MixedGrad: An O(1/T) Convergence Rate Algorithm for Stochastic Smooth Optimization

Abstract

It is well known that the optimal convergence rate for stochastic optimization of smooth functions is $O(1/\sqrt{T})$, which is same as stochastic optimization of Lipschitz continuous convex functions. This is in contrast to optimizing smooth functions using full gradients, which yields a convergence rate of $O(1/T^2)$. In this work, we consider a new setup for optimizing smooth functions, termed as {\bf Mixed Optimization}, which allows to access both a stochastic oracle and a full gradient oracle. Our goal is to significantly improve the convergence rate of stochastic optimization of smooth functions by having an additional small number of accesses to the full gradient oracle. We show that, with an $O(\ln T)$ calls to the full gradient oracle and an $O(T)$ calls to the stochastic oracle, the proposed mixed optimization algorithm is able to achieve an optimization error of $O(1/T)$.

References in corpus (4)

• A Stochastic Gradient Method with an Exponential Convergence Rate for Finite Training Sets
• Stochastic Gradient Descent for Non-smooth Optimization: Convergence Results and Optimal Averaging Schemes
• Better Mini-Batch Algorithms via Accelerated Gradient Methods
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Cited by in corpus (2)

• Coordinate Descent with Arbitrary Sampling I: Algorithms and Complexity
• Randomized First-Order Methods for Saddle Point Optimization