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Many machine learning problems optimize an objective that must be measured with noise. The primary method is a first order stochastic gradient descent using one or more Monte Carlo (MC) samples at each step. There are settings where ill-conditioning makes second order methods such as limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) more effective. We study the use of randomized quasi-Monte Carlo (RQMC) sampling for such problems. When MC sampling has a root mean squared error (RMSE) of O(n−1/2) then RQMC has an RMSE of o(n−1/2) that can be close to O(n−3/2) in favorable settings. We prove that improved sampling accuracy translates directly to improved optimization. In our empirical investigations for variational Bayes, using RQMC with stochastic quasi-Newton method greatly speeds up the optimization, and sometimes finds a better parameter value than MC does. 

Random projections or sketching are widely used in many algorithmic and learning contexts. Here we study the performance of iterative Hessian sketch for leastsquares problems. By leveraging and extending recent results from random matrix theory on the limiting spectrum of matrices randomly projected with the subsampled randomized Hadamard transform, and truncated Haar matrices, we can study and compare the resulting algorithms to a level of precision that has not been possible before. Our technical contributions include a novel formula for the second moment of the inverse of projected matrices. We also find simple closed-form expressions for asymptotically optimal step-sizes and convergence rates. These show that the convergence rate for Haar and randomized Hadamard matrices are identical, and asymptotically improve upon Gaussian random projections. These techniques may be applied to other algorithms that employ randomized dimension reduction.