arXiv subjects

Dang-Khoa Nguyen

Publications and source records attributed to Dang-Khoa Nguyen.

At least 19 recordsLinked to original sources

arXiv2026

The Iterates of Nesterov's Accelerated Algorithm Converge in The Critical Regimes

In this paper, we prove that the iterates of the accelerated Nesterov's algorithm in the critical regime do converge in the weak topology to a global minimizer of an $L$-smooth function in a real Hilbert space, hence answering positively a conjecture posed by H. Attouch and co-authors a decade ago. This result is the algorithmic case of a very recent result on the continuous-time system posted by E. Ryu on X, with assistance from ChatGPT.

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arXiv2026

Taint-Based Code Slicing for LLMs-based Malicious NPM Package Detection

Software supply chain attacks on the npm ecosystem have grown increasingly sophisticated, exploiting obfuscation and complex logic to evade detection. Large Language Models (LLMs) offer strong semantic understanding of code but face practical constraints: limited context windows and high inference costs make full-package analysis infeasible, while naive token-based splitting fragments semantic context and degrades accuracy. This paper introduces an LLM-based framework for malicious npm package detection built on code-slicing techniques. We propose an adaptation of taint-based slicing for the npm ecosystem, guided by a curated inventory of JavaScript-specific sensitive APIs, to isolate security-relevant data flows from benign boilerplate. The approach reduces the mean input token count by 99.75% and the median by 93.7% while preserving critical malicious behaviors. Packages relying on dynamic code generation or obfuscation yield empty slices under static analysis and require deobfuscation preprocessing, a limitation we explicitly discuss. The framework is evaluated on a dataset of more than 7000 malicious and benign npm packages using DeepSeek-Coder6.7B. On the 2537 packages amenable to static taint analysis, taint-based slicing achieves 87.04% detection accuracy, outperforming both a naive token-splitting baseline at 75.41% and a CFG-only static slicing approach at 75.65%. These results demonstrate that semantically targeted input representations improve LLM-based detection performance beyond what is achievable through simple input-size reduction, providing an effective and computationally practical defense against evolving open-source supply-chain threats.

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arXiv2025

Fast Reflected Forward-Backward algorithm: achieving fast convergence rates for convex optimization with linear cone constraints

In this paper, we derive a Fast Reflected Forward-Backward (Fast RFB) algorithm to solve the problem of finding a zero of the sum of a maximally monotone operator and a monotone and Lipschitz continuous operator in a real Hilbert space. Our approach extends the class of reflected forward-backward methods by introducing a Nesterov momentum term and a correction term, resulting in enhanced convergence performance. The iterative sequence of the proposed algorithm is proven to converge weakly, and the Fast RFB algorithm demonstrates impressive convergence rates, achieving $o\left( \frac{1}{k} \right)$ as $k \to +\infty$ for both the discrete velocity and the tangent residual at the \emph{last-iterate}. When applied to minimax problems with a smooth coupling term and nonsmooth convex regularizers, the resulting algorithm demonstrates significantly improved convergence properties compared to the current state of the art in the literature. For convex optimization problems with linear cone constraints, our approach yields a fully splitting primal-dual algorithm that ensures not only the convergence of iterates to a primal-dual solution, but also a \emph{last-iterate} convergence rate of $o\left( \frac{1}{k} \right)$ as $k \to +\infty$ for the objective function value, feasibility measure, and complementarity condition. This represents the most competitive theoretical result currently known for algorithms addressing this class of optimization problems. Numerical experiments are performed to illustrate the convergence behavior of Fast RFB.

