arXiv subjects

Quoc Tran-Dinh

Publications and source records attributed to Quoc Tran-Dinh.

At least 19 recordsLinked to original sources

arXiv2026

Distributed Fast Fixed-Point Algorithms for Composite Monotone Inclusions over Networks

This paper aims to develop new and efficient distributed algorithms for solving a class of monotone inclusions, $0 \in \sum_{i=1}^n (G_ix + T_ix)$, over a connected network of $n$ agents, where the single-valued operator $G_i$ and the possibly multivalued operator $T_i$ remain private to agent $i$. Existing distributed algorithms for this problem class are primarily non-accelerated, and their exact convergence rates in the original primal space are largely unexplored. To bridge this gap, we propose two Decentralized Fast Fixed-Point-based algorithms, \texttt{ND-DFFP} and \texttt{NI-DFFP}, which integrate Nesterov-type acceleration with primal-dual techniques under two prominent settings: (i) \textit{Lipschitz continuity of $G_i$ and maximal monotonicity of $G_i+T_i$}; and (ii) \textit{co-coercivity of $G_i$ and maximal monotonicity of $T_i$}. While \texttt{ND-DFFP} utilizes a homogeneous network-dependent stepsize, \texttt{NI-DFFP} reformulates the problem into a three-operator inclusion to decouple the network topology, enabling heterogeneous network-independent stepsizes. Under appropriate assumptions, we establish an $\mathcal{O}(1/k)$ convergence rate for the consensus error and an $\mathcal{O}(1/k)$ rate for both the restricted gap function and the squared forward-backward splitting residual, with the latter two metrics evaluated at the network-average iterate or its projection onto the effective domain. Finally, numerical experiments on distributed bilinear matrix games and a virtual power plant problem demonstrate the competitive performance and computational efficiency of our methods over recent decentralized baselines in the literature.

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arXiv2026

Variance-Reduced Fast Krasnoselkii-Mann Methods for Finite-Sum Root-Finding Problems

We propose a new class of fast Krasnoselkii--Mann methods with variance reduction to solve a finite-sum co-coercive equation $Gx = 0$. Our algorithm is single-loop and leverages a new family of unbiased variance-reduced estimators specifically designed for a wider class of root-finding algorithms. Our method achieves both $\mathcal{O}(1/k^2)$ and $o(1/k^2)$ last-iterate convergence rates in terms of $\mathbb{E}[\| Gx^k\|^2]$, where $k$ is the iteration counter and $\mathbb{E}[\cdot]$ is the total expectation. We also establish almost sure $o(1/k^2)$ convergence rates and the almost sure convergence of iterates $\{x^k\}$ to a solution of $Gx=0$. We instantiate our framework for two prominent estimators: SVRG and SAGA. By an appropriate choice of parameters, both variants attain an oracle complexity of $\mathcal{O}(n + n^{2/3}ε^{-1})$ to reach an $ε$-solution, where $n$ represents the number of summands in the finite-sum operator $G$. Furthermore, under $σ$-strong quasi-monotonicity, our method achieves a linear convergence rate and an oracle complexity of $\mathcal{O}(n+ \max\{n, n^{2/3}κ\} \log(\frac{1}ε))$, where $κ:= L/σ$. We extend our approach to solve a class of finite-sum inclusions (possibly nonmonotone), demonstrating that our schemes retain the same theoretical guarantees as in the equation setting. Finally, numerical experiments validate our algorithms and demonstrate their promising performance compared to state-of-the-art methods.

