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Learning Representations for Neural Network-Based Classification Using the Information Bottleneck Principle

Rana Ali Amjad, Bernhard C. Geiger

arXiv:1802.09766v6cs.LGcs.CVcs.IT

TL;DR

The paper examines whether the information bottleneck functional is suitable for training and analyzing DNN classifiers. It shows fundamental failures for deterministic networks and argues that reported successes rely on modifications such as stochasticity, decision rules, or replacement cost functions.

  • Problem

    The IB functional is evaluated as a possible way to encourage classification representations that are robust to noise and distortions and simple for decision-making.

  • Method

    The paper analyzes the IB functional for deterministic DNNs and examines remedies including stochastic networks, soft decision rules, and quantized or noisy representations.

  • Results

    For deterministic DNNs, the IB functional is either infinite for almost all parameters or piecewise constant, while related works' successes partly reflect additional modifications rather than the unmodified functional.

  • Takeaways & Limitations

    The IB functional cannot be expected to characterize robustness, simple decision functions, or generalization during DNN training.

  • Takeaways & Limitations

    A good IB-functional result does not establish robustness or simplicity, and the functional is ill-posed or piecewise constant for the classification task.

Abstract

from arXiv · show

In this theory paper, we investigate training deep neural networks (DNNs) for classification via minimizing the information bottleneck (IB) functional. We show that the resulting optimization problem suffers from two severe issues: First, for deterministic DNNs, either the IB functional is infinite for almost all values of network parameters, making the optimization problem ill-posed, or it is piecewise constant, hence not admitting gradient-based optimization methods. Second, the invariance of the IB functional under bijections prevents it from capturing properties of the learned representation that are desirable for classification, such as robustness and simplicity. We argue that these issues are partly resolved for stochastic DNNs, DNNs that include a (hard or soft) decision rule, or by replacing the IB functional with related, but more well-behaved cost functions. We conclude that recent successes reported about training DNNs using the IB framework must be attributed to such solutions. As a side effect, our results indicate limitations of the IB framework for the analysis of DNNs. We also note that rather than trying to repair the inherent problems in the IB functional, a better approach may be to design regularizers on latent representation enforcing the desired properties directly.

1 INTRODUCTION

The paper examines whether minimizing the information bottleneck functional can train DNN classifiers with desirable representations. It identifies fundamental problems for deterministic networks and explains why reported successes rely on modifications to the original framework.

  • 1 INTRODUCTION: The IB framework trains representations by compressing information about X while preserving information about class variable Y.The functional uses I(X; L) for compression and I(Y; L) for class information.
  • 1 INTRODUCTION: The paper tests whether representation-dependent IB regularization captures robustness to noise and distortions and simplicity of the learned representation.These properties motivate using data-dependent regularization rather than regularization based only on network parameters.
  • 1 INTRODUCTION: For deterministic DNNs, the IB optimization is ill-posed when the functional is infinite for almost all parameters or inaccessible to gradient methods when it is piecewise constant.These are the paper’s two principal objections to directly training deterministic networks with the IB functional.
  • 1 INTRODUCTION: The IB functional captures only a small subset of classification-relevant representation properties, limiting its suitability as a deterministic-DNN training cost.The paper specifically connects this limitation to robustness and representational simplicity.
  • 1 INTRODUCTION: Reported successes from IB-based DNN training follow departures such as replacing the functional, using stochastic networks, or including a decision rule.Earlier works replace mutual-information terms with bounds to obtain computable objectives suitable for gradient optimization.
  • 1 INTRODUCTION: The analysis assumes the joint distribution of X and Y is known, while also commenting on how conclusions change for finite datasets.This assumption separates the analysis from the optimization heuristic and represents a best-case training scenario.
  • 1 INTRODUCTION: Without the proposed remedies, the IB framework cannot establish robustness, classification performance, or representational simplicity for a given DNN.The paper states that information-theoretic compression and class-information claims may remain possible when the functional is finite.

2 SETUP AND PRELIMINARIES

The paper formalizes feature-based classification with feed-forward DNNs that transform input features into latent or output representations. It distinguishes deterministic and stochastic layer maps and reviews the relevant information quantities.

  • 2 SETUP AND PRELIMINARIES: The setup assumes a joint distribution P_X,Y over an N-dimensional feature vector X and a discrete class label Y.Realizations are denoted by lowercase letters, and the class labels lie in a finite set.
  • 2 SETUP AND PRELIMINARIES: A feed-forward DNN maps X to an output random variable Ỹ, from which the class label Y is estimated using a decision rule.Ỹ denotes the final-layer output, not the post-decision estimate of Y.
  • 2 SETUP AND PRELIMINARIES: In deterministic DNNs, adjacent representations are connected by parameterized functions of weights, biases, and coordinate-wise activation functions.The paper’s conclusions do not depend on whether the activation is sigmoid, ReLU, leaky ReLU, tanh, or softplus unless stated otherwise.
  • 2 SETUP AND PRELIMINARIES: Each latent representation L_i is produced by an encoder f_i from X and mapped onward by a decoder h_i to the DNN output.For stochastic DNNs, the layer maps are stochastic, while encoder and decoder composition is defined analogously.
  • 2 SETUP AND PRELIMINARIES: The paper uses H for entropy, h for differential entropy, and I for mutual information.For continuous X and discrete Y, mutual information can be expressed through either entropy or differential-entropy differences when the terms are finite.
  • 2 SETUP AND PRELIMINARIES: Differential and discrete entropy have different finiteness behavior: H(X) is infinite for nondiscrete X, while h(Y) is negative infinity for noncontinuous Y.This distinction underlies later computability issues involving mutual information in continuous representations.

