Online Newton Method for Bandit Convex Optimisation
Hidde Fokkema, Dirk van der Hoeven, Tor Lattimore, Jack J. Mayo
TL;DR
This work presents a computationally efficient approach to zeroth-order bandit convex optimisation by embedding constrained problems in an unconstrained online Newton framework. It builds a convex extension of the constrained losses via the Minkowski functional and couples it with a surrogate Gaussian-based learning procedure, complemented by a refined restart mechanism to handle adversarial losses. The method achieves a high-probability regret of $Reg_n\le d^{3.5}\sqrt{n}\,\mathrm{polylog}(n,d,1/\delta)$ in the adversarial setting and $Reg_n\le M d^{2}\sqrt{n}\,\mathrm{polylog}(n,d,1/\delta)$ in the stochastic setting, with computational efficiency under membership and sampling oracles for $K$. It also extends to bandit submodular minimisation via Lovász extensions and discusses practical and theoretical trade-offs between geometry, extension schemes, and restart strategies. Overall, the paper advances a constructive, geometry-aware framework for bandit convex optimisation that integrates convex extensions, surrogate losses, and adaptive restarts to achieve near-optimal regret bounds with polynomial-time implementability.
Abstract
We introduce a computationally efficient algorithm for zeroth-order bandit convex optimisation and prove that in the adversarial setting its regret is at most $d^{3.5} \sqrt{n} \mathrm{polylog}(n, d)$ with high probability where $d$ is the dimension and $n$ is the time horizon. In the stochastic setting the bound improves to $M d^{2} \sqrt{n} \mathrm{polylog}(n, d)$ where $M \in [d^{-1/2}, d^{-1 / 4}]$ is a constant that depends on the geometry of the constraint set and the desired computational properties.