This paper introduces a novel approach to solving multi-block nonconvex composite optimization problems through a proximal linearized Alternating Direction Method of Multipliers (ADMM). This method incorporates an Increasing Penalization and Decreasing Smoothing (IPDS) strategy. Distinguishing itself from existing ADMM-style algorithms, our approach (denoted IPDS-ADMM) imposes a less stringent condition, specifically requiring continuity in just one block of the objective function. IPDS-ADMM requires that the penalty increases and the smoothing parameter decreases, both at a controlled pace. When the associated linear operator is bijective, IPDS-ADMM uses an over-relaxation stepsize for faster convergence; however, when the linear operator is surjective, IPDS-ADMM uses an under-relaxation stepsize for global convergence. We devise a novel potential function to facilitate our convergence analysis and prove an oracle complexity $\mathcal{O}(\epsilon^{-3})$ to achieve an $\epsilon$-approximate critical point. To the best of our knowledge, this is the first complexity result for using ADMM to solve this class of nonsmooth nonconvex problems. Finally, some experiments on the sparse PCA problem are conducted to demonstrate the effectiveness of our approach.

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 $\epsilon$-approximate critical points of the problem within an oracle complexity of $\mathcal{O}(1/\epsilon^{3})$. 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.

Efficient optimization is essential for training large language models. Although intra-layer selective updates have been explored, a general mechanism that enables fine-grained control while ensuring convergence guarantees is still lacking. To bridge this gap, we propose \textbf{MGUP}, a novel mechanism for selective updates. \textbf{MGUP} augments standard momentum-based optimizers by applying larger step-sizes to a selected fixed proportion of parameters in each iteration, while applying smaller, non-zero step-sizes to the rest. As a nearly {plug-and-play} module, \textbf{MGUP} seamlessly integrates with optimizers such as AdamW, Lion, and Muon. This yields powerful variants such as \textbf{MGUP-AdamW}, \textbf{MGUP-Lion}, and \textbf{MGUP-Muon}. Under standard assumptions, we provide theoretical convergence guarantees for \textbf{MGUP-AdamW} (without weight decay) in stochastic optimization. Extensive experiments across diverse tasks, including MAE pretraining, LLM pretraining, and downstream fine-tuning, demonstrate that our \textbf{MGUP}-enhanced optimizers achieve superior or more stable performance compared to their original base optimizers. We offer a principled, versatile, and theoretically grounded strategy for efficient intra-layer selective updates, accelerating and stabilizing the training of large-scale models. The code is publicly available at https://github.com/MaeChd/MGUP.

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. The code is available at https://github.com/hed-ucas/AlphaDecay.

Nonsmooth sparsity constrained optimization encompasses a broad spectrum of applications in machine learning. This problem is generally non-convex and NP-hard. Existing solutions to this problem exhibit several notable limitations, including their inability to address general nonsmooth problems, tendency to yield weaker optimality conditions, and lack of comprehensive convergence analysis. This paper considers Smoothing Proximal Gradient Methods (SPGM) as solutions to nonsmooth sparsity constrained optimization problems. Two specific variants of SPGM are explored: one based on Iterative Hard Thresholding (SPGM-IHT) and the other on Block Coordinate Decomposition (SPGM-BCD). It is shown that the SPGM-BCD algorithm finds stronger stationary points compared to previous methods. Additionally, novel theories for analyzing the convergence rates to approximate global optimal solutions of both the SPGM-IHT and SPGM-BCD algorithms are developed. Our theoretical bounds, capitalizing on the intrinsic sparsity of the optimization problem, are on par with the best-known error bounds available to date. Finally, numerical experiments reveal that SPGM-IHT performs comparably to current IHT-style methods, while SPGM-BCD consistently surpasses them.

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 mildlocally 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 globally bounded nonconvexity assumption and Luo-Tseng error bound assumption, coordinate descent methods achieve 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.

