦𝐚𝐬𝐤</ci><ci id="algorithm2.6.6.m1.2.2.4.2.3.cmml" xref="algorithm2.6.6.m1.2.2.4.2.3">⋆</ci></apply><ci id="algorithm2.6.6.m1.2.2.4.3.cmml" xref="algorithm2.6.6.m1.2.2.4.3">𝑖</ci></apply><list id="algorithm2.6.6.m1.2.2.2.3.cmml" xref="algorithm2.6.6.m1.2.2.2.2"><apply id="algorithm2.6.6.m1.1.1.1.1.1.cmml" xref="algorithm2.6.6.m1.1.1.1.1.1"><or id="algorithm2.6.6.m1.1.1.1.1.1.1.cmml" xref="algorithm2.6.6.m1.1.1.1.1.1.1"></or><apply id="algorithm2.6.6.m1.1.1.1.1.1.2.cmml" xref="algorithm2.6.6.m1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="algorithm2.6.6.m1.1.1.1.1.1.2.1.cmml" xref="algorithm2.6.6.m1.1.1.1.1.1.2">subscript</csymbol><apply id="algorithm2.6.6.m1.1.1.1.1.1.2.2.cmml" xref="algorithm2.6.6.m1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="algorithm2.6.6.m1.1.1.1.1.1.2.2.1.cmml" xref="algorithm2.6.6.m1.1.1.1.1.1.2">superscript</csymbol><ci id="algorithm2.6.6.m1.1.1.1.1.1.2.2.2.cmml" xref="algorithm2.6.6.m1.1.1.1.1.1.2.2.2">𝐦𝐚𝐬𝐤</ci><ci id="algorithm2.6.6.m1.1.1.1.1.1.2.2.3.cmml" xref="algorithm2.6.6.m1.1.1.1.1.1.2.2.3">𝑘</ci></apply><ci id="algorithm2.6.6.m1.1.1.1.1.1.2.3.cmml" xref="algorithm2.6.6.m1.1.1.1.1.1.2.3">𝑖</ci></apply><apply id="algorithm2.6.6.m1.1.1.1.1.1.3.cmml" xref="algorithm2.6.6.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="algorithm2.6.6.m1.1.1.1.1.1.3.1.cmml" xref="algorithm2.6.6.m1.1.1.1.1.1.3">subscript</csymbol><apply id="algorithm2.6.6.m1.1.1.1.1.1.3.2.cmml" xref="algorithm2.6.6.m1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="algorithm2.6.6.m1.1.1.1.1.1.3.2.1.cmml" xref="algorithm2.6.6.m1.1.1.1.1.1.3">superscript</csymbol><ci id="algorithm2.6.6.m1.1.1.1.1.1.3.2.2.cmml" xref="algorithm2.6.6.m1.1.1.1.1.1.3.2.2">𝐦𝐚𝐬𝐤</ci><ci id="algorithm2.6.6.m1.1.1.1.1.1.3.2.3.cmml" xref="algorithm2.6.6.m1.1.1.1.1.1.3.2.3">𝑙</ci></apply><ci id="algorithm2.6.6.m1.1.1.1.1.1.3.3.cmml" xref="algorithm2.6.6.m1.1.1.1.1.1.3.3">𝑖</ci></apply></apply><apply id="algorithm2.6.6.m1.2.2.2.2.2.cmml" xref="algorithm2.6.6.m1.2.2.2.2.2"><csymbol cd="latexml" id="algorithm2.6.6.m1.2.2.2.2.2.1.cmml" xref="algorithm2.6.6.m1.2.2.2.2.2.1">for-all</csymbol><ci id="algorithm2.6.6.m1.2.2.2.2.2.2.cmml" xref="algorithm2.6.6.m1.2.2.2.2.2.2">𝑖</ci></apply></list></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm2.6.6.m1.2c">{\boldsymbol{\mathrm{mask}}}^{\star}_{i}\leftarrow{\boldsymbol{\mathrm{mask}}}^{k}_{i}\,\lor\,{\boldsymbol{\mathrm{mask}}}^{l}_{i}\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\ \forall i</annotation></semantics></math><span id="algorithm2.6.6.4" class="ltx_text" style="font-size:90%;">  </span><span id="algorithm2.6.6.5" class="ltx_text" style="font-size:90%;color:#8C8C8C;">// <span id="algorithm2.6.6.5.1" class="ltx_text" style="font-size:89%;">Combine masks</span></span>
</div>
<div id="algorithm2.7.7" class="ltx_listingline">
<span id="algorithm2.7.7.1" class="ltx_text" style="font-size:90%;">  </span><span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span><span id="algorithm2.7.7.2" class="ltx_text" style="font-size:90%;">   </span><span id="algorithm2.7.7.3" class="ltx_text" style="font-size:90%;">
</span><math id="algorithm2.7.7.m1.1" class="ltx_Math" alttext="\mathcal{D}_{\mathrm{tr}}^{\star}\leftarrow\{({\boldsymbol{x}}_{i}\odot{\boldsymbol{\mathrm{mask}}}^{\star}_{i},y_{i})\}_{i}" display="inline"><semantics id="algorithm2.7.7.m1.1a"><mrow id="algorithm2.7.7.m1.1.1" xref="algorithm2.7.7.m1.1.1.cmml"><msubsup id="algorithm2.7.7.m1.1.1.3" xref="algorithm2.7.7.m1.1.1.3.cmml"><mi class="ltx_font_mathcaligraphic" mathsize="90%" id="algorithm2.7.7.m1.1.1.3.2.2" xref="algorithm2.7.7.m1.1.1.3.2.2.cmml">𝒟</mi><mi mathsize="90%" id="algorithm2.7.7.m1.1.1.3.2.3" xref="algorithm2.7.7.m1.1.1.3.2.3.cmml">tr</mi><mo mathsize="90%" id="algorithm2.7.7.m1.1.1.3.3" xref="algorithm2.7.7.m1.1.1.3.3.cmml">⋆</mo></msubsup><mo mathsize="90%" stretchy="false" id="algorithm2.7.7.m1.1.1.2" xref="algorithm2.7.7.m1.1.1.2.cmml">←</mo><msub id="algorithm2.7.7.m1.1.1.1" xref="algorithm2.7.7.m1.1.1.1.cmml"><mrow id="algorithm2.7.7.m1.1.1.1.1.1" xref="algorithm2.7.7.m1.1.1.1.1.2.cmml"><mo maxsize="90%" minsize="90%" id="algorithm2.7.7.m1.1.1.1.1.1.2" xref="algorithm2.7.7.m1.1.1.1.1.2.cmml">{</mo><mrow id="algorithm2.7.7.m1.1.1.1.1.1.1.2" xref="algorithm2.7.7.m1.1.1.1.1.1.1.3.cmml"><mo maxsize="90%" minsize="90%" id="algorithm2.7.7.m1.1.1.1.1.1.1.2.3" xref="algorithm2.7.7.m1.1.1.1.1.1.1.3.cmml">(</mo><mrow id="algorithm2.7.7.m1.1.1.1.1.1.1.1.1" xref="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.cmml"><msub id="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.2" xref="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.2.cmml"><mi mathsize="90%" id="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.2.2" xref="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.2.2.cmml">𝒙</mi><mi mathsize="90%" id="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.2.3" xref="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.2.3.cmml">i</mi></msub><mo lspace="0.222em" mathsize="90%" rspace="0.222em" id="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.1" xref="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.1.cmml">⊙</mo><msubsup id="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.3" xref="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.3.cmml"><mi mathsize="90%" id="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.3.2.2" xref="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.3.2.2.cmml">𝐦𝐚𝐬𝐤</mi><mi mathsize="90%" id="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.3.3" xref="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.3.3.cmml">i</mi><mo mathsize="90%" id="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.3.2.3" xref="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.3.2.3.cmml">⋆</mo></msubsup></mrow><mo mathsize="90%" id="algorithm2.7.7.m1.1.1.1.1.1.1.2.4" xref="algorithm2.7.7.m1.1.1.1.1.1.1.3.cmml">,</mo><msub id="algorithm2.7.7.m1.1.1.1.1.1.1.2.2" xref="algorithm2.7.7.m1.1.1.1.1.1.1.2.2.cmml"><mi mathsize="90%" id="algorithm2.7.7.m1.1.1.1.1.1.1.2.2.2" xref="algorithm2.7.7.m1.1.1.1.1.1.1.2.2.2.cmml">y</mi><mi mathsize="90%" id="algorithm2.7.7.m1.1.1.1.1.1.1.2.2.3" xref="algorithm2.7.7.m1.1.1.1.1.1.1.2.2.3.cmml">i</mi></msub><mo maxsize="90%" minsize="90%" id="algorithm2.7.7.m1.1.1.1.1.1.1.2.5" xref="algorithm2.7.7.m1.1.1.1.1.1.1.3.cmml">)</mo></mrow><mo maxsize="90%" minsize="90%" id="algorithm2.7.7.m1.1.1.1.1.1.3" xref="algorithm2.7.7.m1.1.1.1.1.2.cmml">}</mo></mrow><mi mathsize="90%" id="algorithm2.7.7.m1.1.1.1.3" xref="algorithm2.7.7.m1.1.1.1.3.cmml">i</mi></msub></mrow><annotation-xml encoding="MathML-Content" id="algorithm2.7.7.m1.1b"><apply id="algorithm2.7.7.m1.1.1.cmml" xref="algorithm2.7.7.m1.1.1"><ci id="algorithm2.7.7.m1.1.1.2.cmml" xref="algorithm2.7.7.m1.1.1.2">←</ci><apply id="algorithm2.7.7.m1.1.1.3.cmml" xref="algorithm2.7.7.m1.1.1.3"><csymbol cd="ambiguous" id="algorithm2.7.7.m1.1.1.3.1.cmml" xref="algorithm2.7.7.m1.1.1.3">superscript</csymbol><apply id="algorithm2.7.7.m1.1.1.3.2.cmml" xref="algorithm2.7.7.m1.1.1.3"><csymbol cd="ambiguous" id="algorithm2.7.7.m1.1.1.3.2.1.cmml" xref="algorithm2.7.7.m1.1.1.3">subscript</csymbol><ci id="algorithm2.7.7.m1.1.1.3.2.2.cmml" xref="algorithm2.7.7.m1.1.1.3.2.2">𝒟</ci><ci id="algorithm2.7.7.m1.1.1.3.2.3.cmml" xref="algorithm2.7.7.m1.1.1.3.2.3">tr</ci></apply><ci id="algorithm2.7.7.m1.1.1.3.3.cmml" xref="algorithm2.7.7.m1.1.1.3.3">⋆</ci></apply><apply id="algorithm2.7.7.m1.1.1.1.cmml" xref="algorithm2.7.7.m1.1.1.1"><csymbol cd="ambiguous" id="algorithm2.7.7.m1.1.1.1.2.cmml" xref="algorithm2.7.7.m1.1.1.1">subscript</csymbol><set id="algorithm2.7.7.m1.1.1.1.1.2.cmml" xref="algorithm2.7.7.m1.1.1.1.1.1"><interval closure="open" id="algorithm2.7.7.m1.1.1.1.1.1.1.3.cmml" xref="algorithm2.7.7.m1.1.1.1.1.1.1.2"><apply id="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.cmml" xref="algorithm2.7.7.m1.1.1.1.1.1.1.1.1"><csymbol cd="latexml" id="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.1.cmml" xref="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.1">direct-product</csymbol><apply id="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.2.cmml" xref="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.2"><csymbol cd="ambiguous" id="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.2.1.cmml" xref="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.2">subscript</csymbol><ci id="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.2.2.cmml" xref="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.2.2">𝒙</ci><ci id="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.2.3.cmml" xref="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.2.3">𝑖</ci></apply><apply id="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.3.cmml" xref="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.3.1.cmml" xref="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.3">subscript</csymbol><apply id="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.3.2.cmml" xref="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.3"><csymbol cd="ambiguous" id="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.3.2.1.cmml" xref="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.3">superscript</csymbol><ci id="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.3.2.2.cmml" xref="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.3.2.2">𝐦𝐚𝐬𝐤</ci><ci id="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.3.2.3.cmml" xref="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.3.2.3">⋆</ci></apply><ci id="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.3.3.cmml" xref="algorithm2.7.7.m1.1.1.1.1.1.1.1.1.3.3">𝑖</ci></apply></apply><apply id="algorithm2.7.7.m1.1.1.1.1.1.1.2.2.cmml" xref="algorithm2.7.7.m1.1.1.1.1.1.1.2.2"><csymbol cd="ambiguous" id="algorithm2.7.7.m1.1.1.1.1.1.1.2.2.1.cmml" xref="algorithm2.7.7.m1.1.1.1.1.1.1.2.2">subscript</csymbol><ci id="algorithm2.7.7.m1.1.1.1.1.1.1.2.2.2.cmml" xref="algorithm2.7.7.m1.1.1.1.1.1.1.2.2.2">𝑦</ci><ci id="algorithm2.7.7.m1.1.1.1.1.1.1.2.2.3.cmml" xref="algorithm2.7.7.m1.1.1.1.1.1.1.2.2.3">𝑖</ci></apply></interval></set><ci id="algorithm2.7.7.m1.1.1.1.3.cmml" xref="algorithm2.7.7.m1.1.1.1.3">𝑖</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm2.7.7.m1.1c">\mathcal{D}_{\mathrm{tr}}^{\star}\leftarrow\{({\boldsymbol{x}}_{i}\odot{\boldsymbol{\mathrm{mask}}}^{\star}_{i},y_{i})\}_{i}</annotation></semantics></math><span id="algorithm2.7.7.4" class="ltx_text" style="font-size:90%;">  </span><span id="algorithm2.7.7.5" class="ltx_text" style="font-size:90%;color:#8C8C8C;">// <span id="algorithm2.7.7.5.1" class="ltx_text" style="font-size:89%;">Prepare masked data</span></span>
</div>
<div id="algorithm2.8.8" class="ltx_listingline">
<span id="algorithm2.8.8.1" class="ltx_text" style="font-size:90%;">  </span><span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span><span id="algorithm2.8.8.2" class="ltx_text" style="font-size:90%;">   </span><span id="algorithm2.8.8.3" class="ltx_text" style="font-size:90%;">
</span><math id="algorithm2.8.8.m1.2" class="ltx_Math" alttext="{\boldsymbol{\theta}}_{\star}\leftarrow\mathop{\textrm{argmin}}\mathcal{L}(\mathcal{D}_{\mathrm{tr}}^{m},{\boldsymbol{\theta}}_{k})" display="inline"><semantics id="algorithm2.8.8.m1.2a"><mrow id="algorithm2.8.8.m1.2.2" xref="algorithm2.8.8.m1.2.2.cmml"><msub id="algorithm2.8.8.m1.2.2.4" xref="algorithm2.8.8.m1.2.2.4.cmml"><mi mathsize="90%" id="algorithm2.8.8.m1.2.2.4.2" xref="algorithm2.8.8.m1.2.2.4.2.cmml">𝜽</mi><mo mathsize="90%" id="algorithm2.8.8.m1.2.2.4.3" xref="algorithm2.8.8.m1.2.2.4.3.cmml">⋆</mo></msub><mo mathsize="90%" stretchy="false" id="algorithm2.8.8.m1.2.2.3" xref="algorithm2.8.8.m1.2.2.3.cmml">←</mo><mrow id="algorithm2.8.8.m1.2.2.2" xref="algorithm2.8.8.m1.2.2.2.cmml"><mtext mathsize="90%" id="algorithm2.8.8.m1.2.2.2.3" xref="algorithm2.8.8.m1.2.2.2.3a.cmml">argmin</mtext><mrow id="algorithm2.8.8.m1.2.2.2.2" xref="algorithm2.8.8.m1.2.2.2.2.cmml"><mi class="ltx_font_mathcaligraphic" mathsize="90%" id="algorithm2.8.8.m1.2.2.2.2.4" xref="algorithm2.8.8.m1.2.2.2.2.4.cmml">ℒ</mi><mo lspace="0em" rspace="0em" id="algorithm2.8.8.m1.2.2.2.2.3" xref="algorithm2.8.8.m1.2.2.2.2.3.cmml">​</mo><mrow id="algorithm2.8.8.m1.2.2.2.2.2.2" xref="algorithm2.8.8.m1.2.2.2.2.2.3.cmml"><mo maxsize="90%" minsize="90%" id="algorithm2.8.8.m1.2.2.2.2.2.2.3" xref="algorithm2.8.8.m1.2.2.2.2.2.3.cmml">(</mo><msubsup id="algorithm2.8.8.m1.1.1.1.1.1.1.1" xref="algorithm2.8.8.m1.1.1.1.1.1.1.1.cmml"><mi class="ltx_font_mathcaligraphic" mathsize="90%" id="algorithm2.8.8.m1.1.1.1.1.1.1.1.2.2" xref="algorithm2.8.8.m1.1.1.1.1.1.1.1.2.2.cmml">𝒟</mi><mi mathsize="90%" id="algorithm2.8.8.m1.1.1.1.1.1.1.1.2.3" xref="algorithm2.8.8.m1.1.1.1.1.1.1.1.2.3.cmml">tr</mi><mi mathsize="90%" id="algorithm2.8.8.m1.1.1.1.1.1.1.1.3" xref="algorithm2.8.8.m1.1.1.1.1.1.1.1.3.cmml">m</mi></msubsup><mo mathsize="90%" id="algorithm2.8.8.m1.2.2.2.2.2.2.4" xref="algorithm2.8.8.m1.2.2.2.2.2.3.cmml">,</mo><msub id="algorithm2.8.8.m1.2.2.2.2.2.2.2" xref="algorithm2.8.8.m1.2.2.2.2.2.2.2.cmml"><mi mathsize="90%" id="algorithm2.8.8.m1.2.2.2.2.2.2.2.2" xref="algorithm2.8.8.m1.2.2.2.2.2.2.2.2.cmml">𝜽</mi><mi mathsize="90%" id="algorithm2.8.8.m1.2.2.2.2.2.2.2.3" xref="algorithm2.8.8.m1.2.2.2.2.2.2.2.3.cmml">k</mi></msub><mo maxsize="90%" minsize="90%" id="algorithm2.8.8.m1.2.2.2.2.2.2.5" xref="algorithm2.8.8.m1.2.2.2.2.2.3.cmml">)</mo></mrow></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm2.8.8.m1.2b"><apply id="algorithm2.8.8.m1.2.2.cmml" xref="algorithm2.8.8.m1.2.2"><ci id="algorithm2.8.8.m1.2.2.3.cmml" xref="algorithm2.8.8.m1.2.2.3">←</ci><apply id="algorithm2.8.8.m1.2.2.4.cmml" xref="algorithm2.8.8.m1.2.2.4"><csymbol cd="ambiguous" id="algorithm2.8.8.m1.2.2.4.1.cmml" xref="algorithm2.8.8.m1.2.2.4">subscript</csymbol><ci id="algorithm2.8.8.m1.2.2.4.2.cmml" xref="algorithm2.8.8.m1.2.2.4.2">𝜽</ci><ci id="algorithm2.8.8.m1.2.2.4.3.cmml" xref="algorithm2.8.8.m1.2.2.4.3">⋆</ci></apply><apply id="algorithm2.8.8.m1.2.2.2.cmml" xref="algorithm2.8.8.m1.2.2.2"><ci id="algorithm2.8.8.m1.2.2.2.3a.cmml" xref="algorithm2.8.8.m1.2.2.2.3"><mtext mathsize="90%" id="algorithm2.8.8.m1.2.2.2.3.cmml" xref="algorithm2.8.8.m1.2.2.2.3">argmin</mtext></ci><apply id="algorithm2.8.8.m1.2.2.2.2.cmml" xref="algorithm2.8.8.m1.2.2.2.2"><times id="algorithm2.8.8.m1.2.2.2.2.3.cmml" xref="algorithm2.8.8.m1.2.2.2.2.3"></times><ci id="algorithm2.8.8.m1.2.2.2.2.4.cmml" xref="algorithm2.8.8.m1.2.2.2.2.4">ℒ</ci><interval closure="open" id="algorithm2.8.8.m1.2.2.2.2.2.3.cmml" xref="algorithm2.8.8.m1.2.2.2.2.2.2"><apply id="algorithm2.8.8.m1.1.1.1.1.1.1.1.cmml" xref="algorithm2.8.8.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="algorithm2.8.8.m1.1.1.1.1.1.1.1.1.cmml" xref="algorithm2.8.8.m1.1.1.1.1.1.1.1">superscript</csymbol><apply id="algorithm2.8.8.m1.1.1.1.1.1.1.1.2.cmml" xref="algorithm2.8.8.m1.1.1.1.1.1.1.1"><csymbol cd="ambiguous" id="algorithm2.8.8.m1.1.1.1.1.1.1.1.2.1.cmml" xref="algorithm2.8.8.m1.1.1.1.1.1.1.1">subscript</csymbol><ci id="algorithm2.8.8.m1.1.1.1.1.1.1.1.2.2.cmml" xref="algorithm2.8.8.m1.1.1.1.1.1.1.1.2.2">𝒟</ci><ci id="algorithm2.8.8.m1.1.1.1.1.1.1.1.2.3.cmml" xref="algorithm2.8.8.m1.1.1.1.1.1.1.1.2.3">tr</ci></apply><ci id="algorithm2.8.8.m1.1.1.1.1.1.1.1.3.cmml" xref="algorithm2.8.8.m1.1.1.1.1.1.1.1.3">𝑚</ci></apply><apply id="algorithm2.8.8.m1.2.2.2.2.2.2.2.cmml" xref="algorithm2.8.8.m1.2.2.2.2.2.2.2"><csymbol cd="ambiguous" id="algorithm2.8.8.m1.2.2.2.2.2.2.2.1.cmml" xref="algorithm2.8.8.m1.2.2.2.2.2.2.2">subscript</csymbol><ci id="algorithm2.8.8.m1.2.2.2.2.2.2.2.2.cmml" xref="algorithm2.8.8.m1.2.2.2.2.2.2.2.2">𝜽</ci><ci id="algorithm2.8.8.m1.2.2.2.2.2.2.2.3.cmml" xref="algorithm2.8.8.m1.2.2.2.2.2.2.2.3">𝑘</ci></apply></interval></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm2.8.8.m1.2c">{\boldsymbol{\theta}}_{\star}\leftarrow\mathop{\textrm{argmin}}\mathcal{L}(\mathcal{D}_{\mathrm{tr}}^{m},{\boldsymbol{\theta}}_{k})</annotation></semantics></math><span id="algorithm2.8.8.4" class="ltx_text" style="font-size:90%;">  </span><span id="algorithm2.8.8.5" class="ltx_text" style="font-size:90%;color:#8C8C8C;">// <span id="algorithm2.8.8.5.1" class="ltx_text" style="font-size:89%;">Fine-tune</span></span>
</div>
<div id="algorithm2.9.9" class="ltx_listingline">
<span id="algorithm2.9.9.1" class="ltx_text" style="font-size:90%;">  </span><span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span><span id="algorithm2.9.9.2" class="ltx_text" style="font-size:90%;">   </span><span id="algorithm2.9.9.3" class="ltx_text" style="font-size:90%;">
</span><math id="algorithm2.9.9.m1.1" class="ltx_Math" alttext="{\boldsymbol{S}}\leftarrow{\boldsymbol{S}}\leavevmode\nobreak\ \bigcup\leavevmode\nobreak\ {\boldsymbol{\theta}}_{\star}" display="inline"><semantics id="algorithm2.9.9.m1.1a"><mrow id="algorithm2.9.9.m1.1.1" xref="algorithm2.9.9.m1.1.1.cmml"><mi mathsize="90%" id="algorithm2.9.9.m1.1.1.2" xref="algorithm2.9.9.m1.1.1.2.cmml">𝑺</mi><mo mathsize="90%" stretchy="false" id="algorithm2.9.9.m1.1.1.1" xref="algorithm2.9.9.m1.1.1.1.cmml">←</mo><mrow id="algorithm2.9.9.m1.1.1.3" xref="algorithm2.9.9.m1.1.1.3.cmml"><mi mathsize="90%" id="algorithm2.9.9.m1.1.1.3.2" xref="algorithm2.9.9.m1.1.1.3.2.cmml">𝑺</mi><mo lspace="0.617em" rspace="0em" id="algorithm2.9.9.m1.1.1.3.1" xref="algorithm2.9.9.m1.1.1.3.1.cmml">​</mo><mrow id="algorithm2.9.9.m1.1.1.3.3" xref="algorithm2.9.9.m1.1.1.3.3.cmml"><mo maxsize="90%" minsize="90%" rspace="0.617em" stretchy="true" id="algorithm2.9.9.m1.1.1.3.3.1" xref="algorithm2.9.9.m1.1.1.3.3.1.cmml">⋃</mo><msub id="algorithm2.9.9.m1.1.1.3.3.2" xref="algorithm2.9.9.m1.1.1.3.3.2.cmml"><mi mathsize="90%" id="algorithm2.9.9.m1.1.1.3.3.2.2" xref="algorithm2.9.9.m1.1.1.3.3.2.2.cmml">𝜽</mi><mo mathsize="90%" id="algorithm2.9.9.m1.1.1.3.3.2.3" xref="algorithm2.9.9.m1.1.1.3.3.2.3.cmml">⋆</mo></msub></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm2.9.9.m1.1b"><apply id="algorithm2.9.9.m1.1.1.cmml" xref="algorithm2.9.9.m1.1.1"><ci id="algorithm2.9.9.m1.1.1.1.cmml" xref="algorithm2.9.9.m1.1.1.1">←</ci><ci id="algorithm2.9.9.m1.1.1.2.cmml" xref="algorithm2.9.9.m1.1.1.2">𝑺</ci><apply id="algorithm2.9.9.m1.1.1.3.cmml" xref="algorithm2.9.9.m1.1.1.3"><times id="algorithm2.9.9.m1.1.1.3.1.cmml" xref="algorithm2.9.9.m1.1.1.3.1"></times><ci id="algorithm2.9.9.m1.1.1.3.2.cmml" xref="algorithm2.9.9.m1.1.1.3.2">𝑺</ci><apply id="algorithm2.9.9.m1.1.1.3.3.cmml" xref="algorithm2.9.9.m1.1.1.3.3"><union id="algorithm2.9.9.m1.1.1.3.3.1.cmml" xref="algorithm2.9.9.m1.1.1.3.3.1"></union><apply id="algorithm2.9.9.m1.1.1.3.3.2.cmml" xref="algorithm2.9.9.m1.1.1.3.3.2"><csymbol cd="ambiguous" id="algorithm2.9.9.m1.1.1.3.3.2.1.cmml" xref="algorithm2.9.9.m1.1.1.3.3.2">subscript</csymbol><ci id="algorithm2.9.9.m1.1.1.3.3.2.2.cmml" xref="algorithm2.9.9.m1.1.1.3.3.2.2">𝜽</ci><ci id="algorithm2.9.9.m1.1.1.3.3.2.3.cmml" xref="algorithm2.9.9.m1.1.1.3.3.2.3">⋆</ci></apply></apply></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm2.9.9.m1.1c">{\boldsymbol{S}}\leftarrow{\boldsymbol{S}}\leavevmode\nobreak\ \bigcup\leavevmode\nobreak\ {\boldsymbol{\theta}}_{\star}</annotation></semantics></math><span id="algorithm2.9.9.4" class="ltx_text" style="font-size:90%;">  </span><span id="algorithm2.9.9.5" class="ltx_text" style="font-size:90%;color:#8C8C8C;">// <span id="algorithm2.9.9.5.1" class="ltx_text" style="font-size:89%;">Append to set of models</span></span>
</div>
<div id="algorithm2.11.16" class="ltx_listingline">
<span id="algorithm2.11.16.1" class="ltx_text" style="font-size:90%;">  </span><span class="ltx_rule" style="width:1px;height:100%;background:black;display:inline-block;"> </span><span id="algorithm2.11.16.2" class="ltx_text" style="font-size:90%;">   </span><span id="algorithm2.11.16.3" class="ltx_text" style="font-size:90%;">
</span>
</div>
<div id="algorithm2.10.10" class="ltx_listingline">
<span id="algorithm2.10.10.2" class="ltx_text ltx_font_bold" style="font-size:90%;">while</span><span id="algorithm2.10.10.3" class="ltx_text" style="font-size:90%;"> </span><em id="algorithm2.10.10.1" class="ltx_emph ltx_font_italic" style="font-size:90%;"><math id="algorithm2.10.10.1.m1.2" class="ltx_Math" alttext="\operatorname{get1BestModel}({\boldsymbol{S}})={\boldsymbol{\theta}}_{\star}" display="inline"><semantics id="algorithm2.10.10.1.m1.2a"><mrow id="algorithm2.10.10.1.m1.2.3" xref="algorithm2.10.10.1.m1.2.3.cmml"><mrow id="algorithm2.10.10.1.m1.2.3.2.2" xref="algorithm2.10.10.1.m1.2.3.2.1.cmml"><mi id="algorithm2.10.10.1.m1.1.1" xref="algorithm2.10.10.1.m1.1.1.cmml">get1BestModel</mi><mo id="algorithm2.10.10.1.m1.2.3.2.2a" xref="algorithm2.10.10.1.m1.2.3.2.1.cmml">⁡</mo><mrow id="algorithm2.10.10.1.m1.2.3.2.2.1" xref="algorithm2.10.10.1.m1.2.3.2.1.cmml"><mo stretchy="false" id="algorithm2.10.10.1.m1.2.3.2.2.1.1" xref="algorithm2.10.10.1.m1.2.3.2.1.cmml">(</mo><mi id="algorithm2.10.10.1.m1.2.2" xref="algorithm2.10.10.1.m1.2.2.cmml">𝐒</mi><mo stretchy="false" id="algorithm2.10.10.1.m1.2.3.2.2.1.2" xref="algorithm2.10.10.1.m1.2.3.2.1.cmml">)</mo></mrow></mrow><mo id="algorithm2.10.10.1.m1.2.3.1" xref="algorithm2.10.10.1.m1.2.3.1.cmml">=</mo><msub id="algorithm2.10.10.1.m1.2.3.3" xref="algorithm2.10.10.1.m1.2.3.3.cmml"><mi id="algorithm2.10.10.1.m1.2.3.3.2" xref="algorithm2.10.10.1.m1.2.3.3.2.cmml">𝛉</mi><mo id="algorithm2.10.10.1.m1.2.3.3.3" xref="algorithm2.10.10.1.m1.2.3.3.3.cmml">⋆</mo></msub></mrow><annotation-xml encoding="MathML-Content" id="algorithm2.10.10.1.m1.2b"><apply id="algorithm2.10.10.1.m1.2.3.cmml" xref="algorithm2.10.10.1.m1.2.3"><eq id="algorithm2.10.10.1.m1.2.3.1.cmml" xref="algorithm2.10.10.1.m1.2.3.1"></eq><apply id="algorithm2.10.10.1.m1.2.3.2.1.cmml" xref="algorithm2.10.10.1.m1.2.3.2.2"><ci id="algorithm2.10.10.1.m1.1.1.cmml" xref="algorithm2.10.10.1.m1.1.1">get1BestModel</ci><ci id="algorithm2.10.10.1.m1.2.2.cmml" xref="algorithm2.10.10.1.m1.2.2">𝐒</ci></apply><apply id="algorithm2.10.10.1.m1.2.3.3.cmml" xref="algorithm2.10.10.1.m1.2.3.3"><csymbol cd="ambiguous" id="algorithm2.10.10.1.m1.2.3.3.1.cmml" xref="algorithm2.10.10.1.m1.2.3.3">subscript</csymbol><ci id="algorithm2.10.10.1.m1.2.3.3.2.cmml" xref="algorithm2.10.10.1.m1.2.3.3.2">𝛉</ci><ci id="algorithm2.10.10.1.m1.2.3.3.3.cmml" xref="algorithm2.10.10.1.m1.2.3.3.3">⋆</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm2.10.10.1.m1.2c">\operatorname{get1BestModel}({\boldsymbol{S}})={\boldsymbol{\theta}}_{\star}</annotation></semantics></math>  <span id="algorithm2.10.10.1.1" class="ltx_text ltx_font_upright" style="color:#8C8C8C;">// <span id="algorithm2.10.10.1.1.1" class="ltx_text" style="font-size:89%;">New combination is best</span></span></em>
</div>
<div id="algorithm2.11.11" class="ltx_listingline">