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arXiv2026

New Accelerated Past-Extragradient Methods with Variance Reduction for Generalized Equations

We develop a novel past-extragradient-type algorithmic framework, combining both Nesterov's \textit{acceleration} and \textit{variance-reduction} techniques, to solve a class of generalized equations involving possibly \textit{nonmonotone operators} in data-driven applications. Our framework covers a wide class of stochastic variance-reduced schemes, including mini-batching and both unbiased and biased control-variate estimators. We establish that our method achieves $\mathcal{O}(1/k^2)$ convergence rates in expectation for the squared norm of the residual under Lipschitz continuity and a ``co-hypomonotonicity-type'' assumption, significantly improving upon non-accelerated counterparts by a factor of $1/k$. We also prove faster $o(1/k^2)$ convergence rates, both in expectation and almost surely. In addition, we show that the sequence of iterates generated by our method almost surely converges to a solution of the underlying problem. We demonstrate the applicability of our method using general error approximation criteria, covering mini-batch stochastic estimators as well as three well-known control variate estimators: Loopless SVRG, SAGA, and Loopless SARAH. The resulting three variants attain significantly better oracle complexities than existing methods. We validate our framework and theoretical results through three numerical examples. The numerical results illustrate promising performance of our accelerated method over its non-accelerated counterparts.

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arXiv2026

Unbiased and Biased Variance-Reduced Forward-Reflected-Backward Splitting Methods for Stochastic Composite Inclusions

This paper develops new variance-reduction techniques for the forward-reflected-backward splitting (FRBS) method to solve a class of possibly nonmonotone stochastic composite inclusions. Unlike unbiased estimators such as mini-batching, developing stochastic biased variants faces a fundamental technical challenge and has not been utilized before for inclusions and fixed-point problems. We fill this gap by designing a new framework that can handle both unbiased and biased estimators. Our main idea is to construct stochastic variance-reduced estimators for the forward-reflected direction and use them to perform iterate updates. First, we propose a class of unbiased variance-reduced estimators and show that increasing mini-batch SGD, loopless-SVRG, and SAGA estimators fall within this class. For these unbiased estimators, we establish a $\mathcal{O}(1/k)$ best-iterate convergence rate for the expected squared residual norm, together with almost-sure convergence of the iterate sequence to a solution. Consequently, we prove that the best oracle complexities for the $n$-finite-sum and expectation settings are $\mathcal{O}(n^{2/3}ε^{-2})$ and $\mathcal{O}(ε^{-10/3})$, respectively, when employing loopless-SVRG or SAGA, where $ε$ is a desired accuracy. Second, we introduce a new class of biased variance-reduced estimators for the forward-reflected direction, which includes SARAH, Hybrid SGD, and Hybrid SVRG as special instances. While the convergence rates remain valid for these biased estimators, the resulting oracle complexities are $\mathcal{O}(n^{3/4}ε^{-2})$ and $\mathcal{O}(ε^{-5})$ for the $n$-finite-sum and expectation settings, respectively. Finally, we conduct two numerical experiments on AUC optimization for imbalanced classification and policy evaluation in reinforcement learning.

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arXiv2026

Provable Parameter-Free Fixed-Point Algorithms with Linear Convergence Rates

In this paper, we develop provable parameter-free and adaptive fixed-point algorithms for contractive mappings, with an emphasis on automatically exploiting hidden contractivity without requiring prior knowledge of the contraction factor. Our first method is a completely parameter-free variant of the Halpern fixed-point iteration. It requires no line search, bisection, or prior estimate of the contraction factor, while retaining essentially the same per-iteration computational cost as classical fixed-point schemes. We establish explicit linear convergence rates for both the fixed-point residual and the distance to the unique fixed point. The second algorithm is an adaptive Halpern method that requires only an upper bound on the contraction factor and reduces to an existing adaptive Halpern scheme in the nonexpansive case. This method also enjoys explicit linear convergence guarantees. We further extend these ideas in two directions. First, by combining the proposed fixed-point schemes with Tikhonov regularization, we obtain a parameter-free method for solving co-coercive equations and establish an iteration complexity of $\mathcal{O}({ε^{-1}\ln(ε^{-1})})$ for computing an $ε$-solution. Second, using the relation between Halpern iterations and Nesterov's accelerated fixed-point schemes, we derive parameter-free Nesterov's accelerated variants that inherit linear convergence in the contractive setting. Numerical experiments on several examples demonstrate that the proposed algorithms are competitive with, and often outperform, existing adaptive fixed-point methods. In particular, the methods successfully exploit contractive behavior when it is present while remaining effective on nonexpansive problems.

