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Question: We present a neural rendering architecture that helps variational autoencoders (VAEs) learn disentangled representations. Instead of the deconvolutional network typically used in the decoder of VAEs, we tile (broadcast) the latent vector across space, concatenate fixed X- and Y-“coordinate” channels, and apply a fully convolutional network with 1x1 stride. This provides an architectural prior for dissociating positional from non-positional features in the latent space, yet without providing any explicit supervision to this effect. We show that this architecture, which we term the Spatial Broadcast decoder, improves disentangling, reconstruction accuracy, and generalization to held-out regions in data space. We show the Spatial Broadcast Decoder is complementary to state-of-the-art (SOTA) disentangling techniques and when incorporated improves their performance. Knowledge transfer and generalization are hallmarks of human intelligence. From grammatical generalization when learning a new language to visual generalization when interpreting a Picasso, humans have an extreme ability to recognize and apply learned patterns in new contexts. Current machine learning algorithms pale in contrast, suffering from overfitting, adversarial attacks, and domain specialization BID12 BID16. We believe that one fruitful approach to improve generalization in machine learning is to learn compositional representations in an unsupervised manner. A compositional representation consists of components that can be recombined, and such recombination underlies generalization. For example, consider a pink elephant. With a representation that composes color and object independently, imagining a pink elephant is trivial. However, a pink elephant may not be within the scope of a representation that mixes color and object. Compositionality comes in a variety of flavors, including feature compositionality (e.g. pink elephant), multi-object compositionality (e.g. elephant next to a penguin), and relational compositionality (e.g. the smallest elephant). In this work we focus on feature compositionality. Input: latents z ∈ R k, width w, height h Output: tiled latents z sb ∈ R h×w×(k+2)1: z b = TILE(z, (h, w, 1))2: x = LINSPACE(−1, 1, w) In the decoder, we broadcast (tile) a latent sample to the image width and height, and concatenate two "coordinate" channels. This is then fed to an unstrided convolutional decoder. (right) Pseudo-code of the spatial broadcast operation, assuming a numpy / Tensorflow-like API.Representations with feature compositionality are sometimes referred to as "disentangled" representations BID2. Learning disentangled representations from images has recently garnered much attention. However, even in the best understood conditions, finding hyperparameters to robustly obtain such representations still proves quite challenging BID15. Here we present the Spatial Broadcast decoder FIG0 ), a simple modification of the variational autoencoder (VAE) BID11 ) decoder architecture that:• Improves reconstruction accuracy and disentangling in a VAE on datasets of simple objects.• Is complementary to (and improves) state-of-the-art disentangling techniques.• Shows particularly significant benefits when the objects in the dataset are small, a regime notoriously difficult for standard VAE architectures (Appendix D).• Improves representational generalization to out-of-distribution test datasets involving both interpolation and extrapolation in latent space (Appendix H). When applying VAEs to image datasets, standard architecture decoders consist of an MLP followed by an upsampling deconvolutional network. However, with this architecture a VAE learns highly entangled representations in an effort to represent the data as closely as possibly to its Gaussian prior (e.g. FIG2). A number of new variations of the VAE objective have been developed to encourage disentanglement, though all of them introduce additional hyperparameters BID6 BID3 BID9 BID4. Furthermore, a recent study found them to be extremely sensitive to these hyperparameters BID15.Meanwhile, upsampling deconvolutional networks like the standard VAE decoder have been found to pose optimization challenges, such as checkerboard artifacts and spatial discontinuities BID14. These effects seem likely to raise problems for latent space representation-learning. Intuitively, asking a deconvolutional network to render an object at a particular position is a tall order: Since the network's filters have no explicit spatial information, the network must learn to propagate spatial asymmetries down from its highest layers and in from the edges of the layers. This is a complicated function to learn, so optimization is difficult. To remedy this, in the Spatial Broadcast decoder FIG0 ) we remove all upsampling deconvolutions from the network, instead tiling the latent vector across space, appending fixed coordinate channels, then applying an unstrided convolutional network. With this architecture, rendering an object at a position becomes a very simple function. Such simplicity of computation gives rise to ease of optimization. In addition to better disentangling, we find that the Spatial Broadcast VAE can yield better reconstructions (Figure 3), all with shallower networks and fewer parameters than a standard deconvolutional architecture. However, it is worth noting that a standard DeConv decoder may in principle more easily place patterns relative to each other or capture more extended spatial correlations. We did not find this to impact performance, even on datasets with extended sparial correlations and no variation of object position (Appendix F), but it is still a possible limitation of our model. The idea of appending coordinate channels to convolutional layers has recently been highlighted (and named CoordConv) in the context of improving positional generalization BID14. However,the CoordConv technique had been used beforehand (; BID13 ; ; ;) and its origin is unclear. While CoordConv VAE BID14 incorporates CoordConv layers into an upsampling deconvolutional network in a VAE (see Appendix E), to our knowledge no prior work has combined CoordConv with spatially tiling a generative model's representation as we do here. The Spatial Broadcast decoder was designed with object-feature representations in mind, so to initially showcase its performance we use a dataset of simple objects: colored 2-dimensional sprites BID3 BID17. This dataset has 8 factors of variation: X-position, Y-position, Size, Shape, Angle, and three-dimensional Color. In FIG1 we compare a standard DeConv VAE (a VAE with an MLP + deconvolutional network decoder) to a Spatial Broadcast VAE (a VAE with the Spatial Broadcast decoder). We see that Figure 3: Rate-distortion curves. We swept β log-linearly from 0.4 to 5.4 and for each value trained 10 replicas each of Deconv β-VAE (blue) and Spatial Broadcast β-VAE (orange) on colored sprites. The dots show the mean over these replicas for each β, and the shaded region shows the hull of one standard deviation. White dots indicate β = 1. (left) Reconstruction (Negative Log-Likelihood, NLL) vs KL. β < 1 yields low NLL and high KL (bottom-right of figure), whereas β > 1 yields high NLL and low KL (top-left of figure). See BID0 for details. Spatial Broadcast β-VAE shows a better rate-distortion curve than Deconv β-VAE.(right) Reconstruction vs MIG metric. β < 1 correspond to lower NLL and low MIG regions (bottom-left of figure), and β > 1 values correspond to high NLL and high MIG scores (towards top-right of figure). Spatial Broadcast β-VAE is better disentangled (higher MIG scores) than Deconv β-VAE.the Spatial Broadcast VAE outperforms the DeConv VAE both in terms of the Mutual Information Gap (MIG) disentangling metric BID4 and traversal visualizations, even though the hyperparameters for the models were chosen to minimize the model's error, not chosen for any disentangling properties explicitly (details in Appendix G).The Spatial Broadcast decoder is complementary to existing disentangling VAE techniques, hence improves not only a vanilla VAE but SOTA models as well. For example, Figure 3 shows that the Spatial Broadcast decoder improves disentangling and yields a lower rate-distortion curve in a β-VAE BID6, hence induces a more efficient representation of the data than a DeConv decoder. See Appendix C for similar in other SOTA models. Evaluating the quality of a representation can be challenging and time-consuming. While a number of metrics have been proposed to quantify disentangling, many of them have serious shortcomings and there is as yet no consensus in the literature which to use BID15 On the top row we show: (left) the data generative factor distribution uniform in X-and Y-position, (middle) a grid of points in generative factor space spanning the data distribution, and (right) 16 sample images from the dataset. The next two rows show a analysis of the DeConv VAE and Spatial Broadcast VAE on this dataset. In each we see a histogram of MIG metric values over 15 independent replicas and a latent space geometry visualization for the replica with the worst MIG and the replica with the best MIG (colored stars in the histograms). These geometry plots show the embedding of the ground truth factor grid in the latent subspace spanned by the two lowest-variance (most informative) latent components. Note that the MIG does not capture this contrast because it is very sensitive to rotation in the latent space (see Appendex H for more discussion about the MIG metric).which we found useful in our research: We plot in latent space the locations corresponding to a grid of points in generative factor space, thereby viewing the embedding of generative factor space in the model's latent space. While this is only shows the latent embedding of a 2-dimensional subspace of generative factor space, it can be very revealing of the latent space geometry. We showcase this analysis method in FIG2 on a dataset of circles varying in X-and Y-position. This shows that a Spatial Broadcast VAE's learned latent representation is a near-perfect Euclidean transformation of the data generative factors, in sharp contrast to a DeConv VAE's representation. This disentangling with the Spatial Broadcast decoder is robust even when the generative factors in the dataset are not independent (Appendix H). Here we present the Spatial Broadcast decoder for Variational Autoencoders. We demonstrate that