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arXiv2025

Tikhonov regularization of monotone operator flows not only ensures strong convergence of the trajectories but also speeds up the vanishing of the residuals

In the framework of real Hilbert spaces, we investigate first-order dynamical systems governed by monotone and continuous operators. We demonstrate that when the monotone operator flow is augmented with a Tikhonov regularization term, the resulting trajectory converges strongly to the element of the set of zeros with minimal norm. In addition, rates of convergence in norm for the trajectory's velocity and the operator along the trajectory can be derived in terms of the regularization function. In some particular cases, these rates of convergence can outperform the ones of the coercive operator flows and can be as fast as $O(\frac{1}{t})$ as $t \rightarrow +\infty$. In this way, we emphasize a surprising acceleration feature of the Tikhonov regularization. Additionally, we explore these properties for monotone operator flows that incorporate time rescaling and an anchor point and show that they are closely linked to second-order dynamics with a vanishing damping term. The convergence and convergence rate results we achieve for these systems complement recent findings for the Fast Optimistic Gradient Descent Ascent (OGDA) dynamics. When the monotone operator is defined as the identity minus a nonexpansive operator, the monotone equations transform into a fixed point problem. In such cases, explicitly discretizing the system with Tikhonov regularization, enhanced by an anchor point, leads to the Halpern fixed point iteration. We identify two regimes for the regularization sequence which ensure that the generated sequence of iterates converges strongly to the fixed point nearest to the anchor point. Furthermore, we establish a general theoretical framework that provides convergence rates for the vanishing of the discrete velocity and the fixed point residual. For certain regularization sequences, we derive specific convergence rates that align with those observed in continuous time.

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arXiv2025

Recovering Nesterov accelerated dynamics from Heavy Ball dynamics via time rescaling

In a real Hilbert space, we consider two classical problems: the global minimization of a smooth and convex function $f$ (i.e., a convex optimization problem) and finding the zeros of a monotone and continuous operator $V$ (i.e., a monotone equation). Attached to the optimization problem, first we study the asymptotic properties of the trajectories generated by a second-order dynamical system which features a constant viscous friction coefficient and a positive, monotonically increasing function $b(\cdot)$ multiplying $\nabla f$. For a generated solution trajectory $y(t)$, we show small $o$ convergence rates dependent on $b(t)$ for $f(y(t)) - \min f$, and the weak convergence of $y(t)$ towards a global minimizer of $f$. In 2015, Su, Boyd and Candés introduced a second-order system which could be seen as the continuous-time counterpart of Nesterov's accelerated gradient. As the first key point of this paper, we show that for a special choice for $b(t)$, these two seemingly unrelated dynamical systems are connected: namely, they are time reparametrizations of each other. Every statement regarding the continuous-time accelerated gradient system may be recovered from its Heavy Ball counterpart. As the second key point of this paper, we observe that this connection extends beyond the optimization setting. Attached to the monotone equation involving the operator $V$, we again consider a Heavy Ball-like system which features an additional correction term which is the time derivative of the operator along the trajectory. We establish a time reparametrization equivalence with the Fast OGDA dynamics introduced by Bot, Csetnek and Nguyen in 2022, which can be seen as an analog of the continuous accelerated gradient dynamics, but for monotone operators. Again, every statement regarding the Fast OGDA system may be recovered from a Heavy Ball-like system.

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arXiv2025

Fast convex optimization via closed-loop time scaling of gradient dynamics

In a Hilbert setting, for convex differentiable optimization, we develop a general framework for adaptive accelerated gradient methods. They are based on damped inertial dynamics where the coefficients are designed in a closed-loop way. Specifically, the damping is a feedback control of the velocity, or of the gradient of the objective function. For this, we develop a closed-loop version of the time scaling and averaging technique introduced by the authors. We thus obtain autonomous inertial dynamics which involve vanishing viscous damping and implicit Hessian driven damping. By simply using the convergence rates for the continuous steepest descent and Jensen's inequality, without the need for further Lyapunov analysis, we show that the trajectories have several remarkable properties at once: they ensure fast convergence of values, fast convergence of the gradients towards zero, and they converge to optimal solutions. Our approach leads to parallel algorithmic results, that we study in the case of proximal algorithms. These are among the very first general results of this type obtained using autonomous dynamics. Since the proposed numerical methods are based on proximal techniques, the results can be extended to a broader class, specifically to the problem of minimizing a proper, lower semicontinuous, and convex function. Numerical experiments are conducted to demonstrate the efficiency of the proposed methods.