<math id="algorithm2.11.11.m1.2" class="ltx_Math" alttext="{\boldsymbol{\theta}}_{\star}\leavevmode\nobreak\ \leftarrow\leavevmode\nobreak\ \operatorname{get1BestModel}({\boldsymbol{S}})" display="inline"><semantics id="algorithm2.11.11.m1.2a"><mrow id="algorithm2.11.11.m1.2.3" xref="algorithm2.11.11.m1.2.3.cmml"><msub id="algorithm2.11.11.m1.2.3.2" xref="algorithm2.11.11.m1.2.3.2.cmml"><mi mathsize="90%" id="algorithm2.11.11.m1.2.3.2.2" xref="algorithm2.11.11.m1.2.3.2.2.cmml">𝜽</mi><mo mathsize="90%" id="algorithm2.11.11.m1.2.3.2.3" xref="algorithm2.11.11.m1.2.3.2.3.cmml">⋆</mo></msub><mo lspace="0.728em" mathsize="90%" rspace="0.728em" stretchy="false" id="algorithm2.11.11.m1.2.3.1" xref="algorithm2.11.11.m1.2.3.1.cmml">←</mo><mrow id="algorithm2.11.11.m1.2.3.3.2" xref="algorithm2.11.11.m1.2.3.3.1.cmml"><mi mathsize="90%" id="algorithm2.11.11.m1.1.1" xref="algorithm2.11.11.m1.1.1.cmml">get1BestModel</mi><mo id="algorithm2.11.11.m1.2.3.3.2a" xref="algorithm2.11.11.m1.2.3.3.1.cmml">⁡</mo><mrow id="algorithm2.11.11.m1.2.3.3.2.1" xref="algorithm2.11.11.m1.2.3.3.1.cmml"><mo maxsize="90%" minsize="90%" id="algorithm2.11.11.m1.2.3.3.2.1.1" xref="algorithm2.11.11.m1.2.3.3.1.cmml">(</mo><mi mathsize="90%" id="algorithm2.11.11.m1.2.2" xref="algorithm2.11.11.m1.2.2.cmml">𝑺</mi><mo maxsize="90%" minsize="90%" id="algorithm2.11.11.m1.2.3.3.2.1.2" xref="algorithm2.11.11.m1.2.3.3.1.cmml">)</mo></mrow></mrow></mrow><annotation-xml encoding="MathML-Content" id="algorithm2.11.11.m1.2b"><apply id="algorithm2.11.11.m1.2.3.cmml" xref="algorithm2.11.11.m1.2.3"><ci id="algorithm2.11.11.m1.2.3.1.cmml" xref="algorithm2.11.11.m1.2.3.1">←</ci><apply id="algorithm2.11.11.m1.2.3.2.cmml" xref="algorithm2.11.11.m1.2.3.2"><csymbol cd="ambiguous" id="algorithm2.11.11.m1.2.3.2.1.cmml" xref="algorithm2.11.11.m1.2.3.2">subscript</csymbol><ci id="algorithm2.11.11.m1.2.3.2.2.cmml" xref="algorithm2.11.11.m1.2.3.2.2">𝜽</ci><ci id="algorithm2.11.11.m1.2.3.2.3.cmml" xref="algorithm2.11.11.m1.2.3.2.3">⋆</ci></apply><apply id="algorithm2.11.11.m1.2.3.3.1.cmml" xref="algorithm2.11.11.m1.2.3.3.2"><ci id="algorithm2.11.11.m1.1.1.cmml" xref="algorithm2.11.11.m1.1.1">get1BestModel</ci><ci id="algorithm2.11.11.m1.2.2.cmml" xref="algorithm2.11.11.m1.2.2">𝑺</ci></apply></apply></annotation-xml><annotation encoding="application/x-tex" id="algorithm2.11.11.m1.2c">{\boldsymbol{\theta}}_{\star}\leavevmode\nobreak\ \leftarrow\leavevmode\nobreak\ \operatorname{get1BestModel}({\boldsymbol{S}})</annotation></semantics></math><span id="algorithm2.11.11.1" class="ltx_text" style="font-size:90%;">  </span><span id="algorithm2.11.11.2" class="ltx_text" style="font-size:90%;color:#8C8C8C;">// <span id="algorithm2.11.11.2.1" class="ltx_text" style="font-size:89%;">Return best model</span></span><span id="algorithm2.11.11.3" class="ltx_text" style="font-size:90%;">
</span>
</div>
</div>
<figcaption class="ltx_caption" style="font-size:90%;"><span class="ltx_tag ltx_tag_float"><span id="algorithm2.15.1.1" class="ltx_text ltx_font_bold">Algorithm 2</span> </span>Distilling multiple models into one with greedy pairwise combinations.</figcaption>
</figure>
</section>
<section id="A4" class="ltx_appendix">
<h2 class="ltx_title ltx_title_appendix">
<span class="ltx_tag ltx_tag_appendix">Appendix D </span>Experimental details: collages</h2>

<div id="A4.p1" class="ltx_para ltx_noindent">
<p id="A4.p1.1" class="ltx_p"><span id="A4.p1.1.1" class="ltx_text ltx_font_bold">Collages dataset, raw pixels as input.</span> 
The <em id="A4.p1.1.2" class="ltx_emph ltx_font_italic">collages</em> dataset is described in <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib72" title="" class="ltx_ref">72</a>]</cite>.
We used the 4-block ordered version (each of the four source datasets appear in the same quadrant in every instance).
We re-generated the collages with bilinear downsampling by 1/4th using the code from the authors.<span id="footnote7" class="ltx_note ltx_role_footnote"><sup class="ltx_note_mark">7</sup><span class="ltx_note_outer"><span class="ltx_note_content"><sup class="ltx_note_mark">7</sup><span class="ltx_tag ltx_tag_note">7</span><a target="_blank" href="https://github.com/dteney/collages-dataset" title="" class="ltx_ref ltx_url ltx_font_typewriter">https://github.com/dteney/collages-dataset</a></span></span></span>
The reason we do not use full-size images is purely computational.
The original data <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib72" title="" class="ltx_ref">72</a>]</cite> used nearest-neighbour downsampling because it preserves the contrast and dynamic range,
but bilinear downsampling produces smoother images that better match the manifold assumption of natural data that is important in this work.
The bilinear downsampling causes a compression of the dynamic range that is uneven across the tiles and mostly affects SVHN and CIFAR.
To compensate for it, we apply a standard local contrast normalization (<span id="A4.p1.1.3" class="ltx_text ltx_font_typewriter" style="font-size:90%;">adapthisteq</span> in Matlab).
We also replace the pixels with a constant value near 0 with small noise. These pixels exist in the MNIST data but are again unrepresentative of natural data.</p>
</div>
<div id="A4.1.p1" class="ltx_para ltx_minipage ltx_align_middle" style="width:433.6pt;">
<p id="A4.1.p1.1" class="ltx_p">In Matlab code:

<span id="A4.1.p1.1.1" class="ltx_inline-block ltx_framed ltx_framed_rectangle" style="border-color: #000000;">
<span id="A4.1.p1.1.1.1" class="ltx_p"><span id="A4.1.p1.1.1.1.1" class="ltx_text ltx_font_typewriter" style="font-size:70%;color:#00BF00;">% Local contrast normalization
<br class="ltx_break"><span id="A4.1.p1.1.1.1.1.1" class="ltx_text" style="color:#000000;">img = adapthisteq(img, ’NumTiles’, [2 2]);
<br class="ltx_break"></span>% Replace dark pixels with small random values
<br class="ltx_break"><span id="A4.1.p1.1.1.1.1.2" class="ltx_text" style="color:#000000;">img(img &lt; .05) = rand(nnz(img &lt; .05), 1) * .05;
<span id="A4.1.p1.1.1.1.1.2.1" class="ltx_text ltx_font_serif"></span></span></span></span>
</span></p>
</div>
<div id="A4.p2" class="ltx_para ltx_noindent">
<p id="A4.p2.1" class="ltx_p"><span id="A4.p2.1.1" class="ltx_text ltx_font_bold">Collages dataset, ResNet features as input.</span> 
We obtain feature maps by passing full-resolution images of collages into a standard frozen ImageNet-pretrained ResNet-18 <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib27" title="" class="ltx_ref">27</a>]</cite>.
We chose to use feature maps from an intermediate layer <span id="A4.p2.1.2" class="ltx_text ltx_font_typewriter" style="font-size:90%;">res3b_relu</span> because it retains a reasonable spatial resolution (8<math id="A4.p2.1.m1.1" class="ltx_Math" alttext="\times" display="inline"><semantics id="A4.p2.1.m1.1a"><mo id="A4.p2.1.m1.1.1" xref="A4.p2.1.m1.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="A4.p2.1.m1.1b"><times id="A4.p2.1.m1.1.1.cmml" xref="A4.p2.1.m1.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="A4.p2.1.m1.1c">\times</annotation></semantics></math>8 with 128 channels).
The independence loss (<a href="#S4.E1" title="In 4.1 Independent models ‣ 4 Proposed method ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">1</span></a>) for these experiments uses a dot product rather than a cosine similarity.
It works better but we don’t know why.</p>
</div>
<div id="A4.p3" class="ltx_para ltx_noindent">
<p id="A4.p3.1" class="ltx_p"><span id="A4.p3.1.1" class="ltx_text ltx_font_bold">Hyperparameters.</span> 
See Table <a href="#A4.T5" title="Table 5 ‣ Appendix D Experimental details: collages ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">5</span></a>.
The classifiers’ hyperparameters are chosen to optimize the baseline reported as “Upper bounds” in the results.
None of these hyperparameters are particularly tuned to the proposed methods.</p>
</div>
<figure id="A4.T5" class="ltx_table">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table"><span id="A4.T5.9.1.1" class="ltx_text" style="font-size:90%;">Table 5</span>: </span><span id="A4.T5.10.2" class="ltx_text" style="font-size:90%;">Hyperparameters used on collages (top), GQA, and WILDS-Camelyon17 (bottom).</span></figcaption>
<table id="A4.T5.7" class="ltx_tabular ltx_centering ltx_align_middle">
<tr id="A4.T5.7.8" class="ltx_tr">
<td id="A4.T5.7.8.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify ltx_border_tt" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.8.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.8.1.1.1" class="ltx_p"><span id="A4.T5.7.8.1.1.1.1" class="ltx_text ltx_font_bold" style="font-size:70%;">Hyperparameter</span></span>
</span>
</td>
<td id="A4.T5.7.8.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left ltx_border_tt" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.8.2.1" class="ltx_text ltx_font_bold" style="font-size:70%;">Collages, pixels as input</span></td>
<td id="A4.T5.7.8.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left ltx_border_tt" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.8.3.1" class="ltx_text ltx_font_bold" style="font-size:70%;">ResNet features as input</span></td>
</tr>
<tr id="A4.T5.7.9" class="ltx_tr">
<td id="A4.T5.7.9.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify ltx_border_t" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.9.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.9.1.1.1" class="ltx_p"><span id="A4.T5.7.9.1.1.1.1" class="ltx_text ltx_font_bold" style="font-size:70%;">Classifier</span></span>
</span>
</td>
<td id="A4.T5.7.9.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_border_t" style="padding:0.05pt 0.0pt;"></td>
<td id="A4.T5.7.9.3" class="ltx_td ltx_border_t" style="padding:0.05pt 0.0pt;"></td>
</tr>
<tr id="A4.T5.4.4" class="ltx_tr">
<td id="A4.T5.4.4.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.4.4.5.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.4.4.5.1.1" class="ltx_p"><span id="A4.T5.4.4.5.1.1.1" class="ltx_text" style="font-size:70%;">Input dimensions</span></span>
</span>
</td>
<td id="A4.T5.2.2.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.2.2.2.1" class="ltx_text" style="font-size:70%;">16</span><math id="A4.T5.1.1.1.m1.1" class="ltx_Math" alttext="\times" display="inline"><semantics id="A4.T5.1.1.1.m1.1a"><mo mathsize="70%" id="A4.T5.1.1.1.m1.1.1" xref="A4.T5.1.1.1.m1.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="A4.T5.1.1.1.m1.1b"><times id="A4.T5.1.1.1.m1.1.1.cmml" xref="A4.T5.1.1.1.m1.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="A4.T5.1.1.1.m1.1c">\times</annotation></semantics></math><span id="A4.T5.2.2.2.2" class="ltx_text" style="font-size:70%;">16</span><math id="A4.T5.2.2.2.m2.1" class="ltx_Math" alttext="\times" display="inline"><semantics id="A4.T5.2.2.2.m2.1a"><mo mathsize="70%" id="A4.T5.2.2.2.m2.1.1" xref="A4.T5.2.2.2.m2.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="A4.T5.2.2.2.m2.1b"><times id="A4.T5.2.2.2.m2.1.1.cmml" xref="A4.T5.2.2.2.m2.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="A4.T5.2.2.2.m2.1c">\times</annotation></semantics></math><span id="A4.T5.2.2.2.3" class="ltx_text" style="font-size:70%;">1  (grayscale)</span>
</td>
<td id="A4.T5.4.4.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.4.4.4.1" class="ltx_text" style="font-size:70%;">8</span><math id="A4.T5.3.3.3.m1.1" class="ltx_Math" alttext="\times" display="inline"><semantics id="A4.T5.3.3.3.m1.1a"><mo mathsize="70%" id="A4.T5.3.3.3.m1.1.1" xref="A4.T5.3.3.3.m1.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="A4.T5.3.3.3.m1.1b"><times id="A4.T5.3.3.3.m1.1.1.cmml" xref="A4.T5.3.3.3.m1.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="A4.T5.3.3.3.m1.1c">\times</annotation></semantics></math><span id="A4.T5.4.4.4.2" class="ltx_text" style="font-size:70%;">8</span><math id="A4.T5.4.4.4.m2.1" class="ltx_Math" alttext="\times" display="inline"><semantics id="A4.T5.4.4.4.m2.1a"><mo mathsize="70%" id="A4.T5.4.4.4.m2.1.1" xref="A4.T5.4.4.4.m2.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="A4.T5.4.4.4.m2.1b"><times id="A4.T5.4.4.4.m2.1.1.cmml" xref="A4.T5.4.4.4.m2.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="A4.T5.4.4.4.m2.1c">\times</annotation></semantics></math><span id="A4.T5.4.4.4.3" class="ltx_text" style="font-size:70%;">128   (res3b_relu layer)</span>
</td>
</tr>
<tr id="A4.T5.7.10" class="ltx_tr">
<td id="A4.T5.7.10.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.10.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.10.1.1.1" class="ltx_p"><span id="A4.T5.7.10.1.1.1.1" class="ltx_text" style="font-size:70%;">Architecture</span></span>
</span>
</td>
<td id="A4.T5.7.10.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.10.2.1" class="ltx_text" style="font-size:70%;">1 Hidden layer (fully-connected)</span></td>
<td id="A4.T5.7.10.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.10.3.1" class="ltx_text" style="font-size:70%;">1 Hidden layer (channel-wise)</span></td>
</tr>
<tr id="A4.T5.7.11" class="ltx_tr">
<td id="A4.T5.7.11.1" class="ltx_td ltx_nopad_l ltx_nopad_r" style="padding:0.05pt 0.0pt;"></td>
<td id="A4.T5.7.11.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.11.2.1" class="ltx_text" style="font-size:70%;">8 neurons</span></td>
<td id="A4.T5.7.11.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.11.3.1" class="ltx_text" style="font-size:70%;">16 neurons</span></td>
</tr>
<tr id="A4.T5.7.12" class="ltx_tr">
<td id="A4.T5.7.12.1" class="ltx_td ltx_nopad_l ltx_nopad_r" style="padding:0.05pt 0.0pt;"></td>
<td id="A4.T5.7.12.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.12.2.1" class="ltx_text" style="font-size:70%;">Output layer (fully-connected).</span></td>
<td id="A4.T5.7.12.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.12.3.1" class="ltx_text" style="font-size:70%;">Output layer (fully-connected)</span></td>
</tr>
<tr id="A4.T5.7.13" class="ltx_tr">
<td id="A4.T5.7.13.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.13.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.13.1.1.1" class="ltx_p"><span id="A4.T5.7.13.1.1.1.1" class="ltx_text" style="font-size:70%;">Hidden Activations</span></span>
</span>
</td>
<td id="A4.T5.7.13.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.13.2.1" class="ltx_text" style="font-size:70%;">Leaky ReLU, leak scale 0.01</span></td>
<td id="A4.T5.7.13.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.13.3.1" class="ltx_text" style="font-size:70%;">Leaky ReLU, leak scale 0.01</span></td>
</tr>
<tr id="A4.T5.7.14" class="ltx_tr">
<td id="A4.T5.7.14.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.14.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.14.1.1.1" class="ltx_p"><span id="A4.T5.7.14.1.1.1.1" class="ltx_text" style="font-size:70%;">Output activation</span></span>
</span>
</td>
<td id="A4.T5.7.14.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.14.2.1" class="ltx_text" style="font-size:70%;">Sigmoid</span></td>
<td id="A4.T5.7.14.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.14.3.1" class="ltx_text" style="font-size:70%;">Sigmoid</span></td>
</tr>
<tr id="A4.T5.7.15" class="ltx_tr">
<td id="A4.T5.7.15.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.15.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.15.1.1.1" class="ltx_p"><span id="A4.T5.7.15.1.1.1.1" class="ltx_text" style="font-size:70%;">Mini-batch size</span></span>
</span>
</td>
<td id="A4.T5.7.15.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.15.2.1" class="ltx_text" style="font-size:70%;">256</span></td>
<td id="A4.T5.7.15.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.15.3.1" class="ltx_text" style="font-size:70%;">256</span></td>
</tr>
<tr id="A4.T5.7.16" class="ltx_tr">
<td id="A4.T5.7.16.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.16.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.16.1.1.1" class="ltx_p"><span id="A4.T5.7.16.1.1.1.1" class="ltx_text" style="font-size:70%;">Optimizer</span></span>
</span>
</td>
<td id="A4.T5.7.16.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.16.2.1" class="ltx_text" style="font-size:70%;">Adam</span></td>
<td id="A4.T5.7.16.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.16.3.1" class="ltx_text" style="font-size:70%;">Adam</span></td>
</tr>
<tr id="A4.T5.7.17" class="ltx_tr">
<td id="A4.T5.7.17.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.17.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.17.1.1.1" class="ltx_p"><span id="A4.T5.7.17.1.1.1.1" class="ltx_text" style="font-size:70%;">Learning rate</span></span>
</span>
</td>
<td id="A4.T5.7.17.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.17.2.1" class="ltx_text" style="font-size:70%;">0.002</span></td>
<td id="A4.T5.7.17.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.17.3.1" class="ltx_text" style="font-size:70%;">0.001</span></td>
</tr>
<tr id="A4.T5.7.18" class="ltx_tr">
<td id="A4.T5.7.18.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.18.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.18.1.1.1" class="ltx_p"><span id="A4.T5.7.18.1.1.1.1" class="ltx_text" style="font-size:70%;">Optimization length</span></span>
</span>
</td>
<td id="A4.T5.7.18.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.18.2.1" class="ltx_text" style="font-size:70%;">10,000 Updates</span></td>
<td id="A4.T5.7.18.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.18.3.1" class="ltx_text" style="font-size:70%;">30,000 Updates</span></td>
</tr>
<tr id="A4.T5.7.19" class="ltx_tr">
<td id="A4.T5.7.19.1" class="ltx_td ltx_nopad_l ltx_nopad_r" style="padding:0.05pt 0.0pt;"></td>
<td id="A4.T5.7.19.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.19.2.1" class="ltx_text" style="font-size:70%;">No early stopping</span></td>
<td id="A4.T5.7.19.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.19.3.1" class="ltx_text" style="font-size:70%;">No early stopping</span></td>
</tr>
<tr id="A4.T5.7.20" class="ltx_tr">
<td id="A4.T5.7.20.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify ltx_border_t" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.20.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.20.1.1.1" class="ltx_p"><span id="A4.T5.7.20.1.1.1.1" class="ltx_text ltx_font_bold" style="font-size:70%;">PCA Manifold model</span></span>
</span>
</td>
<td id="A4.T5.7.20.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_border_t" style="padding:0.05pt 0.0pt;"></td>
<td id="A4.T5.7.20.3" class="ltx_td ltx_border_t" style="padding:0.05pt 0.0pt;"></td>
</tr>
<tr id="A4.T5.7.21" class="ltx_tr">
<td id="A4.T5.7.21.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.21.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.21.1.1.1" class="ltx_p"><span id="A4.T5.7.21.1.1.1.1" class="ltx_text" style="font-size:70%;">Number of components</span></span>
</span>
</td>
<td id="A4.T5.7.21.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.21.2.1" class="ltx_text" style="font-size:70%;">24</span></td>
<td id="A4.T5.7.21.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.21.3.1" class="ltx_text" style="font-size:70%;">–</span></td>
</tr>
<tr id="A4.T5.7.22" class="ltx_tr">
<td id="A4.T5.7.22.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.22.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.22.1.1.1" class="ltx_p"><span id="A4.T5.7.22.1.1.1.1" class="ltx_text" style="font-size:70%;">Retained variance</span></span>
</span>
</td>
<td id="A4.T5.7.22.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.22.2.1" class="ltx_text" style="font-size:70%;">85%</span></td>
<td id="A4.T5.7.22.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.22.3.1" class="ltx_text" style="font-size:70%;">–</span></td>
</tr>
<tr id="A4.T5.7.23" class="ltx_tr">
<td id="A4.T5.7.23.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify ltx_border_t" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.23.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.23.1.1.1" class="ltx_p"><span id="A4.T5.7.23.1.1.1.1" class="ltx_text ltx_font_bold" style="font-size:70%;">VAE Manifold model</span></span>
</span>
</td>
<td id="A4.T5.7.23.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_border_t" style="padding:0.05pt 0.0pt;"></td>
<td id="A4.T5.7.23.3" class="ltx_td ltx_border_t" style="padding:0.05pt 0.0pt;"></td>
</tr>
<tr id="A4.T5.7.24" class="ltx_tr">
<td id="A4.T5.7.24.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.24.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.24.1.1.1" class="ltx_p"><span id="A4.T5.7.24.1.1.1.1" class="ltx_text" style="font-size:70%;">Architecture</span></span>
</span>
</td>
<td id="A4.T5.7.24.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.24.2.1" class="ltx_text" style="font-size:70%;">2-Hidden layer MLP</span></td>
<td id="A4.T5.7.24.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.24.3.1" class="ltx_text" style="font-size:70%;">2-Hidden layer MLP</span></td>
</tr>
<tr id="A4.T5.7.25" class="ltx_tr">
<td id="A4.T5.7.25.1" class="ltx_td ltx_nopad_l ltx_nopad_r" style="padding:0.05pt 0.0pt;"></td>
<td id="A4.T5.7.25.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.25.2.1" class="ltx_text" style="font-size:70%;">128 neurons per layer</span></td>
<td id="A4.T5.7.25.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.25.3.1" class="ltx_text" style="font-size:70%;">128 neurons per layer</span></td>
</tr>
<tr id="A4.T5.7.26" class="ltx_tr">
<td id="A4.T5.7.26.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.26.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.26.1.1.1" class="ltx_p"><span id="A4.T5.7.26.1.1.1.1" class="ltx_text" style="font-size:70%;">Latent dimensions</span></span>
</span>
</td>
<td id="A4.T5.7.26.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.26.2.1" class="ltx_text" style="font-size:70%;">24</span></td>
<td id="A4.T5.7.26.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.26.3.1" class="ltx_text" style="font-size:70%;">24</span></td>
</tr>
<tr id="A4.T5.7.27" class="ltx_tr">
<td id="A4.T5.7.27.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.27.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.27.1.1.1" class="ltx_p"><span id="A4.T5.7.27.1.1.1.1" class="ltx_text" style="font-size:70%;">Hidden activations</span></span>
</span>
</td>
<td id="A4.T5.7.27.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.27.2.1" class="ltx_text" style="font-size:70%;">ReLU</span></td>
<td id="A4.T5.7.27.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.27.3.1" class="ltx_text" style="font-size:70%;">ReLU</span></td>
</tr>
<tr id="A4.T5.7.28" class="ltx_tr">
<td id="A4.T5.7.28.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.28.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.28.1.1.1" class="ltx_p"><span id="A4.T5.7.28.1.1.1.1" class="ltx_text" style="font-size:70%;">Output activations</span></span>