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 *globally*, and they are guaranteed to converge to coordinate-wise stationary points. In the case of a convex denominator, under a weak *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.

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.

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 effective 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 on synthetic data and real-world data have shown that our method significantly and consistently outperforms existing solutions in term of accuracy.

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)[1], 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.

We address the problem of correcting the exposure of underexposed photos. Previous methods have tackled this problem from many different perspectives and achieved remarkable progress. However, they usually fail to produce natural-looking results due to the existence of visual artifacts such as color distortion, loss of detail, exposure inconsistency, etc. We find that the main reason why existing methods induce these artifacts is because they break a perceptually similarity between the input and output. Based on this observation, an effective criterion, termed as perceptually bidirectional similarity (PBS) is proposed. Based on this criterion and the Retinex theory, we cast the exposure correction problem as an illumination estimation optimization, where PBS is defined as three constraints for estimating illumination that can generate the desired result with even exposure, vivid color and clear textures. Qualitative and quantitative comparisons, and the user study demonstrate the superiority of our method over the state-of-the-art methods.

Binary optimization is a central problem in mathematical optimization and its applications are abundant. To solve this problem, we propose a new class of continuous optimization techniques, which is based on Mathematical Programming with Equilibrium Constraints (MPECs). We first reformulate the binary program as an equivalent augmented biconvex optimization problem with a bilinear equality constraint, then we propose an exact penalty method to solve it. The resulting algorithm seeks a desirable solution to the original problem via solving a sequence of linear programming convex relaxation subproblems. In addition, we prove that the penalty function, induced by adding the complementarity constraint to the objective, is exact, i.e., it has the same local and global minima with those of the original binary program when the penalty parameter is over some threshold. The convergence of the algorithm can be guaranteed, since it essentially reduces to block coordinate descent in the literature. Finally, we demonstrate the effectiveness of our method on the problem of dense subgraph discovery. Extensive experiments show that our method outperforms existing techniques, such as iterative hard thresholding and linear programming relaxation.

Composite function minimization captures a wide spectrum of applications in both computer vision and machine learning. It includes bound constrained optimization and cardinality regularized optimization as special cases. This paper proposes and analyzes a new Matrix Splitting Method (MSM) 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. Incorporating a new Gaussian elimination procedure, the matrix splitting method achieves state-of-the-art performance. For convex problems, we establish the global convergence, convergence rate, and iteration complexity of MSM, while for non-convex problems, we prove its global convergence. Finally, we validate the performance of our matrix splitting method on two particular applications: nonnegative matrix factorization and cardinality regularized sparse coding. Extensive experiments show that our method outperforms existing composite function minimization techniques in term of both efficiency and efficacy.

In this paper, we propose a fast Gauss-Seidel Operator Splitting (GSOS) algorithm for addressing multi-term nonsmooth convex composite optimization, which has wide applications in machine learning, signal processing and statistics. The proposed GSOS algorithm inherits the advantage of the Gauss-Seidel technique to accelerate the optimization procedure, and leverages the operator splitting technique to reduce the computational complexity. In addition, we develop a new technique to establish the global convergence of the GSOS algorithm. To be specific, we first reformulate the iterations of GSOS as a two-step iterations algorithm by employing the tool of operator optimization theory. Subsequently, we establish the convergence of GSOS based on the two-step iterations algorithm reformulation. At last, we apply the proposed GSOS algorithm to solve overlapping group Lasso and graph-guided fused Lasso problems. Numerical experiments show that our proposed GSOS algorithm is superior to the state-of-the-art algorithms in terms of both efficiency and effectiveness.

