arXiv subjects

Anthony Man-Cho So

Publications and source records attributed to Anthony Man-Cho So.

At least 37 records · Page 2Linked to original sources

arXiv2025

Accelerated Price Adjustment for Fisher Markets with Exact Recovery of Competitive Equilibrium

The canonical price-adjustment process, tâtonnement, typically fails to converge to the exact competitive equilibrium (CE) and requires a high iteration complexity of $\tilde{\mathcal{O}}(1/ε)$ to compute $ε$-CE prices in widely studied linear and quasi-linear Fisher markets. This paper proposes refined price-adjustment processes to overcome these limitations. By formulating the task of finding CE of a (quasi-)linear Fisher market as a strongly convex nonsmooth minimization problem, we develop a novel accelerated price-adjustment method (APM) that finds an $ε$-CE price in $\tilde{\mathcal{O}}(1/\sqrtε)$ lightweight iterations, which significantly improves upon the iteration complexities of tâtonnement methods. Furthermore, through our new formulation, we construct a recovery oracle that maps approximate CE prices to exact CE prices at a low computational cost. By coupling this recovery oracle with APM, we obtain an adaptive price-adjustment method whose iterates converge to CE prices in finite steps. To the best of our knowledge, this is the first convergence guarantee to exact CE for price-adjustment methods in linear and quasi-linear Fisher markets. Our developments pave the way for efficient lightweight computation of CE prices. We also present numerical results to demonstrate the fast convergence of the proposed methods and the efficient recovery of CE prices.

math.OC↗

arXiv2025

Doubly Smoothed Optimistic Gradients: A Universal Approach for Smooth Minimax Problems

Smooth minimax optimization problems play a central role in a wide range of applications, including machine learning, game theory, and operations research. However, existing algorithmic frameworks vary significantly depending on the problem structure -- whether it is convex-concave, nonconvex-concave, convex-nonconcave, or even nonconvex-nonconcave with additional regularity conditions. In particular, this diversity complicates the tuning of step-sizes since even verifying convexity (or concavity) assumptions is challenging and problem-dependent. We introduce a universal and single-loop algorithm, Doubly Smoothed Optimistic Gradient Descent Ascent (DS-OGDA), that applies to a broad class of smooth minimax problems. Specifically, this class includes convex-concave, nonconvex-concave, convex-nonconcave, and nonconvex-nonconcave minimax optimization problems satisfying a one-sided Kurdyka-Lojasiewicz (KL) property. DS-OGDA works with a universal single set of parameters for all problems in this class, eliminating the need for prior structural knowledge to determine step-sizes. Moreover, when a particular problem structure in our class is specified, DS-OGDA achieves optimal or best-known performance guarantees. Overall, our results provide a comprehensive and versatile framework for smooth minimax optimization, bridging the gap between convex and nonconvex problem structures and simplifying the choice of algorithmic strategies across diverse applications.

math.OC↗

arXiv2025

Probe-Free Low-Rank Activation Intervention

Language models (LMs) can produce texts that appear accurate and coherent but contain untruthful or toxic content. Inference-time interventions that edit the hidden activations have shown promising results in steering the LMs towards desirable generations. Existing activation intervention methods often comprise an activation probe to detect undesirable generation, triggering the activation modification to steer subsequent generation. This paper proposes a probe-free intervention method FLORAIN for all attention heads in a specific activation layer. It eliminates the need to train classifiers for probing purposes. The intervention function is parametrized by a sample-wise nonlinear low-rank mapping, which is trained by minimizing the distance between the modified activations and their projection onto the manifold of desirable content. Under specific constructions of the manifold and projection distance, we show that the intervention strategy can be computed efficiently by solving a smooth optimization problem. The empirical results, benchmarked on multiple base models, demonstrate that FLORAIN consistently outperforms several baseline methods in enhancing model truthfulness and quality across generation and multiple-choice tasks.

