Limitations of Information-Theoretic Generalization Bounds for Gradient Descent Methods in Stochastic Convex Optimization
This work addresses a foundational limitation in machine learning theory for researchers, showing that current information-theoretic approaches are insufficient for analyzing gradient descent, which is incremental in highlighting gaps rather than providing solutions.
The paper tackles the problem of establishing minimax rates for gradient descent in stochastic convex optimization using information-theoretic generalization bounds, and proves that existing frameworks, including noisy surrogate methods, fail to achieve this.
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.