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Title: On the Information Bottleneck Theory of Deep Learning

Paper content:
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 results 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 (Schmidhuber, 2015; are the tool of choice for real-world tasks ranging from visual object recognition BID16 , to unsupervised learning BID11 BID19 and reinforcement learning (Silver et al., 2016) .
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 (Saxe et al., 2014; BID8 BID1 , or the classes of functions that can be efficiently represented by deep networks BID20 Poggio et al., 2017) .
This paper analyzes a recent inventive proposal to study the dynamics of learning through the lens of information theory (Tishby & Zaslavsky, 2015; Shwartz-Ziv & Tishby, 2017) .
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 (Tishby et al., 1999) .
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 (Shwartz-Ziv & Tishby, 2017) .
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 Shwartz-Ziv & Tishby (2017) 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 results 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 results 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 results 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.
Our results 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 results 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

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