In this paper, we consider the decentralized optimization problems with generalized orthogonality constraints, where both the objective function and the constraint exhibit a distributed structure. Such optimization problems, albeit ubiquitous in practical applications, remain unsolvable by existing algorithms in the presence of distributed constraints. To address this issue, we convert the original problem into an unconstrained penalty model by resorting to the recently proposed constraint-dissolving operator. However, this transformation compromises the essential property of separability in the resulting penalty function, rendering it impossible to employ existing algorithms to solve. We overcome this difficulty by introducing a novel algorithm that tracks the gradient of the objective function and the Jacobian of the constraint mapping simultaneously. The global convergence guarantee is rigorously established with an iteration complexity. To substantiate the effectiveness and efficiency of our proposed algorithm, we present numerical results on both synthetic and real-world datasets.

In this paper, we concentrate on decentralized optimization problems with nonconvex and nonsmooth objective functions, especially on the decentralized training of nonsmooth neural networks. We introduce a unified framework to analyze the global convergence of decentralized stochastic subgradient-based methods. We prove the global convergence of our proposed framework under mild conditions, by establishing that the generated sequence asymptotically approximates the trajectories of its associated differential inclusion. Furthermore, we establish that our proposed framework covers a wide range of existing efficient decentralized subgradient-based methods, including decentralized stochastic subgradient descent (DSGD), DSGD with gradient-tracking technique (DSGD-T), and DSGD with momentum (DSGD-M). In addition, we introduce the sign map to regularize the update directions in DSGD-M, and show it is enclosed in our proposed framework. Consequently, our convergence results establish, for the first time, global convergence of these methods when applied to nonsmooth nonconvex objectives. Preliminary numerical experiments demonstrate that our proposed framework yields highly efficient decentralized subgradient-based methods with convergence guarantees in the training of nonsmooth neural networks.

In this paper, we consider the minimization of a nonsmooth nonconvex objective function $f(x)$ over a closed convex subset $\mathcal{X}$ of $\mathbb{R}^n$, with additional nonsmooth nonconvex constraints $c(x) = 0$. We develop a unified framework for developing Lagrangian-based methods, which takes a single-step update to the primal variables by some subgradient methods in each iteration. These subgradient methods are ``embedded'' into our framework, in the sense that they are incorporated as black-box updates to the primal variables. We prove that our proposed framework inherits the global convergence guarantees from these embedded subgradient methods under mild conditions. In addition, we show that our framework can be extended to solve constrained optimization problems with expectation constraints. Based on the proposed framework, we show that a wide range of existing stochastic subgradient methods, including the proximal SGD, proximal momentum SGD, and proximal ADAM, can be embedded into Lagrangian-based methods. Preliminary numerical experiments on deep learning tasks illustrate that our proposed framework yields efficient variants of Lagrangian-based methods with convergence guarantees for nonconvex nonsmooth constrained optimization problems.

In this paper, we investigate the convergence properties of a wide class of Adam-family methods for minimizing quadratically regularized nonsmooth nonconvex optimization problems, especially in the context of training nonsmooth neural networks with weight decay. Motivated by the AdamW method, we propose a novel framework for Adam-family methods with decoupled weight decay. Within our framework, the estimators for the first-order and second-order moments of stochastic subgradients are updated independently of the weight decay term. Under mild assumptions and with non-diminishing stepsizes for updating the primary optimization variables, we establish the convergence properties of our proposed framework. In addition, we show that our proposed framework encompasses a wide variety of well-known Adam-family methods, hence offering convergence guarantees for these methods in the training of nonsmooth neural networks. More importantly, we show that our proposed framework asymptotically approximates the SGD method, thereby providing an explanation for the empirical observation that decoupled weight decay enhances generalization performance for Adam-family methods. As a practical application of our proposed framework, we propose a novel Adam-family method named Adam with Decoupled Weight Decay (AdamD), and establish its convergence properties under mild conditions. Numerical experiments demonstrate that AdamD outperforms Adam and is comparable to AdamW, in the aspects of both generalization performance and efficiency.

In this paper, we investigate the convergence properties of the stochastic gradient descent (SGD) method and its variants, especially in training neural networks built from nonsmooth activation functions. We develop a novel framework that assigns different timescales to stepsizes for updating the momentum terms and variables, respectively. Under mild conditions, we prove the global convergence of our proposed framework in both single-timescale and two-timescale cases. We show that our proposed framework encompasses a wide range of well-known SGD-type methods, including heavy-ball SGD, SignSGD, Lion, normalized SGD and clipped SGD. Furthermore, when the objective function adopts a finite-sum formulation, we prove the convergence properties for these SGD-type methods based on our proposed framework. In particular, we prove that these SGD-type methods find the Clarke stationary points of the objective function with randomly chosen stepsizes and initial points under mild assumptions. Preliminary numerical experiments demonstrate the high efficiency of our analyzed SGD-type methods.

The optimization problem in the form of UNP has numerous important applications in machine learning and data science, especially in training deep neural networks. In these applications of UNP, we usually only have access to the stochastic evaluations of the exact gradients of f. The stochastic gradient descent (SGD) is one of the most popular methods for solving UNP, and incorporating the momentum terms to SGD for acceleration is also very popular in practice. In SGD, the updating rule depends on the stepsizes (i.e., learning rates), where all of the coordinates of the variable x are equipped with the same stepsize. Recently, a variety of accelerated versions for SGD are proposed. In particular, the widely used Adam algorithm (Kingma and Ba, 2015) is developed based on the adaptive adjustment of the coordinate-wise stepsizes and the incorporation of momentum terms in each iteration. These enhancements have led to its high efficiency in practice. Motivated by Adam, a number of efficient Adam-family methods are developed, such as AdaBelief (Zhuang et al., 2020), AMSGrad (Reddi et al., 2018), NAdam (Dozat), Yogi (Zaheer et al., 2018), etc. Towards the convergence properties of these Adam-family methods, Kingma and Ba (2015) shows the convergence properties for Adam with constant stepsize and global Lipschitz objective function f.