Experimental design is crucial in evidence-based decision-making with multiple treatment arms, such as online advertisements and medical treatments. This study investigates the problem of identifying the treatment arm with the highest expected outcome, referred to as the best treatment arm, while minimizing the probability of misidentification. This problem has been studied across numerous research fields, including best arm identification (BAI) and ordinal optimization. In our experiments, the number of treatment-allocation rounds is fixed. During each round, a decision-maker allocates a treatment arm to an experimental unit and observes a corresponding outcome, which follows a Gaussian distribution with variances that can differ among the treatment arms. At the end of the experiment, we recommend one of the treatment arms as an estimate of the best treatment arm based on the observations. To design an experiment, we first discuss the worst-case lower bound for the probability of misidentification through an information-theoretic approach. Then, under the assumption that the variances are known, we propose the Generalized-Neyman-Allocation (GNA)-empirical-best-arm (EBA) strategy, an extension of the Neyman allocation proposed by Neyman (1934). We show that the GNA-EBA strategy is asymptotically optimal in the sense that its probability of misidentification aligns with the lower bounds as the sample size increases indefinitely and the differences between the expected outcomes of the best and other suboptimal arms converge to a uniform value. We refer to such strategies as asymptotically worst-case optimal.

The Perceptron algorithm, despite its simplicity, often performs well on online classification tasks. The Perceptron becomes especially effective when it is used in conjunction with kernels. However, a common difficulty encountered when implementing kernel-based online algorithms is the amount of memory required to store the online hypothesis, which may grow unboundedly. In this paper we present and analyze the Forgetron algorithm for kernel-based online learning on a fixed memory budget. To our knowledge, this is the first online learning algorithm which, on one hand, maintains a strict limit on the number of examples it stores while, on the other hand, entertains a relative mistake bound.

We consider the fixed-budget best arm identification problem in two-armed Gaussian bandits with unknown variances. The tightest lower bound on the complexity and an algorithm whose performance guarantee matches the lower bound have long been open problems when the variances are unknown and when the algorithm is agnostic to the optimal proportion of the arm draws. In this paper, we propose a strategy comprising a sampling rule with randomized sampling (RS) following the estimated target allocation probabilities of arm draws and a recommendation rule using the augmented inverse probability weighting (AIPW) estimator, which is often used in the causal inference literature. We refer to our strategy as the RS-AIPW strategy. In the theoretical analysis, we first derive a large deviation principle for martingales, which can be used when the second moment converges in mean, and apply it to our proposed strategy. Then, we show that the proposed strategy is asymptotically optimal in the sense that the probability of misidentification achieves the lower bound by Kaufmann et al. (2016) when the sample size becomes infinitely large and the gap between the two arms goes to zero.