Supermodular f-divergences and bounds on lossy compression and generalization error with mutual f-information

In this paper, we introduce super-modular $\mf$-divergences and provide three applications for them: (i) we introduce Sanov's upper bound on the tail probability of sum of independent random variables based on super-modular $\mf$-divergence and show that our generalized Sanov's bound strictly improves over ordinary one, (ii) we consider the lossy compression problem which studies the set of achievable rates for a given distortion and code length. We extend the rate-distortion function using mutual $\mf$-information and provide new and strictly better bounds on achievable rates in the finite blocklength regime using super-modular $\mf$-divergences, and (iii) we provide a connection between the generalization error of algorithms with bounded input/output mutual $\mf$-information and a generalized rate-distortion problem. This connection allows us to bound the generalization error of learning algorithms using lower bounds on the rate-distortion function. Our bound is based on a new lower bound on the rate-distortion function that (for some examples) strictly improves over previously best-known bounds. Moreover, super-modular $\mf$-divergences are utilized to reduce the dimension of the problem and obtain single-letter bounds.

Sequential Estimation under Multiple Resources: a Bandit Point of View

The problem of Sequential Estimation under Multiple Resources (SEMR) is defined in a federated setting. SEMR could be considered as the intersection of statistical estimation and bandit theory. In this problem, an agent is confronting with k resources to estimate a parameter $\theta$. The agent should continuously learn the quality of the resources by wisely choosing them and at the end, proposes an estimator based on the collected data. In this paper, we assume that the resources' distributions are Gaussian. The quality of the final estimator is evaluated by its mean squared error. Also, we restrict our class of estimators to unbiased estimators in order to define a meaningful notion of regret. The regret measures the performance of the agent by the variance of the final estimator in comparison to the optimal variance. We propose a lower bound to determine the fundamental limit of the setting even in the case that the distributions are not Gaussian. Also, we offer an order-optimal algorithm to achieve this lower bound.