</span>
</td>
<td id="A4.T5.7.28.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.28.2.1" class="ltx_text" style="font-size:70%;">Sigmoid</span></td>
<td id="A4.T5.7.28.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.28.3.1" class="ltx_text" style="font-size:70%;">Sigmoid</span></td>
</tr>
<tr id="A4.T5.7.29" class="ltx_tr">
<td id="A4.T5.7.29.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.29.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.29.1.1.1" class="ltx_p"><span id="A4.T5.7.29.1.1.1.1" class="ltx_text" style="font-size:70%;">Mini-batch size</span></span>
</span>
</td>
<td id="A4.T5.7.29.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.29.2.1" class="ltx_text" style="font-size:70%;">256</span></td>
<td id="A4.T5.7.29.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.29.3.1" class="ltx_text" style="font-size:70%;">128</span></td>
</tr>
<tr id="A4.T5.7.30" class="ltx_tr">
<td id="A4.T5.7.30.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.30.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.30.1.1.1" class="ltx_p"><span id="A4.T5.7.30.1.1.1.1" class="ltx_text" style="font-size:70%;">Optimizer</span></span>
</span>
</td>
<td id="A4.T5.7.30.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.30.2.1" class="ltx_text" style="font-size:70%;">Adam</span></td>
<td id="A4.T5.7.30.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.30.3.1" class="ltx_text" style="font-size:70%;">Adam</span></td>
</tr>
<tr id="A4.T5.7.31" class="ltx_tr">
<td id="A4.T5.7.31.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.31.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.31.1.1.1" class="ltx_p"><span id="A4.T5.7.31.1.1.1.1" class="ltx_text" style="font-size:70%;">Learning rate</span></span>
</span>
</td>
<td id="A4.T5.7.31.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.31.2.1" class="ltx_text" style="font-size:70%;">0.001</span></td>
<td id="A4.T5.7.31.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.31.3.1" class="ltx_text" style="font-size:70%;">0.001</span></td>
</tr>
<tr id="A4.T5.7.32" class="ltx_tr">
<td id="A4.T5.7.32.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.32.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.32.1.1.1" class="ltx_p"><span id="A4.T5.7.32.1.1.1.1" class="ltx_text" style="font-size:70%;">Optimization length</span></span>
</span>
</td>
<td id="A4.T5.7.32.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.32.2.1" class="ltx_text" style="font-size:70%;">100 Epochs</span></td>
<td id="A4.T5.7.32.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.32.3.1" class="ltx_text" style="font-size:70%;">300 Epochs</span></td>
</tr>
<tr id="A4.T5.7.33" class="ltx_tr">
<td id="A4.T5.7.33.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.33.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.33.1.1.1" class="ltx_p"><span id="A4.T5.7.33.1.1.1.1" class="ltx_text" style="font-size:70%;">Weight of KL loss</span></span>
</span>
</td>
<td id="A4.T5.7.33.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.33.2.1" class="ltx_text" style="font-size:70%;">0.01</span></td>
<td id="A4.T5.7.33.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.33.3.1" class="ltx_text" style="font-size:70%;">0.01</span></td>
</tr>
<tr id="A4.T5.7.34" class="ltx_tr">
<td id="A4.T5.7.34.1" class="ltx_td ltx_nopad_l ltx_nopad_r" style="padding:0.05pt 0.0pt;"></td>
<td id="A4.T5.7.34.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;" colspan="2"><span id="A4.T5.7.34.2.1" class="ltx_text" style="font-size:70%;">(small to prioritize accurate reconstructions)</span></td>
</tr>
<tr id="A4.T5.7.35" class="ltx_tr">
<td id="A4.T5.7.35.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify ltx_border_tt ltx_border_tt" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.35.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.35.1.1.1" class="ltx_p"><span id="A4.T5.7.35.1.1.1.1" class="ltx_text ltx_font_bold" style="font-size:70%;">Hyperparameter</span></span>
</span>
</td>
<td id="A4.T5.7.35.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left ltx_border_tt ltx_border_tt" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.35.2.1" class="ltx_text ltx_font_bold" style="font-size:70%;">GQA</span></td>
<td id="A4.T5.7.35.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left ltx_border_tt ltx_border_tt" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.35.3.1" class="ltx_text ltx_font_bold" style="font-size:70%;">WILDS-Camelyon17</span></td>
</tr>
<tr id="A4.T5.7.36" class="ltx_tr">
<td id="A4.T5.7.36.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify ltx_border_t" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.36.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.36.1.1.1" class="ltx_p"><span id="A4.T5.7.36.1.1.1.1" class="ltx_text ltx_font_bold" style="font-size:70%;">Classifier</span></span>
</span>
</td>
<td id="A4.T5.7.36.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_border_t" style="padding:0.05pt 0.0pt;"></td>
<td id="A4.T5.7.36.3" class="ltx_td ltx_border_t" style="padding:0.05pt 0.0pt;"></td>
</tr>
<tr id="A4.T5.6.6" class="ltx_tr">
<td id="A4.T5.6.6.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.6.6.3.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.6.6.3.1.1" class="ltx_p"><span id="A4.T5.6.6.3.1.1.1" class="ltx_text" style="font-size:70%;">Input dimensions</span></span>
</span>
</td>
<td id="A4.T5.5.5.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.5.5.1.1" class="ltx_text" style="font-size:70%;">1</span><math id="A4.T5.5.5.1.m1.1" class="ltx_Math" alttext="\times" display="inline"><semantics id="A4.T5.5.5.1.m1.1a"><mo mathsize="70%" id="A4.T5.5.5.1.m1.1.1" xref="A4.T5.5.5.1.m1.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="A4.T5.5.5.1.m1.1b"><times id="A4.T5.5.5.1.m1.1.1.cmml" xref="A4.T5.5.5.1.m1.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="A4.T5.5.5.1.m1.1c">\times</annotation></semantics></math><span id="A4.T5.5.5.1.2" class="ltx_text" style="font-size:70%;">300 (text)</span>
</td>
<td id="A4.T5.6.6.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.6.6.2.1" class="ltx_text" style="font-size:70%;">1</span><math id="A4.T5.6.6.2.m1.1" class="ltx_Math" alttext="\times" display="inline"><semantics id="A4.T5.6.6.2.m1.1a"><mo mathsize="70%" id="A4.T5.6.6.2.m1.1.1" xref="A4.T5.6.6.2.m1.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="A4.T5.6.6.2.m1.1b"><times id="A4.T5.6.6.2.m1.1.1.cmml" xref="A4.T5.6.6.2.m1.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="A4.T5.6.6.2.m1.1c">\times</annotation></semantics></math><span id="A4.T5.6.6.2.2" class="ltx_text" style="font-size:70%;">1024 (DenseNet features)</span>
</td>
</tr>
<tr id="A4.T5.7.7" class="ltx_tr">
<td id="A4.T5.7.7.2" class="ltx_td ltx_nopad_l ltx_nopad_r" style="padding:0.05pt 0.0pt;"></td>
<td id="A4.T5.7.7.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.7.1.1" class="ltx_text" style="font-size:70%;">1</span><math id="A4.T5.7.7.1.m1.1" class="ltx_Math" alttext="\times" display="inline"><semantics id="A4.T5.7.7.1.m1.1a"><mo mathsize="70%" id="A4.T5.7.7.1.m1.1.1" xref="A4.T5.7.7.1.m1.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="A4.T5.7.7.1.m1.1b"><times id="A4.T5.7.7.1.m1.1.1.cmml" xref="A4.T5.7.7.1.m1.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="A4.T5.7.7.1.m1.1c">\times</annotation></semantics></math><span id="A4.T5.7.7.1.2" class="ltx_text" style="font-size:70%;">2048 (image)</span>
</td>
<td id="A4.T5.7.7.3" class="ltx_td ltx_nopad_l ltx_nopad_r" style="padding:0.05pt 0.0pt;"></td>
</tr>
<tr id="A4.T5.7.37" class="ltx_tr">
<td id="A4.T5.7.37.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.37.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.37.1.1.1" class="ltx_p"><span id="A4.T5.7.37.1.1.1.1" class="ltx_text" style="font-size:70%;">Hidden layers dimension</span></span>
</span>
</td>
<td id="A4.T5.7.37.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.37.2.1" class="ltx_text" style="font-size:70%;">256</span></td>
<td id="A4.T5.7.37.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.37.3.1" class="ltx_text" style="font-size:70%;">–</span></td>
</tr>
<tr id="A4.T5.7.38" class="ltx_tr">
<td id="A4.T5.7.38.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.38.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.38.1.1.1" class="ltx_p"><span id="A4.T5.7.38.1.1.1.1" class="ltx_text" style="font-size:70%;">Hidden activations</span></span>
</span>
</td>
<td id="A4.T5.7.38.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.38.2.1" class="ltx_text" style="font-size:70%;">Leaky ReLU</span></td>
<td id="A4.T5.7.38.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.38.3.1" class="ltx_text" style="font-size:70%;">–</span></td>
</tr>
<tr id="A4.T5.7.39" class="ltx_tr">
<td id="A4.T5.7.39.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.39.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.39.1.1.1" class="ltx_p"><span id="A4.T5.7.39.1.1.1.1" class="ltx_text" style="font-size:70%;">Output activations</span></span>
</span>
</td>
<td id="A4.T5.7.39.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.39.2.1" class="ltx_text" style="font-size:70%;">Sigmoid</span></td>
<td id="A4.T5.7.39.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.39.3.1" class="ltx_text" style="font-size:70%;">Sigmoid</span></td>
</tr>
<tr id="A4.T5.7.40" class="ltx_tr">
<td id="A4.T5.7.40.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.40.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.40.1.1.1" class="ltx_p"><span id="A4.T5.7.40.1.1.1.1" class="ltx_text" style="font-size:70%;">Mini-batch size</span></span>
</span>
</td>
<td id="A4.T5.7.40.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.40.2.1" class="ltx_text" style="font-size:70%;">256</span></td>
<td id="A4.T5.7.40.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.40.3.1" class="ltx_text" style="font-size:70%;">256</span></td>
</tr>
<tr id="A4.T5.7.41" class="ltx_tr">
<td id="A4.T5.7.41.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.41.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.41.1.1.1" class="ltx_p"><span id="A4.T5.7.41.1.1.1.1" class="ltx_text" style="font-size:70%;">Optimizer</span></span>
</span>
</td>
<td id="A4.T5.7.41.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.41.2.1" class="ltx_text" style="font-size:70%;">Adam</span></td>
<td id="A4.T5.7.41.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.41.3.1" class="ltx_text" style="font-size:70%;">Adam</span></td>
</tr>
<tr id="A4.T5.7.42" class="ltx_tr">
<td id="A4.T5.7.42.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.42.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.42.1.1.1" class="ltx_p"><span id="A4.T5.7.42.1.1.1.1" class="ltx_text" style="font-size:70%;">Learning rate</span></span>
</span>
</td>
<td id="A4.T5.7.42.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.42.2.1" class="ltx_text" style="font-size:70%;">0.002</span></td>
<td id="A4.T5.7.42.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.42.3.1" class="ltx_text" style="font-size:70%;">0.001</span></td>
</tr>
<tr id="A4.T5.7.43" class="ltx_tr">
<td id="A4.T5.7.43.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.43.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.43.1.1.1" class="ltx_p"><span id="A4.T5.7.43.1.1.1.1" class="ltx_text" style="font-size:70%;">Optimization length</span></span>
</span>
</td>
<td id="A4.T5.7.43.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.43.2.1" class="ltx_text" style="font-size:70%;">50,000 Updates (40 epochs)</span></td>
<td id="A4.T5.7.43.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.43.3.1" class="ltx_text" style="font-size:70%;">12,000 Updates (10 epochs)</span></td>
</tr>
<tr id="A4.T5.7.44" class="ltx_tr">
<td id="A4.T5.7.44.1" class="ltx_td ltx_nopad_l ltx_nopad_r" style="padding:0.05pt 0.0pt;"></td>
<td id="A4.T5.7.44.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.44.2.1" class="ltx_text" style="font-size:70%;">No early stopping</span></td>
<td id="A4.T5.7.44.3" class="ltx_td ltx_nopad_l ltx_nopad_r" style="padding:0.05pt 0.0pt;"></td>
</tr>
<tr id="A4.T5.7.45" class="ltx_tr">
<td id="A4.T5.7.45.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify ltx_border_t" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.45.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.45.1.1.1" class="ltx_p"><span id="A4.T5.7.45.1.1.1.1" class="ltx_text ltx_font_bold" style="font-size:70%;">PCA Manifold model</span></span>
</span>
</td>
<td id="A4.T5.7.45.2" class="ltx_td ltx_border_t" style="padding:0.05pt 0.0pt;"></td>
<td id="A4.T5.7.45.3" class="ltx_td ltx_border_t" style="padding:0.05pt 0.0pt;"></td>
</tr>
<tr id="A4.T5.7.46" class="ltx_tr">
<td id="A4.T5.7.46.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.46.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.46.1.1.1" class="ltx_p"><span id="A4.T5.7.46.1.1.1.1" class="ltx_text" style="font-size:70%;">Number of components</span></span>
</span>
</td>
<td id="A4.T5.7.46.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.46.2.1" class="ltx_text" style="font-size:70%;">168 (chosen for best performance)</span></td>
<td id="A4.T5.7.46.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.46.3.1" class="ltx_text" style="font-size:70%;">Same as number of models</span></td>
</tr>
<tr id="A4.T5.7.47" class="ltx_tr">
<td id="A4.T5.7.47.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.47.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.47.1.1.1" class="ltx_p"><span id="A4.T5.7.47.1.1.1.1" class="ltx_text" style="font-size:70%;">Retained variance</span></span>
</span>
</td>
<td id="A4.T5.7.47.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.47.2.1" class="ltx_text" style="font-size:70%;">85%</span></td>
<td id="A4.T5.7.47.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.47.3.1" class="ltx_text" style="font-size:70%;">Varies: 75% with 2 components, 85% with 3,</span></td>
</tr>
<tr id="A4.T5.7.48" class="ltx_tr">
<td id="A4.T5.7.48.1" class="ltx_td ltx_nopad_l ltx_nopad_r" style="padding:0.05pt 0.0pt;"></td>
<td id="A4.T5.7.48.2" class="ltx_td ltx_nopad_l ltx_nopad_r" style="padding:0.05pt 0.0pt;"></td>
<td id="A4.T5.7.48.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.48.3.1" class="ltx_text" style="font-size:70%;">90% with 6, 95% with 13, 97% with 25</span></td>
</tr>
<tr id="A4.T5.7.49" class="ltx_tr">
<td id="A4.T5.7.49.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify ltx_border_t" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.49.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.49.1.1.1" class="ltx_p"><span id="A4.T5.7.49.1.1.1.1" class="ltx_text ltx_font_bold" style="font-size:70%;">VAE Manifold model</span></span>
</span>
</td>
<td id="A4.T5.7.49.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_border_t" style="padding:0.05pt 0.0pt;"></td>
<td id="A4.T5.7.49.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_border_t" style="padding:0.05pt 0.0pt;"></td>
</tr>
<tr id="A4.T5.7.50" class="ltx_tr">
<td id="A4.T5.7.50.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.50.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.50.1.1.1" class="ltx_p"><span id="A4.T5.7.50.1.1.1.1" class="ltx_text" style="font-size:70%;">Architecture</span></span>
</span>
</td>
<td id="A4.T5.7.50.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.50.2.1" class="ltx_text" style="font-size:70%;">2-Hidden layer MLP</span></td>
<td id="A4.T5.7.50.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.50.3.1" class="ltx_text" style="font-size:70%;">1-Hidden layer MLP</span></td>
</tr>
<tr id="A4.T5.7.51" class="ltx_tr">
<td id="A4.T5.7.51.1" class="ltx_td ltx_nopad_l ltx_nopad_r" style="padding:0.05pt 0.0pt;"></td>
<td id="A4.T5.7.51.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.51.2.1" class="ltx_text" style="font-size:70%;">512 neurons per layer</span></td>
<td id="A4.T5.7.51.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.51.3.1" class="ltx_text" style="font-size:70%;">128 neurons per layer</span></td>
</tr>
<tr id="A4.T5.7.52" class="ltx_tr">
<td id="A4.T5.7.52.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.52.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.52.1.1.1" class="ltx_p"><span id="A4.T5.7.52.1.1.1.1" class="ltx_text" style="font-size:70%;">Latent dimensions</span></span>
</span>
</td>
<td id="A4.T5.7.52.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.52.2.1" class="ltx_text" style="font-size:70%;">32</span></td>
<td id="A4.T5.7.52.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.52.3.1" class="ltx_text" style="font-size:70%;">14</span></td>
</tr>
<tr id="A4.T5.7.53" class="ltx_tr">
<td id="A4.T5.7.53.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.53.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.53.1.1.1" class="ltx_p"><span id="A4.T5.7.53.1.1.1.1" class="ltx_text" style="font-size:70%;">Hidden Activations</span></span>
</span>
</td>
<td id="A4.T5.7.53.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.53.2.1" class="ltx_text" style="font-size:70%;">ReLU</span></td>
<td id="A4.T5.7.53.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.53.3.1" class="ltx_text" style="font-size:70%;">ReLU</span></td>
</tr>
<tr id="A4.T5.7.54" class="ltx_tr">
<td id="A4.T5.7.54.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.54.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.54.1.1.1" class="ltx_p"><span id="A4.T5.7.54.1.1.1.1" class="ltx_text" style="font-size:70%;">Output activation</span></span>
</span>
</td>
<td id="A4.T5.7.54.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.54.2.1" class="ltx_text" style="font-size:70%;">None</span></td>
<td id="A4.T5.7.54.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.54.3.1" class="ltx_text" style="font-size:70%;">None</span></td>
</tr>
<tr id="A4.T5.7.55" class="ltx_tr">
<td id="A4.T5.7.55.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.55.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.55.1.1.1" class="ltx_p"><span id="A4.T5.7.55.1.1.1.1" class="ltx_text" style="font-size:70%;">Mini-batch size</span></span>
</span>
</td>
<td id="A4.T5.7.55.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.55.2.1" class="ltx_text" style="font-size:70%;">256</span></td>
<td id="A4.T5.7.55.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.55.3.1" class="ltx_text" style="font-size:70%;">256</span></td>
</tr>
<tr id="A4.T5.7.56" class="ltx_tr">
<td id="A4.T5.7.56.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.56.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.56.1.1.1" class="ltx_p"><span id="A4.T5.7.56.1.1.1.1" class="ltx_text" style="font-size:70%;">Optimizer</span></span>
</span>
</td>
<td id="A4.T5.7.56.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.56.2.1" class="ltx_text" style="font-size:70%;">Adam</span></td>
<td id="A4.T5.7.56.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.56.3.1" class="ltx_text" style="font-size:70%;">Adam</span></td>
</tr>
<tr id="A4.T5.7.57" class="ltx_tr">
<td id="A4.T5.7.57.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.57.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.57.1.1.1" class="ltx_p"><span id="A4.T5.7.57.1.1.1.1" class="ltx_text" style="font-size:70%;">Learning rate</span></span>
</span>
</td>
<td id="A4.T5.7.57.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.57.2.1" class="ltx_text" style="font-size:70%;">0.001</span></td>
<td id="A4.T5.7.57.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.57.3.1" class="ltx_text" style="font-size:70%;">0.001</span></td>
</tr>
<tr id="A4.T5.7.58" class="ltx_tr">
<td id="A4.T5.7.58.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.58.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.58.1.1.1" class="ltx_p"><span id="A4.T5.7.58.1.1.1.1" class="ltx_text" style="font-size:70%;">Optimization length</span></span>
</span>
</td>
<td id="A4.T5.7.58.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.58.2.1" class="ltx_text" style="font-size:70%;">100 Epochs</span></td>
<td id="A4.T5.7.58.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.58.3.1" class="ltx_text" style="font-size:70%;">20 Epochs</span></td>
</tr>
<tr id="A4.T5.7.59" class="ltx_tr">
<td id="A4.T5.7.59.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify ltx_border_bb" style="padding:0.05pt 0.0pt;">
<span id="A4.T5.7.59.1.1" class="ltx_inline-block ltx_align_top">
<span id="A4.T5.7.59.1.1.1" class="ltx_p"><span id="A4.T5.7.59.1.1.1.1" class="ltx_text" style="font-size:70%;">Weight of KL loss</span></span>
</span>
</td>
<td id="A4.T5.7.59.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left ltx_border_bb" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.59.2.1" class="ltx_text" style="font-size:70%;">0.01</span></td>
<td id="A4.T5.7.59.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_left ltx_border_bb" style="padding:0.05pt 0.0pt;"><span id="A4.T5.7.59.3.1" class="ltx_text" style="font-size:70%;">0.01</span></td>
</tr>
</table>
</figure>
<div id="A4.p4" class="ltx_para ltx_noindent">
<p id="A4.p4.1" class="ltx_p"><span id="A4.p4.1.1" class="ltx_text ltx_font_bold">Baselines.</span> 
The “dropout” baseline in Table <a href="#S6.T1" title="Table 1 ‣ 6.1 Experiments on controlled data: collages ‣ 6 Experiments ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">1</span></a> is implemented with as a standard dropout on the input pixels, with a different dropout mask for each model. The idea is that each model sees a different (dropped-out) version of the input, and therefore might pick up different features.</p>
</div>
</section>
<section id="A5" class="ltx_appendix">
<h2 class="ltx_title ltx_title_appendix">
<span class="ltx_tag ltx_tag_appendix">Appendix E </span>Experimental details: GQA</h2>

<div id="A5.p1" class="ltx_para ltx_noindent">
<p id="A5.p1.1" class="ltx_p"><span id="A5.p1.1.1" class="ltx_text ltx_font_bold">VQA Model.</span> 
For our experiments on visual question answering, we use a simplified version of the classical BUTD model <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib73" title="" class="ltx_ref">73</a>]</cite>. For the text input (question), we use standard 300-dimensional GloVE embeddings <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib53" title="" class="ltx_ref">53</a>]</cite> (frozen) averaged over the sequence. For the image input, we use the object features provided with the GQA dataset. These are 2048-dimensional from a “bottom-up” object detector <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib3" title="" class="ltx_ref">3</a>]</cite>. We average and L2-normalize these features to obtain a single 2048D vector representing each image.
