A Stochastic Operator Framework for Inexact Static and Online Optimization

Abstract

This paper provides a unified stochastic operator framework to analyze the convergence of iterative optimization algorithms for both static problems and online optimization and learning. In particular, the framework is well suited for algorithms that are implemented in an inexact or stochastic fashion because of (i) stochastic errors emerging in algorithmic steps, and because (ii) the algorithm may feature random coordinate updates. To this end, the paper focuses on separable operators of the form T x = (T 1 x, . . . , Tnx), defined over the direct sum of possibly infinite-dimensional Hilbert spaces, and investigates the convergence of the associated stochastic Banach-Picard iteration. Results in terms of convergence in mean and in high-probability are presented when the errors affecting the operator follow a sub-Weibull distribution and when updates T i x are performed based on a Bernoulli random variable. In particular, the results are derived for the cases where T is contractive and averaged in terms of convergence to the unique fixed point and cumulative fixed-point residual, respectively. The results do not assume vanishing errors or vanishing parameters of the operator, as typical in the literature (this case is subsumed by the proposed framework), and links with exiting results in terms of almost sure convergence are provided. In the online optimization context, the operator changes at each iteration to reflect changes in the underlying optimization problem. This leads to an online Banach-Picard iteration, and similar results are derived where the bounds for the convergence in mean and high-probability further depend on the evolution of fixed points (i.e., optimal solutions of the time-varying optimization problem).

Stochastic Saddle Point Problems with Decision-Dependent Distributions

“…We include it here as it allows to better describe the behavior of distributions generated from the closure properties. This approach has been used in the study of stochastic gradient methods in [5,56].…”

“…Convergence rates of Lasso estimates using Sub-Weibull covariates and additive noise assumptions are provided in [30]. High probability bounds for the tracking error of time-varying minimization problems have been established using sub-Weibull distributions [5,56].…”

Online Stochastic Gradient Methods Under Sub-Weibull Noise and the Polyak-Łojasiewicz Condition

“…Regarding the error e t , the sub-Weibull distribution allows one to consider a variety of cases in a unified manner; in fact, the sub-Weibull distribution includes sub-Gaussian distributions and sub-exponential distributions, and random variables whose probability density function has finite support as sub-cases [41], [42]. The sub-Weibull also allows one to consider distributions with arbitrary tail decay rates (via the parameter θ).…”

“…To show (34), recall first that if e t ∼ subW(θ, K t ), then the error is also e t ∼ subW(2θ, K t ) because of the inclusion property of the sub-Weibull random variable [40,Prop. 1], [42,Prop. 2.14].…”

“…Illustrative simulations are provided for a real-time demand response problem in the context of a power systems. We conclude this section by mentioning that the sub-Weibull distribution allows one to consider a variety of error models in a unified manner; in fact, the sub-Weibull distribution includes sub-Gaussian distributions and sub-exponential distributions, and random variables whose probability density function has finite support as sub-cases [41], [42]. Furthermore, while the majority of the high-probability bounds in the context of stochastic optimization are derived using a sub-Gaussian assumption, recent works suggest that stochastic gradient descent may exhibit errors with tails that are heavier than a sub-Gaussian, especially for small mini-batch sizes (see, e.g., [43], [44]).…”