# Parallel algorithms and probability of large deviation for stochastic optimization problems

## TL;DR

This paper investigates parallel stochastic optimization algorithms under limited function value information, demonstrating that parallelization with averaging maintains low deviation probability and reduces computational time.

## Contribution

It introduces bounds for large deviation probabilities in parallel stochastic optimization without access to objective function values, extending previous results to more practical scenarios.

## Key findings

- Parallelization with averaging achieves similar deviation bounds as serial methods.
- Under light-tail assumptions, parallel algorithms reduce computational time.
- Results apply even when objective function values are unavailable or costly to compute.

## Abstract

We consider convex stochastic optimization problems under different assumptions on the properties of available stochastic subgradient. It is known that, if the value of the objective function is available, one can obtain, in parallel, several independent approximate solutions in terms of the objective residual expectation. Then, choosing the solution with the minimum function value, one can control the probability of large deviation of the objective residual. On the contrary, in this short paper, we address the situation, when the value of the objective function is unavailable or is too expensive to calculate. Under "`light-tail"' assumption for stochastic subgradient and in general case with moderate large deviation probability, we show that parallelization combined with averaging gives bounds for probability of large deviation similar to a serial method. Thus, in these cases, one can benefit from parallel computations and reduce the computational time without loss in the solution quality.