Separable sublinear functions are used to provide upper bounds on the recourse function of a stochastic program. The resulting problem's objective involves the inf-convolution of convex functions. A dual of this problem is formulated to obtain an implementable procedure to calculate the bound. Function evaluations for the resulting convex program only require a small number of single integratio...

We consider the problem of sampling from a strongly log-concave density in Rd, and prove an information theoretic lower bound on number stochastic gradient queries log needed. Several popular algorithms (including many Markov chain Monte Carlo methods) operate by using gradients to generate sample; our results establish limit for all these algorithms. show that every algorithm, there exists wel...