arXiv subjects

Jalal Fadili

Publications and source records attributed to Jalal Fadili.

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.

math.OC↗

arXiv2026

Inexact and Stochastic Gradient Optimization Algorithms with Inertia and Hessian Driven Damping

In a real Hilbert space setting, we study the convergence properties of an inexact gradient algorithm featuring both viscous and Hessian driven damping for convex differentiable optimization. In this algorithm, the gradient evaluation can be subject to deterministic and stochastic perturbations. In the deterministic case, we show that under appropriate summability assumptions on the perturbation, our algorithm enjoys fast convergence of the objective values, of the gradients and weak convergence of the iterates toward a minimizer of the objective. In the stochastic case, assuming the perturbation is zero-mean, we can weaken our summability assumptions on the error variance and provide fast convergence of the values both in expectation and almost surely. We also improve the convergence rates from $\mathcal{O}(\cdot)$ to $o(\cdot)$ in almost sure sense. We also prove almost sure summability property of the gradients, which implies the almost sure fast convergence of the gradients towards zero. We will highlight the trade-off between fast convergence and the applicable regime on the sequence of errors in the gradient computations. We finally report some numerical results to support our findings.

math.OC↗

arXiv2026

Differentiating Minimal-Norm Solutions to Parametric Optimization Problems

Differentiating through parametric optimization problems is central to bilevel programming and meta-learning, often accomplished using approximate implicit differentiation. The implicit function theorem requires inverting a partial Jacobian of the optimality condition, which fails when there are many solutions. Nonetheless, in such cases it is possible to relax invertibility to a strictly weaker uniform range condition, under which it is shown that the minimal-norm solution mapping admits generalized derivatives by using a limiting Tikhonov regularization argument and conservative set-valued field theory. With additional control on the eigenvalues of the generalized Hessians, a pseudoinverse formula is justified. This is established for a class of smooth convex objectives and extended to nonsmooth composite problems. These assumptions are verified for Least-Squares, Huber regression and LASSO. The resulting extension of nonsmooth implicit differentiation to ill-posed settings is examined experimentally on data poisoning and data hypercleaning problems.

math.OC↗

arXiv2026

Physics Matters in PnP: Recovery Guarantees with the MMSE and NN Denoisers

We investigate the forward-backward-splitting version of the Plug and Play (PnP) method for linear ill-posed problems with MMSE estimators as denoisers. In contrast to existing literature, we consider estimators which are specialized for (degenerate) Gaussian noise with possibly non-diagonal covariance matrices. We further deviate from the classical iteration by replacing parts of the descent step with a linear operator that relates the observation noise to that of the MMSE estimator. Under mild assumptions, we derive several properties of the denoiser and prove recovery guarantees of the iteration both pointwise and in the Wasserstein distance of the underlying probability distributions. Crucially, our analysis shows that the denoiser cannot be chosen in a physics-agnostic way, that is, independently of the forward model. We extend our results to the case where the MMSE denoiser is parametrized by a neural network and derive the corresponding recovery bounds.

math.OC↗

arXiv2026

A plug-and-play approach with fast uncertainty quantification for weak lensing mass mapping

Upcoming stage-IV surveys such as Euclid and Rubin will deliver vast amounts of high-precision data, opening new opportunities to constrain cosmological models with unprecedented accuracy. A key step in this process is the reconstruction of the dark matter distribution from noisy weak-lensing shear measurements. Current deep-learning-based mass-mapping methods achieve high reconstruction accuracy, but either require retraining a model for each new observed sky region (limiting practicality) or rely on slow Markov chain Monte Carlo sampling. Efficient exploitation of future survey data therefore calls for a new method that is accurate, flexible, and fast at inference. In addition, an uncertainty quantification with coverage guarantees is essential for a reliable cosmological parameter estimation. We introduce PnPMass, a plug-and-play approach for weak-lensing mass mapping. The algorithm produces point estimates by alternating between a gradient descent step with a carefully chosen data fidelity term and a denoising step implemented with a single deep-learning model trained on simulated data corrupted by Gaussian white noise. We also propose a fast sampling-free uncertainty quantification scheme based on moment networks, with calibrated error bars obtained through conformal prediction to ensure coverage guarantees. Finally, we benchmark PnPMass against model-driven and data-driven mass-mapping techniques. PnPMass achieves a performance close to that of the currently best deep-learning methods while offering fast inference. It converges in just a few iterations, and it requires only a single training phase, regardless of the noise covariance of the observations. It therefore combines flexibility, efficiency, and reconstruction accuracy while delivering tighter error bars than existing approaches, making it well suited for upcoming weak-lensing surveys.

