arXiv subjects

Barnabás Póczos

Publications and source records attributed to Barnabás Póczos.

44 records · Page 3Linked to original sources

arXiv2016

The Multi-fidelity Multi-armed Bandit

We study a variant of the classical stochastic $K$-armed bandit where observing the outcome of each arm is expensive, but cheap approximations to this outcome are available. For example, in online advertising the performance of an ad can be approximated by displaying it for shorter time periods or to narrower audiences. We formalise this task as a multi-fidelity bandit, where, at each time step, the forecaster may choose to play an arm at any one of $M$ fidelities. The highest fidelity (desired outcome) expends cost $λ^{(m)}$. The $m^{\text{th}}$ fidelity (an approximation) expends $λ^{(m)} < λ^{(M)}$ and returns a biased estimate of the highest fidelity. We develop MF-UCB, a novel upper confidence bound procedure for this setting and prove that it naturally adapts to the sequence of available approximations and costs thus attaining better regret than naive strategies which ignore the approximations. For instance, in the above online advertising example, MF-UCB would use the lower fidelities to quickly eliminate suboptimal ads and reserve the larger expensive experiments on a small set of promising candidates. We complement this result with a lower bound and show that MF-UCB is nearly optimal under certain conditions.

cs.LG↗

arXiv2016

Efficient Nonparametric Smoothness Estimation

Sobolev quantities (norms, inner products, and distances) of probability density functions are important in the theory of nonparametric statistics, but have rarely been used in practice, partly due to a lack of practical estimators. They also include, as special cases, $L^2$ quantities which are used in many applications. We propose and analyze a family of estimators for Sobolev quantities of unknown probability density functions. We bound the bias and variance of our estimators over finite samples, finding that they are generally minimax rate-optimal. Our estimators are significantly more computationally tractable than previous estimators, and exhibit a statistical/computational trade-off allowing them to adapt to computational constraints. We also draw theoretical connections to recent work on fast two-sample testing. Finally, we empirically validate our estimators on synthetic data.

math.ST↗

arXiv2016

Analysis of k-Nearest Neighbor Distances with Application to Entropy Estimation

Estimating entropy and mutual information consistently is important for many machine learning applications. The Kozachenko-Leonenko (KL) estimator (Kozachenko & Leonenko, 1987) is a widely used nonparametric estimator for the entropy of multivariate continuous random variables, as well as the basis of the mutual information estimator of Kraskov et al. (2004), perhaps the most widely used estimator of mutual information in this setting. Despite the practical importance of these estimators, major theoretical questions regarding their finite-sample behavior remain open. This paper proves finite-sample bounds on the bias and variance of the KL estimator, showing that it achieves the minimax convergence rate for certain classes of smooth functions. In proving these bounds, we analyze finite-sample behavior of k-nearest neighbors (k-NN) distance statistics (on which the KL estimator is based). We derive concentration inequalities for k-NN distances and a general expectation bound for statistics of k-NN distances, which may be useful for other analyses of k-NN methods.

math.ST↗

arXiv2016

Finite-Sample Analysis of Fixed-k Nearest Neighbor Density Functional Estimators

We provide finite-sample analysis of a general framework for using k-nearest neighbor statistics to estimate functionals of a nonparametric continuous probability density, including entropies and divergences. Rather than plugging a consistent density estimate (which requires $k \to \infty$ as the sample size $n \to \infty$) into the functional of interest, the estimators we consider fix k and perform a bias correction. This is more efficient computationally, and, as we show in certain cases, statistically, leading to faster convergence rates. Our framework unifies several previous estimators, for most of which ours are the first finite sample guarantees.

math.ST↗

arXiv2016

Generalized Exponential Concentration Inequality for Rényi Divergence Estimation

Estimating divergences in a consistent way is of great importance in many machine learning tasks. Although this is a fundamental problem in nonparametric statistics, to the best of our knowledge there has been no finite sample exponential inequality convergence bound derived for any divergence estimators. The main contribution of our work is to provide such a bound for an estimator of Rényi-$α$ divergence for a smooth Hölder class of densities on the $d$-dimensional unit cube $[0, 1]^d$. We also illustrate our theoretical results with a numerical experiment.

cs.IT↗

arXiv2016

On Variance Reduction in Stochastic Gradient Descent and its Asynchronous Variants

We study optimization algorithms based on variance reduction for stochastic gradient descent (SGD). Remarkable recent progress has been made in this direction through development of algorithms like SAG, SVRG, SAGA. These algorithms have been shown to outperform SGD, both theoretically and empirically. However, asynchronous versions of these algorithms---a crucial requirement for modern large-scale applications---have not been studied. We bridge this gap by presenting a unifying framework for many variance reduction techniques. Subsequently, we propose an asynchronous algorithm grounded in our framework, and prove its fast convergence. An important consequence of our general approach is that it yields asynchronous versions of variance reduction algorithms such as SVRG and SAGA as a byproduct. Our method achieves near linear speedup in sparse settings common to machine learning. We demonstrate the empirical performance of our method through a concrete realization of asynchronous SVRG.

cs.LG↗

arXiv2014

On the Decreasing Power of Kernel and Distance based Nonparametric Hypothesis Tests in High Dimensions

This paper is about two related decision theoretic problems, nonparametric two-sample testing and independence testing. There is a belief that two recently proposed solutions, based on kernels and distances between pairs of points, behave well in high-dimensional settings. We identify different sources of misconception that give rise to the above belief. Specifically, we differentiate the hardness of estimation of test statistics from the hardness of testing whether these statistics are zero or not, and explicitly discuss a notion of "fair" alternative hypotheses for these problems as dimension increases. We then demonstrate that the power of these tests actually drops polynomially with increasing dimension against fair alternatives. We end with some theoretical insights and shed light on the \textit{median heuristic} for kernel bandwidth selection. Our work advances the current understanding of the power of modern nonparametric hypothesis tests in high dimensions.

stat.ML↗

arXiv2010

Estimation of Rényi Entropy and Mutual Information Based on Generalized Nearest-Neighbor Graphs

We present simple and computationally efficient nonparametric estimators of Rényi entropy and mutual information based on an i.i.d. sample drawn from an unknown, absolutely continuous distribution over $\R^d$. The estimators are calculated as the sum of $p$-th powers of the Euclidean lengths of the edges of the `generalized nearest-neighbor' graph of the sample and the empirical copula of the sample respectively. For the first time, we prove the almost sure consistency of these estimators and upper bounds on their rates of convergence, the latter of which under the assumption that the density underlying the sample is Lipschitz continuous. Experiments demonstrate their usefulness in independent subspace analysis.

stat.ML↗