We consider the fixed-budget best arm identification problem where the goal is to find the arm of the largest mean with a fixed number of samples. It is known that the probability of misidentifying the best arm is exponentially small to the number of rounds. However, limited characterizations have been discussed on the rate (exponent) of this value. In this paper, we characterize the minimax optimal rate as a result of an optimization over all possible parameters. We introduce two rates, Rgo and Rgo, corresponding to lower bounds on the probability of misidentification, each of which is associated with a proposed algorithm. The rate Rgo is associated with Rgo-tracking, which can be efficiently implemented by a neural network and is shown to outperform existing algorithms. However, this rate requires a nontrivial condition to be achievable. To address this issue, we introduce the second rate Rgo . We show that this rate is indeed achievable by introducing a conceptual algorithm called delayed optimal tracking (DOT).

We study the problem of active pure exploration with fixed confidence in generic stochastic bandit environments. The goal of the learner is to answer a query about the environment with a given level of certainty while minimizing her sampling budget. For this problem, instance-specific lower bounds on the expected sample complexity reveal the optimal proportions of arm draws an Oracle algorithm would apply. These proportions solve an optimization problem whose tractability strongly depends on the structural properties of the environment, but may be instrumental in the design of efficient learning algorithms. We devise Frank-Wolfe-based Sampling (FWS), a simple algorithm whose sample complexity matches the lower bounds for a wide class of pure exploration problems. The algorithm is computationally efficient as, to learn and track the optimal proportion of arm draws, it relies on a single iteration of Frank-Wolfe algorithm applied to the lower-bound optimization problem. We apply FWS to various pure exploration tasks, including best arm identification in unstructured, thresholded, linear, and Lipschitz bandits. Despite its simplicity, FWSis competitive compared to state-of-art algorithms.

Pure exploration in multi-armed bandits has emerged as an important framework for modeling decision making and search under uncertainty. In modern applications however, one is often faced with a tremendously large number of options and even obtaining one observation per option may be too costly rendering traditional pure exploration algorithms ineffective. Fortunately, one often has access to similarity relationships amongst the options that can be leveraged. In this paper, we consider the pure exploration problem in stochastic multi-armed bandits where the similarities between the arms is captured by a graph and the rewards may be represented as a smooth signal on this graph. In particular, we consider the problem of finding the arm with the maximum reward (i.e., the maximizing problem) or one that has sufficiently high reward (i.e., the satisficing problem) under this model. We propose novel algorithms GRUB (GRaph based UcB) and ζ-GRUB for these problems and provide theoretical characterization of their performance which specifically elicits the benefit of the graph side information. We also prove a lower bound on the data requirement that shows a large class of problems where these algorithms are near-optimal. We complement our theory with experimental results that show the benefit of capitalizing on such side information.

The learner's goal is to identify the optimal arm

The knowledge gradient (KG) algorithm is a popular policy for the best arm identification (BAI) problem. It is built on the simple idea of always choosing the measurement that yields the greatest expected one-step improvement in the estimate of the best mean of the arms.