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Question: In this paper, we propose Continuous Graph Flow, a generative continuous flow based method that aims to model complex distributions of graph-structured data. Once learned, the model can be applied to an arbitrary graph, defining a probability density over the random variables represented by the graph. It is formulated as an ordinary differential equation system with shared and reusable functions that operate over the graphs. This leads to a new type of neural graph message passing scheme that performs continuous message passing over time. This class of models offers several advantages: a flexible representation that can generalize to variable data dimensions; ability to model dependencies in complex data distributions; reversible and memory-efficient; and exact and efficient computation of the likelihood of the data. We demonstrate the effectiveness of our model on a diverse set of generation tasks across different domains: graph generation, image puzzle generation, and layout generation from scene graphs. Our proposed model achieves significantly better performance compared to state-of-the-art models. Modeling and generating graph-structured data has important applications in various scientific fields such as building knowledge graphs , inventing new molecular structures and generating diverse images from scene graphs . Being able to train expressive graph generative models is an integral part of AI research. Significant research effort has been devoted in this direction. Traditional graph generative methods (Erdős & Rényi, 1959; ; Albert & Barabási, 2002;) are based on rigid structural assumptions and lack the capability to learn from observed data. Modern deep learning frameworks within the variational autoencoder (VAE) formalism offer promise of learning distributions from data. Specifially, for structured data, research efforts have focused on bestowing VAE based generative models with the ability to learn structured latent space models (; ;). Nevertheless, their capacity is still limited mainly because of the assumptions placed on the form of distributions. Another class of graph generative models are based on autoregressive methods . These models construct graph nodes sequentially wherein each iteration involves generation of edges connecting a generated node in that iteration with the previously generated set of nodes. Such autoregressive models have been proven to be the most successful so far. However, due to the sequential nature of the generation process, the generation suffers from the inability to maintain long-term dependencies in larger graphs. Therefore, existing methods for graph generation are yet to realize the full potential of their generative power, particularly, the ability to model complex distributions with the flexibility to address variable data dimensions. Alternatively, for modeling the relational structure in data, graph neural networks (GNNs) or message passing neural networks (MPNNs) (; ; ; ; ;) have been shown to be effective in learning generalizable representations over variable input data dimensions. These models operate on the underlying principle of iterative neural message passing wherein the node representations are updated iteratively for a fixed number of steps. Hereafter, we use the term message passing to refer to this neural message passing in GNNs. We leverage this representational ability towards graph generation. In this paper, we introduce a new class of models -Continuous Graph Flow (CGF): a graph generative model based on continuous normalizing flows ) that Figure 1: Illustration of evolution of message passing mechanisms from discrete updates (a) to our proposed continuous updates (b). Continuous Graph Flow leverages normalizing flows to transform simple distributions (e.g. Gaussian) at t 0 to the target distributions at t 1. The distribution of only one graph node is shown here for visualization, but, all the node distributions transform over time. generalizes the message passing mechanism in GNNs to continuous time. Specifically, to model continuous time dynamics of the graph variables, we adopt a neural ordinary different equation (ODE) formulation. Our CGF model has both the flexibility to handle variable data dimensions (by using GNNs) and the ability to model arbitrarily complex data distributions due to free-form model architectures enabled by the neural ODE formulation. Inherently, the ODE formulation also imbues the model with following properties: reversibility and exact likelihood computation. Concurrent work on Graph Normalizing Flows (GNF) also proposes a reversible graph neural network using normalizing flows. However, their model requires a fixed number of transformations. In contrast, while our proposed CGF is also reversible and memory efficient, the underlying flow model relies on continuous message passing scheme. Moreover, the message passing in GNF involves partitioning of data dimensions into two halves and employs coupling layers to couple them back. This leads to several constraints on function forms and model architectures that have a significant impact on performance . In contrast, our CGF model has unconstrained (free-form) Jacobians, enabling it to learn more expressive transformations. Moreover, other similar work is also based on normalizing flows as compared to CGF that models continuous time dynamics. We demonstrate the effectiveness of our CGF-based models on three diverse tasks: graph generation, image puzzle generation, and layout generation based on scene graphs. Experimental show that our proposed model achieves significantly better performance than state-of-the-art models. Graph neural networks. Relational networks such as Graph Neural Networks (GNNs) facilitate learning of non-linear interactions using neural networks. In every layer ∈ N of a GNN, the embedding h i corresponding to a graph node accumulates information from its neighbors of the previous layer recursively as described below. where the function g is an aggregator function, S(i) is the set of neighbour nodes of node i, and f ij is the message function from node j to node i, x i (−1) and x j (−1) represent the node features corresponding to node i and j at layer (− 1) respectively. Our model uses a restricted form of GNNs where embeddings of the graph nodes are updated in-place (x i ← h i), thus, we denote graph node as x and ignore h hereafter. These in-place updates allow using x i in the flow-based models while maintaining the same dimensionality across subsequent transformations. Normalizing flows and change of variables. Flow-based models enable construction of complex distributions from simple distributions (e.g. Gaussian) through a sequence of invertible mappings . For instance, a random variable z is transformed from an initial state density z 0 to the final state z K using a chain of K invertible functions f k described as: The computation of log-likelihood of a random variable uses change of variables rule formulated as: where ∂f k (z k−1)/∂z k is the Jacobian of f k for k ∈ {1, 2, ..., K}. Continuous normalizing flows. Continuous normalizing flows (CNFs) ) model the continuous-time dynamics by pushing the limit on number of transformations. Given a random variable z, the following ordinary differential equation (ODE) defines the change in the state of the variable. extended the change of variables rule described in Eq. 3 to continuous version. The dynamics of the log-likelihood of a random variable is then defined as the following ODE. Following the above equation, the log likelihood of the variable z at time t 1 starting from time t 0 is where the trace computation is more computationally efficient than computation of the Jacobian in Equation. Building on CNFs, we present continuous graph flow which effectively models continuous time dynamics over graph-structured data. Given a set of random variables X containing n related variables, the goal is to learn the joint distribution p(X) of the set of variables X. Each element of set X is x i ∈ R m where i = 1, 2..., n and m represents the number of dimensions of the variable. For continuous time dynamics of the set of variables X, we formulate an ordinary differential equation system as follows:... whereẋ i = dx i /dt and X(t) is the set of variables at time t. The random variable x i at time t 0 follows a base distribution that can have simple forms, e.g. Gaussian distributions. The function f i implicitly defines the interaction among the variables. Following this formulation, the transformation of the individual graph variable is defined as This provides transformation of the value of the variable x i from time t 0 to time t 1. The form in Eq. 8 represents a generic multi-variate update where interaction functions are defined over all the variables in the set X. However, the functions do not take into account the relational structure between the graph variables. To address this, we define a neural message passing process that operates over a graph by defining the update functions over variables according to the graph structure. This process begins from time t 0 where each variable x i (t 0) only contain local information. At time t, these variables are updated based on the information gathered from other neighboring variables. For such updates, the function f i in Eq. 8 is defined as: wheref ij (·) is a reusable message function and used for