Limitations of Information-Theoretic Generalization Bounds for Gradient Descent Methods in Stochastic Convex Optimization

TL;DR

This paper demonstrates that existing information-theoretic bounds are insufficient for establishing minimax rates in stochastic convex optimization with gradient descent, highlighting the need for new analytical approaches.

Contribution

The paper shows the limitations of current information-theoretic frameworks in deriving minimax rates for gradient descent in stochastic convex optimization.

Findings

Existing bounds cannot establish minimax rates.

No analysis of noisy surrogate algorithms yields minimax rates.

New methods are needed for information-theoretic analysis of gradient descent.

Abstract

To date, no "information-theoretic" frameworks for reasoning about generalization error have been shown to establish minimax rates for gradient descent in the setting of stochastic convex optimization. In this work, we consider the prospect of establishing such rates via several existing information-theoretic frameworks: input-output mutual information bounds, conditional mutual information bounds and variants, PAC-Bayes bounds, and recent conditional variants thereof. We prove that none of these bounds are able to establish minimax rates. We then consider a common tactic employed in studying gradient methods, whereby the final iterate is corrupted by Gaussian noise, producing a noisy "surrogate" algorithm. We prove that minimax rates cannot be established via the analysis of such surrogates. Our results suggest that new ideas are required to analyze gradient descent using information-theoretic techniques.