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Optimization over Stiefel manifolds has found wide applications in many scientific and engineering domains. Despite considerable research effort, high-dimensional optimization problems over Stiefel manifolds remain challenging, and the situation is exacerbated by nonsmooth objective functions. The purpose of this paper is to propose and study a novel coordinate-type algorithm for weakly convex (possibly nonsmooth) optimization problems over high-dimensional Stiefel manifolds, named randomized submanifold subgradient method (RSSM). Similar to coordinate-type algorithms in the Euclidean setting, RSSM exhibits low per-iteration cost and is suitable for high-dimensional problems. We prove that RSSM converges to the set of stationary points and attains $\varepsilon$-stationary points with respect to a natural stationarity measure in $\mathcal{O}(\varepsilon^{-4})$ iterations in both expectation and the almost-sure senses. To the best of our knowledge, these are the first convergence guarantees for coordinate-type algorithms to optimize nonconvex nonsmooth functions over Stiefel manifolds. An important technical tool in our convergence analysis is a new Riemannian subgradient inequality for weakly convex functions on proximally smooth matrix manifolds, which could be of independent interest.

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.

Despite the considerable success of Bregman proximal-type algorithms, such as mirror descent, in machine learning, a critical question remains: Can existing stationarity measures, often based on Bregman divergence, reliably distinguish between stationary and non-stationary points? In this paper, we present a groundbreaking finding: All existing stationarity measures necessarily imply the existence of spurious stationary points. We further establish an algorithmic independent hardness result: Bregman proximal-type algorithms are unable to escape from a spurious stationary point in finite steps when the initial point is unfavorable, even for convex problems. Our hardness result points out the inherent distinction between Euclidean and Bregman geometries, and introduces both fundamental theoretical and numerical challenges to both machine learning and optimization communities.

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.

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.

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., Comment: Accepted by AISTATS 2024

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., Comment: 39 pages, 5 figures, accepted for publication in IEEE Journal on Selected Areas in Communications

The heteroscedastic probabilistic principal component analysis (PCA) technique, a variant of the classic PCA that considers data heterogeneity, is receiving more and more attention in the data science and signal processing communities. In this paper, to estimate the underlying low-dimensional linear subspace (simply called \emph{ground truth}) from available heterogeneous data samples, we consider the associated non-convex maximum-likelihood estimation problem, which involves maximizing a sum of heterogeneous quadratic forms over an orthogonality constraint (HQPOC). We propose a first-order method -- generalized power method (GPM) -- to tackle the problem and establish its \emph{estimation performance} guarantee. Specifically, we show that, given a suitable initialization, the distances between the iterates generated by GPM and the ground truth decrease at least geometrically to some threshold associated with the residual part of certain "population-residual decomposition". In establishing the estimation performance result, we prove a novel local error bound property of another closely related optimization problem, namely quadratic optimization with orthogonality constraint (QPOC), which is new and can be of independent interest. Numerical experiments are conducted to demonstrate the superior performance of GPM in both Gaussian noise and sub-Gaussian noise settings., Comment: 22 pages

This paper investigates the problem of exact community recovery in the symmetric $d$-uniform $(d \geq 2)$ hypergraph stochastic block model ($d$-HSBM). In this model, a $d$-uniform hypergraph with $n$ nodes is generated by first partitioning the $n$ nodes into $K\geq 2$ equal-sized disjoint communities and then generating hyperedges with a probability that depends on the community memberships of $d$ nodes. Despite the non-convex and discrete nature of the maximum likelihood estimation problem, we develop a simple yet efficient iterative method, called the \emph{projected tensor power method}, to tackle it. As long as the initialization satisfies a partial recovery condition in the logarithmic degree regime of the problem, we show that our proposed method can exactly recover the hidden community structure down to the information-theoretic limit with high probability. Moreover, our proposed method exhibits a competitive time complexity of $\mathcal{O}(n\log^2n/\log\log n)$ when the aforementioned initialization condition is met. We also conduct numerical experiments to validate our theoretical findings.

