Ganzhao Yuan

Ganzhao Yuan, Bernard Ghanem

Total Variation (TV) is an effective and popular prior model in the field of regularization-based image processing. This paper focuses on total variation for removing impulse noise in image restoration. This type of noise frequently arises in data acquisition and transmission due to many reasons, e.g. a faulty sensor or analog-to-digital converter errors. Removing this noise is an important task in image restoration. State-of-the-art methods such as Adaptive Outlier Pursuit(AOP) \cite{yan2013restoration}, which is based on TV with $\ell_{02}$-norm data fidelity, only give sub-optimal performance. In this paper, we propose a new sparse optimization method, called $\ell_0TV$-PADMM, which solves the TV-based restoration problem with $\ell_0$-norm data fidelity. To effectively deal with the resulting non-convex non-smooth optimization problem, we first reformulate it as an equivalent biconvex Mathematical Program with Equilibrium Constraints (MPEC), and then solve it using a proximal Alternating Direction Method of Multipliers (PADMM). Our $\ell_0TV$-PADMM method finds a desirable solution to the original $\ell_0$-norm optimization problem and is proven to be convergent under mild conditions. We apply $\ell_0TV$-PADMM to the problems of image denoising and deblurring in the presence of impulse noise. Our extensive experiments demonstrate that $\ell_0TV$-PADMM outperforms state-of-the-art image restoration methods.

Ganzhao Yuan, Li Shen, Wei-Shi Zheng

The sparse generalized eigenvalue problem arises in a number of standard and modern statistical learning models, including sparse principal component analysis, sparse Fisher discriminant analysis, and sparse canonical correlation analysis. However, this problem is difficult to solve since it is NP-hard. In this paper, we consider a new decomposition method to tackle this problem. Specifically, we use random or/and swapping strategies to find a working set and perform global combinatorial search over the small subset of variables. We consider a bisection search method and a coordinate descent method for solving the quadratic fractional programming subproblem. In addition, we provide some theoretical analysis for the proposed method. Our experiments have shown that the proposed method significantly and consistently outperforms existing solutions in term of accuracy.

Ganzhao Yuan, Li Shen, Wei-Shi Zheng

Sparse optimization is a central problem in machine learning and computer vision. However, this problem is inherently NP-hard and thus difficult to solve in general. Combinatorial search methods find the global optimal solution but are confined to small-sized problems, while coordinate descent methods are efficient but often suffer from poor local minima. This paper considers a new block decomposition algorithm that combines the effectiveness of combinatorial search methods and the efficiency of coordinate descent methods. Specifically, we consider a random strategy or/and a greedy strategy to select a subset of coordinates as the working set, and then perform a global combinatorial search over the working set based on the original objective function. We show that our method finds stronger stationary points than Amir Beck et al.'s coordinate-wise optimization method. In addition, we establish the convergence rate of our algorithm. Our experiments on solving sparse regularized and sparsity constrained least squares optimization problems demonstrate that our method achieves state-of-the-art performance in terms of accuracy. For example, our method generally outperforms the well-known greedy pursuit method.

Ganzhao Yuan, Wei-Shi Zheng, Li Shen et al.

Composite function minimization captures a wide spectrum of applications in both computer vision and machine learning. It includes bound constrained optimization, $\ell_1$ norm regularized optimization, and $\ell_0$ norm regularized optimization as special cases. This paper proposes and analyzes a new Generalized Matrix Splitting Algorithm (GMSA) for minimizing composite functions. It can be viewed as a generalization of the classical Gauss-Seidel method and the Successive Over-Relaxation method for solving linear systems in the literature. Our algorithm is derived from a novel triangle operator mapping, which can be computed exactly using a new generalized Gaussian elimination procedure. We establish the global convergence, convergence rate, and iteration complexity of GMSA for convex problems. In addition, we also discuss several important extensions of GMSA. Finally, we validate the performance of our proposed method on three particular applications: nonnegative matrix factorization, $\ell_0$ norm regularized sparse coding, and $\ell_1$ norm regularized Dantzig selector problem. Extensive experiments show that our method achieves state-of-the-art performance in term of both efficiency and efficacy.

