Junchi Yang

Junchi Yang, Murat Yildirim, Qiu Feng

In distributed machine learning, efficient training across multiple agents with different data distributions poses significant challenges. Even with a centralized coordinator, current algorithms that achieve optimal communication complexity typically require either large minibatches or compromise on gradient complexity. In this work, we tackle both centralized and decentralized settings across strongly convex, convex, and nonconvex objectives. We first demonstrate that a basic primal-dual method, (Accelerated) Gradient Ascent Multiple Stochastic Gradient Descent (GA-MSGD), applied to the Lagrangian of distributed optimization inherently incorporates local updates, because the inner loops of running Stochastic Gradient Descent on the primal variable require no inter-agent communication. Notably, for strongly convex objectives, (Accelerated) GA-MSGD achieves linear convergence in communication rounds despite the Lagrangian being only linear in the dual variables. This is due to a structural property where the dual variable is confined to the span of the coupling matrix, rendering the dual problem strongly concave. When integrated with the Catalyst framework, our approach achieves nearly optimal communication complexity across various settings without the need for minibatches.

Junchi Yang, Antonio Orvieto, Aurelien Lucchi et al.

Gradient descent ascent (GDA), the simplest single-loop algorithm for nonconvex minimax optimization, is widely used in practical applications such as generative adversarial networks (GANs) and adversarial training. Albeit its desirable simplicity, recent work shows inferior convergence rates of GDA in theory even assuming strong concavity of the objective on one side. This paper establishes new convergence results for two alternative single-loop algorithms -- alternating GDA and smoothed GDA -- under the mild assumption that the objective satisfies the Polyak-Lojasiewicz (PL) condition about one variable. We prove that, to find an $ε$-stationary point, (i) alternating GDA and its stochastic variant (without mini batch) respectively require $O(κ^{2} ε^{-2})$ and $O(κ^{4} ε^{-4})$ iterations, while (ii) smoothed GDA and its stochastic variant (without mini batch) respectively require $O(κε^{-2})$ and $O(κ^{2} ε^{-4})$ iterations. The latter greatly improves over the vanilla GDA and gives the hitherto best known complexity results among single-loop algorithms under similar settings. We further showcase the empirical efficiency of these algorithms in training GANs and robust nonlinear regression.

Siqi Zhang, Junchi Yang, Cristóbal Guzmán et al.

This paper studies the complexity for finding approximate stationary points of nonconvex-strongly-concave (NC-SC) smooth minimax problems, in both general and averaged smooth finite-sum settings. We establish nontrivial lower complexity bounds of $Ω(\sqrtκΔLε^{-2})$ and $Ω(n+\sqrt{nκ}ΔLε^{-2})$ for the two settings, respectively, where $κ$ is the condition number, $L$ is the smoothness constant, and $Δ$ is the initial gap. Our result reveals substantial gaps between these limits and best-known upper bounds in the literature. To close these gaps, we introduce a generic acceleration scheme that deploys existing gradient-based methods to solve a sequence of crafted strongly-convex-strongly-concave subproblems. In the general setting, the complexity of our proposed algorithm nearly matches the lower bound; in particular, it removes an additional poly-logarithmic dependence on accuracy present in previous works. In the averaged smooth finite-sum setting, our proposed algorithm improves over previous algorithms by providing a nearly-tight dependence on the condition number.

Junchi Yang, Negar Kiyavash, Niao He

Nonconvex minimax problems appear frequently in emerging machine learning applications, such as generative adversarial networks and adversarial learning. Simple algorithms such as the gradient descent ascent (GDA) are the common practice for solving these nonconvex games and receive lots of empirical success. Yet, it is known that these vanilla GDA algorithms with constant step size can potentially diverge even in the convex setting. In this work, we show that for a subclass of nonconvex-nonconcave objectives satisfying a so-called two-sided Polyak-Łojasiewicz inequality, the alternating gradient descent ascent (AGDA) algorithm converges globally at a linear rate and the stochastic AGDA achieves a sublinear rate. We further develop a variance reduced algorithm that attains a provably faster rate than AGDA when the problem has the finite-sum structure.