Rotation group $\mathcal{SO}(d)$ synchronization is an important inverse problem and has attracted intense attention from numerous application fields such as graph realization, computer vision, and robotics. In this paper, we focus on the least-squares estimator of rotation group synchronization with general additive noise models, which is a nonconvex optimization problem with manifold constraints. Unlike the phase/orthogonal group synchronization, there are limited provable approaches for solving rotation group synchronization. First, we derive improved estimation results of the least-squares/spectral estimator, illustrating the tightness and validating the existing relaxation methods of solving rotation group synchronization through the optimum of relaxed orthogonal group version under near-optimal noise level for exact recovery. Moreover, departing from the standard approach of utilizing the geometry of the ambient Euclidean space, we adopt an intrinsic Riemannian approach to study orthogonal/rotation group synchronization. Benefiting from a quotient geometric view, we prove the positive definite condition of quotient Riemannian Hessian around the optimum of orthogonal group synchronization problem, and consequently the Riemannian local error bound property is established to analyze the convergence rate properties of various Riemannian algorithms. As a simple and feasible method, the sequential convergence guarantee of the (quotient) Riemannian gradient method for solving orthogonal/rotation group synchronization problem is studied, and we derive its global linear convergence rate to the optimum with the spectral initialization. All results are deterministic without any probabilistic model.

This work presents ReSync, a Riemannian subgradient-based algorithm for solving the robust rotation synchronization problem, which arises in various engineering applications. ReSync solves a least-unsquared minimization formulation over the rotation group, which is nonsmooth and nonconvex, and aims at recovering the underlying rotations directly. We provide strong theoretical guarantees for ReSync under the random corruption setting. Specifically, we first show that the initialization procedure of ReSync yields a proper initial point that lies in a local region around the ground-truth rotations. We next establish the weak sharpness property of the aforementioned formulation and then utilize this property to derive the local linear convergence of ReSync to the ground-truth rotations. By combining these guarantees, we conclude that ReSync converges linearly to the ground-truth rotations under appropriate conditions. Experiment results demonstrate the effectiveness of ReSync., Comment: Accepted for publication in NeurIPS 2023

The subgradient method is one of the most fundamental algorithmic schemes for nonsmooth optimization. The existing complexity and convergence results for this algorithm are mainly derived for Lipschitz continuous objective functions. In this work, we first extend the typical complexity results for the subgradient method to convex and weakly convex minimization without assuming Lipschitz continuity. Specifically, we establish $\mathcal{O}(1/\sqrt{T})$ bound in terms of the suboptimality gap ``$f(x) - f^*$'' for convex case and $\mathcal{O}(1/{T}^{1/4})$ bound in terms of the gradient of the Moreau envelope function for weakly convex case. Furthermore, we provide convergence results for non-Lipschitz convex and weakly convex objective functions using proper diminishing rules on the step sizes. In particular, when $f$ is convex, we show $\mathcal{O}(\log(k)/\sqrt{k})$ rate of convergence in terms of the suboptimality gap. With an additional quadratic growth condition, 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 derived. The central idea is that the dynamics of properly chosen step sizes rule fully controls the movement of the subgradient method, which leads to boundedness of the iterates, and then a trajectory-based analysis can be conducted to establish the desired results. To further illustrate the wide applicability of our framework, we extend the complexity results to the truncated subgradient, the stochastic subgradient, the incremental subgradient, and the proximal subgradient methods for non-Lipschitz functions.

Graph learning from signals is a core task in Graph Signal Processing (GSP). One of the most commonly used models to learn graphs from stationary signals is SpecT. However, its practical formulation rSpecT is known to be sensitive to hyperparameter selection and, even worse, to suffer from infeasibility. In this paper, we give the first condition that guarantees the infeasibility of rSpecT and design a novel model (LogSpecT) and its practical formulation (rLogSpecT) to overcome this issue. Contrary to rSpecT, the novel practical model rLogSpecT is always feasible. Furthermore, we provide recovery guarantees of rLogSpecT, which are derived from modern optimization tools related to epi-convergence. These tools could be of independent interest and significant for various learning problems. To demonstrate the advantages of rLogSpecT in practice, a highly efficient algorithm based on the linearized alternating direction method of multipliers (L-ADMM) is proposed. The subproblems of L-ADMM admit closed-form solutions and the convergence is guaranteed. Extensive numerical results on both synthetic and real networks corroborate the stability and superiority of our proposed methods, underscoring their potential for various graph learning applications.