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arXiv2024

Fast second-order dynamics with slow vanishing damping approaching the zeros of a monotone and continuous operator

In this work, we approach the problem of finding the zeros of a continuous and monotone operator through a second-order dynamical system with a damping term of the form $1/t^{r}$, where $r\in [0, 1]$. The system features the time derivative of the operator evaluated along the trajectory, which is a Hessian-driven type damping term when the governing operator comes from a potential. Also entering the system is a time rescaling parameter $β(t)$ which satisfies a certain growth condition. We derive $o\left(\frac{1}{t^{2r}β(t)}\right)$ convergence rates for the norm of the operator evaluated along the generated trajectories as well as for a gap function which serves as a measure of optimality for the associated variational inequality. The parameter $r$ enters the growth condition for $β(t)$: when $r < 1$, the damping $1/t^{r}$ approaches zero at a slower speed than Nesterov's $1/t$ damping; in this case, we are allowed to choose $β(t)$ to be an exponential function, thus having linear convergence rates for the involved quantities. We also show weak convergence of the trajectories towards zeros of the governing operator. Through a particular choice for the operator, we establish a connection with the problem of minimizing a smooth and convex function with linear constraints. The convergence rates we derived in the operator case are inherited by the objective function evaluated at the trajectories and for the feasibility gap. We also prove weak convergence of the trajectories towards primal-dual solutions of the problem. A discretization of the dynamical system yields an implicit algorithm that exhibits analogous convergence properties to its continuous counterpart. We complement our theoretical findings with two numerical experiments.

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arXiv2024

Fast Optimistic Gradient Descent Ascent (OGDA) method in continuous and discrete time

In the framework of real Hilbert spaces we study continuous in time dynamics as well as numerical algorithms for the problem of approaching the set of zeros of a single-valued monotone and continuous operator $V$. The starting poin is a second order dynamical system that combines a vanishing damping term with the time derivative of $V$ along the trajectory. Our method exhibits fast convergence rates of order $o \left( \frac{1}{tβ(t)} \right)$ for $\|V(z(t))\|$, wher $β(\cdot)$ is a positive nondecreasing function satisfying a growth condition, and also for the restricted gap function. We also prove the weak convergence of the trajectory to a zero of $V$. Temporal discretizations of the dynamical system generate implicit and explicit numerical algorithms, which can be both seen as accelerated versions of the Optimistic Gradient Descent Ascent (OGDA) method, for which we prove that the generated sequence of iterates shares the asymptotic features of the continuous dynamics. In particular we show for the implicit numerical algorithm convergence rates of order $o \left( \frac{1}{kβ_k} \right)$ for $\|V(z^k)\|$ and the restricted gap function, where $(β_k)_{k \geq 0}$ is a positive nondecreasing sequence satisfying a growth condition. For the explicit numerical algorithm we show by additionally assuming that the operator $V$ is Lipschitz continuous convergence rates of order $o \left( \frac{1}{k} \right)$ for $\|V(z^k)\|$ and the restricted gap function. All convergence rate statements are last iterate convergence results; in addition we prove for both algorithms the convergence of the iterates to a zero of $V$. To our knowledge, our study exhibits the best known convergence rate results for monotone equations. Numerical experiments indicate the overwhelming superiority of our explicit numerical algorithm over other methods for monotone equations.

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arXiv2023

Fast Forward-Backward splitting for monotone inclusions with a convergence rate of the tangent residual of $o(1/k)$

We address the problem of finding the zeros of the sum of a maximally monotone operator and a cocoercive operator. Our approach introduces a modification to the forward-backward method by integrating an inertial/momentum term alongside a correction term. We demonstrate that the sequence of iterations thus generated converges weakly towards a solution for the monotone inclusion problem. Furthermore, our analysis reveals an outstanding attribute of our algorithm: it displays rates of convergence of the order $o(1/k)$ for the discrete velocity and the tangent residual approaching zero. These rates for tangent residuals can be extended to fixed-point residuals frequently discussed in the existing literature. Specifically, when applied to minimize a nonsmooth convex function subject to linear constraints, our method evolves into a primal-dual full splitting algorithm. Notably, alongside the convergence of iterates, this algorithm possesses a remarkable characteristic of nonergodic/last iterate $o(1/k)$ convergence rates for both the function value and the feasibility measure. Our algorithm showcases the most advanced convergence and convergence rate outcomes among primal-dual full splitting algorithms when minimizing nonsmooth convex functions with linear constraints.

