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We consider stochastic convex optimization problems where the objective is anexpectation over smooth functions. For this setting we suggest a novel gradientestimate that combines two recent mechanism that are related to notion ofmomentum. Then, we design an SGD-style algorithm as well as an acceleratedversion that make use of this new estimator, and demonstrate the robustness ofthese new approaches to the choice of the learning rate. Concretely, we showthat these approaches obtain the optimal convergence rates for both noiselessand noisy case with the same choice of fixed learning rate. Moreover, for thenoisy case we show that these approaches achieve the same optimal bound for avery wide range of learning rates.

$\mu^2$-SGD: Stable Stochastic Optimization via a Double Momentum Mechanism