From Non-Convex to Strongly Convex: Curvature-Adaptive FTPL for Online Optimization

2026-06-01

AI Summary

This work addresses curvature-adaptive online optimization for non-convex loss functions, aiming to unify sublinear regret in general non-convex settings with logarithmic regret under strong convexity. The authors propose a novel curvature-adaptive Follow-the-Perturbed-Leader (FTPL) algorithm that, for the first time, incorporates a dynamic perturbation scale into the non-convex FTPL framework, automatically adapting to the local geometry of the loss without requiring prior knowledge of cumulative curvature. By combining time-varying perturbations, a Follow-the-Leader tuning rule, and an approximate offline optimization oracle, the method achieves an $O(\sqrt{T})$ regret bound for arbitrary Lipschitz non-convex losses. When cumulative curvature grows linearly—including the strongly convex case—and the oracle is sufficiently accurate, it further attains an $O(\log T)$ regret bound, which is shown to be information-theoretically optimal.

Abstract

Curvature adaptivity is a classical theme in online optimization: for convex Lipschitz losses, adaptive methods interpolate between the optimal $O(\sqrt{T})$ regret for general convex losses and $O(\log T)$ regret under strong convexity. Recent work has shown that Follow-the-Perturbed-Leader (FTPL) achieves optimal $O(\sqrt{T})$ regret even for online non-convex Lipschitz losses, assuming access to an approximate offline-optimization oracle, but these guarantees do not exploit curvature. We show that FTPL can be made curvature-adaptive in the non-convex setting, without knowing in advance how curvature will accumulate over time. Our algorithm replaces the fixed perturbation scale of standard FTPL with a time-varying scale chosen using only past information. We give a simple follow-the-leader tuning rule for this scale and show that it competes, up to constants, with the best choice in hindsight. The resulting method achieves $O(\sqrt{T})$ regret for arbitrary non-convex Lipschitz losses and improves as cumulative curvature grows; with sufficiently accurate oracle calls, it achieves $O(\log T)$ regret when cumulative curvature grows linearly, which includes the classical strongly convex regime. We complement these upper bounds with matching lower bounds for prescribed cumulative-curvature sequences, already for one-dimensional convex losses, showing that the tradeoff between worst-case non-convex regret and curvature-driven fast rates is intrinsic.

Problem

• online optimization
• non-convex losses
• curvature adaptivity
• regret bounds
• strong convexity

Innovation

• curvature adaptivity
• non-convex online optimization
• Follow-the-Perturbed-Leader
• dynamic perturbation scaling
• regret bounds