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How Does Information Bottleneck Help Deep Learning?

Kenji Kawaguchi, Zhun Deng, Xu Ji, Jiaoyang Huang

arXiv:2305.18887v1cs.LGcs.AIcs.CLcs.CVcs.ITmath.IT

TL;DR

The paper addresses the lack of rigorous, valid sample-complexity bounds explaining why information bottlenecks may benefit deep learning. It proves bounds relating information bottlenecks to generalization for fixed and learned representations, and concludes that controlling them is one way—but not the only or necessary way—to control generalization.

  • Problem

    The literature lacked rigorous, valid sample-complexity bounds justifying why controlling information bottlenecks is desirable.

    Source: §1 · p. 2

  • Method

    The paper proves generalization bounds for fixed representations and for representations learned end-to-end, relating information bottleneck quantities to generalization.

    Source: §1 · p. 2

  • Results

    The paper completes the fixed-representation conjecture with an improved bound and establishes the first rigorous generalization bound for learned representations.

    Source: §1 · p. 2

  • Takeaways & Limitations

    Controlling information bottlenecks is one way to control deep-learning generalization, but it is not necessary.

    Source: §8 · p. 9

    The earlier conjecture assumes the encoder is fixed and independent of the training data; minimizing representation compression alone does not constrain a learned encoder's complexity.

    Source: §1 · p. 2

Abstract

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Numerous deep learning algorithms have been inspired by and understood via the notion of information bottleneck, where unnecessary information is (often implicitly) minimized while task-relevant information is maximized. However, a rigorous argument for justifying why it is desirable to control information bottlenecks has been elusive. In this paper, we provide the first rigorous learning theory for justifying the benefit of information bottleneck in deep learning by mathematically relating information bottleneck to generalization errors. Our theory proves that controlling information bottleneck is one way to control generalization errors in deep learning, although it is not the only or necessary way. We investigate the merit of our new mathematical findings with experiments across a range of architectures and learning settings. In many cases, generalization errors are shown to correlate with the degree of information bottleneck: i.e., the amount of the unnecessary information at hidden layers. This paper provides a theoretical foundation for current and future methods through the lens of information bottleneck. Our new generalization bounds scale with the degree of information bottleneck, unlike the previous bounds that scale with the number of parameters, VC dimension, Rademacher complexity, stability or robustness. Our code is publicly available at: https://github.com/xu-ji/information-bottleneck

1. Introduction

The introduction asks for a rigorous learning-theoretic justification of information bottlenecks and presents generalization bounds for both fixed and learned representations. It argues that representation simplicity and representation-function simplicity support generalization, while experiments find bound factors predict generalization gaps.

1. Introduction

  • The earlier conjecture is not valid for learned encoders: minimizing I(X; Zs_l) alone leaves encoder complexity unconstrained and permits overfitting.The cited counterexample concerns the case where encoder parameters are learned from the training data.Source: §1 · p. 2
  • The paper addresses the lack of rigorous, valid sample-complexity bounds explaining why controlling information bottlenecks is desirable..Source: §1 · p. 2
  • The paper gives the first rigorous generalization bound for learned representations, with simplicity of both the representation and its function supporting generalization.The authors describe the proofs as complete for end-to-end learning of intermediate representations.Source: §1 · p. 2
  • For fixed representations, the paper completes the earlier conjecture and improves its bound from exponential to linear dependence on mutual information.The revised bound also uses a smaller information quantity, I(X; Zs_l | Y).Source: §1 · p. 2
  • Experiments find that empirical estimates of the main factors in the paper's bounds are strong predictors of the generalization gap.Source: §1 · p. 2

2. Preliminaries

The preliminaries set up generalization analysis for a deep network learned from i.i.d. data and define representations at successive layers. They also assume finite input and layer-function spaces so the mutual information and resulting bounds are finite.