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arXiv2026

From Halpern's Fixed-Point Iterations to Nesterov's Accelerated Interpretations for Root-Finding Problems

We derive an equivalent form of Halpern's fixed-point iteration scheme for solving a co-coercive equation (also called a root-finding problem), which can be viewed as a Nesterov's accelerated interpretation. We show that one method is equivalent to another via a simple transformation, leading to a straightforward convergence proof for Nesterov's accelerated scheme. Alternatively, we directly establish convergence rates of Nesterov's accelerated variant, and as a consequence, we obtain a new convergence rate of Halpern's fixed-point iteration. Next, we apply our results to different methods to solve monotone inclusions, where our convergence guarantees are applied. Since the gradient/forward scheme requires the co-coerciveness of the underlying operator, we derive new Nesterov's accelerated variants for both recent extra-anchored gradient and past-extra anchored gradient methods in the literature. These variants alleviate the co-coerciveness condition by only assuming the monotonicity and Lipschitz continuity of the underlying operator. Interestingly, our new Nesterov's accelerated interpretation of the past-extra anchored gradient method involves two past-iterate correction terms. This formulation is expected to guide us developing new Nesterov's accelerated methods for minimax problems and their continuous views without co-coericiveness. We test our theoretical results on two numerical examples, where the actual convergence rates match well the theoretical ones up to a constant factor.

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arXiv2026

Lagrange Multipliers and Duality with Applications to Constrained Support Vector Machine

In this paper, we employ the concept of quasi-relative interior to analyze the method of Lagrange multipliers and establish strong Lagrangian duality for nonsmooth convex optimization problems in Hilbert spaces. Then, we generalize the classical support vector machine (SVM) model by incorporating a new geometric constraint or a regularizer on the separating hyperplane, serving as a regularization mechanism for the SVM model. This new SVM model is examined using Lagrangian duality and other convex optimization techniques in both theoretical and numerical aspects via a new subgradient algorithm as well as a primal-dual method.

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arXiv2025

A Class of Accelerated Fixed-Point-Based Methods with Delayed Inexact Oracles and Its Applications

In this paper, we develop a novel accelerated fixed-point-based framework using delayed inexact oracles to approximate a fixed point of a nonexpansive operator (or equivalently, a root of a co-coercive operator), a central problem in scientific computing. Our approach leverages both Nesterov's acceleration technique and the Krasnosel'skii-Mann (KM) iteration, while accounting for delayed inexact oracles, a key mechanism in asynchronous algorithms. We also introduce a unified approximate error condition for delayed inexact oracles, which can cover various practical scenarios. Under mild conditions and appropriate parameter updates, we establish both $\mathcal{O}(1/k^2)$ non-asymptotic and $o(1/k^2)$ asymptotic convergence rates in expectation for the squared norm of residual. Our rate significantly improves the $\mathcal{O}(1/k)$ rates in classical KM-type methods, including their asynchronous variants. We also establish $o(1/k^2)$ almost sure convergence rates and the almost sure convergence of iterates to a solution of the problem. Within our framework, we instantiate three settings for the underlying operator: (i) a deterministic universal delayed oracle; (ii) a stochastic delayed oracle; and (iii) a finite-sum structure with asynchronous updates. For each case, we instantiate our framework to obtain a concrete algorithmic variant for which our convergence results still apply, and whose iteration complexity depends linearly on the maximum delay. Finally, we verify our algorithms and theoretical results through two numerical examples on both matrix game and shallow neural network training problems.

