Unbiased least squares regression via averaged stochastic gradient descent

We consider an on-line least squares regression problem with optimal solution θ^* and Hessian matrix H, and study a time-average stochastic gradient descent estimator of θ^*. For k≥2, we provide an unbiased estimator of θ^* that is a modification of the time-average estimator, runs with an expected number of time-steps of order k, with O(1/k) expected excess risk. The constant behind the O notation depends on parameters of the regression and is a poly-logarithmic function of the smallest eigenvalue of H. We provide both a biased and unbiased estimator of the expected excess risk of the time-average estimator and of its unbiased counterpart, without requiring knowledge of either H or θ^*. We describe an "average-start" version of our estimators with similar properties. Our approach is based on randomized multilevel Monte Carlo. Our numerical experiments confirm our theoretical findings.