passing information between variables x i and x j, S(i) is the set of neighboring variables that interacts with variable x i, and g(·) is a function that aggregates the information passed to a variable. The above formulation describes the case of pairwise message functions, though this can be generalized to higher-order interactions. We formulate it as a continuous process which eliminates the requirement of having a predetermined number of steps of message passing. By further pushing the message passing process to update at infinitesimally smaller steps and continuing the updates for an arbitrarily large number of steps, the update associated with each variable can be represented using shared and reusable functions as the following ordinary differential equation (ODE) system.... Performing message passing to derive final states is equivalent to solving an initial value problem for an ODE system. Following the ODE formulation, the final states of variables can be computed as follows. This formulation can be solved with an ODE solver. Continuous graph flow leverages the continuous message passing mechanism (described in Sec. 3.1) and formulates the message passing as implicit density transformations of the variables (illustrated in Figure 1). Given a set of variables X with dependencies among them, the goal is to learn a model that captures the distribution from which the data were sampled. Assume the joint distribution p(X) at time t 0 has a simple form such as independent Gaussian distribution for each variable x i (t 0). The continuous message passing process allows the transformation of the set of variables from X(t 0) to X(t 1). Moreover, this process also converts the distributions over variables from simple base distributions to complex data distributions. Building on the independent variable continuous time dynamics described in Eq. 5, we define the dynamics corresponding to related graph variables as: where F represents a set of reusable functions incorporating aggregated messages. Therefore, the joint distribution of set of variables X can be described as: Here we use two types of density transformations for message passing: generic message transformations -transformations with generic update functions where trace in Eq. 13 can be approximated instead of computing by brute force method, and multi-scale message transformationstransformations with generic update functions at multiple scales of information. Generic message transformations. The trace of Jacobian matrix in Eq. 13 is modeled using a generic neural network function. The likelihood is defined as: where F denotes a neural network for message functions, and is a noise vector and usually can be sampled from standard Gaussian or Rademacher distributions. Multi-scale message transformations. As a generalization of generic message transformations, we design a model with multi-scale message passing to encode different levels of information in the variables. Similar to , we construct our multi-scale CGF model by stacking several blocks wherein each flow block performs message passing based on generic message transformations. After passing the input through a block, we factor out a portion of the output and feed it as input to the subsequent block. The likelihood is defined as: where b = 1, 2,..., (B − 1) with B as the total number of blocks in the design of the multi-scale architecture. Assume at time t b (t 0 < t b < t 1), X(t b) is factored out into two. We use one of these (denoted asX(t b)) as the input to the (b + 1) th block. LetX(t b) be the input to the next block, the density transformation is formulated as: To demonstrate the generality and effectiveness of our Continuous Graph Flow (CGF), we evaluate our model on three diverse tasks: graph generation, image puzzle generation, and layout generation based on scene graphs. Graph generation requires the model to learn complex distributions over the graph structure. Image puzzle generation requires the model to learn local and global correlations in the puzzle pieces. Layout generation has a diverse set of possible nodes and edges. These tasks have high complexity in the distributions of graph variables and diverse potential function types. Together these tasks pose a challenging evaluation for our proposed method. Datasets and Baselines. We evaluate our model on graph generation on two benchmark datasets EGO-SMALL and COMMUNITY-SMALL against four strong state-of-the-art baselines: VAE-based method , autoregressive graph generative model GraphRNN and DeepGMG, and Graph normalizing flows . Evaluation. We conduct a quantitative evaluation of the generated graphs using Maximum Mean Discrepancy (MMD) measures proposed in GRAPHRNN . The MMD evaluation in GRAPHRNN was performed using a test set of N ground truth graphs, computing their distribution over the nodes, and then searching for a set of N generated graphs from a larger set of samples generated from the model that best matches this distribution. As mentioned by , this evaluation process would likely have high variance as the graphs are very small. Therefore, we also performed an evaluation by generating 1024 graphs for each model and computing the MMD distance between this generated set of graphs and the ground truth test set. Baseline are from. Implementation details refer to Appendix A. Results and Analysis. Table 1 shows the in terms of MMD. Our CGF outperforms the baselines by a significant margin and also the concurrent work GNF. We believe our CGF outperforms GNF because it employs free-flow functions forms unlike GNF that has some contraints necessitated by the coupling layers. Fig. 2 visualizes the graphs generated by our model. Our model can capture the characteristics of datasets and generate diverse graphs that are not seen during the training. For additional visualizations and comparisons, refer to the Appendix A. Task description. We design image puzzles for image datasets to test model's ability on fitting very complex node distributions in graphs. Given an image of size W × W, we design a puzzle by dividing the original image into non-overlapping unique patches. A puzzle patch is of size w × w, in which w represents the width of the puzzle. Each image is divided into p = W/w puzzle patches both horizontally and vertically, and therefore we obtain P = p × p patches in total. Each patch corresponds to a node in the graph. To evaluate the performance of our model on dynamic graph sizes, instead of training the model with all nodes, we samplep adjacent patches wherep is uniformly sampled from {1, . . ., P} as input to the model during training and test. In our experiments, we use Figure 2: Visualization of generated graphs from our model. Our model can capture the characteristic of datasets and generate diverse graphs not appearing in the training set. patch size w = 16, p ∈ {2, 3, 4} and edge function for each direction (left, right, up, down) within a neighbourhood of a node. Additional details are in Appendix A. Datasets and baselines. We design the image puzzle generation task for three datasets: MNIST , CIFAR10 , and CelebA. CelebA dataset does not have a validation set, thus, we split the original dataset into a training set of 27,000 images and test set of 3,000 images as in . We compare our model with six state-of-the-art VAE based models: StructuredVAE , Graphite , Variational message passing using structured inference networks (VMP-SIN) , BiLSTM + VAE: a bidirectional LSTM used to model the interaction between node latent variables (obtained after serializing the graph) in an autoregressive manner similar to , Variational graph autoencoder (GAE) , and Neural relational inference (NRI) : we adapt this to model data for single time instance and model interactions between the nodes. Results and analysis. We report the negative log likelihood (NLL) in bits/dimension (lower is better). The in Table 2 indicate that CGF significantly outperforms the baselines. In addition to the quantitative , we also conduct sampling based evaluation and perform two types of generation experiments: Unconditional Generation: Given a puzzle size p, p 2 puzzle patches are generated using a vector z sampled from Gaussian distribution (refer Fig. 3(a) ); and Conditional Generation: Given p 1 patches from an image puzzle having p 2 patches, we generate the remaining (p 2 − p 1) patches of the puzzle using our model (see Fig. 3(b) ). We believe the task of conditional generation is easier than unconditional generation as there is more relevant information in the input during flow based transformations. For unconditional generation, samples from a base distribution (e.g. Gaussian) are transformed into learnt data distribution using the CGF model. For conditional generation, we map x a ∈ X a where X a ⊂ X to the points in base distribution to obtain z a and (a) Unconditional Generation (b) Conditional Generation Figure 3: Qualitative for image puzzle generation. Samples generated using our model for 2x2 MNIST puzzles (above horizontal line) and 3x3 CelebA-HQ puzzles (below horizontal line) in (a) unconditional generation and (b) conditional generation settings. For setting (b), generated patches (highlighted in green boxes) are conditioned on the remaining patches (from ground truth). subsequently concatenate the samples from Gaussian distribution to z a to obtain z that match the dimensions of desired graph and generate samples by transforming from z to x ∈ X