In this work, we explore online convex optimization (OCO) and introduce a new condition and analysis that provides fast rates by exploiting the curvature of feasible sets. In online linear optimization, it is known that if the average gradient of loss functions exceeds a certain threshold, the curvature of feasible sets can be exploited by the follow-the-leader (FTL) algorithm to achieve a logarithmic regret. This study reveals that algorithms adaptive to the curvature of loss functions can also leverage the curvature of feasible sets. In particular, we first prove that if an optimal decision is on the boundary of a feasible set and the gradient of an underlying loss function is non-zero, then the algorithm achieves a regret bound of O ( ρ ln T) in stochastic environments. Here, ρ > 0 is the radius of the smallest sphere that includes the optimal decision and encloses the feasible set. Our approach, unlike existing ones, can work directly with convex loss functions, exploiting the curvature of loss functions simultaneously, and can achieve the logarithmic regret only with a local property of feasible sets. Additionally, the algorithm achieves an O ( T) regret even in adversarial environments, in which FTL suffers an Ω( T) regret, and achieves an O ( ρ ln T + Cρ ln T) regret in corrupted stochastic environments with corruption level C .

Despite its popularity in practice, theoretical guarantees for high-dimensional problems are only available under strong structural assumptions (e.g., sparsity).

We develop a deeper understanding of its relationship with more conventional analyses of extragradient methods [Nem04, Nes07]. We also give improved solvers for the subproblems required by variants of the [She17] algorithm, designed through the lens of relative smoothness [BBT17, LFN18].

Work completed at the University of Washington. Work completed at Microsoft Research.

This approach can also be integrated into various change-of-variable-based models, such as V AEs or normalizing flows.

In many machine learning tasks, there are commonly sources of exogenous and endogenous distribution shift, necessitating that the algorithm be retrained repeatedly over time.