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Generalization Error Bounds for Noisy, Iterative Algorithms

Ankit Pensia, Varun Jog, Po-Ling Loh

arXiv:1801.04295v1cs.LGcs.ITstat.ML

TL;DR

The paper addresses how to bound generalization error for noisy iterative ERM algorithms beyond standard mutual-information results for empirical risk minimization. It derives bounds for bounded, noisy, Markovian updates and applies them to broad algorithmic outputs and sampling schemes. The framework covers SGLD and related noisy methods, but excludes ordinary SGD and certain degenerate-noise settings.

  • Problem

    Generalization error must be controlled when empirical risk minimization uses stochastic iterative algorithms whose outputs depend on training data.

    Source: §1 · p. 1

  • Method

    The paper combines mutual-information bounds with a chain-rule analysis of bounded, noisy Markovian updates and arbitrary functions of their iterates.

    Source: §1 · pp. 1–2, §3 · p. 5

  • Results

    The resulting bounds apply to SGLD and related noisy algorithms, arbitrary iterate-based outputs, and non-uniform data sampling.

    Source: §1 · p. 2, §2.3 · p. 4, §3.2 · p. 6

  • Takeaways & Limitations

    Mutual information provides a broad framework for analyzing generalization in noisy iterative ERM beyond stability-based settings.

    Source: §1 · p. 2, §5 · p. 9

    The bounds may be inapplicable or weak for ordinary SGD and degenerate noise such as uniform noise on a sphere.

    Source: §4.2 · p. 7, §5 · p. 9

Abstract

from arXiv · show

In statistical learning theory, generalization error is used to quantify the degree to which a supervised machine learning algorithm may overfit to training data. Recent work [Xu and Raginsky (2017)] has established a bound on the generalization error of empirical risk minimization based on the mutual information $I(S;W)$ between the algorithm input $S$ and the algorithm output $W$, when the loss function is sub-Gaussian. We leverage these results to derive generalization error bounds for a broad class of iterative algorithms that are characterized by bounded, noisy updates with Markovian structure. Our bounds are very general and are applicable to numerous settings of interest, including stochastic gradient Langevin dynamics (SGLD) and variants of the stochastic gradient Hamiltonian Monte Carlo (SGHMC) algorithm. Furthermore, our error bounds hold for any output function computed over the path of iterates, including the last iterate of the algorithm or the average of subsets of iterates, and also allow for non-uniform sampling of data in successive updates of the algorithm.

1 Introduction

The paper develops mutual-information generalization bounds for noisy, bounded iterative ERM algorithms with Markovian iterates. The framework covers flexible output functions and data-sampling strategies, including SGLD and related methods.

  • The paper derives generalization error bounds for iterative ERM algorithms whose updates add noise, have Markovian dependence, and remain bounded apart from noise.SGLD with bounded gradients is a principal example.Source: §1 · p. 1
  • Mutual-information analysis extends to arbitrary functions of the iterates, including final estimates and averages, while allowing data-dependent sampling that is agnostic to parameters.Source: §1 · p. 2
  • Compared with stability-based analyses, the framework covers broader update rules, averaged iterates, and non-uniform sampling schemes.The paper notes that stability-based SGLD bounds can be tighter in some cases.Source: §1 · p. 2
  • The paper’s analysis proceeds from mutual-information bounds to expected and high-probability generalization results, followed by examples for noisy algorithms.Source: §1 · p. 2

2 Problem setting

The paper formulates stochastic ERM as an iterative noisy update process and connects generalization to mutual information between the data and algorithm output. Its assumptions permit broad update maps, output functions, and sampling schemes while requiring bounded directions and parameter-agnostic sampling.

2.1 Preliminaries

  • The learning algorithm outputs an estimate from a conditional distribution given the training set, and generalization error measures expected empirical-risk deviation from actual risk.Source: §2.1 · p. 3

2.2 Generalization error bounds

  • Under sub-Gaussian losses, generalization error is controlled by mutual information between the training data and algorithm output [20].The paper uses this information-theoretic result as the basis for its iterative analysis.Source: §2.2 · p. 3

2.3 Class of learning algorithms

  • The algorithm iteratively samples data, applies a scaled direction, and adds isotropic Gaussian noise to update the parameter estimate.The update is represented by an equation whose preceding iterate is transformed by a deterministic map g.Source: §2.3 · p. 4
  • The framework assumes bounded update directions and sampling that is agnostic to previous parameter iterates, but it does not require convexity or a specific relationship between F and the loss.Source: §2.3 · pp. 4–5
  • Sampling may be non-uniform, time-varying, or dependent on earlier sampled data, but it must remain independent of previous parameter iterates.Adaptive sampling based on current parameter iterates can invalidate the paper’s conditional-independence relation and may promote overfitting.Source: §2.3 · pp. 4–5
  • The output may be any function of the iterates, including their mean, the last iterate, or suffix averages.Source: §2.3 · p. 5

3 Main results

The paper bounds mutual information between dataset inputs and outputs for noisy iterative algorithms, then converts that bound into expected and high-probability generalization guarantees.