using the trained graph flow. Task description and evaluation metrics. Layout generation from scene graphs is a crucial task in computer vision and bridges the gap between the symbolic graph-based scene description and the object layouts in the scene (; ;). Scene graphs represent scenes as directed graphs, where nodes are objects and edges give relationships between objects. Object layouts are described by the set of corresponding bounding box annotations . Our model uses scene graph as inputs (nodes correspond to objects and edges represent relations). An edge function is defined for each relationship type. The output contains a set of object bounding boxes described by, where x i, y i are the top-left coordinates, and w i, h i are the bounding box width and height respectively. We use negative log likelihood per node (lower is better) for evaluating models on scene layout generation. Datasets and baselines. Two large-scale challenging datasets are used to evaluate the proposed model: Visual Genome and COCO-Stuff datasets. Visual Genome contains 175 object and 45 relation types. The training, validation and test set contain 62565, 5506 and 5088 images respectively. COCO-Stuff dataset contains 24972 train, 1024 validation, and 2048 test scene graphs. We use the same baselines as in Sec. 4.2. Results and analysis. We show quantitative in Table 3 against several state-of-the-art baselines. Our CGF model significantly outperforms these baselines in terms of negative log likelihood. Moreover, we show some qualitative in Fig. 4. Our model can learn the correct relations defined in scene graphs for both conditional and unconditional generation, Furthermore, our model is capable to learn one-to-many mappings and generate diverse of layouts for the same scene graph. Table 3: Quantitative for layout generation for scene graph in negative log-likelihood. These are for unconditional generation using CGF with generic message transformations. Method Visual Genome COCO-Stuff BiLSTM + VAE -1.20 -1.60 StructuredVAE -1.05 -1.36 Graphite -1.17 -0.93 VMP-SIN -0.61 -0.85 GAE -1.85 -1.92 NRI -0.76 -0.91 CGF -4.24 -6.21 To test the generalizability of our model to variable graph sizes, we design three different evaluation settings and test it on image puzzle task: odd to even: training with graphs having odd graph sizes and testing on graphs with even numbers of nodes, less to more: training on graphs with smaller sizes and testing on graphs with larger sizes, and more to less: training on graphs with larger sizes and testing on graphs with smaller. In the less to more setting, we test the model's ability to use the functions learned from small graphs on more complicated ones, whereas the more to less setting evaluates the model's ability to learn disentangled functions without explicitly seeing them during training. In our experiments, for the less to more setting, we use sizes less than G/2 for training and more than G/2 for testing where G is the size of the full graph. Similarly, for the less to more setting, we use sizes less than G/2 for training and more than G/2 for testing. Table 4 reports the NLL for these settings. The NLL of these models are close to the performance on the models trained on full dataset demonstrating that our model is able to generalize to unseen graph sizes. In this paper, we presented continuous graph flow, a generative model that generalizes the neural message passing in graphs to continuous time. We formulated the model as an neural ordinary differential equation system with shared and reusable functions that operate over the graph structure. We conducted evaluation for a diverse set of generation tasks across different domains: graph generation, image puzzle generation, and layout generation for scene graph. Experimental showed that continuous graph flow achieves significant performance improvement over various of state-ofthe-art baselines. For future work, we will focus on generation tasks for large-scale graphs which is promising as our model is reversible and memory-efficient. We provide supplementary materials to support the contents of the main paper. In this part, we describe implementation details of our model. We also provide additional qualitative for the generation tasks: Graph generation, image puzzle generation and layout generation from scene graphs. The ODE formulation for continuous graph flow (CGF) model was solved using ODE solver provided by NeuralODE . In this section, we provide specific details of the configuration of our CGF model used in our experiments on two different generation tasks used for evaluation in the paper. Graph Generation. For each graph, we firstly generate its line graph with edges switched to nodes and nodes switched to edges. Then the graph generation problem is now generating the current nodes values which represents the adjacency matrix in the original graph. Each node value is binary (0 or 1) and is dequantized to continuous values through variational dequantization, with a global learnable Gaussian distribution as variational distribution. For our architecture, we use two blocks of continuous graph flow with two fully connected layers in Community-small dataset, and one block of continuous graph flow with one fully connected layer in Citeseer-small dataset. The hidden dimensions are all 32. Image puzzle generation. Each graph for this task comprise nodes corresponding to the puzzle pieces. The pieces that share an edge in the puzzle grid are considered to be connected and an edge function is defined over those connections. In our experiments, each node is transformed to an embedding of size 64 using convolutional layer. The graph message passing is performed over these node embeddings. The image puzzle generation model is designed using a multi-scale continuous graph flow architecture. We use two levels of downscaling in our model each of which factors out the channel dimension of the random variable by 2. We have two blocks of continuous graph flow before each downscaling wth four convolutional message passing blocks in each of them. Each message passing block has a unary message passing function and binary passing functions based on the edge types -all containing hidden dimensions of 64. Layout generation for scene graphs. For scene graph layout generation, a graph comprises node corresponding to object bounding boxes described by, where x i, y i represents the top-left coordinates, and w i, h i represents the bounding box width and height respectively and edge functions are defined based on the relation types. In our experiments, the layout generation model uses two blocks of continuous graph flow units, with four linear graph message passing blocks in each of them. The message passing function uses 64 hidden dimensions, and takes the embedding of node label and edge label in unary message passing function and binary message passing function respectively. The embedding dimension is also set to 64 dimensions. For binary message passing function, we pass the messages both through the direction of edge and the reverse direction of edge to increase the model capacity. A.2 IMAGE PUZZLE GENERATION: ADDITIONAL QUALITATIVE FOR CELEBA-HQ Fig. 5 and Fig. 6 presents the image puzzles generated using unconditional generation and conditional generation respectively. A.3 LAYOUT GENERATION FROM SCENE GRAPH: QUALITATIVE Fig. 7 and Fig. 8 show qualitative on unconditional and conditional layout generation from scene graphs for COCO-stuff dataset respectively. Fig. 9 and Fig. 10 show qualitative on unconditional and conditional layout generation from scene graphs for Visual Genome dataset respectively. The generated have diverse layouts corresponding to a single scene graph. Figure 5: Qualitative on CelebA-HQ for image puzzle generation. Samples generated using our model for 3x3 CelebA-HQ puzzles in unconditional generation setting. Best viewed in color. Figure 6: Qualitative on CelebA-HQ for image puzzle generation. Samples generated using our model for 3x3 CelebA-HQ puzzles in conditional generation setting. Generated patches are highlighted in green. Best viewed in color. Figure 9: Unconditional generation of layouts from scene graphs for Visual Genome dataset. We sample 4 layouts for each scene graph. The generated have different layouts, but sharing the same scene graph. Best viewed in color. Please zoom in to see the category of each object. A.4 GRAPH GENERATION: ADDITIONAL QUALITATIVE Fig. 11 and Fig. 12 present the generated graphs for EGO-SMALL and COMMUNITY-SMALL respectively. We sample 4 layouts for each scene graph. The generated have different layouts except the conditional layout objects in (b), but sharing the same scene graph. Best viewed in color. Please zoom in to see the category of each object. We analyze the variation in number of function evaluations (NFE) required to solve the ODE as the number of nodes in the graph changes. Refer to Figure 13. The interestingly show that the average number of function evaluation does not increase linearly with the increment in number of graph nodes, which would be the case if the variables were independent. (a) CGF samples

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Graph generative models based on generalization of message passing to continuous time using ordinary differential equations

We propose a new class of inference models that iteratively encode gradients to estimate approximate posterior distributions.


