Accelerated Gradient Descent for Faster Convergence with Minimal Overhead

We introduce PF-AGD, the first parameter-free, deterministic, accelerated first-order method to achieve $O(ε^{-5/3}\log(1/ε))$ oracle complexity bound when minimizing sufficiently smooth, non-convex functions; this is the best-known bound for first-order methods on smooth non-convex objectives. Unlike existing methods possessing this rate that require a priori knowledge of smoothness constants, we use an adaptive backtracking scheme and a gradient-based restart mechanism to estimate local curvature. This yields a practical algorithm that matches best-known theoretical rates. Empirically, PF-AGD outperforms the practical variant of AGD-Until-Guilty (Carmon et al., 2017), as well as other parameter-free variants, and is a viable alternative to nonlinear conjugate gradient methods.

We propose a robust gradient estimator based on per-sample gradient clipping and analyze its properties both theoretically and empirically. We show that the resulting method, per-sample clipped SGD (PS-Clip-SGD), achieves optimal in-expectation convergence rates for non-convex optimization problems under heavy-tailed gradient noise. Moreover, we establish high-probability convergence guarantees that match the in-expectation rates up to polylogarithmic factors in the failure probability. We complement our theoretical results with multiple numerical experiments. In particular, we demonstrate that PS-Clip-SGD outperforms both vanilla SGD with momentum and standard gradient clipping when training AlexNet on the CIFAR-100 dataset, even after accounting for the additional computational time caused by per-sample clipping. We also empirically show that, in the presence of gradient accumulation, applying clipping at the mini-batch level can improve training performance while incurring virtually no additional computational cost. This finding is particularly interesting, as it contradicts the common practice of applying clipping only after all accumulation steps have been completed.

Gradient based optimization methods are nowadays the methods of choice for training deep neural networks (DNNs) in artificial intelligence (AI) systems. In practically relevant DNN training problems, one does usually not apply the standard gradient descent (GD) optimization method but instead one employs suitable sophisticated GD optimization methods, which incorporate adaptivity and/or acceleration techniques, such as the famous Adam optimizer. It is a key contribution of this work to provide a general unified convergence analysis for GD optimization methods in the training of DNNs with analytic activations such as the softplus and the popular Gaussian error linear unit (GeLU) activation. Our general unified convergence result applies to a large class of gradient based optimization methods such as the standard GD, the momentum, the Nesterov accelerated gradient (NAG), the RMSprop, the Adam, the Adamax, the Nadam, the Nadamax, the Adan, the AdaBelief, the AMSGrad, and the Yogi optimizers. Our analysis employs the theory of Kurdyka-Łojasiewicz (KL) inequalities to establish convergence to critical points in the training of DNNs. To the best of our knowledge, the generality of our convergence analysis is also just in the special situation of the Adam optimizer a new contribution to the literature on the analysis of AI optimization algorithms.