2. Preliminaries

  • The analysis concerns the gap between expected loss and training loss for a deep network learned from i.i.d. training data.The loss is bounded, and the examples are i.i.d. from a joint distribution over inputs and labels.Source: §2 · p. 2
  • Each layer’s representation is described using an encoder for the first layers and a decoder for the remaining layers.The encoder-decoder terminology does not require an explicit encoder-decoder architecture.Source: §2 · p. 3
  • The analysis presumes finite input and layer-function spaces, making the mutual information finite and the bounds nontrivial.The paper says these restrictions can be relaxed in later results.Source: §2 · p. 3

3. Main Results

The section develops sample-complexity bounds linking information bottlenecks to generalization error, first for fixed encoders and then for encoders learned from the training data. The main theorem treats the learned-encoder setting by relating input information to both label-conditioned representation information and encoder dependence on the training data.

3.1. Encoder independent with the training data

  • The fixed-encoder result also applies when representations are learned from data independent of the task’s training set, including some transfer-learning and unsupervised-learning settings.Fixed-feature learning is given as a practical example.Source: §3.1 · pp. 3–4
  • For fixed encoders independent of the training data, Theorem 1 completes the earlier conjecture and improves its bound by replacing exponential dependence with linear dependence.The bound uses I(X; Z_s |Y), the superfluous input information conditioned on the label.Source: §3.1 · p. 3

3.2. Encoder learned with the training data

  • The main theorem gives a generalization bound for encoders learned with the same training data, incorporating information terms for the representation and the encoder’s dependence on the sample.The bound holds with probability at least 1 − δ over the training set.Source: §3.2 · p. 4
  • The learned-encoder bound accounts for both input information retained by the representation and information the encoder obtains from the training dataset.The paper identifies encoder–dataset mutual information as necessary to address the counterexample to the fixed-encoder conjecture.Source: §3.2 · p. 4
  • Theorem 2 applies when the encoder uses the training set and potentially additional independent data, covering supervised, semi-supervised, unsupervised, and transfer learning.The paper notes that transfer learning may reduce encoder dependence on the target dataset.Source: §3.2 · pp. 4–5

4. Extensions

The extensions address settings where the input domain is infinite or mutual information is infinite. They establish finite-information cases for ReLU networks and adapt the bounds to handle infinite mutual information through finite approximations with a trade-off.

4. Extensions

  • The extensions broaden the argument beyond finite input domains, where mutual information may be infinite and require separate treatment.The paper says the finite-domain assumption ensures finite mutual information and nontrivial bounds.Source: §4 · p. 5

4.1. Neural networks with ReLU activation functions

  • For ReLU networks, infinitely many continuous input distributions yield finite I(X; Z|Y).This result concerns deterministic neural networks with ReLU activations.Source: §4.1 · p. 5

4.2. Modification for valid bounds in the case of infinite mutual information

  • When the original model has infinite mutual information, the theorems can still give finite bounds using a finite-information representation, provided the approximation error is finite.The paper notes that increasing bin size can lower estimated mutual information but raises the correction term, preventing arbitrary changes to the bounds through binning choices.Source: §4.2 · p. 5

5. Experiments

The experiments test whether representation compression predicts generalization alone, whether combining it with model compression improves prediction, and how layer choice affects these measures. Across the reported settings, combined compression metrics are more predictive than representation compression alone.

5.1. On the Representation Compression Bound

  • In constrained 2D classification experiments, combining model and representation compression yielded the best generalization predictor, outperforming representation compression alone.The study trained 216 models across architectures, weight-decay rates, dataset draws, and random seeds.Source: §5.1 · p. 6

5.2. Image Classification with DNNs

  • For CIFAR10, increasing layer index coincided with higher encoder–dataset mutual information and lower representation–input mutual information, revealing a trade-off between the measures.The figure reports layerwise metrics averaged across models and marks each metric’s minimum.Source: §5.2 · p. 7, Figure 2 · p. 7
  • On CIFAR10, metrics combining model and representation compression correlated more strongly with the generalization gap than either type alone, and the combined metric performed best overall.The minimum-over-layers summary also outperformed other layer summarization methods.Source: §5.2 · p. 7, Table 2 · p. 7, Table 3 · p. 7

5.3. Feature binning experiments

  • In deterministic-feature binning experiments, imposing the information bottleneck reduced the predictive performance of feature-compression metrics, while adding model compression increased it.The experiments covered 216 models across MLP architectures and hyperparameters.Source: §5.3 · p. 8, §A.2 · p. 15

6. Proof Sketch

The proof sketches outline how the paper derives a generalization bound for a fixed representation and extends the argument to learned representations and flexible layer choices.

