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Question: Survival function estimation is used in many disciplines, but it is most common in medical analytics in the form of the Kaplan-Meier estimator. Sensitive data (patient records) is used in the estimation without any explicit control on the information leakage, which is a significant privacy concern. We propose a first differentially private estimator of the survival function and show that it can be easily extended to provide differentially private confidence intervals and test statistics without spending any extra privacy budget. We further provide extensions for differentially private estimation of the competing risk cumulative incidence function. Using nine real-life clinical datasets, we provide empirical evidence that our proposed method provides good utility while simultaneously providing strong privacy guarantees. A patient progresses from HIV infection to AIDS after 4.5 years. A study using the patient's data publishes the survival function estimates (a standard practice in clinical research). An adversary, with only access to the published estimates (even in the form of survival function plots), can reconstruct user-level data . Effectively leading to the disclosure of sensitive information. This is just one scenario. The survival function is used for modeling any time to an event, taking into account that some subjects will not experience the event at the time of data collection. The survival function is used in many domains, some examples are the duration of unemployment (in economics); time until the failure of a machine part (in engineering); time to disease recurrence, time to infection, time to death (in healthcare); etc. Our personal healthcare information is the most sensitive private attribute, protected by law, violations of which carry severe penalties. And as the initial example suggests, of all application areas, information leakage in the healthcare domain is the most serious issue and is our focus in this study. For estimation of the survival function, we focus on the Kaplan-Meier's (KM) non-parametric method. KM's method is ubiquitous in clinical research. A quick search of the term on PubMed 1 yields 109,421 . It is not an overstatement to say that almost every clinical study uses KM's method to report summary statistics on their cohort's survival. Statistical agencies around the world use this method to report on the survival of the general population or specific disease-related survival estimates. To best of our knowledge, there does not exist any model that can provide formal privacy guarantees for estimation of survival function using the KM method. The only related work is by Nguyên & , which uses the output and objective perturbation for regression modeling of discrete time to event data. The approach is limited to "multivariate" regression models and cannot be directly used to estimate survival function in a differentially private fashion. One can argue that generative models such as the differentially private generative adversarial networks (; ; ; ;) can be trained to generate differentially private synthetic data. Which can then be used to estimate the survival function. But, GANs do not generalize well to the datasets typically encountered for our use-case (very small sample size (can be less than a hundred), highly constrained dimensionality (d ∈ ), a mixture of categorical and continuous variables, no data pre-processing allowed, etc.). We propose the first differentially private method for estimating the survival function based on the KM method. Grounded by the core principles of differential privacy, our method guarantees the differentially private estimation of the survival function. Also, we show that our method easily extends to provide differentially private confidence intervals and differentially private test statistics (for comparison of survival function between multiple groups) without any extra privacy cost. We further extend our method for differentially private estimation of the competing risk cumulative incidence function (another popular estimate in clinical research). Using nine real-life clinical datasets, we provide empirical evidence that our proposed method provides good utility while simultaneously providing strong privacy guarantees. Lastly, we release our method as an R 2 (R) package for rapid accessibility and adoption. We use this section to introduce the concepts central to the understanding of our method. The survival function is used to model time to event data, where the event may not have yet occurred (but the probability of occurrence is non-zero). Such as for HIV infection to AIDS timeline data, at the end of the follow-up period, some patients would have progressed (our event of interest), while others would not have yet progressed (censored observations). Accounting for censored observations (patients that never experience the event during our follow-up) is the central component in the estimation of the survival function. Formally, this gives the probability of not having an event just before time t, or more generally, the probability that the event of interest has not occurred by time t. In practice, survival function can be estimated using more than one approach. Several parametric methods (that make assumptions on the distribution of survival times) such as the ones based on the exponential, Weibull, Gompertz, and log-normal distributions are available. Or one can opt for the most famous and most often used non-parametric method (Kaplan-Meier's method ), which does not assume how the probability of an event changes over time. Our focus in this paper is the latter, which has become synonymous with survival models in clinical literature. KM estimator of the survival function is defined as followŝ where t j, (j ∈ 1, · · ·, k) is the set of k distinct failure times (not censored), d j is the number of failures at t j, and r j are the number of individuals "at risk" before the j-th failure time. We can see that the functionŜ(t) only changes at each failure time, not for censored observations, ing in a "step" function (the characteristic feature of KM estimate). Differential privacy provides provable privacy notion, with the intuition that a randomized algorithm behaves similarly on similar input datasets. Formally, Definition 1. (Differential privacy ) A randomized algorithm M with domain N |X | preserves (, δ)-differentially privacy if for all S ⊆ Range(M) and for all x, y ∈ N |X | such that ||x − y|| 1 ≤ 1, we have where the two datasets (x, y) only differ in any one row (neighboring datasets) and the probability space is over the coin flips of the mechanism M. If δ = 0, we have "pure -differential privacy". Smaller (, δ) provide stronger privacy guarantees. Differential privacy has some interesting properties. We briefly introduce the main property that is crucial to our proposal of differentially private survival function estimation. That is, the postprocessing, formally Theorem 1. (Post processing ) Let M: N |X | → R be a randomized algorithm that is (, δ)-differentially private. Let f: R → R be an arbitrary randomized mapping. Then Theorem 1 states that differential privacy is immune to post-processing. That is, an adversary acting only on the output of M, cannot increase the privacy loss. This notion is central to our approach and we will revisit it in the following sections. Now we introduce our method for differentially private estimation of the survival function using the Kaplan-Meier's method. We follow the basic principles of differential privacy to ensure that our estimate of the survival function is differentially private. We subsequently show that following our simple approach, it is possible to estimate a wide variety of accompanying statistics (such as the confidence intervals, comparison test statistics, etc.) in a differentially private way without spending any extra privacy budget. Before we begin, we recap some of the notations introduced in Section 2.1. We have a vector of time points (t j, j ∈ {1, · · ·, k}), and for each time point, we have a corresponding number of subjects at risk r j (number of subjects not experiencing a progression up to that time point), and we have the number of subjects experiencing the event at that time point (number of progressions), which we denote as d j. We first create a dataset (a matrix) where each row has the data on the number of events (d j) and the number at risk (r j) for each unique time point (t j). Let's denote this matrix by M. Then using the L 1 sensitivity (S) of M, we draw a noise matrix Z from the Laplace distribution (Lap( S /)), where is the privacy parameter and Z is of the same size as M. We then create a differentially private version of M by adding Z, that is, M = M + Z. All subsequent calculations use M. We succinctly present our method as Algorithm 1. 1: procedure DP(Ŝ(t)) 2: Create a matrix M; [r j, d j] ∈ M; for every t j 3: returnŜ (t) 6: end procedure We use this paragraph to briefly discuss Algorithm 1. We begin with the noticeable simplicity of the procedure, that is, the minimal changes required to the original estimation procedure to make it differentially private. This further boosts the accessibility of our differentially private version (it can be implemented using any readily available software package). Also, the required changes for differential privacy come with no computational overhead compared to the original estimation (our method is computationally cheap). Below we provide the formal privacy guarantees and further details on how this method can be easily extended for differentially private estimation of "other" associated statistics. Now we are ready to formally state the differential privacy guarantees of our proposed method. Before we state our main theorem, we start with a supporting Lemma for establishing the global L 1 sensitivity (S) of our method. Lemma 1. L 1 sensitivity (S) of M is two. Proof. As M only contains count variables for the number of events and number at risk for each unique time point. Adding or removing any single individual can change the counts by at most two (that is being in at-risk group and having an event). Theorem 2. Algorithm 1 is -differentially private. Proof. Sketch: We have established the L 1 sensitivity of M. Using it to add Laplace noise (M = M + Lap( 2 /)) makes sure M is differentially private and so are its individual components (that are r j, d j). Using (r j, d j) to calculate the survival function (Eqn. 2) ensures that the estimated function is differentially private by the post-processing theorem . Complete formal proof is provided in the Appendix. As mentioned in the introduction, one of the advantages of our approach is its easy extension to other essential statistics often required and reported along with the estimates of the survival function. Such as the confidence intervals, test statistics for comparing the survival function distributions, etc. Here we formally define the extensions with their privacy guarantees. When reporting survival function estimates, it is often required to include the related confidence intervals, reported to reflect the uncertainty of the estimate. And for group comparison, such as comparing the infection rates between two treatment arms of a clinical trial, hypothesis testing is used with the help of test statistic. So, it is of paramount interest to provide the differentially private counterparts of both (confidence intervals and test statistics). We start with the confidence intervals. Confidence Intervals for survival function estimates are calculated in a "point-wise" fashion, that is, they are calculated at discrete time-points whenever an event is observed (for the same time points at which the survival function changes its value). We start with proving that the calculations required for obtaining confidence intervals are differentially private following the changes made to the data in Algorithm 1. Theorem 3. Confidence Intervals forŜ (t) are -differentially private. Proof. There are more than one type of confidence intervals available for the survival function. Here we focus on the most often used Greenwood's linear-point-wise confidence intervals. Greenwood's formula for the confidence intervals is given aŝ where Replacing by their respective differentially private counterparts from Algorithm 1. estimate forV [Ŝ(t)] is now differentially private, using it in conjunction withŜ (t) makes the confidence intervals differentially private by the post-processing theorem . As