Ganzhao Yuan

Nonsmooth composite optimization with orthogonality constraints has a wide range of applications in statistical learning and data science. However, this problem is challenging due to its nonsmooth objective and computationally expensive, non-convex constraints. In this paper, we propose a new approach called \textbf{OBCD}, which leverages Block Coordinate Descent to address these challenges. \textbf{OBCD} is a feasible method with a small computational footprint. In each iteration, it updates $k$ rows of the solution matrix, where $k \geq 2$, by globally solving a small nonsmooth optimization problem under orthogonality constraints. We prove that the limiting points of \textbf{OBCD}, referred to as (global) block-$k$ stationary points, offer stronger optimality than standard critical points. Furthermore, we show that \textbf{OBCD} converges to $ε$-block-$k$ stationary points with an ergodic convergence rate of $\mathcal{O}(1/ε)$. Additionally, under the Kurdyka-Lojasiewicz (KL) inequality, we establish the non-ergodic convergence rate of \textbf{OBCD}. We also extend \textbf{OBCD} by incorporating breakpoint searching methods for subproblem solving and greedy strategies for working set selection. Comprehensive experiments demonstrate the superior performance of our approach across various tasks.

Di He, Songjun Tu, Ajay Jaiswal et al.

Weight decay is a standard regularization technique for training large language models (LLMs). While it is common to assign a uniform decay rate to every layer, this approach overlooks the structural diversity of LLMs and the varying spectral properties across modules. In this paper, we introduce AlphaDecay, a simple yet effective method that adaptively assigns different weight decay strengths to each module of an LLM. Our approach is guided by Heavy-Tailed Self-Regularization (HT-SR) theory, which analyzes the empirical spectral density (ESD) of weight correlation matrices to quantify "heavy-tailedness." Modules exhibiting more pronounced heavy-tailed ESDs, reflecting stronger feature learning, are assigned weaker decay, while modules with lighter-tailed spectra receive stronger decay. Our method leverages tailored weight decay assignments to balance the module-wise differences in spectral properties, leading to improved performance. Extensive pre-training tasks with various model sizes from 60M to 1B demonstrate that AlphaDecay achieves better perplexity and generalization than conventional uniform decay and other adaptive decay baselines. Our code is available at https://github.com/hed-ucas/AlphaDecay.

Da Chang, Yu Li, Ganzhao Yuan

In the training of large language models (LLMs), updating parameters more efficiently and stably has always been an important challenge. To achieve efficient parameter updates, existing methods usually achieve performance comparable to full parameter updates through methods such as low-dimensional decomposition or layer-wise selective updates. In this work, we propose AlphaAdam, an optimization framework for LLM from the perspective of intra-layer parameter updates. By decoupling parameter updates and dynamically adjusting their strength, AlphaAdam accelerates convergence and improves training stability. We construct parameter masks based on the consistency of historical momentum and gradient direction and combine them with an adaptive mask strength strategy to ensure efficient optimization and theoretical convergence guarantees, which is also applicable to most momentum-based optimizers. Extensive experiments show that AlphaAdam outperforms state-of-the-art methods such as AdamW in terms of convergence speed and computational efficiency across tasks, including GPT-2 pre-trained and fine-tuned RoBERTa and Llama-7B. Our AlphaAdam implements an optimizer enhancement framework for LLMs through intra-layer asynchronous masked adaptive updates. Our code is available in this https://github.com/MaeChd/AlphaAdam.

Da Chang, Qiankun Shi, Lvgang Zhang et al.

Orthogonalized-update optimizers such as Muon improve training of matrix-valued parameters, but existing extensions mostly act either after orthogonalization by rescaling updates or before it with heavier whitening-based preconditioners. We introduce {\method}, a lightweight family of pre-orthogonalization equilibration schemes for Muon in three forms: two-sided row/column normalization (RC), row normalization (R), and column normalization (C). These variants rebalance the momentum matrix before finite-step Newton--Schulz using row/column squared-norm statistics and only $\mathcal{O}(m+n)$ auxiliary state. We show that finite-step orthogonalization is governed by input spectral properties, especially stable rank and condition number, and that row/column normalization is a zeroth-order whitening surrogate that removes marginal scale mismatch. For the hidden matrix weights targeted by {\method}, the row-normalized variant R is the natural default and preserves the $\widetilde{\mathcal{O}}(T^{-1/4})$ stationarity guarantee of Muon-type methods. In LLaMA2 pretraining on C4, the default R variant consistently outperforms Muon on 130M and 350M models, yielding faster convergence and lower validation perplexity.

