No abstract

The contextual bandit literature has traditionally focused on algorithms that address the exploration–exploitation tradeoff. In particular, greedy algorithms that exploit current estimates without any exploration may be suboptimal in general. However, exploration-free greedy algorithms are desirable in practical settings where exploration may be costly or unethical (e.g., clinical trials). Surprisingly, we find that a simple greedy algorithm can be rate optimal (achieves asymptotically optimal regret) if there is sufficient randomness in the observed contexts (covariates). We prove that this is always the case for a two-armed bandit under a general class of context distributions that satisfy a condition we term covariate diversity. Furthermore, even absent this condition, we show that a greedy algorithm can be rate optimal with positive probability. Thus, standard bandit algorithms may unnecessarily explore. Motivated by these results, we introduce Greedy-First, a new algorithm that uses only observed contexts and rewards to determine whether to follow a greedy algorithm or to explore. We prove that this algorithm is rate optimal without any additional assumptions on the context distribution or the number of arms. Extensive simulations demonstrate that Greedy-First successfully reduces exploration and outperforms existing (exploration-based) contextual bandit algorithms such as Thompson sampling or upper confidence bound. This paper was accepted by J. George Shanthikumar, big data analytics.

In this paper we study methods for estimating causal effects in settings with panel data, where a subset of units are exposed to a treatment during a subset of periods, and the goal is estimating counterfactual (untreated) outcomes for the treated unit/period combinations. We develop a class of matrix completion estimators that uses the observed elements of the matrix of control outcomes corresponding to untreated unit/periods to predict the "missing" elements of the matrix, corresponding to treated units/periods. The approach estimates a matrix that well-approximates the original (incomplete) matrix, but has lower complexity according to the nuclear norm for matrices. From a technical perspective, we generalize results from the matrix completion literature by allowing the patterns of missing data to have a time series dependency structure. We also present novel insights concerning the connections between the matrix completion literature, the literature on interactive fixed effects models and the literatures on program evaluation under unconfoundedness and synthetic control methods.

The contextual bandit literature has traditionally focused on algorithms that address the explorationexploitation tradeoff. In particular, greedy algorithms that exploit current estimates without any exploration may be sub-optimal in general. However, exploration-free greedy algorithms are desirable in practical settings where exploration may be costly or unethical (e.g., clinical trials). Surprisingly, we find that a simple greedy algorithm can be rate-optimal (achieves asymptotically optimal regret) if there is sufficient randomness in the observed contexts (covariates). We prove that this is always the case for a two-armed bandit under a general class of context distributions that satisfy a condition we term covariate diversity. Furthermore, even absent this condition, we show that a greedy algorithm can be rate optimal with positive probability. Thus, standard bandit algorithms may unnecessarily explore. Motivated by these results, we introduce Greedy-First, a new algorithm that uses only observed contexts and rewards to determine whether to follow a greedy algorithm or to explore. We prove that this algorithm is rate-optimal without any additional assumptions on the context distribution or the number of arms. Extensive simulations demonstrate that Greedy-First successfully reduces exploration and outperforms existing (exploration-based) contextual bandit algorithms such as Thompson sampling or upper confidence bound (UCB).

We provide a unifying view of statistical information measures, multiway Bayesian hypothesis testing, loss functions for multiclass classification problems and multidistribution f -divergences, elaborating equivalence results between all of these objects, and extending existing results for binary outcome spaces to more general ones. We consider a generalization of f -divergences to multiple distributions, and we provide a constructive equivalence between divergences, statistical information (in the sense of DeGroot) and losses for multiclass classification. A major application of our results is in multiclass classification problems in which we must both infer a discriminant function γ -for making predictions on a label Y from datum X-and a data representation (or, in the setting of a hypothesis testing problem, an experimental design), represented as a quantizer q from a family of possible quantizers Q. In this setting, we characterize the equivalence between loss functions, meaning that optimizing either of two losses yields an optimal discriminant and quantizer q, complementing and extending earlier results of Nguyen et al. [Ann. Statist. 37 (2009) 876-904] to the multiclass case. Our results provide a more substantial basis than standard classification calibration results for comparing different losses: we describe the convex losses that are consistent for jointly choosing a data representation and minimizing the (weighted) probability of error in multiclass classification problems.