we don't need any additional access to the sensitive data for calculating confidence intervals. Hence, calculating and providing differentially private confidence intervals with the differentially private survival function estimates does not incur any additional privacy cost. In other words, we get the differentially private confidence intervals for free. Hypothesis tests are often used to compare the distribution of survival function estimates between groups. For example: To compare infection rates between two treatment arms of a clinical trial. Most often used statistical test in such scenarios is the Logrank test . Below we show that using our method (Algorithm 6), the hypothesis testing using the Logrank test is differentially private. Theorem 4. Hypothesis test forŜ (t) is -differentially private. Proof. Logrank test statistic (Z) is given as where O 1j are observed number of failures (events) (d 1j) and E 1j are the expected number of failures at time j in group 1, we have and V j is the variance, given as Replacing the corresponding quantities by their differentially private counterparts using Algorithm 1, we get which makes V j differentially private as no other sensitive information is required for its estimation. Using it in conjunction with O 1j and E 1j, which can be made differentially private following the same argument, makes the test statistic Z differentially private by the post-processing theorem . The calculation, again being the case of standard post-processing on differentially private data does not add to our overall privacy budget. Hence, after using Algorithm 1, we can output the related confidence intervals and the test statistic without spending any additional privacy budget. In certain scenarios, we can have more than one type of event. Using our prior example of HIV infection, we might have a scenario where patients die before progression to AIDS, making the observation of progression impossible. Such events (death) that preclude any possibility of our event of interest (progression) are known as competing events. Competing events are a frequent occurrence in clinical data and require specialized estimates that take this phenomenon into account, without which our estimates will be biased. One such estimate is the competing risk cumulative incidence, which is also the most widely used and reported estimate in the literature, akin to the KM estimate, but for competing events. Here we show that using Algorithm 1, we can easily extend differential privacy to competing risk scenarios. Theorem 5. Competing risk cumulative incidence using our method is -differentially private. Proof. Cumulative incidence extends Kaplan-Meier estimator and is given bŷ where d ij is the number of events of type j at time t (i) andŜ(t i) is the standard Kaplan-Meier estimator at time t (i). Replacing associated quantities with their differentially private counterparts (using same reasoning as Algorithm 1).Î Its not hard to see thatÎ j (t) is differentially private by the post-processing theorem. Further statistics associated with the cumulative incidence such as the confidence intervals and hypothesis tests, hazard function and hazard ratios, etc. that directly depend on the quantities made differentially private using Algorithm 1 can be similarly argued to be differentially private. Another popular extension that we easily get from our method is the differentially private version of the Nelson-Aalen estimate of the cumulative hazard (; 1969;). Which is simplyĤ (t) = tj ≤t d j /r j, or can be estimated directly from its relationship with the survival function (Ŝ (t) = exp(−Ĥ (t))). Here we present the empirical evaluation of our method on nine real-life clinical datasets of varying properties. We start with the dataset description. Nine real-life clinical datasets with time to event information are used to evaluate our proposed method. Dataset details are provided in Table 1. For space constraints, we provide further details (dataset properties, pre-processing, group comparison details for hypothesis tests, etc.) in the Appendix. As there is no current method for producing differentially private estimates of the survival function. We compare our approach to the original "non-private" version. This provides us with a comparison to the upper bound (we cannot get better than the non-noisy version). Good utility in comparison with the original non-perturbed version will add credibility to our claim of high utility and will encourage practitioners to adopt our method for practical use. Now we present the outcome of our evaluation on nine real-life datasets. We start with the estimation of the differentially private survival function and then move on to the evaluation of the extensions (confidence intervals, test statistic, etc.). For the differentially private estimation of the survival function (our primary goal), Figure 1 shows the . We can see that our privacy-preserving estimation (green line) faithfully estimates the survival function (black line), with little utility loss. As expected, estimation deteriorates with increased privacy budget (orange line). Followup time is on the X-axis and the probability of survival is on the Y-axis. The black line is the original function estimate, the green line is the differentially private estimate with = 2, and the orange line is the differentially private estimate with = 1. We observe that our method provides good utility while protecting an individual's privacy. Small sample sized datasets fare worse compared to larger datasets. An observation worth making is that as the dataset size gets smaller (a small number of events; as in ovarian, Leukemia, Gehan), the utility of our differentially private estimation gets worse. Which is intuitive from the differential privacy point of view, because to protect an individual's privacy in a small dataset, we will need to add large noise (large perturbation). Whereas for moderate to medium-sized datasets, our differentially private estimation provides good , even for the high privacy regime. An important estimate often reported with survival function is the median survival time and its associated confidence intervals. Median survival time is defined as the time point when the survival function attains the value of 0.5, confidence intervals for the survival function at that time point serve as the confidence intervals of the median survival. Table 2 shows the . For "Median Survival (95% CI)", we see that our method estimates the median with high precision, even for high privacy regime. For some cases due to high survival (as is the case with Myeloid and Ovarian datasets), it is not possible to estimate the upper bounds on the confidence intervals, that is why they are marked as "NA". We see a similar trend as we saw with in Figure 1, our precision increases with increasing dataset size, an acceptable trade-off for individual-level privacy protection. Table 2, we observe that our differentially private estimation performs at par with the original "non-noisy" estimation, even for the high privacy regime. The test statistic (Z) follows the χ 2 distribution with one degree of freedom. Using it to derive the p-values, we observe that none of the differentially private estimates change statistical significance threshold (at 0.05 level). That is, none of the differentially private estimates make the "non-noisy" statistically significant non-significant or vice-versa. For cumulative incidence, we use two new datasets with competing risk information. Results are similar with the estimation of competing risk cumulative incidence, that is, our proposed method provides good utility while protecting an individual's privacy. Our method provides faithful estimation even at high privacy regime. For space constraints, detailed are presented in the Appendix. Much work has been done in the intersection of statistical modeling and differential privacy, including many works proposing different differentially private methods for regression modeling (; ; ; ;). Using the same principles, Nguyên & further developed a differentially private regression model for survival analysis. This approach is limited to the "multivariate" regression models and cannot be used for direct differentially private estimation of the survival function. Differentially private generative models such as the differentially private generative adversarial networks (; ; ; ;) have been recently proposed. But, as discussed in the introduction, they are not suitable for generating data for survival function estimation. We have presented the first method for differentially private estimation of the survival function and we have shown that our proposed method can be easily extended to differentially private estimation of "other" often used statistics such as the associated confidence intervals, test statistics, and the competing risk cumulative incidence. With extensive empirical evaluation on nine real-life datasets, we have shown that our proposed method provides good privacy-utility trade-off. Here we provide details on the datasets used for evaluation. 1. Cancer: It pertains to the data on survival in patients with advanced lung cancer from the. Survival time in days is converted into months. Groups compared are males and females. 2. Gehan: This is the dataset from a trial of 42 leukemia patients. Groups compared are the control and treatment groups. 3. Kidney: This dataset is on the recurrence times to infection, at the point of insertion of the catheter, for kidney patients using portable dialysis equipment. Time is converted into months and groups compared are males and females.. Time is converted into months and groups compared are whether maintenance chemotherapy was given or not. This dataset is about natural history of subjects with monoclonal gammopathy of undetermined significance (MGUS). Time is converted into months and groups compared are males and females. 6. Myeloid: Dataset is based on a trial in acute myeloid leukemia. Time is converted into months and groups compared are the two treatment arms. 7. Ovarian: This dataset pertains to survival in a randomized trial comparing two treatments for ovarian cancer. Time is converted into months and groups compared are the different treatment groups. 8. Stanford: This dataset contains the Stanford Heart Transplant data. Time is converted into months and groups compared are the age groups (above and below median). 9. Veteran: This dataset has information from randomized trial of two treatment regimens for lung cancer. Time is converted into months and groups compared are the treatment arms. For empirical evaluation in a competing risk scenario, we use two datasets that have more than one type of event. First is from a clinical trial for primary biliary cirrhosis (PBC) of the liver . With the event variable being receipt of a liver transplant, censor, or death; our event of interest is the transplant, and death here is a competing event. The second dataset has the data on the subjects on a liver transplant waiting list from 1990-1999, and their disposition: received a transplant (event of interest), died while waiting (competing risk), or censored . Figure 2 shows the (cumulative incidence is the opposite of survival function, so the plots go upward). We observe that our differentially private extension does an excellent job of differentially private estimation of the competing risk cumulative incidence function while providing strong privacy guarantees. A.3 PROOFS Theorem 6. Algorithm 1 is -differentially private. Proof. Let M ∈ R d and M * ∈ R d, such that the L 1 sensitivity, S, is ||M − M * || 1 ≤ 1, and let f denote some function, f: R d → R k. Let p M denote the probability density function of Z(M, f,), and let p M * denote the probability density function of Z(M *, f,), we compare both at some arbitrary point q ∈ R k. Figure 2: Extending differentially private estimation to competing risk cumulative incidence (cumulative incidence is the opposite of survival function, so the plots go upward). Black is the original, unperturbed estimate. Green is with = 2 and orange is with = 1. We can see that our method does a good job of estimating competing risk cumulative incidence while providing strong privacy guarantees. last inequality follows from the definition of sensitivity S As our function estimation uses everything from M (our differentially private version of M) and nothing else from the sensitive data, our survival function estimation is differentially private by the post-processing Theorem .