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Question: The practical successes of deep neural networks have not been matched by theoretical progress that satisfyingly explains their behavior. In this work, we study the information bottleneck (IB) theory of deep learning, which makes three specific claims: first, that deep networks undergo two distinct phases consisting of an initial fitting phase and a subsequent compression phase; second, that the compression phase is causally related to the excellent generalization performance of deep networks; and third, that the compression phase occurs due to the diffusion-like behavior of stochastic gradient descent. Here we show that none of these claims hold true in the general case. Through a combination of analytical and simulation, we demonstrate that the information plane trajectory is predominantly a function of the neural nonlinearity employed: double-sided saturating nonlinearities like tanh yield a compression phase as neural activations enter the saturation regime, but linear activation functions and single-sided saturating nonlinearities like the widely used ReLU in fact do not. Moreover, we find that there is no evident causal connection between compression and generalization: networks that do not compress are still capable of generalization, and vice versa. Next, we show that the compression phase, when it exists, does not arise from stochasticity in training by demonstrating that we can replicate the IB findings using full batch gradient descent rather than stochastic gradient descent. Finally, we show that when an input domain consists of a subset of task-relevant and task-irrelevant information, hidden representations do compress the task-irrelevant information, although the overall information about the input may monotonically increase with training time, and that this compression happens concurrently with the fitting process rather than during a subsequent compression period. Deep neural networks (; are the tool of choice for real-world tasks ranging from visual object recognition BID16, to unsupervised learning BID11 BID19 and reinforcement learning . These practical successes have spawned many attempts to explain the performance of deep learning systems BID12, mostly in terms of the properties and dynamics of the optimization problem in the space of weights (; BID8 BID1, or the classes of functions that can be efficiently represented by deep networks BID20). This paper analyzes a recent inventive proposal to study the dynamics of learning through the lens of information theory . In this view, deep learning is a question of representation learning: each layer of a deep neural network can be seen as a set of summary statistics which contain some but not all of the information present in the input, while retaining as much information about the target output as possible. The amount of information in a hidden layer regarding the input and output can then be measured over the course of learning, yielding a picture of the optimization process in the information plane. Crucially, this method holds the promise to serve as a general analysis that can be used to compare different architectures, using the common currency of mutual information. Moreover, the elegant information bottleneck (IB) theory provides a fundamental bound on the amount of input compression and target output information that any representation can achieve . The IB bound thus serves as a method-agnostic ideal to which different architectures and algorithms may be compared. A preliminary empirical exploration of these ideas in deep neural networks has yielded striking findings . Most saliently, trajectories in the information plane appear to consist of two distinct phases: an initial "fitting" phase where mutual information between the hidden layers and both the input and output increases, and a subsequent "compression" phase where mutual information between the hidden layers and the input decreases. It has been hypothesized that this compression phase is responsible for the excellent generalization performance of deep networks, and further, that this compression phase occurs due to the random diffusion-like behavior of stochastic gradient descent. Here we study these phenomena using a combination of analytical methods and simulation. In Section 2, we show that the compression observed by arises primarily due to the double-saturating tanh activation function used. Using simple models, we elucidate the effect of neural nonlinearity on the compression phase. Importantly, we demonstrate that the ReLU activation function, often the nonlinearity of choice in practice, does not exhibit a compression phase. We discuss how this compression via nonlinearity is related to the assumption of binning or noise in the hidden layer representation. To better understand the dynamics of learning in the information plane, in Section 3 we study deep linear networks in a tractable setting where the mutual information can be calculated exactly. We find that deep linear networks do not compress over the course of training for the setting we examine. Further, we show a dissociation between generalization and compression. In Section 4, we investigate whether stochasticity in the training process causes compression in the information plane. We train networks with full batch gradient descent, and compare the to those obtained with stochastic gradient descent. We find comparable compression in both cases, indicating that the stochasticity of SGD is not a primary factor in the observed compression phase. Moreover, we show that the two phases of SGD occur even in networks that do not compress, demonstrating that the phases are not causally related to compression. These may seem difficult to reconcile with the intuition that compression can be necessary to attain good performance: if some input channels primarily convey noise, good generalization requires excluding them. Therefore, in Section 5 we study a situation with explicitly task-relevant and task-irrelevant input dimensions. We show that the hidden-layer mutual information with the task-irrelevant subspace does indeed drop during training, though the overall information with the input increases. However, instead of a secondary compression phase, this task-irrelevant information is compressed at the same time that the taskrelevant information is boosted. Our highlight the importance of noise assumptions in applying information theoretic analyses to deep learning systems, and put in doubt the generality of the IB theory of deep learning as an explanation of generalization performance in deep architectures. The starting point for our analysis is the observation that changing the activation function can markedly change the trajectory of a network in the information plane. In FIG0, we show our replication of the reported by for networks with the tanh nonlinearity.1 This replication was performed with the code supplied by the authors of , and closely follows the experimental setup described therein. Briefly, a neural network with 7 fully connected hidden layers of width 12-10-7-5-4-3-2 is trained with stochastic gradient descent to produce a binary classification from a 12-dimensional input. In our replication we used 256 randomly selected samples per batch. The mutual information of the network layers with respect to the input and output variables is calculated by binning the neuron's tanh output activations into 30 equal intervals between -1 and 1. Discretized values for each neuron in each layer are then used to directly calculate the joint distributions, over the 4096 equally likely input patterns and true output labels. In line with prior work , the dynamics in FIG0 show a;, no compression is observed except in the final classification layer with sigmoidal neurons. See Appendix B for the KDE MI method applied to the original Tishby dataset; additional using a second popular nonparametric k-NN-based method BID15; and for other neural nonlinearities.transition between an initial fitting phase, during which information about the input increases, and a subsequent compression phase, during which information about the input decreases. We then modified the code to train deep networks using rectified linear activation functions (f (x) = max(0, x)). While the activities of tanh networks are bounded in the range [−1, 1], ReLU networks have potentially unbounded positive activities. To calculate mutual information, we first trained the ReLU networks, next identified their largest activity value over the course of training, and finally chose 100 evenly spaced bins between the minimum and maximum activity values to discretize the hidden layer activity. The ing information plane dynamics are shown in FIG0. The mutual information with the input monotonically increases in all ReLU layers, with no apparent compression phase. To see whether our were an artifact of the small network size, toy dataset, or simple binning-based mutual information estimator we employed, we also trained larger networks on the MNIST dataset and computed mutual information using a state-of-the-art nonparametric kernel density estimator which assumes hidden activity is distributed as a mixture of Gaussians (see Appendix B for details). Fig. C-D show that, again, tanh networks compressed but ReLU networks did not. Appendix B shows that similar also obtain with the popular nonparametric k-nearest-neighbor estimator of BID15, and for other neural nonlinearities. Thus, the choice of nonlinearity substantively affects the dynamics in the information plane. To understand the impact of neural nonlinearity on the mutual information dynamics, we develop a minimal model that exhibits this phenomenon. In particular, consider the simple three neuron network shown in FIG1. We assume a scalar Gaussian input distribution X ∼ N, which is fed through the scalar first layer weight w 1, and passed through a neural nonlinearity f (·), yielding the hidden unit activity h = f (w 1 X). To calculate the mutual information with the input, this hidden unit activity is then binned yielding the new discrete variable T = bin(h) (for instance, into 30 evenly spaced bins from -1 to 1 for the tanh nonlinearity). This binning process is depicted in FIG1. In this simple setting, the mutual information I(T ; X) between the binned hidden layer activity T and the input X can be calculated exactly. In particular, DISPLAYFORM0 where H(·) denotes entropy, and we have used the fact that H(T |X) = 0 since T is a deterministic function of X. Here the probabilities p i = P (h ≥ b i and h < b i+1) are simply the probability that an input X produces a hidden unit activity that lands in bin i, defined by lower and upper bin limits b i and b i+1 respectively. This probability can be calculated exactly for monotonic nonlinearities f (·) using the cumulative density of X, DISPLAYFORM1 where f −1 (·) is the inverse function of f (·).As shown in FIG1 -D, as a function of the weight w 1, mutual information with the input first increases and then decreases for the tanh nonlinearity, but always increases for the ReLU nonlinearity. Intuitively, for small weights w 1 ≈ 0, neural activities lie near zero on the approximately linear part of the tanh function. Therefore f (w 1 X) ≈ w 1 X, yielding a rescaled Gaussian with information that grows with the size of the weights. However for very large weights w 1 → ∞, the tanh hidden unit nearly always saturates, yielding a discrete variable that concentrates in just two bins. This is more or less a coin flip, containing mutual information with the input of approximately 1 bit. Hence the distribution of T collapses to a much lower entropy distribution, yielding compression for large weight values. With the ReLU nonlinearity, half of the inputs are negative and land in the bin containing a hidden activity of zero. The other half are Gaussian distributed, and thus have entropy that increases with the size of the weight. Hence double-saturating nonlinearities can lead to compression of information about the input, as hidden units enter their saturation regime, due to the binning procedure used to calculate mutual information. The crux of the issue is that the actual I(h; X) is infinite, unless the network itself adds noise to the hidden layers. In particular, without added noise, the transformation from X to the continuous hidden activity h is deterministic and the mutual information I(h; X) would generally be infinite (see Appendix C for extended discussion). Networks that include noise in their processing (e.g.,) can have finite I(T ; X). Otherwise, to obtain a finite MI, one must compute mutual information as though there were binning or added noise in the activations. But this binning/noise is not actually a part of the operation of the network, and is therefore somewhat arbitrary (different binning schemes can in different mutual information with the input, as shown in Fig. 14 of Appendix C).We note that the binning procedure can be viewed as implicitly adding noise to the hidden layer activity: a range of X values map to a single bin, such that the mapping between X and T is no longer perfectly invertible BID17. The binning procedure is therefore crucial to obtaining a finite MI value, and corresponds approximately to a model where noise enters the system after the calculation of h, that is, T = h +, where is noise of fixed variance independent from h and X. This approach is common in information theoretic analyses of deterministic systems, and can serve as a measure of the complexity of a system's representation (see Sec 2.4 of). However, neither binning nor noise is present in the networks that considered, nor the ones in FIG1, either during training or testing. It therefore remains unclear whether robustness of a representation to this sort of noise in fact influences generalization performance in deep learning systems. Furthermore, the addition of noise means that different architectures may no longer be compared in a common currency of mutual information: the binning/noise structure is arbitrary, and architectures that implement an identical input-output map can nevertheless have different robustness to noise added in their internal representation. For instance, Appendix C describes a family of linear networks that compute exactly the same input-output map and therefore generalize identically, but yield different mutual information with respect to the input. Finally, we note that approaches which view the weights obtained from the training process as the random variables of interest may sidestep this issue BID0. Hence when a tanh network is initialized with small weights and over the course of training comes to saturate its nonlinear units (as it must to compute most functions of practical interest, see discussion in Appendix D), it will enter a compression period where mutual information decreases. FIG0 of Appendix E show histograms of neural activity over the course of training, demonstrating that activities in the tanh network enter the saturation regime during training. This nonlinearity-based compression furnishes another explanation for the observation that training slows down as tanh networks enter their compression phase : some fraction of inputs have saturated the nonlinearities, reducing backpropagated error gradients. The preceding section investigates the role of nonlinearity in the observed compression behavior, tracing the source to double-saturating nonlinearities and the binning methodology used to calculate mutual information. However, other mechanisms could lead to compression as well. Even without nonlinearity, neurons could converge to highly correlated activations, or project out irrelevant directions of the input. These phenomena are not possible to observe in our simple three neuron minimal model, as they require multiple inputs and hidden layer activities. To search for these mechanisms, we turn to a tractable model system: deep linear neural networks BID3;; Saxe et al. FORMULA1 ). In particular, we exploit recent on the generalization dynamics in simple linear networks trained in a student-teacher setup (; BID1 . In a student-teacher setting, one "student" neural network learns to approximate the output of another "teacher" neural network. This setting is a way of generating a dataset with interesting structure that nevertheless allows exact calculation of the generalization performance of the network, exact calculation of the mutual information of the representation (without any binning procedure), and, though we do not do so here, direct comparison to the IB bound which is already known for linear Gaussian problems BID6.We consider a scenario where a linear teacher neural network generates input and output examples which are then fed to a deep linear student network to learn FIG2. Following the formulation of BID1, we assume multivariate Gaussian inputs X ∼ N (0, 1 Ni I Ni) and a scalar output Y. The output is generated by the teacher network according to DISPLAYFORM0 represents aspects of the target function which cannot be represented by a neural network (that is, the approximation error or bias in statistical learning theory), and the teacher weights W o are drawn independently from N (0, σ 2 w). Here, the weights of the teacher define the rule to be learned. The signal to noise ratio SNR = σ 2 w /σ 2 o determines the strength of the rule linking inputs to outputs relative to the inevitable approximation error. We emphasize that the "noise" added to the teacher's output is fundamentally different from the noise added for the purpose of calculating mutual information: o models the approximation error for the task-even the best possible neural network may still make errors because the target function is not representable exactly as a neural network-and is part of the construction of the dataset, not part of the analysis of the student network. To train the student network, a dataset of P examples is generated using the teacher. The student network is then trained to minimize the mean squared error between its output and the target output using standard (batch or stochastic) gradient descent on this dataset. Here the student is a deep linear neural network consisting of potentially many layers, but where the the activation function of each neuron is simply f (u) = u. That is, a depth D deep linear network computes the output DISPLAYFORM1 While linear activation functions stop the network from computing complex nonlinear functions of the input, deep linear networks nevertheless show complicated nonlinear learning trajectories , the optimization problem remains nonconvex BID3, and the generalization dynamics can exhibit substantial overtraining BID10 BID1.Importantly, because of the simplified setting considered here, the true generalization error is easily shown to be DISPLAYFORM2 where W tot (t) is the overall linear map implemented by the network at training epoch t (that is, DISPLAYFORM3 Furthermore, the mutual information with the input and output may be calculated exactly, because the distribution of the activity of any hidden layer is Gaussian. Let T be the activity of a specific hidden layer, and letW be the linear map from the input to this activity (that is, for layer l,W = W l · · · W 2 W 1). Since T =W X, the mutual information of X and T calculated using differential entropy is infinite. For the purpose of calculating the mutual information, therefore, we assume that Gaussian noise is added to the hidden layer activity, T =W X + M I, with mean 0 and variance σ 2 M I = 1.0. This allows the analysis to apply to networks of any size, including overcomplete layers, but as before we emphasize that we do not add this noise either during training or testing. With these assumptions, T and X are jointly Gaussian and we have DISPLAYFORM4 where |·| denotes the determinant of a matrix. Finally the mutual information with the output Y, also jointly Gaussian, can be calculated similarly (see Eqns. FORMULA22 - FORMULA25 of Appendix G). Here the network has an input layer of 100 units, 1 hidden layer of 100 units each and one output unit. The network was trained with batch gradient descent on a dataset of 100 examples drawn from the teacher with signal to noise ratio of 1.0. The linear network behaves qualitatively like the ReLU network, and does not exhibit compression. Nevertheless, it learns a map that generalizes well on this task and shows minimal overtraining. Hence, in the setting we study here, generalization performance can be acceptable without any compression phase. The in BID1 ) show that, for the case of linear networks, overtraining is worst when the number of inputs matches the number of training samples, and is reduced by making the number of samples smaller or larger. FIG4 shows learning dynamics with the number of samples matched to the size of the network. Here overfitting is substantial, and again no compression is seen in the information plane. Comparing to the in FIG2, both networks exhibit similar information dynamics with respect to the input (no compression), but yield different generalization performance. Hence, in this linear analysis of a generic setting, there do not appear to be additional mechanisms that cause compression over the course of learning; and generalization behavior can be widely different for networks with the same dynamics of information compression regarding the input. We note that, in the setting considered here, all input dimensions have the same variance, and the weights of the teacher are drawn independently. Because of this, there are no special directions in the input, and each subspace of the input contains