Non-asymptotic bounds for stochastic optimization with biased noisy gradient oracles

We introduce biased gradient oracles to capture a setting where the function measurements have an estimation error that can be controlled through a batch size parameter. Our proposed oracles are appealing in several practical contexts, for instance, risk measure estimation from a batch of independent and identically distributed (i.i.d.) samples, or simulation optimization, where the function measurements are `biased' due to computational constraints. In either case, increasing the batch size reduces the estimation error. We highlight the applicability of our biased gradient oracles in a risk-sensitive reinforcement learning setting. In the stochastic non-convex optimization context, we analyze a variant of the randomized stochastic gradient (RSG) algorithm with a biased gradient oracle. We quantify the convergence rate of this algorithm by deriving non-asymptotic bounds on its performance. Next, in the stochastic convex optimization setting, we derive non-asymptotic bounds for the last iterate of a stochastic gradient descent (SGD) algorithm with a biased gradient oracle.