The text and question vectors are linearly projected to a common dimension (256), combined with an element-wise product, then passed through a 1-layer MLP to obtain scores over candidate answers.</p>
</div>
<div id="A5.p2" class="ltx_para ltx_noindent">
<p id="A5.p2.1" class="ltx_p"><span id="A5.p2.1.1" class="ltx_text ltx_font_bold">Grad-CAM Rank correlation.</span> 
The rank correlation reported in Table <a href="#S6.T3" title="Table 3 ‣ 6.3 Experiments on visual question answering: GQA dataset ‣ 6 Experiments ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">3</span></a> is the Spearman rank correlation of grad-CAM scores on the validation set, average over questions and pairs of models. The grad-CAM scores correspond to the 2048D input gradients multiplied by the unpooled (36 <math id="A5.p2.1.m1.1" class="ltx_Math" alttext="\times" display="inline"><semantics id="A5.p2.1.m1.1a"><mo id="A5.p2.1.m1.1.1" xref="A5.p2.1.m1.1.1.cmml">×</mo><annotation-xml encoding="MathML-Content" id="A5.p2.1.m1.1b"><times id="A5.p2.1.m1.1.1.cmml" xref="A5.p2.1.m1.1.1"></times></annotation-xml><annotation encoding="application/x-tex" id="A5.p2.1.m1.1c">\times</annotation></semantics></math> 2048D) object features. The grad-CAM scores are used because they provide a spatial importance map over the image despite the model using globally-pooled features (hence an input gradient that is spatially uniform over the image).</p>
</div>
<div id="A5.p3" class="ltx_para ltx_noindent">
<p id="A5.p3.1" class="ltx_p">The proposed method is applied in the same way as with the collages.
Almost all hyperparameters are remarkably identical to those for the collages (see Table <a href="#A4.T5" title="Table 5 ‣ Appendix D Experimental details: collages ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">5</span></a>).</p>
<div class="ltx_pagination ltx_role_newpage"></div>
</div>
</section>
<section id="A6" class="ltx_appendix">
<h2 class="ltx_title ltx_title_appendix">
<span class="ltx_tag ltx_tag_appendix">Appendix F </span>Additional results: collages</h2>

<div id="A6.p1" class="ltx_para ltx_noindent">
<p id="A6.p1.1" class="ltx_p">We provide below additional results on the <em id="A6.p1.1.1" class="ltx_emph ltx_font_italic">collages</em> dataset.

We also include experiments using <span id="A6.p1.1.2" class="ltx_text ltx_font_bold">features from a ResNet-18 as inputs</span> rather than raw pixels.
We use a standard ResNet-18 pretrained on ImageNet then kept frozen.
Our method is applied identically as in other experiments, except that the independence and on-manifold constraints are now applied in the space of ResNet feature maps rather than in pixel space.
The manifold model is a VAE trained on such feature maps.
<span id="A6.p1.1.3" class="ltx_text ltx_font_bold">The behaviour of our method with ResNet features is qualitatively the same as in other experiments.</span>
This demonstrates the applicability of the method on features from deep architectures.</p>
</div>
<figure id="A6.F10" class="ltx_figure">
<div class="ltx_flex_figure">
<div class="ltx_flex_cell ltx_flex_size_many">
<figure id="A6.F10.sf1" class="ltx_figure ltx_figure_panel ltx_align_center"><img src="/html/2207.02598/assets/x10.png" id="A6.F10.sf1.g1" class="ltx_graphics ltx_centering ltx_img_square" width="415" height="410" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span id="A6.F10.sf1.3.1.1" class="ltx_text" style="font-size:90%;">(a)</span> </span><span id="A6.F10.sf1.4.2" class="ltx_text" style="font-size:50%;">Spectral decoupling.</span></figcaption>
</figure>
</div>
<div class="ltx_flex_cell ltx_flex_size_many">
<figure id="A6.F10.sf2" class="ltx_figure ltx_figure_panel ltx_align_center"><img src="/html/2207.02598/assets/x11.png" id="A6.F10.sf2.g1" class="ltx_graphics ltx_centering ltx_img_square" width="415" height="410" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span id="A6.F10.sf2.3.1.1" class="ltx_text" style="font-size:90%;">(b)</span> </span><span id="A6.F10.sf2.4.2" class="ltx_text" style="font-size:50%;">Input grad. L1 norm.</span></figcaption>
</figure>
</div>
<div class="ltx_flex_cell ltx_flex_size_many">
<figure id="A6.F10.sf3" class="ltx_figure ltx_figure_panel ltx_align_center"><img src="/html/2207.02598/assets/x12.png" id="A6.F10.sf3.g1" class="ltx_graphics ltx_centering ltx_img_square" width="415" height="410" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span id="A6.F10.sf3.3.1.1" class="ltx_text" style="font-size:90%;">(c)</span> </span><span id="A6.F10.sf3.4.2" class="ltx_text" style="font-size:50%;">Input grad. L2 norm.</span></figcaption>
</figure>
</div>
<div class="ltx_flex_cell ltx_flex_size_many">
<figure id="A6.F10.sf4" class="ltx_figure ltx_figure_panel ltx_align_center"><img src="/html/2207.02598/assets/x13.png" id="A6.F10.sf4.g1" class="ltx_graphics ltx_centering ltx_img_square" width="415" height="420" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span id="A6.F10.sf4.3.1.1" class="ltx_text" style="font-size:90%;">(d)</span> </span><span id="A6.F10.sf4.4.2" class="ltx_text" style="font-size:50%;">Indep. loss (cos. sim.)</span></figcaption>
</figure>
</div>
<div class="ltx_flex_cell ltx_flex_size_many">
<figure id="A6.F10.sf5" class="ltx_figure ltx_figure_panel ltx_align_center"><img src="/html/2207.02598/assets/x14.png" id="A6.F10.sf5.g1" class="ltx_graphics ltx_centering ltx_img_square" width="415" height="425" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span id="A6.F10.sf5.3.1.1" class="ltx_text" style="font-size:90%;">(e)</span> </span><span id="A6.F10.sf5.4.2" class="ltx_text" style="font-size:50%;">Dot prod. on Grad-CAM.</span></figcaption>
</figure>
</div>
<div class="ltx_flex_break"></div>
<div class="ltx_flex_cell ltx_flex_size_many">
<figure id="A6.F10.sf6" class="ltx_figure ltx_figure_panel ltx_align_center"><img src="/html/2207.02598/assets/x15.png" id="A6.F10.sf6.g1" class="ltx_graphics ltx_centering ltx_img_square" width="415" height="407" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span id="A6.F10.sf6.3.1.1" class="ltx_text" style="font-size:90%;">(f)</span> </span><span id="A6.F10.sf6.4.2" class="ltx_text" style="font-size:50%;">Indep. loss (dot prod.)</span></figcaption>
</figure>
</div>
<div class="ltx_flex_cell ltx_flex_size_many">
<figure id="A6.F10.sf7" class="ltx_figure ltx_figure_panel ltx_align_center"><img src="/html/2207.02598/assets/x16.png" id="A6.F10.sf7.g1" class="ltx_graphics ltx_centering ltx_img_square" width="415" height="407" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span id="A6.F10.sf7.3.1.1" class="ltx_text" style="font-size:90%;">(g)</span> </span><span id="A6.F10.sf7.4.2" class="ltx_text" style="font-size:50%;">Indep. loss (cos. sim.) + on-manifold (PCA).</span></figcaption>
</figure>
</div>
<div class="ltx_flex_cell ltx_flex_size_many">
<figure id="A6.F10.sf8" class="ltx_figure ltx_figure_panel ltx_align_center"><img src="/html/2207.02598/assets/x17.png" id="A6.F10.sf8.g1" class="ltx_graphics ltx_centering ltx_img_square" width="415" height="407" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span id="A6.F10.sf8.3.1.1" class="ltx_text" style="font-size:90%;">(h)</span> </span><span id="A6.F10.sf8.4.2" class="ltx_text" style="font-size:50%;">Indep. loss (cos. sim.) + on-manifold (VAE).</span></figcaption>
</figure>
</div>
<div class="ltx_flex_cell ltx_flex_size_many">
<figure id="A6.F10.sf9" class="ltx_figure ltx_figure_panel ltx_align_center"><img src="/html/2207.02598/assets/x18.png" id="A6.F10.sf9.g1" class="ltx_graphics ltx_centering ltx_img_square" width="415" height="407" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span id="A6.F10.sf9.3.1.1" class="ltx_text" style="font-size:90%;">(i)</span> </span><span id="A6.F10.sf9.4.2" class="ltx_text" style="font-size:50%;">Indep. loss (cos. sim.) + on-manifold (VAE) + fine-tuning.</span></figcaption>
</figure>
</div>
<div class="ltx_flex_cell ltx_flex_size_many">
<figure id="A6.F10.sf10" class="ltx_figure ltx_figure_panel ltx_align_center"><img src="/html/2207.02598/assets/x19.png" id="A6.F10.sf10.g1" class="ltx_graphics ltx_centering ltx_img_square" width="415" height="407" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span id="A6.F10.sf10.3.1.1" class="ltx_text" style="font-size:90%;">(j)</span> </span><span id="A6.F10.sf10.4.2" class="ltx_text" style="font-size:50%;">Indep. loss (cos. sim.) + on-manifold (VAE) + fine-tuning + combinations.</span></figcaption>
</figure>
</div>
</div>
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span id="A6.F10.7.1.1" class="ltx_text" style="font-size:90%;">Figure 10</span>: </span><span id="A6.F10.8.2" class="ltx_text ltx_font_bold" style="font-size:90%;">(Collages, pixels as input)<span id="A6.F10.8.2.1" class="ltx_text ltx_font_medium"> 
Accuracy (average over the 4 test sets) of </span>existing methods (first row)<span id="A6.F10.8.2.2" class="ltx_text ltx_font_medium"> and </span>ablations (second row)<span id="A6.F10.8.2.3" class="ltx_text ltx_font_medium"> for various hyperparameters and numbers of models.
The only existing methods with non-trivial performance (<em id="A6.F10.8.2.3.1" class="ltx_emph ltx_font_italic">e.g</em>.<span id="A6.F10.8.2.3.2" class="ltx_text"></span> (f)) require training at least an order of magnitude more models than ours (j).
</span></span></figcaption>
</figure>
<figure id="A6.F11" class="ltx_figure">
<div class="ltx_flex_figure">
<div class="ltx_flex_cell ltx_flex_size_2">
<figure id="A6.F11.sf1" class="ltx_figure ltx_figure_panel ltx_align_center"><img src="/html/2207.02598/assets/x20.png" id="A6.F11.sf1.g1" class="ltx_graphics ltx_centering ltx_img_square" width="276" height="276" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span id="A6.F11.sf1.2.1.1" class="ltx_text" style="font-size:90%;">(a)</span> </span><span id="A6.F11.sf1.3.2" class="ltx_text" style="font-size:90%;">Pixels as input.</span></figcaption>
</figure>
</div>
<div class="ltx_flex_cell ltx_flex_size_2">
<figure id="A6.F11.sf2" class="ltx_figure ltx_figure_panel ltx_align_center"><img src="/html/2207.02598/assets/x21.png" id="A6.F11.sf2.g1" class="ltx_graphics ltx_centering ltx_img_square" width="276" height="272" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span id="A6.F11.sf2.2.1.1" class="ltx_text" style="font-size:90%;">(b)</span> </span><span id="A6.F11.sf2.3.2" class="ltx_text" style="font-size:90%;">ResNet features as input.</span></figcaption>
</figure>
</div>
<div class="ltx_flex_break"></div>
</div>
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span id="A6.F11.6.1.1" class="ltx_text" style="font-size:90%;">Figure 11</span>: </span><span id="A6.F11.7.2" class="ltx_text ltx_font_bold" style="font-size:90%;">(Collages)<span id="A6.F11.7.2.1" class="ltx_text ltx_font_medium"> 
Accuracy of the proposed method (average over the 4 test sets) for various loss weights of the independence and on-manifold losses (with 12 models).
</span>Performance is stable over a range of values<span id="A6.F11.7.2.2" class="ltx_text ltx_font_medium"> and both losses are important, as seen from the leftmost &amp; lowermost cells.
Performance is also repeatable: each cell reports a single run <em id="A6.F11.7.2.2.1" class="ltx_emph ltx_font_italic">i.e</em>.<span id="A6.F11.7.2.2.2" class="ltx_text"></span>not averaged over multiple random seeds.
</span></span></figcaption>
</figure>
<figure id="A6.F12" class="ltx_figure">
<div class="ltx_flex_figure">
<div class="ltx_flex_cell ltx_flex_size_3">
<figure id="A6.F12.sf1" class="ltx_figure ltx_figure_panel ltx_align_center"><img src="/html/2207.02598/assets/x22.png" id="A6.F12.sf1.g1" class="ltx_graphics ltx_centering ltx_img_square" width="322" height="326" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span id="A6.F12.sf1.2.1.1" class="ltx_text" style="font-size:90%;">(a)</span> </span><span id="A6.F12.sf1.3.2" class="ltx_text" style="font-size:90%;">Pixels as input.</span></figcaption>
</figure>
</div>
<div class="ltx_flex_cell ltx_flex_size_3">
<figure id="A6.F12.sf2" class="ltx_figure ltx_figure_panel ltx_align_center"><img src="/html/2207.02598/assets/x23.png" id="A6.F12.sf2.g1" class="ltx_graphics ltx_centering ltx_img_square" width="322" height="326" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span id="A6.F12.sf2.2.1.1" class="ltx_text" style="font-size:90%;">(b)</span> </span><span id="A6.F12.sf2.3.2" class="ltx_text" style="font-size:90%;">Pixels as input + fine-tuning.</span></figcaption>
</figure>
</div>
<div class="ltx_flex_cell ltx_flex_size_3">
<figure id="A6.F12.sf3" class="ltx_figure ltx_figure_panel ltx_align_center"><img src="/html/2207.02598/assets/x24.png" id="A6.F12.sf3.g1" class="ltx_graphics ltx_centering ltx_img_square" width="322" height="326" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span id="A6.F12.sf3.2.1.1" class="ltx_text" style="font-size:90%;">(c)</span> </span><span id="A6.F12.sf3.3.2" class="ltx_text" style="font-size:90%;">ResNet features as input.</span></figcaption>
</figure>
</div>
</div>
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span id="A6.F12.5.2.1" class="ltx_text" style="font-size:90%;">Figure 12</span>: </span><span id="A6.F12.2.1" class="ltx_text ltx_font_bold" style="font-size:90%;">(Collages)<span id="A6.F12.2.1.1" class="ltx_text ltx_font_medium"> 
Accuracy of the proposed method as a function of the number of models trained.
Non-trivial accuracy is obtained on all 4 test sets with as few as 4 models.
In line with our theoretical predictions, the accuracy decreases after <math id="A6.F12.2.1.1.m1.1" class="ltx_Math" alttext="&gt;" display="inline"><semantics id="A6.F12.2.1.1.m1.1b"><mo id="A6.F12.2.1.1.m1.1.1" xref="A6.F12.2.1.1.m1.1.1.cmml">&gt;</mo><annotation-xml encoding="MathML-Content" id="A6.F12.2.1.1.m1.1c"><gt id="A6.F12.2.1.1.m1.1.1.cmml" xref="A6.F12.2.1.1.m1.1.1"></gt></annotation-xml><annotation encoding="application/x-tex" id="A6.F12.2.1.1.m1.1d">&gt;</annotation></semantics></math>24 models, which is the recommended value based on the intrinsic dimensionality of the dataset (Section <a href="#S5" title="5 Practical considerations ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">5</span></a>).
</span></span></figcaption>
</figure>
<figure id="A6.F13" class="ltx_figure">
<div class="ltx_flex_figure">
<div class="ltx_flex_cell ltx_flex_size_1">
<figure id="A6.F13.sf1" class="ltx_figure ltx_figure_panel ltx_align_center"><img src="/html/2207.02598/assets/x25.png" id="A6.F13.sf1.g1" class="ltx_graphics ltx_centering ltx_img_landscape" width="461" height="68" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span id="A6.F13.sf1.3.1.1" class="ltx_text" style="font-size:90%;">(a)</span> </span><span id="A6.F13.sf1.4.2" class="ltx_text ltx_font_bold" style="font-size:90%;">Pixels<span id="A6.F13.sf1.4.2.1" class="ltx_text ltx_font_medium"> as input.</span></span></figcaption>
</figure>
</div>
<div class="ltx_flex_break"></div>
<div class="ltx_flex_cell ltx_flex_size_1">
<figure id="A6.F13.sf2" class="ltx_figure ltx_figure_panel ltx_align_center"><img src="/html/2207.02598/assets/x26.png" id="A6.F13.sf2.g1" class="ltx_graphics ltx_centering ltx_img_landscape" width="461" height="68" alt="Refer to caption">
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span id="A6.F13.sf2.3.1.1" class="ltx_text" style="font-size:90%;">(b)</span> </span><span id="A6.F13.sf2.4.2" class="ltx_text ltx_font_bold" style="font-size:90%;">ResNet<span id="A6.F13.sf2.4.2.1" class="ltx_text ltx_font_medium"> features as input.</span></span></figcaption>
</figure>
</div>
<div class="ltx_flex_break"></div>
</div>
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span id="A6.F13.3.1.1" class="ltx_text" style="font-size:90%;">Figure 13</span>: </span><span id="A6.F13.4.2" class="ltx_text ltx_font_bold" style="font-size:90%;">(Collages)<span id="A6.F13.4.2.1" class="ltx_text ltx_font_medium"> 
Accuracy on the 4 test sets (columns) of models with the best accuracy on each set (rows).
The diagonal pattern indicates that the models specialize and learn different, non-overlapping features.