Ganzhao Yuan

The Exponential Moving Average (EMA) is a cornerstone of widely used optimizers such as Adam. However, existing theoretical analyses of Adam-style methods have notable limitations: their guarantees can remain suboptimal in the zero-noise regime, rely on restrictive boundedness conditions (e.g., bounded gradients or objective gaps), use constant or open-loop stepsizes, or require prior knowledge of Lipschitz constants. To overcome these bottlenecks, we introduce OptEMA and analyze two novel variants: OptEMA-M, which applies an adaptive, decreasing EMA coefficient to the first-order moment with a fixed second-order decay, and OptEMA-V, which swaps these roles. Crucially, OptEMA is closed-loop and Lipschitz-free in the sense that its effective stepsizes are trajectory-dependent and do not require the Lipschitz constant for parameterization. Under standard stochastic gradient descent (SGD) assumptions, namely smoothness, a lower-bounded objective, and unbiased gradients with bounded variance, we establish rigorous convergence guarantees. Both variants achieve a noise-adaptive convergence rate of $\widetilde{\mathcal{O}}(T^{-1/2}+σ^{1/2} T^{-1/4})$ for the average gradient norm, where $σ$ is the noise level. In particular, in the zero-noise regime where $σ=0$, our bounds reduce to the nearly optimal deterministic rate $\widetilde{\mathcal{O}}(T^{-1/2})$ without manual hyperparameter retuning.

Da Chang, Yongxiang Liu, Ganzhao Yuan

The Muon optimizer has demonstrated remarkable empirical success in handling matrix-structured parameters for training neural networks. However, a significant gap remains between its practical performance and theoretical understanding. Existing analyses show that the Muon variants achieve only a suboptimal iteration complexity of $\mathcal{O}(T^{-1/4})$ in stochastic non-convex settings, where $T$ denotes the number of iterations. To explore the theoretical limits of the Muon framework, we analyze two Momentum-based Variance-Reduced variants: a one-batch version (Muon-MVR1) and a two-batch version (Muon-MVR2). We provide the first rigorous proof that incorporating variance reduction enables Muon-MVR2 to attain the optimal iteration complexity of $\tilde{\mathcal{O}}(T^{-1/3})$, thereby matching the theoretical lower bound for this class of problems. Furthermore, our analysis establishes last-iterate convergence guarantees for Muon variants under the Polyak-Łojasiewicz (PŁ) condition. Extensive experiments on vision (CIFAR-10) and language (C4) benchmarks corroborate our theoretical findings on per-iteration convergence. Overall, this work offers the first proof of optimality for a Muon-style optimizer and clarifies the path toward developing more practically efficient, accelerated variants.

Ganzhao Yuan

We propose ALFCG (Adaptive Lipschitz-Free Conditional Gradient), the first \textit{adaptive} projection-free framework for stochastic composite nonconvex minimization that \textit{requires neither global smoothness constants nor line search}. Unlike prior conditional gradient methods that use openloop diminishing stepsizes, conservative Lipschitz constants, or costly backtracking, ALFCG maintains a self-normalized accumulator of historical iterate differences to estimate local smoothness and minimize a quadratic surrogate model at each step. This retains the simplicity of Frank-Wolfe while adapting to unknown geometry. We study three variants. ALFCG-FS addresses finite-sum problems with a SPIDER estimator. ALFCG-MVR1 and ALFCG-MVR2 handle stochastic expectation problems by using momentum-based variance reduction with single-batch and two-batch updates, and operate under average and individual smoothness, respectively. To reach an $ε$-stationary point, ALFCG-FS attains $\mathcal{O}(N+\sqrt{N}ε^{-2})$ iteration complexity, while ALFCG-MVR1 and ALFCG-MVR2 achieve $\tilde{\mathcal{O}}(σ^2ε^{-4}+ε^{-2})$ and $\tilde{\mathcal{O}}(σε^{-3}+ε^{-2})$, where $N$ is the number of components and $σ$ is the noise level. In contrast to typical $\mathcal{O}(ε^{-4})$ or $\mathcal{O}(ε^{-3})$ rates, our bounds reduce to the optimal rate up to logarithmic factors $\tilde{\mathcal{O}}(ε^{-2})$ as the noise level $σ\to 0$. Extensive experiments on multiclass classification over nuclear norm balls and $\ell_p$ balls show that ALFCG generally outperforms state-of-the-art conditional gradient baselines.

