lil UCB : An Optimal Exploration Algorithm for Multi-Armed Bandits

The paper proposes a novel upper confidence bound (UCB) procedure for identifying the arm with the largest mean in a multi-armed bandit game in the fixed confidence setting using a small number of total samples. The procedure cannot be improved in the sense that the number of samples required to identify the best arm is within a constant factor of a lower bound based on the law of the iterated logarithm (LIL). Inspired by the LIL, we construct our confidence bounds to explicitly account for the infinite time horizon of the algorithm. In addition, by using a novel stopping time for the algorithm we avoid a union bound over the arms that has been observed in other UCB-type algorithms. We prove that the algorithm is optimal up to constants and also show through simulations that it provides superior performance with respect to the state-of-the-art.

On Finding the Largest Mean Among Many

Sampling from distributions to find the one with the largest mean arises in a broad range of applications, and it can be mathematically modeled as a multi-armed bandit problem in which each distribution is associated with an arm. This paper studies the sample complexity of identifying the best arm (largest mean) in a multi-armed bandit problem. Motivated by large-scale applications, we are especially interested in identifying situations where the total number of samples that are necessary and sufficient to find the best arm scale linearly with the number of arms. We present a single-parameter multi-armed bandit model that spans the range from linear to superlinear sample complexity. We also give a new algorithm for best arm identification, called PRISM, with linear sample complexity for a wide range of mean distributions. The algorithm, like most exploration procedures for multi-armed bandits, is adaptive in the sense that the next arms to sample are selected based on previous samples. We compare the sample complexity of adaptive procedures with simpler non-adaptive procedures using new lower bounds. For many problem instances, the increased sample complexity required by non-adaptive procedures is a polynomial factor of the number of arms.

On the Limits of Sequential Testing in High Dimensions

This paper presents results pertaining to sequential methods for support recovery of sparse signals in noise. Specifically, we show that any sequential measurement procedure fails provided the average number of measurements per dimension grows slower then log s / D(f0||f1) where s is the level of sparsity, and D(f0||f1) the Kullback-Leibler divergence between the underlying distributions. For comparison, we show any non-sequential procedure fails provided the number of measurements grows at a rate less than log n / D(f1||f0), where n is the total dimension of the problem. Lastly, we show that a simple procedure termed sequential thresholding guarantees exact support recovery provided the average number of measurements per dimension grows faster than (log s + log log n) / D(f0||f1), a mere additive factor more than the lower bound.

Sequential Analysis in High Dimensional Multiple Testing and Sparse Recovery

This paper studies the problem of high-dimensional multiple testing and sparse recovery from the perspective of sequential analysis. In this setting, the probability of error is a function of the dimension of the problem. A simple sequential testing procedure is proposed. We derive necessary conditions for reliable recovery in the non-sequential setting and contrast them with sufficient conditions for reliable recovery using the proposed sequential testing procedure. Applications of the main results to several commonly encountered models show that sequential testing can be exponentially more sensitive to the difference between the null and alternative distributions (in terms of the dependence on dimension), implying that subtle cases can be much more reliably determined using sequential methods.