it improves learned latent representations, most dramatically for datasets with objects varying in position. It also improves generalization in latent space and can be incorporated into SOTA models to boost their performance in terms of both disentangling and reconstruction accuracy. We believe that learning compositional representations is an important ingredient for flexibility and generalization in many contexts, from supervised learning to reinforcement learning, and the Spatial Broadcast decoder is one step towards robust compositional visual representation learning. For all VAE models we used a Bernoulli decoder distribution, parametrized by its logits. It is with respect to this distribution that the reconstruction error (negative log likelihood) was computed. This could accomodate our datasets, since they were normalized to have pixel values in. We also explored using a Gaussian distribution with fixed variance (for which the NLL is equivalent to scaled MSE), and found that this produces qualitatively similar and in fact improves stability. Hence while a Bernoulli distribution usually works, we suggest the reader wishing to experiment with these models starts with a Gaussian decoder distribution with mean parameterized by the decoder network output and variance constant at 0.3.In all networks we used ReLU activations, weights initialized by a truncated normal BID8, and biases initialized to zero. We use no other neural network tricks (no BatchNorm or dropout), and all models were trained with the Adam optimizer BID10. See below for learning rate details. For all VAE models except β-VAE (shown only in Figure 3), we use a standard VAE loss, namely with a KL term coefficient β = 1. For FactorVAE we also use β = 1, as in BID9.For the VAE, β-VAE, CoordConv VAE and ablation study we used the network parameters in Table 1. We note that, while the Spatial Broadcast decoder uses fewer parameters than the DeConv decoder, it does require about 50% more memory to store the weights. However, for the 3D Object-in-Room dataset we included three additional deconv layers in the Spatial Droadcast decoder (without these additional layers the decoder was not powerful enough to give good reconstructions on that dataset).All of these models were trained using a learning rate of 3 · 10 −4 on with batch size 16. All convolutional and deconvolutional layers have "same" padding, i.e. have zero-padded input so that the output shape is input_shape×stride in the case of convolution and input_shape/stride in the case of deconvolution. DISPLAYFORM0 BROADCAST DECODER Output Logits Conv(k=4, s=1, c=C) Conv(k=4, s=1, c=64) Conv(k=4, s=1, c=64) append coord channels tile(64 × 64 × 10) Input Vector Table 1: Network architectures for Vanilla VAE, β-VAE, CoordConv VAE and ablation study. The numbers of layers were selected to minimize the ELBO loss of a VAE on the colored sprites data (see Appendix G). Note that for 3D Object-in-Room, we include three additional convolutional layers in the spatial broadcast decoder. Here C refers to the number of channels of the input image, either 1 or 3 depending on whether the dataset has color. For the FactorVAE model, we used the hyperparameters described in the FactorVAE paper BID9. Those network parameters are reiterated in TAB3. Note that the Spatial Broadcast parameters are the same as for the other models in Table 1. For the optimization hyperparameters we used γ = 35, a learning rate of 10 −4 for the VAE updates, a learning rate of 2 · 10 −5 for the discriminator updates, and batch size 32. These parameters generally gave the best . However, when training the FactorVAE model on colored sprites we encountered instability during training. We subsequently did a number of hyperparameter sweeps attempting to improve stability, but to no avail. Ultimately, we used the hyperparameters in BID9, and the Spatial Broadcast decoder architecture is the same as for the other models (Table 1.) A.3 DATASETS All datasets were rendered in images of size 64 × 64 and normalized to. For this dataset, we use the binary dsprites dataset open-sourced in BID17, but multiplied by colors sampled in HSV space uniformly within the region H ∈ [0.0, DISPLAYFORM0 Sans color, there are 737,280 images in this dataset. However, we sample the colors online from a continuous distribution, effectively making the dataset size infinite. This dataset is open-sourced in BID1 . This dataset, unlike all others we use, has only a single channel in its images. It contains 86,366 images. This dataset was used extensively in the FactorVAE paper BID9 . It consists of an object in a room and has 6 factors of variation: Camera angle, object size, object shape, object color, wall color, and floor color. The colors are sampled uniformly from a continuous set of hues in the range [0.0, 0.9]. This dataset contains 480,000 images, procedurally generated as all combinations of 10 floor hues, 10 wall hues, 10 object hues, 8 object sizes, 4 object shapes, and 15 camera angles. To more thoroughly explore datasets with a variety of distributions, factors of variation, and held-out test sets we wrote our own procedural image generator for circular objects in PyGame (rendered with an anti-aliasing factor of 5). We used this to generate the data for in FIG2 and Appendix H. In these datasets we control subsets of the following factors of variation: X-position, Y-position, Size, Color. We generated five datasets in this way, which we call X-Y, X-H, R-G, X-Y-H Small, and X-Y-H Tiny, and can be seen in FIG0, FIG0, FIG0, FIG8, and FIG9 respectively. TAB5 shows the values of these factors for each dataset. Note that for some datasets we define the color distribution in RGB space, and for others we define it in HSV space. To create the datasets with dependent factors, we hold out one quarter of the dataset (the intersection of half of the ranges of each of the two factors), either centered within the data distribution or in the corner. For each dataset we generate 500,000 randomly sampled training images. The number of training steps for each model on each dataset can be found in TAB6. In general, for each dataset we used enough training steps so that all models converged. Note that while the training iterations is different for FactorVAE than for the other models on colored sprites (due to instability of FactorVAE), this has no bearing on our because we do not compare across models. We only compare across decoder architectures, and we always used the same training steps for both DeConv and Spatial Broadcast decoders within each model. One aspect of the Spatial Broadcast decoder is the concatenation of constant coordinate channels to its tiled input latent vector. While our justification of its performance emphasizes the simplicity of computation it affords, it may seem possible that the coordinate channels are only used to provide positional information and the simplicity of this positional information (linear meshgrid) is irrelevant. Here we perform an ablation study to demonstrate that this is not the case; the organization of the coordinate channels is important. For this experiment, we randomly permute the coordinate channels through space. Specifically, we take the [h × w × 2]-shape coordinate channels and randomly permute the h · w entries. We keep each (i, j) pair together to ensure that after the shuffling each location does still have a unique pair of coordinates in the coordinate channels. Importantly, we only shuffle the coordinate channels once, then keep them constant throughout training. As shown in Figure 3, the Spatial Broadcast decoder improves disentangling and the rate-distortion trade-off for β-VAE BID6, indicating it improves state-of-the-art models. To further support this indication, we introduce the Spatial Broadcast decoder into the recently developed FactorVAE BID9.Indeed, FIG7 shows the Spatial Broadcast decoder improves disentangling in FactorVAE. See Appendix H for further to this effect. In exploring datasets with objects varying in position, we often find a (standard) DeConv VAE learns a representation that is discontinuous with respect to object position. This effect is amplified as the size of the object decreases. This makes sense, because the pressure for a VAE to represent position continuously comes from the fact that an object and a position-perturbed version of itself overlap in pixel space (hence it is economical for the VAE to map noise in its latent samples to local translations of an object). However, as an object's size decreases, this pixel overlap decreases, hence the pressure for a VAE to represent position continuously weakens. In this small-object regime the Spatial Broadcast decoder's architectural bias towards representing positional variation continuously proves extremely useful. We see this in FIG8 and FIG9.We were surprised to see disentangling of such tiny objects in FIG9 and have not explored the lower object size limit for disentangling with the Spatial Broadcast decoder. CoordConv VAE BID14 has been proposed as a decoder architecture to improve the continuity of VAE representations. CoordConv VAE appends coordinate channels to every feature layer of the standard deconvolutional decoder, yet does not spatially tile the latent vector, hence retains upsampling deconvolutions. FIG10 shows analysis of this model on the colored sprites dataset. While the latent space does appear to be continuous with respect to object position, it is quite entangled (far more so than a Spatial Broadcast VAE). This is not very surprising, since CoordConv VAE uses upscale deconvolutions to go all they way from spatial shape 1 × 1 to spatial shape 64 × 64, while in Table 10 we see that introducing upscaling hurts disentangling in a Spatial Broadcast VAE. The sprites and circles datasets discussed in Section 3 seem particularly well-suited for the Spatial Broadcast decoder because X-and Y-position are factors of variation. However, we also evaluate the architecture on datasets that have no positional factors of variation: Chairs and 3D Object-in-Room datasets BID1 BID9. In the latter, the factors of variation are highly non-local, affecting multiple regions spanning nearly the entire image. We find that on both datasets a Spatial Broadcast VAE learns representations that look as well disentangled as SOTA methods on FIG1, we see that this achieves far lower scores than a Spatial Broadcast VAE, though about the same as (or maybe slightly better than) a DeConv VAE. (right) Latent space traversals are entangled. Note that in contrast to traversal plots in the main text, we show the effect of traversing all 10 latent components (sorted by smallest to largest mean variance), including the non-coding ones (in the bottom rows).these datasets and without any modification of the standard VAE objective BID9 BID6. See Figures 10 for . We attribute the good performance on these datasets to the Spatial Broadcast Decoder's use of a shallower network and no upsampling deconvolutions (which have been observed to cause optimization difficulties in a variety of settings BID14) ). BID1 and the 3D Object-in-Room dataset BID9. In order to remain objective when selecting model hyperparameters for the Spatial Broadcast and Deconv decoders, we chose hyperparameters based on minimizing the ELBO loss, not considering any information about disentangling. After finding reasonable encoder hyperparameters, we performed large-scale (25 replicas each) sweeps over a few decoder hyperparameters for both the DeConv and Spatial Broadcast decoder on the colored sprites dataset. These sweeps are revealing of hyperparameter sensitivity, so we report the following quantities for them:• ELBO. This is the evidence lower bound (total VAE loss). It is the sum of the negative log likelihood (NLL) and KL-divergence.