The sequence-to-sequence (seq2seq) task aims at generating the target sequence based on the given input source sequence. Traditionally, most of the seq2seq task is resolved by the Encoder-Decoder framework which requires an encoder to encode the source sequence and a decoder to generate the target text. Recently, a bunch of new approaches have emerged that apply decoder-only language models directly to the seq2seq task. Despite the significant advancements in applying language models to the seq2seq task, there is still a lack of thorough analysis on the effectiveness of the decoder-only language model architecture. This paper aims to address this gap by conducting a detailed comparison between the encoder-decoder architecture and the decoder-only language model framework through the analysis of a regularized encoder-decoder structure. This structure is designed to replicate all behaviors in the classical decoder-only language model but has an encoder and a decoder making it easier to be compared with the classical encoder-decoder structure. Based on the analysis, we unveil the attention degeneration problem in the language model, namely, as the generation step number grows, less and less attention is focused on the source sequence. To give a quantitative understanding of this problem, we conduct a theoretical sensitivity analysis of the attention output with respect to the source input. Grounded on our analysis, we propose a novel partial attention language model to solve the attention degeneration problem. Experimental results on machine translation, summarization, and data-to-text generation tasks support our analysis and demonstrate the effectiveness of our proposed model.

We focus on a class of non-smooth optimization problems over the Stiefel manifold in the decentralized setting, where a connected network of $n$ agents cooperatively minimize a finite-sum objective function with each component being weakly convex in the ambient Euclidean space. Such optimization problems, albeit frequently encountered in applications, are quite challenging due to their non-smoothness and non-convexity. To tackle them, we propose an iterative method called the decentralized Riemannian subgradient method (DRSM). The global convergence and an iteration complexity of $\mathcal{O}(\varepsilon^{-2} \log^2(\varepsilon^{-1}))$ for forcing a natural stationarity measure below $\varepsilon$ are established via the powerful tool of proximal smoothness from variational analysis, which could be of independent interest. Besides, we show the local linear convergence of the DRSM using geometrically diminishing stepsizes when the problem at hand further possesses a sharpness property. Numerical experiments are conducted to corroborate our theoretical findings., Comment: 27 pages, 6 figures, 1 table

In this work, we present the Bregman Alternating Projected Gradient (BAPG) method, a single-loop algorithm that offers an approximate solution to the Gromov-Wasserstein (GW) distance. We introduce a novel relaxation technique that balances accuracy and computational efficiency, albeit with some compromises in the feasibility of the coupling map. Our analysis is based on the observation that the GW problem satisfies the Luo-Tseng error bound condition, which relates to estimating the distance of a point to the critical point set of the GW problem based on the optimality residual. This observation allows us to provide an approximation bound for the distance between the fixed-point set of BAPG and the critical point set of GW. Moreover, under a mild technical assumption, we can show that BAPG converges to its fixed point set. The effectiveness of BAPG has been validated through comprehensive numerical experiments in graph alignment and partition tasks, where it outperforms existing methods in terms of both solution quality and wall-clock time., Comment: Accepted by ICLR 2023

We study the computational problem of the stationarity test for the empirical loss of neural networks with ReLU activation functions. Our contributions are: Hardness: We show that checking a certain first-order approximate stationarity concept for a piecewise linear function is co-NP-hard. This implies that testing a certain stationarity concept for a modern nonsmooth neural network is in general computationally intractable. As a corollary, we prove that testing so-called first-order minimality for functions in abs-normal form is co-NP-complete, which was conjectured by Griewank and Walther (2019, SIAM J. Optim., vol. 29, p284). Regularity: We establish a necessary and sufficient condition for the validity of an equality-type subdifferential chain rule in terms of Clarke, Fr\'echet, and limiting subdifferentials of the empirical loss of two-layer ReLU networks. This new condition is simple and efficiently checkable. Robust algorithms: We introduce an algorithmic scheme to test near-approximate stationarity in terms of both Clarke and Fr\'echet subdifferentials. Our scheme makes no false positive or false negative error when the tested point is sufficiently close to a stationary one and a certain qualification is satisfied. This is the first practical and robust stationarity test approach for two-layer ReLU networks., Comment: 42 pages