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A first differentially private estimate of the survival function

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We introduce a statistical approach to assessing neural network robustness that provides an informative notion of how robust a network is, rather than just the conventional binary assertion of whether or not of property is violated.


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Question: Neural networks can learn to extract statistical properties from data, but they seldom make use of structured information from the label space to help representation learning. Although some label structure can implicitly be obtained when training on huge amounts of data, in a few-shot learning context where little data is available, making explicit use of the label structure can inform the model to reshape the representation space to reflect a global sense of class dependencies. We propose a meta-learning framework, Conditional class-Aware Meta-Learning (CAML), that conditionally transforms feature representations based on a metric space that is trained to capture inter-class dependencies. This enables a conditional modulation of the feature representations of the base-learner to impose regularities informed by the label space. Experiments show that the conditional transformation in CAML leads to more disentangled representations and achieves competitive on the miniImageNet benchmark. In machine learning, the objective of classification is to train a model to categorize inputs into various classes. We usually assume a categorical distribution over the label space, and thus effectively ignore dependencies among them. However, class structure does exist in real world and is also present in most datasets. Although class structure can be implicitly obtained as a by-product during learning, it is not commonly exploited in an explicit manner to develop better learning systems. The use of label structure might not be of prime importance when having access to huge amounts of data, such the full ImageNet dataset. However, in the case of few-shot learning where little data is available, meta-information such as dependencies in the label space can be crucial. In recent years, few-shot learning-learning from few examples across many tasks-has received considerable attention BID23 BID28 BID6 BID30. In particular, the concept of meta-learning has been shown to provide effective tools for few-shot learning tasks. In contrast to common transfer learning methods that aim to fine-tune a pre-trained model, meta-learning systems are trained by being exposed to a large number of tasks and evaluated in their ability to learn new tasks effectively. In meta-training, learning happens at two levels: a meta-learner that learns across many tasks, and a base-learner that optimizes for each task. Model-Agnostic Meta-Learning (MAML) is a gradient-based meta-learning algorithm that provides a mechanism for rapid adaptation by optimizing only for the initial parameters of the base-learner BID6.Our motivation stems from a core challenge in gradient-based meta-learning, wherein the quality of gradient information is key to fast generalization: it is known that gradient-based optimization fails to converge adequately when trained from only a few examples BID23, hampering the effectiveness of gradient-based meta-learning techniques. We hypothesize that under such circumstances, introducing a metric space trained to encode regularities of the label structure can impose global class dependencies on the model. This class structure can then provide a high-level view of the input examples, in turn leading to learning more disentangled representations. We propose a meta-learning framework taking advantage of this class structure information, which is available in a number of applications. The Conditional class-Aware Meta-Learning (CAML) model is tasked with producing activations in a manner similar to a standard neural network, but with the additional flexibility to shift and scale those activations conditioned on some auxiliary meta-information. While there are no restrictions on the nature of the conditioning factor, in this work we model class dependencies by means of a metric space. We aim to learn a function mapping inputs to a metric space where semantic distances between instances follow an Euclidean geometry-classes that are semantically close lie in close proximity in an p sense. The goal of the conditional class-aware transformation is to make explicit use of the label structure to inform the model to reshape the representation landscape in a manner that incorporates a global sense of class structure. The contributions of this work are threefold: (i) We provide a meta-learning framework that makes use of structured class information in the form of a metric space to modulate representations in few-shot learning tasks; (ii) We introduce class-aware grouping to improve the statistical strength of few-shot learning tasks; (iii) We show experimentally that our proposed algorithm learns more disentangled representation and achieves competitive on the miniImageNet benchmark. We start by describing the meta-learning formulation proposed by BID30 and BID23, and review MAML BID6, of which CAML is an instance. The goal of meta-learning is to learn from a distribution of tasks. The learning happens on two levels: (i) a meta-level model, or meta-learner, that learns across many tasks, and (ii) a base-level model, or base-learner, that operates within each specific task. Meta-learning happens in task space, where each task can be treated as one meta-example. In the meta-learning formulation, we define a collection of regular tasks as meta-sets D, and each task D ∈ D has its own D train and D test split. D train is often denoted as the "support set" and D test the "query set". The ing meta-learner objective is to choose parameters θ that minimize the expected loss L(·;θ) across all tasks in D, DISPLAYFORM0 At the meta-level, the meta-sets D can be further split into disjoint meta-training set D meta−train, meta-validation set D meta−valid and meta-test set D meta−test. The meta-learner is trained on D meta−train, validated on D meta−valid and finally evaluated on D meta−test. Model-Agnostic Meta-Learning BID6 ) is a meta-learning algorithm that aims to learn representations that encourage fast adaptation across different tasks. The meta-learner and base-learner share the same network structure, and the parameters learned by the meta-learner are used to initialize the base-learner on a new task. To optimize the meta-learner, we first sample a set of tasks {D 1,D 2,...,D S} from the meta-training set D meta−train. For a meta-learner parameterized by θ, we compute its adapted parameters θ i for each sampled task D i. The adapted parameters θ i are task-specific and tell us the effectiveness of θ as to whether it can achieve generalization through one or a few additional gradient steps. The objective of the meta-learner is to optimize the representation θ such that it leads to good task-specific adaptations θ i with only a few gradient steps. The meta-learner performs slow learning at the meta-level across many tasks to support fast learning on new tasks. At meta-test time, we initialize the base-learner with the meta-learned representation θ * followed by gradient-based fine-tuning. 3.1 CONDITIONAL CLASS-AWARE META-LEARNING As shown in Figure 1, the proposed Conditional class-Aware Meta-Learning (CAML) is composed of four components: an embedding function f φ that maps inputs to a metric space, a base-learner f θ that learns each individual task, an adaptation function f c that conditionally modulates the representations of Figure 1: Overview of Conditional class-Aware Meta-Learning. Inputs to the model are mapped onto an embedding space using f φ which are then used to modulate the base-learner f θ through a conditional transformation f c. We use MAML (not shown) to meta-learn f c, f θ, and a metric loss to pre-train f φ the base-learner, and a meta-learner that learns across different tasks. Figure 1 depicts a toy illustration of the task inference procedure where examples from three classes are mapped onto a metric space using f φ, which are further used to modulate the base-learner f θ through a conditional transformation function f c.The main contribution of this paper is to incorporate metric-based conditional transformations (f c) into the meta-learning framework at the instance level. A notable feature of the proposed method is that the model has a global sense of the label space through the embedding function f φ by mapping examples onto the semantically meaningful metric space. The embeddings on the metric space inform the base-learner f θ about the label structure which in turn helps disentangle representations from different classes. This structured information can also provide a global view of the input examples to improve gradient-based meta-learning. In a simplistic form, our proposed model makes predictions usinĝ DISPLAYFORM0 where the base-learner f θ is conditioned on the embedding space f φ (x) through the conditional transformation f c. This is in contrast with a regular base-learner whereŷ =f θ (x). In our framework, we use MAML to meta-learn f c and f θ. The metric space is