astro-ph.CO↗

arXiv2026

A New Perspective on Precision and Recall for Generative Models

With the recent success of generative models in image and text, the question of their evaluation has recently gained a lot of attention. While most methods from the state of the art rely on scalar metrics, the introduction of Precision and Recall (PR) for generative model has opened up a new avenue of research. The associated PR curve allows for a richer analysis, but their estimation poses several challenges. In this paper, we present a new framework for estimating entire PR curves based on a binary classification standpoint. We conduct a thorough statistical analysis of the proposed estimates. As a byproduct, we obtain a minimax upper bound on the PR estimation risk. We also show that our framework extends several landmark PR metrics of the literature which by design are restrained to the extreme values of the curve. Finally, we study the different behaviors of the curves obtained experimentally in various settings.

cs.AI↗

arXiv2026

Second order splitting dynamics for stochastic monotone inclusions with closed loop distribution

In this paper, we investigate the problem of finding a zero of the sum of a maximal monotone operator $A$ and a cocoercive operator $\Bm$ in a Hilbert space. This formulation naturally captures stochastic optimization problems with decision-dependent distributions, often referred to as performative prediction. We propose and analyze continuous-time second-order dynamics governed by a distributionally evaluated forward-backward splitting operator. We establish the existence and uniqueness of the equilibrium point under a general uniform monotonicity assumption. In this setting, employing a vanishing viscous damping coefficient, we prove the strong convergence of the trajectories to the equilibrium, accompanied by fast asymptotic convergence rates for the velocities. Furthermore, when the regularizing operator is strongly monotone, we consider a constant Polyak-type damping coefficient and we establish global exponential convergence rates for the dynamical system.

math.OC↗

arXiv2026

Neutrino Oscillation Parameter Estimation Using Structured Hierarchical Transformers

Neutrino oscillations encode fundamental information about neutrino masses and mixing parameters, offering a unique window into physics beyond the Standard Model. Estimating these parameters from oscillation probability maps is, however, computationally challenging due to the maps' high dimensionality and nonlinear dependence on the underlying physics. Traditional inference methods, such as likelihood-based or Monte Carlo sampling approaches, require extensive simulations to explore the parameter space, creating major bottlenecks for large-scale analyses. In this work, we introduce a data-driven framework that reformulates atmospheric neutrino oscillation parameter inference as a supervised regression task over structured oscillation maps. We propose a hierarchical transformer architecture that explicitly models the two-dimensional structure of these maps, capturing angular dependencies at fixed energies and global correlations across the energy spectrum. To improve physical consistency, the model is trained using a surrogate simulation constraint that enforces agreement between the predicted parameters and the reconstructed oscillation patterns. Furthermore, we introduce a neural network-based uncertainty quantification mechanism that produces distribution-free prediction intervals with formal coverage guarantees. Experiments on simulated oscillation maps under Earth-matter conditions demonstrate that the proposed method is comparable to a Markov Chain Monte Carlo baseline in estimation accuracy, with substantial improvements in computational cost (around 240$\times$ fewer FLOPs and 33$\times$ faster in average processing time). Moreover, the conformally calibrated prediction intervals remain narrow while achieving the target nominal coverage of 90%, confirming both the reliability and efficiency of our approach.

hep-ph↗

arXiv2025

Towards Uncertainty Quantification in Generative Model Learning

While generative models have become increasingly prevalent across various domains, fundamental concerns regarding their reliability persist. A crucial yet understudied aspect of these models is the uncertainty quantification surrounding their distribution approximation capabilities. Current evaluation methodologies focus predominantly on measuring the closeness between the learned and the target distributions, neglecting the inherent uncertainty in these measurements. In this position paper, we formalize the problem of uncertainty quantification in generative model learning. We discuss potential research directions, including the use of ensemble-based precision-recall curves. Our preliminary experiments on synthetic datasets demonstrate the effectiveness of aggregated precision-recall curves in capturing model approximation uncertainty, enabling systematic comparison among different model architectures based on their uncertainty characteristics.

