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arXiv · 1907.06550

A Dimension-free Algorithm for Contextual Continuum-armed Bandits

Wenhao Li·Ningyuan Chen·L. Jeff Hong
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Abstract

In contextual continuum-armed bandits, the contexts $x$ and the arms $y$ are both continuous and drawn from high-dimensional spaces. The payoff function to learn $f(x,y)$ does not have a particular parametric form. The literature has shown that for Lipschitz-continuous functions, the optimal regret is $\tilde{O}(T^{\frac{d_x+d_y+1}{d_x+d_y+2}})$, where $d_x$ and $d_y$ are the dimensions of contexts and arms, and thus suffers from the curse of dimensionality. We develop an algorithm that achieves regret $\tilde{O}(T^{\frac{d_x+1}{d_x+2}})$ when $f$ is globally concave in $y$. The global concavity is a common assumption in many applications. The algorithm is based on stochastic approximation and estimates the gradient information in an online fashion. Our results generate a valuable insight that the curse of dimensionality of the arms can be overcome with some mild structures of the payoff function.

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Wenhao Li, Ningyuan Chen, L. Jeff Hong. 2022-10-03. A Dimension-free Algorithm for Contextual Continuum-armed Bandits. https://arxiv.org/abs/1907.06550

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