cs.LG↗

arXiv2025

Nonsmooth Nonconvex-Nonconcave Minimax Optimization: Primal-Dual Balancing and Iteration Complexity Analysis

Nonconvex-nonconcave minimax optimization has gained widespread interest over the last decade. However, most existing works focus on variants of gradient descent-ascent (GDA) algorithms, which are only applicable to smooth nonconvex-concave settings. To address this limitation, we propose a novel algorithm named smoothed proximal linear descent-ascent (smoothed PLDA), which can effectively handle a broad range of structured nonsmooth nonconvex-nonconcave minimax problems. Specifically, we consider the setting where the primal function has a nonsmooth composite structure and the dual function possesses the Kurdyka-Lojasiewicz (KL) property with exponent $θ\in [0,1)$. We introduce a novel convergence analysis framework for smoothed PLDA, the key components of which are our newly developed nonsmooth primal error bound and dual error bound. Using this framework, we show that smoothed PLDA can find both $ε$-game-stationary points and $ε$-optimization-stationary points of the problems of interest in $\mathcal{O}(ε^{-2\max\{2θ,1\}})$ iterations. Furthermore, when $θ\in [0,\frac{1}{2}]$, smoothed PLDA achieves the optimal iteration complexity of $\mathcal{O}(ε^{-2})$. To further demonstrate the effectiveness and wide applicability of our analysis framework, we show that certain max-structured problem possesses the KL property with exponent $θ=0$ under mild assumptions. As a by-product, we establish algorithm-independent quantitative relationships among various stationarity concepts, which may be of independent interest.

math.OC↗

arXiv2025

Testing Approximate Stationarity Concepts for Piecewise Affine Functions

We study the basic computational problem of detecting approximate stationary points for continuous piecewise affine (PA) functions. Our contributions span multiple aspects, including complexity, regularity, and algorithms. Specifically, we show that testing first-order approximate stationarity concepts, as defined by commonly used generalized subdifferentials, is computationally intractable unless P=NP. To facilitate computability, we consider a polynomial-time solvable relaxation by abusing the convex subdifferential sum rule and establish a tight characterization of its exactness. Furthermore, addressing an open issue motivated by the need to terminate the subgradient method in finite time, we introduce the first oracle-polynomial-time algorithm to detect so-called near-approximate stationary points for PA functions. A notable byproduct of our development in regularity is the first necessary and sufficient condition for the validity of an equality-type (Clarke) subdifferential sum rule. Our techniques revolve around two new geometric notions for convex polytopes and may be of independent interest in nonsmooth analysis. Moreover, some corollaries of our work on complexity and algorithms for stationarity testing address open questions in the literature. To demonstrate the versatility of our results, we complement our findings with applications to a series of structured piecewise smooth functions, including $ρ$-margin-loss SVM, piecewise affine regression, and nonsmooth neural networks.

math.OC↗

arXiv2024

Extreme Point Pursuit -- Part I: A Framework for Constant Modulus Optimization

This study develops a framework for a class of constant modulus (CM) optimization problems, which covers binary constraints, discrete phase constraints, semi-orthogonal matrix constraints, non-negative semi-orthogonal matrix constraints, and several types of binary assignment constraints. Capitalizing on the basic principles of concave minimization and error bounds, we study a convex-constrained penalized formulation for general CM problems. The advantage of such formulation is that it allows us to leverage non-convex optimization techniques, such as the simple projected gradient method, to build algorithms. As the first part of this study, we explore the theory of this framework. We study conditions under which the formulation provides exact penalization results. We also examine computational aspects relating to the use of the projected gradient method for each type of CM constraint. Our study suggests that the proposed framework has a broad scope of applicability.