• NLL. This is the negative log likelihood of an image with respect to the model's reconstructed distribution of that image. It is a measure of reconstruction accuracy.• KL. This is the KL divergence of the VAE's latent distribution with its Gaussian prior. It measures how much information is being encoded in the latent space.• Latents Used. This is the mean number of latent coordinates with standard deviation less than 0.5. Typically, a VAE will have some unused latent coordinates (with standard deviation near 1) and some used latent coordinates. The threshold 0.5 is arbitrary, but this quantity does provide a rough idea of how many factors of variation the model may be representing.• MIG. The MIG metric.• Factor VAE. This is the metric described in the FactorVAE paper BID9. We found this metric to be less consistent than the MIG metric (and equally flawed with respect to rotated coordinates), but it qualitatively agrees with the MIG metric most of the time. G.1 CONVNET DEPTH TAB8 shows of sweeping over ConvNet depth in the Spatial Broadcast decoder. This reveals a consistent trend: As the ConvNet deepens, the model moves towards lower rate/higher distortion. Consequently, latent space information and reconstruction accuracy drop. Traversals with deeper nets show the model dropping factors of variation (the dataset has 8 factors of variation). The Spatial Broadcast decoder as presented in this work is fully convolutional. It contains no MLP. However, motivated by the need for more depth on the 3D Object-in-Room dataset, we did explore applying an MLP to the input vector prior to the broadcast operation. We found that including this MLP had a qualitatively similar effect as increasing the number of convolutional layers on the colored sprited dataset, decreasing latent capacity and giving poorer reconstructions. These are shown in TAB10.However, on the 3D Object-in-Room dataset adding the MLP did improve the model when using ConvNet depth 3 (the same as for colored sprites). Results of a sweep over depth of a pre-broadcast MLP are shown in TAB12. As mentioned in Section F, we were able to achieve the same effect by instead increasing the ConvNet depth to 6, but for those interested in computational efficiency using a pre-broadcast MLP may be a better choice for datasets of this sort. In the DeConv decoder, increasing the MLP layers again has a broadly similar effect as increasing the ConvNet layers, as shown in TAB11. TAB10, except using the 3D Object-in-Room dataset. Here the model seems to over-representation the dataset generative factors (or which there are 6 for this dataset) without a pre-broadcast MLP (top row). However, adding a pre-broadcast MLP with 2 or 3 layers gives rise to accurate reconstructions with the appropriate number of used latents and good disentangling. Adding a pre-broadcast MLP like this is an alternative to increasing the ConvNet depth in the model (shown in FIG0). We acknowledge that there is a continuum of models between the Spatial Broadcast decoder and the Deconv decoder. One could interpolate from one to the other by incrementally replacing the convolutional layers in the Spatial Broadcast decoder's network by deconvolutional layers with stride 2 (and simultaneously decreasing the height and width of the tiling operation). Table 10 shows a few steps of such a progression, where (starting from the bottom) 1, 2, and all 3 of the convolutional layers in the Spatial Broadcast decoder are replaced by a deconvolutional layer with stride 2. We see that this hurts disentangling without affecting the other metrics, further evidence that upscaling deconvolutional layers are bad for representation learning. Table 10: Effect of upscale deconvolution on the Spatial Broadcast VAE's performance. These use the colored sprites dataset. The columns show the effect of repeatedly replacing the convolutional, stride-1 layers in the decoder by deconvolutional, stride-2 layers (starting at the bottom-most layer). This incrementally reduces performance without affecting the other statistics much, testament to the negative impact of upscaling deconvolutional layer on VAE representations. We showed visualization of latent space geometry on the circles datasets in FIG2. However, we also conducted the same style experiments on many more datasets and on FactorVAE models. In this section we will present these additional . We consider three generative factor pairs: (X-Position, Y-Position), (X-Position, Hue), and (Redness, Greenness). For each such pair we generate a dataset with independently sampled generative factors and two datasets with dependent generative factors (one with a hole in the center and another with a hole in the corner of generative factor space). For each dataset we run VAE and FactorVAE models with both DeConv and Spatial Broadcast decoder. Broadly, our in the following figures show that the Spatial Broadcast decoder nearly always helps disentangling. It helps most dramatically on the most positional variation (X-Position, Y-Position) and least significantly when there is no positional variation (Redness, Greenness).Note, however, that even with no position variation, the Spatial Broadcast decoder does seem to improve latent space geometry in the generalization experiments FIG0. We believe this may be due in part to the fact that the Spatial Broadcast decoder is shallower than the DeConv decoder. Finally, we explore one completely different dataset with dependent factors: A dataset where half the images have no object (are entirely black). This we do to simulate conditions like that in a multi-entity VAE such as BID18 when the dataset has a variable number of entities. These conditions pose a challenge for disentangling, because the VAE objective will wish to allocate a large (low-KL) region of latent space to representing a blank image when there is a large proportion of blank images in the dataset. However, we do see a stark improvement by using the Spatial Broadcast decoder in this case. The reader will note that in many of the figures in this section the MIG does not capture the intuitive notion of disentangling very well. We believe this is because:• It depends on a choice of basis for the ground truth factors, and heavily penalizes rotation of the representation with respect to this basis. Yet it is often unclear what the correct basis for the ground truth factors is (e.g. RGB vs HSV vs HSL). For example, see the bottom row of FIG2.• It is invariant to a folding of the representation space, as long as the folds align with the axes of variation. See the middle row of FIG2 for an example of a double-fold in the latent space which isn't penalized by the MIG metric. More broadly, while a number of metrics have been proposed to quantify disentangling, many of them have serious shortcomings and there is as yet no consensus in the literature which to use BID15. We believe it is impossible to quantify how good a representation is with a single scalar, because there is a fundamental trade-off between how much information a representation contains and how well-structured the representation is. This has been noted by others in the disentangling literature (; BID5). This disentangling-distortion trade-off is a recapitulation of the rate-distortion trade-off BID0 and can be seen first-hand in Figure 3. We would like representations that both reconstruct well and disentangle well, but exactly how to balance these two factors is a matter of subjective preference (and surely depends on the dataset). Any scalar disentangling metric will implicitly favor some arbitrary disentangling-reconstruction potential. Due to the subjective nature of disentangling and the difficulty in defining appropriate metrics, we put heavy emphasis on latent space visualization as a means for representational analysis. Latent space traversals have been extensively used in the literature and can be quite revealing BID6. However, in our experience, traversals suffer two shortcomings:• Some latent space entanglement can be difficult for the eye to perceive in traversals. For example, a slight change in brightness in a latent traversal that represents changing position can easily go unnoticed.