Abstract
An usual problem in statistics consists in estimating the minimizer of a convex function. When we have to deal with large samples taking values in high dimensional spaces, stochastic gradient algorithms and their averaged versions are efficient candidates. Indeed, (1) they do not need too much computational efforts, (2) they do not need to store all the data, which is crucial when we deal with big data, (3) they allow to simply update the estimates, which is important when data arrive sequentially. The aim of this work is to give asymptotic and non asymptotic rates of convergence of stochastic gradient estimates as well as of their averaged versions when the function we would like to minimize is only locally strongly convex.
Highlights
With the development of automatic sensors, it is more and more important to think about methods able to deal with large samples of observations taking values in high dimensional spaces such as functional spaces
We focus here on an usual stochastic optimization problem which consists in estimating m := arg min E [g(X, h)], h∈H
[22] and [1] give some general conditions to get the rate of convergence in quadratic mean of averaged stochastic gradient algorithms, while [13], for instance, focus on non asymptotic rates for strongly convex stochastic composite optimization
Summary
With the development of automatic sensors, it is more and more important to think about methods able to deal with large samples of observations taking values in high dimensional spaces such as functional spaces. [22] and [1] give some general conditions to get the rate of convergence in quadratic mean of averaged stochastic gradient algorithms, while [13], for instance, focus on non asymptotic rates for strongly convex stochastic composite optimization. Two examples of application are given in Section 3: we first focus on the estimation of geometric quantiles, which are a generalization of the real quantiles introduced by [8]. They are robust indicators which can be useful in statistical depth and outliers detection (see [30], [9] or [17]).
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