pre-trained using distance-based loss function. We encode information of the label structure through f φ in the form of an M-dimensional metric space, where each input example is reduced to a point in the metric space. The goal of the metric learning step is to optimize parameter φ such that distances between examples in the metric space are semantically meaningful. Given the parameters of the metric space φ, which is represented by a convolutional network, we calculate a centroid c t for each class t, DISPLAYFORM0 where K denotes the number of examples for class t, 1 {yi=t} denotes an indicator function of y i which takes value 1 when y i =t and 0 otherwise. The centroid c t is the sample mean among all instances from the same class which is treated as a prototype representation of the class t. The mapping function f φ is optimized to minimize the negative log-probability defined in Eq. by minimizing the Euclidean distance d between an example and its corresponding class centroid c t while maximizing its Euclidean distance to other class centroids c t: DISPLAYFORM1 In relation to prototypical networks BID28, we use the same loss function for metric learning. However, these frameworks differ in the test mode: we are not interested in example-centroid distances for label assignment, but rather in the projection f φ (x i) from the input space to the metric space that encapsulates inferred class regularities given the input example x i. In relation to other pre-training methods that use the meta-train classes to train a 64-way classifier, our use of the metric space imposes distance-based constraints to learn embeddings that follow semantically meaningful distance measures. We empirically find it difficult to optimize both the metric space and base-learner end-to-end. The metric space is pre-trained on the meta-train data and it is not updated during meta-learning. This also ensures the metric space is trained on a large number of classes to capture the global class dependencies. We now turn to describing the conditionally transformed convolutional block, shown in FIG0, which uses the metric space described in Section 3.2 to inform the base-learner about the label structure of a task. The conditional transformation f c receives embeddings from the metric space and produces transformation operations to modulate convolutional representations of the base-learner f θ.Our conditional transformation has close relation to Batch Normalization (BN) BID11 that normalizes the input to every layer of a neural network. In order to conditionally modulate feature representations, we use Conditional Batch Normalization (CBN) BID22 to predict scale and shift operators from conditional input s i: DISPLAYFORM0 where f c,γ and f c,β can be any differentiable function. This gives our model the flexibility to shift or scale the intermediate representations based on some source information in s i. Since examples belonging to the same class are conceptually close, we exploit this inherent relationship in the metric space to modulate the feature maps at the example level in a way that encodes the label structure. Once we obtained the embedding function f φ, we use two auxiliary networks, learned end-to-end together with the meta-learner, to predict the shift and scale factors of the convolutional feature map: DISPLAYFORM1 Having computedγ i,c andβ i,c, Conditional Batch Normalization (CBN) is applied as follows: DISPLAYFORM2 where R i,c refers to the c th feature map from the i th example, is a small constant, β c and γ c are learnable parameters shared within a task. E[R c] and Var[R c] are batch mean and variance of R c.It is worthwhile to note the effect of conditional transformation. The conditional bias transformation witĥ β i,c is analogous to concatenation-based conditioning where the conditional information is concatenated to the feature maps BID5. The conditional scaling factor provides multiplicative interactions between the metric space and the feature maps to aggregate information. Furthermore, the goal of the conditionally transformed convolutional block is to simultaneously capture the two views of a classification task: a global view that is aware of the relationships among all classes, and a local views of the current N-way K-shot classification task. The metric space, or the global view, is pre-trained in a way that is independent of the current N-way K-shot task; while the base-learner, or the local view, attempts to develop representations for the current classification task. Although the metric space is never trained on the meta-test classes, we expect the learned metric space to generalize from the meta-train tasks to the meta-test tasks. We further describe parameter sharing for CBN learning in Section 3.3.1, and class-aware grouping in Section 3.3.2 which provides more statistical strength for more effective few-shot learning. Although one can predictγ c andβ c using two separate functions, we find it beneficial to use shared parameters as shown in Figure 3. The shared representations are more efficient at producing conditional transformations which also provide a strong inductive bias to help learning BID2. We propose class-aware grouping, as shown in Figure 4 (b), to further exploit properties of metric space. The motivation stems from a lack of statistical strength when learning from only a few examples. As an example, in N-way 1-shot learning, the model is required to find the most meaningful way to distinguish different classes. However, gradient-based optimization may lead to the discovery of irrelevant features which coincide with the class labels, such as colors. We address this problem by class-aware grouping that is guided by our metric space. This is related to "transduction", which is a standard technique in MAML-based methods. Transduction as discussed in BID20, makes use of the channel mean E[R c] and variance Var[R c], defined in Eq., of query examples when evaluating a base-learner. In contrast to standard transduction methods that calculate mean and variance over all examples of the current batch, we introduce class-aware grouping that clusters examples into different groups and use group-based mean and variance to normalize different channels. The grouping is determined by distance measures in the metric space where examples are grouped together based on their nearest centroid c t defined in Section 3.2. Class-aware grouping is integrated into CBN as: DISPLAYFORM0 where 1 {xi∈t} indicates if an example x i belongs to cluster t, and E[R i,c ·1 {xi∈t} ] represents the average of channel R c among examples clustered at c t. This is depicted in Figure 4 where the channel mean and variance are calculated for every group. This approach informs the base-learner about what to expect from the query examples at the class level through channel mean and variance, which provides more explicit guidance to the meta-learning procedure. The base-learner (f θ) is composed of 4 layers of 3×3 convolutions with a 4×4 skip connections from the input to the final convolutional layer. The use of skip connections is to improve the gradient flow as MAML unfolds the inner loop into one computational graph. The use of skip connections is empirically important to the proposed model. Each convolutional layer has 30 channels and is followed by CBN, ReLU and 2×2 max-pooling operations. The output of the final convolution is flattened and fed to a 1-layer dense classifier. For learning the metric space (f φ), we use the same residual network (ResNet-12) as BID21. The metric space is pre-trained on the same meta-training dataset for 30,000 episodes and not updated while learning the base-learner. The meta-learner is trained for 50,000 episodes. We empirically observe that training the metric space and meta-learner end-to-end is overly complex and prone to over-fitting. For CBN functions (f c), we use 3 dense layers with 30 hidden units each. Every layer is followed by a ReLU except for the last layer where no activation is used. For the meta-learner, we use MAML with 1 gradient step for 1-shot learning and 5 gradient steps for 5-shot learning. We use the Adam optimizer and clip the L2 norm of gradients with an upper bound of 5. Similar to MAML, we use transduction where the statistics of the current batch is used for E and Var in Eq. FORMULA6 for both training and testing.4 RELATED WORK 4.1 META-LEARNING Meta-learning or "learning-to-learn" BID27 BID1 BID17 BID29 has been studied as a means to acquire meta-knowledge across many tasks. In recent years, meta-learning has become an important approach for few-shot learning. A number of approaches aim to learn universal learning procedure approximators by supplying training examples to the meta-learner that outputs predictions on testing examples BID10 BID30 BID26 BID15. Other approaches learn to generate model parameters conditioned on training examples BID8 BID18 BID9 BID7, or learning optimization algorithms across different tasks BID23 BID0 BID14. Our work is more inline with gradient-based meta-learning that aims to learn representations that encourage fast adaptation on new tasks. These methods are based on model-agnostic meta-learning (MAML) introduced by BID6. While the original MAML requires second-order gradients in meta-optimization, REPTILE BID20 only uses first-order gradient