6. Proof Sketch

  • The proof sketches give an overview of Theorems 1–2; complete proofs are provided in Appendix F.The sketches use probabilistic bounds over different sample spaces, which the authors say enable exponential improvement over previous bounds.Source: §6 · p. 8, §6.1 · p. 8

6.1. Proof Sketch of Theorem 1

  • For Theorem 1, the generalization gap is decomposed into two terms: one bounded using a typical set and one using concentration over that set.The sketch bounds the second term at roughly O(1/√n) and the first at roughly O((ln |T|)/n), with high probability.Source: §6.1 · p. 8

6.2. Proof Sketch of Theorem 2

  • Theorem 2 extends the bound to learned representations by constructing hypothesis spaces for candidate representations, applying Theorem 1, and taking union bounds across representations and layers.For the final layer, the sketch uses Hoeffding’s inequality instead.Source: §6.2 · pp. 8–9

7. Related Works

The related-work discussion distinguishes this paper’s information-bottleneck generalization bounds from both earlier information-bottleneck analyses and bounds using other information-theoretic quantities. It also contrasts the paper’s target with prior bounds for different losses or model-level information.

7. Related Works

  • Earlier information-bottleneck studies bounded estimation error for mutual information or a different encoder–decoder cross-entropy, rather than deep learning’s learned-network cross-entropy (Tishby & Zaslavsky, 2015).The paper states that the earlier cross-entropy bound requires n≫|X|², making it inapplicable in the stated regime.Source: §7 · p. 9
  • Prior bounds using I(f^S; S) concern information between the training set and the entire model, not the representation-level information bottleneck used here.The paper says those bounds therefore do not justify the information-bottleneck principle.Source: §7 · p. 9
  • The paper argues that Achille and Soatto’s noise-based quantity ignores dependence between learned parameters and the training set, which is central to the generalization question (Achille & Soatto, 2018).The authors note that setting the artificial noise variance to 1 makes that quantity zero even when learned parameters depend on the dataset.Source: §7 · p. 9
  • The authors present their work as the first generalization bound via the information bottleneck, distinct from prior bounds using other approaches.The cited approaches include VC dimension, Rademacher complexity, stability, robustness, and PAC-Bayes.Source: §7 · p. 9

8. Conclusion

The conclusion presents theoretical and empirical support for controlling information bottlenecks as one approach to understanding generalization in deep learning.

8. Conclusion

  • The paper completes the previous conjecture with near-exponential improvements for fixed representations and gives the first rigorous generalization bound for learned representations.The authors also report experiments supporting these findings.Source: §8 · p. 9
  • The authors frame the work as a technical contribution to deep-learning methods viewed through information bottlenecks.They note that information bottlenecks appear explicitly in some algorithms and may also arise implicitly through architectures and training.Source: §8 · p. 9
  • The conclusion identifies implicit effects of architectures such as transformers and convolution, and of training such as SGD and self-supervised objectives, as possible motivations for future methods.Source: §8 · p. 9

A.1. Training

The experiments use constrained optimization and several estimators to examine information bottleneck metrics, with additional results showing that larger samples improve baseline representation-compression metrics’ predictive ability.

