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Information-theoretic analysis of generalization capability of learning algorithms

Aolin Xu, Maxim Raginsky

arXiv:1705.07809v2cs.LGcs.ITstat.ML

TL;DR

The paper asks how to understand and control learning generalization through the information exchanged between a training dataset and an algorithm’s output. It derives mutual-information upper bounds, extends prior results, and proposes regularized algorithms that balance empirical fit with generalization. The framework also yields concentration results and applies to algorithm design and composition.

  • Problem

    Learning algorithms must achieve low population risk despite observing only empirical risk, creating a generalization problem and a fit-versus-generalization trade-off.

  • Method

    The paper models learning algorithms as randomized channels and relates their generalization error to input-output mutual information, using this relation to design regularized algorithms.

  • Results

    The analysis extends Russo and Zou to uncountable hypothesis spaces, improves expected absolute-generalization-error bounds, and derives concentration inequalities.

  • Takeaways & Limitations

    Controlling input-output mutual information provides theoretical guidance for balancing data fit and generalization, including through Gibbs and noisy ERM algorithms.

  • Takeaways & Limitations

    The initial theorems provide only upper bounds on expected generalization error, motivating stronger tools for absolute-error analysis.

Abstract

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We derive upper bounds on the generalization error of a learning algorithm in terms of the mutual information between its input and output. The bounds provide an information-theoretic understanding of generalization in learning problems, and give theoretical guidelines for striking the right balance between data fit and generalization by controlling the input-output mutual information. We propose a number of methods for this purpose, among which are algorithms that regularize the ERM algorithm with relative entropy or with random noise. Our work extends and leads to nontrivial improvements on the recent results of Russo and Zou.

1 Introduction

The paper frames learning algorithms as randomized channels and studies generalization through the information exchanged between training data and output hypotheses. It develops mutual-information bounds and design principles for balancing empirical fit against generalization.

  • Problem setup: Learning algorithms map training datasets to output hypotheses, and generalization error measures the difference between population and empirical risk.This difference quantifies how much the learned hypothesis overfits the training data.
  • Contributions: The paper derives upper bounds on generalization error using mutual information between the input dataset and algorithm output, extending Russo and Zou to uncountable hypothesis spaces.It also provides improved bounds on expected absolute generalization error and concentration inequalities.
  • Contributions: Controlling input-output mutual information formalizes the intuition that algorithms extracting less information from training data overfit less.The quantity depends on the dataset distribution, hypothesis space, algorithm, and potentially the loss function.
  • Algorithm design: Mutual-information regularization of ERM yields the Gibbs algorithm, while random-noise regularization provides another way to control input-output mutual information.The paper discusses calibrating both methods using prior knowledge of population risks.
  • Formal framework: The framework uses a Markov kernel to map an i.i.d. dataset from an unknown distribution to a random output hypothesis and evaluates population and empirical risks.Learning seeks low population risk in expectation or with high probability, while empirical risk serves as a computable proxy.
  • Problem setup: The expected population risk decomposes into an empirical-fit term and a generalization term, creating a trade-off that cannot generally be optimized simultaneously.Reducing population risk therefore requires balancing data fit and generalization.

2 Algorithmic stability in input-output mutual information

The paper defines stability through how much information an algorithm’s output reveals about its input dataset. Viewing the algorithm as a channel makes this stability equivalent to controlling its information capacity under product-form inputs.

  • Stability concepts: Traditional algorithmic stability means that small changes to an input produce small changes in the output.The paper contrasts this view with stability defined through input-output mutual information.
  • Information stability: ε-stability in input-output mutual information measures the algorithm through the information its output provides about the input dataset.Less output information about the dataset corresponds to greater stability.
  • Information stability: For the channel PW|S from Z^n to W, supµ I(S; W) is the channel’s information capacity when input distributions are constrained to product form.The stability definition therefore treats smaller constrained information capacity as greater algorithmic stability.

3 Upper-bounding generalization error via I(S; W)

The paper bounds generalization through information shared between the training dataset and learned hypothesis, extending prior results to uncountable hypothesis spaces and concentration guarantees. Controlling this information yields high-probability and expected-absolute-error bounds under subgaussian losses.

