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We consider the problem of minimizing a strongly convex function that dependson an uncertain parameter $\theta$. The uncertainty in the objective functionmeans that the optimum, $x^*(\theta)$, is also a function of $\theta$. Wepropose an efficient method to compute $x^*(\theta)$ and its statistics. We usea chaos expansion of $x^*(\theta)$ along a truncated basis and studyfirst-order methods that compute the optimal coefficients. We establish theconvergence rate of the method as the number of basis functions, and hence thedimensionality of the optimization problem is increased. We give the firstnon-asymptotic rates for the gradient descent and the accelerated gradientdescent methods. Our analysis exploits convexity and does not rely on adiminishing step-size strategy. As a result, it is much faster than thestate-of-the-art both in theory and in our preliminary numerical experiments. Asurprising side-effect of our analysis is that the proposed method also acts asa variance reduction technique to the problem of estimating $x^*(\theta)$.

Uncertainty Quantification for Gradient and Accelerated Gradient Descent Methods on Strongly Convex Functions