Descent with Misaligned Gradients and Applications to Hidden Convexity

We consider the problem of minimizing a convex objective given access to an oracle that outputs "misaligned" stochastic gradients, where the expected value of the output is guaranteed to be correlated with, but not necessarily equal to the true gradient of the objective. In the case where the misalignment (or bias) of the oracle changes slowly, we obtain an optimization algorithm that achieves the optimum iteration complexity of $\tilde O(\epsilon^{-2})$; for the more general case where the changes need not be slow, we obtain an algorithm with $\tilde O(\epsilon^{-3})$ iteration complexity. As an application of our framework, we consider optimization problems with a "hidden convexity" property, and obtain an algorithm with $O(\epsilon^{-3})$ iteration complexity.