eess.SP↗

arXiv2024

Extreme Point Pursuit -- Part II: Further Error Bound Analysis and Applications

In the first part of this study, a convex-constrained penalized formulation was studied for a class of constant modulus (CM) problems. In particular, the error bound techniques were shown to play a vital role in providing exact penalization results. In this second part of the study, we continue our error bound analysis for the cases of partial permutation matrices, size-constrained assignment matrices and non-negative semi-orthogonal matrices. We develop new error bounds and penalized formulations for these three cases, and the new formulations possess good structures for building computationally efficient algorithms. Moreover, we provide numerical results to demonstrate our framework in a variety of applications such as the densest k-subgraph problem, graph matching, size-constrained clustering, non-negative orthogonal matrix factorization and sparse fair principal component analysis.

eess.SP↗

arXiv2024

Network Games Induced Prior for Graph Topology Learning

Learning the graph topology of a complex network is challenging due to limited data availability and imprecise data models. A common remedy in existing works is to incorporate priors such as sparsity or modularity which highlight on the structural property of graph topology. We depart from these approaches to develop priors that are directly inspired by complex network dynamics. Focusing on social networks with actions modeled by equilibriums of linear quadratic games, we postulate that the social network topologies are optimized with respect to a social welfare function. Utilizing this prior knowledge, we propose a network games induced regularizer to assist graph learning. We then formulate the graph topology learning problem as a bilevel program. We develop a two-timescale gradient algorithm to tackle the latter. We draw theoretical insights on the optimal graph structure of the bilevel program and show that they agree with the topology in several man-made networks. Empirically, we demonstrate the proposed formulation gives rise to reliable estimate of graph topology.

eess.SP↗

arXiv2024

Revisiting Subgradient Method: Complexity and Convergence Beyond Lipschitz Continuity

The subgradient method is one of the most fundamental algorithmic schemes for nonsmooth optimization. The existing complexity and convergence results for this method are mainly derived for Lipschitz continuous objective functions. In this work, we first extend the typical iteration complexity results for the subgradient method to cover non-Lipschitz convex and weakly convex minimization. Specifically, for the convex case, we can drive the suboptimality gap to below $\varepsilon$ in $\mathcal{O}( \varepsilon^{-2} )$ iterations; for the weakly convex case, we can drive the gradient norm of the Moreau envelope of the objective function to below $\varepsilon$ in $\mathcal{O}( \varepsilon^{-4} )$ iterations. Then, we provide convergence results for the subgradient method in the non-Lipschitz setting when proper diminishing rules on the step size are used. In particular, when $f$ is convex, we establish an $\mathcal{O}(\log(k)/\sqrt{k})$ rate of convergence in terms of the suboptimality gap, where $k$ represents the iteration count. With an additional quadratic growth property, the rate is improved to $\mathcal{O}(1/k)$ in terms of the squared distance to the optimal solution set. When $f$ is weakly convex, asymptotic convergence is established. Our results neither require any modification to the subgradient method nor impose any growth condition on the subgradients, while our analysis is surprisingly simple. To further illustrate the wide applicability of our framework, we extend the aforementioned iteration complexity results to cover the truncated subgradient, the stochastic subgradient, and the proximal subgradient methods for non-Lipschitz convex / weakly convex objective functions.

math.OC↗

arXiv2024

Shuffling Gradient Descent-Ascent with Variance Reduction for Nonconvex-Strongly Concave Smooth Minimax Problems

In recent years, there has been considerable interest in designing stochastic first-order algorithms to tackle finite-sum smooth minimax problems. To obtain the gradient estimates, one typically relies on the uniform sampling-with-replacement scheme or various sampling-without-replacement (also known as shuffling) schemes. While the former is easier to analyze, the latter often have better empirical performance. In this paper, we propose a novel single-loop stochastic gradient descent-ascent (GDA) algorithm that employs both shuffling schemes and variance reduction to solve nonconvex-strongly concave smooth minimax problems. We show that the proposed algorithm achieves $ε$-stationarity in expectation in $\mathcal{O}(κ^2 ε^{-2})$ iterations, where $κ$ is the condition number of the problem. This outperforms existing shuffling schemes and matches the complexity of the best-known sampling-with-replacement algorithms. Our proposed algorithm also achieves the same complexity as that of its deterministic counterpart, the two-timescale GDA algorithm. Our numerical experiments demonstrate the superior performance of the proposed algorithm.

