Dynamic Regret via Discounted-to-Dynamic Reduction with Applications to Curved Losses and Adam Optimizer

We study dynamic regret minimization in non-stationary online learning, with a primary focus on follow-the-regularized-leader (FTRL) methods. FTRL is important for curved losses and for understanding adaptive optimizers such as Adam, yet existing dynamic regret analyses are less explored for FTRL. To address this, we build on the discounted-to-dynamic reduction and present a modular way to obtain dynamic regret bounds of FTRL-related problems. Specifically, we focus on two representative curved losses: linear regression and logistic regression. Our method not only simplifies existing proofs for the optimal dynamic regret of online linear regression, but also yields new dynamic regret guarantees for online logistic regression. Beyond online convex optimization, we apply the reduction to analyze the Adam optimizers, obtaining optimal convergence rates in stochastic, non-convex, and non-smooth settings. The reduction also enables a more detailed treatment of Adam with two discount parameters $(β_1,β_2)$, leading to new results for both clipped and clip-free variants of Adam optimizers.

Gradient-Variation Online Learning under Generalized Smoothness

Gradient-variation online learning aims to achieve regret guarantees that scale with the variations in the gradients of online functions, which has been shown to be crucial for attaining fast convergence in games and robustness in stochastic optimization, hence receiving increased attention. Existing results often require the smoothness condition by imposing a fixed bound on the gradient Lipschitzness, but this may not hold in practice. Recent efforts in neural network optimization suggest a generalized smoothness condition, allowing smoothness to correlate with gradient norms. In this paper, we systematically study gradient-variation online learning under generalized smoothness. To this end, we extend the classic optimistic mirror descent algorithm to derive gradient-variation bounds by conducting stability analysis over the optimization trajectory and exploiting smoothness locally. Furthermore, we explore universal online learning, designing a single algorithm enjoying optimal gradient-variation regrets for convex and strongly convex functions simultaneously without knowing curvature information. The algorithm adopts a two-layer structure with a meta-algorithm running over a group of base-learners. To ensure favorable guarantees, we have designed a new meta-algorithm that is Lipschitz-adaptive to handle potentially unbounded gradients and meanwhile ensures second-order regret to cooperate with base-learners. Finally, we provide implications of our findings and obtain new results in fast-rate games and stochastic extended adversarial optimization.

Efficient Methods for Non-stationary Online Learning

Non-stationary online learning has drawn much attention in recent years. In particular, dynamic regret and adaptive regret are proposed as two principled performance measures for online convex optimization in non-stationary environments. To optimize them, a two-layer online ensemble is usually deployed due to the inherent uncertainty of the non-stationarity, in which a group of base-learners are maintained and a meta-algorithm is employed to track the best one on the fly. However, the two-layer structure raises the concern about the computational complexity -- those methods typically maintain $\mathcal{O}(\log T)$ base-learners simultaneously for a $T$-round online game and thus perform multiple projections onto the feasible domain per round, which becomes the computational bottleneck when the domain is complicated. In this paper, we present efficient methods for optimizing dynamic regret and adaptive regret, which reduce the number of projections per round from $\mathcal{O}(\log T)$ to $1$. Moreover, our obtained algorithms require only one gradient query and one function evaluation at each round. Our technique hinges on the reduction mechanism developed in parameter-free online learning and requires non-trivial twists on non-stationary online methods. Empirical studies verify our theoretical findings.