Kuangyu Ding

Kuangyu Ding, Jingyang Li, Kim-Chuan Toh

Stochastic gradient methods for minimizing nonconvex composite objective functions typically rely on the Lipschitz smoothness of the differentiable part, but this assumption fails in many important problem classes like quadratic inverse problems and neural network training, leading to instability of the algorithms in both theory and practice. To address this, we propose a family of stochastic Bregman proximal gradient (SBPG) methods that only require smooth adaptivity. SBPG replaces the quadratic approximation in SGD with a Bregman proximity measure, offering a better approximation model that handles non-Lipschitz gradients in nonconvex objectives. We establish the convergence properties of vanilla SBPG and show it achieves optimal sample complexity in the nonconvex setting. Experimental results on quadratic inverse problems demonstrate SBPG's robustness in terms of stepsize selection and sensitivity to the initial point. Furthermore, we introduce a momentum-based variant, MSBPG, which enhances convergence by relaxing the mini-batch size requirement while preserving the optimal oracle complexity. We apply MSBPG to the training of deep neural networks, utilizing a polynomial kernel function to ensure smooth adaptivity of the loss function. Experimental results on benchmark datasets confirm the effectiveness and robustness of MSBPG in training neural networks. Given its negligible additional computational cost compared to SGD in large-scale optimization, MSBPG shows promise as a universal open-source optimizer for future applications.

Kuangyu Ding, Marie Maros, Gesualdo Scutari

We study finite-sum nonlinear programs with localized variable coupling encoded by a (hyper)graph. We introduce a graph-compliant decomposition framework that brings message passing into continuous optimization in a rigorous, implementable, and provable way. The (hyper)graph is partitioned into tree clusters (hypertree factor graphs). At each iteration, agents update in parallel by solving local subproblems whose objective splits into an {\it intra}-cluster term summarized by cost-to-go messages from one min-sum sweep on the cluster tree, and an {\it inter}-cluster coupling term handled Jacobi-style using the latest out-of-cluster variables. To reduce computation/communication, the method supports graph-compliant surrogates that replace exact messages/local solves with compact low-dimensional parametrizations; in hypergraphs, the same principle enables surrogate hyperedge splitting, to tame heavy hyperedge overlaps while retaining finite-time intra-cluster message updates and efficient computation/communication. We establish convergence for (strongly) convex and nonconvex objectives, with topology- and partition-explicit rates that quantify curvature/coupling effects and guide clustering and scalability. To our knowledge, this is the first convergent message-passing method on loopy graphs.

Kuangyu Ding, Nachuan Xiao, Kim-Chuan Toh

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.

Xingyu Xie, Kuangyu Ding, Shuicheng Yan et al.

Large Language Models have driven significant AI advancements, yet their training is resource-intensive and highly sensitive to hyper-parameter selection. While scaling laws provide valuable guidance on model size and data requirements, they fall short in choosing dynamic hyper-parameters, such as learning-rate (LR) schedules, that evolve during training. To bridge this gap, we present Optimization Hyper-parameter Laws (Opt-Laws), a framework that effectively captures the relationship between hyper-parameters and training outcomes, enabling the pre-selection of potential optimal schedules. Grounded in stochastic differential equations, Opt-Laws introduce novel mathematical interpretability and offer a robust theoretical foundation for some popular LR schedules. Our extensive validation across diverse model sizes and data scales demonstrates Opt-Laws' ability to accurately predict training loss and identify optimal LR schedule candidates in pre-training, continual training, and fine-tuning scenarios. This approach significantly reduces computational costs while enhancing overall model performance.

Nachuan Xiao, Kuangyu Ding, Xiaoyin Hu et al.

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.

Kuangyu Ding, Kim-Chuan Toh

We study a nonsmooth nonconvex optimization problem defined over nonconvex constraints, where the feasible set is given by the intersection of the closure of an open set and a smooth manifold. By endowing the open set with a Riemannian metric induced by a barrier function, we obtain a Riemannian subgradient flow formulated as a differential inclusion, which remains strictly within the interior of the feasible set. This continuous dynamical system unifies two classes of iterative optimization methods, namely the Hessian barrier method and mirror descent scheme, by revealing that these methods can be interpreted as discrete approximations of the continuous flow. We explore the long-term behavior of the trajectories generated by this dynamical system and show that the existing deficient convergence properties of the Hessian barrier and mirror descent scheme can be unifily and more insightfully interpreted through these of the continuous trajectory. For instance, the notorious spurious stationary points \cite{chen2024spurious} observed in Hessian barrier method and mirror descent scheme are interpreted as stable equilibria of the dynamical system that do not correspond to real stationary points of the original optimization problem. We provide two sufficient condition such that these spurious stationary points can be avoided if the strict complementarity conditions holds. In the absence of these regularity condition, we propose a random perturbation strategy that ensures the trajectory converges (subsequentially) to an approximate stationary point. Building on these insights, we introduce two iterative Riemannian subgradient methods, form of interior point methods, that generalizes the existing Hessian barrier method and mirror descent scheme for solving nonsmooth nonconvex optimization problems.

Jingyang Li, Kuangyu Ding, Kim-Chuan Toh et al.

Preconditioned stochastic optimization algorithms, exemplified by Shampoo, outperform first-order optimizers by offering theoretical convergence benefits and practical gains in large-scale neural network training. However, they incur substantial memory overhead due to the storage demands of non-diagonal preconditioning matrices. To address this, we introduce 4-bit quantization for Shampoo's preconditioners. We introduce two key methods: First, we apply Cholesky decomposition followed by quantization of the Cholesky factors, reducing memory usage by leveraging their lower triangular structure while better preserving spectral properties to minimize information loss. To our knowledge, this is the first quantization approach applied to Cholesky factors of preconditioners. Second, we incorporate error feedback in the quantization process, efficiently storing Cholesky factor and error state in the lower and upper triangular parts of the same matrix. Through extensive experiments, we demonstrate that combining Cholesky quantization with error feedback enhances memory efficiency and algorithm performance in large-scale deep-learning tasks. Theoretically, we also provide convergence proofs for quantized Shampoo under both smooth and non-smooth stochastic optimization settings.