A.1. Training

  • The constrained-optimization experiments used 5-way classification on clustered 2D inputs and retained 180 models after filtering low-training-accuracy models.The 216-model sweep varied MLP architecture, weight decay, dataset draw, seed, and sample-set size; models below 85% training accuracy were discarded.Source: §A.1 · p. 13
  • During constrained optimization, updating multiplier λ adds negligible cost because its gradient is already a term in the parameter gradient.Figure 4 gives example plots of estimated I(X; Z_s^l) and λ during training at ρ = 1.5.Source: §A.1 · p. 13, Figure 4 · p. 14

A.2. Metrics

  • The metric procedure averages in the log domain using a Jensen upper bound, improving numerical stability over probability-domain averaging.The passage attributes the improvement to large magnitudes of log p(w_j|s).Source: §A.2 · p. 14

A.3. Additional results

  • Using 500 generator samples instead of 50 training points improved the predictive ability of baseline representation-compression metrics.The larger evaluation sample was drawn from the known toy-data generator.Source: §A.3 · p. 15

B. MNIST and Fashion MNIST

Experiments on stochastic feature models for MNIST and Fashion MNIST found that combining model and representation compression produced more predictive metrics than representation compression alone.

B. MNIST and Fashion MNIST

  • Metrics combining model compression with representation compression outperformed metrics based on representation compression alone on MNIST and Fashion MNIST.The experiments used unconstrained stochastic feature models.Source: B. MNIST and Fashion MNIST · p. 16
  • The study trained 144 models across two datasets, three architectures, two weight-decay rates, two batch sizes, three dataset draws, and two random seeds.Each sampled training set contained 8K examples; each test set was the original 10K test set.Source: B. MNIST and Fashion MNIST · p. 16
  • The reported comparison concerns metric correlations with generalization-gap loss for MNIST and Fashion MNIST models.Tables 9 and 10 report those correlations, with the penultimate layer fixed.Source: B. MNIST and Fashion MNIST · p. 16, Table 9 · p. 17, Table 10 · p. 17

C. Feature binning

The experiments test compression metrics with binned deterministic features and convolutional-network representations, including adaptive kernel-density estimation for continuous features.

C. Feature binning

  • Feature binning discretized each node’s activity into 10 buckets to estimate mutual information for deterministic-feature MLPs.The experiments reused the binning implementation of Saxe et al. (2019) and trained 216 models across architectures, weight decay, dataset draws, seeds, and regularization settings.Source: C. Feature binning · pp. 17–18
  • With information-bottleneck regularization, feature-compression-only metrics performed significantly worse, while feature and model compression both performed reasonably without it.The comparison used feature binning on the toy clustering task.Source: C. Feature binning · p. 18

D.1. Training

  • The PreResNet study trained 540 models across three architectures, three weight-decay rates, three batch sizes, five dataset draws, and four seeds.Models were trained for 200 epochs with SGD at a learning rate of 1e−2.Source: §D.1 · pp. 18–19, Table 13 · p. 19

D.2. Metrics

  • The metrics analysis also tested defining the learning algorithm without the seed by averaging across seeds.The seed variable and set of seeds were denoted Γ and G, respectively.Source: §D.2 · p. 19

D.3. Kernel Density Estimation

  • Adaptive kernel-density estimation adds layer-scaled Gaussian noise and selects its variance to keep estimated mutual information monotonic across layers.The base noise variance was 1e−3; layer-wise variance selection used maximum likelihood under the monotonicity constraint.Source: §D.3 · p. 19

D.4. Further results

Further results include a reported clustering pattern attributed mostly to batch size and weight decay, alongside additional metric results and theoretical explanations in the appendices.

D.4. Further results

  • The illustrated performance-based clustering behavior is attributed mostly to batch size and weight decay.Figure 6 presents the illustration.Source: Figure 6 · p. 20
  • The further-results tables cover prediction metrics, model-compression metrics, and representation-compression metrics with alternative layer summaries.The listed representation summaries include l ∈ {1, D} and mean, minimum, and maximum across layers.Source: Table 14 · p. 20, Table 16 · p. 21, Table 17 · p. 21

E. Additional Results and Explanations for Theory

E.1. On Theorems 1–2

This section discusses the bounds and proofs for Theorems 1–2, including encoder overfitting, randomness models, and connections to earlier results. It outlines typical-set proof techniques and lemmas for bounding atypical mass, decomposing expected loss, and completing the theorem proofs.