Gromov-Wasserstein (GW) distance is a powerful tool for comparing and aligning probability distributions supported on different metric spaces. Recently, GW has become the main modeling technique for aligning heterogeneous data for a wide range of graph learning tasks. However, the GW distance is known to be highly sensitive to outliers, which can result in large inaccuracies if the outliers are given the same weight as other samples in the objective function. To mitigate this issue, we introduce a new and robust version of the GW distance called RGW. RGW features optimistically perturbed marginal constraints within a Kullback-Leibler divergence-based ambiguity set. To make the benefits of RGW more accessible in practice, we develop a computationally efficient and theoretically provable procedure using Bregman proximal alternating linearized minimization algorithm. Through extensive experimentation, we validate our theoretical results and demonstrate the effectiveness of RGW on real-world graph learning tasks, such as subgraph matching and partial shape correspondence.

Fine-tuning a pre-trained model (such as BERT, ALBERT, RoBERTa, T5, GPT, etc.) has proven to be one of the most promising paradigms in recent NLP research. However, numerous recent works indicate that fine-tuning suffers from the instability problem, i.e., tuning the same model under the same setting results in significantly different performance. Many recent works have proposed different methods to solve this problem, but there is no theoretical understanding of why and how these methods work. In this paper, we propose a novel theoretical stability analysis of fine-tuning that focuses on two commonly used settings, namely, full fine-tuning and head tuning. We define the stability under each setting and prove the corresponding stability bounds. The theoretical bounds explain why and how several existing methods can stabilize the fine-tuning procedure. In addition to being able to explain most of the observed empirical discoveries, our proposed theoretical analysis framework can also help in the design of effective and provable methods. Based on our theory, we propose three novel strategies to stabilize the fine-tuning procedure, namely, Maximal Margin Regularizer (MMR), Multi-Head Loss (MHLoss), and Self Unsupervised Re-Training (SURT). We extensively evaluate our proposed approaches on 11 widely used real-world benchmark datasets, as well as hundreds of synthetic classification datasets. The experiment results show that our proposed methods significantly stabilize the fine-tuning procedure and also corroborate our theoretical analysis.

Nonconvex-nonconcave minimax optimization has received intense attention over the last decade due to its broad applications in machine learning. Most existing algorithms rely on one-sided information, such as the convexity (resp. concavity) of the primal (resp. dual) functions, or other specific structures, such as the Polyak-\L{}ojasiewicz (P\L{}) and Kurdyka-\L{}ojasiewicz (K\L{}) conditions. However, verifying these regularity conditions is challenging in practice. To meet this challenge, we propose a novel universally applicable single-loop algorithm, the doubly smoothed gradient descent ascent method (DS-GDA), which naturally balances the primal and dual updates. That is, DS-GDA with the same hyperparameters is able to uniformly solve nonconvex-concave, convex-nonconcave, and nonconvex-nonconcave problems with one-sided K\L{} properties, achieving convergence with $\mathcal{O}(\epsilon^{-4})$ complexity. Sharper (even optimal) iteration complexity can be obtained when the K\L{} exponent is known. Specifically, under the one-sided K\L{} condition with exponent $\theta\in(0,1)$, DS-GDA converges with an iteration complexity of $\mathcal{O}(\epsilon^{-2\max\{2\theta,1\}})$. They all match the corresponding best results in the literature. Moreover, we show that DS-GDA is practically applicable to general nonconvex-nonconcave problems even without any regularity conditions, such as the P\L{} condition, K\L{} condition, or weak Minty variational inequalities condition. For various challenging nonconvex-nonconcave examples in the literature, including ``Forsaken'', ``Bilinearly-coupled minimax'', ``Sixth-order polynomial'', and ``PolarGame'', the proposed DS-GDA can all get rid of limit cycles. To the best of our knowledge, this is the first first-order algorithm to achieve convergence on all of these formidable problems.