Ganzhao Yuan

This paper proposes {\sf AEPG-SPIDER}, an Adaptive Extrapolated Proximal Gradient (AEPG) method with variance reduction for minimizing composite nonconvex finite-sum functions. It integrates three acceleration techniques: adaptive stepsizes, Nesterov's extrapolation, and the recursive stochastic path-integrated estimator SPIDER. Unlike existing methods that adjust the stepsize factor using historical gradients, {\sf AEPG-SPIDER} relies on past iterate differences for its update. While targeting stochastic finite-sum problems, {\sf AEPG-SPIDER} simplifies to {\sf AEPG} in the full-batch, non-stochastic setting, which is also of independent interest. To our knowledge, {\sf AEPG-SPIDER} and {\sf AEPG} are the first Lipschitz-free methods to achieve optimal iteration complexity for this class of \textit{composite} minimization problems. Specifically, {\sf AEPG} achieves the optimal iteration complexity of $\mathcal{O}(N ε^{-2})$, while {\sf AEPG-SPIDER} achieves $\mathcal{O}(N + \sqrt{N} ε^{-2})$ for finding $ε$-approximate stationary points, where $N$ is the number of component functions. Under the Kurdyka-Lojasiewicz (KL) assumption, we establish non-ergodic convergence rates for both methods. Preliminary experiments on sparse phase retrieval and linear eigenvalue problems demonstrate the superior performance of {\sf AEPG-SPIDER} and {\sf AEPG} compared to existing methods.

Ganzhao Yuan

This paper considers a class of structured fractional minimization problems. The numerator consists of a differentiable function, a simple nonconvex nonsmooth function, a concave nonsmooth function, and a convex nonsmooth function composed with a linear operator. The denominator is a continuous function that is either weakly convex or has a weakly convex square root. These problems are prevalent in various important applications in machine learning and data science. Existing methods, primarily based on subgradient methods and smoothing proximal gradient methods, often suffer from slow convergence and numerical stability issues. In this paper, we introduce {\sf FADMM}, the first Alternating Direction Method of Multipliers tailored for this class of problems. {\sf FADMM} decouples the original problem into linearized proximal subproblems, featuring two variants: one using Dinkelbach's parametric method ({\sf FADMM-D}) and the other using the quadratic transform method ({\sf FADMM-Q}). By introducing a novel Lyapunov function, we establish that {\sf FADMM} converges to $ε$-approximate critical points of the problem within an oracle complexity of $\mathcal{O}(1/ε^{3})$. Extensive experiments on synthetic and real-world datasets, including sparse Fisher discriminant analysis, robust Sharpe ratio minimization, and robust sparse recovery, demonstrate the effectiveness of our approach. Keywords: Fractional Minimization, Nonconvex Optimization, Proximal Linearized ADMM, Nonsmooth Optimization, Convergence Analysis

