On the Stability and Convergence of Stochastic Gradient Descent with Momentum

While momentum-based methods, in conjunction with the stochastic gradient descent, are widely used when training machine learning models, there is little theoretical understanding on the generalization error of such methods. In practice, the momentum parameter is often chosen in a heuristic fashion with little theoretical guidance. In the first part of this paper, for the case of general loss functions, we analyze a modified momentum-based update rule, i.e., the method of early momentum, and develop an upper-bound on the generalization error using the framework of algorithmic stability. Our results show that machine learning models can be trained for multiple epochs of this method while their generalization errors are bounded. We also study the convergence of the method of early momentum by establishing an upper-bound on the expected norm of the gradient. In the second part of the paper, we focus on the case of strongly convex loss functions and the classical heavy-ball momentum update rule. We use the framework of algorithmic stability to provide an upper-bound on the generalization error of the stochastic gradient method with momentum. We also develop an upper-bound on the expected true risk, in terms of the number of training steps, the size of the training set, and the momentum parameter. Experimental evaluations verify the consistency between the numerical results and our theoretical bounds and the effectiveness of the method of early momentum for the case of non-convex loss functions

Differentially Private Accelerated Optimization Algorithms

We present two classes of differentially private optimization algorithms derived from the well-known accelerated first-order methods. The first algorithm is inspired by Polyak's heavy ball method and employs a smoothing approach to decrease the accumulated noise on the gradient steps required for differential privacy. The second class of algorithms are based on Nesterov's accelerated gradient method and its recent multi-stage variant. We propose a noise dividing mechanism for the iterations of Nesterov's method in order to improve the error behavior of the algorithm. The convergence rate analyses are provided for both the heavy ball and the Nesterov's accelerated gradient method with the help of the dynamical system analysis techniques. Finally, we conclude with our numerical experiments showing that the presented algorithms have advantages over the well-known differentially private algorithms.Comment: 28 pages, 4 figure