Fine-tuning pre-trained models has been ubiquitously proven to be effective in a wide range of NLP tasks. However, fine-tuning the whole model is parameter inefficient as it always yields an entirely new model for each task. Currently, many research works propose to only fine-tune a small portion of the parameters while keeping most of the parameters shared across different tasks. These methods achieve surprisingly good performance and are shown to be more stable than their corresponding fully fine-tuned counterparts. However, such kind of methods is still not well understood. Some natural questions arise: How does the parameter sparsity lead to promising performance? Why is the model more stable than the fully fine-tuned models? How to choose the tunable parameters? In this paper, we first categorize the existing methods into random approaches, rule-based approaches, and projection-based approaches based on how they choose which parameters to tune. Then, we show that all of the methods are actually sparse fine-tuned models and conduct a novel theoretical analysis of them. We indicate that the sparsity is actually imposing a regularization on the original model by controlling the upper bound of the stability. Such stability leads to better generalization capability which has been empirically observed in a lot of recent research works. Despite the effectiveness of sparsity grounded by our theory, it still remains an open problem of how to choose the tunable parameters. To better choose the tunable parameters, we propose a novel Second-order Approximation Method (SAM) which approximates the original problem with an analytically solvable optimization function. The tunable parameters are determined by directly optimizing the approximation function. The experimental results show that our proposed SAM model outperforms many strong baseline models and it also verifies our theoretical 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., Comment: 5 pages, 8 figures, accepted for publication in IEEE ICASSP 2024

We consider the computation of an approximately stationary point for a Lipschitz and semialgebraic function $f$ with a local oracle. If $f$ is smooth, simple deterministic methods have dimension-free finite oracle complexities. For the general Lipschitz setting, only recently, Zhang et al. [47] introduced a randomized algorithm that computes Goldstein's approximate stationarity [25] to arbitrary precision with a dimension-free polynomial oracle complexity. In this paper, we show that no deterministic algorithm can do the same. Even without the dimension-free requirement, we show that any finite time guaranteed deterministic method cannot be general zero-respecting, which rules out most of the oracle-based methods in smooth optimization and any trivial derandomization of Zhang et al. [47]. Our results reveal a fundamental hurdle of nonconvex nonsmooth problems in the modern large-scale setting and their infinite-dimensional extension., Comment: 17 pages, submitted to SODA 2023

To do dimensionality reduction on the datasets with outliers, the $\ell_1$-norm principal component analysis (L1-PCA) as a typical robust alternative of the conventional PCA has enjoyed great popularity over the past years. In this work, we consider a rotationally invariant L1-PCA, which is hardly studied in the literature. To tackle it, we propose a proximal alternating linearized minimization method with a nonlinear extrapolation for solving its two-block reformulation. Moreover, we show that the proposed method converges at least linearly to a limiting critical point of the reformulated problem. Such a point is proved to be a critical point of the original problem under a condition imposed on the step size. Finally, we conduct numerical experiments on both synthetic and real datasets to support our theoretical developments and demonstrate the efficacy of our approach., Comment: 11 pages, 3 figures

In this paper, we consider a strongly convex finite-sum minimization problem over a decentralized network and propose a communication-efficient decentralized Newton's method for solving it. We first apply dynamic average consensus (DAC) so that each node is able to use a local gradient approximation and a local Hessian approximation to track the global gradient and Hessian, respectively. Second, since exchanging Hessian approximations is far from communication-efficient, we require the nodes to exchange the compressed ones instead and then apply an error compensation mechanism to correct for the compression noise. Third, we introduce multi-step consensus for exchanging local variables and local gradient approximations to balance between computation and communication. To avoid each node transmitting the entire local Hessian approximation, we design a compression procedure with error compensation to estimate the global Hessian in a communication-efficient way. With novel analysis, we establish the globally linear (resp., asymptotically super-linear) convergence rate of the proposed method when m is constant (resp., tends to infinity), where m is the number of consensus inner steps. To the best of our knowledge, this is the first super-linear convergence result for a communication-efficient decentralized Newton's method. Moreover, the rate we establish is provably faster than those of first-order methods. Our numerical results on various applications corroborate the theoretical findings.

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 $\theta \in [0,1)$. 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 $\epsilon$-game-stationary points and $\epsilon$-optimization-stationary points of the problems of interest in $\mathcal{O}(\epsilon^{-2\max\{2\theta,1\}})$ iterations. Furthermore, when $\theta \in [0,\frac{1}{2}]$, smoothed PLDA achieves the optimal iteration complexity of $\mathcal{O}(\epsilon^{-2})$. 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 $\theta=0$ under mild assumptions. As a by-product, we establish algorithm-independent quantitative relationships among various stationarity concepts, which may be of independent interest.

