SGD and Hogwild! Convergence Without the Bounded Gradients Assumption
This work addresses theoretical limitations in optimization algorithms for machine learning practitioners, offering more general convergence proofs.
The paper tackles the convergence analysis of SGD and Hogwild! by relaxing the assumption of bounded gradients, showing that a bound on stochastic gradients relative to true gradients always holds in machine learning problems, and provides new convergence results for diminished learning rates.
Stochastic gradient descent (SGD) is the optimization algorithm of choice in many machine learning applications such as regularized empirical risk minimization and training deep neural networks. The classical convergence analysis of SGD is carried out under the assumption that the norm of the stochastic gradient is uniformly bounded. While this might hold for some loss functions, it is always violated for cases where the objective function is strongly convex. In (Bottou et al.,2016), a new analysis of convergence of SGD is performed under the assumption that stochastic gradients are bounded with respect to the true gradient norm. Here we show that for stochastic problems arising in machine learning such bound always holds; and we also propose an alternative convergence analysis of SGD with diminishing learning rate regime, which results in more relaxed conditions than those in (Bottou et al.,2016). We then move on the asynchronous parallel setting, and prove convergence of Hogwild! algorithm in the same regime, obtaining the first convergence results for this method in the case of diminished learning rate.