  • Information-theoretic framework: The learning problem is modeled as a channel from dataset S to hypothesis W, with mutual information linking dependence on data to generalization.Generalization error is population risk minus empirical risk, measuring overfitting.
  • Expected generalization error: Under σ-subgaussian losses, expected generalization is bounded using input-output mutual information, while bounded losses satisfy the condition automatically.The bound permits unbounded losses when the subgaussian assumption holds.
  • Information-theoretic framework: The framework extends Russo and Zou’s finite-space result to uncountable hypothesis spaces and provides improved expected absolute-error bounds.It also derives concentration inequalities not provided in the earlier result.
  • Concentration inequality: Theorem 3 gives high-probability guarantees when I(ΛW(S); W) ≤ ε, with sample complexity polynomial in 1/α and logarithmic in 1/β.The proof adapts the monitor technique and requires subgaussian rather than bounded losses.
  • Concentration inequality: For g(n) = 2, ε ≤ β log(2/β) yields n = (16σ^2/α^2) log(2/β), matching the order of the independent S and W case.A second setting gives n = (64σ^4/α^4)(log(2/β))^2.
  • Expected absolute error: Theorem 4 additionally upper-bounds expected absolute generalization error and improves a prior Russo–Zou bound.Markov’s inequality converts this result into a sufficient sample-complexity guarantee.

4 Learning algorithms with input-output mutual information stability

This section interprets learning algorithms through input-output mutual information and develops methods that control it while balancing empirical fit and generalization. It analyzes empirical-cover methods, Gibbs regularization, noisy ERM, and composition-based stability.

  • Information-theoretic stability: Input-output mutual information captures information extracted from the dataset and can guide the trade-off between empirical risk and generalization.The section contrasts this quantity with measures depending only on the hypothesis space or algorithm.
  • Binary Classification: The two-stage binary-classification algorithm controls conditional mutual information by constructing an empirical cover from one data split and selecting by empirical risk on the other.Its analysis uses I(S2; W|S1) ≤ log S_n1 ≤ V log(n1 + 1).
  • Gibbs algorithm: The Gibbs algorithm solves empirical-risk minimization regularized by an upper bound on input-output mutual information that uses a reference distribution Q.The resulting Gibbs distribution does not depend on the unknown data distribution µ.
  • Gibbs algorithm: For bounded loss, the Gibbs algorithm has an input-output mutual-information bound I(S; W) ≤ 2β, while tighter expected-generalization bounds are also available.The parameter β balances fitting and generalization, and the tighter bound is attributed to prior work.
  • Gibbs algorithm: The reference distribution Q expresses prior preferences over hypotheses, and better prior knowledge can reduce the sample complexity needed to achieve a target expected excess risk.For countable spaces, examples include ordered priors and uniform Q when no preference exists.
  • Noisy ERM: Noisy ERM adds independent hypothesis-specific noise to empirical risks, allowing preferred hypotheses to be selected more often when empirical risks are similar.Noisy ERM may be beneficial with high-quality prior knowledge of the optimal hypothesis and large hypothesis spaces.
  • Adaptive composition and other methods: For composed algorithms, the final generalization error can be controlled through conditional mutual information at each step, using local guarantees of constituent algorithms.Strong data-processing inequalities and preprocessing or postprocessing are identified as additional routes to sharper or induced stability.

A Proof of Lemma 1

The proof derives the mutual-information generalization bound from the variational representation of relative entropy and subgaussian control. It then identifies mutual information as relative entropy between the joint and product distributions.

  • Proof strategy: The proof uses the Donsker–Varadhan variational representation of relative entropy over measurable functions with integrable exponentials.This representation is attributed to the framework of Russo and Zou and is used as the main proof tool.
  • Proof strategy: The subgaussian assumption yields a quadratic inequality in λ whose nonpositive discriminant produces the desired bound.The argument converts concentration of the loss into an information-theoretic inequality.
  • Proof strategy: The proof concludes by using I(X; Y) = D(P_X,Y ∥ P_X ⊗ P_Y), identifying mutual information with relative entropy.