cs.LG↗

arXiv2025

Convergence rates of regularized quasi-Newton methods without strong convexity

In this paper, we study convergence rates of the cubic regularized proximal quasi-Newton method (\csr) for solving non-smooth additive composite problems that satisfy the so-called Kurdyka-Łojasiewicz (KŁ) property with respect to some desingularization function $ϕ$ rather than strong convexity. After a number of iterations $k_0$, Cubic SR1 PQN exhibits non-asymptotic explicit super-linear convergence rates for any $k\geq k_0$. In particular, when $ϕ(t)=ct^{1/2}$, Cubic SR1 PQN has a convergence rate of order $\left(\frac{C}{(k-k_0)^{1/2}}\right)^{(k-k_0)/2}$, where $k$ is the number of iterations and $C>0$ is a constant. For the special case, i.e. functions which satisfy Łojasiewicz inequality, the rate becomes global and non-asymptotic. This work presents, for the first time, non-asymptotic explicit convergence rates of regularized (proximal) SR1 quasi-Newton methods applied to non-convex non-smooth problems with KŁ property. Actually, the rates are novel even in the smooth non-convex case. Notably, we achieve this without employing line search or trust region strategies, without assuming the Dennis-Moré condition, without any assumptions on quasi-Newton metrics and without assuming strong convexity. Furthermore, for convex problems, we focus on a more tractable gradient regularized quasi-Newton method (Grad SR1 PQN) which can achieve results similar to those obtained with cubic regularization. We also demonstrate, for the first time, the non-asymptotic super-linear convergence rate of Grad SR1 PQN for solving convex problems with the help of the Łojasiewicz inequality instead of strong convexity.

math.OC↗

arXiv2025

An SDE Perspective on Stochastic Inertial Gradient Dynamics with Time-Dependent Viscosity and Geometric Damping

Our approach is part of the close link between continuous dissipative dynamical systems and optimization algorithms. We aim to solve convex minimization problems by means of stochastic inertial differential equations which are driven by the gradient of the objective function. This will provide a general mathematical framework for analyzing fast optimization algorithms with stochastic gradient input. Our study is a natural extension of our previous work devoted to the first-order in time stochastic steepest descent. Our goal is to develop these results further by considering second-order stochastic differential equations in time, incorporating a viscous time-dependent damping and a Hessian-driven damping. To develop this program, we rely on stochastic Lyapunov analysis. Assuming a square-integrability condition on the diffusion term times a function dependant on the viscous damping, and that the Hessian-driven damping is a positive constant, our first main result shows that almost surely, there is convergence of the values, and states fast convergence of the values in expectation. Besides, in the case where the Hessian-driven damping is zero, we conclude with the fast convergence of the values in expectation and in almost sure sense, we also managed to prove almost sure weak convergence of the trajectory. We provide a comprehensive complexity analysis by establishing several new pointwise and ergodic convergence rates in expectation for the convex and strongly convex case.

math.OC↗

arXiv2025

Inertial Methods with Viscous and Hessian driven Damping for Non-Convex Optimization

In this paper, we aim to study non-convex minimization problems via second-order (in-time) dynamics, including a non-vanishing viscous damping and a geometric Hessian-driven damping. Second-order systems that only rely on a viscous damping may suffer from oscillation problems towards the minima, while the inclusion of a Hessian-driven damping term is known to reduce this effect without explicit construction of the Hessian in practice. There are essentially two ways to introduce the Hessian-driven damping term: explicitly or implicitly. For each setting, we provide conditions on the damping coefficients to ensure convergence of the gradient towards zero. Moreover, if the objective function is definable, we show global convergence of the trajectory towards a critical point as well as convergence rates. Besides, in the autonomous case, if the objective function is Morse, we conclude that the trajectory converges to a local minimum of the objective for almost all initializations. We also study algorithmic schemes for both dynamics and prove discrete analogues of the previous properties under appropriate stepsize conditions. In particular, we consider the case where the objective is only locally Lipschitz smooth and propose a backtracking strategy for which we establish convergence guarantees. Our work is the first one that handles this situation.