math.OC↗

arXiv2024

On subdifferential chain rule of matrix factorization and beyond

In this paper, we study equality-type Clarke subdifferential chain rules of matrix factorization and factorization machine. Specifically, we show for these problems that provided the latent dimension is larger than some multiple of the problem size (i.e., slightly overparameterized) and the loss function is locally Lipschitz, the subdifferential chain rules hold everywhere. In addition, we examine the tightness of the analysis through some interesting constructions and make some important observations from the perspective of optimization; e.g., we show that for all this type of problems, computing a stationary point is trivial. Some tensor generalizations and neural extensions are also discussed, albeit they remain mostly open.

math.OC↗

arXiv2024

$\ell_1$-norm rank-one symmetric matrix factorization has no spurious second-order stationary points

This paper studies the nonsmooth optimization landscape of the $\ell_1$-norm rank-one symmetric matrix factorization problem using tools from second-order variational analysis. Specifically, as the main finding of this paper, we show that any second-order stationary point (and thus local minimizer) of the problem is actually globally optimal. Besides, some other results concerning the landscape of the problem, such as a complete characterization of the set of stationary points, are also developed, which should be interesting in their own rights. Furthermore, with the above theories, we revisit existing results on the generic minimizing behavior of simple algorithms for nonsmooth optimization and showcase the potential risk of their applications to our problem through several examples. Our techniques can potentially be applied to analyze the optimization landscapes of a variety of other more sophisticated nonsmooth learning problems, such as robust low-rank matrix recovery.

math.OC↗

arXiv2024

Computing Competitive Equilibrium for Chores: Linear Convergence and Lightweight Iteration

Competitive equilibrium (CE) for chores has recently attracted significant attention, with many algorithms proposed to approximately compute it. However, existing algorithms either lack iterate convergence guarantees to an exact CE or require solving high-dimensional linear or quadratic programming subproblems. This paper overcomes these issues by proposing a novel unconstrained difference-of-convex formulation, whose stationary points correspond precisely to the CE for chores. We show that the new formulation possesses the local error bound property and the Kurdyka-Łojasiewicz property with an exponent of $1/2$. Consequently, we present the first algorithm whose iterates provably converge linearly to an exact CE for chores. Furthermore, by exploiting the max structure within our formulation and applying smoothing techniques, we develop a subproblem-free algorithm that finds an approximate CE in polynomial time. Numerical experiments demonstrate that the proposed algorithms outperform the state-of-the-art method.

math.OC↗

arXiv2024

On the Oracle Complexity of a Riemannian Inexact Augmented Lagrangian Method for Riemannian Nonsmooth Composite Problems

In this paper, we establish for the first time the oracle complexity of a Riemannian inexact augmented Lagrangian (RiAL) method with the classical dual update for solving a class of Riemannian nonsmooth composite problems. By using the Riemannian gradient descent method with a specified stopping criterion for solving the inner subproblem, we show that the RiAL method can find an $\varepsilon$-stationary point of the considered problem with $\mathcal{O}(\varepsilon^{-3})$ calls to the first-order oracle. This achieves the best oracle complexity known to date. Numerical results demonstrate that the use of the classical dual stepsize is crucial to the high efficiency of the RiAL method.

math.OC↗

arXiv2024

Nonconvex Federated Learning on Compact Smooth Submanifolds With Heterogeneous Data

