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Register a new userA Note on KL-UCB+ Policy for the Stochastic Bandit
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Junya Honda
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A classic setting of the stochastic K-armed bandit problem is considered in this note. In this problem it has been known that KL-UCB policy achieves the asymptotically optimal regret bound and KL-UCB+ policy empirically performs better than the KL-UCB policy although the regret bound for the original form of the KL-UCB+ policy has been unknown. This note demonstrates that a simple proof of the asymptotic optimality of the KL-UCB+ policy can be given by the same technique as those used for analyses of other known policies.
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We propose $tt RandUCB$, a bandit strategy that builds on theoretically derived confidence intervals similar to upper confidence bound (UCB) algorithms, but akin to Thompson sampling (TS), it uses randomization to trade off exploration and exploitation. In the $K$-armed bandit setting, we show that there are infinitely many variants of $tt RandUCB$, all of which achieve the minimax-optimal $widetilde{O}(sqrt{K T})$ regret after $T$ rounds. Moreover, for a specific multi-armed bandit setting, we show that both UCB and TS can be recovered as special cases of $tt RandUCB$. For structured bandits, where each arm is associated with a $d$-dimensional feature vector and rewards are distributed according to a linear or generalized linear model, we prove that $tt RandUCB$ achieves the minimax-optimal $widetilde{O}(d sqrt{T})$ regret even in the case of infinitely many arms. Through experiments in both the multi-armed and structured bandit settings, we demonstrate that $tt RandUCB$ matches or outperforms TS and other randomized exploration strategies. Our theoretical and empirical results together imply that $tt RandUCB$ achieves the best of both worlds.
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 algorithm. We give two general algorithms for this setting, which we show benefit from favorable regret guarantees. We show that the regret of the corralling algorithms is no worse than that of the best algorithm containing the arm with the highest reward, and depends on the gap between the highest reward and other rewards.
Many reinforcement learning algorithms can be seen
Balancing exploration and exploitation (EE) is a fundamental problem in contex-tual bandit. One powerful principle for EE trade-off isOptimism in Face of Uncer-tainty(OFU), in which the agent takes the action according to an upper confidencebound (UCB) of reward. OFU has achieved (near-)optimal regret bound for lin-ear/kernel contextual bandits. However, it is in general unknown how to deriveefficient and effective EE trade-off methods for non-linearcomplex tasks, suchas contextual bandit with deep neural network as the reward function. In thispaper, we propose a novel OFU algorithm namedregularized OFU(ROFU). InROFU, we measure the uncertainty of the reward by a differentiable function andcompute the upper confidence bound by solving a regularized optimization prob-lem. We prove that, for multi-armed bandit, kernel contextual bandit and neuraltangent kernel bandit, ROFU achieves (near-)optimal regret bounds with certainuncertainty measure, which theoretically justifies its effectiveness on EE trade-off.Importantly, ROFU admits a very efficient implementation with gradient-basedoptimizer, which easily extends to general deep neural network models beyondneural tangent kernel, in sharp contrast with previous OFU methods. The em-pirical evaluation demonstrates that ROFU works extremelywell for contextualbandits under various settings.
Out of the rich family of generalized linear bandits, perhaps the most well studied ones are logisitc bandits that are used in problems with binary rewards: for instance, when the learner/agent tries to maximize the profit over a user that can select one of two possible outcomes (e.g., `click vs `no-click). Despite remarkable recent progress and improved algorithms for logistic bandits, existing works do not address practical situations where the number of outcomes that can be selected by the user is larger than two (e.g., `click, `show me later, `never show again, `no click). In this paper, we study such an extension. We use multinomial logit (MNL) to model the probability of each one of $K+1geq 2$ possible outcomes (+1 stands for the `not click outcome): we assume that for a learners action $mathbf{x}_t$, the user selects one of $K+1geq 2$ outcomes, say outcome $i$, with a multinomial logit (MNL) probabilistic model with corresponding unknown parameter $bar{boldsymboltheta}_{ast i}$. Each outcome $i$ is also associated with a revenue parameter $rho_i$ and the goal is to maximize the expected revenue. For this problem, we present MNL-UCB, an upper confidence bound (UCB)-based algorithm, that achieves regret $tilde{mathcal{O}}(dKsqrt{T})$ with small dependency on problem-dependent constants that can otherwise be arbitrarily large and lead to loose regret bounds. We present numerical simulations that corroborate our theoretical results.
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