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

Mahdi Haghifam, Borja Rodríguez-Gálvez, Ragnar Thobaben, Mikael Skoglund, Daniel M. Roy, Gintare Karolina Dziugaite
Proceedings of The 34th International Conference on Algorithmic Learning Theory, PMLR 201:663-706, 2023.

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v201-haghifam23a, title = {Limitations of Information-Theoretic Generalization Bounds for Gradient Descent Methods in Stochastic Convex Optimization}, author = {Haghifam, Mahdi and Rodr\'iguez-G\'alvez, Borja and Thobaben, Ragnar and Skoglund, Mikael and M. Roy, Daniel and Karolina Dziugaite, Gintare}, booktitle = {Proceedings of The 34th International Conference on Algorithmic Learning Theory}, pages = {663--706}, year = {2023}, editor = {Agrawal, Shipra and Orabona, Francesco}, volume = {201}, series = {Proceedings of Machine Learning Research}, month = {20 Feb--23 Feb}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v201/haghifam23a/haghifam23a.pdf}, url = {https://proceedings.mlr.press/v201/haghifam23a.html}, 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.} }
Endnote
%0 Conference Paper %T Limitations of Information-Theoretic Generalization Bounds for Gradient Descent Methods in Stochastic Convex Optimization %A Mahdi Haghifam %A Borja Rodríguez-Gálvez %A Ragnar Thobaben %A Mikael Skoglund %A Daniel M. Roy %A Gintare Karolina Dziugaite %B Proceedings of The 34th International Conference on Algorithmic Learning Theory %C Proceedings of Machine Learning Research %D 2023 %E Shipra Agrawal %E Francesco Orabona %F pmlr-v201-haghifam23a %I PMLR %P 663--706 %U https://proceedings.mlr.press/v201/haghifam23a.html %V 201 %X 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.
APA
Haghifam, M., Rodríguez-Gálvez, B., Thobaben, R., Skoglund, M., M. Roy, D. & Karolina Dziugaite, G.. (2023). Limitations of Information-Theoretic Generalization Bounds for Gradient Descent Methods in Stochastic Convex Optimization. Proceedings of The 34th International Conference on Algorithmic Learning Theory, in Proceedings of Machine Learning Research 201:663-706 Available from https://proceedings.mlr.press/v201/haghifam23a.html.

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