SGD for Structured Nonconvex Functions: Learning Rates, Minibatching and Interpolation

Robert Gower, Othmane Sebbouh, Nicolas Loizou
Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, PMLR 130:1315-1323, 2021.

Abstract

Stochastic Gradient Descent (SGD) is being used routinely for optimizing non-convex functions. Yet, the standard convergence theory for SGD in the smooth non-convex setting gives a slow sublinear convergence to a stationary point. In this work, we provide several convergence theorems for SGD showing convergence to a global minimum for non-convex problems satisfying some extra structural assumptions. In particular, we focus on two large classes of structured non-convex functions: (i) Quasar (Strongly) Convex functions (a generalization of convex functions) and (ii) functions satisfying the Polyak-Łojasiewicz condition (a generalization of strongly-convex functions). Our analysis relies on an Expected Residual condition which we show is a strictly weaker assumption than previously used growth conditions, expected smoothness or bounded variance assumptions. We provide theoretical guarantees for the convergence of SGD for different step-size selections including constant, decreasing and the recently proposed stochastic Polyak step-size. In addition, all of our analysis holds for the arbitrary sampling paradigm, and as such, we give insights into the complexity of minibatching and determine an optimal minibatch size. Finally, we show that for models that interpolate the training data, we can dispense of our Expected Residual condition and give state-of-the-art results in this setting.

Cite this Paper

BibTeX
@InProceedings{pmlr-v130-gower21a, title = { SGD for Structured Nonconvex Functions: Learning Rates, Minibatching and Interpolation }, author = {Gower, Robert and Sebbouh, Othmane and Loizou, Nicolas}, booktitle = {Proceedings of The 24th International Conference on Artificial Intelligence and Statistics}, pages = {1315--1323}, year = {2021}, editor = {Banerjee, Arindam and Fukumizu, Kenji}, volume = {130}, series = {Proceedings of Machine Learning Research}, month = {13--15 Apr}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v130/gower21a/gower21a.pdf}, url = {https://proceedings.mlr.press/v130/gower21a.html}, abstract = { Stochastic Gradient Descent (SGD) is being used routinely for optimizing non-convex functions. Yet, the standard convergence theory for SGD in the smooth non-convex setting gives a slow sublinear convergence to a stationary point. In this work, we provide several convergence theorems for SGD showing convergence to a global minimum for non-convex problems satisfying some extra structural assumptions. In particular, we focus on two large classes of structured non-convex functions: (i) Quasar (Strongly) Convex functions (a generalization of convex functions) and (ii) functions satisfying the Polyak-Łojasiewicz condition (a generalization of strongly-convex functions). Our analysis relies on an Expected Residual condition which we show is a strictly weaker assumption than previously used growth conditions, expected smoothness or bounded variance assumptions. We provide theoretical guarantees for the convergence of SGD for different step-size selections including constant, decreasing and the recently proposed stochastic Polyak step-size. In addition, all of our analysis holds for the arbitrary sampling paradigm, and as such, we give insights into the complexity of minibatching and determine an optimal minibatch size. Finally, we show that for models that interpolate the training data, we can dispense of our Expected Residual condition and give state-of-the-art results in this setting. } }
Endnote
%0 Conference Paper %T SGD for Structured Nonconvex Functions: Learning Rates, Minibatching and Interpolation %A Robert Gower %A Othmane Sebbouh %A Nicolas Loizou %B Proceedings of The 24th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2021 %E Arindam Banerjee %E Kenji Fukumizu %F pmlr-v130-gower21a %I PMLR %P 1315--1323 %U https://proceedings.mlr.press/v130/gower21a.html %V 130 %X Stochastic Gradient Descent (SGD) is being used routinely for optimizing non-convex functions. Yet, the standard convergence theory for SGD in the smooth non-convex setting gives a slow sublinear convergence to a stationary point. In this work, we provide several convergence theorems for SGD showing convergence to a global minimum for non-convex problems satisfying some extra structural assumptions. In particular, we focus on two large classes of structured non-convex functions: (i) Quasar (Strongly) Convex functions (a generalization of convex functions) and (ii) functions satisfying the Polyak-Łojasiewicz condition (a generalization of strongly-convex functions). Our analysis relies on an Expected Residual condition which we show is a strictly weaker assumption than previously used growth conditions, expected smoothness or bounded variance assumptions. We provide theoretical guarantees for the convergence of SGD for different step-size selections including constant, decreasing and the recently proposed stochastic Polyak step-size. In addition, all of our analysis holds for the arbitrary sampling paradigm, and as such, we give insights into the complexity of minibatching and determine an optimal minibatch size. Finally, we show that for models that interpolate the training data, we can dispense of our Expected Residual condition and give state-of-the-art results in this setting.
APA
Gower, R., Sebbouh, O. & Loizou, N.. (2021). SGD for Structured Nonconvex Functions: Learning Rates, Minibatching and Interpolation . Proceedings of The 24th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 130:1315-1323 Available from https://proceedings.mlr.press/v130/gower21a.html.

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