</span></span></figcaption>
</figure>
<figure id="A6.T6" class="ltx_table">
<table id="A6.T6.2" class="ltx_tabular ltx_centering ltx_align_middle">
<tr id="A6.T6.2.1" class="ltx_tr">
<td id="A6.T6.2.1.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify ltx_border_tt" style="padding:0.05pt 0.0pt;">
<span id="A6.T6.2.1.1.1" class="ltx_inline-block ltx_align_top">
<span id="A6.T6.2.1.1.1.1" class="ltx_p"><span id="A6.T6.2.1.1.1.1.1" class="ltx_text ltx_font_bold" style="font-size:70%;">Collages</span><span id="A6.T6.2.1.1.1.1.2" class="ltx_text" style="font-size:70%;"> dataset, </span><span id="A6.T6.2.1.1.1.1.3" class="ltx_text ltx_font_bold" style="font-size:70%;">ResNet</span><span id="A6.T6.2.1.1.1.1.4" class="ltx_text" style="font-size:70%;"> features as input (accuracy in %)</span></span>
</span>
</td>
<td id="A6.T6.2.1.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_tt" style="padding:0.05pt 0.0pt;" colspan="4"><span id="A6.T6.2.1.2.1" class="ltx_text" style="font-size:70%;">Best model on</span></td>
<td id="A6.T6.2.1.3" class="ltx_td ltx_border_tt" style="padding:0.05pt 0.0pt;"></td>
</tr>
<tr id="A6.T6.2.2" class="ltx_tr">
<td id="A6.T6.2.2.1" class="ltx_td ltx_nopad_l ltx_nopad_r" style="padding:0.05pt 0.0pt;"></td>
<td id="A6.T6.2.2.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding:0.05pt 0.0pt;">
<div id="A6.T6.2.2.2.1" class="ltx_inline-block ltx_transformed_outer" style="width:4.8pt;height:23.1pt;vertical-align:-0.0pt;"><span class="ltx_transformed_inner" style="width:23.1pt;transform:translate(-9.18pt,-9.18pt) rotate(-90deg) ;">
<p id="A6.T6.2.2.2.1.1" class="ltx_p"><span id="A6.T6.2.2.2.1.1.1" class="ltx_text" style="font-size:70%;">MNIST</span></p>
</span></div>
</td>
<td id="A6.T6.2.2.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding:0.05pt 0.0pt;">
<div id="A6.T6.2.2.3.1" class="ltx_inline-block ltx_transformed_outer" style="width:4.8pt;height:26.6pt;vertical-align:-0.0pt;"><span class="ltx_transformed_inner" style="width:26.6pt;transform:translate(-10.93pt,-10.93pt) rotate(-90deg) ;">
<p id="A6.T6.2.2.3.1.1" class="ltx_p"><span id="A6.T6.2.2.3.1.1.1" class="ltx_text" style="font-size:70%;">SVHN</span></p>
</span></div>
</td>
<td id="A6.T6.2.2.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding:0.05pt 0.0pt;">
<div id="A6.T6.2.2.4.1" class="ltx_inline-block ltx_transformed_outer" style="width:4.9pt;height:26.6pt;vertical-align:-0.0pt;"><span class="ltx_transformed_inner" style="width:26.6pt;transform:translate(-10.86pt,-10.86pt) rotate(-90deg) ;">
<p id="A6.T6.2.2.4.1.1" class="ltx_p"><span id="A6.T6.2.2.4.1.1.1" class="ltx_text" style="font-size:70%;">Fashion</span><span id="A6.T6.2.2.4.1.1.2" class="ltx_text ltx_phantom" style="font-size:70%;"><span style="visibility:hidden">e</span></span></p>
</span></div>
</td>
<td id="A6.T6.2.2.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding:0.05pt 0.0pt;">
<div id="A6.T6.2.2.5.1" class="ltx_inline-block ltx_transformed_outer" style="width:4.8pt;height:34.2pt;vertical-align:-0.0pt;"><span class="ltx_transformed_inner" style="width:34.2pt;transform:translate(-14.72pt,-14.72pt) rotate(-90deg) ;">
<p id="A6.T6.2.2.5.1.1" class="ltx_p"><span id="A6.T6.2.2.5.1.1.1" class="ltx_text" style="font-size:70%;">CIFAR-10</span><span id="A6.T6.2.2.5.1.1.2" class="ltx_text ltx_phantom" style="font-size:70%;"><span style="visibility:hidden">e</span></span></p>
</span></div>
</td>
<td id="A6.T6.2.2.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.05pt 0.0pt;">
<div id="A6.T6.2.2.6.1" class="ltx_inline-block ltx_transformed_outer" style="width:6.1pt;height:24.7pt;vertical-align:-0.0pt;"><span class="ltx_transformed_inner" style="width:24.7pt;transform:translate(-9.28pt,-8.6pt) rotate(-90deg) ;">
<p id="A6.T6.2.2.6.1.1" class="ltx_p"><span id="A6.T6.2.2.6.1.1.1" class="ltx_text" style="font-size:70%;">Average</span></p>
</span></div>
</td>
</tr>
<tr id="A6.T6.2.3" class="ltx_tr">
<td id="A6.T6.2.3.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify ltx_border_t" style="padding:0.05pt 0.0pt;">
<span id="A6.T6.2.3.1.1" class="ltx_inline-block ltx_align_top">
<span id="A6.T6.2.3.1.1.1" class="ltx_p"><span id="A6.T6.2.3.1.1.1.1" class="ltx_text" style="font-size:70%;">Upper bounds (training on OOD data)</span></span>
</span>
</td>
<td id="A6.T6.2.3.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.3.2.1" class="ltx_text" style="font-size:70%;">100.0</span></td>
<td id="A6.T6.2.3.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.3.3.1" class="ltx_text" style="font-size:70%;">98.0</span></td>
<td id="A6.T6.2.3.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.3.4.1" class="ltx_text" style="font-size:70%;">92.3</span></td>
<td id="A6.T6.2.3.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.3.5.1" class="ltx_text" style="font-size:70%;">89.9</span></td>
<td id="A6.T6.2.3.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.3.6.1" class="ltx_text" style="font-size:70%;">95.1</span></td>
</tr>
<tr id="A6.T6.2.4" class="ltx_tr">
<td id="A6.T6.2.4.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify ltx_border_t" style="padding:0.05pt 0.0pt;">
<span id="A6.T6.2.4.1.1" class="ltx_inline-block ltx_align_top">
<span id="A6.T6.2.4.1.1.1" class="ltx_p"><span id="A6.T6.2.4.1.1.1.1" class="ltx_text" style="font-size:70%;">Baseline</span></span>
</span>
</td>
<td id="A6.T6.2.4.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.4.2.1" class="ltx_text" style="font-size:70%;">99.8</span></td>
<td id="A6.T6.2.4.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.4.3.1" class="ltx_text" style="font-size:70%;">49.9</span></td>
<td id="A6.T6.2.4.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.4.4.1" class="ltx_text" style="font-size:70%;">50.7</span></td>
<td id="A6.T6.2.4.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.4.5.1" class="ltx_text" style="font-size:70%;">49.9</span></td>
<td id="A6.T6.2.4.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.4.6.1" class="ltx_text" style="font-size:70%;">62.6</span></td>
</tr>
<tr id="A6.T6.2.5" class="ltx_tr">
<td id="A6.T6.2.5.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify ltx_border_t" style="padding:0.05pt 0.0pt;">
<span id="A6.T6.2.5.1.1" class="ltx_inline-block ltx_align_top">
<span id="A6.T6.2.5.1.1.1" class="ltx_p"><span id="A6.T6.2.5.1.1.1.1" class="ltx_text" style="font-size:70%;">Independence loss</span></span>
</span>
</td>
<td id="A6.T6.2.5.2" class="ltx_td ltx_border_t" style="padding:0.05pt 0.0pt;"></td>
<td id="A6.T6.2.5.3" class="ltx_td ltx_border_t" style="padding:0.05pt 0.0pt;"></td>
<td id="A6.T6.2.5.4" class="ltx_td ltx_border_t" style="padding:0.05pt 0.0pt;"></td>
<td id="A6.T6.2.5.5" class="ltx_td ltx_border_t" style="padding:0.05pt 0.0pt;"></td>
<td id="A6.T6.2.5.6" class="ltx_td ltx_border_t" style="padding:0.05pt 0.0pt;"></td>
</tr>
<tr id="A6.T6.2.6" class="ltx_tr">
<td id="A6.T6.2.6.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A6.T6.2.6.1.1" class="ltx_inline-block ltx_align_top">
<span id="A6.T6.2.6.1.1.1" class="ltx_p"><span id="A6.T6.2.6.1.1.1.1" class="ltx_text" style="font-size:70%;">4 models</span></span>
</span>
</td>
<td id="A6.T6.2.6.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.6.2.1" class="ltx_text" style="font-size:70%;">86.8</span></td>
<td id="A6.T6.2.6.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.6.3.1" class="ltx_text" style="font-size:70%;">60.2</span></td>
<td id="A6.T6.2.6.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.6.4.1" class="ltx_text" style="font-size:70%;">62.1</span></td>
<td id="A6.T6.2.6.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.6.5.1" class="ltx_text" style="font-size:70%;">51.6</span></td>
<td id="A6.T6.2.6.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.6.6.1" class="ltx_text" style="font-size:70%;">65.2</span></td>
</tr>
<tr id="A6.T6.2.7" class="ltx_tr">
<td id="A6.T6.2.7.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A6.T6.2.7.1.1" class="ltx_inline-block ltx_align_top">
<span id="A6.T6.2.7.1.1.1" class="ltx_p"><span id="A6.T6.2.7.1.1.1.1" class="ltx_text" style="font-size:70%;">8 models</span></span>
</span>
</td>
<td id="A6.T6.2.7.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.7.2.1" class="ltx_text" style="font-size:70%;">88.4</span></td>
<td id="A6.T6.2.7.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.7.3.1" class="ltx_text" style="font-size:70%;">61.5</span></td>
<td id="A6.T6.2.7.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.7.4.1" class="ltx_text" style="font-size:70%;">63.3</span></td>
<td id="A6.T6.2.7.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.7.5.1" class="ltx_text" style="font-size:70%;">51.9</span></td>
<td id="A6.T6.2.7.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.7.6.1" class="ltx_text" style="font-size:70%;">66.3</span></td>
</tr>
<tr id="A6.T6.2.8" class="ltx_tr">
<td id="A6.T6.2.8.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A6.T6.2.8.1.1" class="ltx_inline-block ltx_align_top">
<span id="A6.T6.2.8.1.1.1" class="ltx_p"><span id="A6.T6.2.8.1.1.1.1" class="ltx_text" style="font-size:70%;">16 models</span></span>
</span>
</td>
<td id="A6.T6.2.8.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.8.2.1" class="ltx_text" style="font-size:70%;">91.6</span></td>
<td id="A6.T6.2.8.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.8.3.1" class="ltx_text" style="font-size:70%;">59.3</span></td>
<td id="A6.T6.2.8.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.8.4.1" class="ltx_text" style="font-size:70%;">59.9</span></td>
<td id="A6.T6.2.8.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.8.5.1" class="ltx_text" style="font-size:70%;">52.1</span></td>
<td id="A6.T6.2.8.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.8.6.1" class="ltx_text" style="font-size:70%;">65.7</span></td>
</tr>
<tr id="A6.T6.2.9" class="ltx_tr">
<td id="A6.T6.2.9.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A6.T6.2.9.1.1" class="ltx_inline-block ltx_align_top">
<span id="A6.T6.2.9.1.1.1" class="ltx_p"><span id="A6.T6.2.9.1.1.1.1" class="ltx_text" style="font-size:70%;">24 models</span></span>
</span>
</td>
<td id="A6.T6.2.9.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.9.2.1" class="ltx_text" style="font-size:70%;">93.9</span></td>
<td id="A6.T6.2.9.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.9.3.1" class="ltx_text" style="font-size:70%;">58.2</span></td>
<td id="A6.T6.2.9.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.9.4.1" class="ltx_text" style="font-size:70%;">57.7</span></td>
<td id="A6.T6.2.9.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.9.5.1" class="ltx_text" style="font-size:70%;">52.1</span></td>
<td id="A6.T6.2.9.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.9.6.1" class="ltx_text" style="font-size:70%;">65.5</span></td>
</tr>
<tr id="A6.T6.2.10" class="ltx_tr">
<td id="A6.T6.2.10.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify ltx_border_t" style="padding:0.05pt 0.0pt;">
<span id="A6.T6.2.10.1.1" class="ltx_inline-block ltx_align_top">
<span id="A6.T6.2.10.1.1.1" class="ltx_p"><span id="A6.T6.2.10.1.1.1.1" class="ltx_text" style="font-size:70%;">Independence loss + on-manifold constraint (VAE)</span></span>
</span>
</td>
<td id="A6.T6.2.10.2" class="ltx_td ltx_border_t" style="padding:0.05pt 0.0pt;"></td>
<td id="A6.T6.2.10.3" class="ltx_td ltx_border_t" style="padding:0.05pt 0.0pt;"></td>
<td id="A6.T6.2.10.4" class="ltx_td ltx_border_t" style="padding:0.05pt 0.0pt;"></td>
<td id="A6.T6.2.10.5" class="ltx_td ltx_border_t" style="padding:0.05pt 0.0pt;"></td>
<td id="A6.T6.2.10.6" class="ltx_td ltx_border_t" style="padding:0.05pt 0.0pt;"></td>
</tr>
<tr id="A6.T6.2.11" class="ltx_tr">
<td id="A6.T6.2.11.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A6.T6.2.11.1.1" class="ltx_inline-block ltx_align_top">
<span id="A6.T6.2.11.1.1.1" class="ltx_p"><span id="A6.T6.2.11.1.1.1.1" class="ltx_text" style="font-size:70%;">4 models</span></span>
</span>
</td>
<td id="A6.T6.2.11.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.11.2.1" class="ltx_text" style="font-size:70%;">99.6</span></td>
<td id="A6.T6.2.11.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.11.3.1" class="ltx_text" style="font-size:70%;">71.7</span></td>
<td id="A6.T6.2.11.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.11.4.1" class="ltx_text" style="font-size:70%;">76.1</span></td>
<td id="A6.T6.2.11.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.11.5.1" class="ltx_text" style="font-size:70%;">68.3</span></td>
<td id="A6.T6.2.11.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.11.6.1" class="ltx_text" style="font-size:70%;">78.9</span></td>
</tr>
<tr id="A6.T6.2.12" class="ltx_tr">
<td id="A6.T6.2.12.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A6.T6.2.12.1.1" class="ltx_inline-block ltx_align_top">
<span id="A6.T6.2.12.1.1.1" class="ltx_p"><span id="A6.T6.2.12.1.1.1.1" class="ltx_text" style="font-size:70%;">8 models</span></span>
</span>
</td>
<td id="A6.T6.2.12.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.12.2.1" class="ltx_text" style="font-size:70%;">99.2</span></td>
<td id="A6.T6.2.12.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.12.3.1" class="ltx_text" style="font-size:70%;">86.8</span></td>
<td id="A6.T6.2.12.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.12.4.1" class="ltx_text" style="font-size:70%;">76.3</span></td>
<td id="A6.T6.2.12.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.12.5.1" class="ltx_text" style="font-size:70%;">67.8</span></td>
<td id="A6.T6.2.12.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.12.6.1" class="ltx_text" style="font-size:70%;">82.5</span></td>
</tr>
<tr id="A6.T6.2.13" class="ltx_tr">
<td id="A6.T6.2.13.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.05pt 0.0pt;">
<span id="A6.T6.2.13.1.1" class="ltx_inline-block ltx_align_top">
<span id="A6.T6.2.13.1.1.1" class="ltx_p"><span id="A6.T6.2.13.1.1.1.1" class="ltx_text" style="font-size:70%;">16 models</span></span>
</span>
</td>
<td id="A6.T6.2.13.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.13.2.1" class="ltx_text" style="font-size:70%;">99.5</span></td>
<td id="A6.T6.2.13.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.13.3.1" class="ltx_text" style="font-size:70%;">90.0</span></td>
<td id="A6.T6.2.13.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.13.4.1" class="ltx_text" style="font-size:70%;">77.5</span></td>
<td id="A6.T6.2.13.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.13.5.1" class="ltx_text" style="font-size:70%;">76.1</span></td>
<td id="A6.T6.2.13.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.13.6.1" class="ltx_text" style="font-size:70%;">85.8</span></td>
</tr>
<tr id="A6.T6.2.14" class="ltx_tr">
<td id="A6.T6.2.14.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify ltx_border_bb" style="padding:0.05pt 0.0pt;">
<span id="A6.T6.2.14.1.1" class="ltx_inline-block ltx_align_top">
<span id="A6.T6.2.14.1.1.1" class="ltx_p"><span id="A6.T6.2.14.1.1.1.1" class="ltx_text ltx_font_bold" style="font-size:70%;">24 models</span></span>
</span>
</td>
<td id="A6.T6.2.14.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_bb" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.14.2.1" class="ltx_text ltx_font_bold" style="font-size:70%;">99.2</span></td>
<td id="A6.T6.2.14.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_bb" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.14.3.1" class="ltx_text ltx_font_bold" style="font-size:70%;">93.0</span></td>
<td id="A6.T6.2.14.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_bb" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.14.4.1" class="ltx_text ltx_font_bold" style="font-size:70%;">80.1</span></td>
<td id="A6.T6.2.14.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_bb" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.14.5.1" class="ltx_text ltx_font_bold" style="font-size:70%;">76.0</span></td>
<td id="A6.T6.2.14.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_bb" style="padding:0.05pt 0.0pt;"><span id="A6.T6.2.14.6.1" class="ltx_text ltx_font_bold" style="font-size:70%;">87.1</span></td>
</tr>
</table>
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_table"><span id="A6.T6.3.1.1" class="ltx_text" style="font-size:90%;">Table 6</span>: </span><span id="A6.T6.4.2" class="ltx_text" style="font-size:90%;">Accuracy of the proposed method and ablations using ResNet features as input.
The independence and on-manifold constraints are both important for high performance.
There remains a small gap with the upper-bound accuracy, but we achieve drastically better results than the baseline.
</span></figcaption>
</figure>
<div class="ltx_pagination ltx_role_newpage"></div>
</section>
<section id="A7" class="ltx_appendix">
<h2 class="ltx_title ltx_title_appendix">
<span class="ltx_tag ltx_tag_appendix">Appendix G </span>Additional results: WILDS-Camelyon17</h2>

<figure id="A7.T7" class="ltx_table">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table"><span id="A7.T7.26.1.1" class="ltx_text" style="font-size:90%;">Table 7</span>: </span><span id="A7.T7.27.2" class="ltx_text" style="font-size:90%;">Best accuracy (%) on the test-OOD split of WILDS-Camelyon17 (average over 6 random seeds). We show the difference in performance between the baseline and various ablations of our method while training 12 models (as in Table <a href="#S6.T2" title="Table 2 ‣ 6.2 Experiments on real data: WILDS-Camelyon17 ‣ 6 Experiments ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">2</span></a>), starting with each of the 10 pretrained models provided by the dataset authors <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib40" title="" class="ltx_ref">40</a>]</cite>. <span id="A7.T7.27.2.1" class="ltx_text ltx_font_bold">While the ranking of methods is roughly similar, the absolute accuracy (first and last rows) is highly variable across pretrained models.</span>
A possible solution for eliminating this variability would be to apply our method during pretraining – at far greater computational expense than when training linear classifiers.</span></figcaption>
<table id="A7.T7.23" class="ltx_tabular ltx_centering ltx_align_middle">
<tr id="A7.T7.23.24" class="ltx_tr">
<td id="A7.T7.23.24.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify ltx_border_tt" style="padding:0.45pt 0.0pt;">
<span id="A7.T7.23.24.1.1" class="ltx_inline-block ltx_align_top">
<span id="A7.T7.23.24.1.1.1" class="ltx_p"><span id="A7.T7.23.24.1.1.1.1" class="ltx_text ltx_font_bold" style="font-size:90%;">WILDS-Camelyon17</span></span>
</span>
</td>
<td id="A7.T7.23.24.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_tt" style="padding:0.45pt 0.0pt;" colspan="10"><span id="A7.T7.23.24.2.1" class="ltx_text" style="font-size:90%;">Pretrained model</span></td>
</tr>
<tr id="A7.T7.23.25" class="ltx_tr">
<td id="A7.T7.23.25.1" class="ltx_td ltx_nopad_l ltx_nopad_r" style="padding:0.45pt 0.0pt;"></td>
<td id="A7.T7.23.25.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.25.2.1" class="ltx_text" style="font-size:90%;">#1</span></td>
<td id="A7.T7.23.25.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.25.3.1" class="ltx_text" style="font-size:90%;">#2</span></td>
<td id="A7.T7.23.25.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.25.4.1" class="ltx_text" style="font-size:90%;">#3</span></td>
<td id="A7.T7.23.25.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.25.5.1" class="ltx_text" style="font-size:90%;">#4</span></td>
<td id="A7.T7.23.25.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.25.6.1" class="ltx_text" style="font-size:90%;">#5</span></td>
<td id="A7.T7.23.25.7" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.25.7.1" class="ltx_text" style="font-size:90%;">#6</span></td>
<td id="A7.T7.23.25.8" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.25.8.1" class="ltx_text" style="font-size:90%;">#7</span></td>
<td id="A7.T7.23.25.9" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.25.9.1" class="ltx_text" style="font-size:90%;">#8</span></td>
<td id="A7.T7.23.25.10" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.25.10.1" class="ltx_text" style="font-size:90%;">#9</span></td>
<td id="A7.T7.23.25.11" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.25.11.1" class="ltx_text" style="font-size:90%;">#10</span></td>
</tr>
<tr id="A7.T7.23.26" class="ltx_tr">
<td id="A7.T7.23.26.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify ltx_border_t" style="padding:0.45pt 0.0pt;">
<span id="A7.T7.23.26.1.1" class="ltx_inline-block ltx_align_top">
<span id="A7.T7.23.26.1.1.1" class="ltx_p"><span id="A7.T7.23.26.1.1.1.1" class="ltx_text" style="font-size:90%;">ERM Baseline</span></span>
</span>
</td>
<td id="A7.T7.23.26.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_t" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.26.2.1" class="ltx_text" style="font-size:90%;">68.4</span></td>
<td id="A7.T7.23.26.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_t" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.26.3.1" class="ltx_text" style="font-size:90%;">78.0</span></td>
<td id="A7.T7.23.26.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_t" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.26.4.1" class="ltx_text" style="font-size:90%;">73.3</span></td>
<td id="A7.T7.23.26.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_t" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.26.5.1" class="ltx_text" style="font-size:90%;">60.4</span></td>
<td id="A7.T7.23.26.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_t" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.26.6.1" class="ltx_text" style="font-size:90%;">78.3</span></td>
<td id="A7.T7.23.26.7" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_t" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.26.7.1" class="ltx_text" style="font-size:90%;">78.5</span></td>
<td id="A7.T7.23.26.8" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_t" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.26.8.1" class="ltx_text" style="font-size:90%;">64.4</span></td>
<td id="A7.T7.23.26.9" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_t" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.26.9.1" class="ltx_text" style="font-size:90%;">74.2</span></td>
<td id="A7.T7.23.26.10" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_t" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.26.10.1" class="ltx_text" style="font-size:90%;">71.9</span></td>
<td id="A7.T7.23.26.11" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_t" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.26.11.1" class="ltx_text" style="font-size:90%;">60.2</span></td>
</tr>
<tr id="A7.T7.10.10" class="ltx_tr">
<td id="A7.T7.10.10.11" class="ltx_td ltx_nopad_l ltx_nopad_r" style="padding:0.45pt 0.0pt;"></td>
<td id="A7.T7.1.1.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;">
<math id="A7.T7.1.1.1.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A7.T7.1.1.1.m1.1a"><mo mathsize="50%" id="A7.T7.1.1.1.m1.1.1" xref="A7.T7.1.1.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A7.T7.1.1.1.m1.1b"><csymbol cd="latexml" id="A7.T7.1.1.1.m1.1.1.cmml" xref="A7.T7.1.1.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A7.T7.1.1.1.m1.1c">\pm</annotation></semantics></math><span id="A7.T7.1.1.1.1" class="ltx_text" style="font-size:50%;">0.1</span>
</td>
<td id="A7.T7.2.2.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;">
<math id="A7.T7.2.2.2.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A7.T7.2.2.2.m1.1a"><mo mathsize="50%" id="A7.T7.2.2.2.m1.1.1" xref="A7.T7.2.2.2.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A7.T7.2.2.2.m1.1b"><csymbol cd="latexml" id="A7.T7.2.2.2.m1.1.1.cmml" xref="A7.T7.2.2.2.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A7.T7.2.2.2.m1.1c">\pm</annotation></semantics></math><span id="A7.T7.2.2.2.1" class="ltx_text" style="font-size:50%;">0.1</span>
</td>
<td id="A7.T7.3.3.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;">
<math id="A7.T7.3.3.3.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A7.T7.3.3.3.m1.1a"><mo mathsize="50%" id="A7.T7.3.3.3.m1.1.1" xref="A7.T7.3.3.3.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A7.T7.3.3.3.m1.1b"><csymbol cd="latexml" id="A7.T7.3.3.3.m1.1.1.cmml" xref="A7.T7.3.3.3.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A7.T7.3.3.3.m1.1c">\pm</annotation></semantics></math><span id="A7.T7.3.3.3.1" class="ltx_text" style="font-size:50%;">0.0</span>
</td>
<td id="A7.T7.4.4.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;">
<math id="A7.T7.4.4.4.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A7.T7.4.4.4.m1.1a"><mo mathsize="50%" id="A7.T7.4.4.4.m1.1.1" xref="A7.T7.4.4.4.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A7.T7.4.4.4.m1.1b"><csymbol cd="latexml" id="A7.T7.4.4.4.m1.1.1.cmml" xref="A7.T7.4.4.4.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A7.T7.4.4.4.m1.1c">\pm</annotation></semantics></math><span id="A7.T7.4.4.4.1" class="ltx_text" style="font-size:50%;">0.1</span>
</td>
<td id="A7.T7.5.5.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;">
<math id="A7.T7.5.5.5.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A7.T7.5.5.5.m1.1a"><mo mathsize="50%" id="A7.T7.5.5.5.m1.1.1" xref="A7.T7.5.5.5.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A7.T7.5.5.5.m1.1b"><csymbol cd="latexml" id="A7.T7.5.5.5.m1.1.1.cmml" xref="A7.T7.5.5.5.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A7.T7.5.5.5.m1.1c">\pm</annotation></semantics></math><span id="A7.T7.5.5.5.1" class="ltx_text" style="font-size:50%;">0.0</span>