3 LEARNING REPRESENTATIONS FOR CLASSIFI-

The paper defines classification-oriented representation goals beyond preserving class information and compressing inputs. These goals include simple decisions, robustness, and practical architectural economy, motivating direct representation regularization.

  • 3 LEARNING REPRESENTATIONS FOR CLASSIFI-: The paper argues that maximizing class information while compressing X is insufficient for classification.It therefore introduces additional desired properties for intermediate representations.
  • 3 LEARNING REPRESENTATIONS FOR CLASSIFI-: P1 requires a representation L to preserve as much information about Y as the input features X, making L sufficient for Y.This is the information-preservation objective for classification.
  • 3 LEARNING REPRESENTATIONS FOR CLASSIFI-: P2 requires maximal compression so L is invariant, in some sense, to nuisance factors irrelevant to Y.Compression may be quantified statistically or geometrically, such as mapping classes to distinct dense clusters.
  • 3 LEARNING REPRESENTATIONS FOR CLASSIFI-: P3 requires successive representations to permit increasingly simple decision functions for estimating Y.For the output, arg max or binary quantization should suffice when appropriate.
  • 3 LEARNING REPRESENTATIONS FOR CLASSIFI-: P4 requires robustness: small input noise or well-behaved transformations should not cause large changes in the representation.Geometrically, class clusters should remain separated under small deformations.
  • 3 LEARNING REPRESENTATIONS FOR CLASSIFI-: Training traditionally targets class information and simple decisions with mean-squared error or cross-entropy, while robustness has been linked to generalization.The paper situates these goals alongside regularization approaches used in learning algorithms.
  • 3 LEARNING REPRESENTATIONS FOR CLASSIFI-: Architectural economy additionally favors few layers, few neurons, sparse weights, or computationally efficient inference, especially for embedded and edge deployment.This goal concerns the network producing the representation, not only the representation itself.
  • 3 LEARNING REPRESENTATIONS FOR CLASSIFI-: Robustness and compression may simplify decisions but can require an encoder with substantial architectural or computational complexity.The representation-level goals are therefore not independent of network complexity.

4 WHY AND HOW IB FAILS FOR TRAINING DETERMINISTIC DNNS

For deterministic DNNs, minimizing the IB functional is ill-posed or difficult to optimize, and equal IB values need not yield simple or robust classifiers.

  • 4.3 Invariance under Bijections: The IB Functional is Insufficient: The IB functional is insufficient for classification because minimizing it does not necessarily produce representations supporting simple decision rules or robustness to noise and distortions.The paper compares DNNs with equivalent IB values whose representations differ in decision-rule complexity and robustness.
  • 4.1 Continuous Features: The IB Functional is Infinite: For feature distributions with an absolutely continuous component, the IB functional is infinite for almost every non-trivial deterministic DNN parameter setting under Theorem 1’s activation assumptions.The result applies to bi-Lipschitz or continuously differentiable activations with strictly positive derivative, including leaky ReLU, sigmoid, and tanh networks.
  • 4.2 Discrete Features or Learning from Data: The IB Functional is Piecewise Constant: For discrete inputs, the deterministic representation is discrete, but the IB functional changes only when parameter changes alter which inputs share representation values.Consequently, the objective is piecewise constant and has zero gradient almost everywhere, including when optimized from a finite dataset.
  • 4.2 Discrete Features or Learning from Data: The IB Functional is Piecewise Constant: Piecewise-constant behavior can also occur with continuous inputs when step or ReLU activations operate on disconnected input supports.Thus, the optimization difficulty is not restricted to discrete feature distributions.
  • 4.3 Invariance under Bijections: The IB Functional is Insufficient: Two DNNs can have identical IB functionals while one admits a simpler threshold decision and another is less robust to noisy inputs.The robustness example shows that a noisy point can be confused by one network but distinguished by the other, despite equal IB values.
  • 4.3 Invariance under Bijections: The IB Functional is Insufficient: The authors conclude that IB can produce compressed, class-informative outputs, but cannot by itself establish simple decisions, robustness, or generalization during training.This conclusion also applies to discrete-valued features, where the continuous-feature infinity issue is absent.

5 HOW TO USE IB-LIKE COST FUNCTIONS FOR TRAINING DNNS

The paper examines remedies for deterministic DNNs' ill-posed or non-optimizable IB objective, including decision rules, stochastic representations, and better-behaved surrogate costs.

