Complexity of Zeroth- and First-order Stochastic Trust-Region Algorithms

Model update (MU) and candidate evaluation (CE) are classical steps incorporated inside many stochastic trust-region (TR) algorithms. The sampling effort exerted within these steps, often decided with the aim of controlling model error, largely determines a stochastic TR algorithm's sample complexity. Given that MU and CE are amenable to variance reduction, we investigate the effect of incorporating common random numbers (CRN) within MU and CE on complexity. Using ASTRO and ASTRO-DF as prototype first-order and zeroth-order families of algorithms, we demonstrate that CRN's effectiveness leads to a range of complexities depending on sample-path regularity and the oracle order. For instance, we find that in first-order oracle settings with smooth sample paths, CRN's effect is pronounced -- ASTRO with CRN achieves $\tilde{O}(\epsilon^{-2})$ a.s. sample complexity compared to $\tilde{O}(\epsilon^{-6})$ a.s. in the generic no-CRN setting. By contrast, CRN's effect is muted when the sample paths are not Lipschitz, with the sample complexity improving from $\tilde{O}(\epsilon^{-6})$ a.s. to $\tilde{O}(\epsilon^{-5})$ and $\tilde{O}(\epsilon^{-4})$ a.s. in the zeroth- and first-order settings, respectively. Since our results imply that improvements in complexity are largely inherited from generic aspects of variance reduction, e.g., finite-differencing for zeroth-order settings and sample-path smoothness for first-order settings within MU, we anticipate similar trends in other contexts.