</td>
<td id="A7.T7.6.6.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;">
<math id="A7.T7.6.6.6.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A7.T7.6.6.6.m1.1a"><mo mathsize="50%" id="A7.T7.6.6.6.m1.1.1" xref="A7.T7.6.6.6.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A7.T7.6.6.6.m1.1b"><csymbol cd="latexml" id="A7.T7.6.6.6.m1.1.1.cmml" xref="A7.T7.6.6.6.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A7.T7.6.6.6.m1.1c">\pm</annotation></semantics></math><span id="A7.T7.6.6.6.1" class="ltx_text" style="font-size:50%;">0.1</span>
</td>
<td id="A7.T7.7.7.7" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;">
<math id="A7.T7.7.7.7.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A7.T7.7.7.7.m1.1a"><mo mathsize="50%" id="A7.T7.7.7.7.m1.1.1" xref="A7.T7.7.7.7.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A7.T7.7.7.7.m1.1b"><csymbol cd="latexml" id="A7.T7.7.7.7.m1.1.1.cmml" xref="A7.T7.7.7.7.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A7.T7.7.7.7.m1.1c">\pm</annotation></semantics></math><span id="A7.T7.7.7.7.1" class="ltx_text" style="font-size:50%;">0.1</span>
</td>
<td id="A7.T7.8.8.8" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;">
<math id="A7.T7.8.8.8.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A7.T7.8.8.8.m1.1a"><mo mathsize="50%" id="A7.T7.8.8.8.m1.1.1" xref="A7.T7.8.8.8.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A7.T7.8.8.8.m1.1b"><csymbol cd="latexml" id="A7.T7.8.8.8.m1.1.1.cmml" xref="A7.T7.8.8.8.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A7.T7.8.8.8.m1.1c">\pm</annotation></semantics></math><span id="A7.T7.8.8.8.1" class="ltx_text" style="font-size:50%;">0.0</span>
</td>
<td id="A7.T7.9.9.9" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;">
<math id="A7.T7.9.9.9.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A7.T7.9.9.9.m1.1a"><mo mathsize="50%" id="A7.T7.9.9.9.m1.1.1" xref="A7.T7.9.9.9.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A7.T7.9.9.9.m1.1b"><csymbol cd="latexml" id="A7.T7.9.9.9.m1.1.1.cmml" xref="A7.T7.9.9.9.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A7.T7.9.9.9.m1.1c">\pm</annotation></semantics></math><span id="A7.T7.9.9.9.1" class="ltx_text" style="font-size:50%;">0.2</span>
</td>
<td id="A7.T7.10.10.10" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;">
<math id="A7.T7.10.10.10.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A7.T7.10.10.10.m1.1a"><mo mathsize="50%" id="A7.T7.10.10.10.m1.1.1" xref="A7.T7.10.10.10.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A7.T7.10.10.10.m1.1b"><csymbol cd="latexml" id="A7.T7.10.10.10.m1.1.1.cmml" xref="A7.T7.10.10.10.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A7.T7.10.10.10.m1.1c">\pm</annotation></semantics></math><span id="A7.T7.10.10.10.1" class="ltx_text" style="font-size:50%;">0.1</span>
</td>
</tr>
<tr id="A7.T7.23.27" class="ltx_tr">
<td id="A7.T7.23.27.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify ltx_border_t" style="padding:0.45pt 0.0pt;">
<span id="A7.T7.23.27.1.1" class="ltx_inline-block ltx_align_top">
<span id="A7.T7.23.27.1.1.1" class="ltx_p"><span id="A7.T7.23.27.1.1.1.1" class="ltx_text" style="font-size:90%;">+ Independence constraint</span></span>
</span>
</td>
<td id="A7.T7.23.27.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_t" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.27.2.1" class="ltx_text" style="font-size:90%;">+6.2</span></td>
<td id="A7.T7.23.27.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_t" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.27.3.1" class="ltx_text" style="font-size:90%;">+4.5</span></td>
<td id="A7.T7.23.27.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_t" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.27.4.1" class="ltx_text" style="font-size:90%;">+0.8</span></td>
<td id="A7.T7.23.27.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_t" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.27.5.1" class="ltx_text" style="font-size:90%;">+4.4</span></td>
<td id="A7.T7.23.27.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_t" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.27.6.1" class="ltx_text" style="font-size:90%;">-1.5</span></td>
<td id="A7.T7.23.27.7" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_t" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.27.7.1" class="ltx_text" style="font-size:90%;">+2.9</span></td>
<td id="A7.T7.23.27.8" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_t" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.27.8.1" class="ltx_text" style="font-size:90%;">+1.3</span></td>
<td id="A7.T7.23.27.9" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_t" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.27.9.1" class="ltx_text" style="font-size:90%;">+0.8</span></td>
<td id="A7.T7.23.27.10" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_t" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.27.10.1" class="ltx_text" style="font-size:90%;">+4.1</span></td>
<td id="A7.T7.23.27.11" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_t" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.27.11.1" class="ltx_text" style="font-size:90%;">+2.3</span></td>
</tr>
<tr id="A7.T7.23.28" class="ltx_tr">
<td id="A7.T7.23.28.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.45pt 0.0pt;">
<span id="A7.T7.23.28.1.1" class="ltx_inline-block ltx_align_top">
<span id="A7.T7.23.28.1.1.1" class="ltx_p"><span id="A7.T7.23.28.1.1.1.1" class="ltx_text" style="font-size:90%;">+ On-manifold soft regularizer, VAE</span></span>
</span>
</td>
<td id="A7.T7.23.28.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.28.2.1" class="ltx_text" style="font-size:90%;">+11.9</span></td>
<td id="A7.T7.23.28.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.28.3.1" class="ltx_text ltx_font_bold" style="font-size:90%;">+7.1</span></td>
<td id="A7.T7.23.28.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.28.4.1" class="ltx_text" style="font-size:90%;">+0.6</span></td>
<td id="A7.T7.23.28.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.28.5.1" class="ltx_text" style="font-size:90%;">+11.5</span></td>
<td id="A7.T7.23.28.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.28.6.1" class="ltx_text" style="font-size:90%;">-1.0</span></td>
<td id="A7.T7.23.28.7" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.28.7.1" class="ltx_text" style="font-size:90%;">+5.1</span></td>
<td id="A7.T7.23.28.8" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.28.8.1" class="ltx_text" style="font-size:90%;">+3.6</span></td>
<td id="A7.T7.23.28.9" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.28.9.1" class="ltx_text" style="font-size:90%;">+2.2</span></td>
<td id="A7.T7.23.28.10" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.28.10.1" class="ltx_text" style="font-size:90%;">+6.2</span></td>
<td id="A7.T7.23.28.11" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.28.11.1" class="ltx_text" style="font-size:90%;">+2.3</span></td>
</tr>
<tr id="A7.T7.23.29" class="ltx_tr">
<td id="A7.T7.23.29.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.45pt 0.0pt;">
<span id="A7.T7.23.29.1.1" class="ltx_inline-block ltx_align_top">
<span id="A7.T7.23.29.1.1.1" class="ltx_p"><span id="A7.T7.23.29.1.1.1.1" class="ltx_text" style="font-size:90%;">+ On-manifold hard projection, VAE</span></span>
</span>
</td>
<td id="A7.T7.23.29.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.29.2.1" class="ltx_text" style="font-size:90%;">+7.9</span></td>
<td id="A7.T7.23.29.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.29.3.1" class="ltx_text" style="font-size:90%;">+4.2</span></td>
<td id="A7.T7.23.29.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.29.4.1" class="ltx_text" style="font-size:90%;">+2.2</span></td>
<td id="A7.T7.23.29.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.29.5.1" class="ltx_text" style="font-size:90%;">+12.9</span></td>
<td id="A7.T7.23.29.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.29.6.1" class="ltx_text" style="font-size:90%;">-0.3</span></td>
<td id="A7.T7.23.29.7" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.29.7.1" class="ltx_text ltx_font_bold" style="font-size:90%;">+7.4</span></td>
<td id="A7.T7.23.29.8" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.29.8.1" class="ltx_text ltx_font_bold" style="font-size:90%;">+9.9</span></td>
<td id="A7.T7.23.29.9" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.29.9.1" class="ltx_text" style="font-size:90%;">+2.6</span></td>
<td id="A7.T7.23.29.10" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.29.10.1" class="ltx_text" style="font-size:90%;">+7.1</span></td>
<td id="A7.T7.23.29.11" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.29.11.1" class="ltx_text ltx_font_bold" style="font-size:90%;">+4.5</span></td>
</tr>
<tr id="A7.T7.23.30" class="ltx_tr">
<td id="A7.T7.23.30.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.45pt 0.0pt;">
<span id="A7.T7.23.30.1.1" class="ltx_inline-block ltx_align_top">
<span id="A7.T7.23.30.1.1.1" class="ltx_p"><span id="A7.T7.23.30.1.1.1.1" class="ltx_text" style="font-size:90%;">+ On-manifold soft regularizer, PCA</span></span>
</span>
</td>
<td id="A7.T7.23.30.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.30.2.1" class="ltx_text" style="font-size:90%;">+10.6</span></td>
<td id="A7.T7.23.30.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.30.3.1" class="ltx_text" style="font-size:90%;">+4.7</span></td>
<td id="A7.T7.23.30.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.30.4.1" class="ltx_text" style="font-size:90%;">+2.6</span></td>
<td id="A7.T7.23.30.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.30.5.1" class="ltx_text" style="font-size:90%;">+8.6</span></td>
<td id="A7.T7.23.30.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.30.6.1" class="ltx_text" style="font-size:90%;">-0.4</span></td>
<td id="A7.T7.23.30.7" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.30.7.1" class="ltx_text" style="font-size:90%;">+6.4</span></td>
<td id="A7.T7.23.30.8" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.30.8.1" class="ltx_text" style="font-size:90%;">+7.4</span></td>
<td id="A7.T7.23.30.9" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.30.9.1" class="ltx_text" style="font-size:90%;">+2.9</span></td>
<td id="A7.T7.23.30.10" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.30.10.1" class="ltx_text ltx_font_bold" style="font-size:90%;">+8.4</span></td>
<td id="A7.T7.23.30.11" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.23.30.11.1" class="ltx_text" style="font-size:90%;">+4.2</span></td>
</tr>
<tr id="A7.T7.11.11" class="ltx_tr">
<td id="A7.T7.11.11.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.45pt 0.0pt;">
<span id="A7.T7.11.11.1.1" class="ltx_inline-block ltx_align_top">
<span id="A7.T7.11.11.1.1.1" class="ltx_p"><span id="A7.T7.11.11.1.1.1.1" class="ltx_text" style="font-size:90%;">+ On-manifold hard projection, PCA (</span><sup id="A7.T7.11.11.1.1.1.2" class="ltx_sup"><span id="A7.T7.11.11.1.1.1.2.1" class="ltx_text" style="font-size:90%;">∗</span></sup><span id="A7.T7.11.11.1.1.1.3" class="ltx_text" style="font-size:90%;">)</span></span>
</span>
</td>
<td id="A7.T7.11.11.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.11.11.2.1" class="ltx_text" style="font-size:90%;">+13.2</span></td>
<td id="A7.T7.11.11.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.11.11.3.1" class="ltx_text" style="font-size:90%;">+5.8</span></td>
<td id="A7.T7.11.11.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.11.11.4.1" class="ltx_text" style="font-size:90%;">+2.6</span></td>
<td id="A7.T7.11.11.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.11.11.5.1" class="ltx_text" style="font-size:90%;">+12.2</span></td>
<td id="A7.T7.11.11.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.11.11.6.1" class="ltx_text" style="font-size:90%;">+0.1</span></td>
<td id="A7.T7.11.11.7" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.11.11.7.1" class="ltx_text" style="font-size:90%;">+5.3</span></td>
<td id="A7.T7.11.11.8" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.11.11.8.1" class="ltx_text" style="font-size:90%;">+7.2</span></td>
<td id="A7.T7.11.11.9" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.11.11.9.1" class="ltx_text" style="font-size:90%;">+3.4</span></td>
<td id="A7.T7.11.11.10" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.11.11.10.1" class="ltx_text" style="font-size:90%;">+6.7</span></td>
<td id="A7.T7.11.11.11" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.11.11.11.1" class="ltx_text" style="font-size:90%;">+3.4</span></td>
</tr>
<tr id="A7.T7.12.12" class="ltx_tr">
<td id="A7.T7.12.12.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.45pt 0.0pt;">
<span id="A7.T7.12.12.1.1" class="ltx_inline-block ltx_align_top">
<span id="A7.T7.12.12.1.1.1" class="ltx_p"><span id="A7.T7.12.12.1.1.1.1" class="ltx_text" style="font-size:90%;">(</span><sup id="A7.T7.12.12.1.1.1.2" class="ltx_sup"><span id="A7.T7.12.12.1.1.1.2.1" class="ltx_text" style="font-size:90%;">∗</span></sup><span id="A7.T7.12.12.1.1.1.3" class="ltx_text" style="font-size:90%;">) + Fine-tuning &amp; distillation</span></span>
</span>
</td>
<td id="A7.T7.12.12.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.12.12.2.1" class="ltx_text ltx_font_bold" style="font-size:90%;">+14.1</span></td>
<td id="A7.T7.12.12.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.12.12.3.1" class="ltx_text" style="font-size:90%;">+6.3</span></td>
<td id="A7.T7.12.12.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.12.12.4.1" class="ltx_text ltx_font_bold" style="font-size:90%;">+2.8</span></td>
<td id="A7.T7.12.12.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.12.12.5.1" class="ltx_text ltx_font_bold" style="font-size:90%;">+15.1</span></td>
<td id="A7.T7.12.12.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.12.12.6.1" class="ltx_text ltx_font_bold" style="font-size:90%;">+0.9</span></td>
<td id="A7.T7.12.12.7" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.12.12.7.1" class="ltx_text" style="font-size:90%;">+5.7</span></td>
<td id="A7.T7.12.12.8" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.12.12.8.1" class="ltx_text" style="font-size:90%;">+9.3</span></td>
<td id="A7.T7.12.12.9" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.12.12.9.1" class="ltx_text ltx_font_bold" style="font-size:90%;">+3.8</span></td>
<td id="A7.T7.12.12.10" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.12.12.10.1" class="ltx_text" style="font-size:90%;">+6.7</span></td>
<td id="A7.T7.12.12.11" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right" style="padding:0.45pt 0.0pt;"><span id="A7.T7.12.12.11.1" class="ltx_text" style="font-size:90%;">+3.5</span></td>
</tr>
<tr id="A7.T7.13.13" class="ltx_tr">
<td id="A7.T7.13.13.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify ltx_border_t" style="padding:0.45pt 0.0pt;">
<span id="A7.T7.13.13.1.1" class="ltx_inline-block ltx_align_top">
<span id="A7.T7.13.13.1.1.1" class="ltx_p"><span id="A7.T7.13.13.1.1.1.1" class="ltx_text" style="font-size:90%;">(</span><sup id="A7.T7.13.13.1.1.1.2" class="ltx_sup"><span id="A7.T7.13.13.1.1.1.2.1" class="ltx_text" style="font-size:90%;">∗</span></sup><span id="A7.T7.13.13.1.1.1.3" class="ltx_text" style="font-size:90%;">) + Fine-tuning &amp; distillation</span></span>
</span>
</td>
<td id="A7.T7.13.13.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_t" style="padding:0.45pt 0.0pt;"><span id="A7.T7.13.13.2.1" class="ltx_text" style="font-size:90%;">82.5</span></td>
<td id="A7.T7.13.13.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_t" style="padding:0.45pt 0.0pt;"><span id="A7.T7.13.13.3.1" class="ltx_text" style="font-size:90%;">84.3</span></td>
<td id="A7.T7.13.13.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_t" style="padding:0.45pt 0.0pt;"><span id="A7.T7.13.13.4.1" class="ltx_text" style="font-size:90%;">76.2</span></td>
<td id="A7.T7.13.13.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_t" style="padding:0.45pt 0.0pt;"><span id="A7.T7.13.13.5.1" class="ltx_text" style="font-size:90%;">75.5</span></td>
<td id="A7.T7.13.13.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_t" style="padding:0.45pt 0.0pt;"><span id="A7.T7.13.13.6.1" class="ltx_text" style="font-size:90%;">79.3</span></td>
<td id="A7.T7.13.13.7" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_t" style="padding:0.45pt 0.0pt;"><span id="A7.T7.13.13.7.1" class="ltx_text" style="font-size:90%;">84.2</span></td>
<td id="A7.T7.13.13.8" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_t" style="padding:0.45pt 0.0pt;"><span id="A7.T7.13.13.8.1" class="ltx_text" style="font-size:90%;">73.8</span></td>
<td id="A7.T7.13.13.9" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_t" style="padding:0.45pt 0.0pt;"><span id="A7.T7.13.13.9.1" class="ltx_text" style="font-size:90%;">78.0</span></td>
<td id="A7.T7.13.13.10" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_t" style="padding:0.45pt 0.0pt;"><span id="A7.T7.13.13.10.1" class="ltx_text" style="font-size:90%;">78.6</span></td>
<td id="A7.T7.13.13.11" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_t" style="padding:0.45pt 0.0pt;"><span id="A7.T7.13.13.11.1" class="ltx_text" style="font-size:90%;">63.7</span></td>
</tr>
<tr id="A7.T7.23.23" class="ltx_tr">
<td id="A7.T7.23.23.11" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_border_bb" style="padding:0.45pt 0.0pt;"></td>
<td id="A7.T7.14.14.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_bb" style="padding:0.45pt 0.0pt;">
<math id="A7.T7.14.14.1.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A7.T7.14.14.1.m1.1a"><mo mathsize="50%" id="A7.T7.14.14.1.m1.1.1" xref="A7.T7.14.14.1.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A7.T7.14.14.1.m1.1b"><csymbol cd="latexml" id="A7.T7.14.14.1.m1.1.1.cmml" xref="A7.T7.14.14.1.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A7.T7.14.14.1.m1.1c">\pm</annotation></semantics></math><span id="A7.T7.14.14.1.1" class="ltx_text" style="font-size:50%;">2.4</span>
</td>
<td id="A7.T7.15.15.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_bb" style="padding:0.45pt 0.0pt;">
<math id="A7.T7.15.15.2.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A7.T7.15.15.2.m1.1a"><mo mathsize="50%" id="A7.T7.15.15.2.m1.1.1" xref="A7.T7.15.15.2.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A7.T7.15.15.2.m1.1b"><csymbol cd="latexml" id="A7.T7.15.15.2.m1.1.1.cmml" xref="A7.T7.15.15.2.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A7.T7.15.15.2.m1.1c">\pm</annotation></semantics></math><span id="A7.T7.15.15.2.1" class="ltx_text" style="font-size:50%;">2.1</span>
</td>
<td id="A7.T7.16.16.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_bb" style="padding:0.45pt 0.0pt;">
<math id="A7.T7.16.16.3.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A7.T7.16.16.3.m1.1a"><mo mathsize="50%" id="A7.T7.16.16.3.m1.1.1" xref="A7.T7.16.16.3.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A7.T7.16.16.3.m1.1b"><csymbol cd="latexml" id="A7.T7.16.16.3.m1.1.1.cmml" xref="A7.T7.16.16.3.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A7.T7.16.16.3.m1.1c">\pm</annotation></semantics></math><span id="A7.T7.16.16.3.1" class="ltx_text" style="font-size:50%;">0.4</span>
</td>
<td id="A7.T7.17.17.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_bb" style="padding:0.45pt 0.0pt;">
<math id="A7.T7.17.17.4.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A7.T7.17.17.4.m1.1a"><mo mathsize="50%" id="A7.T7.17.17.4.m1.1.1" xref="A7.T7.17.17.4.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A7.T7.17.17.4.m1.1b"><csymbol cd="latexml" id="A7.T7.17.17.4.m1.1.1.cmml" xref="A7.T7.17.17.4.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A7.T7.17.17.4.m1.1c">\pm</annotation></semantics></math><span id="A7.T7.17.17.4.1" class="ltx_text" style="font-size:50%;">2.1</span>
</td>
<td id="A7.T7.18.18.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_bb" style="padding:0.45pt 0.0pt;">
<math id="A7.T7.18.18.5.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A7.T7.18.18.5.m1.1a"><mo mathsize="50%" id="A7.T7.18.18.5.m1.1.1" xref="A7.T7.18.18.5.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A7.T7.18.18.5.m1.1b"><csymbol cd="latexml" id="A7.T7.18.18.5.m1.1.1.cmml" xref="A7.T7.18.18.5.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A7.T7.18.18.5.m1.1c">\pm</annotation></semantics></math><span id="A7.T7.18.18.5.1" class="ltx_text" style="font-size:50%;">0.4</span>
</td>
<td id="A7.T7.19.19.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_bb" style="padding:0.45pt 0.0pt;">
<math id="A7.T7.19.19.6.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A7.T7.19.19.6.m1.1a"><mo mathsize="50%" id="A7.T7.19.19.6.m1.1.1" xref="A7.T7.19.19.6.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A7.T7.19.19.6.m1.1b"><csymbol cd="latexml" id="A7.T7.19.19.6.m1.1.1.cmml" xref="A7.T7.19.19.6.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A7.T7.19.19.6.m1.1c">\pm</annotation></semantics></math><span id="A7.T7.19.19.6.1" class="ltx_text" style="font-size:50%;">0.6</span>
</td>
<td id="A7.T7.20.20.7" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_bb" style="padding:0.45pt 0.0pt;">
<math id="A7.T7.20.20.7.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A7.T7.20.20.7.m1.1a"><mo mathsize="50%" id="A7.T7.20.20.7.m1.1.1" xref="A7.T7.20.20.7.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A7.T7.20.20.7.m1.1b"><csymbol cd="latexml" id="A7.T7.20.20.7.m1.1.1.cmml" xref="A7.T7.20.20.7.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A7.T7.20.20.7.m1.1c">\pm</annotation></semantics></math><span id="A7.T7.20.20.7.1" class="ltx_text" style="font-size:50%;">2.5</span>
</td>
<td id="A7.T7.21.21.8" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_bb" style="padding:0.45pt 0.0pt;">
<math id="A7.T7.21.21.8.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A7.T7.21.21.8.m1.1a"><mo mathsize="50%" id="A7.T7.21.21.8.m1.1.1" xref="A7.T7.21.21.8.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A7.T7.21.21.8.m1.1b"><csymbol cd="latexml" id="A7.T7.21.21.8.m1.1.1.cmml" xref="A7.T7.21.21.8.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A7.T7.21.21.8.m1.1c">\pm</annotation></semantics></math><span id="A7.T7.21.21.8.1" class="ltx_text" style="font-size:50%;">1.8</span>
</td>
<td id="A7.T7.22.22.9" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_bb" style="padding:0.45pt 0.0pt;">
<math id="A7.T7.22.22.9.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A7.T7.22.22.9.m1.1a"><mo mathsize="50%" id="A7.T7.22.22.9.m1.1.1" xref="A7.T7.22.22.9.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A7.T7.22.22.9.m1.1b"><csymbol cd="latexml" id="A7.T7.22.22.9.m1.1.1.cmml" xref="A7.T7.22.22.9.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A7.T7.22.22.9.m1.1c">\pm</annotation></semantics></math><span id="A7.T7.22.22.9.1" class="ltx_text" style="font-size:50%;">1.0</span>
</td>
<td id="A7.T7.23.23.10" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_right ltx_border_bb" style="padding:0.45pt 0.0pt;">
<math id="A7.T7.23.23.10.m1.1" class="ltx_Math" alttext="\pm" display="inline"><semantics id="A7.T7.23.23.10.m1.1a"><mo mathsize="50%" id="A7.T7.23.23.10.m1.1.1" xref="A7.T7.23.23.10.m1.1.1.cmml">±</mo><annotation-xml encoding="MathML-Content" id="A7.T7.23.23.10.m1.1b"><csymbol cd="latexml" id="A7.T7.23.23.10.m1.1.1.cmml" xref="A7.T7.23.23.10.m1.1.1">plus-or-minus</csymbol></annotation-xml><annotation encoding="application/x-tex" id="A7.T7.23.23.10.m1.1c">\pm</annotation></semantics></math><span id="A7.T7.23.23.10.1" class="ltx_text" style="font-size:50%;">0.4</span>
</td>
</tr>
</table>
</figure>
</section>
<section id="A8" class="ltx_appendix">
<h2 class="ltx_title ltx_title_appendix">
<span class="ltx_tag ltx_tag_appendix">Appendix H </span>Additional results: VQA</h2>

<div id="A8.p1" class="ltx_para">
<p id="A8.p1.1" class="ltx_p">The experiments of Section <a href="#S6.SS3" title="6.3 Experiments on visual question answering: GQA dataset ‣ 6 Experiments ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">6.3</span></a> use only the binary (yes/no) question of the GQA dataset <cite class="ltx_cite ltx_citemacro_cite">[<a href="#bib.bib32" title="" class="ltx_ref">32</a>]</cite>.
We repeated them with the full dataset.
The setup, model, and hyperparameters are unchanged.
The models’ accuracy remains close to that of the baseline (Table <a href="#A8.T8" title="Table 8 ‣ Appendix H Additional results: VQA ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">8</span></a>) even when the independence constraint induces the models to focus on different input features.
In Figure <a href="#A8.F14" title="Figure 14 ‣ Appendix H Additional results: VQA ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">14</span></a>, we include examples from the validation set that illustrate this effect.
Contrary to Section <a href="#S6.SS3" title="6.3 Experiments on visual question answering: GQA dataset ‣ 6 Experiments ‣ Predicting is not Understanding: Recognizing and Addressing Underspecification in Machine Learning" class="ltx_ref"><span class="ltx_text ltx_ref_tag">6.3</span></a>, we did not observe improvements in accuracy when training three models with our method.
