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We consider the least-squares regression problem and provide a detailedasymptotic analysis of the performance of averaged constant-step-sizestochastic gradient descent (a.k.a. least-mean-squares). In the strongly-convexcase, we provide an asymptotic expansion up to explicit exponentially decayingterms. Our analysis leads to new insights into stochastic approximationalgorithms: (a) it gives a tighter bound on the allowed step-size; (b) thegeneralization error may be divided into a variance term which is decaying asO(1/n), independently of the step-size $\gamma$, and a bias term that decays asO(1/$\gamma$ 2 n 2); (c) when allowing non-uniform sampling, the choice of agood sampling density depends on whether the variance or bias terms dominate.In particular, when the variance term dominates, optimal sampling densities donot lead to much gain, while when the bias term dominates, we can choose largerstep-sizes that leads to significant improvements.

Constant Step Size Least-Mean-Square: Bias-Variance Trade-offs and Optimal Sampling Distributions