We study the best-arm identification problem in multi-armed bandits with stochastic, potentially private rewards, when the goal is to identify the arm with the highest quantile at a fixed, prescribed level. First, we propose a (non-private) successive elimination algorithm for strictly optimal best-arm identification, we show that our algorithm is $\delta$-PAC and we characterize its sample complexity. Further, we provide a lower bound on the expected number of pulls, showing that the proposed algorithm is essentially optimal up to logarithmic factors. Both upper and lower complexity bounds depend on a special definition of the associated suboptimality gap, designed in particular for the quantile bandit problem, as we show when the gap approaches zero, best-arm identification is impossible. Second, motivated by applications where the rewards are private, we provide a differentially private successive elimination algorithm whose sample complexity is finite even for distributions with infinite support-size, and we characterize its sample complexity as well. Our algorithms do not require prior knowledge of either the suboptimality gap or other statistical information related to the bandit problem at hand.

We study the contextual linear bandit problem, a version of the standard stochastic multi-armed bandit (MAB) problem where a learner sequentially selects actions to maximize a reward which depends also on a user provided per-round context. Though the context is chosen arbitrarily or adversarially, the reward is assumed to be a stochastic function of a feature vector that encodes the context and selected action. Our goal is to devise private learners for the contextual linear bandit problem. So instead, we adopt the notion of joint differential privacy, where we assume that the action chosen on day t is only revealed to user t and thus needn't be kept private that day, only on following days. We give a general scheme converting the classic linear-UCB algorithm into a joint differentially private algorithm using the tree-based algorithm.

Learning the parameters of a Gaussian mixtures models is a fundamental and widely studied problem with numerous applications. In this work, we give new algorithms for learning the parameters of a high-dimensional, well separated, Gaussian mixture model subject to the strong constraint of differential privacy. In particular, we give a differentially private analogue of the algorithm of Achlioptas and McSherry. Our algorithm has two key properties not achieved by prior work: (1) The algorithm's sample complexity matches that of the corresponding non-private algorithm up to lower order terms in a wide range of parameters. (2) The algorithm does not require strong a priori bounds on the parameters of the mixture components.

We present a provably optimal differentially private algorithm for the stochastic multi-arm bandit problem, as opposed to the private analogue of the UCB-algorithm [Mishra and Thakurta, 2015; Tossou and Dimitrakakis, 2016] which doesn't meet the recently discovered lower-bound of $\Omega \left(\frac{K\log(T)}{\epsilon} \right)$ [Shariff and Sheffet, 2018]. Our construction is based on a different algorithm, Successive Elimination [Even-Dar et al. 2002], that repeatedly pulls all remaining arms until an arm is found to be suboptimal and is then eliminated. In order to devise a private analogue of Successive Elimination we visit the problem of private stopping rule, that takes as input a stream of i.i.d samples from an unknown distribution and returns a multiplicative $(1 \pm \alpha)$-approximation of the distribution's mean, and prove the optimality of our private stopping rule. We then present the private Successive Elimination algorithm which meets both the non-private lower bound [Lai and Robbins, 1985] and the above-mentioned private lower bound. We also compare empirically the performance of our algorithm with the private UCB algorithm.

We study the contextual linear bandit problem, a version of the standard stochastic multi-armed bandit (MAB) problem where a learner sequentially selects actions to maximize a reward which depends also on a user provided per-round context. Though the context is chosen arbitrarily or adversarially, the reward is assumed to be a stochastic function of a feature vector that encodes the context and selected action. Our goal is to devise private learners for the contextual linear bandit problem. We first show that using the standard definition of differential privacy results in linear regret. So instead, we adopt the notion of joint differential privacy, where we assume that the action chosen on day t is only revealed to user t and thus needn't be kept private that day, only on following days. We give a general scheme converting the classic linear-UCB algorithm into a joint differentially private algorithm using the tree-based algorithm. We then apply either Gaussian noise or Wishart noise to achieve joint-differentially private algorithms and bound the resulting algorithms' regrets. In addition, we give the first lower bound on the additional regret any private algorithms for the MAB problem must incur.

We study the contextual linear bandit problem, a version of the standard stochastic multi-armed bandit (MAB) problem where a learner sequentially selects actions to maximize a reward which depends also on a user provided per-round context. Though the context is chosen arbitrarily or adversarially, the reward is assumed to be a stochastic function of a feature vector that encodes the context and selected action. Our goal is to devise private learners for the contextual linear bandit problem. We first show that using the standard definition of differential privacy results in linear regret. So instead, we adopt the notion of joint differential privacy, where we assume that the action chosen on day t is only revealed to user t and thus needn't be kept private that day, only on following days. We give a general scheme converting the classic linear-UCB algorithm into a joint differentially private algorithm using the tree-based algorithm. We then apply either Gaussian noise or Wishart noise to achieve joint-differentially private algorithms and bound the resulting algorithms' regrets. In addition, we give the first lower bound on the additional regret any private algorithms for the MAB problem must incur.

We study the contextual linear bandit problem, a version of the standard stochastic multi-armed bandit (MAB) problem where a learner sequentially selects actions to maximize a reward which depends also on a user provided per-round context. Though the context is chosen arbitrarily or adversarially, the reward is assumed to be a stochastic function of a feature vector that encodes the context and selected action. Our goal is to devise private learners for the contextual linear bandit problem. We first show that using the standard definition of differential privacy results in linear regret. So instead, we adopt the notion of joint differential privacy, where we assume that the action chosen on day $t$ is only revealed to user $t$ and thus needn't be kept private that day, only on following days. We give a general scheme converting the classic linear-UCB algorithm into a joint differentially private algorithm using the tree-based algorithm. We then apply either Gaussian noise or Wishart noise to achieve joint-differentially private algorithms and bound the resulting algorithms' regrets. In addition, we give the first lower bound on the additional regret any private algorithms for the MAB problem must incur.

This work concerns learning probabilistic models for ranking data in a heterogeneous population.The specific problem we study is learning the parameters of a Mallows Mixture Model. Despite being widely studied, current heuristics for this problem do not have theoretical guarantees and can get stuck in bad local optima. We present the first polynomial time algorithm which provably learns the parameters ofa mixture of two Mallows models. A key component of our algorithm is a novel use of tensor decomposition techniques to learn the top-k prefix in both the rankings. Before this work, even the question of identifiability in the case of a mixture of two Mallows models was unresolved.

Traditional approaches to ranking in web search follow the paradigm of rank-by-score: a learned function gives each query-URL combination an absolute score and URLs are ranked according to this score. This paradigm ensures that if the score of one URL is better than another then one will always be ranked higher than the other. Scoring contradicts prior work in behavioral economics that showed that users' preferences between two items depend not only on the items but also on the presented alternatives. Thus, for the same query, users' preference between items A and B depends on the presence/absence of item C. We propose a new model of ranking, the Random Shopper Model, that allows and explains such behavior. In this model, each feature is viewed as a Markov chain over the items to be ranked, and the goal is to find a weighting of the features that best reflects their importance. We show that our model can be learned under the empirical risk minimization framework, and give an efficient learning algorithm. Experiments on commerce search logs demonstrate that our algorithm outperforms scoring-based approaches including regression and listwise ranking.