EVEN supervised learning is subject to the famous NoFree Lunch Theorems [1]-[3], which say that, in combinatorial optimization, there is no universal algorithm that works better than its competitors for every objective function [4]-[6]. Indeed, David Wolpert has recently proven that, on average, cross-validation performs as well as anti-crossvalidation (choosing among a set of candidate algorithms based on which has the worst out-of-sample behavior) for supervised learning. Still, he acknowledges that "it is hard to imagine any scientist who would not prefer to use [crossvalidation] to using anti-cross-validation" [7]. On the other hand, unsupervised learning has seldom been studied from the perspective of the NFLTs. This may be because the adjective "unsupervised" suggests that no human input is needed, which is misleading as many unsupervised tasks are combinatorial optimization problems that depend on the choice of the objective function. For instance, it is well known that, among the eigenvectors of the covariance matrix, Principal Components Analysis selects those with the largest variances [8]. However, mode-hunting techniques that rely on spectral manipulation aim at the opposite objective: selecting the eigenvectors of the covariance matrix with the smallest variances [9], [10]. Therefore, unlike in supervised learning, where it is difficult to identify reasons to optimize with respect to anti-cross-validation, in unsupervised learning there are strong reasons to reduce dimensionality for variance minimization. D. A. D ıaz-Pach on and T. Liu are with the Division of Biostatistics, University of Miami, Miami, FL, 33136 USA (e-mail: ddiaz3@miami.edu,

We study the following learning problem with dependent data: Given a trajectory of length $n$ from a stationary Markov chain with $k$ states, the goal is to predict the distribution of the next state.

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In the most basic setting of this problem, an agent has to pull one among a finite set of arms (or actions), and she receives a reward that depends on the chosen action.

We propose a new exact minibatch MH method, TunaMH, which exposes a tunable trade-off between its batch size and its theoretically guaranteed convergence rate.

Given a convex function f: Rd R, the problem of sampling from a distribution e f(x) is called log-concave sampling. This task has wide applications in machine learning, physics, statistics, etc.