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arXiv2025

Variance-Reduced Fast Operator Splitting Methods for Generalized Equations

We develop two variance-reduced fast operator splitting methods to approximate solutions of a class of generalized equations, covering fundamental problems such as \rvs{minimization}, minimax problems, and variational inequalities as special cases. Our approach integrates recent advances in accelerated operator splitting and fixed-point methods, co-hypomonotonicity, and variance reduction. First, we introduce a class of variance-reduced estimators and establish their variance-reduction bounds. This class includes both unbiased and biased instances and comprises common estimators as special cases, including SVRG, SAGA, SARAH, and Hybrid-SGD. Second, we design a novel accelerated variance-reduced forward-backward splitting (FBS) method using these estimators to solve generalized equations in both finite-sum and expectation settings. Our algorithm achieves both $\mathcal{O}(1/k^2)$ and $o(1/k^2)$ convergence rates on the expected squared norm $\mathbb{E}[ \| G_λx^k\|^2]$ of the FBS residual $G_λ$, where $k$ is the iteration counter. Additionally, we establish almost sure convergence rates and the almost sure convergence of iterates to a solution of the underlying generalized equation. Unlike existing stochastic operator splitting algorithms, our methods accommodate co-hypomonotone operators, which can include nonmonotone problems arising in recent applications. Third, we specify our method for each concrete estimator mentioned above and derive the corresponding oracle complexity, demonstrating that these variants achieve the best-known oracle complexity bounds without requiring additional enhancement techniques. Fourth, we develop a variance-reduced fast backward-forward splitting (BFS) method, which attains similar convergence results and oracle complexity bounds as our FBS-based algorithm.

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arXiv2025

Randomized Block-Coordinate Optimistic Gradient Algorithms for Root-Finding Problems

In this paper, we develop two new randomized block-coordinate optimistic gradient algorithms to approximate a solution of nonlinear equations in large-scale settings, which are called root-finding problems. Our first algorithm is non-accelerated with constant stepsizes, and achieves $\mathcal{O}(1/k)$ best-iterate convergence rate on $\mathbb{E}[ \Vert Gx^k\Vert^2]$ when the underlying operator $G$ is Lipschitz continuous and satisfies a weak Minty solution condition, where $\mathbb{E}[\cdot]$ is the expectation and $k$ is the iteration counter. Our second method is a new accelerated randomized block-coordinate optimistic gradient algorithm. We establish both $\mathcal{O}(1/k^2)$ and $o(1/k^2)$ last-iterate convergence rates on both $\mathbb{E}[ \Vert Gx^k\Vert^2]$ and $\mathbb{E}[ \Vert x^{k+1} - x^{k}\Vert^2]$ for this algorithm under the co-coerciveness of $G$. In addition, we prove that the iterate sequence $\{x^k\}$ converges to a solution almost surely, and $k\Vert Gx^k\Vert$ attains a $o(1/k)$ almost sure convergence rate. Then, we apply our methods to a class of large-scale finite-sum inclusions, which covers prominent applications in machine learning, statistical learning, and network optimization, especially in federated learning. We obtain two new federated learning-type algorithms and their convergence rate guarantees for solving this problem class.

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arXiv2025

Stochastic Variance-Reduced Forward-Reflected-Backward Splitting Methods for Nonmonotone Generalized Equations

We develop two novel stochastic variance-reduction methods to approximate solutions of a class of nonmonotone [generalized] equations. Our algorithms leverage a new combination of ideas from the forward-reflected-backward splitting method and a class of unbiased variance-reduced estimators. We construct two new stochastic estimators within this class, inspired by the well-known SVRG and SAGA estimators. These estimators significantly differ from existing approaches used in minimax and variational inequality problems. By appropriately choosing parameters, both algorithms achieve a state-of-the-art oracle complexity of $\mathcal{O}(n + n^{2/3}ε^{-2})$ for obtaining an $ε$-solution in terms of the operator residual norm for a class of nonmonotone problems, where $n$ is the number of summands and $ε$ signifies the desired accuracy. This complexity aligns with the best-known results in SVRG and SAGA methods for stochastic nonconvex optimization. We test our algorithms on some numerical examples and compare them with existing methods. The results demonstrate promising improvements offered by the new methods compared to their competitors.

