Decentralized Learning Dynamics in the Gossip Model

We study a distributed multi-armed bandit setting among a population of $n$ memory-constrained nodes in the gossip model: at each round, every node locally adopts one of $m$ arms, observes a reward drawn from the arm's (adversarially chosen) distribution, and then communicates with a randomly sampled neighbor, exchanging information to determine its policy in the next round. We introduce and analyze several families of dynamics for this task that are decentralized: each node's decision is entirely local and depends only on its most recently obtained reward and that of the neighbor it sampled. We show a connection between the global evolution of these decentralized dynamics with a certain class of "zero-sum" multiplicative weight update algorithms, and we develop a general framework for analyzing the population-level regret of these natural protocols. Using this framework, we derive sublinear regret bounds under a wide range of parameter regimes (i.e., the size of the population and number of arms) for both the stationary reward setting (where the mean of each arm's distribution is fixed over time) and the adversarial reward setting (where means can vary over time). Further, we show that these protocols can approximately optimize convex functions over the simplex when the reward distributions are generated from a stochastic gradient oracle.

Differentially Private Maximal Information Coefficients

The Maximal Information Coefficient (MIC) is a powerful statistic to identify dependencies between variables. However, it may be applied to sensitive data, and publishing it could leak private information. As a solution, we present algorithms to approximate MIC in a way that provides differential privacy. We show that the natural application of the classic Laplace mechanism yields insufficient accuracy. We therefore introduce the MICr statistic, which is a new MIC approximation that is more compatible with differential privacy. We prove MICr is a consistent estimator for MIC, and we provide two differentially private versions of it. We perform experiments on a variety of real and synthetic datasets. The results show that the private MICr statistics significantly outperform direct application of the Laplace mechanism. Moreover, experiments on real-world datasets show accuracy that is usable when the sample size is at least moderately large.

Consistency of the Maximal Information Coefficient Estimator

The Maximal Information Coefficient (MIC) of Reshef et al. (Science, 2011) is a statistic for measuring dependence between variable pairs in large datasets. In this note, we prove that MIC is a consistent estimator of the corresponding population statistic MIC$_*$. This corrects an error in an argument of Reshef et al. (JMLR, 2016), which we describe.