Tracking the gradients using the Hessian: A new look at variance reducing stochastic methods

Our goal is to improve variance reducing stochastic methods through better control variates. We first propose a modification of SVRG which uses the Hessian to track gradients over time, rather than to recondition, increasing the correlation of the control variates and leading to faster theoretical convergence close to the optimum. We then propose accurate and computationally efficient approximations to the Hessian, both using a diagonal and a low-rank matrix. Finally, we demonstrate the effectiveness of our method on a wide range of problems.Comment: 17 pages, 2 figures, 1 tabl

A Comparison of Relaxations of Multiset Cannonical Correlation Analysis and Applications

Canonical correlation analysis is a statistical technique that is used to find relations between two sets of variables. An important extension in pattern analysis is to consider more than two sets of variables. This problem can be expressed as a quadratically constrained quadratic program (QCQP), commonly referred to Multi-set Canonical Correlation Analysis (MCCA). This is a non-convex problem and so greedy algorithms converge to local optima without any guarantees on global optimality. In this paper, we show that despite being highly structured, finding the optimal solution is NP-Hard. This motivates our relaxation of the QCQP to a semidefinite program (SDP). The SDP is convex, can be solved reasonably efficiently and comes with both absolute and output-sensitive approximation quality. In addition to theoretical guarantees, we do an extensive comparison of the QCQP method and the SDP relaxation on a variety of synthetic and real world data. Finally, we present two useful extensions: we incorporate kernel methods and computing multiple sets of canonical vectors

Local antithetic sampling with scrambled nets

We consider the problem of computing an approximation to the integral $I=\int_{[0,1]^d}f(x) dx$. Monte Carlo (MC) sampling typically attains a root mean squared error (RMSE) of $O(n^{-1/2})$ from $n$ independent random function evaluations. By contrast, quasi-Monte Carlo (QMC) sampling using carefully equispaced evaluation points can attain the rate $O(n^{-1+\varepsilon})$ for any $\varepsilon>0$ and randomized QMC (RQMC) can attain the RMSE $O(n^{-3/2+\varepsilon})$, both under mild conditions on $f$. Classical variance reduction methods for MC can be adapted to QMC. Published results combining QMC with importance sampling and with control variates have found worthwhile improvements, but no change in the error rate. This paper extends the classical variance reduction method of antithetic sampling and combines it with RQMC. One such method is shown to bring a modest improvement in the RMSE rate, attaining $O(n^{-3/2-1/d+\varepsilon})$ for any $\varepsilon>0$, for smooth enough $f$.Comment: Published in at http://dx.doi.org/10.1214/07-AOS548 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org