E.1. On Theorems 1–2

  • Theorem 2’s I(ϕ_S; S) term measures encoder overfitting, which is needed to avoid the Hafez-Kolahi et al. (2020, Example 3.1) counterexample; Conjecture 1’s sample-complexity bound is invalid in this setting.The bound accounts for overfitting of the encoder, unlike the conjectured bound discussed here.Source: §E.1 · p. 24

E.1.1. ADDITIONAL NOTATION

  • The appendix models each input as a label-dependent function of nuisance variables and defines layer sensitivity with respect to those variables.It also introduces random variables for the nuisance variables and associated input representations.Source: §E.1.1 · p. 24

E.1.2. DETAILS OF OTHER TERMS

  • Proposition 2 bounds G_3 by a constant independent of the size of the latent-variable set.The passage also characterizes this constant as much smaller than that set’s size.Source: §E.1.2 · p. 24
  • Proposition 3 shows that under fast probability decay, entropy and C_{λ_l,l} are bounded independently of N, while under slow decay entropy grows as ln(N) + O(1) but C_{λ_l,l} remains bounded.The fast-decay condition is p_i ≤ C/i^α for α > 1; the slow-decay case has p_i = c_i/(Z i^α) for 0 ≤ α < 1.Source: §E.1.2 · p. 25
  • The theorems allow stochastic or deterministic models of the same real-world learning algorithm, but changing the model trades algorithmic randomness against mutual information.For example, fixing a random seed gives a deterministic view, whereas treating seeds as random gives a stochastic view.Source: §E.1.2 · pp. 25–26

E.2. On the application to the case of infinite mutual information

  • Bounds for an original model with infinite mutual information can use a nearby finite-mutual-information model plus a distance penalty, formalizing binning and noise-injection approaches.The authors describe these mutual-information estimation methods as theoretically valid even when the original model’s mutual information is infinite.Source: §E.2 · pp. 26–27
  • For Gaussian noise injection, the bound trades lower mutual information against a larger error penalty as noise increases.The cited construction adds λϑ to the original encoder output and uses a distance term proportional to λL∥ϑ∥₂.Source: §E.2 · p. 27

E.3. On comparisons with previous information-theoretic bounds

  • Setting D = {D + 1} in Theorem 2 recovers earlier bounds, which consider singleton decoder hypothesis spaces; the paper instead uses encoder information bottlenecks for non-singleton decoder spaces.The earlier bounds do not require a bottleneck to control decoder hypothesis-space complexity when the decoder space is a singleton.Source: §E.3 · p. 27
  • The bound’s I(X; Zs_l | Y) is a better predictor than the previous bound’s I(ϕs_l; S) and captures a tradeoff between the two information quantities.Minimizing the information bottleneck while minimizing training loss can come at the cost of increasing I(ϕs_l; S).Source: §E.3 · pp. 27–28

E.4. On the standard arguments for proving the conjecture

  • The standard PAC finite-hypothesis-space argument for the conjecture incurs an exponential factor 2^I(X;Zs_l).Source: §E.4 · p. 28

F.1. Overview of Proofs of Theorems

  • The proof combines typical-set properties, McDiarmid’s inequality, and a four-term expected-loss decomposition to avoid an exponential factor and dependence on the full hypothesis space.The bound’s I(X; Zs_l) is independent of the hypothesis space once the final hypothesis ϕs_l is given.Source: §F.1 · p. 28

F.2.1. PROBABILITY OF GOING OUTSIDE OF THE TYPICAL SUBSET

  • For any γ > 0, the conditional probability of leaving the typical subset is bounded by γ√n.Source: §F.2.1 · p. 29

F.2.2. SIZE OF THE TYPICAL SUBSET

  • A lemma bounds the size of the typical subset, though the supplied passage gives no bound’s expression.Source: §F.2.2 · p. 30