Ganzhao Yuan

We consider a class of structured fractional minimization problems, in which the numerator part of the objective is the sum of a differentiable convex function and a convex non-smooth function, while the denominator part is a convex or concave function. This problem is difficult to solve since it is non-convex. By exploiting the structure of the problem, we propose two Coordinate Descent (CD) methods for solving this problem. The proposed methods iteratively solve a one-dimensional subproblem \textit{globally}, and they are guaranteed to converge to coordinate-wise stationary points. In the case of a convex denominator, under a weak \textit{locally bounded non-convexity condition}, we prove that the optimality of coordinate-wise stationary point is stronger than that of the standard critical point and directional point. Under additional suitable conditions, CD methods converge Q-linearly to coordinate-wise stationary points. In the case of a concave denominator, we show that any critical point is a global minimum, and CD methods converge to the global minimum with a sublinear convergence rate. We demonstrate the applicability of the proposed methods to some machine learning and signal processing models. Our experiments on real-world data have shown that our method significantly and consistently outperforms existing methods in terms of accuracy.

Ganzhao Yuan

Difference-of-Convex (DC) minimization, referring to the problem of minimizing the difference of two convex functions, has been found rich applications in statistical learning and studied extensively for decades. However, existing methods are primarily based on multi-stage convex relaxation, only leading to weak optimality of critical points. This paper proposes a coordinate descent method for minimizing a class of DC functions based on sequential nonconvex approximation. Our approach iteratively solves a nonconvex one-dimensional subproblem globally, and it is guaranteed to converge to a coordinate-wise stationary point. We prove that this new optimality condition is always stronger than the standard critical point condition and directional point condition under a mild \textit{locally bounded nonconvexity assumption}. For comparisons, we also include a naive variant of coordinate descent methods based on sequential convex approximation in our study. When the objective function satisfies a \textit{globally bounded nonconvexity assumption} and \textit{Luo-Tseng error bound assumption}, coordinate descent methods achieve \textit{Q-linear} convergence rate. Also, for many applications of interest, we show that the nonconvex one-dimensional subproblem can be computed exactly and efficiently using a breakpoint searching method. Finally, we have conducted extensive experiments on several statistical learning tasks to show the superiority of our approach. Keywords: Coordinate Descent, DC Minimization, DC Programming, Difference-of-Convex Programs, Nonconvex Optimization, Sparse Optimization, Binary Optimization.

Ganzhao Yuan, Haoxian Tan, Wei-Shi Zheng

Sparse inverse covariance selection is a fundamental problem for analyzing dependencies in high dimensional data. However, such a problem is difficult to solve since it is NP-hard. Existing solutions are primarily based on convex approximation and iterative hard thresholding, which only lead to sub-optimal solutions. In this work, we propose a coordinate-wise optimization algorithm to solve this problem which is guaranteed to converge to a coordinate-wise minimum point. The algorithm iteratively and greedily selects one variable or swaps two variables to identify the support set, and then solves a reduced convex optimization problem over the support set to achieve the greatest descent. As a side contribution of this paper, we propose a Newton-like algorithm to solve the reduced convex sub-problem, which is proven to always converge to the optimal solution with global linear convergence rate and local quadratic convergence rate. Finally, we demonstrate the efficacy of our method on synthetic data and real-world data sets. As a result, the proposed method consistently outperforms existing solutions in terms of accuracy.

Ganzhao Yuan, Yin Yang, Zhenjie Zhang et al.

Differential privacy enables organizations to collect accurate aggregates over sensitive data with strong, rigorous guarantees on individuals' privacy. Previous work has found that under differential privacy, computing multiple correlated aggregates as a batch, using an appropriate \emph{strategy}, may yield higher accuracy than computing each of them independently. However, finding the best strategy that maximizes result accuracy is non-trivial, as it involves solving a complex constrained optimization program that appears to be non-linear and non-convex. Hence, in the past much effort has been devoted in solving this non-convex optimization program. Existing approaches include various sophisticated heuristics and expensive numerical solutions. None of them, however, guarantees to find the optimal solution of this optimization problem. This paper points out that under ($ε$, $δ$)-differential privacy, the optimal solution of the above constrained optimization problem in search of a suitable strategy can be found, rather surprisingly, by solving a simple and elegant convex optimization program. Then, we propose an efficient algorithm based on Newton's method, which we prove to always converge to the optimal solution with linear global convergence rate and quadratic local convergence rate. Empirical evaluations demonstrate the accuracy and efficiency of the proposed solution.