Many machine learning tasks, such as principal component analysis and low-rank matrix completion, give rise to manifold optimization problems. Although there is a large body of work studying the design and analysis of algorithms for manifold optimization in the centralized setting, there are currently very few works addressing the federated setting. In this paper, we consider nonconvex federated learning over a compact smooth submanifold in the setting of heterogeneous client data. We propose an algorithm that leverages stochastic Riemannian gradients and a manifold projection operator to improve computational efficiency, uses local updates to improve communication efficiency, and avoids client drift. Theoretically, we show that our proposed algorithm converges sub-linearly to a neighborhood of a first-order optimal solution by using a novel analysis that jointly exploits the manifold structure and properties of the loss functions. Numerical experiments demonstrate that our algorithm has significantly smaller computational and communication overhead than existing methods.

cs.LG↗

arXiv2024

A Survey of Recent Advances in Optimization Methods for Wireless Communications

Mathematical optimization is now widely regarded as an indispensable modeling and solution tool for the design of wireless communications systems. While optimization has played a significant role in the revolutionary progress in wireless communication and networking technologies from 1G to 5G and onto the future 6G, the innovations in wireless technologies have also substantially transformed the nature of the underlying mathematical optimization problems upon which the system designs are based and have sparked significant innovations in the development of methodologies to understand, to analyze, and to solve those problems. In this paper, we provide a comprehensive survey of recent advances in mathematical optimization theory and algorithms for wireless communication system design. We begin by illustrating common features of mathematical optimization problems arising in wireless communication system design. We discuss various scenarios and use cases and their associated mathematical structures from an optimization perspective. We then provide an overview of recently developed optimization techniques in areas ranging from nonconvex optimization, global optimization, and integer programming, to distributed optimization and learning-based optimization. The key to successful solution of mathematical optimization problems is in carefully choosing or developing suitable algorithms (or neural network architectures) that can exploit the underlying problem structure. We conclude the paper by identifying several open research challenges and outlining future research directions.

cs.IT↗

arXiv2024

Lower-level Duality Based Reformulation and Majorization Minimization Algorithm for Hyperparameter Optimization

Hyperparameter tuning is an important task of machine learning, which can be formulated as a bilevel program (BLP). However, most existing algorithms are not applicable for BLP with non-smooth lower-level problems. To address this, we propose a single-level reformulation of the BLP based on lower-level duality without involving any implicit value function. To solve the reformulation, we propose a majorization minimization algorithm that marjorizes the constraint in each iteration. Furthermore, we show that the subproblems of the proposed algorithm for several widely used hyperparameter turning models can be reformulated into conic programs that can be efficiently solved by the off-the-shelf solvers. We theoretically prove the convergence of the proposed algorithm and demonstrate its superiority through numerical experiments.

math.OC↗

arXiv2023

An Efficient Alternating Riemannian/Projected Gradient Descent Ascent Algorithm for Fair Principal Component Analysis

Fair principal component analysis (FPCA), a ubiquitous dimensionality reduction technique in signal processing and machine learning, aims to find a low-dimensional representation for a high-dimensional dataset in view of fairness. The FPCA problem involves optimizing a non-convex and non-smooth function over the Stiefel manifold. The state-of-the-art methods for solving the problem are subgradient methods and semidefinite relaxation-based methods. However, these two types of methods have their obvious limitations and thus are only suitable for efficiently solving the FPCA problem in special scenarios. This paper aims at developing efficient algorithms for solving the FPCA problem in general, especially large-scale, settings. In this paper, we first transform FPCA into a smooth non-convex linear minimax optimization problem over the Stiefel manifold. To solve the above general problem, we propose an efficient alternating Riemannian/projected gradient descent ascent (ARPGDA) algorithm, which performs a Riemannian gradient descent step and an ordinary projected gradient ascent step at each iteration. We prove that ARPGDA can find an $\varepsilon$-stationary point of the above problem within $\mathcal{O}(\varepsilon^{-3})$ iterations. Simulation results show that, compared with the state-of-the-art methods, our proposed ARPGDA algorithm can achieve a better performance in terms of solution quality and speed for solving the FPCA problems.

math.OC↗