B Proof of Theorem 3

The proof of Theorem 3 combines independent parallel executions, a monitor construction, mutual-information chain rules, and data processing. The resulting argument transfers local information stability into a generalization guarantee under subgaussian losses.

  • Parallel execution: Parallel execution of m independent copies multiplies the input-output mutual-information bound from ε to mε.The bound applies when each copy satisfies I(Λ_W(S); W) ≤ ε.
  • Subgaussian control: Lemma B.2 supplies a bound independent of m when the loss is σ-subgaussian for every hypothesis.The proof obtains this through subgaussian scaling and Lemma 1.
  • Monitor technique: The monitor selects a hypothesis, an execution index, and a sign from the parallel outputs, with at most 2m possible conditional outcomes.This gives the auxiliary information bound I(Λ_W(S1), …, Λ_W(Sm); W*, T*, R*|W^m) ≤ log(2m).
  • Information bounds: The proof combines the monitor bound with chain-rule and data-processing inequalities to control information about the monitored output.The argument uses the assumed stability of each constituent algorithm through the parallel-execution lemma.
  • Contradiction argument: The contradiction argument sets m = ⌊1/β⌋ and shows that violating the claimed generalization property contradicts the information condition.The proof concludes when the resulting inequality conflicts with condition (16).

C Proof of Theorem 5

The proof solves the relaxed mutual-information-regularized empirical-risk problem by optimizing over conditional output distributions. Its solution is the Gibbs algorithm and does not depend on the unknown data distribution.

  • Optimization: The relaxed objective combines expected empirical risk with a relative-entropy penalty D(P_W|S=s∥Q) weighted by 1/β.The optimization is performed separately for each dataset s.
  • Gibbs solution: The optimization is convex, and its solution for each dataset s is the Gibbs algorithm.The Gibbs solution is expressed using the reference distribution Q and empirical loss.
  • Gibbs solution: Because the Gibbs solution depends on the empirical dataset and Q rather than µ, the relaxed optimization does not require the unknown data distribution.

D Proof of Corollary 2

The proof bounds the Gibbs algorithm’s expected empirical risk by comparing it with a dataset-independent point-mass algorithm, then derives the stated result using an upper bound on expected generalization error.

  • The Gibbs algorithm’s expected empirical risk is bounded using a point-mass algorithm that ignores the dataset and always outputs w.The comparison relies on Theorem 5.
  • Setting w = wo and using E[LS(wo)] = Lµ(wo) connects the dataset-independent comparison to the population risk.
  • For countable W, D(δwo∥Q) = −log Q(wo), yielding the divergence term used in (29).

E Proof of Corollary 3

The proof analyzes the Gibbs algorithm by comparing it with a Gaussian dataset-independent hypothesis distribution, applying nonnegative relative entropy, Lipschitzness, and the generalization bound to obtain (31).

  • A Gaussian distribution N(wo, a2Id) is treated as a dataset-independent learning algorithm for bounding the Gibbs algorithm’s expected empirical risk.The distribution draws hypotheses without using the dataset.
  • Nonnegativity of relative entropy and Theorem 5 provide the comparison inequalities for the Gaussian reference distribution.
  • The proof combines empirical-risk and population-risk integrals against N(w; wo, a2Id) with the expected generalization bound (28).
  • Because ℓ(·, z) is ρ-Lipschitz for every z, the resulting inequality can be substituted into (E.19) to obtain (31).

F Proof of Corollary 4

The proof of Corollary 4 first bounds expected generalization error through I(S; W), then bounds expected empirical risk and combines the two inequalities to obtain (34).

  • The argument assumes |W| = k; for countably infinite W, the proof replaces k with ∞.
  • Expected generalization error is upper-bounded through a chain of inequalities involving I(S; W).
  • The mutual-information step uses data processing, product-channel bounds, additive exponential noise-channel capacity, and E[LS(wi)] = Lµ(wi).
  • Since ℓ takes values in [0, 1], ℓ(w, Z) is 1/2-subgaussian, supporting the generalization analysis.
  • The proof separately upper-bounds expected empirical risk, combines the resulting inequalities, and uses log(1 + x) ≤ x to reach (34).
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