This paper studies large-scale optimization problems on Riemannian manifolds whose objective function is a finite sum of negative log-probability losses. Such problems arise in various machine learning and signal processing applications. By introducing the notion of Fisher information matrix in the manifold setting, we propose a novel Riemannian natural gradient method, which can be viewed as a natural extension of the natural gradient method from the Euclidean setting to the manifold setting. We establish the almost-sure global convergence of our proposed method under standard assumptions. Moreover, we show that if the loss function satisfies certain convexity and smoothness conditions and the input-output map satisfies a Riemannian Jacobian stability condition, then our proposed method enjoys a local linear -- or, under the Lipschitz continuity of the Riemannian Jacobian of the input-output map, even quadratic -- rate of convergence. We then prove that the Riemannian Jacobian stability condition will be satisfied by a two-layer fully connected neural network with batch normalization with high probability, provided that the width of the network is sufficiently large. This demonstrates the practical relevance of our convergence rate result. Numerical experiments on applications arising from machine learning demonstrate the advantages of the proposed method over state-of-the-art ones.

The K-subspaces (KSS) method is a generalization of the K-means method for subspace clustering. In this work, we present local convergence analysis and a recovery guarantee for KSS, assuming data are generated by the semi-random union of subspaces model, where $N$ points are randomly sampled from $K \ge 2$ overlapping subspaces. We show that if the initial assignment of the KSS method lies within a neighborhood of a true clustering, it converges at a superlinear rate and finds the correct clustering within $\Theta(\log\log N)$ iterations with high probability. Moreover, we propose a thresholding inner-product based spectral method for initialization and prove that it produces a point in this neighborhood. We also present numerical results of the studied method to support our theoretical developments., Comment: This paper is accepted by ICML 2022

In this paper, we study the design and analysis of a class of efficient algorithms for computing the Gromov-Wasserstein (GW) distance tailored to large-scale graph learning tasks. Armed with the Luo-Tseng error bound condition~\citep{luo1992error}, two proposed algorithms, called Bregman Alternating Projected Gradient (BAPG) and hybrid Bregman Proximal Gradient (hBPG) enjoy the convergence guarantees. Upon task-specific properties, our analysis further provides novel theoretical insights to guide how to select the best-fit method. As a result, we are able to provide comprehensive experiments to validate the effectiveness of our methods on a host of tasks, including graph alignment, graph partition, and shape matching. In terms of both wall-clock time and modeling performance, the proposed methods achieve state-of-the-art results.

Signed graphs encode similarity and dissimilarity relationships among different entities with positive and negative edges. In this paper, we study the problem of community recovery over signed graphs generated by the signed stochastic block model (SSBM) with two equal-sized communities. Our approach is based on the maximum likelihood estimation (MLE) of the SSBM. Unlike many existing approaches, our formulation reveals that the positive and negative edges of a signed graph should be treated unequally. We then propose a simple two-stage iterative algorithm for solving the regularized MLE. It is shown that in the logarithmic degree regime, the proposed algorithm can exactly recover the underlying communities in nearly-linear time at the information-theoretic limit. Numerical results on both synthetic and real data are reported to validate and complement our theoretical developments and demonstrate the efficacy of the proposed method.

In Part I of this work, we have proposed a general framework of decentralized stochastic quasi-Newton methods, which converge linearly to the optimal solution under the assumption that the local Hessian inverse approximations have bounded positive eigenvalues. In Part II, we specify two fully decentralized stochastic quasi-Newton methods, damped regularized limited-memory DFP (Davidon-Fletcher-Powell) and damped limited-memory BFGS (Broyden-Fletcher-Goldfarb-Shanno), to locally construct such Hessian inverse approximations without extra sampling or communication. Both of the methods use a fixed moving window of $M$ past local gradient approximations and local decision variables to adaptively construct positive definite Hessian inverse approximations with bounded eigenvalues, satisfying the assumption in Part I for the linear convergence. For the proposed damped regularized limited-memory DFP, a regularization term is added to improve the performance. For the proposed damped limited-memory BFGS, a two-loop recursion is applied, leading to low storage and computation complexity. Numerical experiments demonstrate that the proposed quasi-Newton methods are much faster than the existing decentralized stochastic first-order algorithms.