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arXiv2023

Fast Krasnosel'skii-Mann algorithm with a convergence rate of the fixed point iteration of $o\left(\frac{1}{k}\right)$

The Krasnosel'skii-Mann (KM) algorithm is the most fundamental iterative scheme designed to find a fixed point of an averaged operator in the framework of a real Hilbert space, since it lies at the heart of various numerical algorithms for solving monotone inclusions and convex optimization problems. We enhance the Krasnosel'skii-Mann algorithm with Nesterov's momentum updates and show that the resulting numerical method exhibits a convergence rate for the fixed point residual of $o(1/k)$ while preserving the weak convergence of the iterates to a fixed point of the operator. Numerical experiments illustrate the superiority of the resulting so-called Fast KM algorithm over various fixed point iterative schemes, and also its oscillatory behavior, which is a specific of Nesterov's momentum optimization algorithms.

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arXiv2023

A Fast Optimistic Method for Monotone Variational Inequalities

We study monotone variational inequalities that can arise as optimality conditions for constrained convex optimisation or convex-concave minimax problems and propose a novel algorithm that uses only one gradient/operator evaluation and one projection onto the constraint set per iteration. The algorithm, which we call fOGDA-VI, achieves a $o \left( \frac{1}{k} \right)$ rate of convergence in terms of the restricted gap function as well as the natural residual for the last iterate. Moreover, we provide a convergence guarantee for the sequence of iterates to a solution of the variational inequality. These are the best theoretical convergence results for numerical methods for (only) monotone variational inequalities reported in the literature. To empirically validate our algorithm we investigate a two-player matrix game with mixed strategies of the two players. Concluding, we show promising results regarding the application of fOGDA-VI to the training of generative adversarial nets.

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arXiv2023

Accelerated Griffin-Lim algorithm: A fast and provably converging numerical method for phase retrieval

The recovery of a signal from the magnitudes of its transformation, like the Fourier transform, is known as the phase retrieval problem and is of big relevance in various fields of engineering and applied physics. In this paper, we present a fast inertial/momentum based algorithm for the phase retrieval problem and we prove a convergence guarantee for the new algorithm and for the Fast Griffin-Lim algorithm, whose convergence remained unproven in the past decade. In the final chapter, we compare the algorithm for the Short Time Fourier transform phase retrieval with the Griffin-Lim algorithm and FGLA and to other iterative algorithms typically used for this type of problem.

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arXiv2023

Fast convex optimization via time scale and averaging of the steepest descent

In a Hilbert setting, we develop a gradient-based dynamic approach for fast solving convex optimization problems. By applying time scaling, averaging, and perturbation techniques to the continuous steepest descent (SD), we obtain high-resolution ODEs of the Nesterov and Ravine methods. These dynamics involve asymptotically vanishing viscous damping and Hessian driven damping (either in explicit or implicit form). Mathematical analysis does not require developing a Lyapunov analysis for inertial systems. We simply exploit classical convergence results for (SD) and its external perturbation version, then use tools of differential and integral calculus, including Jensen's inequality. The method is flexible and by way of illustration we show how it applies starting from other important dynamics in optimization. We consider the case where the initial dynamics is the regularized Newton method, then the case where the starting dynamics is the differential inclusion associated with a convex lower semicontinuous potential, and finally we show that the technique can be naturally extended to the case of a monotone cocoercive operator. Our approach leads to parallel algorithmic results, which we study in the case of fast gradient and proximal algorithms. Our averaging technique shows new links between the Nesterov and Ravine methods.

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arXiv2023

Faster than Fast: Accelerating the Griffin-Lim Algorithm

The phase retrieval problem is found in various areas of applications of engineering and applied physics. It is also a very active field of research in mathematics, signal processing and machine learning. In this paper, we present an accelerated version of the well known Fast Griffin-Lim algorithm (FGLA) for the phase retrieval problem in a general setting. It has increased the speed of convergence, and most importantly, the limit points of the generated sequence can reach a significantly smaller error than the ones generated by FGLA. We will give a motivation of the acceleration and compare it numerically to its predecessors and other algorithms typically used to solve similar problems.

