Minimax Regret for Bandit Convex Optimisation of Ridge Functions

• URL: http://arxiv.org/abs/2106.00444v1
• Date: Tue, 1 Jun 2021 12:51:48 GMT
• Title: Minimax Regret for Bandit Convex Optimisation of Ridge Functions
• Authors: Tor Lattimore
• Abstract summary: We analyse adversarial bandit convex optimisation with an adversary that is restricted to playing functions of the form $f(x) = g(langle x, thetarangle)$ for convex $g : mathbb R to mathbb R$ and $theta in mathbb Rd$. We provide a short information-theoretic proof that the minimax regret is at most $O(dsqrtn log(operatornamediammathcal K))$ where $n
• Score: 34.686687996497525
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We analyse adversarial bandit convex optimisation with an adversary that is restricted to playing functions of the form $f(x) = g(\langle x, \theta\rangle)$ for convex $g : \mathbb R \to \mathbb R$ and $\theta \in \mathbb R^d$. We provide a short information-theoretic proof that the minimax regret is at most $O(d\sqrt{n} \log(\operatorname{diam}\mathcal K))$ where $n$ is the number of interactions, $d$ the dimension and $\operatorname{diam}(\mathcal K)$ is the diameter of the constraint set. Hence, this class of functions is at most logarithmically harder than the linear case.

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