Adaptive SGD with Polyak stepsize and Line-search: Robust Convergence and Variance Reduction

• URL: http://arxiv.org/abs/2308.06058v2
• Date: Mon, 21 Aug 2023 16:28:13 GMT
• Title: Adaptive SGD with Polyak stepsize and Line-search: Robust Convergence and Variance Reduction
• Authors: Xiaowen Jiang and Sebastian U. Stich
• Abstract summary: We propose two new variants of SPS and SLS, called AdaSPS and AdaSLS, which guarantee convergence in non-interpolation settings. We equip AdaSPS and AdaSLS with a novel variance reduction technique and obtain algorithms that require $smashwidetildemathcalO(n+1/epsilon)$ gradient evaluations.
• Score: 26.9632099249085
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The recently proposed stochastic Polyak stepsize (SPS) and stochastic line-search (SLS) for SGD have shown remarkable effectiveness when training over-parameterized models. However, in non-interpolation settings, both algorithms only guarantee convergence to a neighborhood of a solution which may result in a worse output than the initial guess. While artificially decreasing the adaptive stepsize has been proposed to address this issue (Orvieto et al. [2022]), this approach results in slower convergence rates for convex and over-parameterized models. In this work, we make two contributions: Firstly, we propose two new variants of SPS and SLS, called AdaSPS and AdaSLS, which guarantee convergence in non-interpolation settings and maintain sub-linear and linear convergence rates for convex and strongly convex functions when training over-parameterized models. AdaSLS requires no knowledge of problem-dependent parameters, and AdaSPS requires only a lower bound of the optimal function value as input. Secondly, we equip AdaSPS and AdaSLS with a novel variance reduction technique and obtain algorithms that require $\smash{\widetilde{\mathcal{O}}}(n+1/\epsilon)$ gradient evaluations to achieve an $\mathcal{O}(\epsilon)$-suboptimality for convex functions, which improves upon the slower $\mathcal{O}(1/\epsilon^2)$ rates of AdaSPS and AdaSLS without variance reduction in the non-interpolation regimes. Moreover, our result matches the fast rates of AdaSVRG but removes the inner-outer-loop structure, which is easier to implement and analyze. Finally, numerical experiments on synthetic and real datasets validate our theory and demonstrate the effectiveness and robustness of our algorithms.

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