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arXiv2022

Time rescaling of a primal-dual dynamical system with asymptotically vanishing damping

In this work, we approach the minimization of a continuously differentiable convex function under linear equality constraints by a second-order dynamical system with an asymptotically vanishing damping term. The system under consideration is a time rescaled version of another system previously found in the literature. We show fast convergence of the primal-dual gap, the feasibility measure, and the objective function value along the generated trajectories. These convergence rates now depend on the rescaling parameter, and thus can be improved by choosing said parameter appropriately. When the objective function has a Lipschitz continuous gradient, we show that the primal-dual trajectory asymptotically converges weakly to a primal-dual optimal solution to the underlying minimization problem. We also exhibit improved rates of convergence of the gradient along the primal trajectories and of the adjoint of the corresponding linear operator along the dual trajectories. Even in the unconstrained case, some trajectory convergence result seems to be new. We illustrate the theoretical outcomes through numerical experiments.

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arXiv2022

Fast Augmented Lagrangian Method in the convex regime with convergence guarantees for the iterates

This work aims to minimize a continuously differentiable convex function with Lipschitz continuous gradient under linear equality constraints. The proposed inertial algorithm results from the discretization of the second-order primal-dual dynamical system with asymptotically vanishing damping term addressed by Bot and Nguyen in [Bot, Nguyen, JDE, 2021], and it is formulated in terms of the Augmented Lagrangian associated with the minimization problem. The general setting we consider for the inertial parameters covers the three classical rules by Nesterov, Chambolle-Dossal and Attouch-Cabot used in the literature to formulate fast gradient methods. For these rules, we obtain in the convex regime convergence rates of order ${\cal O}(1/k^{2})$ for the primal-dual gap, the feasibility measure, and the objective function value. In addition, we prove that the generated sequence of primal-dual iterates converges to a primal-dual solution in a general setting that covers the two latter rules. This is the first result which provides the convergence of the sequence of iterates generated by a fast algorithm for linearly constrained convex optimization problems without additional assumptions such as strong convexity. We also emphasize that all convergence results of this paper are compatible with the ones obtained in [Bot, Nguyen, JDE, 2021] in the continuous setting.

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arXiv2021

Improved convergence rates and trajectory convergence for primal-dual dynamical systems with vanishing damping

In this work, we approach the minimization of a continuously differentiable convex function under linear equality constraints by a second-order dynamical system with asymptotically vanishing damping term. The system is formulated in terms of the augmented Lagrangian associated to the minimization problem. We show fast convergence of the primal-dual gap, the feasibility measure, and the objective function value along the generated trajectories. In case the objective function has Lipschitz continuous gradient, we show that the primal-dual trajectory asymptotically weakly converges to a primal-dual optimal solution of the underlying minimization problem. To the best of our knowledge, this is the first result which guarantees the convergence of the trajectory generated by a primal-dual dynamical system with asymptotic vanishing damping. Moreover, we will rediscover in case of the unconstrained minimization of a convex differentiable function with Lipschitz continuous gradient all convergence statements obtained in the literature for Nesterov's accelerated gradient method.

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arXiv2020

The proximal alternating direction method of multipliers in the nonconvex setting: convergence analysis and rates

We propose two numerical algorithms in the fully nonconvex setting for the minimization of the sum of a smooth function and the composition of a nonsmooth function with a linear operator. The iterative schemes are formulated in the spirit of the proximal alternating direction method of multipliers and its linearized variant, respectively. The proximal terms are introduced via variable metrics, a fact which allows us to derive new proximal splitting algorithms for nonconvex structured optimization problems, as particular instances of the general schemes. Under mild conditions on the sequence of variable metrics and by assuming that a regularization of the associated augmented Lagrangian has the Kurdyka-Lojasiewicz property, we prove that the iterates converge to a KKT point of the objective function. By assuming that the augmented Lagrangian has the Lojasiewicz property, we also derive convergence rates for both the augmented Lagrangian and the iterates.

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