This paper proposes a Generalized Power Method (GPM) to tackle the problem of community detection and group synchronization simultaneously in a direct non-convex manner. Under the stochastic group block model (SGBM), theoretical analysis indicates that the algorithm is able to exactly recover the ground truth in $O(n\log^2n)$ time, sharply outperforming the benchmark method of semidefinite programming (SDP) in $O(n^{3.5})$ time. Moreover, a lower bound of parameters is given as a necessary condition for exact recovery of GPM. The new bound breaches the information-theoretic threshold for pure community detection under the stochastic block model (SBM), thus demonstrating the superiority of our simultaneous optimization algorithm over the trivial two-stage method which performs the two tasks in succession. We also conduct numerical experiments on GPM and SDP to evidence and complement our theoretical analysis., Comment: 29 pages

In this paper, we consider a time-varying optimization approach to the problem of tracking a moving target using noisy time-of-arrival (TOA) measurements. Specifically, we formulate the problem as that of sequential TOA-based source localization and apply online gradient descent (OGD) to it to generate the position estimates of the target. To analyze the tracking performance of OGD, we first revisit the classic least-squares formulation of the (static) TOA-based source localization problem and elucidate its estimation and geometric properties. In particular, under standard assumptions on the TOA measurement model, we establish a bound on the distance between an optimal solution to the least-squares formulation and the true target position. Using this bound, we show that the loss function in the formulation, albeit non-convex in general, is locally strongly convex at its global minima. To the best of our knowledge, these results are new and can be of independent interest. By combining them with existing techniques from online strongly convex optimization, we then establish the first non-trivial bound on the cumulative target tracking error of OGD. Our numerical results corroborate the theoretical findings and show that OGD can effectively track the target at different noise levels., Comment: Accepted for publication in IEEE Transactions on Signal Processing

We report a practical finite-time algorithmic scheme to compute approximately stationary points for nonconvex nonsmooth Lipschitz functions. In particular, we are interested in two kinds of approximate stationarity notions for nonconvex nonsmooth problems, i.e., Goldstein approximate stationarity (GAS) and near-approximate stationarity (NAS). For GAS, our scheme removes the unrealistic subgradient selection oracle assumption in (Zhang et al., 2020, Assumption 1) and computes GAS with the same finite-time complexity. For NAS, Davis & Drusvyatskiy (2019) showed that $\rho$-weakly convex functions admit finite-time computation, while Tian & So (2021) provided the matching impossibility results of dimension-free finite-time complexity for first-order methods. Complement to these developments, in this paper, we isolate a new class of functions that could be Clarke irregular (and thus not weakly convex anymore) and show that our new algorithmic scheme can compute NAS points for functions in that class within finite time. To demonstrate the wide applicability of our new theoretical framework, we show that $\rho$-margin SVM, $1$-layer, and $2$-layer ReLU neural networks, all being Clarke irregular, satisfy our new conditions., Comment: 20 pages, 3 figures, ICML 2022

Group synchronization refers to estimating a collection of group elements from the noisy pairwise measurements. Such a nonconvex problem has received much attention from numerous scientific fields including computer vision, robotics, and cryo-electron microscopy. In this paper, we focus on the orthogonal group synchronization problem with general additive noise models under incomplete measurements, which is much more general than the commonly considered setting of complete measurements. Characterizations of the orthogonal group synchronization problem are given from perspectives of optimality conditions as well as fixed points of the projected gradient ascent method which is also known as the generalized power method (GPM). It is well worth noting that these results still hold even without generative models. In the meantime, we derive the local error bound property for the orthogonal group synchronization problem which is useful for the convergence rate analysis of different algorithms and can be of independent interest. Finally, we prove the linear convergence result of the GPM to a global maximizer under a general additive noise model based on the established local error bound property. Our theoretical convergence result holds under several deterministic conditions which can cover certain cases with adversarial noise, and as an example we specialize it to the setting of the Erd\"os-R\'enyi measurement graph and Gaussian noise.

