A Simple, Optimal and Efficient Algorithm for Online Exp-Concave Optimization

Online eXp-concave Optimization (OXO) is a fundamental problem in online learning, where the goal is to minimize regret when loss functions are exponentially concave. The standard algorithm, Online Newton Step (ONS), guarantees an optimal $O(d \log T)$ regret, where $d$ is the dimension and $T$ is the time horizon. Despite its simplicity, ONS may face a computational bottleneck due to the Mahalanobis projection at each round. This step costs $Ω(d^ω)$ arithmetic operations for bounded domains, even for simple domains such as the unit ball, where $ω\in (2,3]$ is the matrix-multiplication exponent. As a result, the total runtime can reach $\tilde{O}(d^ωT)$, particularly when iterates frequently oscillate near the domain boundary. This paper proposes a simple variant of ONS, called LightONS, which reduces the total runtime to $O(d^2 T + d^ω\sqrt{T \log T})$ while preserving the optimal regret. Deploying LightONS with the online-to-batch conversion implies a method for stochastic exp-concave optimization with runtime $\tilde{O}(d^3/ε)$, thereby answering an open problem posed by Koren [2013]. The design leverages domain-conversion techniques from parameter-free online learning and defers expensive Mahalanobis projections until necessary, thereby preserving the elegant structure of ONS and enabling LightONS to act as an efficient plug-in replacement in broader scenarios, including gradient-norm adaptivity, parametric stochastic bandits, and memory-efficient OXO.

💡 Research Summary

The paper addresses a long‑standing computational bottleneck in Online eXp‑concave Optimization (OXO), where the classic Online Newton Step (ONS) algorithm, despite achieving the minimax‑optimal regret of (O(d\log T)), requires a Mahalanobis projection at every round. For bounded domains such as the unit ball, each projection costs (\Omega(d^{\omega})) arithmetic operations (with (\omega\in(2,3]) the matrix‑multiplication exponent), leading to a total runtime of (\tilde O(d^{\omega}T)). This becomes prohibitive when the iterates frequently touch the boundary of the feasible set.

The authors propose LightONS, a simple variant of ONS that dramatically reduces the computational cost while preserving the optimal regret. The key idea is to defer expensive Mahalanobis projections until they are truly necessary. LightONS defines an expanded ball (eX_k = {x\in\mathbb R^d : |x|_2 \le kD/2}) that contains the original domain (X) (with diameter (D)) and a deferral coefficient (k>1). At each round the algorithm computes the Newton‑style update \