• Traversals only represent the latent space geometry around one point in space, and crossreferencing corresponding traversals between multiple points is quite time-consuming. Consequently, we caution the reader against relying too heavily on traversals when evaluating latent space geometry. In many cases, we found the latent factor visualizations in this section to be much more informative of representational quality. FIG0: Latent space analysis for independent X-Y dataset. This is the same as in FIG2, except with the additional FactorVAE . We see that the Spatial Broadcast decoder improves FactorVAE as well as a VAE (and in the FactorVAE case seems to always be axis-aligned). Note that DeConv FactorVAE has a surprisingly messy latent space -we found that using a fixed-variance Normal (instead of Bernoulli) decoder distribution improved this significantly, though still not to the level of the Spatial Broadcast FactorVAE. We also noticed in small-scale experiments that including shape or size variability in the dataset helped FactorVAE disentangle as well. However, FactorVAE does seem to be generally quite sensitive to hyperparameters BID15, as adversarial models often are. FIG0: Latent space analysis for dependent X-Y dataset. Analogous to FIG0, except the dataset has a large held-out hole in generative factor space (see dataset distribution in top-left), hence the generative factors are not independent. For the latent geometry visualizations, we do evaluate the representation of images in this held-out hole, which are shown as black dots. This tests for generalization, namely extrapolation in pixel space and interpolation in generative factor space. Again, the Spatial Broadcast decoder dramatically improves the representation -its representation looks nearly linear with respect to the ground truth factors, even through the extrapolation region. FIG0: Latent space analysis for dependent X-Y dataset. This is similar to FIG0, except the "hole" in the dataset is in the corner of generative factor space rather than the middle. Hence this tests not only extrapolation in pixel space, but also extrapolation in generative factor space. As usual, the Spatial Broadcast decoder helps a lot, though in this case the extrapolation is naturally more difficult than in FIG0. FIG0: Latent space analysis for independent X-H dataset. In this dataset the circle varies in X-Position and Hue. As expected given this has less positional variation than the X-Y datasets, we see the relative improvement of the Spatial Broadcast decoder to be lower, though still quite significant. Interestingly, the representation with the Spatial Broadcast decoder is always axis-aligned and nearly linear in the positional direction, though non-linear in the hue direction. While this is not what the VAE objective is pressuring the model to do (the VAE objective would like to balance mean and variance in its latent space), we attribute this to the fact that a linear representation is much easier to compute from the coordinate channels with ReLU layers than a non-linear effect, particularly with only three convolutional ReLU layers. In a sense, the inductive bias of the architecture is overriding the inductive bias of the objective function. FIG0: Latent space analysis for dependent X-H dataset. This is the same as FIG0 except the dataset has a held-out "hole" in the middle, hence tests the model's generalization ability. This generalization is extrapolation in pixel space yet interpolation in generative factor space. This poses a serious challenge for the DeConv decoder, and again the Spatial Broadcast decoder helps a lot. Interestingly, note the severe contraction by FactorVAE of the "hole" in latent space. The independence pressure in FactorVAE strongly tries to eliminate unused regions of latent space. FIG0: Latent space analysis for dependent X-H dataset. This is the same as FIG0 except the held-out "hole" is in the corner of generative factor space, testing extrapolation in both pixel space and generative factor space. The Spatial Broadcast decoder again yields significant improvements, and as in FIG0 we see FactorVAE clearly sacrificing latent space geometry to remove the "hole" from the latent space prior distribution (see bottom row). FIG0: Latent space analysis for dependent R-G dataset. This is the same as FIG0 except with a held-out "hole" in the center of generative factor space, testing extrapolation in pixel space (interpolation in generative factor space). Here the benefit of the Spatial Broadcast decoder is more clear than in FIG0. We attribute it's benefit here in part to it being a shallower network. Under review as a workshop paper at ICLR 2019 FIG0: Latent space analysis for dependent R-G dataset. This is the same as FIG0 except the "hole" is in the corner of generative factor space, testing extrapolation in both pixel space and generative factor space. While the Spatial Broadcast decoder seems to give rise to slightly lower MIG scores, this appears to be from rotation in the latent space. FIG1: Latent space analysis for X-H dataset with blank images. This is the same as FIG0 except the dataset consists half of images with no objects (entirely black images, as can be seen in the dataset samples in the top-right). This simulates the data distribution produced by the VAE of a multi-entity VAE BID18 on a dataset with a variable number of objects. Again the Spatial Broadcast decoder improves latent space geometry, according to both the MIG metric and the traversals. In the latent geometry plots, the black star represents the encoding of a blank image. We noticed that the Spatial Broadcast VAE always allocates a third relevant latent to indicate the (binary) presence or absence of an object, a very natural representation of this dataset.

We introduce a neural rendering architecture that helps VAEs learn disentangled latent representations.

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We show that posterior collapse in linear VAEs is caused entirely by marginal log-likelihood (not ELBO). Experiments on deep VAEs suggest a similar phenomenon is at play.


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Question: Similar to humans and animals, deep artificial neural networks exhibit critical periods during which a temporary stimulus deficit can impair the development of a skill. The extent of the impairment depends on the onset and length of the deficit window, as in animal models, and on the size of the neural network. Deficits that do not affect low-level statistics, such as vertical flipping of the images, have no lasting effect on performance and can be overcome with further training. To better understand this phenomenon, we use the Fisher Information of the weights to measure the effective connectivity between layers of a network during training. Counterintuitively, information rises rapidly in the early phases of training, and then decreases, preventing redistribution of information resources in a phenomenon we refer to as a loss of "Information Plasticity". Our analysis suggests that the first few epochs are critical for the creation of strong connections that are optimal relative to the input data distribution. Once such strong connections are created, they do not appear to change during additional training. These findings suggest that the initial learning transient, under-scrutinized compared to asymptotic behavior, plays a key role in determining the outcome of the training process. Our findings, combined with recent theoretical in the literature, also suggest that forgetting (decrease of information in the weights) is critical to achieving invariance and disentanglement in representation learning. Finally, critical periods are not restricted to biological systems, but can emerge naturally in learning systems, whether biological or artificial, due to fundamental constrains arising from learning dynamics and information processing. Critical periods are time windows of early post-natal development during which sensory deficits can lead to permanent skill impairment BID12. Researchers have documented critical periods affecting a range of species and systems, from visual acuity in kittens BID35 BID33 to song learning in birds BID18. Uncorrected eye defects (e.g., strabismus, cataracts) during the critical period for visual development lead to amblyopia in one in fifty adults. The cause of critical periods is ascribed to the biochemical modulation of windows of neuronal plasticity BID10. In this paper, however, we show that deep neural networks (DNNs), while completely devoid of such regulations, respond to sensory deficits in ways similar to those observed in humans and animal models. This surprising suggests that critical periods may arise from information processing, rather than biochemical, phenomena. We propose using the information in the weights, measured by an efficient approximation of the Fisher Information, to study critical period phenomena in DNNs. We show that, counterintuitively, the information in the weights does not increase monotonically during training. Instead, a rapid growth in information ("memorization phase") is followed by a reduction of information ("reorganization" or "forgetting" phase), even as classification performance keeps increasing. This behavior is consistent across different tasks and network architectures. Critical periods are centered in the memorization phase. Under review as a conference paper at ICLR 2019 Performance is permanently impaired if the deficit is not corrected early enough, regardless of how much additional training is performed. As in animal models, critical periods coincide with the early learning phase during which test accuracy would rapidly increase in the absence of deficits (dashed). (B) For comparison, we report acuity for kittens monocularly deprived since birth and tested at the time of eye-opening (solid), and normal development visual acuity in kittens as a function of age (dashed) BID7 BID23.artificial neural networks (ANNs) are only loosely inspired by biological systems .Most studies to date have focused either on the behavior of networks at convergence (Representation Learning) or on the asymptotic properties of the numerical scheme used to get there (Optimization). The role of the initial transient, especially its effect in biasing the network towards "good" regions of the complex and high-dimensional optimization problem, is rarely addressed. To study this initial learning phase of ANNs, we replicate experiments performed in animal models and find that the responses to early deficits are remarkably similar, despite the large underlying differences between the two systems. In particular, we show that the quality of the solution depends only minimally on the final, relatively well-understood, phase of the training process or on its very first epochs; instead, it depends critically on the period prior to initial convergence. In animals, sensory deficits introduced during critical periods induce changes in the architecture of the corresponding areas BID4 BID34 BID9. To determine whether a similar phenomenon exists in ANNs, we compute the Fisher Information of the weights of the network as a proxy to measure its "effective connectivity", that is, the density of connections that are effectively used by the network in order to solve the task. Like others before us BID28, we observe two distinct phases during the training, first a "learning phase" in which the Fisher