Anytime Online-to-Batch Conversions, Optimism, and Acceleration
This work provides a solution for practitioners needing reliable convergence guarantees in stochastic optimization, though it is incremental as it builds on existing online learning methods.
The paper tackles the problem of obtaining convergence guarantees for individual iterates in stochastic convex optimization by introducing a black-box modification to online learning algorithms, achieving a fast rate of O(L/T^{3/2} + σ/√T) for smooth losses and an optimal accelerated rate of Õ(L/T^2 + σ/√T) with automatic adaptation to parameters.
A standard way to obtain convergence guarantees in stochastic convex optimization is to run an online learning algorithm and then output the average of its iterates: the actual iterates of the online learning algorithm do not come with individual guarantees. We close this gap by introducing a black-box modification to any online learning algorithm whose iterates converge to the optimum in stochastic scenarios. We then consider the case of smooth losses, and show that combining our approach with optimistic online learning algorithms immediately yields a fast convergence rate of $O(L/T^{3/2}+σ/\sqrt{T})$ on $L$-smooth problems with $σ^2$ variance in the gradients. Finally, we provide a reduction that converts any adaptive online algorithm into one that obtains the optimal accelerated rate of $\tilde O(L/T^2 + σ/\sqrt{T})$, while still maintaining $\tilde O(1/\sqrt{T})$ convergence in the non-smooth setting. Importantly, our algorithms adapt to $L$ and $σ$ automatically: they do not need to know either to obtain these rates.