Efficient Stochastic Approximation of Minimax Excess Risk Optimization

Lijun Zhang, Haomin Bai, Wei-Wei Tu, Ping Yang, Yao Hu
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:58599-58630, 2024.

Abstract

While traditional distributionally robust optimization (DRO) aims to minimize the maximal risk over a set of distributions, Agarwal & Zhang (2022) recently proposed a variant that replaces risk with excess risk. Compared to DRO, the new formulation—minimax excess risk optimization (MERO) has the advantage of suppressing the effect of heterogeneous noise in different distributions. However, the choice of excess risk leads to a very challenging minimax optimization problem, and currently there exists only an inefficient algorithm for empirical MERO. In this paper, we develop efficient stochastic approximation approaches which directly target MERO. Specifically, we leverage techniques from stochastic convex optimization to estimate the minimal risk of every distribution, and solve MERO as a stochastic convex-concave optimization (SCCO) problem with biased gradients. The presence of bias makes existing theoretical guarantees of SCCO inapplicable, and fortunately, we demonstrate that the bias, caused by the estimation error of the minimal risk, is under-control. Thus, MERO can still be optimized with a nearly optimal convergence rate. Moreover, we investigate a practical scenario where the quantity of samples drawn from each distribution may differ, and propose a stochastic approach that delivers distribution-dependent convergence rates.

Cite this Paper

BibTeX
@InProceedings{pmlr-v235-zhang24d, title = {Efficient Stochastic Approximation of Minimax Excess Risk Optimization}, author = {Zhang, Lijun and Bai, Haomin and Tu, Wei-Wei and Yang, Ping and Hu, Yao}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, pages = {58599--58630}, year = {2024}, editor = {Salakhutdinov, Ruslan and Kolter, Zico and Heller, Katherine and Weller, Adrian and Oliver, Nuria and Scarlett, Jonathan and Berkenkamp, Felix}, volume = {235}, series = {Proceedings of Machine Learning Research}, month = {21--27 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v235/main/assets/zhang24d/zhang24d.pdf}, url = {https://proceedings.mlr.press/v235/zhang24d.html}, abstract = {While traditional distributionally robust optimization (DRO) aims to minimize the maximal risk over a set of distributions, Agarwal & Zhang (2022) recently proposed a variant that replaces risk with excess risk. Compared to DRO, the new formulation—minimax excess risk optimization (MERO) has the advantage of suppressing the effect of heterogeneous noise in different distributions. However, the choice of excess risk leads to a very challenging minimax optimization problem, and currently there exists only an inefficient algorithm for empirical MERO. In this paper, we develop efficient stochastic approximation approaches which directly target MERO. Specifically, we leverage techniques from stochastic convex optimization to estimate the minimal risk of every distribution, and solve MERO as a stochastic convex-concave optimization (SCCO) problem with biased gradients. The presence of bias makes existing theoretical guarantees of SCCO inapplicable, and fortunately, we demonstrate that the bias, caused by the estimation error of the minimal risk, is under-control. Thus, MERO can still be optimized with a nearly optimal convergence rate. Moreover, we investigate a practical scenario where the quantity of samples drawn from each distribution may differ, and propose a stochastic approach that delivers distribution-dependent convergence rates.} }
Endnote
%0 Conference Paper %T Efficient Stochastic Approximation of Minimax Excess Risk Optimization %A Lijun Zhang %A Haomin Bai %A Wei-Wei Tu %A Ping Yang %A Yao Hu %B Proceedings of the 41st International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2024 %E Ruslan Salakhutdinov %E Zico Kolter %E Katherine Heller %E Adrian Weller %E Nuria Oliver %E Jonathan Scarlett %E Felix Berkenkamp %F pmlr-v235-zhang24d %I PMLR %P 58599--58630 %U https://proceedings.mlr.press/v235/zhang24d.html %V 235 %X While traditional distributionally robust optimization (DRO) aims to minimize the maximal risk over a set of distributions, Agarwal & Zhang (2022) recently proposed a variant that replaces risk with excess risk. Compared to DRO, the new formulation—minimax excess risk optimization (MERO) has the advantage of suppressing the effect of heterogeneous noise in different distributions. However, the choice of excess risk leads to a very challenging minimax optimization problem, and currently there exists only an inefficient algorithm for empirical MERO. In this paper, we develop efficient stochastic approximation approaches which directly target MERO. Specifically, we leverage techniques from stochastic convex optimization to estimate the minimal risk of every distribution, and solve MERO as a stochastic convex-concave optimization (SCCO) problem with biased gradients. The presence of bias makes existing theoretical guarantees of SCCO inapplicable, and fortunately, we demonstrate that the bias, caused by the estimation error of the minimal risk, is under-control. Thus, MERO can still be optimized with a nearly optimal convergence rate. Moreover, we investigate a practical scenario where the quantity of samples drawn from each distribution may differ, and propose a stochastic approach that delivers distribution-dependent convergence rates.
APA
Zhang, L., Bai, H., Tu, W., Yang, P. & Hu, Y.. (2024). Efficient Stochastic Approximation of Minimax Excess Risk Optimization. Proceedings of the 41st International Conference on Machine Learning, in Proceedings of Machine Learning Research 235:58599-58630 Available from https://proceedings.mlr.press/v235/zhang24d.html.

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