Our ongoing work is exploring the range of hyperparameters (larger number of models, different regularizer weights) to further investigate these observations.</p>
</div>
<figure id="A8.T8" class="ltx_table">
<figcaption class="ltx_caption"><span class="ltx_tag ltx_tag_table"><span id="A8.T8.2.1.1" class="ltx_text" style="font-size:90%;">Table 8</span>: </span><span id="A8.T8.3.2" class="ltx_text" style="font-size:90%;">Experiments on the full GQA dataset. Models maintain very similar accuracy across the board while focusing on different image features.</span></figcaption>
<table id="A8.T8.4" class="ltx_tabular ltx_centering ltx_align_middle">
<tr id="A8.T8.4.1" class="ltx_tr">
<td id="A8.T8.4.1.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify ltx_border_tt" style="padding:0.35pt 0.0pt;">
<span id="A8.T8.4.1.1.1" class="ltx_inline-block ltx_align_top">
<span id="A8.T8.4.1.1.1.1" class="ltx_p"><span id="A8.T8.4.1.1.1.1.1" class="ltx_text ltx_font_bold" style="font-size:70%;">GQA full</span></span>
</span>
</td>
<td id="A8.T8.4.1.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_tt" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.1.2.1" class="ltx_text" style="font-size:70%;">N. of</span></td>
<td id="A8.T8.4.1.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_tt" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.1.3.1" class="ltx_text" style="font-size:70%;">GQA Val.</span></td>
<td id="A8.T8.4.1.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_tt" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.1.4.1" class="ltx_text" style="font-size:70%;">GQA Val.</span></td>
<td id="A8.T8.4.1.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_tt" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.1.5.1" class="ltx_text" style="font-size:70%;">GQA-OOD</span></td>
<td id="A8.T8.4.1.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_tt" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.1.6.1" class="ltx_text" style="font-size:70%;">GQA-OOD</span></td>
<td id="A8.T8.4.1.7" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_tt" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.1.7.1" class="ltx_text" style="font-size:70%;">Grad-CAM</span></td>
</tr>
<tr id="A8.T8.4.2" class="ltx_tr">
<td id="A8.T8.4.2.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.35pt 0.0pt;">
<span id="A8.T8.4.2.1.1" class="ltx_inline-block ltx_align_top">
<span id="A8.T8.4.2.1.1.1" class="ltx_p"><span id="A8.T8.4.2.1.1.1.1" class="ltx_text" style="font-size:70%;">(accuracy in %)</span></span>
</span>
</td>
<td id="A8.T8.4.2.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.2.2.1" class="ltx_text" style="font-size:70%;">models</span></td>
<td id="A8.T8.4.2.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.2.3.1" class="ltx_text" style="font-size:70%;">(best)</span></td>
<td id="A8.T8.4.2.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.2.4.1" class="ltx_text" style="font-size:70%;">(ensemble)</span></td>
<td id="A8.T8.4.2.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.2.5.1" class="ltx_text" style="font-size:70%;">Val-head (best)</span></td>
<td id="A8.T8.4.2.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.2.6.1" class="ltx_text" style="font-size:70%;">Val-tail (best)</span></td>
<td id="A8.T8.4.2.7" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.2.7.1" class="ltx_text" style="font-size:70%;">rank corr.</span></td>
</tr>
<tr id="A8.T8.4.3" class="ltx_tr">
<td id="A8.T8.4.3.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify ltx_border_t" style="padding:0.35pt 0.0pt;">
<span id="A8.T8.4.3.1.1" class="ltx_inline-block ltx_align_top">
<span id="A8.T8.4.3.1.1.1" class="ltx_p"><span id="A8.T8.4.3.1.1.1.1" class="ltx_text" style="font-size:70%;">Baseline</span></span>
</span>
</td>
<td id="A8.T8.4.3.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.3.2.1" class="ltx_text" style="font-size:70%;">3</span></td>
<td id="A8.T8.4.3.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.3.3.1" class="ltx_text" style="font-size:70%;">49.7</span></td>
<td id="A8.T8.4.3.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.3.4.1" class="ltx_text" style="font-size:70%;">52.3</span></td>
<td id="A8.T8.4.3.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.3.5.1" class="ltx_text" style="font-size:70%;">53.3</span></td>
<td id="A8.T8.4.3.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.3.6.1" class="ltx_text ltx_font_bold" style="font-size:70%;">35.6</span></td>
<td id="A8.T8.4.3.7" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_t" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.3.7.1" class="ltx_text" style="font-size:70%;">0.1404</span></td>
</tr>
<tr id="A8.T8.4.4" class="ltx_tr">
<td id="A8.T8.4.4.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify ltx_border_t" style="padding:0.35pt 0.0pt;" colspan="5"><span id="A8.T8.4.4.1.1" class="ltx_text" style="font-size:70%;">With PCA manifold model</span></td>
<td id="A8.T8.4.4.2" class="ltx_td ltx_border_t" style="padding:0.35pt 0.0pt;"></td>
<td id="A8.T8.4.4.3" class="ltx_td ltx_border_t" style="padding:0.35pt 0.0pt;"></td>
</tr>
<tr id="A8.T8.4.5" class="ltx_tr">
<td id="A8.T8.4.5.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.35pt 0.0pt;">
<span id="A8.T8.4.5.1.1" class="ltx_inline-block ltx_align_top">
<span id="A8.T8.4.5.1.1.1" class="ltx_p"><span id="A8.T8.4.5.1.1.1.1" class="ltx_text" style="font-size:70%;">+ Independence</span></span>
</span>
</td>
<td id="A8.T8.4.5.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.5.2.1" class="ltx_text" style="font-size:70%;">3</span></td>
<td id="A8.T8.4.5.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.5.3.1" class="ltx_text" style="font-size:70%;">50.9</span></td>
<td id="A8.T8.4.5.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.5.4.1" class="ltx_text ltx_font_bold" style="font-size:70%;">52.7</span></td>
<td id="A8.T8.4.5.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.5.5.1" class="ltx_text" style="font-size:70%;">54.4</span></td>
<td id="A8.T8.4.5.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.5.6.1" class="ltx_text" style="font-size:70%;">35.0</span></td>
<td id="A8.T8.4.5.7" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.5.7.1" class="ltx_text" style="font-size:70%;">0.2230</span></td>
</tr>
<tr id="A8.T8.4.6" class="ltx_tr">
<td id="A8.T8.4.6.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.35pt 0.0pt;">
<span id="A8.T8.4.6.1.1" class="ltx_inline-block ltx_align_top">
<span id="A8.T8.4.6.1.1.1" class="ltx_p"><span id="A8.T8.4.6.1.1.1.1" class="ltx_text" style="font-size:70%;">+ Ind. + on-manifold</span></span>
</span>
</td>
<td id="A8.T8.4.6.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.6.2.1" class="ltx_text" style="font-size:70%;">3</span></td>
<td id="A8.T8.4.6.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.6.3.1" class="ltx_text ltx_font_bold" style="font-size:70%;">51.0</span></td>
<td id="A8.T8.4.6.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.6.4.1" class="ltx_text" style="font-size:70%;">52.5</span></td>
<td id="A8.T8.4.6.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.6.5.1" class="ltx_text" style="font-size:70%;">54.7</span></td>
<td id="A8.T8.4.6.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.6.6.1" class="ltx_text" style="font-size:70%;">35.2</span></td>
<td id="A8.T8.4.6.7" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.6.7.1" class="ltx_text" style="font-size:70%;">0.1868</span></td>
</tr>
<tr id="A8.T8.4.7" class="ltx_tr">
<td id="A8.T8.4.7.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify ltx_border_t" style="padding:0.35pt 0.0pt;" colspan="5"><span id="A8.T8.4.7.1.1" class="ltx_text" style="font-size:70%;">With PCA manifold model</span></td>
<td id="A8.T8.4.7.2" class="ltx_td ltx_border_t" style="padding:0.35pt 0.0pt;"></td>
<td id="A8.T8.4.7.3" class="ltx_td ltx_border_t" style="padding:0.35pt 0.0pt;"></td>
</tr>
<tr id="A8.T8.4.8" class="ltx_tr">
<td id="A8.T8.4.8.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify" style="padding:0.35pt 0.0pt;">
<span id="A8.T8.4.8.1.1" class="ltx_inline-block ltx_align_top">
<span id="A8.T8.4.8.1.1.1" class="ltx_p"><span id="A8.T8.4.8.1.1.1.1" class="ltx_text" style="font-size:70%;">+ Independence</span></span>
</span>
</td>
<td id="A8.T8.4.8.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.8.2.1" class="ltx_text" style="font-size:70%;">3</span></td>
<td id="A8.T8.4.8.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.8.3.1" class="ltx_text" style="font-size:70%;">50.7</span></td>
<td id="A8.T8.4.8.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.8.4.1" class="ltx_text" style="font-size:70%;">51.7</span></td>
<td id="A8.T8.4.8.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.8.5.1" class="ltx_text" style="font-size:70%;">55.1</span></td>
<td id="A8.T8.4.8.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.8.6.1" class="ltx_text" style="font-size:70%;">34.0</span></td>
<td id="A8.T8.4.8.7" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.8.7.1" class="ltx_text" style="font-size:70%;">0.1626</span></td>
</tr>
<tr id="A8.T8.4.9" class="ltx_tr">
<td id="A8.T8.4.9.1" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_justify ltx_border_bb" style="padding:0.35pt 0.0pt;">
<span id="A8.T8.4.9.1.1" class="ltx_inline-block ltx_align_top">
<span id="A8.T8.4.9.1.1.1" class="ltx_p"><span id="A8.T8.4.9.1.1.1.1" class="ltx_text" style="font-size:70%;">+ Ind. + on-manifold</span></span>
</span>
</td>
<td id="A8.T8.4.9.2" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_bb" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.9.2.1" class="ltx_text" style="font-size:70%;">3</span></td>
<td id="A8.T8.4.9.3" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_bb" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.9.3.1" class="ltx_text" style="font-size:70%;">50.7</span></td>
<td id="A8.T8.4.9.4" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_bb" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.9.4.1" class="ltx_text" style="font-size:70%;">51.6</span></td>
<td id="A8.T8.4.9.5" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_bb" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.9.5.1" class="ltx_text ltx_font_bold" style="font-size:70%;">55.8</span></td>
<td id="A8.T8.4.9.6" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_bb" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.9.6.1" class="ltx_text" style="font-size:70%;">33.8</span></td>
<td id="A8.T8.4.9.7" class="ltx_td ltx_nopad_l ltx_nopad_r ltx_align_center ltx_border_bb" style="padding:0.35pt 0.0pt;"><span id="A8.T8.4.9.7.1" class="ltx_text ltx_font_bold" style="font-size:70%;">0.1012</span></td>
</tr>
</table>
</figure>
<figure id="A8.F14" class="ltx_figure">
<div class="ltx_flex_figure">
<div class="ltx_flex_cell ltx_flex_size_1"><img src="/html/2207.02598/assets/x27.png" id="A8.F14.g1" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_landscape" width="438" height="102" alt="Refer to caption"></div>
<div class="ltx_flex_break"></div>
<div class="ltx_flex_cell ltx_flex_size_1"><img src="/html/2207.02598/assets/x28.png" id="A8.F14.g2" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_landscape" width="438" height="222" alt="Refer to caption"></div>
<div class="ltx_flex_break"></div>
<div class="ltx_flex_cell ltx_flex_size_1"><img src="/html/2207.02598/assets/x29.png" id="A8.F14.g3" class="ltx_graphics ltx_centering ltx_figure_panel ltx_img_landscape" width="438" height="131" alt="Refer to caption"></div>
<div class="ltx_flex_break"></div>
</div>
<figcaption class="ltx_caption ltx_centering"><span class="ltx_tag ltx_tag_figure"><span id="A8.F14.3.1.1" class="ltx_text" style="font-size:90%;">Figure 14</span>: </span><span id="A8.F14.4.2" class="ltx_text" style="font-size:90%;">
Examples from the GQA validation set. We show the input question, ground truth answer, and predicted answer from 3 models trained with our method. The input images are weighted with grad-CAM scores over object detections.
The models often predict the same answer while focusing on different regions.
This diversity in spatial locations emerges naturally: our independence constraint is applied across the <em id="A8.F14.4.2.1" class="ltx_emph ltx_font_italic">channels</em> of globally-pooled visual features.
</span></figcaption>
</figure>
<div class="ltx_pagination ltx_role_newpage"></div>
</section>
</article>
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2207.02598
Machine learning (ML)  models are typically optimized for their accuracy on a given dataset.
However, this predictive criterion rarely captures all desirable properties of a model, in particular how well it matches a domain expert’sunderstandingof a task.
Underspecificationrefers to the existence of multiple models that are indistinguishable in their in-domain accuracy, even though they differ in other desirable properties such as out-of-distribution (OOD)  performance.
Identifying these situations is critical for assessing the reliability of ML models. We formalize the concept of underspecification and propose a method to identify and partially address it.
We train multiple models with an independence constraint that forces them to implement different functions.
They discover predictive features that are otherwise ignored by standard empirical risk minimization (ERM) , which we then distill into a global model with superior OOD performance.
Importantly, we constrain the models to align with the data manifold to ensure that they discover meaningful features.
We demonstrate the method on multiple datasets in computer vision (collages, WILDS-Camelyon17, GQA)  and discuss general implications of underspecification.
Most notably, in-domain performance cannot serve for OOD model selection without additional assumptions.
abstract
Predicting is not Understanding:Recognizing and Addressing Underspecificationin Machine Learning
Abstract
2207.02598
1IntroductionIs data all you need?A finite set of i.i.d. examples is almost never sufficient to learn a task.
Inductive biases have long been known to be necessary for in-domain generalization.
Out-of-distribution (OOD) 111In this paper, OOD refers to test data that has undergone a covariate shiftcompared to the training data.generalization complicates things further since one also needs to determine which predictive patterns of the training data will remain relevant at test time.
Correlations between inputs and labels that are
important for the task may be indistinguishable from spurious ones that result from dataset-specific artefacts such as selection biases.An example in image recognition.Image labels are often correlated with objects and the backgrounds they appear in (e.g.cars in cities, birds in nature) .
Recognizing either often suffice to predict correct labels.
However, robust OOD generalization (e.g.correctly labeling images of birds in street scenes)  requires to rely on shapes and to ignore the background.
When this requirement cannot be deduced from the data
(because both features leave a similar signature in the joint training distribution) ,
the task is said to be underspecified.
In this example, the task requires the additional knowledge that labels refer to object shapes rather than background textures.
Such knowledge is often task-specific.
For example, the opposite choice of prioritizing color or texture over shape would be sensible for recognizing traffic signs or segmenting medical images.Underspecification gap:
the difference between the information provided in a dataset and the information required to performas desiredon a task.The qualifier “as desired” captures the fact that different use cases require different properties such as adversarial robustness, interpretability, fairness, or OOD generalization. The latter is the focus of this paper.
Underspecification arises because these properties do not necessarily correlate with the ERM objectivetypically used to train models.This paper argues thatidentifying underspecificationis important for
assessing the reliability of ML models,
their reliance on hidden assumptions,
and for identifying the information missing for OOD robustness.
We identify underspecification bydiscovering multiple understandingsof the data.
We learn multiple predictive models compatible with a given dataset and hypothesis class (low in-domain risk) .
We force them to rely on different predictive features by encouraging orthogonality of their input gradients. We also ensure that these features remain semantically meaningful by constraining the input gradients to the data manifold.
Training multiple models stands in contrast with the standard practice of optimizing a single solution to a learning problem – which hides the existence of underspecification.
With our method, we discover predictive features otherwise ignored by standard ERM.
This alone produces candidate models with superior OOD performance.
In addition, we show how to distill selected features from multiple candidate models into one that is robust across a wider range of distribution shifts.
In all cases, a selection strategy must be provided (see Section5)  such as an OOD validation set, domain expertise, task-specific heuristics, etc.Experiments.We apply the method to controlled data (collages)  and computer vision benchmarks (WILDS-Camelyon17, GQA) .
On visual question answering, we show that multiple models can produce similar answers while relying on different visual features (Figure1) .Implications.Our work complements other studiesin formalizing underspecification as a root cause of multiple challenges in ML including shortcut learning, distribution shifts, and even adversarial vulnerabilities (an extreme case of OOD inputs) .
Our formalization of underspecification makes it obvious that ID and OOD performance are not necessarily coupled.
Therefore, without further assumptions, in-domain validation performance is not a reliable model selection strategy for OOD performance despite contradictory suggestions made in the literature.
The prevalence of underspecificationalso suggests that task-specific knowledge and assumptions are often necessary to build robust ML models, since they cannot emerge from simply scaling up data and architectures.Summary of contributions.1.We propose a mathematical framework for quantifying and addressing underspecification.2.We derive a method to learn a set of models compatible with a given dataset that exhibit distinct OOD behaviour. We force the models to rely on different features (independence objective)  that are nonetheless semantically meaningful (on-manifold constraint) .3.We use the method for (1)  highlighting underspecification in given dataset/architecture pairs, and (2)  building models with superior OOD performance. We use controlled and real-world datasets (collages, WILDS-Camelyon17, GQA) .Figure 1:Example of underspecification in visual question answering.
Our method trains multiple models that each discover different predictive features.
We obtain three models producing identical answers on most training and validation data, even though they rely on different visual clues (evidenced by grad-CAM visualizations over object proposals) .
Each model reflects adifferent understanding of the taskcompatible with the data (possibly incomprehensible to humans)  which reveals ambiguity in its specification.
introduction
Predicting is not Understanding:Recognizing and Addressing Underspecificationin Machine Learning
1Introduction
2207.02598
2Related workUnderspecification in ML.An empirical study by d’Amouret al.inshowed that models trained with different random seeds have wide variations in OOD performance despite similar in-domain performance.
The effect was surprising in its prevalence but it echoed earlier observations.
It results from the well-known impossibility of achieving OOD generalization solely through ERM and i.i.d. data.
Indeed, since this learning objective does not constrain the model’s behaviour outside the training distribution (i.e.multiple hypotheses are plausible) , additional assumptions/expertise is necessarye.g.to tweak the network architecture to the task.
Data-driven OOD generalization is also possible but it requires heterogeneous (non-i.i.d.)  data such as multiple environments, counterfactual examples, or non-stationary data.
The impossibility of OOD generalization from i.i.d. data is intimately related to identifiability in causal discoveryand non-linear ICA, and the impossibility of unsupervised disentanglement in representation learning.Multiple hypotheses compatible with the data.TheRashomon effectis almost synonymous with underspecfication but it is agnostic to OOD data.
The Rashomon setis the set of models of a given class whose training loss or in-domain (ID)  risk lies below a threshold.Predictive multiplicityrefers to the existence of multiple models compatible with the data that make conflicting predictions (on ID data again, rather than OOD) .
This paper is about models with low ID risk,
yet different OOD behaviour.
We propose a method tolearn multiple modelscompatible with the data.
Related approaches were described for ensembling, interpretability, and control of inductive biases.
The number of models required inwas very large.
We solve this issue by constraining models to align with the data manifold.
Other works concerned with the identification of multiple solutions to a learning problem include, Bayesian deep learning, and classical feature selection,
none of which are suitable to large-scale models and datasets.Ensembles.Our method for combining robust features resembles traditional ensembling, which lowers variance by combining predictors with uncorrelated errors.
In comparison however, we lower prediction bias by selecting features causally related to the target.
The key in our approach is to discover predictive features that are otherwise missed by ERM.Diversity in feature space.Allen-Zhu and Liexplained the success of deep ensembles with the use of different features by different networks simply because of different initializations.
Their argument is also based on the ubiquity of underspecification which they call “multi-view structure” of data.
Concurrently to our work,used this explanation to build better ensembles by encouraging diversity infeature spacewith a Bayesian framework.
Even more recently, Heljakkaet al.showed how to quantify representational (i.e.feature-wise)  multiplicity.Diversity in prediction space.Concurrently to our work,andproposed to learn models that maximally disagree in OOD predictions.
Both learn only two models.
Obviously, neither would do well on ourcollagedataset (Section6.1)  which requires discovering 4 predictive signals.
On real vision datasets, our experiments also suggest that many predictive features (2)  can be discovered.
body
Predicting is not Understanding:Recognizing and Addressing Underspecificationin Machine Learning
2Related work
2207.02598
3Formalizing underspecificationLet us focus on binary classification tasks.
A dataset provideslabeled exampleswith,.
The goal of a learning algorithm is to identify apredictorto estimate labels222We defineto output logits. A binary predictionis obtained as.of examples from a test set.
While the input datais typically high-dimensional (e.g.vectorized images) ,
natural data (e.g.photographs)  occupies only a fraction of the input space assumed to form a low-dimensionalmanifold.
The dimensionality()  is known as theintrinsic dimensionalityof the data.
Training and test data are drawn from a distribution on this manifold(in-domain examples)  while unbiased natural data (free of dataset-specific selection biases)  is drawn from a distributionof typically broader support.Inductive biasesare the properties of a learning algorithm that determine what modelis returned for a datasetfrom a hypothesis classwhereis a model with free parameters.
Inductive biases enable generalization from finite databy encoding assumptions on the relation betweenand.
In particular, classical learning theory assumes thatandcontain i.i.d. samples from the same distribution. For completeness, we summarize a standard training workflow.1.Randomly split the data into training and validation sets:.2.Ahypothesis classis chosene.g.by defining a neural architecture.3.Empirical risk minimizationserves to optimize the free parameters ofaswhere the empirical risk is defined as,andis a predictive loss such as binary cross-entropy.4.Validation performanceserves to refine various choices (architecture, regularizers, …)  by trial and error,i.e.loosely solvingwhereis often substituted with a task-specific metric such as the error rate.There is often a multitude of models satisfying the above procedure, not all are equally desirable because they differ in properties that the procedure does not constrain.
The degree of underspecification indicates the importance of arbitrary and stochastic factors in the outcome of the learning process.This paper focuses ondifferences in OOD performanceamong predictive models.
OOD performance is the predictive performance of a model (in terms of risk, accuracy, or another task-specific metric)  on test data drawn from a distribution.
On OOD data, features that were predictive in the training data may become irrelevant or misleading, causing a drop in performance of a model that relies on them.
By definition, OOD performance is underspecified by the ERM objective, since the empirical risk is estimated on in-domain data.To capture variability in OOD performance, we propose a definition of underspecification based on the number of ways to fit the data with the above procedure and produce different OOD predictions.333Seefor related uses of thevolumeof a hypothesis spaces in the definition of Rashomon sets.Definition 1.Thedegree of underspecificationof a dataset, input manifold, and hypothesis classis the ratio of volumesof the largest subset of modelssuch that its elementsall have, for small constants,:•A low training risk:,•A low validation risk:,•Distinct OOD predictions:.In the next section, we derive an algorithm to learn a set of models with these properties.
body
Predicting is not Understanding:Recognizing and Addressing Underspecificationin Machine Learning
3Formalizing underspecification
2207.02598
4Proposed methodOverview.We train multiple models with the same architecture and data while enforcing them to represent different functions and use different features.
The models use different initializations, but this does not always suffice to produce significantly-different models.
We add two regularizers that enforce (1)  independence of the models (mutually-orthogonal input gradients) 
and (2)  alignment with the data manifold such that the models learn meaningful features.Since the constraints directly follow from Definition1, the number of models trainable to satisfy them indicates the degree of underspecification.
The only existence of multiple such models thus highlights cases of underspecification.
The models also discover some predictive features missed by standard ERM, which can be combined by distillation into a predictor with superior OOD performance.
The next sections describe how implement the two constraints as differentiable regularizers.4.1Independent modelsTo optimize for distinct OOD predictions, we turn the criteria of Definition1into a differentiable objective.
We reuse the concept of independent models from Rosset al..Definition 2.A pair of predictors,arelocally independentatiff their predictions are statistically independent for Gaussian perturbations around:,.Definition 3.A set of predictorsareglobally independenton a datasetiff every pair of them are locally independent at every.This formalizes the notion that models can rely on different features.
In our case, we seek a set of models globally independent from one another.
We obtain a tractable objective using the relation between statistical independence and geometric orthogonality developed in.Proposition 1.A pair of predictors,are locally independent at(Definition2)  iff the mutual informationwith.For infinitesimally small perturbations () , the samplescan be approximated through linearization by the input gradients.
These are 1D Gaussian random variables whose correlation is given byand their mutual information:.
Therefore, the statistical independence between the models’ outputs as their inputs are perturbed by small Gaussian variations can be enforced by making their input gradients orthogonal.
Ourlocal independence lossfor a pair of models is:(1) with.
To enforceglobalindependence, this loss will be applied to all training points and pairs of models in the overall objective of Eq. (4.2) .4.2On-manifold constraintFigure 2:Method overviewduring training.The independence constraint (1)  makes models’ input gradients orthogonal to one another.
The number of models satisfying it grows exponentially with the input dimension ()  but many are practically irrelevant because the natural data manifold usually occupies much fewer dimensions.
Intuitively, when the constraint affects a model’s gradients in dimensions pointing outward the manifold, it does not affect its predictions on natural data.
Consequently, the independence constraint could be satisfied by models that produce identical predictions on every natural input (thus defeating its purpose)  because their decision boundaries are identical when projected on the manifold.
The issue stems from theisotropicperturbations in Eq. (1) . Only perturbationsonthe manifold are meaningful.One straightforward solution would be to enforce independence after projecting the data on a learned approximation of the manifold.
This approach, proposed infailed in our early experiments because of the difficulty of optimizing the independence objective under such a strict on-manifold constraint.
Instead, we implement a soft constraint as a regularizer that proved easy to train and resilient to imperfect models of the manifold.To learn the data manifold, we need unlabeled examples, ideally containing the type of OOD data expected at test timee.g.a broad collection of natural images:.
We use this data off-line to prepare a functionthat projects an arbitrary vectoratin the input space onto the manifold (Figure3) .
During training, we penalize each model with the distance between its input gradients and their projection on the manifold.
Theon-manifold lossis defined as(2) We describe several implementations ofin AppendixBusing a variational auto-encoder (VAE)  or a simple PCA.
In summary, the on-manifold loss encourages a model to be sensitive to variations in the input that are likely to be encountered in natural test data.
