Multilevel Monte-Carlo (MLMC) technique and establish its sample and computational complexities.

In this paper we study the second-order optimality of decentralized stochastic algorithm that escapes saddle point efficiently for nonconvex optimization problems.

Then momentum acceleration is also introduced into our dual accelerating update step.

Then momentum acceleration is also introduced into our dual accelerating update step.

Despite its success in practice, theoretical analyses of Adam are still limited.

In this paper, we provide a rigorous proof of convergence of the Adaptive Moment Estimate (Adam) algorithm for a wide class of optimization objectives. Despite the popularity and efficiency of the Adam algorithm in training deep neural networks, its theoretical properties are not yet fully understood, and existing convergence proofs require unrealistically strong assumptions, such as globally bounded gradients, to show the convergence to stationary points. In this paper, we show that Adam provably converges to $\epsilon$-stationary points with $\mathcal{O}(\epsilon^{-4})$ gradient complexity under far more realistic conditions. The key to our analysis is a new proof of boundedness of gradients along the optimization trajectory of Adam, under a generalized smoothness assumption according to which the local smoothness (i.e., Hessian norm when it exists) is bounded by a sub-quadratic function of the gradient norm. Moreover, we propose a variance-reduced version of Adam with an accelerated gradient complexity of $\mathcal{O}(\epsilon^{-3})$.

We analyze stochastic gradient algorithms for optimizing nonconvex problems. In particular, our goal is to find local minima (second-order stationary points) instead of just finding first-order stationary points which may be some bad unstable saddle points.

We study finite-sum nonconvex optimization problems, where the objective function is an average of $n$ nonconvex functions. We propose a new stochastic gradient descent algorithm based on nested variance reduction. Compared with conventional stochastic variance reduced gradient (SVRG) algorithm that uses two reference points to construct a semi-stochastic gradient with diminishing variance in each iteration, our algorithm uses $K+1$ nested reference points to build a semi-stochastic gradient to further reduce its variance in each iteration.

Algorithmic reproducibility measures the deviation in outputs of machine learning algorithms upon minor changes in the training process. Previous work suggests that first-order methods would need to trade-off convergence rate (gradient complexity) for better reproducibility. In this work, we challenge this perception and demonstrate that both optimal reproducibility and near-optimal convergence guarantees can be achieved for smooth convex minimization and smooth convex-concave minimax problems under various error-prone oracle settings. Particularly, given the inexact initialization oracle, our regularization-based algorithms achieve the best of both worlds -- optimal reproducibility and near-optimal gradient complexity -- for minimization and minimax optimization. With the inexact gradient oracle, the near-optimal guarantees also hold for minimax optimization. Additionally, with the stochastic gradient oracle, we show that stochastic gradient descent ascent is optimal in terms of both reproducibility and gradient complexity. We believe our results contribute to an enhanced understanding of the reproducibility-convergence trade-off in the context of convex optimization.

In the paper, we propose a class of efficient mirror descent ascent methods to solve the nonsmooth nonconvex-strongly-concave minimax problems by using dynamic mirror functions, and introduce a convergence analysis framework to conduct rigorous theoretical analysis for our mirror descent ascent methods. For our stochastic algorithms, we first prove that the mini-batch stochastic mirror descent ascent (SMDA) method obtains a gradient complexity of $O(\kappa^3\epsilon^{-4})$ for finding an $\epsilon$-stationary point, where $\kappa$ denotes the condition number. Further, we propose an accelerated stochastic mirror descent ascent (VR-SMDA) method based on the variance reduced technique. We prove that our VR-SMDA method achieves a lower gradient complexity of $O(\kappa^3\epsilon^{-3})$. For our deterministic algorithm, we prove that our deterministic mirror descent ascent (MDA) achieves a lower gradient complexity of $O(\sqrt{\kappa}\epsilon^{-2})$ under mild conditions, which matches the best known complexity in solving smooth nonconvex-strongly-concave minimax optimization. We conduct the experiments on fair classifier and robust neural network training tasks to demonstrate the efficiency of our new algorithms.