MixedGrad: An O(1/T) Convergence Rate Algorithm for Stochastic Smooth Optimization
This work addresses the slow convergence rate in stochastic optimization for smooth functions, offering a significant improvement for machine learning and optimization practitioners, though it is incremental as it builds on existing oracle setups.
The paper tackles the problem of improving the convergence rate for stochastic optimization of smooth functions by introducing a mixed optimization setup that uses both stochastic and full gradient oracles, achieving an optimization error of O(1/T) with O(ln T) calls to the full gradient oracle and O(T) calls to the stochastic oracle.
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 $O(1/T)$.