Information of the weights increases as the network learns from the data, followed by a "consolidation" or "compression" phase in which the Fisher Information decreases and stabilizes. Sensitivity to critical-period-inducing deficits is maximal exactly when the Fisher Information peaks. A layer-wise analysis of the network's effective connectivity shows that, in the tasks and deficits we consider, the hierarchy of low-level and high-level features in the training data is a key aspect behind the observed phenomena. In particular, our experiments suggest that the existence of critical periods in deep neural networks depends on the inability of the network to change its effective connectivity pattern in order to process different information (in response to deficit removal). We call this phenomenon, which is not mediated by any external factors, a loss of the "Information Plasticity" of the network. A notable example of critical period-inducing deficit, which also commonly affects humans, is amblyopia (reduced visual acuity in one eye) caused unilateral cataracts during infancy or childhood (; BID32 : Even after surgical correction of the cataracts, the Figure 2 : Sensitivity of learning phase: (C) Final test accuracy of a DNN as a function of the onset of a short 40-epoch deficit. The decrease in the final performance can be used to measure the sensitivity to deficits. The most sensitive epochs corresponds to the early rapid learning phase, before the test error (dashed line) begins to plateau. Afterwards, the network is largely unaffected by the temporary deficit. (D) This can be compared with changes in the degree of functional disconnection (normalized numbers of V1 monocular cells disconnected from the contralateral eye) as a function of the kittens' age at the onset of a 10-12-day deficit window BID26. Dashed lines are as in A and B respectively. ability of the patients to regain normal acuity in the affected eye depends both on the duration of the deficit and on its age of onset, with earlier and longer deficits causing more severe effects. In order to replicate this experimental setup in ANNs, we train a standard convolutional network (CNN) to classify objects in small 32 ⇥ 32 RGB images from the CIFAR-10 dataset BID19 ) in 10 classes. To simulate the effect of cataracts, for the first t 0 epochs the images in the dataset are downsampled to 8⇥8 and then upsampled back to 32⇥32 using bilinear interpolation, in practice blurring the image and destroying small-scale details.1 After that, the training continues for 300 more epochs, giving the network enough time to converge and ensuring it is exposed to the same number of uncorrupted images as in the control (t 0 = 0) experiment. In FIG0, we graph the final performance of the network (described in Materials and Methods) as a function of the epoch at which the deficit is corrected (t 0).We clearly observe the existence of a critical period for this deficit in the ANN: if the blur is not removed within the first 60 epochs, the final performance is severely decreased when compared to the baseline (from a test error of ⇠6.4%, [In this plot it is 8%, it is 6.4% for the resnet later. We can swap the plots or change the text] in the absence of a deficit, to more than 18% when the blur is present over 140 epochs, a ⇠300% increase). The profile of the curve is also strikingly similar to the one obtained in kittens monocularly deprived from near birth and whose visual acuity upon eye-opening was tested and plotted against the length of the deficit window BID23. Just like in humans and animal models (where critical periods are characteristic of early development), the critical period in the DNN also arises during the initial rapid learning phase. At this stage, the network is quickly learning a solution before the test error plateaus and the longer asymptotic convergence phase begins. Sensitivity to deficit. To quantify more accurately the sensitivity of the ANN to image blurring throughout its early learning phase, we introduced the deficit in a short constant window (40 epochs), starting at different epochs, and then measured the decrease in the ANN's final performance compared to the baseline. In Figure 2, we plot the final testing error of the network against the epoch of onset of the deficit. We observe that the network's sensitivity to blurring peaks in the central part of the early rapid learning phase (around 30 epochs), while later deficits produce little or no effect. A similar experiment was also performed on kittens by Olson and Freeman, using a window of 10-12 days during which the animals were monocularly deprived and using it to "scan" the first 4 months after birth to obtain a sensitivity profile BID26. We subsequently evaluated the effect of other training data modifications: a more drastic deprivation where the input is substituted with random noise, simulating complete sensory deprivation, and two "high-level" modifications of the training data: vertical flipping of the input image and permutation. Performance is permanently impaired if the deficit is not corrected early enough, regardless of how much additional training is performed. As in animal models, critical periods coincide with the early learning phase during which, in the absence of deficits, test accuracy would rapidly increase (dashed). (B) For comparison, we report acuity for kittens monocularly deprived since birth and tested at the time of eye-opening (solid), and normal visual acuity development (in kittens) as a function of their age (dashed) BID7 BID23. Sensitivity during learning: (C) Final test accuracy of a DNN as a function of the onset of a short 40-epoch deficit. The decrease in the final performance can be used to measure the sensitivity to deficits. The most sensitive epochs corresponds to the early rapid learning phase, before the test error (dashed line) begins to plateau. Afterwards, the network is largely unaffected by the temporary deficit. (D) This can be compared with changes in the degree of functional disconnection (normalized numbers of V1 monocular cells disconnected from the contralateral eye) as a function of the kittens' age at the onset of a 10-12-day deficit window BID26. Dashed lines are as in A and B respectively, up to a re-scaling of the y-axis. Our findings, described in Section 2, indicate that the early transient is critical in determining the final solution of the optimization associated with training an artificial neural network. In particular, the effects of sensory deficits during a critical period cannot be overcome, no matter how much additional training is performed. Yet most theoretical studies have focused on the network behavior after convergence (Representation Learning) or on the asymptotic properties of the optimization scheme used for training (SGD).To study this early phase, in Section 3, we use the Fisher Information to quantify the effective connectivity of a network during training, and introduce the notion of Information Plasticity in learning. Information Plasticity is maximal during the memorization phase, and decreases in the reorganization phase. We show that deficit sensitivity during critical periods correlates strongly with the effective connectivity. In Section 4 we discuss our contribution in relation to previous work. When considered in conjunction with recent on representation learning BID0, our findings indicate that forgetting (reducing information in the weights) is critical to achieving invariance to nuisance variability as well as independence of the components of the representation, but comes at the price of reduced adaptability later in the training. We also hypothesize that the loss of physical connectivity in biology (neural plasticity) could be a consequence, rather than a cause, of the loss of Information Plasticity, which depends on how the information is distributed throughout a network during the early stages of learning. These also shed light on the common practice of pre-training a model on a task and then fine-tune it for another, one of the most rudimentary forms of transfer learning. Our experiments show that, rather than helpful, pre-training can be detrimental, even if the tasks are similar (e.g., same labels, slightly blurred images). A notable example of critical period-related deficit, commonly affecting humans, is amblyopia (reduced visual acuity in one eye) caused by cataracts during infancy or childhood BID31 Published as a conference paper at ICLR 2019 DISPLAYFORM0 Figure 2: (Left) High-level perturbations do not induce a critical period. When the deficit only affects high-level features (vertical flip of the image) or the last layer of the CNN (label permutation), the network does not exhibit critical periods (test accuracy remains largely flat). On the other hand, a sensory deprivation-like deficit (image is replaced by random noise) does cause a deficit, but the effect is less severe than in the case of image blur. (Right) Dependence of the critical period profile on the network's depth. Adding more convolutional layers increases the effect of the deficit during its critical period (shown here is the decrease in test accuracy due to the deficit with respect to the test accuracy reached without deficits).von BID32. Even after surgical correction of cataracts, the ability of the patients to regain normal acuity in the affected eye depends both on the duration of the deficit and on its age of onset, with earlier and longer deficits causing more severe effects. In this section, we aim to study the effects of similar deficits in DNNs. To do so, we train a standard All-CNN architecture based on (see Appendix A) to classify objects in small 32 × 32 images from the CIFAR-10 dataset BID19 ). We train with SGD using an exponential annealing schedule for the learning rate. To simulate the effect of cataracts, for the first t 0 epochs the images in the dataset are downsampled to 8 × 8 and then upsampled back to 32 × 32 using bilinear interpolation, in practice blurring the image and destroying small-scale details. After that, the training continues for 160 more epochs, giving the network time to converge and ensuring it is exposed to the same number of uncorrupted images as in the control (t 0 = 0) experiment. In FIG0, we plot the final performance of a network affected by the deficit as a function of the epoch t 0 at which the deficit is corrected. We can readily observe the existence of a critical period: If the blur is not removed within the first 40-60 epochs, the final performance is severely decreased when compared to the baseline (up to a threefold increase in error). The decrease in performance follows trends commonly observed in animals, and may be qualitatively compared, for example, to the loss of visual acuity observed in kittens monocularly deprived from birth as a function of the length of the deficit BID23. 