It typically has no effect on in-domain performance (Figure6(c) )  since it only removes a model’s sensitivity to “unnatural” inputs such as variations of isolated pixels, which are unlikely to appear in natural images.The overall learning objective combines the predictive, independence, and on-manifold losses:(3) \begin{overpic}[width=138.76157pt]{figManifold.pdf}
\put(60.0,39.6) {$\blacksquare$}
\put(68.0,35.0) {$\blacksquare$}
\put(66.0,29.6) {$\blacksquare$}
\put(36.0,22.6) {$\bullet$}
\put(31.5,39.2) {$\bullet$}
\put(38.8,31.35) {$\bullet$} \put(60.5,20.5) {$\bigvarstar$}
\put(74.0,64.0) {\footnotesize Input space $\mathbb{R}^{d_{\textrm{in}}}$}
\put(85.0,46.0) {\footnotesize Data manifold $\mathcal{M}$}
\put(-94.0,64.0) {\scriptsize Two possible decision boundaries ({\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}red}/{\color[rgb]{0,0,0.75}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0.75}blue}) ,}
\put(-94.0,58.65) {\scriptsize input gradients at one training point,}
\put(-94.0,53.0) {\scriptsize and their projection on the manifold.}
\put(56.0,4.7) {{\scriptsize$\bullet$} {\tiny$\blacksquare$}\scriptsize\leavevmode\nobreak\ \leavevmode\nobreak\ Training points of two classes.}
\put(55.0,-1.0) {\scriptsize\leavevmode\nobreak\ \leavevmode\nobreak\ \,$\bigvarstar$\leavevmode\nobreak\ \leavevmode\nobreak\ \, Test point, OOD, underspecified.}
\end{overpic}Figure 3:Effect of the proposed method in input space.
Data such as natural images is assumed to lie on a low-dimensional manifold.
The training set covers a subset of this manifold (gray ellipse) . OOD test data ()  lies outside this subset.
In this example, our method discovers two models (redandbluedecision boundaries)  whose input gradients are orthogonal (shown at one training point, in colors matching the boundary) .
Even though a third model (greenvector)  could satisfy the orthogonality constraint, its input gradient would point outside the manifold.
This would violate theon-manifoldconstraint, which requires gradients to closely match their projection on the manifold.4.3Fine-tuningAfter training a set of models with (4.2) , we propose to relax the independence and on-manifold constraints (,)  then fine-tune the models.
This eases the optimization and typically allows the models to reach a higher predictive accuracy.
Concretely, we apply binary masks on the data such that each model is fine-tuned only on the elements most relevant to itself:444In our implementation, masked elements are not replaced with zeros, but rater with random values from other instances in the current mini-batch.(4) with.
The masks are computed before starting the fine-tuning to highlight the data most relevant to each model.
Each element (i.e. pixel or channel)  is unmasked only for the model with the largest corresponding gradient magnitude:(5) We fine-tune each model on its own masked version of the data.555We obtain very similar results between fine-tuning and retraining models from scratch on the masked data.This ensures that the models remain distinct despite disabling the regularizers (,) .
See Algorithm1for a summary.Inputs:Labeled examples.
Unlabeled examples(typically) . Architecture.Result:Set of independent models.Method:With, estimate dimensionalityand set the number of models.With, prepare functionby PCA decomp. or by training a VAE.With, traininstances ofin parallel (Eq.4.2) :Determine masks on input data (Eq.5) :foreachdo// Optional fine-tuning on masked data// Prepare masked data// Use only predictive loss// Fine-tuneAlgorithm 1Training and fine-tuning models.4.4Distilling multiple models into oneFinally, after training/fine-tuning a set of models, we propose to combine the best of them into a global one that uses all of the most relevant features.666Our combination of models is not equivalent to an ensemble: we train a global model to use/ignore specific, chosen features. In comparison, a traditional ensemble combines models in prediction space, relying on their uncorrelated errors to lower the prediction variance. See also AppendixAfor an the analogy of our method with disentanglement in predictor space.We train this global model from scratch, without regularizers, on masked data as described above, using masks frommultipleselected models combined with a logical OR.
In our experiments, we combine the two models with the highest accuracies on an OOD validation set.
We repeat this pairwise combination as long the accuracy of the global model increases, usually for 2–3 iterations (as formalized in Algorithm2in the Appendix) .
body
Predicting is not Understanding:Recognizing and Addressing Underspecificationin Machine Learning
4Proposed method
2207.02598
5Practical considerationsNumber of models to train.We initially estimate the dimensionality of the manifold ()  from.
This is the upper bound on the number of predictors aligned with the manifold that can be mutually independent.
The dimensionality is estimated with a simple methodbased on distances between nearest neighbours.
Only the worst case (highest degree of underspecification per Definition1)  should allow training a number of modelswhile satisfying the independence and on-manifold constraints.
In our experiments, underspecification is usually less severe.
We observed that the value of the predictive loss ()  on training and validation data would remain high for some models, while it would converge normally for others.
Therefore, monitoring the loss per model could potentially serve for evaluating the degree of underspecification of a given setting (dataset and architecture)  by noting the number of models that converge.
A thorough investigation and validation of this technique is an important topic left for future work.Is the data manifold necessary in the definition of underspecification ?Our definition without the reference to the data manifold would be pointless because virtually all conceivable models would be severely underspecified.
The manifold (e.g.natural photographs)  only fills a fraction of the ambient input space (e.g.all possible RGB pixel grids) .
Our definition captures underspecification with respect to this smaller range of inputs that are expected at test time.How to select models to combine into a global predictor?Our approach is agnostic to the model selection strategy.
Many options are possible to identify models that rely on “robust” features.
The simplest is an evaluation on an OOD validation set.
Alternatively, interpretability methods allow domain experts to inspect models and the features they rely on.
The selection may also be semi-automated with task-specific heuristicsor additional annotationse.g.human attention and rationales.Computational cost.With parallelization, our method scales sublinearly in computing time and linearly in memory w.r.t. the number of models.
Models share mini-batches.
And since they use the same architecture, most computations can be parallelized withgrouped convolutions(one group per model) .
The method is also applicable on a frozen feature extractor (Section6.2and AppendixF) .
However, fine-tuning a shared extractor with the independence loss is not possible. The model could simply dispatch redundant features on multiple channels and satisfy the independence constraint without the desired increase in diversity.
Compared to, we significantly reduce the number of models needed to discover useful features.
body
Predicting is not Understanding:Recognizing and Addressing Underspecificationin Machine Learning
5Practical considerations
2207.02598
6ExperimentsWe first present experiments that validate the method on controlled data with multiple known predictive features (collages, Section6.1) .
We then demonstrate applications to existing datasets: WILDS-Camelyon17,6.2and GQA, Section6.3.6.1Experiments on controlled data: collagesThis diagnostic dataset contains images with binary labels that are constructed to contain multiple predictive features.
Each image contains four tiles representing one of two classes respectively from MNIST (0/1) , CIFAR-10 (automobile/truck) , Fashion-MNIST (pullover/coat) , and SVHN (0/1) .•Attraining time, the labels are perfectly correlated with the four tiles (0/1 respectively for the first/second possible class in each tile) .
There are (at least)  four equally-valid ways of understanding the task (i.e.relying on any of the four tiles) .•Attest time, we evaluate a model on four test sets that represent different OOD conditions. In each, only one tile is correlated with the correct label while others tiles are randomized.
By examining the performance on the four test sets, we can identify which tile(s)  the model relies on\floatsetup[figure]capposition=beside,capbesideposition=center,right,floatwidth=0.54capbesidewidth=0.42Class 0Zero, pulloverautomobile, zero.Class 1One, coattruck, one.Figure 4:Examples of collages. Training labels are correlated with all four tiles. Each tile contains features of different complexity.Task difficulty.This dataset is surprisingly challenging because the tiles vary greatly in learning difficulty (e.g.MNIST 0s/1s are very distinct while Fashion-MNIST pullovers/coats look extremely similar) .
It would be reasonable to learn a model that relies on all four tiles. However, an ERM-trained baseline surprisingly uses only a few MNIST pixels (achieving99% accuracy on the MNIST test set and50% on the others) , as shown in previous work on the simplicity bias of neural networks.We followand use our method to learn multiple models compatible with the data. We then report the accuracy of the best model on each test set,i.e.the best accuracy assuming perfect model selection.
This avoids confounding the performance of the learning algorithm and with that of the selection strategy.Applying the proposed method.We follow Algorithm1.
We prepare unlabeled data to defines the data manifold as the union of the training and test sets, thus covering all combinations of contents of the four tiles.
With this data, we estimate the dimensionality of the manifold withas about 23.8 (over 10 runs) .
We prepare two generative models of the manifold: a PCA with 24 components (capturing85% of the variance) 
and a VAE with 24 latent dimensions (details in AppendixD) .
We define a simple architecture (2-layer MLPs)  and train multiple instances in parallel with the proposed objective.
The only hyperparameters are the number of models and weights of independence/on-manifold constraints.
We plot a range of values in the appendix (Figure11) .Results.Our method learns a set of models that focus on different parts of the images. Remarkably, learning as few as 4 models is sufficient to obtain models with high accuracy on all of the four test sets.
Let us examine several ablations.•The baseline ()  only learns about MNIST.•The independence constraint (,)  is crucial for learning distinct models. On its own, it requires training a very large number of models (32)  before picking up features outside the MNIST tiles.
Visualizations of input gradients (Figure6)  reveal that these models each rely only on a single or a few pixels.
These trivial solutions to the independence constraint, akin to adversarial examples, are avoided with the on-manifold constraint.•In the full method (,)  the models discover distinct features that align with the semantic contents of images. The effect of the on-manifold constraint on input gradients is striking (Figure6) .
It forces models to be sensitive to natural variations of the data – rather than unlikely single-pixel patterns.
Remarkably,image regions emerge as meaningful features without inductive bias for spatial locality(e.g.no convolutions) .\floatsetup[figure]capposition=beside,capbesideposition=center,right,floatwidth=0.35capbesidewidth=0.61Figure 5:Collages dataset: accuracy on the four test sets (columns)  of models with best accuracy on each set (rows) .
Diagonal patterns indicate thatmodels specialize and learn different, non-overlapping features.
The baseline only learns features relevant to MNIST.Hyperparameters.A number of models between 4 and 24 give excellent results. As expected, the larger this number, the more granular the features these models learn (Figure6) .
The effect breaks down for24 models, matching theoretical expectations since the dimensionality of the manifold was estimated at24.
The method is stable over a range of regularizer weights.
Additional comparisons in Table1show that a VAE is better than PCA to represent the manifold.
This agrees with the general expectation that natural images form a non-linear manifold in pixel space.
We also found overall results to be robust to variations in architecture and hyperparameters of the VAE.\floatsetup[figure]capposition=below,floatwidth=Baseline4 Models8 Models12 Models(a) With standard training, all models rely on a small, identical region of the image, despite the fact that predictive features are present all over.+ Independence(b) Independence produces distinct gradients, but many models are needed to discover new features and they are sensitive to isolated pixels.+ On-manifold(c) The on-manifold constraint forces gradients to align with natural variations of the data. Accuracy is virtually identical to the baseline.+ Indep. + on-manifold(d) With both constraints,we learn semantically relevant features in all image regionswith as few as 4 models.Figure 6:Input gradients for a random test image from the collages dataset.
It is remarkable that, with the proposed method (d) image regions emerge as meaningful features without any inductive bias for spatial localitysuch as convolutions (models in these experiments are fully-connected MLPs) .Fine-tuning.We report the accuracy of models fine-tuned on masked inputs as proposed in Section4.3.
This optional step relaxes the independence constraint to maximize each model’s predictive performance.
The accuracy jumps significantly and almost reaches the upper-bound on each test set (Table1) .
We experimented with relaxing both the independence and on-manifold constraints. Disabling the former has a significant effect. But the latter has no significant effect on accuracy on its own as expected and discussed in Section4.2.Distilling multiple models into one.We report the performance of combinations of features described in Section4.4.
This procedure is most effective after training a large number of models (24 here) .
This is unsurprising since models then discover finer-grained features.
Each combination selects features relevant to only one specific tile to achieve near-maximal accuracy on the test set of that tile.
Simple traditional ensembling of models completely failed in our experiments.Comparison with existing methods.No other method reported in Table1performed well on this dataset.
The method of Teneyet al.is technically the most similar to ours, but it requires training a much larger number of models and still achieves much lower accuracy.
While all experiments of this section used a model taking raw pixels as input, we repeated the whole evaluation using a shared, frozen ResNet to extract features in AppendixF.
This implementation is computationally appealing for larger-scale applications, and gave essentially similar findings with higher overall accuracy thanks to the deeper architecture.\floatsetup[table]capposition=beside,capbesideposition=center,right,floatwidth=0.55capbesidewidth=0.4Table 1:Accuracy oncollagesof existing and proposed methods (8 models per method unless specified) .
The 4 test sets simulate different OOD conditions: only one tile in each set is correlated with the labels.Standard training only learns a fraction of predictive features.
Existing methods cannot do better than chance except on MNIST, or they require training a large number of models.
Ours learns a variety of features and give near-optimal predictions on every test set (last row) .Collagesdataset (accuracy in %) Best model onMNISTSVHNFashioneCIFAR-10eAverageUpper bound (training on test-domain data) 99.992.480.868.685.5ERM Baseline99.850.050.050.062.5Spectral decoupling99.949.850.649.962.5With penalty on L1 norm of gradients98.549.650.550.062.1With penalty on L2 norm of gradients96.652.152.354.363.8Input dropout (best ratio: 0.9) 97.450.756.152.164.1Independence loss (cosine similarity) 99.750.451.550.263.0Independence loss (dot product) 99.553.553.350.564.2With many more modelsIndependence loss (cosine similarity) ,1024models99.558.166.863.071.9Independence loss (dot product) ,128models98.784.971.661.579.2Proposed method (only 8 models) Independence + on-manifold constraints, PCA97.369.862.260.072.3Independence + on-manifold constraints, VAE (∗) 96.585.161.162.176.2(∗)   +  FT  (fine-tuning) 99.790.981.467.484.8(∗)   +  FT  +  pairwise combinations (1) 99.992.279.366.384.4(∗)   +  FT  +  pairwise combinations (2) 99.992.580.267.585.0(∗)   +  FT  +  pairwise combinations (3) 99.992.380.868.585.46.2Experiments on real data: WILDS-Camelyon17Dataset.The WILDS-Camelyon17 benchmarkprovides histopathology images to classify as “tumor” or “normal”.
The images come from different sets of hospitals in the training, validation (val-OOD) , and test splits (test-OOD) .
The challenge is to learn a model that generalizes from the training hospitals to those of the test set.
The original authors oftrained a Densenet-121 model from scratch on this data with 10 random seeds. They showed that the performance on val-OOD and test-OOD varies wildly across seeds, demonstrating that the task is severely underspecified with only the standard training images (the dataset provides additional hospital labels that could enable generalization; neither ERM nor our method uses them) .Implementation of our method.We use frozen features (last-layer activations)  from one of the pretrained models fromas input. We will show that we can recover even more variability in performance than the complete models trained on different random seeds, even while keeping the model frozen (i.e.retraining only a classifier) .
We first determined that the best ERM-trained classifier on frozen features is a simple linear one, rather than an MLP.
Our method simplifies in two ways with a linear classifier.
First, input gradients are equal to the classifier weights, and the proposed regularizers do not require second-order derivatives anymore during back-propagation.
Second, we found empirically that the soft on-manifold regularizer can be replaced with a hard constraint: we explicit project the input gradients onto the manifold and apply the independence regularizer on these projections, as proposed in.
As noted in Section4.2, this option completely failed in our early experiments with MLPs, but it seems viable with linear classifiers. This further simplifies the implementation and removes the hyperparameter.Results.We plot in Figure7the spread of accuracies of models trained with different methods (using features from the first pretrained model from, see AppendixGfor similar results with the others) .
TheERM baselinesimply recovers the accuracy of the original complete Densenet, with essentially no variation across random seeds.
With ourindependence constraint, the spread of accuracies significantly widens, both below and above the baseline.
In Figure8, we see that the models span various trade-offs in accuracy on val-OOD and on test-OOD, neither of which is correlated with the accuracy on in-domain data (val-ID) , thus showing evidence of underspecification.
Back to Figure7, with our additionalon-manifold constraint, the best models reach higher accuracies. This also tops out when training a handful of models (about 10–14, near the intrinsic dimensionality of the data estimated at 12 with) .
Keeping in mind that we use frozen features, these results show that the ERM-pretrained model extracts features useful for OOD performance but that are ignored by the pretrained classifier.
Similar findings were recently reported in.
Our method recovers these features and produces alternative classifiers with a variety of trade-offs in performance across various OOD conditions.Ablations.In Table2, we compare additional ablations of our method, using a fixed number of 10 models.
Theessential components are the independence and on-manifold constraints.
The fine-tuning and distillation steps contribute to a marginal improvement.
We report similarrelativeimprovements in AppendixGwith other pretrained models, but the absolute performance is very muchdependent on a “good” pretrained model.Model selection.Our method brings similar relative improvements on either val-OOD or test-OOD but typically with different models for each (see Figure7)  despite both being OOD relative to the training data.Model selection absent labelled target-domain data therefore remains an issueon this dataset.
Fortunately, only little such data may be sufficient.
We repeated a few experiments while holding out 1% of test-OOD (less than 1,000 instances)  and we observed a 99.87% correlation coefficient between the accuracy on test-OOD and this held-out data.
While all our results assume perfect “oracle” model selection, it seems reasonable that real applications could provide a small amount of labelled test data to achieve similar results.Figure 7:Spread of accuracies on WILDS-Camelyon17 of models trained with different ablations of our method.
The upper/lower bounds of theshaded areasshow the highest/lowest accuracy of any model from one run, averaged over 6 seeds.
Theviolinsshow the distribution of accuracies over all seeds (hence some values outside the shaded areas) .Take-aways.The independence constraint (gray)  produces a wide variety of models compared to the baseline (red) .
However, the highest accuracy in each run grows slowly with the number of models.
With the on-manifold constraint (blue) , the improvement is clearly larger and only requires a handful of models. The fine-tuning and distillation (green)  bring an additional marginal improvement.Figure 8:On WILDS-Camelyon17, each model (same colors as Figure7)  achieves a different trade-off in performance on different splits, as seen with test-OODvs.val-ID/val-OOD (left/right, respectively) .
Each dot represents one model at one training epoch. We plot every epoch since each model moves on these charts as training progresses.
The proposed method (green)  produces models with much better performance on all splits (toward the upper right)  than the baseline (red) .\floatsetup[table]capposition=beside,capbesideposition=center,right,floatwidth=0.58capbesidewidth=0.40Table 2:Accuracy on WILDS-Camelyon17 (average over 6 random seeds)  while training 12 models with various ablations.
Each component of our method contributes to improving the accuracy of the best model from each run. The data also appears simple enough that a PCA approximates the manifold well enough. This allows implementing the on-manifold constraint as a hard projection rather than a soft regularizer.WILDS-Camelyon17Best accuracy (%)  onval-OODtest-OODPseudo-Label–67.78.2DANN–68.49.2FixMatch–71.04.9CORAL–77.96.6NoisyStudent–86.71.7ERM Baseline84.90.168.40.1+ Independence constraint85.30.574.60.9+ On-manifold soft regularizer, VAE85.40.480.31.7+ On-manifold hard projection, VAE88.22.176.32.8+ On-manifold soft regularizer, PCA87.80.379.02.9+ On-manifold hard projection, PCA∗88.40.781.61.4(∗)  + Fine-tuning & distillation88.40.782.52.46.3Experiments on visual question answering: GQA datasetWe now demonstrate the applicability of our method on a more complex task.
We choose visual question answering (VQA)  because it is notorious for dataset biasesthat cause shortcut learning.
We train a simple architecture (details in AppendixE)  on the GQA datasetand report accuracy on the GQA and GQA-OODvalidation sets.
We focus on binary questions to lighten the computational expense.
As input the model, we use frozen 2048-dimensional image features (globally-pooled bottom-up features) .
Our method is applied in the space of these features.
In Table3, we observe that our method produces models with slightly better accuracy on all test sets.
The independence regularizer alone is not sufficient, most likely because it is unable to discover meaningful features in the large 2048-dimensional space.
The on-manifold constraint does solve this difficulty.
Interestingly, the individual models are better as well as a simple ensemble of all of these models.
This suggests a benefit from the greater variety of features learned collectively by the models.
We show in Figure1and in AppendixHthat the models typically focus each on different regions of images (with grad-CAM-weighted visualizations as in) .
It is important to note that our method is applied across channels of image features, and that the spatial diversity emerges naturally.
We quantitatively verify this increase in diversity in Table3with the average Spearman rank correlation of grad-CAM scores across models.\floatsetup[table]capposition=beside,capbesideposition=center,right,floatwidth=0.61capbesidewidth=0.36Table 3:Application to visual question answering. Models trained with the proposed method achieve higher accuracy.
They also show higher diversity in the image regions they rely on (last column, lower correlation of grad-CAM scores) .GQA yes/noN. ofGQA Val.GQA Val.GQA-OODGQA-OODGrad-CAM(accuracy in %) models(best) (ensemble) Val-head(best) Val-tail(best) rank corr.Baseline367.969.370.466.50.68Baseline1668.668.971.267.3–+ Independence367.669.270.567.10.57+ Independence467.669.470.166.2–+ Independence867.870.070.766.7–+ Independence1268.170.370.669.8–+ Independence1668.170.571.169.6–+ Ind. + on-manifold368.769.771.972.50.59+ Ind. + on-manifold469.270.472.970.5–+ Ind. + on-manifold868.870.071.569.1–+ Ind. + on-manifold1269.070.272.669.4–+ Ind. + on-manifold1669.370.372.471.5–Figure 9:Examples from the GQA validation set. We show the input question, ground truth answer, and predicted answer from 3 models trained with our method. The input images are weighted with grad-CAM scores over object detections.
The models often predict the same answer while focusing on different regions.
This diversity in spatial locations emerges naturally: our independence constraint is applied across thechannelsof globally-pooled visual features.
body
Predicting is not Understanding:Recognizing and Addressing Underspecificationin Machine Learning
6Experiments
2207.02598
7DiscussionWe presented a method that highlights cases of underspecification by training multiple models with similar in-domain performance yet different OOD behaviour.
This method offers a partial solution to building robust models since it discovers features that are otherwise missed by standard ERM due to shortcut learning or other implicit inductive biases.What do we gain from identifying cases of underspecification?The level of underspecification (indicated by the number of models that can be trained with the proposed constraints)  shows how far from unique a solution to a learning problem is.
Diagnosing underspecification is not a pass-or-fail test: all but the simplest tasks and models are underspecified to some extent.
Measuring underspecification should help determining the level of trust attributed to an ML model.
Our constructive approach has the added advantage of exposing the range of predictive features present in the data.Importance to both engineering and science.There is a continuing source of research questions in the apparent mismatch between empirical practices in ML and some hard limitations of learning methods.
The concept of underspecification has the potential to unify phenomena including shortcut learning, distribution shifts, and adversarial robustness.
These are important for ML as an engineering discipline (improving reliability and applicability of ML methods)  as well as a scientific endeavour (understanding the structure of real-world data and how/why existing methods work) .A first implicationof underspecification is that ERM is insufficient to guarantee OOD generalization.
Identified cases of underspecification point at the need for additional task-specific information in the design of reliable learning methods.
If such information cannot be integrated, learned models are at risk of unexpected behaviour when deployed on OOD data, because the depend on stochastic or arbitrary factors (e.g.reliance on texturevs.shape in image classification) .A second implicationis that ID and OOD performance are not necessarily coupled.
Without further assumptions, in-domain validation is not a reliable model selection strategy for OOD performance despite contradictory suggestions made in the literature.
This strategy might still be useful as a heuristic owing to some inherent structure in real-world data, but its limits of applicability are yet to be understood.A third implicationis that high OOD performance of a model is no guarantee for its reliability. High apparent performance might happen by accident in an underspecified setting. In such cases, the model behaviour depends on hidden assumptions and it could still fail unexpectedly.
Identifying underspecification remains important to identify these hidden assumptions, which is
particularly important for high-stakes applications such as medical imaging.The proposed analysis also corroborates existing explanations for techniques that successfully improve generalization, such as data augmentation and contrastive learning.
Both were indeed shown to depend on the injection of additional knowledge, respectively in the design of the augmentationsand pair selection strategy.
And this extra knowledge is often task-specific. For example, augmenting images with rotations may help in identifying flowers but not traffic signs.
Injecting task-specific knowledge is sometimes vilified in a “data-driven” culture.
This study suggests that we would rather benefit from highlighting this practice and making assumptions more explicit, thus helping one to identify the limits of applicability of various methods.Conclusion.This paper made theoretical and methodological steps on the study of underspecification. It complements an observational studywith a method to diagnose and address the problem.Limitations.The proposed method for building models with better generalization is only a partial solution since it requires an external model selection procedure.
New methods for model selection, robust evaluation, and explainabilityare all suitable to implement this selection.
Interactive approachesare another option that injects expert knowledge.
Another possible extension is to apply the method to the end-to-end training of larger models.
Finally, this work focused on i.i.d. training data.
We hope to extend the analysis to forms of data known to be valuable for OOD generalization such as multiple environments, counterfactual examples, and non-stationary data.
The analysis of multi-environment training, as used in domain generalization, may elucidate why these methods are often ineffective in practice.
conclusion
Predicting is not Understanding:Recognizing and Addressing Underspecificationin Machine Learning
7Discussion