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Stochastic Second-Order Methods Improve Best-Known Sample Complexity of SGD for Gradient-Dominated Functions
Saeed Masiha · Saber Salehkaleybar · Niao He · Negar Kiyavash · Patrick Thiran

Tue Nov 29 09:00 AM -- 11:00 AM (PST) @ Hall J #826
We study the performance of Stochastic Cubic Regularized Newton (SCRN) on a class of functions satisfying gradient dominance property with $1\le\alpha\le2$ 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 $\epsilon$-global optimum is $\mathcal{O}(\epsilon^{-7/(2\alpha)+1})$ for $1\le\alpha< 3/2$ and $\mathcal{\tilde{O}}(\epsilon^{-2/(\alpha)})$ for $3/2\le\alpha\le 2$. 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 ${\mathcal{O}}(\epsilon^{-2/\alpha})$ for $1\le\alpha< 3/2$ 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.

Author Information

Saeed Masiha (Epfl)
Saber Salehkaleybar (EPFL)
Niao He (ETH Zurich)
Negar Kiyavash (École Polytechnique Fédérale de Lausanne)
Patrick Thiran (EPFL)

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