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arXiv · 2205.12856

Stochastic Second-Order Methods Improve Best-Known Sample Complexity of SGD for Gradient-Dominated Function

Saeed Masiha·Saber Salehkaleybar·Niao He·Negar Kiyavash·Patrick Thiran
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Abstract

We study the performance of Stochastic Cubic Regularized Newton (SCRN) on a class of functions satisfying gradient dominance property with $1\leα\le2$ which holds in a wide range of applications in machine learning and signal processing. This condition ensures that any first-order stationary point is a global optimum. We prove that the total sample complexity of SCRN in achieving $ε$-global optimum is $\mathcal{O}(ε^{-7/(2α)+1})$ for $1\leα< 3/2$ and $\mathcal{\tilde{O}}(ε^{-2/(α)})$ for $3/2\leα\le 2$. SCRN improves the best-known sample complexity of stochastic gradient descent. Even under a weak version of gradient dominance property, which is applicable to policy-based reinforcement learning (RL), SCRN achieves the same improvement over stochastic policy gradient methods. Additionally, we show that the average sample complexity of SCRN can be reduced to ${\mathcal{O}}(ε^{-2})$ for $α=1$ using a variance reduction method with time-varying batch sizes. Experimental results in various RL settings showcase the remarkable performance of SCRN compared to first-order methods.

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Saeed Masiha, Saber Salehkaleybar, Niao He, Negar Kiyavash, Patrick Thiran. 2023-01-20. Stochastic Second-Order Methods Improve Best-Known Sample Complexity of SGD for Gradient-Dominated Function. https://arxiv.org/abs/2205.12856

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