On the Convergence of SGD with Biased Gradients

We analyze the complexity of biased stochastic gradient methods (SGD), where individual updates are corrupted by deterministic, i.e. biased error terms. We derive convergence results for smooth (non-convex) functions and give improved rates under the Polyak-Łojasiewicz condition. We quantify how the magnitude of the bias impacts the attainable accuracy and the convergence rates (sometimes leading to divergence). Our framework covers many applications where either only biased gradient updates are available, or preferred, over unbiased ones for performance reasons. For instance, in the domain of distributed learning, biased gradient compression techniques such as top- k compression have been proposed as a tool to alleviate the communication bottleneck and in derivative-free optimization, only biased gradient estimators can be queried. We discuss a few guiding examples that show the broad appli-cability of our analysis.