Semi-definite rank minimization problems model a wide range of applications in both signal processing and machine learning fields. This class of problem is NP-hard in general. In this paper, we propose a proximal Alternating Direction Method (ADM) for the well-known semi-definite rank regularized minimization problem. Specifically, we first reformulate this NP-hard problem as an equivalent biconvex MPEC (Mathematical Program with Equilibrium Constraints), and then solve it using proximal ADM, which involves solving a sequence of structured convex semi-definite subproblems to find a desirable solution to the original rank regularized optimization problem. Moreover, based on the Kurdyka-Lojasiewicz inequality, we prove that the proposed method always converges to a KKT stationary point under mild conditions. We apply the proposed method to the widely studied and popular sensor network localization problem. Our extensive experiments demonstrate that the proposed algorithm outperforms state-of-the-art low-rank semi-definite minimization algorithms in terms of solution quality.

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 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-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.

Total Variation (TV) is an effective and popular prior model in the field of regularization-based image processing. This paper focuses on TV for image restoration in the presence of impulse noise. 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) [42], which is based on TV with ℓ02-norm data fidelity, only give sub-optimal performance. In this paper, we propose a new method, called ℓ0TV -PADMM, which solves the TV-based restoration problem with ℓ0-norm data fidelity. To effectively deal with the resulting non-convex non-smooth optimization problem, we first reformulate it as an equivalent MPEC (Mathematical Program with Equilibrium Constraints), and then solve it using a proximal Alternating Direction Method of Multipliers (PADMM). Our ℓ0TV -PADMM method finds a desirable solution to the original ℓ0-norm optimization problem and is proven to be convergent under mild conditions. We apply ℓ0TV -PADMM to the problems of image denoising and deblurring in the presence of impulse noise. Our extensive experiments demonstrate that ℓ0TV -PADMM outperforms state-of-the-art image restoration methods.

Differential privacy is a promising privacy-preserving paradigm for statistical query processing over sensitive data. It works by injecting random noise into each query result such that it is provably hard for the adversary to infer the presence or absence of any individual record from the published noisy results. The main objective in differentially private query processing is to maximize the accuracy of the query results while satisfying the privacy guarantees. Previous work, notably Li et al. [2010], has suggested that, with an appropriate strategy, processing a batch of correlated queries as a whole achieves considerably higher accuracy than answering them individually. However, to our knowledge there is currently no practical solution to find such a strategy for an arbitrary query batch; existing methods either return strategies of poor quality (often worse than naive methods) or require prohibitively expensive computations for even moderately large domains. Motivated by this, we propose a low-rank mechanism (LRM), the first practical differentially private technique for answering batch linear queries with high accuracy. LRM works for both exact (i.e., ϵ-) and approximate (i.e., (ϵ, δ)-) differential privacy definitions. We derive the utility guarantees of LRM and provide guidance on how to set the privacy parameters, given the user's utility expectation. Extensive experiments using real data demonstrate that our proposed method consistently outperforms state-of-the-art query processing solutions under differential privacy, by large margins.

Differential privacy is a promising privacy-preserving paradigm for statistical query processing over sensitive data. It works by injecting random noise into each query result, such that it is provably hard for the adversary to infer the presence or absence of any individual record from the published noisy results. The main objective in differentially private query processing is to maximize the accuracy of the query results, while satisfying the privacy guarantees. Previous work, notably the matrix mechanism [16], has suggested that processing a batch of correlated queries as a whole can potentially achieve considerable accuracy gains, compared to answering them individually. However, as we point out in this paper, the matrix mechanism is mainly of theoretical interest; in particular, several inherent problems in its design limit its accuracy in practice, which almost never exceeds that of naïve methods. In fact, we are not aware of any existing solution that can effectively optimize a query batch under differential privacy. Motivated by this, we propose the Low-Rank Mechanism (LRM), the first practical differentially private technique for answering batch queries with high accuracy, based on a low rank approximation of the workload matrix. We prove that the accuracy provided by LRM is close to the theoretical lower bound for any mechanism to answer a batch of queries under differential privacy. Extensive experiments using real data demonstrate that LRM consistently outperforms state-of-the-art query processing solutions under differential privacy, by large margins.