F.2.3. DECOMPOSITION OF EXPECTED LOSS USING THE TYPICAL SUBSET

  • Lemma 3 deterministically decomposes expected loss using the typical subset.The proof overview describes four terms: A for inputs in the typical set, and B, C, and D for inputs outside it.Source: §F.1 · p. 28, §F.2.3 · p. 31

F.2.4. BOUNDING THE THIRD AND FORTH TERMS IN THE DECOMPOSITION

  • Lemma 4 bounds the third and fourth decomposition terms by γ√n for any γ > 0.Source: §F.2.4 · pp. 33–34

F.2.5. BOUNDING THE FIRST AND SECOND TERM IN THE DECOMPOSITION

  • With Φ fixed independently of the training sample, Lemma 5 bounds the first two terms with probability at least 1−δ, and Lemma 6 combines Lemmas 3–5.Source: §F.2.5 · p. 34, §F.2.6 · p. 37

F.2.6. COMBINE LEMMAS

  • Combining Lemmas 3–5 yields the desired high-probability statement for each layer.The statement holds for any γ > 0 and δ > 0 when the specified encoder condition is met.Source: §F.2.6 · pp. 37, 39

F.3. Completing the Proof of Theorem 1 with Key Lemmas

  • Lemma 6 completes Theorem 1’s proof, and Theorem 2 follows directly from that lemma.For Theorem 1, the encoder is fixed independently of the training dataset, allowing Lemma 6 to be invoked with a singleton encoder class.Source: §F.2.6 · p. 37, §F.3 · p. 39

F.4. Completing the Proof of Theorem 2 with Key Lemmas

  • Theorem 2’s proof proceeds by extending Lemma 6.Source: §F.4 · p. 39

F.4.1. FINDING A LIKELY SPACE OF ENCODER

  • For ε = (1/λ) ln(Cλ/δ), Lemma 7 bounds by δ the probability of falling outside the typical encoder subset.The bound is stated for any λ > 0.Source: §F.4.1 · p. 40

F.4.2. RESULT WITH FIXED LAYER INDEX

For a fixed layer index, combining Lemmas 6 and 7 yields Lemma 8, a high-probability result. The following proof sections complete the theorem and address related propositions.

F.4.2. RESULT WITH FIXED LAYER INDEX

  • Lemma 8 gives a probability-at-least-1−δ bound for a fixed layer, with its terms including the encoder’s information about the sample and G4.The supplied bound also includes a term involving G1 and factors involving n, δ, and the label set.Source: §F.4.2 · pp. 40–41

F.4.3. COMPLETING THE PROOF

  • Union bounds extend the fixed-layer result across the layer indices, while the final layer is handled using a finite encoder hypothesis space.The proof also gives a term involving ln(|Φ_{D+1,ϵ}|/δ)/(2n) for that final-layer case.Source: §F.4.3 · pp. 42–44

F.5. Proof of Remark 1

  • The Remark 1 proof distinguishes an encoder function from its parameter-vector representation.The text identifies the latter as the version of A_l that outputs the parameter vector.Source: §F.5 · p. 44

F.6. Proof of Corollary 1

  • Theorem 1–2 statements retain their original forms under the encoder choice considered in the corollary proof.The proof also states that the resulting Q quantities are finite when CE is finite.Source: §F.6 · p. 44

F.7. Proof of Proposition 1

  • For separable class components, the constructed ReLU network maps them to distinct outputs with bounded entropy, and the construction extends to infinitely many distributions.The text also connects mappings of classes to distinct points with the neural-collapse phenomenon observed in practice.Source: §F.7 · p. 45

F.8. Proof of Proposition 2

  • The Proposition 2 proof invokes a sequence bound v_k≤Ce^{−(k/β)^α}, with α≥1.This is the stated condition for applying Lemma 9 in the proof.Source: §F.8 · pp. 45–46

F.9. Proof of Proposition 3

  • The Proposition 3 proof states that H(A_s) diverges as ln N.A displayed expression in the proof contains a λ ln N term.Source: §F.9 · pp. 47–48
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