We show that stochastic acceleration can be achieved under the perturbed iterate framework (Mania et al., 2017) in asynchronous lock-free optimization, which leads to the optimal incremental gradient complexity for finite-sum objectives. We prove that our new accelerated method requires the same linear speed-up condition as the existing non-accelerated methods. Our core algorithmic discovery is a new accelerated SVRG variant with sparse updates. Empirical results are presented to verify our theoretical findings., Comment: 21 pages, 22 figures

A popular robust alternative of the classic principal component analysis (PCA) is the $\ell_1$-norm PCA (L1-PCA), which aims to find a subspace that captures the most variation in a dataset as measured by the $\ell_1$-norm. L1-PCA has shown great promise in alleviating the effect of outliers in data analytic applications. However, it gives rise to a challenging non-smooth non-convex optimization problem, for which existing algorithms are either not scalable or lack strong theoretical guarantees on their convergence behavior. In this paper, we propose a proximal alternating minimization method with extrapolation (PAMe) for solving a two-block reformulation of the L1-PCA problem. We then show that for both the L1-PCA problem and its two-block reformulation, the Kurdyka-\L ojasiewicz exponent at any of the limiting critical points is $1/2$. This allows us to establish the linear convergence of the sequence of iterates generated by PAMe and to determine the criticality of the limit of the sequence with respect to both the L1-PCA problem and its two-block reformulation. To complement our theoretical development, we show via numerical experiments on both synthetic and real-world datasets that PAMe is competitive with a host of existing methods. Our results not only significantly advance the convergence theory of iterative methods for L1-PCA but also demonstrate the potential of our proposed method in applications., Comment: A preliminary version of this work has appeared in the Proceedings of the 2019 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2019)

Simplex identification via split augmented Lagrangian (SISAL) is a popularly-used algorithm in blind unmixing of hyperspectral images. Developed by Jos\'{e} M. Bioucas-Dias in 2009, the algorithm is fundamentally relevant to tackling simplex-structured matrix factorization, and by extension, non-negative matrix factorization, which have many applications under their umbrellas. In this article, we revisit SISAL and provide new meanings to this quintessential algorithm. The formulation of SISAL was motivated from a geometric perspective, with no noise. We show that SISAL can be explained as an approximation scheme from a probabilistic simplex component analysis framework, which is statistical and is principally more powerful in accommodating the presence of noise. The algorithm for SISAL was designed based on a successive convex approximation method, with a focus on practical utility. It was not known, by analyses, whether the SISAL algorithm has any kind of guarantee of convergence to a stationary point. By establishing associations between the SISAL algorithm and a line-search-based proximal gradient method, we confirm that SISAL can indeed guarantee convergence to a stationary point. Our re-explanation of SISAL also reveals new formulations and algorithms. The performance of these new possibilities is demonstrated by numerical experiments.

In this paper, we study the problem of exact community recovery in the symmetric stochastic block model, where a graph of $n$ vertices is randomly generated by partitioning the vertices into $K \ge 2$ equal-sized communities and then connecting each pair of vertices with probability that depends on their community memberships. Although the maximum-likelihood formulation of this problem is discrete and non-convex, we propose to tackle it directly using projected power iterations with an initialization that satisfies a partial recovery condition. Such an initialization can be obtained by a host of existing methods. We show that in the logarithmic degree regime of the considered problem, the proposed method can exactly recover the underlying communities at the information-theoretic limit. Moreover, with a qualified initialization, it runs in $\mathcal{O}(n\log^2n/\log\log n)$ time, which is competitive with existing state-of-the-art methods. We also present numerical results of the proposed method to support and complement our theoretical development., Comment: This paper has been accepted to ICML 2021

In convex optimization, the problem of finding near-stationary points has not been adequately studied yet, unlike other optimality measures such as the function value. Even in the deterministic case, the optimal method (OGM-G, due to Kim and Fessler (2021)) has just been discovered recently. In this work, we conduct a systematic study of algorithmic techniques for finding near-stationary points of convex finite-sums. Our main contributions are several algorithmic discoveries: (1) we discover a memory-saving variant of OGM-G based on the performance estimation problem approach (Drori and Teboulle, 2014); (2) we design a new accelerated SVRG variant that can simultaneously achieve fast rates for minimizing both the gradient norm and function value; (3) we propose an adaptively regularized accelerated SVRG variant, which does not require the knowledge of some unknown initial constants and achieves near-optimal complexities. We put an emphasis on the simplicity and practicality of the new schemes, which could facilitate future work., Comment: 29 pages, 4 figures