information. Furthermore, Latent Embedding Optimization (LEO) BID25 ) is proposed to perform gradient-based optimization on a lowdimensional latent space instead of the original high-dimensional parameter space. We emphasize that all those methods do not make explicit use of structured label information, which is a main novelty in this paper. Our work also relates closely to metric-based meta-learning that learns a metric space across different tasks. Siamese networks BID13 learn a similarity measure between inputs using a shared network architecture that outputs high probability when paired examples are from the same class. Matching networks BID30 use full context embeddings to encode examples to the metric space and use attention as a similarity measure for predictions. Prototypical networks BID28 ) compute a centroid, or prototype, for every class that are later used for distance-based queries of new examples. Task dependent adaptive metric (TADAM) BID21 uses metric scaling based on tasks representations to learn a task-dependent metric space. A notable difference between the metric-based methods and our approach is that, the metric space in our model is not aimed for distance-based classification. Rather, we use the metric space to represent class structure which facilitates the gradient-based meta learning towards better generalization. Another difference between our method and TADAM is that, TADAM scales the metric space at the task level where all examples within a task are scaled in the same manner. In contrast, our method provides instance-based conditioning that makes use of the precise representation of each example. Put another way, TADAM modulates the inference on a metric space from a task perspective, while CAML uses example-level representation to modulate the representation at the content level. In style transfer, conditional instance normalization is proposed by BID22 The notion that is common to all these methods is the use of an additional input source, e.g., style or language, to conditionally transform intermediate representations of a network. In few-shot learning, BID31 suggested that it is easier to operate in the concept space in the form of a lower dimensional representation. This is compatible with our proposed approach that uses the metric space as concept-level representation to modulate intermediate features of the base-learner. We use miniImageNet to evaluate the proposed Conditional class-Aware Meta-Learning algorithm. miniImageNet BID30 ) is composed of 84×84 colored images from 100 classes, with 600 examples in each class. We adopt the class split by BID23 that uses 64 classes for training, 16 for validation, and 20 for test. For N-way K-shot training, we randomly sample N classes from the meta-train classes each containing K examples for training and 20 examples for testing. At meta-testing time, we randomly sample 600 N-way K-shot tasks from the test classes. The presented in TAB0 show that our proposed algorithm has comparable performance on the state-of-the-art miniImageNet 5-way 1-shot classification task, and competitive on the 5-way 5-shot task. Unlike LEO that applies meta-learning on pre-trained representations, our meta-learner is able to effectively operate on the high-dimensional parameter space. Our method also does not require co-training compared with TADAM BID21. Figure 5 shows the t-SNE plot of the learned metric space for both meta-train and meta-validation classes. As seen in Figure 4b, examples from the meta-validation set form clusters consistent with their class membership, even though the metric space is not trained on these classes. For example, "mierkat", "tundrarum" and "podenco" are all animals and they are clustered close together. The first main baseline we report is MAML. CAML improves upon MAML by about 10% on both 1-shot and 5-shot tasks. This means incorporating class dependencies in the form of a metric space can greatly facilitate gradient-based meta-learning. We also compare with MAML using our base-learner architecture equipped with skip connections from the input to the last convolutional layer. MAML trained with our base-learner's architecture yields similar performance as the original MAML, suggesting the improvement is ed from the proposed CAML framework, rather than changes in the base-learner's architecture. BID23 43.44% ± 0.77% 60.60% ± 0.71% Matching Networks BID30 46.6% 60.0% Prototypical Network with Soft k-Means BID24 50.41% ± 0.31% 69.88% ± 0.20% MetaNet BID18 49.21% ± 0.96% − TCML BID16 55.71% ± 0.99% 68.88% ± 0.92% adaResNet BID19 56.88% ± 0.62% 71.94 ± 0.57% Cosine Classifier BID7 56.20% ± 0.86% 73.00% ± 0.64% TADAM BID21 58.5% 76.7% LEO BID25 61.76% ± 0.08% 77.59% ± 0.12%MAML BID6 48.7% ± 1.84% 63.11% ± 0.92% MAML on our architecture 48.26% ± 1.04% 64.25% ± 0.78% Prototypical Network BID28 49.42% ± 0.78% 68.2% ± 0.66% Prototypical Network on our metric space 55.96% ± 0.91% 71.64% ± 0.70% CAML (with multitask learning alone) 52.56% ± 0.83% 71.35% ± 1.13% CAML (with class-aware grouping alone) 55.28% ± 0.90% 71.14% ± 0.81% CAML (full model) 59.23% ± 0.99% 72.35% ± 0.71%The confidence intervals are constructed by sampling 600 evaluation tasks from the meta-test classes. The second baseline we use is prototypical network. We measure the classification ability of our metric space using prototypical network as a classifier, shown in TAB0 (Prototypical Network in our metric space). These suggest that making predictions on the metric space alone is inferior to CAML.This can be explained by CAML's ability to fast-adapt representations even when the metric space does not provide good separations. We also find that CAML has larger improvements in 1-shot tasks than 5-shot ones. This is because, in 1-shot learning, metric-based methods estimate class representations from a single example, making it difficult to provide a robust class estimation. We compare activations before and after the conditional transformation to better understand how conditional transformation modulates the feature representations. FIG3 shows the PCA projections of the last convolutional layer in the base-learner. We observe in Figure 5a that, before conditional transformation, examples from three classes ("parallel bars", "tile roof" and "reel") are mixed together. In Figure 5b, after the conditional transformation is applied, one of the previously cluttered classes ("tile roof") become separated from the rest classes. This confirms that metric space can alleviate the difficulty in few-shot learning by means of conditional transformations. We undertake ablation studies to show the impact of multitask learning and class-aware grouping. Empirical in TAB0 suggest that, while 1-shot learning is sensitive to multitask learning and class-aware grouping, 5-shot learning is not affected by those techniques. This is owing to a lack of statistical strength in 1-shot learning, which requires more explicit guidance in the training procedure. This means exploiting metric-based channel mean and variance can provide valuable information to improve meta-learning. More detailed ablation studies are included in Appendix A. In this work, we propose Conditional class-Aware Meta-Learning (CAML) that incorporates class information by means of an embedding space to conditionally modulate representations of the base-learner. By conditionally transforming the intermediate representations of the base-learner, our goal is to reshape the representation with a global sense of class structure. Experiments reveal that the proposed conditional transformation can modulate the convolutional feature maps towards a more disentangled representation. We also introduce class-aware grouping to address a lack of statistical strength in few-shot learning. The proposed approach obtains competitive with the current state-of-the-art performance on 5-way 1-shot and 5-shot miniImageNet benchmark. TAB1 suggest that, while 1-shot learning is sensitive to multitask learning and class-aware grouping, 5-shot learning is less sensitive those techniques. This is owing to a lack of sufficient training examples in 1-shot learning tasks, which requires more explicit guidance in the training procedure. We further note that, in 1-shot learning, using class-aware grouping alone can improve CBN's performance by 3%. This means exploiting metric-based channel mean and variance can provide valuable information for gradient-based meta-learning. For CBN parameters, we observe that more than half of the predictedβ c are negative. This is inline with findings from BID22 that CBN selectively suppresses activations of a feature map when followed by a ReLU. To further examine the impact of the scale and shift operators, we train CBN with each operator alone. TAB2 shows CBN works the best when bothγ c andβ c are used, andγ c contributes more thanβ c, owing to its multiplicative interactions between the metric space and convolutional feature representations.

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CAML is an instance of MAML with conditional class dependencies.

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We leverage deterministic autoencoders as generative models by proposing mixing functions which combine hidden states from pairs of images. These mixes are made to look realistic through an adversarial framework.


