  First-order methods for stochastic optimization have undeniable relevance, in
part due to their pivotal role in machine learning. Variance reduction for
these algorithms has become an important research topic. In contrast to common
approaches, which rarely leverage global models of the objective function, we
exploit convexity and L-smoothness to improve the noisy estimates outputted by
the stochastic gradient oracle. Our method, named COCO denoiser, is the joint
maximum likelihood estimator of multiple function gradients from their noisy
observations, subject to co-coercivity constraints between them. The resulting
estimate is the solution of a convex Quadratically Constrained Quadratic
Problem. Although this problem is expensive to solve by interior point methods,
we exploit its structure to apply an accelerated first-order algorithm, the
Fast Dual Proximal Gradient method. Besides analytically characterizing the
proposed estimator, we show empirically that increasing the number and
proximity of the queried points leads to better gradient estimates. We also
apply COCO in stochastic settings by plugging it in existing algorithms, such
as SGD, Adam or STRSAGA, outperforming their vanilla versions, even in
scenarios where our modelling assumptions are mismatched.
