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

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