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Question: We study the problem of multiset prediction. The goal of multiset prediction is to train a predictor that maps an input to a multiset consisting of multiple items. Unlike existing problems in supervised learning, such as classification, ranking and sequence generation, there is no known order among items in a target multiset, and each item in the multiset may appear more than once, making this problem extremely challenging. In this paper, we propose a novel multiset loss function by viewing this problem from the perspective of sequential decision making. The proposed multiset loss function is empirically evaluated on two families of datasets, one synthetic and the other real, with varying levels of difficulty, against various baseline loss functions including reinforcement learning, sequence, and aggregated distribution matching loss functions. The experiments reveal the effectiveness of the proposed loss function over the others. A relatively less studied problem in machine learning, particularly supervised learning, is the problem of multiset prediction. The goal of this problem is to learn a mapping from an arbitrary input to a multiset 1 of items. This problem appears in a variety of contexts. For instance, in the context of high-energy physics, one of the important problems in a particle physics data analysis is to count how many physics objects, such as electrons, muons, photons, taus, and jets, are in a collision event BID4. In computer vision, automatic alt-text, such as the one available on Facebook, 2 is a representative example of multiset prediction BID16 BID9. 3 In multiset prediction, a learner is presented with an arbitrary input and the associated multiset of items. It is assumed that there is no predefined order among the items, and that there are no further annotations containing information about the relationship between the input and each of the items in the multiset. These properties make the problem of multiset prediction unique from other wellstudied problems. It is different from sequence prediction, because there is no known order among the items. It is not a ranking problem, since each item may appear more than once. It cannot be transformed into classification, because the number of possible multisets grows exponentially with respect to the maximum multiset size. In this paper, we view multiset prediction as a sequential decision making process. Under this view, the problem reduces to finding a policy that sequentially predicts one item at a time, while the outcome is still evaluated based on the aggregate multiset of the predicted items. We first propose an oracle policy that assigns non-zero probabilities only to prediction sequences that exactly in the target, ground-truth multiset given an input. This oracle is optimal in the sense that its prediction never decreases the precision and recall regardless of previous predictions. That is, its decision is optimal in any state (i.e., prediction prefix). We then propose a novel multiset loss which minimizes the KL divergence between the oracle policy and a parametrized policy at every point in a decision trajectory of the parametrized policy. 1 A set that allows multiple instances, e.g. {x, y, x}. See Appendix A for a detailed definition. https://newsroom.fb.com/news/2016/04/using-artificial-intelligenceto-help-blind-people-see-facebook/ 3 We however note that such a multiset prediction problem in computer vision can also be solved as segmentation, if fine-grained annotation is available. See, e.g., BID6.We compare the proposed multiset loss against an extensive set of baselines. They include a sequential loss with an arbitrary rank function, sequential loss with an input-dependent rank function, and an aggregated distribution matching loss and its one-step variant. We also test policy gradient, as was done by BID16 recently for multiset prediction. Our evaluation is conducted on two sets of datasets with varying difficulties and properties. According to the experiments, we find that the proposed multiset loss outperforms all the other loss functions. The paper is structured as follows. We first define multiset prediction at the beginning of Section 2, and compare it to existing problems in supervised learning in 2.1. Then we propose the multiset loss in Section 2.2, followed by alternative baseline losses in Section 3. The multiset loss and baselines are then empirically evaluated in Section 4. A multiset prediction problem is a generalization of classification, where a target is not a single class but a multiset of classes. The goal is to find a mapping from an input x to a multiset Y = y 1,..., y |Y|, where y k ∈ C. Some of the core properties of multiset prediction are 1. the input x is an arbitrary vector.2. there is no predefined order among the items y i in the target multiset Y.3. the size of Y may vary depending on the input x.4. each item in the class set C may appear more than once in Y.Refer to Appendix A for definitions related to multiset prediction. As is typical in supervised learning, in multiset prediction a model DISPLAYFORM0 and computing evaluation metrics m(·) that compare the predicted and target multisets, DISPLAYFORM1 Here, F1 score and exact match (defined in Appendix A), are used as evaluation metrics. Variants of this multiset prediction problem have been extensively studied. However, they differ from our definition of the problem. Here, we go over each variant and discuss how it differs from our definition of multiset prediction. Power Multiset Classification Perhaps the most naive approach to multiset prediction is to transform the class set C into a set M (C) of all possible multisets. This transformation, or the size of M (C), is not well defined unless some constraints are put in place. If the maximum size of a target multiset is set to K, the number of all possible multisets is DISPLAYFORM0 With some constant |C|, we notice that this grows exponentially in the maximum size of the target multiset. Once the class set C is transformed, we can train a multi-class classifier π that maps an input x to one of the elements in M (C). However, this is infeasible in practice and generally intractable. For instance, for the COCO Medium dataset used later in the experiments (see section 4.1), M (C) has roughly 20 thousand elements while the dataset only contains roughly 40 thousand training examples. For the full MS COCO dataset, |M (C)| is on the order of 10 49, making it infeasible to learn a classifier using this method. Ranking A ranking problem can be considered as learning a mapping from a pair of input x and one of the items c ∈ C to its score s(x, c). All the items in the class set are then sorted according to the score, and this sorted order determines the rank of each item. By taking the top-K items from this sorted list, we can turn this problem of ranking into set prediction. Similarly to multiset prediction, the input x is arbitrary, and the target is a set without any prespecific order. However, ranking differs from multiset prediction in that it is unable to handle multiple occurrences of a single item in the target set. Aggregated Distribution Matching Instead of considering the target multiset as an actual multiset, one can convert it into a distribution by computing the frequency of each item from the class set in the target multiset. That is, DISPLAYFORM1 where I · is an indicator function. Then, we can simply minimize a divergence between this distribution and the predictive distribution from a model. This loss function works only when the conditional distribution p(y|x) substantially differs from the marginal distribution p(y), since the model would resort to a trivial solution of predicting the marginal distribution regardless of the input x. We describe this approach in more detail in Sec. 3.1, and test it against our proposal in the experiments. Sequence prediction A sequence prediction problem is characterized as finding a mapping from an input x to a sequence of classes Y = y 1,..., y |Y|. Representative examples of sequence prediction include machine translation, automatic speech recognition and other tagging problems, such as part-of-speech tagging, in natural language processing. Similarly to multiset prediction, the input x is arbitrary, and an item in the class set C may appear more than once in the target sequence. It is, however, different from multiset prediction in that there is a clear, predetermined order of items in the target sequence. We detail this sequence prediction approach later in Sec. 3.2. In this paper, we propose a novel loss function, called multiset loss, for the problem of multiset prediction. This loss function is best motivated by treating the multiset prediction problem as a sequential decision making process with a model being considered a policy π. This policy takes as input the input x and all the previously predicted classesŷ <t at time t, and outputs the distribution over the next class to be predicted. That is, π θ (y t |ŷ <t, x). This policy is parametrized with a set θ of parameters. We first define a free label multiset at time t as Definition 1 (Free Label Multiset).Y t ← Y t−1 \ {ŷ t−1} y t−1 is the prediction made by the policy at time t − 1.This free label multiset Y t contains all the items that remain to be predicted after t − 1 predictions by the policy. We then construct an oracle policy π *. This oracle policy takes as input a sequence of predicted labelsŷ <t, the input x, and the free label multiset with respect to its predictions, Y t = Y\ {ŷ <t}. It outputs a distribution whose entire probability is evenly distributed over all the items in the free label multiset Y t. In other words, Definition 2 (Oracle). DISPLAYFORM0 An interesting and important property of this oracle is that it is optimal given any prefixŷ <t with respect to both precision and recall. This is intuitively clear by noticing that the oracle policy allows only a correct item to be selected. We call this property the optimality of the oracle. Remark 1. Given an arbitrary prefixŷ <t, DISPLAYFORM1 The proof is given in Appendix B. See Appendix A for definitions of precision and recall for multisets. From the remark above, it follows that the oracle policy is an optimal solution to the problem of multiset prediction in terms of precision and recall. Remark 2. DISPLAYFORM2 The proof can be found in Appendix C.It is trivial to show that sampling from such an oracle policy would never in an incorrect prediction. That is, this oracle policy assigns zero probability to any sequence of predictions that is not a permutation of the target multiset. Remark 3. DISPLAYFORM3 where multiset equality refers to exact match, defined in Appendix A. In short, this oracle policy tells us at each time step t which of all the items in the class set C must be selected. This optimality allows us to consider a step-wise loss between a parametrized policy π θ and the oracle policy π *, because the oracle policy provides us with an optimal decision regardless of the quality of the prefix generated so far. We thus propose to minimize the KL divergence from the oracle policy to the parametrized policy at each step separately. This divergence is defined as DISPLAYFORM4 where Y t is formed using predictionsŷ <t from π θ, and H(π t *) is the entropy of the oracle policy at time step t. This entropy term can be safely ignored when learning π θ, since it is constant with respect to θ. We define DISPLAYFORM5 and call it a per-step loss function. We note that it is indeed possible to use another divergence in the place of the KL divergence. It is intractable to minimize the