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arXiv2025

Accelerated Extragradient-Type Methods -- Part 2: Generalization and Sublinear Convergence Rates under Co-Hypomonotonicity

Following the first part of our project, this paper comprehensively studies two types of extragradient-based methods: anchored extragradient and Nesterov's accelerated extragradient for solving [non]linear inclusions (and, in particular, equations), primarily under the Lipschitz continuity and the co-hypomonotonicity assumptions. We unify and generalize a class of anchored extragradient methods for monotone inclusions to a wider range of schemes encompassing existing algorithms as special cases. We establish $\mathcal{O}(1/k)$ last-iterate convergence rates on the residual norm of the underlying mapping for this general framework and then specialize it to obtain convergence guarantees for specific instances, where $k$ denotes the iteration counter. We extend our approach to a class of anchored Tseng's forward-backward-forward splitting methods to obtain a broader class of algorithms for solving co-hypomonotone inclusions. Again, we analyze $\mathcal{O}(1/k)$ last-iterate convergence rates for this general scheme and specialize it to obtain convergence results for existing and new variants. We generalize and unify Nesterov's accelerated extra-gradient method to a new class of algorithms that covers existing schemes as special instances while generating new variants. For these schemes, we can prove $\mathcal{O}(1/k)$ last-iterate convergence rates for the residual norm under co-hypomonotonicity, covering a class of nonmonotone problems. We propose another novel class of Nesterov's accelerated extragradient methods to solve inclusions. Interestingly, these algorithms achieve both $\mathcal{O}(1/k)$ and $o(1/k)$ last-iterate convergence rates, and also the convergence of iterate sequences under co-hypomonotonicity and Lipschitz continuity. Finally, we provide a set of numerical experiments encompassing different scenarios to validate our algorithms and theoretical guarantees.

math.OC↗

arXiv2025

Complexity of Linearized Perturbed Augmented Lagrangian Methods for Nonsmooth Nonconvex Optimization with Nonlinear Equality Constraints

This paper addresses a class of general nonsmooth and nonconvex composite optimization problems subject to nonlinear equality constraints. We assume that a part of the objective function and the functional constraints exhibit local smoothness. To tackle this challenging class of problems, we propose a novel linearized perturbed augmented Lagrangian method. This method incorporates a perturbation in the augmented Lagrangian function by scaling the dual variable with a sub-unitary parameter. Furthermore, we linearize the smooth components of the objective and the constraints within the perturbed Lagrangian function at the current iterate, while preserving the nonsmooth components. This approach, inspired by prox-linear (or Gauss-Newton) methods, results in a convex subproblem that is typically easy to solve. The solution of this subproblem then serves as the next primal iterate, followed by a perturbed ascent step to update the dual variables. Under a newly introduced constraint qualification condition, we establish the boundedness of the dual iterates. We derive convergence guarantees for the primal iterates, proving convergence to an $ε$-first-order optimal solution within $\mathcal{O}(ε^{-3})$ evaluations of the problem's functions and their first derivatives. Moreover, when the problem exhibits for example a semialgebraic property, we derive improved local convergence results. Finally, we validate the theoretical findings and assess the practical performance of our proposed algorithm through numerical comparisons with existing state-of-the-art methods.

math.OC↗

arXiv2024

Shuffling Gradient-Based Methods for Nonconvex-Concave Minimax Optimization

This paper aims at developing novel shuffling gradient-based methods for tackling two classes of minimax problems: nonconvex-linear and nonconvex-strongly concave settings. The first algorithm addresses the nonconvex-linear minimax model and achieves the state-of-the-art oracle complexity typically observed in nonconvex optimization. It also employs a new shuffling estimator for the "hyper-gradient", departing from standard shuffling techniques in optimization. The second method consists of two variants: semi-shuffling and full-shuffling schemes. These variants tackle the nonconvex-strongly concave minimax setting. We establish their oracle complexity bounds under standard assumptions, which, to our best knowledge, are the best-known for this specific setting. Numerical examples demonstrate the performance of our algorithms and compare them with two other methods. Our results show that the new methods achieve comparable performance with SGD, supporting the potential of incorporating shuffling strategies into minimax algorithms.

math.OC↗

arXiv2024

Revisiting Extragradient-Type Methods -- Part 1: Generalizations and Sublinear Convergence Rates