as much information as any other. It is possible that, in real world tasks, higher variance inputs are also the most likely to be relevant to the task (here, have large weights in the teacher). We have not investigated this possibility here. To see whether similar behavior arises in nonlinear networks, we trained tanh networks in the same setting as Section 2, but with 30% of the data, which we found to lead to modest overtraining. Next, we test a core theoretical claim of the information bottleneck theory of deep learning, namely that randomness in stochastic gradient descent is responsible for the compression phase. In particular, because the choice of input samples in SGD is random, the weights evolve in a stochastic way during training. distinguish two phases of SGD optimization: in the first "drift" phase, the mean of the gradients over training samples is large relative to the standard deviation of the gradients; in the second "diffusion" phase, the mean becomes smaller than the standard deviation of the gradients. The authors propose that compression should commence following the transition from a high to a low gradient signal-to-noise ratio (SNR), i.e., the onset of the diffusion phase. The proposed mechanism behind this diffusion-driven compression is as follows. The authors state that during the diffusion phase, the stochastic evolution of the weights can be described as a Fokker-Planck equation under the constraint of small training error. Then, the stationary distribution over weights for this process will have maximum entropy, again subject to the training error constraint. Finally, the authors claim that weights drawn from this stationary distribution will maximize the entropy of inputs given hidden layer activity, H(X|T), subject to a training error constraint, and that this training error constraint is equivalent to a constraint on the mutual information I(T ; Y) for small training error. Since the entropy of the input, H(X), is fixed, the of the diffusion dynamics will be to minimize I(X; T):= H(X) − H(X|T) for a given value of I(T ; Y) reached at the end of the drift phase. However, this explanation does not hold up to either theoretical or empirical investigation. Let us assume that the diffusion phase does drive the distribution of weights to a maximum entropy distribution subject to a training error constraint. Note that this distribution reflects stochasticity of weights across different training runs. There is no general reason that a given set of weights sampled from this distribution (i.e., the weight parameters found in one particular training run) will maximize H(X|T), the entropy of inputs given hidden layer activity. In particular, H(X|T) reflects (conditional) uncertainty about inputs drawn from the data-generating distribution, rather than uncertainty about any kind of distribution across different training runs. We also show empirically that the stochasticity of the SGD is not necessary for compression. To do so, we consider two distinct training procedures: offline stochastic gradient descent (SGD), which learns from a fixed-size dataset, and updates weights by repeatedly sampling a single example from the dataset and calculating the gradient of the error with respect to that single sample (the typical procedure used in practice); and batch gradient descent (BGD), which learns from a fixed-size dataset, and updates weights using the gradient of the total error across all examples. Batch gradient descent uses the full training dataset and, crucially, therefore has no randomness or diffusion-like behavior in its updates. We trained tanh and ReLU networks with SGD and BGD and compare their information plane dynamics in FIG5 (see Appendix H for a linear network). We find largely consistent information dynamics in both instances, with robust compression in tanh networks for both methods. Thus randomness in the training process does not appear to contribute substantially to compression of information about the input. This finding is consistent with the view presented in Section 2 that compression arises predominantly from the double saturating nonlinearity. Finally, we look at the gradient signal-to-noise ratio (SNR) to analyze the relationship between compression and the transition from high to low gradient SNR. FIG1 of Appendix I shows the gradient SNR over training, which in all cases shows a phase transition during learning. Hence the gradient SNR transition is a general phenomenon, but is not causally related to compression. Appendix I offers an extended discussion and shows gradient SNR transitions without compression on the MNIST dataset and for linear networks. Our finding that generalization can occur without compression may seem difficult to reconcile with the intuition that certain tasks involve suppressing irrelevant directions of the input. In the extreme, if certain inputs contribute nothing but noise, then good generalization requires ignoring them. To study this, we consider a variant on the linear student-teacher setup of Section 3: we partition the input X into a set of task-relevant inputs X rel and a set of task-irrelevant inputs X irrel, and alter the teacher network so that the teacher's weights to the task-irrelevant inputs are all zero. Hence the inputs X irrel contribute only noise, while the X rel contain signal. We then calculate the information plane dynamics for the whole layer, and for the task-relevant and task-irrelevant inputs separately. FIG7 shows information plane dynamics for a deep linear neural network trained using SGD (5 samples/batch) on a task with 30 task-relevant inputs and 70 task-irrelevant inputs. While the overall dynamics show no compression phase, the information specifically about the task-irrelevant subspace does compress over the course of training. This compression process occurs at the same time as the fitting to the task-relevant information. Thus, when a task requires ignoring some inputs, the The information with the task-relevant subspace increases robustly over training. (C) However, the information specifically about the task-irrelevant subspace does compress after initially growing as the network is trained.information with these inputs specifically will indeed be reduced; but overall mutual information with the input in general may still increase. Our suggest that compression dynamics in the information plane are not a general feature of deep networks, but are critically influenced by the nonlinearities employed by the network. Doublesaturating nonlinearities lead to compression, if mutual information is estimated by binning activations or by adding homoscedastic noise, while single-sided saturating nonlinearities like ReLUs do not compress in general. Consistent with this view, we find that stochasticity in the training process does not contribute to compression in the cases we investigate. Furthermore, we have found instances where generalization performance does not clearly track information plane behavior, questioning the causal link between compression and generalization. Hence information compression may parallel the situation with sharp minima: although empirical evidence has shown a correlation with generalization error in certain settings and architectures, further theoretical analysis has shown that sharp minima can in fact generalize well BID9. We emphasize that compression still may occur within a subset of the input dimensions if the task demands it. This compression, however, is interleaved rather than in a secondary phase and may not be visible by information metrics that track the overall information between a hidden layer and the input. Finally, we note that our address the specific claims of one scheme to link the information bottleneck principle with current practice in deep networks. The information bottleneck principle itself is more general and may yet offer important insights into deep networks BID0. Moreover, the information bottleneck principle could yield fundamentally new training algorithms for networks that are inherently stochastic and where compression is explicitly encouraged with appropriate regularization terms BID5 BID2 This Appendix investigates the generality of the finding that compression is not observed in neural network layers with certain activation functions. FIG0 of the main text shows example using a binning-based MI estimator and a nonparametric KDE estimator, for both the tanh and ReLU activation functions. Here we describe the KDE MI estimator in detail, and present extended on other datasets. We also show for other activation functions. Finally, we provide entropy estimates based on another nonparametric estimator, the popular k-nearest neighbor approach of BID15. Our findings consistently show that double-saturating nonlinearities can yield compression, while single-sided nonlinearities do not. The KDE approach of; estimates the mutual information between the input and the hidden layer activity by assuming that the hidden activity is distributed as a mixture of Gaussians. This assumption is well-suited to the present setting under the following interpretation: we take the input activity to be distributed as delta functions at each example in the dataset, corresponding to a uniform distribution over these specific samples. In other words, we assume that the empirical distribution of input samples is the true distribution. Next, the hidden layer activity h is a deterministic function of the input. As mentioned in the main text and discussed in more detail in Appendix C, without the assumption of noise, this would have infinite mutual information with the input. We therefore assume for the purposes of analysis that Gaussian noise of variance σ 2 is added, that is, T = h + where ∼ N (0, σ 2 I). Under these assumptions, the distribution of T is genuinely a mixture of Gaussians, with a Gaussian centered on the hidden activity corresponding to each input sample. We emphasize again that the noise is added solely for the purposes of analysis, and is not present during training or testing the network. In this setting, an upper bound for the mutual information with the input is ) DISPLAYFORM0 where P is the number of training samples and h i denotes the hidden activity vector in response to input sample i. Similarly, the mutual information with respect to the output can be calculated as DISPLAYFORM1 DISPLAYFORM2 where L is the number of output labels, P l denotes the number of data samples with output label l, p l = P l /P denotes the probability of output label l, and the sums over i, Y i = l indicate a sum over all examples with output label l. FIG10 -B shows the of applying this MI estimation method on the dataset and network architecture of , with MI estimated on the full dataset and averaged over 50 repetitions. Mutual information was estimated using data