per-step loss from Eq. for every possible state (ŷ <t, x), since the size of the state space grows exponentially with respect to the size of a target multiset. We thus propose here to minimize the per-step loss only for the state, defined as a pair of the input x and the prefixŷ <t, visited by the parametrized policy π θ. That is, we generate an entire trajectory (ŷ 1, . . .,ŷ T) by executing the parametrized policy until either all the items in the target multiset have been predicted or the predefined maximum number of steps have passed. Then, we compute the loss function at each time t based on (x,ŷ <t), for all t = 1,..., T. The final loss function is then the sum of all these per-step loss functions. DISPLAYFORM6 where T is the smaller of the smallest t for which Y t = ∅ and the predefined maximum number of steps allowed. Note that as a consequence of Remarks 2 and 3, minimizing the multiset loss function in maximizing F1 and exact match. As was shown by BID12, the use of the parametrized policy π θ instead of the oracle policy π * allows the upper bound on the learned policy's error to be linear with respect to the size of the target multiset. If the oracle policy had been used, the upper bound would have grown quadratically with respect to the size of the target multiset. To confirm this empirically, we test the following three alternative strategies for executing the parametrized policy π θ in the experiments:1. Greedy search:ŷ t = arg max y log π θ (y|ŷ <t, x) 2. Stochastic sampling: DISPLAYFORM7 Once the proposed multiset loss is minimized, we evaluate the learned policy by greedily selecting each item from the policy. We have defined the proposed loss function for multiset prediction while assuming that the size of the target multiset was known. However, this is a major limitation, and we introduce two different methods for relaxing this constraint. Termination Policy The termination policy π s outputs a stop distribution given the predicted sequence of itemsŷ <t and the input x. Because the size of the target multiset is known during training, we simply train this termination policy in a supervised way using a binary cross-entropy loss. At evaluation time, we simply threshold the predicted stop probability at a predefined threshold (0.5).Special Class An alternative strategy is to introduce a special item to the class set, called END, and add it to the final free label multiset Y |Y|+1 = {END}. Thus, the parametrized policy is trained to predict this special item END once all the items in the target multiset have been predicted. This is analogous to NLP sequence models which predict an end of sentence token BID14 BID0, and was used in BID16 to predict variable-sized multisets. In addition to the proposed multiset loss function, we propose three more loss functions for multiset prediction. They serve as baselines in our experiments later. In the case of distribution matching, we consider the target multiset Y as a set of samples from a single, underlying distribution q * over the class set C. This underlying distribution can be empirically estimated by counting the number of occurrences of each item c ∈ C in Y. That is, DISPLAYFORM0 where I is the indicator function as before. Similarly, we can construct an aggregated distribution computed by the parametrized policy π θ. As with the proposed multiset loss in Def. 3, we first execute π θ to predict a multisetŶ. This is converted into an aggregated distribution q θ in the same way as we turned the target multiset into the oracle aggregate distribution. Learning is equivalent to minimizing the divergence between these two distributions. In this paper, we test two types of divergences. The first one is from a family of L p distances defined as DISPLAYFORM1 where q * and q are the vectors representing the corresponding categorical distributions. The other is a usual KL divergence defined earlier in Eq.: DISPLAYFORM2 One major issue with this approach is that minimizing the divergence between the aggregated distributions does not necessarily in the optimal policy (see the oracle policy in Def. 2.) That is, a policy that minimizes this loss function may assign non-zero probability to an incorrect sequence of predictions, unlike the oracle policy. This is due to the invariance of the aggregated distribution to the order of predictions. Later when analyzing this loss function, we empirically notice that a learned policy often has a different behaviour from the oracle policy, for instance, reflected by the increasing entropy of the action distribution over time. We can train an one-step predictor with this aggregate distribution matching criterion, instead of learning a policy π θ. That is, a predictor outputs both a point q θ (·|x) in a |C|-dimensional simplex and the sizel θ (x) of the target multiset. Then, for each unique item c ∈ C, the number of its occurrences in the predicted multisetŶ is DISPLAYFORM0 where λ > 0 is a coefficient for balancing the contributions from the two terms. A major weakness of this one-step variant, compared to the approaches based on sequential decision making, is the lack of modelling dependencies among the items in the predicted multiset. We test this approach in the experiments later and observe this lack of output dependency modelling in substantially worse prediction accuracy. All the loss functions defined so far have not relied on the availability of an existing order of items in a target multiset. However, by turning the problem of multiset prediction into sequential decision making, minimizing such a loss function is equivalent to capturing an order of items in the target multiset implicitly. Here, we instead describe an approach based on explicitly defining an order in advance. This will serve as a baseline later in the experiments. We first define a rank function r that maps from one of the unique items in the class set c ∈ C to a unique integer. That is, r: C → Z. This function assigns the rank of each item and is used to order items y i in a target multiset Y. This in a sequence S = (s 1, . . ., s |Y|), where r(s i) ≥ r(s j) for all j > i, and s i ∈ Y. With this target sequence S created from Y using the rank function r, we define a sequence loss function as DISPLAYFORM0 Minimizing this loss function is equivalent to maximizing the conditional log-probability of the sequence S given x. This sequence loss function has two clear disadvantages. First, it does not take into account the actual behaviour of the policy π θ (see, e.g., BID1 BID2 BID12 . This makes a learned policy potentially vulnerable to cascading error at test time. Second and more importantly, this loss function requires a pre-specified rank function r. Because multiset prediction does not come with such a rank function by definition, we must design an arbitrary rank function, and the final performance varies significantly based on the choice. We demonstrate this variation in section 4.3.Input-Dependent Rank Function When the input x has a well-known structure, and an object within the input for each item in the target multiset is annotated, it is possible to devise a rank function per input. A representative example is an image input with bounding box annotations. Here, we present two input-dependent rank functions in such a case. First, a spatial rank function r spatial assigns an integer rank to each item in a given target multiset Y such that where x i and x j are the objects corresponding to the items y i and y j .Second, an area rank function r area decides the rank of each label in a target multiset according to the size of the corresponding object inside the input image: DISPLAYFORM1 The area may be determined based on the size of a bounding box or the number of pixels, depending on the level of annotation. We test these two image-specific input-dependent rank functions against a random rank function in the experiments. In BID16, an approach based on reinforcement learning was proposed for multiset prediction. Instead of assuming the existence of an oracle policy, this approach solely relies on a reward function r designed specifically for multiset prediction. The reward function is defined as DISPLAYFORM0 The goal is then to maximize the sum of rewards over a trajectory of predictions from a parametrized policy π θ . The final loss function is DISPLAYFORM1 where the second term inside the expectation is the negative entropy multiplied with a regularization coefficient λ. The second term encourages the exploration during training. As in BID16, we use REINFORCE to stochastically minimize the loss function above with respect to π θ.This loss function is optimal in that the return, i.e., the sum of the step-wise rewards, is maximized when both the precision and recall are maximal (= 1). In other words, the oracle policy, defined in Def. 2, maximizes the expected return. However, this approach of reinforcement learning is known to be difficult, with a high variance BID11. This is especially true here, as the size of the state space grows exponentially with respect to the size of the target multiset, and the action space of each step is as large as the number of unique items in the class set. In this section, we extensively evaluate the proposed multiset loss function against various baseline loss functions presented throughout this paper. More specifically, we focus on its applicability and performance on image-based multiset prediction. MNIST Multi MNIST Multi is a class of synthetic datasets. Each dataset consists of multiple 100x100 images, each of which contains a varying number of digits from the original MNIST . We vary the size of each digit and also add clutters. In the experiments, we consider the following variants of MNIST Multi:• MNIST Multi: |Y| = 4, 20-50 pixel digits • MNIST Multi: |Y| ∈ {1, . . ., 4}, 20-50 pixel digits • MNIST Multi: |Y| = 10, 20 pixel digits Each dataset has a training set with 70,000 examples and a test set with 10,000 examples. We randomly sample 7,000 examples from the training set to use as a validation set, and train with the remaining 63,000 examples. MS COCO As a real-world dataset, we use Microsoft COCO BID10 which includes natural images with multiple objects. Compared to MNIST Multi, each image in MS COCO has objects of more varying sizes and shapes, and there is a large variation in the number of object instances per image which spans from 1 to 91. The problem is made even more challenging with many overlapping and occluded objects. To control the difficulty in order to better study the loss functions, we create the following two variants:• COCO Easy: |Y| = 2, 10,230 training examples, 24 classes• COCO Medium: |Y| ∈ {1, . . ., 4}, 44,121 training examples, 23 classesIn both of the variants, we only include images whose |Y| objects are large and of common classes. An object is defined to be large if the object's area is above the 40-th percentile across the train set of MS COCO. After reducing the dataset to have |Y| large objects per image, we remove images containing only objects of rare classes. A class is considered rare if its frequency is less than 1 |C|, where C is the class set. These two stages ensure that only images with a proper number of large objects are kept. We do not use fine-grained annotation (pixel-level segmentation and bounding boxes) except for creating input-dependent rank functions from Sec. 3.2.For each variant, we hold out a randomly sampled 15% of the training examples as a validation set. We form separate test sets by applying the same filters to the COCO validation set. The test set sizes are 5,107 for COCO Easy and 21,944 for COCO Medium. MNIST Multi We use three convolutional layers of channel sizes 10, 10 