2 We can measure more accurately the sensitivity to a blur deficit during learning by introducing the deficit in a short window of constant length (40 epochs), starting at different epochs, and then measure the decrease in the DNN's final performance compared to the baseline FIG0 ). Doing this, we observe that the sensitivity to the deficit peaks in the central part of the early rapid learning phase (at around 30 epochs), while introducing the deficit later produces little or no effect. A similar experiment performed on kittens, using a window of 10-12 days during which the animals are monocularly deprived, again shows a remarkable similarity between the profiles of the sensitivity curves BID26. High-level deficits are not associated with a critical period: A natural question is whether any change in the input data distribution will have a corresponding critical period for learning. This is not the case for neuronal networks, which remain plastic enough to adapt to high-level changes in sensory processing BID4. For example, it is well-reported that even adult humans can rapidly adapt to certain drastic changes, such as the inversion of the visual field BID30 BID17. In Figure 2, we observe that DNNs are also largely unaffected by high-level deficits -such as vertical flipping of the image, or random permutation of the output labels: After deficit correction, the network quickly recovers its baseline performance. This hints at a finer interplay between the structure of the data distribution and the optimization algorithm, ing in the existence of a critical period. We now apply to the network a more drastic deficit, where each image is replaced by white noise. Figure 2 shows hows this extreme deficit exhibits a remarkably less severe effect than the one obtained by only blurring images: Training the network with white noise does not provide any information on the natural images, and in milder effects than those caused by a deficit (e.g., image blur), which instead conveys some information, but leads the network to (incorrectly) learn that no fine structure is present in the images. A similar effect has been observed in animals, where a period of early sensory deprivation (dark-rearing) can lengthen the critical period and thus cause less severe effects than those documented in light-reared animals . We refer the reader to Appendix C for a more detailed comparison between sensory deprivation and training on white noise. Architecture, depth, and learning rate annealing: FIG2 shows that a fully-connected network trained on the MNIST digit classification dataset also shows a critical period for the image blur deficit. Therefore, the convolutional structure is not necessary, nor is the use of natural images. Similarly, a ResNet-18 trained on CIFAR-10 also has a critical period, which is also remarkably sharper than the one found in a standard convolutional network FIG0 ). This is especially interesting, since ResNets allow for easier backpropagation of gradients to the lower layers, thus suggesting that the critical period is not caused by vanishing gradients. However, Figure 2 (Right) shows that the presence of a critical period does indeed depend critically on the depth of the network. In FIG2, we confirm that a critical period exists even when the network is trained with a constant learning rate, and therefore cannot be explained by an annealed learning rate in later epochs. Optimization method and weight decay: FIG2 (Bottom Right) shows that when using Adam as the optimization scheme, which renormalizes the gradients using a running mean of their first two moments, we still observe a critical period similar to that of standard SGD. However, changing the, showing two distinct phases of training: First, information sharply increases, but once test performance starts to plateau (green line), the information in the weights decreases during a "consolidation" phase. Eventually less information is stored, yet test accuracy improves slightly (green line). The weights' Fisher Information correlates strongly with the networks sensitivity to critical periods, computed as in FIG0 using both a window size of 40 and 60, and fitted here to the Fisher Information using a simple exponential fit. (Center) Recalling the connection between FIM ad connectivity, we may compare it to synaptic density during development in the visual cortex of macaques BID27. Here too, a rapid increase in connectivity is followed by elimination of synapses (pruning) continuing throughout life. (Right) Effects of critical period-inducing blurring on the Fisher Information: The impaired network uses more information to solve the task, compared to training in the absence of a deficit, since it is forced to memorize the labels case by case.hyperparameters of the optimization can change the shape of the critical period: In FIG2 (Bottom Left) we show that increasing weight decay makes critical periods longer and less sharp. This can be explained as it both slows the convergence of the network, and it limits the ability of higher layers to change to overcome the deficit, thus encouraging lower layers to also learn new features. We have established empirically that, in animals and DNNs alike, the initial phases of training are critical to the outcome of the training process. In animals, this strongly relates to changes in the brain architecture of the areas associated with the deficit BID4. This is inevitably different in artificial networks, since their connectivity is formally fixed at all times during training. However, not all the connections are equally useful to the network: Consider a network encoding the approximate posterior distribution p w (y|x), parameterized by the weights w, of the task variable y given an input image x. The dependency of the final output from a specific connection can be estimated by perturbing the corresponding weight and looking at the magnitude of the change in the final distribution. Specifically, given a perturbation w = w + δw of the weights, the discrepancy between the p w (y|x) and the perturbed network output p w (y|x) can be measured by their KullbackLeibler divergence, which, to second-order approximation, is given by: DISPLAYFORM0 where the expectation over x is computed using the empirical data distributionQ(x) given by the dataset, and DISPLAYFORM1 is the Fisher Information Matrix (FIM). The FIM can thus be considered a local metric measuring how much the perturbation of a single weight (or a combination of weights) affects the output of the network BID1. In particular, weights with low Fisher Information can be changed or "pruned" with little effect on the network's performance. This suggests that the Fisher Information can be used as a measure of the effective connectivity of a DNN, or, more generally, of the "synaptic strength" of a connection BID15. Finally, the FIM is also a semidefinite approximation of the Hessian of the loss function BID21 and hence of the curvature of the loss landscape at a particular point w during training, providing an elegant connection between the FIM and the optimization procedure BID1, which we will also employ later. Published as a conference paper at ICLR 2019Unfortunately, the full FIM is too large to compute. Rather, we use its trace to measure the global or layer-wise connection strength, which we can compute efficiently using (Appendix A): DISPLAYFORM2 In order to capture the behavior of the off-diagonal terms, we also tried computing the logdeterminant of the full matrix using the Kronecker-Factorized approximation of BID22, but we observed the same qualitative trend as the trace. Since the FIM is a local measure, it is very sensitive to the irregularities of the loss landscape. Therefore, in this section we mainly use ResNets, which have a relatively smooth landscape BID20. For other architectures we use instead a more robust estimator of the FIM based on the injection of noise in the weights BID0, also described in Appendix A.Two phases of learning: As its name suggests, the FIM can be thought as a measure of the quantity of information about the training data that is contained in the model BID6. Based on this, one would expect the overall strength of the connections to increase monotonically as we acquire information from experience. However, this is not the case: While during an initial phase the network acquires information about the data, which in a large increase in the strength of the connections, once the performance in the task begins to plateau, the network starts decreasing the overall strength of its connections. However, this does not correspond to a reduction in performance, rather, performance keeps slowly improving. This can be seen as a "forgetting, or "compression" phase, during which redundant connections are eliminated and non-relevant variability in the data is discarded. It is well-established how the elimination ("pruning") of unnecessary synapses is a fundamental process during learning and brain development BID27 ) (FIG3, Center); in FIG3 (Left) an analogous phenomenon is clearly and quantitatively shown for DNNs. Strikingly, these changes in the connection strength are closely related to the sensitivity to criticalperiod-inducing deficits such as image blur, computed using the "sliding window" method as in FIG0. In FIG3 we see that the sensitivity closely follows the trend of the FIM. This is remarkable since the FIM is a local quantity computed at a single point during the training of a network in the absence of deficit, while sensitivity during a critical period is computed, using test data, at the end of the impaired network training. FIG3 (Right) further emphasizes the effect of deficits on the FIM: in the presence of a deficit, the FIM grows and remains substantially higher even after the deficit is removed. This may be attributed to the fact that, when the data are so corrupted that classification is impossible, the network is forced to memorize the labels, therefore increasing the quantity of information needed to perform the same task. Layer-wise effects of deficits: A layer-wise analysis of the FIM