This paper presents a comprehensive analysis of the well-known extragradient (EG) method for solving both equations and inclusions. First, we unify and generalize EG for [non]linear equations to a wider class of algorithms, encompassing various existing schemes and potentially new variants. Next, we analyze both sublinear ``best-iterate'' and ``last-iterate'' convergence rates for the entire class of algorithms, and derive new convergence results for two well-known instances. Second, we extend our EG framework above to ``monotone'' inclusions, introducing a new class of algorithms and its corresponding convergence results. Third, we also unify and generalize Tseng's forward-backward-forward splitting (FBFS) method to a broader class of algorithms to solve [non]linear inclusions when a weak-Minty solution exists, and establish its ``best-iterate'' convergence rate. Fourth, to complete our picture, we also investigate sublinear rates of two other common variants of EG using our EG analysis framework developed here: the reflected forward-backward splitting and the golden ratio methods. Finally, we conduct an extensive numerical experiment to validate our theoretical findings. Our results demonstrate that several new variants of our proposed algorithms outperform existing schemes in the majority of examples.

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arXiv2024

Shuffling Momentum Gradient Algorithm for Convex Optimization

The Stochastic Gradient Descent method (SGD) and its stochastic variants have become methods of choice for solving finite-sum optimization problems arising from machine learning and data science thanks to their ability to handle large-scale applications and big datasets. In the last decades, researchers have made substantial effort to study the theoretical performance of SGD and its shuffling variants. However, only limited work has investigated its shuffling momentum variants, including shuffling heavy-ball momentum schemes for non-convex problems and Nesterov's momentum for convex settings. In this work, we extend the analysis of the shuffling momentum gradient method developed in [Tran et al (2021)] to both finite-sum convex and strongly convex optimization problems. We provide the first analysis of shuffling momentum-based methods for the strongly convex setting, attaining a convergence rate of $O(1/nT^2)$, where $n$ is the number of samples and $T$ is the number of training epochs. Our analysis is a state-of-the-art, matching the best rates of existing shuffling stochastic gradient algorithms in the literature.

math.OC↗

arXiv2024

Data Integration Via Analysis of Subspaces (DIVAS)

Modern data collection in many data paradigms, including bioinformatics, often incorporates multiple traits derived from different data types (i.e. platforms). We call this data multi-block, multi-view, or multi-omics data. The emergent field of data integration develops and applies new methods for studying multi-block data and identifying how different data types relate and differ. One major frontier in contemporary data integration research is methodology that can identify partially-shared structure between sub-collections of data types. This work presents a new approach: Data Integration Via Analysis of Subspaces (DIVAS). DIVAS combines new insights in angular subspace perturbation theory with recent developments in matrix signal processing and convex-concave optimization into one algorithm for exploring partially-shared structure. Based on principal angles between subspaces, DIVAS provides built-in inference on the results of the analysis, and is effective even in high-dimension-low-sample-size (HDLSS) situations.

stat.ME↗

arXiv2023

Extragradient-Type Methods with $\mathcal{O} (1/k)$ Last-Iterate Convergence Rates for Co-Hypomonotone Inclusions

We develop two "Nesterov's accelerated" variants of the well-known extragradient method to approximate a solution of a co-hypomonotone inclusion constituted by the sum of two operators, where one is Lipschitz continuous and the other is possibly multivalued. The first scheme can be viewed as an accelerated variant of Tseng's forward-backward-forward splitting (FBFS) method, while the second one is a Nesterov's accelerated variant of the "past" FBFS scheme, which requires only one evaluation of the Lipschitz operator and one resolvent of the multivalued mapping. Under appropriate conditions on the parameters, we theoretically prove that both algorithms achieve $\mathcal{O}(1/k)$ last-iterate convergence rates on the residual norm, where $k$ is the iteration counter. Our results can be viewed as alternatives of a recent class of Halpern-type methods for root-finding problems. For comparison, we also provide a new convergence analysis of the two recent extra-anchored gradient-type methods for solving co-hypomonotone inclusions.

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