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We revisit the classic regret-minimization problem in the stochastic multi-armed bandit setting when the arm-distributions are allowed to be heavy-tailed. Regret minimization has been well studied in simpler settings of either bounded support reward distributions or distributions that belong to a single parameter exponential family. We work under the much weaker assumption that the moments of order $(1+epsilon)$ are uniformly bounded by a known constant B, for some given $epsilon > 0$. We propose an optimal algorithm that matches the lower bound exactly in the first-order term. We also give a finite-time bound on its regret. We show that our index concentrates faster than the well known truncated or trimmed empirical mean estimators for the mean of heavy-tailed distributions. Computing our index can be computationally demanding. To address this, we develop a batch-based algorithm that is optimal up to a multiplicative constant depending on the batch size. We hence provide a controlled trade-off between statistical optimality and computational cost.
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Most bandit algorithm designs are purely theoretical. Therefore, they have strong regret guarantees, but also are often too conservative in practice. In this work, we pioneer the idea of algorithm design by minimizing the empirical Bayes regret, the average regret over problem instances sampled from a known distribution. We focus on a tractable instance of this problem, the confidence interval and posterior width tuning, and propose an efficient algorithm for solving it. The tuning algorithm is analyzed and evaluated in multi-armed, linear, and generalized linear bandits. We report several-fold reductions in Bayes regret for state-of-the-art bandit algorithms, simply by optimizing over a small sample from a distribution.
This paper analyses the problem of Gaussian process (GP) bandits with deterministic observations. The analysis uses a branch and bound algorithm that is related to the UCB algorithm of (Srinivas et al., 2010). For GPs with Gaussian observation noise, with variance strictly greater than zero, (Srinivas et al., 2010) proved that the regret vanishes at the approximate rate of $O(frac{1}{sqrt{t}})$, where t is the number of observations. To complement their result, we attack the deterministic case and attain a much faster exponential convergence rate. Under some regularity assumptions, we show that the regret decreases asymptotically according to $O(e^{-frac{tau t}{(ln t)^{d/4}}})$ with high probability. Here, d is the dimension of the search space and $tau$ is a constant that depends on the behaviour of the objective function near its global maximum.
We design decentralized algorithms for regret minimization in the two-sided matching market with one-sided bandit feedback that significantly improves upon the prior works (Liu et al. 2020a, 2020b, Sankararaman et al. 2020). First, for general markets, for any $varepsilon > 0$, we design an algorithm that achieves a $O(log^{1+varepsilon}(T))$ regret to the agent-optimal stable matching, with unknown time horizon $T$, improving upon the $O(log^{2}(T))$ regret achieved in (Liu et al. 2020b). Second, we provide the optimal $Theta(log(T))$ agent-optimal regret for markets satisfying uniqueness consistency -- markets where leaving participants dont alter the original stable matching. Previously, $Theta(log(T))$ regret was achievable (Sankararaman et al. 2020, Liu et al. 2020b) in the much restricted serial dictatorship setting, when all arms have the same preference over the agents. We propose a phase-based algorithm, wherein each phase, besides deleting the globally communicated dominated arms the agents locally delete arms with which they collide often. This local deletion is pivotal in breaking deadlocks arising from rank heterogeneity of agents across arms. We further demonstrate the superiority of our algorithm over existing works through simulations.
We consider black box optimization of an unknown function in the nonparametric Gaussian process setting when the noise in the observed function values can be heavy tailed. This is in contrast to existing literature that typically assumes sub-Gaussian noise distributions for queries. Under the assumption that the unknown function belongs to the Reproducing Kernel Hilbert Space (RKHS) induced by a kernel, we first show that an adaptation of the well-known GP-UCB algorithm with reward truncation enjoys sublinear $tilde{O}(T^{frac{2 + alpha}{2(1+alpha)}})$ regret even with only the $(1+alpha)$-th moments, $alpha in (0,1]$, of the reward distribution being bounded ($tilde{O}$ hides logarithmic factors). However, for the common squared exponential (SE) and Mat{e}rn kernels, this is seen to be significantly larger than a fundamental $Omega(T^{frac{1}{1+alpha}})$ lower bound on regret. We resolve this gap by developing novel Bayesian optimization algorithms, based on kernel approximation techniques, with regret bounds matching the lower bound in order for the SE kernel. We numerically benchmark the algorithms on environments based on both synthetic models and real-world data sets.
This paper analyzes the problem of Gaussian process (GP) bandits with deterministic observations. The analysis uses a branch and bound algorithm that is related to the UCB algorithm of (Srinivas et al, 2010). For GPs with Gaussian observation noise, with variance strictly greater than zero, Srinivas et al proved that the regret vanishes at the approximate rate of $O(1/sqrt{t})$, where t is the number of observations. To complement their result, we attack the deterministic case and attain a much faster exponential convergence rate. Under some regularity assumptions, we show that the regret decreases asymptotically according to $O(e^{-frac{tau t}{(ln t)^{d/4}}})$ with high probability. Here, d is the dimension of the search space and tau is a constant that depends on the behaviour of the objective function near its global maximum.
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