Stochastic trust-region and direct-search methods: A weak tail bound condition and reduced sample sizing

TL;DR

This paper introduces a new probabilistic tail bound condition for stochastic derivative-free optimization that reduces the required sample size per iteration, improving efficiency while maintaining convergence guarantees.

Contribution

It develops stochastic direct-search and trust-region methods with a novel tail bound condition that lowers sample complexity from $O(\Delta_k^{-4})$ to near $O(\Delta_k^{-2})$, enhancing theoretical analysis.

Findings

Sample complexity per iteration is reduced to $O(\Delta_k^{-2 - \varepsilon})$.

Global convergence of the proposed algorithms is established under the new tail bound condition.

The methods are applicable to non-smooth functions with stochastic observations.

Abstract

Using tail bounds, we introduce a new probabilistic condition for function estimation in stochastic derivative-free optimization which leads to a reduction in the number of samples and eases algorithmic analyses. Moreover, we develop simple stochastic direct-search and trust-region methods for the optimization of a potentially non-smooth function whose values can only be estimated via stochastic observations. For trial points to be accepted, these algorithms require the estimated function values to yield a sufficient decrease measured in terms of a power larger than 1 of the algoritmic stepsize. Our new tail bound condition is precisely imposed on the reduction estimate used to achieve such a sufficient decrease. This condition allows us to select the stepsize power used for sufficient decrease in such a way to reduce the number of samples needed per iteration. In previous works, the number of samples necessary for global convergence at every iteration $k$ of this type of algorithms was $O(\Delta_{k}^{-4})$, where $\Delta_k$ is the stepsize or trust-region radius. However, using the new tail bound condition, and under mild assumptions on the noise, one can prove that such a number of samples is only $O(\Delta_k^{-2 - \varepsilon})$, where $\varepsilon > 0$ can be made arbitrarily small by selecting the power of the stepsize in the sufficient decrease test arbitrarily close to $1$. The global convergence properties of the stochastic direct-search and trust-region algorithms are established under the new tail bound condition.