samples from the test set, and we took the noise variance σ 2 = 0.1. These look similar to the estimate derived from binning, with compression in tanh networks but no compression in ReLU networks. Relative to the binning estimate, it appears that compression is less pronounced in the KDE method. The network was trained using SGD with minibatches of size 128. As before, mutual information was estimated using data samples from the test set, and we took the noise variance σ 2 = 0.1. The smaller layer sizes in the top three hidden layers were selected to ensure the quality of the kernel density estimator given the amount of data in the test set, since the estimates are more accurate for smaller-dimensional data. Because of computational expense, the MNIST are from a single training run. More detailed for the MNIST dataset are provided in FIG12 for the tanh activation function, and in FIG0 for the ReLU activation function. In these figures, the first row shows the evolution of the cross entropy loss (on both training and testing data sets) during training. The second row shows the mutual information between input and the activity of different hidden layers, using the nonparametric KDE estimator described above. The blue region in the second row shows the range of possible MI values, ranging from the upper bound described above (Eq. 10) to the following lower bound, DISPLAYFORM3 The third row shows the mutual information between input and activity of different hidden layers, estimated using the binning method (here, the activity of each neuron was discretized into bins of size 0.5). For both the second and third rows, we also plot the entropy of the inputs, H(X), as a dashed line. H(X) is an upper bound on the mutual information I(X; T), and is computed using the assumption of a uniform distribution over the 10,000 testing points in the MNIST dataset, giving H(X) = log 2 10000.Finally, the fourth row visualizes the dynamics of the SGD updates during training. For each layer and epoch, the green line shows the 2 norm of the weights. We also compute the vector of mean updates across SGD minibatches (this vector has one dimension for each weight parameter), as well as the vector of the standard deviation of the updates across SGD minibatches. The 2 norm of the mean update vector is shown in blue, and the 2 norm of the standard deviation vector is shown in orange. The gradient SNR, computed as the ratio of the norm of the mean vector to the norm of the standard deviation vector, is shown in red. For both the tanh and ReLU networks, the gradient SNR shows a phase transition during training, and the norm of the weights in each layer increases. Importantly, this phase transition occurs despite a lack of compression in the ReLU network, indicating that noise in SGD updates does not yield compression in this setting. Upper and lower bounds for the mutual information I(X; T) between the input (X) and each layer's activity (T), using the nonparametric KDE estimator. Dotted line indicates H(X) = log 2 10000, the entropy of a uniform distribution over 10,000 testing samples. Row 3: Binning-based estimate of the mutual information I(X; T), with each neuron's activity discretized using a bin size of 0.5. Row 4: Gradient SNR and weight norm dynamics. The gradient SNR shows a phase transition during training, and the norm of the weights in each layer increases. Upper and lower bounds for the mutual information I(X; T) between the input (X) and each layer's activity (T), using the nonparametric KDE estimator. Dotted line indicates H(X) = log 2 10000, the entropy of a uniform distribution over 10,000 testing samples. Row 3: Binning-based estimate of the mutual information I(X; T), with each neuron's activity discretized using a bin size of 0.5. Row 4: Gradient SNR and weight norm dynamics. The gradient SNR shows a phase transition during training, and the norm of the weights in each layer increases. Importantly, this phase transition occurs despite a lack of compression in the ReLU network, indicating that noise in SGD updates does not yield compression in this setting. Next, in FIG10 -D, we show from the kernel MI estimator from two additional nonlinear activation functions, the softsign function DISPLAYFORM0 and the softplus function DISPLAYFORM1 These functions are plotted next to tanh and ReLU in FIG0. The softsign function is similar to tanh but saturates more slowly, and yields less compression than tanh. The softplus function is a smoothed version of the ReLU, and yields similar dynamics with no compression. Because softplus never saturates fully to zero, it retains more information with respect to the input than ReLUs in general. We additionally investigated the widely-used nonparametric MI estimator of BID15. This estimator uses nearest neighbor distances between samples to compute an estimate of the entropy of a continuous random variable. Here we focused for simplicity only on the compression phenomenon in the mutual information between the input and hidden layer activity, leaving aside the information with respect to the output (as this is not relevant to the compression phenomenon). Again, without additional noise assumptions, the MI between the hidden representation and the input would be infinite because the mapping is deterministic. Rather than make specific noise assumptions, we instead use the Kraskov method to estimate the entropy of the hidden representations T. Note that the entropy of T is the mutual information up to an unknown constant so long as the noise assumption is homoscedastic, that is, T = h + Z where the random variable Z is independent of X. To see this, note that DISPLAYFORM0 where the constant c = H(Z). Hence observing compression in the layer entropy H(T) is enough to establish that compression occurs in the mutual information. The Kraskov estimate is given by DISPLAYFORM1 where d is the dimension of the hidden representation, P is the number of samples, r i is the distance to the k-th nearest neighbor of sample i, is a small constant for numerical stability, Γ(·) is the Gamma function, and ψ(·) is the digamma function. Here the parameter prevents infinite terms when the nearest neighbor distance r i = 0 for some sample. We took = 10 −16. FIG0 shows the entropy over training for tanh and ReLU networks trained on the dataset of and with the network architecture in , averaged over 50 repeats. In these experiments, we used k = 2. Compression would correspond to decreasing entropy over the course of training, while a lack of compression would correspond to increasing entropy. Several tanh layers exhibit compression, while the ReLU layers do not. Hence qualitatively, the Kraskov estimator returns similar to the binning and KDE strategies. A B FIG0: Entropy dynamics over training for the network architecture and training dataset of Shwartz-Ziv & Tishby FORMULA7, estimated with the nonparametric k-nearest-neighbor-based method of BID15. Here the x-axis is epochs of training time, and the y-axis plots the entropy of the hidden representation, as calculated using nearest-neighbor distances. Note that in this setting, if T is considered to be the hidden activity plus independent noise, the entropy is equal to the mutual information up to a constant (see derivation in text). Layers 0-4 correspond to the hidden layers of size 10-7-5-4-3. (A) tanh neural network layers can show compression over the course of training.(B) ReLU neural network layers show no compression. A recurring theme in the reported in this paper is the necessity of noise assumptions to yield a nontrivial information theoretic analysis. Here we give an extended discussion of this phenomenon, and of issues relating to discrete entropy as opposed to continuous (differential) entropy. The activity of a neural network is often a continuous deterministic function of its input. That is, in response to an input X, a specific hidden layer might produce activity h = f (X) for some function f. The mutual information between h and X is given by DISPLAYFORM0 If h were a discrete variable, then the entropy would be given by DISPLAYFORM1 where p i is the probability of the discrete symbol i, as mentioned in the main text. Then H(h|X) = 0 because the mapping is deterministic and we have I(h; X) = H(h).However h is typically continuous. The continuous entropy, defined for a continuous random variable Z with density p Z by analogy to Eqn. FORMULA8 as DISPLAYFORM2 can be negative and possibly infinite. In particular, note that if p Z is a delta function, then H(Z) = −∞. The mutual information between hidden layer activity h and the input X for continuous h, X is DISPLAYFORM3 Now H(h|X) = −∞ since given the input X, the hidden activity h is distributed as a delta function at f (X). The mutual information is thus generally infinite, so long as the hidden layer activity has finite entropy (H(h) is finite).Figure 13: Effect of binning strategy on minimal three neuron model. Mutual information for the simple three neuron model shown in FIG1 with bin edges b i ∈ tanh(linspace(−50, 50, N)). In contrast to linear binning, the mutual information continues to increase as weights grow. To yield a finite mutual information, some noise in the mapping is required such that H(h|X) remains finite. A common choice (and one adopted here for the linear network, the nonparametric kernel density estimator, and the k-nearest neighbor estimator) is to analyze a new variable with additive noise, T = h + Z, where Z is a random variable independent of X. Then H(T |X) = H(Z) which allows the overall information I(T ; X) = H(T) − H(Z) to remain finite. This noise assumption is not present in the actual neural networks either during training or testing, and is made solely for the purpose of calculating the mutual information. Another strategy is to partition the continuous variable h into a discrete variable T, for instance by binning the values (the approach taken in). This allows use of the discrete entropy, which remains finite. Again, however, in practice the network does not operate on the binned variables T but on the continuous variables h, and the binning is solely for the purpose of calculating the mutual information. Moreover, there are many possible binning strategies, which yield different discrete random variables, and different mutual information with respect to the input. The choice of binning strategy is an assumption analogous to choosing a type of noise to add to the representation in the continuous case: because there is in fact no binning in the operation of the network, there is no clear choice for binning methodology. The strategy we use in binning-based experiments reported here is the following: for bounded activations like the tanh activation, we use evenly spaced bins between the minimum and maximum limits of the function. For unbounded activations