and 32, followed by a convolutional long short-term memory (LSTM) layer BID18. At each step, the feature map from the convolutional LSTM layer is average-pooled spatially and fed to a softmax classifier. In the case of the one-step variant of aggregate distribution matching, the LSTM layer is skipped. MS COCO We use a ResNet-34 BID5 ) pretrained on ImageNet BID3 ) as a feature extractor. The final feature map from this ResNet-34 is fed to a convolutional LSTM layer, as described for MNIST Multi above. We do not finetune the ResNet-34 based feature extractor. In all experiments, for predicting variable-sized multisets we use the termination policy approach since it is easily applicable to all of the baselines, thus ensuring a fair comparison. Conversely, it is unclear how to extend the special class approach to the distribution matching baselines. When evaluating a trained policy, we use greedy decoding and the termination policy for determining the size of a predicted multiset. Each predicted multiset is compared against the ground-truth target multiset, and we report both the accuracy based on the exact match (EM) and F-1 score (F1), as defined in Appendix A.More details about the model architectures and training are in Appendix D. We test three alternatives: a random rank function 4 r and two input-dependent rank functions r spatial and r area. We compare these rank functions on MNIST Multi and COCO Easy validation sets. We present the in TAB0. It is clear from the that the performance of the sequence prediction loss function is dependent on the choice of a rank function. In the case of MNIST Multi, the area-based rank function was far worse than the other choices. However, this was not true on COCO Easy, where the spatial rank function was worst among the three. In both cases, we have observed that the random rank function performed best, and from here on, we use the random rank function in the remaining experiments. This set of experiments firmly suggests the need of an order-invariant multiset loss function, such as the multiset loss function proposed in this paper. In this set of experiments, we compare the three execution strategies for the proposed multiset loss function, illustrated in Sec. 3. They are greedy decoding, stochastic sampling and oracle sampling. We test them on MNIST Multi and COCO Easy. As shown in TAB2, greedy decoding and stochastic sampling, both of which consider states that are likely to be visited by the parametrized policy, outperform the oracle sampling. This is consistent with the theory by BID12. Although the first two strategies perform comparably to each other, across both of the datasets and the two evaluation metrics, greedy decoding tends to outperform stochastic sampling. We conjecture this is due to better matching between training and testing in the case of greedy decoding. Thus, from here on, we use greedy decoding when training a model with the proposed multiset loss function. We now compare the proposed multiset loss function against the five baseline loss functions: reinforcement learning L RL, aggregate distribution matching-L 1 dm and L KL dm -, its one-step variant L 1-step, and sequence prediction L seq.MNIST Multi We present the on the MNIST Multi variants in TAB1. On all three variants and according to both metrics, the proposed multiset loss function outperforms all the others. The reinforcement learning based approach closely follows behind. Its performance, however, drops as the number of items in a target multiset increases. This is understandable, as the variance of policy gradient grows as the length of an episode grows. A similar behaviour was observed with sequence prediction as well as aggregate distribution matching. We were not able to train any decent models with the one-step variant of aggregate distribution matching. This was true especially in terms of exact match (EM), which we attribute to the one-step variant not being capable of modelling dependencies among the predicted items. TAB3. On COCO Easy, with only two objects to predict per example, both aggregated distribution matching (with KL divergence) and the sequence loss functions are as competitive as the proposed multiset loss. The other loss functions significantly underperform these three loss functions, as they did on MNIST Multi. The performance gap between the proposed loss and the others, however, grows substantially on the more challenging COCO Medium, which has more objects per example. The proposed multiset loss outperforms the aggregated distribution matching with KL divergence by 3.7 percentage points on exact match and 4.8 on F1. This is analogous to the experiments on the MNIST Multi variants, where the performance gap increased when moving from four to ten digits. One property of the oracle policy defined in Sec. 2.2 is that the entropy of the predictive distribution strictly decreases over time, i.e., H π * (y|ŷ <t, x) > H π * (y|ŷ ≤t, x). This is a natural consequence from the fact that there is no pre-specified rank function, because the oracle policy cannot prefer any item from the others in a free label multiset. Hence, we examine here how the policy learned based on each loss function compares to the oracle policy in terms of per-step entropy. We consider the policies trained on MNIST Multi, where the differences among them were most clear. As shown in FIG1, the policy trained on MNIST Multi using the proposed multiset loss closely follows the oracle policy. The entropy decreases as the predictions are made. The decreases can be interpreted as concentrating probability mass on progressively smaller free labels sets. The variance is quite small, indicating that this strategy is uniformly applied for any input. The policy trained with reinforcement learning retains a relatively low entropy across steps, with a decreasing trend in the second half. We carefully suspect the low entropy in the earlier steps is due to the greedy nature of policy gradient. The policy receives a high reward more easily by choosing one of many possible choices in an earlier step than in a later step. This effectively discourages the policy from exploring all possible trajectories during training. On the other hand, the policy found by aggregated distribution matching (L KL dm) has the opposite behaviour. The entropy in general grows as more predictions are made. To see why this is suboptimal, consider the final (10th) step. Assuming the first nine predictions {ŷ 1, ...,ŷ 9} were correct (i.e. they form a subset of Y), there is only one correct class left for the final predictionŷ 10. The high entropy, however, indicates that the model is placing a significant amount of probability on incorrect sequences. We believe such a policy is found by minimizing the aggregated distribution matching loss function because it cannot properly distinguish between policies with increasing and decreasing entropies. The increasing entropy also indicates that the policy has learned a rank function implicitly and is fully relying on it. Given some unknown free label multiset, inferred from the input, this policy uses the implicitly learned rank function to choose one item from this set. We conjecture this reliance on an inferred rank function, which is by definition sub-optimal, 5 ed in lower performance of aggregate distribution matching. We have extensively investigated the problem of multiset prediction in this paper. We rigorously defined the problem, and proposed to approach it from the perspective of sequential decision making. In doing so, an oracle policy was defined and shown to be optimal, and a new loss function, called multiset loss, was introduced as a means to train a parametrized policy for multiset prediction. The experiments on two families of datasets, MNIST Multi variants and MS COCO variants, have revealed the effectiveness of the proposed loss function over other loss functions including reinforcement learning, sequence, and aggregated distribution matching loss functions. The success of the proposed multiset loss brings in new opportunities of applying machine learning to various new domains, including high-energy physics. Precision Precision gives the ratio of correctly predicted elements to the number of predicted elements. Specifically, letŶ = (C, µŶ), Y = (C, µ Y) be multisets. Then DISPLAYFORM0 The summation and membership are done by enumerating the multiset. For example, the multisetŝ Y = {a, a, b} and Y = {a, b} are enumerated asŶ = {a DISPLAYFORM1 Formally, precision can be defined as DISPLAYFORM2 where the summation is now over the ground set C. Intuitively, precision decreases by 1 |Ŷ| each time an extra class label is predicted. Recall Recall gives the ratio of correctly predicted elements to the number of ground-truth elements. Recall is defined analogously to precision, as: Similarly, we start with the definition of the recall: DISPLAYFORM3 Rec(ŷ <t, Y) = y∈ŷ<t I y∈Y |Y|.turned into a conditional distribution over the next item after affine transformation followed by a softmax function. When the one-step variant of aggregated distribution matching is used, we skip the convolutional LSTM layers, i.e., c = DISPLAYFORM4 See Fig. 2 for the graphical illustration of the entire network. See TAB4 for the details of the network for each dataset. conv 5 × 5 max-pool 2 × 2 feat 10 81 conv 3 × 3 feat 32 conv 5 × 5 max-pool 2 × 2 feat 10 conv 3 × 3 feat 32 conv 5 × 5 max-pool 2 × 2 feat 32 ResNet-34 361 conv 3 × 3 feat 512 conv 3 × 3 feat 512Preprocessing For MNIST Multi, we do not preprocess the input at all. In the case of MS COCO, input images are of different sizes. Each image is first resized so that its larger dimension has 600 pixels, then along its other dimension is zero-padded to 600 pixels and centered, ing in a 600x600 image. Training The model is trained end-to-end, except ResNet-34 which remains fixed after being pretrained on ImageNet. For all the experiments, we train a neural network using Adam BID7 ) with a fixed learning rate of 0.001, β of (0.9, 0.999) and of 1e-8. The learning rate was selected based on the validation performance during the preliminary experiments, and the other parameters are the default values. For MNIST Multi, the batch size was 64, and for COCO was 32.Feedforward Alternative While we use a recurrent model in the experiments, the multiset loss can be used with a feedforward model as follows. A key use of the recurrent hidden state is to retain the previously predicted labels, i.e. to remember the full conditioning setŷ 1,...,ŷ t−1 in p(y t |ŷ 1, ...,ŷ t−1). Therefore, the proposed loss can be used in a feedforward model by encodinĝ y 1,...,ŷ t−1 in the input x t, and running the feedforward model for |Ŷ| steps, where |Ŷ| is determined with a method from section 2.3. Note that compared to the recurrent model, this approach involves additional feature engineering.

We study the problem of multiset prediction and propose a novel multiset loss function, providing analysis and empirical evidence that demonstrates its effectiveness.

This work aims to provide quantitative answers to the relative importance of concepts of interest via concept activation vectors (CAV). In particular, this framework enables non-machine learning experts to express concepts of interest and test hypotheses using examples (e.g., a set of pictures that illustrate  the concept). We show that CAV can be learned given a relatively small set of examples. Hypothesis testing with CAV can answer whether a particular concept (e.g., gender) is more important in predicting a given class (e.g., doctor) than other sets of concepts. Interpreting networks with CAV does not require any retraining or modification of the network.