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

• URL: http://arxiv.org/abs/2212.13556v1
• Date: Tue, 27 Dec 2022 17:16:48 GMT
• Title: Limitations of Information-Theoretic Generalization Bounds for Gradient Descent Methods in Stochastic Convex Optimization
• Authors: Mahdi Haghifam, Borja Rodr\'iguez-G\'alvez, Ragnar Thobaben, Mikael Skoglund, Daniel M. Roy, Gintare Karolina Dziugaite
• Abstract summary: We consider the prospect of establishing minimax rates for gradient descent in the setting of convex optimization. We consider a common tactic employed in studying gradient methods, whereby the final iterate is corrupted by Gaussian noise, producing a noisy "surrogate" algorithm. Our results suggest that new ideas are required to analyze gradient descent using information-theoretic techniques.
• Score: 48.12845778927164
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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