Kernel-based methods for bandit convex optimization

TL;DR

This paper introduces a novel kernel-based algorithm for adversarial convex bandit problems, achieving improved regret bounds and polynomial-time complexity, advancing the state-of-the-art in derivative-free optimization.

Contribution

The paper presents the first polynomial-time algorithm with sublinear regret for adversarial convex bandit problems, using new kernel methods and annealing schedules.

Findings

Achieves $ ilde{O}(n^{9.5} \sqrt{T})$-regret with polynomial time

A variant runs in polynomial time with additional regret factors

Improves upon previous regret and time complexity bounds

Abstract

We consider the adversarial convex bandit problem and we build the first $\mathrm{poly}(T)$-time algorithm with $\mathrm{poly}(n) \sqrt{T}$-regret for this problem. To do so we introduce three new ideas in the derivative-free optimization literature: (i) kernel methods, (ii) a generalization of Bernoulli convolutions, and (iii) a new annealing schedule for exponential weights (with increasing learning rate). The basic version of our algorithm achieves $\tilde{O}(n^{9.5} \sqrt{T})$-regret, and we show that a simple variant of this algorithm can be run in $\mathrm{poly}(n \log(T))$-time per step at the cost of an additional $\mathrm{poly}(n) T^{o(1)}$ factor in the regret. These results improve upon the $\tilde{O}(n^{11} \sqrt{T})$-regret and $\exp(\mathrm{poly}(T))$-time result of the first two authors, and the $\log(T)^{\mathrm{poly}(n)} \sqrt{T}$-regret and $\log(T)^{\mathrm{poly}(n)}$-time result of Hazan and Li. Furthermore we conjecture that another variant of the algorithm could achieve $\tilde{O}(n^{1.5} \sqrt{T})$-regret, and moreover that this regret is unimprovable (the current best lower bound being $\Omega(n \sqrt{T})$ and it is achieved with linear functions). For the simpler situation of zeroth order stochastic convex optimization this corresponds to the conjecture that the optimal query complexity is of order $n^3 / \epsilon^2$.