sheds further light on how the deficit affects the network. When the network (in this case All-CNN, which has a clearer division among layers than ResNet) is trained without deficits, the most important connections are in the intermediate layers (FIG5, Left), which can process the input CIFAR-10 image at the most informative intermediate scale. However, if the network is initially trained on blurred data (FIG5, top right), the strength of the connections is dominated by the top layer (Layer 6). This is to be expected, since the low-level and mid-level structures of the images are destroyed, making the lower layers ineffective. However, if the deficit is removed early in the training (FIG5, top center), the network manages to "reorganize", reducing the information contained in the last layer, and, at the same time, increasing the information in the intermediate layers. We refer to these phenomena as changes in "Information Plasticity". If, however, the data change occurs after the consolidation phase, the network is unable to change its effective connectivity: The connection strength of each layer remains substantially constant. The network has lost its Information Plasticity and is past its critical period. The analysis of the FIM also sheds light on the geometry of the loss function and the learning dynamics. Since the FIM can be interpreted as the local curvature of the residual landscape, FIG3 shows that learning entails crossing bottlenecks: In the initial phase the network enters regions of high curvature (high Fisher Information), and once consolidation begins, the curvature decreases, allowing it to cross the bottleneck and enter the valley below. If the statistics change after crossing the bottleneck, the network is trapped. In this interpretation, the early phases of convergence are critical in leading the network towards the "right" final valley. The end of critical periods comes after the network has crossed all bottlenecks (and thus learned the features) and entered a wide valley (region of the weight space with low curvature, or low Fisher Information). Blur deficit until epoch 100 In the presence of an image blur deficit until epoch 100, more resources are allocated to the higher layers rather than to the middle layers. The blur deficit destroys low-and mid-level features processed by those layers, leaving only the global features of the image, which are processed by the higher layers. Even if the deficit is removed, the middle layers remain underdeveloped. (Top Center) When the deficit is removed at an earlier epoch, the layers can partially reconfigure (notice, e.g., the fast loss of information of layer 6), ing in less severe long-term consequences. We refer to the redistribution of information and the relative changes in effective connectivity as "Information Plasticity". (Bottom row) Same plots, but using a vertical flip deficit, which does not induce a critical period. As expected, the quantity of information in the layers is not affected. Critical periods have thus far been considered an exclusively biological phenomenon. At the same time, the analysis of DNNs has focused on asymptotic properties and neglected the initial transient behavior. To the best of our knowledge, we are the first to show that artificial neural networks exhibit critical period phenomena, and to highlight the critical role of the transient in determining the asymptotic performance of the network. Inspired by the role of synaptic connectivity in modulating critical periods, we introduce the use of Fisher Information to study this initial phase. We show that the initial sensitivity to deficits closely follows changes in the FIM, both global, as the network first rapidly increases and then decreases the amount of stored information, and layer-wise, as the network "reorganizes" its effective connectivity in order to optimally process information. Our work naturally relates to the extensive literature on critical periods in biology. Despite artificial networks being an extremely reductionist approximation of neuronal networks, they exhibit behaviors that are qualitatively similar to the critical periods observed in human and animal models. Our information analysis shows that the initial rapid memorization phase is followed by a loss of Information Plasticity which, counterintuitively, further improves the performance. On the other hand, when combined with the analysis of BID0 this suggests that a "forgetting" phase may be desirable, or even necessary, in order to learn robust, nuisance-invariant representations. The existence of two distinct phases of training has been observed and discussed by BID28, although their analysis builds on the (Shannon) information of the activations, rather than the (Fisher) information in the weights. On a multi-layer perceptron (MLP), BID28 empirically link the two phases to a sudden increase in the gradients' covariance. It may be tempting to compare these with our Fisher Information analysis. However, it must be noted that the FIM is computed using the gradients with respect to the model prediction, not to the ground truth label, leading to important qualitative differences. In Figure 6, we show that the covariance and norm of the gradients exhibit no clear trends during training with and without deficits, and, therefore, unlike the FIM, do not correlate with the sensitivity to critical periods. However, Published as a conference paper at ICLR 2019 a connection between our FIM analysis and the information in the activations can be established based on the work of BID0, which shows that the FIM of the weights can be used to bound the information in the activations. In fact, we may intuitively expect that pruning of connections naturally leads to loss of information in the corresponding activations. Thus, our analysis corroborates and expands on some of the claims of BID28, while using an independent framework. Aside from being more closely related to the deficit sensitivity during critical periods, Fisher's Information also has a number of technical advantages: Its diagonal is simple to estimate, even on modern state-of-the-art architectures and compelling datasets, and it is less sensitive to the choice estimator of mutual information, avoiding some of the common criticisms to the use of information quantities in the analysis of deep learning models. Finally, the FIM allows us to probe fine changes in the effective connectivity across the layers of the network FIG5 ), which are not visible in BID28.A complete analysis of the activations should account not only for the amount of information (both task-and nuisance-related), but also for its accessibility, e.g., how easily task-related information can be extracted by a linear classifier. Following a similar idea, BID24 aim to study the layer-wise, or "spatial" (but not temporal) evolution of the simplicity of the representation by performing a principal component analysis (PCA) of a radial basis function (RBF) kernel embedding of each layer representation. They show that, on a multi-layer perceptron, task-relevant information increasingly concentrate on the first principal components of the representation's embedding, implying that they become more easily "accessible" layer after layer, while nuisance information (when it is codified at all) is encoded in the remaining components. In our work we instead focus on the temporal evolution of the weights. However, it's important to notice that a network with simpler weights (as measured by the FIM) also requires a simpler smooth representation (as measured, e.g., by the RBF embedding) in order to operate properly, since it needs to be resistant to perturbations of the weights. Thus our analysis is wholly compatible with the intuitions of BID24. It would also be interesting to study the joint spatio-temporal evolution of the network using both frameworks at once. One advantage of focusing on the information of the weights rather than on the activations, or behavior of the network, is to have a readout of the "effective connectivity" during critical periods, which can be compared to similar readouts in animals. In fact, "behavioral" readouts upon deficit removal, both in artificial and neuronal networks, can potentially be confounded by deficit-coping changes at different levels of the visual pathways BID4 BID16. On the other hand, deficits in deprived animals are mirrored by abnormalities in the circuitry of the visual pathways, which we characterize in DNNs using the FIM to study its "effective connectivity", i.e., the connections that are actually employed by the network to solve the task. Sensitivity to critical periods and the trace of the Fisher Information peak at the same epochs, in accord with the evidence that skill development and critical periods in neuronal networks are modulated by changes (generally experience-dependent) in synaptic plasticity BID16 BID10. Our layer-wise analysis of the Fisher Information FIG5 ) also shows that visual deficits reinforce higher layers to the detriment of intermediate layers, leaving low-level layers virtually untouched. If the deficit is removed after the critical period ends, the network is not able to reverse these effects. Although the two systems are radically different, a similar response can be found in the visual pathways of animal models: Lower levels (e.g., retina, lateral geniculate nucleus) and higher-level visual areas (e.g., V2 and post-V2) show little remodeling upon deprivation, while most changes happen in different layers of V1 BID34 BID9 ).An insightful interpretation of critical periods in animal models was proposed by BID16: The initial connections of neuronal networks are unstable and easily modified (highly plastic), but as more "samples" are observed, they change and reach a more stable configuration which is difficult to modify. Learning can, however, still happen within the newly created connectivity pattern. This is largely compatible with our findings: Sensitivity to critical-period-inducing deficits peaks when connections are remodeled (Figure 4, Left), and different connectivity profiles are observed in networks trained with and without a deficit (FIG5). Moreover, high-level deficits such as imageflipping and label permutation, which do not require restructuring of the network's connections in order to be corrected, do not exhibit a critical period. Applying a deficit at the beginning of the training may be compared to the common practice of pretraining, which