The Geometry of Sign Gradient Descent
This work provides theoretical insights into sign-based optimization methods, which are incremental but important for improving efficiency in distributed machine learning and neural network training.
The paper unified existing analyses of sign-based optimization methods by connecting separable smoothness to ℓ∞-smoothness, arguing the latter is weaker and more natural, and identified geometric properties of the objective function that affect performance, finding sign-based methods preferable over gradient descent when the Hessian is concentrated on its diagonal and its maximal eigenvalue is much larger than the average, which are common in deep networks.
Sign-based optimization methods have become popular in machine learning due to their favorable communication cost in distributed optimization and their surprisingly good performance in neural network training. Furthermore, they are closely connected to so-called adaptive gradient methods like Adam. Recent works on signSGD have used a non-standard "separable smoothness" assumption, whereas some older works study sign gradient descent as steepest descent with respect to the $\ell_\infty$-norm. In this work, we unify these existing results by showing a close connection between separable smoothness and $\ell_\infty$-smoothness and argue that the latter is the weaker and more natural assumption. We then proceed to study the smoothness constant with respect to the $\ell_\infty$-norm and thereby isolate geometric properties of the objective function which affect the performance of sign-based methods. In short, we find sign-based methods to be preferable over gradient descent if (i) the Hessian is to some degree concentrated on its diagonal, and (ii) its maximal eigenvalue is much larger than the average eigenvalue. Both properties are common in deep networks.