3 Main results

  • Theorem 1 upper-bounds I(S; W) for the paper’s class of iterative algorithms using a chain-rule decomposition over the iterates.The derivation uses Lemmas 2–5 to justify the decomposition and intermediate bounds.Source: §3.1 · p. 5

3.1 Bound on mutual information

  • The mutual-information analysis can be simplified to remove a logarithmic term using log(1+x) ≤ x for x > 0.This follows from the inequality stated in the paper for positive x.Source: §3.1 · p. 6

3.2 Consequences

  • The mutual-information bound directly yields an expected generalization-error bound for the considered iterative algorithms.The paper obtains this result by combining Theorem 1 with Lemma 1.Source: §3.2 · p. 6
  • The resulting bounds connect the noisy iterative-algorithm analysis to the mutual-information framework of Xu and Raginsky [20].The high-probability consequence is obtained by invoking Theorem 3 in Xu and Raginsky [20].Source: §3 · p. 5, §3.2 · p. 6
  • The same framework also provides a high-probability generalization bound when I(S; W) ≤ ϵ.The probability is taken over both the dataset S ∼µ⊗n and the algorithm output W.Source: §3.2 · p. 6

4 Examples

The examples apply the paper’s mutual-information framework to SGLD, perturbed SGD, noisy momentum, and noisy accelerated gradient methods, deriving generalization bounds under their respective noise and bounded-update conditions.

Examples

  • The framework yields generalization-error bounds for a broad class of noisy iterative algorithms, including SGLD and momentum-like methods.The analysis treats iterates as a Markov process with bounded updates and uses mutual-information bounds to control generalization.Source: §4 · p. 6, §4.3 · p. 8

SGLD

  • For SGLD, the bounds apply under common multi-epoch schedules with time-varying stepsizes, while tighter prior bounds cover only the last iterate and uniform sampling.Mou et al. [9] obtain a tighter bound, but only for the last iterate WT and uniform sampling.Source: §4.1 · p. 6, Bounds in expectation · p. 7

Perturbed SGD

  • The results exclude ordinary SGD because every update must contain independent noise, but apply to perturbed SGD with continuous noise such as uniform noise in a ball.Uniform noise on the sphere is degenerate and makes the exact setting inapplicable, whereas uniform noise in the unit ball yields a bound.Source: §4.2 · p. 7

Noisy momentum and acceleration

  • Noisy momentum and noisy accelerated gradient methods receive the same mutual-information and generalization upper bound because both use bounded gradients and noisy updates in an expanded state space.The state is represented by the pair (Vt, Wt) in R2d; accelerated gradient changes only where the gradient is evaluated.Source: §4.3 · p. 8, §4.4 · p. 8

5 Conclusion

The paper presents mutual information as an effective tool for bounding generalization in noisy iterative ERM algorithms, while identifying important limitations for degenerate-noise settings and loose upper bounds.

Conclusion

  • Mutual information provides effective generalization-error bounds for a large class of iterative ERM algorithms.The analysis relies on the data processing inequality and the chain rule of mutual information.Source: §5 · p. 9

Limitations

  • Degenerate noise can make mutual-information bounds infinite or meaningless, excluding settings such as ordinary SGD and sphere-supported noise.The conclusion highlights the sensitivity of entropy and mutual information to degenerate random variables.Source: §5 · p. 9
  • The framework supplies only upper bounds, so large I(W; S) does not necessarily imply that an algorithm is overfitting.A loose upper bound can be large even when the algorithm is not overfitting.Source: §5 · p. 9

A Proofs of supporting lemmas to Theorem 1

The supporting lemmas bound the mutual information between the dataset and algorithm output through sampled data and Markovian iterate dependencies. They also derive entropy-based bounds under Gaussian noise and note the adjustment required for non-Gaussian noise.

B Details for the SGLD algorithm

This appendix provides additional derivation details for the SGLD analysis in Section 4.

B.1 Generalization error bounds

This subsection states a generalization-error lemma for a specified choice of β and α, under the condition n ≥64R4, and indicates that the result follows from Corollary 2.

  • For a given choice of {β, α}, Lemma 6 assumes n ≥64R4.Source: §B.1 · p. 13
  • The stated conclusion is obtained from Corollary 2.Source: §B.1 · p. 13
  • The derivation is presented as implying the desired result.Source: §B.1 · p. 13

B.2 Optimization error bounds

This subsection derives optimization-error bounds for averaged SGLD iterates on convex, L-Lipschitz losses, using the algorithm's unbiased gradient estimator and standard convex-optimization arguments.

B.2 Optimization error bounds

  • Lemma 7 considers SGLD on an L-Lipschitz convex loss for T time steps with parameters {η, σ}.Source: §B.2 · p. 13
  • The resulting bound concerns the empirical risk of the average of the iterates.Source: §B.2 · pp. 13–14
  • Convexity, Lipschitzness, and independent noise are combined to obtain the desired result.Source: §B.2 · pp. 13–14
  • SGLD's update uses Vt as an unbiased estimator of the empirical-risk gradient, E[Vt|Wt] = ∇LS(Wt).Source: §B.2 · p. 13
  • The optimization argument treats SGLD as a variant of SGD and invokes standard bounds for averaged iterates.Source: §B.2 · pp. 13–14

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