  • 5.1 Including the Decision Rule: Decision rules make output representations discrete, bounding I(X; ˆY) by log |Y| and making the output-layer IB functional computable.Compression is supplied by the decision rule, while a simple decision function also promotes representation simplicity.
  • 5.1 Including the Decision Rule: Hard decision rules can create large equivalence classes of DNNs with identical objective values, and they do not necessarily preserve robustness.The paper gives examples where identical decision-rule outputs yield identical IB functionals despite different robustness.
  • 5.2 Probabilistic Interpretation of the Neuron Outputs: Soft decision rules make the functional finite, support gradient-based optimization on finite datasets, and make the precision term sensitive to simplification.Softmax outputs are interpreted as class probabilities, and the precision term also encourages dense class clusters.
  • 5.3 Stochastic DNNs: Stochastic DNNs add noise to intermediate representations, allowing the compression term to encourage compact representations and providing a form of data augmentation.Sampling noisy intermediate representations multiple times during training may improve robustness without separately training an autoencoder for perturbation.
  • 5.4 Replacing the IB Functional: Replacing mutual information with quantized or otherwise better-behaved quantities can make the objective finite and trainable, but quantizer design remains difficult.Quantizers may be adapted to representation statistics and should also be chosen to instill desired representation properties, not merely estimate mutual information.
  • 5.5 Discussion: Across these remedies, compression is encouraged geometrically through compact, separated class clusters, enabling simple decision rules rather than minimal sufficient statistics.The paper characterizes this as geometric compression and links it to simpler classification decisions.

6 CRITICAL DISCUSSION OF AND EXPERIMENTAL EVIDENCE FROM RELATED WORK

The related work shows that successful IB-based DNN methods generally modify the original functional, while supporting the paper’s analysis of its limitations for deterministic networks.

  • IB-based DNN research spans analysis, theory and performance-oriented training, with this section connecting each category to the paper’s results.
  • Analytical work recognizes that I(X; L) can be infinite for continuous intermediate representations, motivating noisy or quantized alternatives.
  • Reported training successes target generalization, adversarial robustness and out-of-distribution detection, but the methods depart from the unmodified IB functional.
  • Stochastic networks, variational bounds and non-parametric bounds make IB-inspired objectives more tractable and can produce geometrically clustered representations.
  • The paper argues that operational gains in prior work can at least partly be attributed to these additional remedies rather than to the IB functional alone.

7 CONCLUDING REMARKS

The paper concludes that deterministic IB training has fundamental optimization and representation shortcomings, while direct representation regularization offers a more promising direction.

  • Deterministic IB training is ill-posed or inaccessible to gradient-based optimization and does not capture simple decisions or robustness to noise.
  • Decision rules and stochastic DNNs resolve some difficulties, while better-behaved noisy objectives can yield robust representations and outputs compatible with simple decisions.
  • IB analysis cannot establish robustness or decision simplicity from a good functional value, and its ill-posed or piecewise-constant behavior further limits analysis.
  • Directly designed representation regularizers can support gradient-based training and target properties such as geometric clustering.
  • Representation-based regularization has been associated with generalization, robustness, fairness, calibration and task-adapted data augmentation.
  • Geometric representation regularization combined with noise or stochasticity is identified as a promising future research direction.

8 PROOF OF THEOREM 1

The proof establishes that deterministic DNN transformations preserve positive representation dimension under stated regularity conditions, leading to an infinite compression term.

  • For an N-dimensional input with continuous density and compact support, the correlation dimension is d2(X) = N.
  • The proof propagates positive correlation dimension through successive DNN layers using compact support and almost-everywhere linear transformations.
  • Bi-Lipschitz mappings do not change correlation dimension, enabling the recursive layer-by-layer argument.
  • The DNN output has positive information dimension under the relaxed input-distribution assumptions.
  • Because the deterministic conditional distribution of the next representation given X is discrete, d(Li+1|X) = 0.
  • Consequently, the mutual information satisfies I(X; Li+1) = ∞.

9 OUR PERSPECTIVE ON BOUNDING I(Y ; L)

This section reframes cross-entropy as a lower bound on I(Y; L), separating information lost by decoding from approximation loss introduced by the decision rule.

  • The setup permits a stochastic encoder L = f(X) and initially uses a deterministic decoder ˜Y = h(L).
  • The variational approximation QY|L is defined through the decoder output, which may be obtained by applying a simple decision rule to ˜Y.
  • The decomposition I(Y; L) = I(Y; ˜Y) + I(Y; L|˜Y) separates information retained by the decoded output from residual information in L.
  • The cross-entropy lower bound has two approximation losses: decoding L to ˜Y and replacing QY|L with QY|˜Y after the decision rule.
  • For stochastic decoding, the cross-entropy terms have different operational interpretations because losses may be averaged over samples of L and ˜Y.
  • For stochastic encoders, the alternative cross-entropy expression provides a tighter lower bound on I(Y; L) than the prevalent bound.
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