like ReLU, we first train the network completely; next identify the minimum and maximum hidden activation over all units and all training epochs; and finally bin into equally spaced bins between these minimum and maximum values. We note that this procedure places no restriction on the magnitude that the unbounded activation function can take during training, and yields the same MI estimate as using infinite equally spaced bins (because bins for activities larger than the maximum are never seen during training).As an example of another binning strategy that can yield markedly different , we consider evenly spaced bins in a neuron's net input, rather than its activity. That is, instead of evenly spaced bins in the neural activity, we determine the bin edges by mapping a set of evenly spaced values through the neural nonlinearity. For tanh, for instance, this spaces bins more tightly in the saturation region as compared to the linear region. FIG0 shows the of applying this binning strategy FIG0: Effect of binning strategy on information plane dynamics. Results for the same tanh network and training regime as 1A, but with bin edges b i ∈ tanh(linspace(−50, 50, N)). Measured with this binning structure, there is no compression in most layers.to the minimal three neuron model with tanh activations. This binning scheme captures more information as the weights of the network grow larger. FIG0 shows information plane dynamics for this binning structure. The tanh network no longer exhibits compression. (We note that the broken DPI in this example is an artifact of performing binning only for analysis, as discussed below).Any implementation of a neural network on digital hardware is ultimately of finite precision, and hence is a binned, discrete representation. However, it is a very high resolution binning compared to that used here or by: single precision would correspond to using roughly 2 32 bins to discretize each hidden unit's activity, as compared to the 30-100 used here. If the binning is fine-grained enough that each input X yields a different binned activity pattern h, then H(h) = log(P) where P is the number of examples in the dataset, and there will be little to no change in information during training. As an example, we show in FIG0 the of binning at full machine precision. Finally, we note two consequences of the assumption of noise/binning for the purposes of analysis. First, this means that the data processing inequality (DPI) does not apply to the noisy/binned mutual information estimates. The DPI states that information can only be destroyed through successive transformations, that is, if X → h 1 → h 2 form a Markov chain, then I(X; h 1) ≥ I(X; h 2) (see, eg,). Because noise is added only for the purpose of analysis, however, this does not apply here. In particular, for the DPI to apply, the noise added at lower layers would have to propagate through the network to higher layers. That is, if the transformation from hidden layer 1 to hidden layer 2 is h 2 = f (h 1) and T 1 = h 1 + Z 1 is the hidden layer activity after adding noise, then the DPI would hold for the variableT 2 = f (T 1) + Z 2 = f (h 1 + Z 1) + Z 2, not the quantity Information in most layers stays pinned to log 2 (P) = 12. Compression is only observed in the highest and smallest layers near the very end of training, when the saturation of tanh is strong enough to saturate machine precision. DISPLAYFORM4 used in the analysis. Said another way, the Markov chain for DISPLAYFORM5, so the DPI states only that I(X; h 1) ≥ I(X; T 2).A second consequence of the noise assumption is the fact that the mutual information is no longer invariant to invertible transformations of the hidden activity h. A potentially attractive feature of a theory based on mutual information is that it can allow for comparisons between different architectures: mutual information is invariant to any invertible transformation of the variables, so two hidden representations could be very different in detail but yield identical mutual information with respect to the input. However, once noise is added to a hidden representation, this is no longer the case: the variable T = h + Z is not invariant to reparametrizations of h. As a simple example, consider a minimal linear network with scalar weights w 1 and w 2 that computes the outputŷ = w 2 w 1 X. The hidden activity is h = w 1 X. Now consider the family of networks in which we scale down w 1 and scale up w 2 by a factor c = 0, that is, these networks have weightsw 1 = w 1 /c andw 2 = cw 2, yielding the exact same input-output mapŷ =w 2w1 X = cw 2 (w 1 /c)X = w 2 w 1 X. Because they compute the same function, they necessarily generalize identically. However after introducing the noise assumption the mutual information is DISPLAYFORM6 where we have taken the setting in Section 3 in which X is normal Gaussian, and independent Gaussian noise of variance σ 2 M I is added for the purpose of MI computation. Clearly, the mutual information is now dependent on the scaling c of the internal layer, even though this is an invertible linear transformation of the representation. Moreover, this shows that networks which generalize identically can nevertheless have very different mutual information with respect to the input when it is measured in this way. Our argument relating neural saturation to compression in mutual information relies on the notion that in typical training regimes, weights begin small and increase in size over the course of training. We note that this is a virtual necessity for a nonlinearity like tanh, which is linear around the origin: when initialized with small weights, the activity of a tanh network will be in this linear regime and the network can only compute a linear function of its input. Hence a real world nonlinear task can only be learned by increasing the norm of the weights so as to engage the tanh nonlinearity on some examples. This point can also be appreciated from norm-based capacity bounds on neural networks, which show that, for instance, the Rademacher complexity of a neural network with small weights must be low BID4 ). Finally, as an empirical matter, the networks trained in this paper do in fact increase the norm of their weights over the course of For the linear setting considered here, the mutual information between a hidden representation T and the output Y may be calculated using the relations DISPLAYFORM0 DISPLAYFORM1 DISPLAYFORM2 H STOCHASTIC VS BATCH TRAINING FIG0 shows information plane dynamics for stochastic and batch gradient descent learning in a linear network. Randomness in the training process does not dramatically alter the information plane dynamics. The proposed mechanism of compression in is noise arising from stochastic gradient descent training. The in Section 4 of the main text show that compression still occurs under batch gradient descent learning, suggesting that in fact noise in the gradient updates is not the cause of compression. Here we investigate a related claim, namely that during training, networks switch between two phases. These phases are defined by the ratio of the mean of the gradient to the standard deviation of the gradient across training examples, called the gradient signal-to-noise ratio. In the first "drift" phase, the SNR is high, while in the second "diffusion" phase the SNR is low. hypothesize that the drift phase corresponds to movement toward the minimum with no compression, while the diffusion phase corresponds to a constrained diffusion in weight configurations that attain the optimal loss, during which representations compress. However, two phases of gradient descent have been described more generally, sometimes known as the transient and stochastic phases or search and convergence phases BID21 BID7, suggesting that these phases might not be related specifically to compression behavior. In FIG1 we plot the gradient SNR over the course of training for the tanh and ReLU networks in the standard setting of. In particular, for each layer l we calculate the mean and standard deviation as DISPLAYFORM0 DISPLAYFORM1 where · denotes the mean and ST D(·) denotes the element-wise standard deviation across all training samples, and · F denotes the Frobenius norm. The gradient SNR is then the ratio m l /s l. We additionally plot the norm of the weights W l F over the course of training. Both tanh and ReLU networks yield a similar qualitative pattern, with SNR undergoing a step-like transition to a lower value during training. Figures 9 and 10, fourth row, show similar plots for MNIST-trained networks. Again, SNR undergoes a transition from high to low over training. Hence the two phase nature of gradient descent appears to hold across the settings that we examine here. Crucially, this finding shows that the SNR transition is not related to the compression phenomenon because ReLU networks, which show the gradient SNR phase transition, do not compress. Finally, to show the generality of the two-phase gradient SNR behavior and its independence from compression, we develop a minimal model of this phenomenon in a three neuron linear network. We consider the student-teacher setting of FIG2 but with N i = N h = 1, such that the input and hidden layers have just a single neuron (as in the setting of FIG1). Here, with just a single hidden neuron, clearly there can be no compression so long as the first layer weight increases over the course of training. FIG0 shows that even in this simple setting, the SNR shows the phase transition but the weight norm increases over training. Hence again, the two phases of the gradient are present even though there is no compression. To intuitively understand the source of this behavior, note that the weights are initialized to be small and hence early in learning all must be increased in magnitude, yielding a consistent mean gradient. Once the network reaches the vicinity of the minimum, the mean weight change across all samples by definition goes to zero. The standard deviation remains finite, however, because on some specific examples error could be improved by increasing or decreasing the weights-even though across the whole dataset the mean error has been minimized. Hence overall, our show that a two-phase structure in the gradient SNR occurs in all settings we consider, even though compression occurs only in a subset. The gradient SNR behavior is therefore not causally related to compression dynamics, consistent with the view that saturating nonlinearities are the primary source of compression.

Represent the short summary for finding the corresponding scientific article (long)!

We show that several claims of the information bottleneck theory of deep learning are not true in the general case.

We propose a hypothesis for why gradient descent generalizes based on how per-example gradients interact with each other.