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We propose to accelerate existing linear bandit algorithms to achieve per-step time complexity sublinear in the number of arms $K$. The key to sublinear complexity is the realization that the arm selection in many linear bandit algorithms reduces to the maximum inner product search (MIPS) problem. Correspondingly, we propose an algorithm that approximately solves the MIPS problem for a sequence of adaptive queries yielding near-linear preprocessing time complexity and sublinear query time complexity. Using the proposed MIPS solver as a sub-routine, we present two bandit algorithms (one based on UCB, and the other based on TS) that achieve sublinear time complexity. We explicitly characterize the tradeoff between the per-step time complexity and regret, and show that our proposed algorithms can achieve $O(K^{1-alpha(T)})$ per-step complexity for some $alpha(T) > 0$ and $widetilde O(sqrt{T})$ regret, where $T$ is the time horizon. Further, we present the theoretical limit of the tradeoff, which provides a lower bound for the per-step time complexity. We also discuss other choices of approximate MIPS algorithms and other applications to linear bandit problems.
We study the problem of stochastic combinatorial pure exploration (CPE), where an agent sequentially pulls a set of single arms (a.k.a. a super arm) and tries to find the best super arm. Among a variety of problem settings of the CPE, we focus on the
We study the problem of corralling stochastic bandit algorithms, that is combining multiple bandit algorithms designed for a stochastic environment, with the goal of devising a corralling algorithm that performs almost as well as the best base algori
We consider the problem of controlling a known linear dynamical system under stochastic noise, adversarially chosen costs, and bandit feedback. Unlike the full feedback setting where the entire cost function is revealed after each decision, here only
In the compressive phase retrieval problem, or phaseless compressed sensing, or compressed sensing from intensity only measurements, the goal is to reconstruct a sparse or approximately $k$-sparse vector $x in mathbb{R}^n$ given access to $y= |Phi x|
We analyze the popular kernel polynomial method (KPM) for approximating the spectral density (eigenvalue distribution) of an $ntimes n$ Hermitian matrix $A$. We prove that a simple and practical variant of the KPM algorithm can approximate the spectr