is generally found to improve the performance of the network. BID5 study the somewhat related, but now seldom used, practice of layer-wise unsupervised pre-training, and suggest that it may act as a regularizer by moving the weights of the network towards an area of the loss landscape closer to the attractors for good solutions, and that early examples have a stronger effect in steering the network towards particular solutions. Here, we have shown that pre-training on blurred data can have the opposite effect; i.e., it can severely decrease the final performance of the network. However, in our case, interpreting the deficits effect as moving the network close to a bad attractor is difficult to reconcile with the smooth transition observed in the critical periods, since the network would either converge to this attractor, and thus have low accuracy, or escape completely. Instead, we reconcile our experiments with the geometry of the loss function by introducing a different explanation based on the interpretation of the FIM as an approximation of the local curvature. FIG3 suggests that SGD encounters two different phases during the network training: At first, the network moves towards high-curvature regions of the loss landscape, while in the second phase the curvature decreases and the network eventually converges to a flat minimum (as observed in BID13). We can interpret these as the network crossing narrow bottlenecks during its training in order to learn useful features, before eventually entering a flat region of the loss surface once learning is completed and ending up trapped there. When combining this assumption with our deficit sensitivity analysis, we can hypothesize that the critical period occurs precisely upon crossing of this bottleneck. It is also worth noticing how there is evidence that convergence to flat minima (minima with low curvature) in a DNN correlates with a good generalization performance BID11 BID20 BID3 BID13. Indeed, using this interpretation, FIG3 (Right) tells us that networks more affected by the deficit converge to sharper minima. However, we have also found that the performance of the network is already mostly determined during the early "sensitive" phase. The final sharpness at convergence may therefore be an epiphenomenon, rather than the cause of good generalization. Our goal in this paper is not so much to investigate the human (or animal) brain through artificial networks, as to understand fundamental information processing phenomena, both in their biological or artificial implementations. It is also not our goal to suggest that, since they both exhibit critical periods, DNNs are necessarily a valid model of neurobiological information processing, although recent work has emphasized this aspect. We engage in an "Artificial Neuroscience" exercise in part to address a technological need to develop "explainable" artificial intelligence systems whose behavior can be understood and predicted. While traditionally well-understood mathematical models were used by neuroscientists to study biological phenomena, information processing in modern artificial networks is often just as poorly understood as in biology, so we chose to exploit well-known biological phenomena as probes to study information processing in artificial networks. Conversely, it would also be interesting to explore ways to test whether biological networks prune connections as a consequences of a loss of Information Plasticity, rather than as a cause. The mechanisms underlying network reconfiguration during learning and development might be an evolutionary outcome obtained under the pressure of fundamental information processing phenomena. DISPLAYFORM0 In order to avoid interferences between the annealing scheme and the architecture, in these experiments we fix the learning rate to 0.001.The Fully Connected network used for the MNIST experiments has hidden layers of size. All hidden layers use batch normalization followed by ReLU activations. We fix the learning rate to 0.005. Weight decay is not used. We use data augmentation with random translations up to 4 pixels and random horizontal flipping. For MNIST, we pad the images with zeros to bring them to size 32 × 32. To compute the trace of the Fisher Information Matrix, we use the following expression derived directly from the definition: DISPLAYFORM0 where the input image x is sampled from the dataset, while the label y is sampled from the output posterior. Expectations are approximated by Monte-Carlo sampling. Notice, however, that this expression depends only on the local gradients of the loss with respect to the weights at a point w = w 0, so it can be noisy when the loss landscape is highly irregular. This is not a problem for ResNets BID20, but for other architectures we use instead a different technique, proposed in BID0. More in detail, let L(w) be the standard cross-entropy loss. Given the current weights w 0 of the network, we find the diagonal matrix Σ that minimizes: DISPLAYFORM1 where β is a parameter that controls the smoothness of the approximation. Notice that L can be minimized efficiently using the method in BID14. To see how this relates to the Fisher Information Matrix, assume that L(w) can be approximated locally in DISPLAYFORM2 Taking the derivative with respect to Σ, and setting it to zero, we obtain Σ ii = β/H ii. We can then use Σ to estimate the trace of the Hessian, and hence of the Fisher information. Published as a conference paper at ICLR 2019 DISPLAYFORM3 Fitting of sensitivity curves and synaptic density profiles from the literature was performed using: DISPLAYFORM4 as the fitting equation, where t is the age at the time of sampling and τ 1, τ 2, k and d are unconstrained parameters BID2.The exponential fit of the sensitivity to the Fisher Information trace uses the expression DISPLAYFORM5 where a, b and c are unconstrained parameters, F (t) is the Fisher Information trace at epoch t of the training of a network without deficits and S k is the sensitivity computed using a window of size k. That is, S k (t) is the increase in the final test error over a baseline when the network is trained in the presence of a deficit between epochs t and t + k. B ADDITIONAL PLOTS Figure 6: Log of the norm of the gradient means (solid line) and standard deviation (dashed line) during training when: (Left) No deficit is present, (Center) A blur deficit is present until epoch 70, and (Right) a deficit is present until the last epoch. Notice that the presence of a deficit does not decrease the magnitude of the gradients propagated to the first layers during the last epochs, rather it seems to increase it, suggesting that vanishing gradients are not the cause of the critical period for the blurring deficit. Noise deficit until epoch 100 Figure 7: Same plot as in FIG5, but for a noise deficit. Unlike with blur, much more resources are allocated to the lower-layers rather than higher-layers. This may explain why it is easier for the network to reconfigure to solve the task after the deficit is removed. No deficit Blurred up to epoch 100 Figure 8: Visualization of the filters of the first layer of the network used for the experiment in FIG0. In absence of a deficit, the network learns high-frequency filters, as seen by the fact that many filters are not smooth (first picture). However, when a blurring deficit is present, the network learns only smooth filters corresponding to low-frequencies of the input (third picture). If the deficit is removed after the end of the critical period, the network does not manage to learn high-frequency filters (second picture). Critical periods are task-and deficit-specific. The specific task we address is visual acuity, but the performance is necessarily measured through different mechanisms in animals and Artificial Neural Networks. In animals, visual acuity is traditionally measured by testing the ability to discriminate between black-and-white contrast gratings (with varying spatial frequency) and a uniform gray field. The outcome of such tests generally correlates well with the ability of the animal to use the eye to solve other visual tasks relying on acuity. Convolutional Neural Networks, on the other hand, have a very different sensory processing mechanism (based on heavily quantized data), which may trivialize such a test. Rather, we directly measure the performance of the network on an high-level task, specifically image classification, for which CNNs are optimized. We chose to simulate cataracts in our DNN experiments, a deficit which allows us to explore its complex interactions with the structure of the data and the architecture of the network. Unfortunately, while the overall trends of cataract-induced critical periods have been studied and understood in animal models, there is not enough data to confidently regress sensibility curves comparable to those obtained in DNNs. For this reason, in FIG0 we compare the performance loss in a DNN trained in the presence of a cataract-like deficit with the obtained from monocularly deprived kittens, which exhibit similar trends and are one of the most common experimental paradigms in the visual neurosciences. Simulating complete visual deprivation in a neural network is not as simple as feeding a constant stimulus: a network presented with a constant blank input will rapidly become trivial and thus unable to train on new data. This is to be expected, since a blank input is a perfectly predictable stimulus and thus the network can quickly learn the (trivial) solution to the task. We instead wanted to model an uninformative stimulus, akin to noise. Moreover, even when the eyes are sutured or maintained in the darkness, there will be excitation of photoreceptors that is best modeled as noise. To account for this, we simulate sensory deprivation by replacing the input images with a dataset composed of (uninformative) random Gaussian noise. This way the network is trained on solving the highly non-trivial task of memorizing the association between the finitely-many noise patterns and their corresponding labels.

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Sensory deficits in early training phases can lead to irreversible performance loss in both artificial and neuronal networks, suggesting information phenomena as the common cause, and point to the importance of the initial transient and forgetting.

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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.