math.OC↗

arXiv2025

Tikhonov Regularization for Stochastic Non-Smooth Convex Optimization in Hilbert Spaces

To solve convex optimization problems with a noisy gradient input, we analyze the global behavior of subgradient-like flows under stochastic errors. The objective function is composite, being equal to the sum of two convex functions, one being differentiable and the other potentially non-smooth. We then use stochastic differential inclusions where the drift term is minus the subgradient of the objective function, and the diffusion term is either bounded or square-integrable. In this context, under Lipschitz's continuity of the differentiable term and a growth condition of the non-smooth term, our first main result shows almost sure weak convergence of the trajectory process towards a minimizer of the objective function. Then, using Tikhonov regularization with a properly tuned vanishing parameter, we can obtain almost sure strong convergence of the trajectory towards the minimum norm solution. We find an explicit tuning of this parameter when our objective function satisfies a local error-bound inequality. We also provide a comprehensive complexity analysis by establishing several new pointwise and ergodic convergence rates in expectation for the convex, strongly convex, and Lojasiewicz case.

math.OC↗

arXiv2025

Learning-to-Optimize with PAC-Bayesian Guarantees: Theoretical Considerations and Practical Implementation

We use the PAC-Bayesian theory for the setting of learning-to-optimize. To the best of our knowledge, we present the first framework to learn optimization algorithms with provable generalization guarantees (PAC-Bayesian bounds) and explicit trade-off between convergence guarantees and convergence speed, which contrasts with the typical worst-case analysis. Our learned optimization algorithms provably outperform related ones derived from a (deterministic) worst-case analysis. The results rely on PAC-Bayesian bounds for general, possibly unbounded loss-functions based on exponential families. Then, we reformulate the learning procedure into a one-dimensional minimization problem and study the possibility to find a global minimum. Furthermore, we provide a concrete algorithmic realization of the framework and new methodologies for learning-to-optimize, and we conduct four practically relevant experiments to support our theory. With this, we showcase that the provided learning framework yields optimization algorithms that provably outperform the state-of-the-art by orders of magnitude.

cs.LG↗

arXiv2025

Stochastic Inertial Dynamics Via Time Scaling and Averaging

Our work is part of the close link between continuous-time dissipative dynamical systems and optimization algorithms, and more precisely here, in the stochastic setting. We aim to study stochastic convex minimization problems through the lens of stochastic inertial differential inclusions that are driven by the subgradient of a convex objective function. This will provide a general mathematical framework for analyzing the convergence properties of stochastic second-order inertial continuous-time dynamics involving vanishing viscous damping and measurable stochastic subgradient selections. Our chief goal in this paper is to develop a systematic and unified way that transfers the properties recently studied for first-order stochastic differential equations to second-order ones involving even subgradients in lieu of gradients. This program will rely on two tenets: time scaling and averaging, following an approach recently developed in the literature by one of the co-authors in the deterministic case. Under a mild integrability assumption involving the diffusion term and the viscous damping, our first main result shows that almost surely, there is weak convergence of the trajectory towards a minimizer of the objective function and fast convergence of the values and gradients. We also provide a comprehensive complexity analysis by establishing several new pointwise and ergodic convergence rates in expectation for the convex, strongly convex, and (local) Polyak-Lojasiewicz case. Finally, using Tikhonov regularization with a properly tuned vanishing parameter, we can obtain almost sure strong convergence of the trajectory towards the minimum norm solution.

math.OC↗

arXiv2025

Distribution-free uncertainty quantification for inverse problems: application to weak lensing mass mapping

In inverse problems, distribution-free uncertainty quantification (UQ) aims to obtain error bars with coverage guarantees that are independent of any prior assumptions about the data distribution. In the context of mass mapping, uncertainties could lead to errors that affects our understanding of the underlying mass distribution, or could propagate to cosmological parameter estimation, thereby impacting the precision and reliability of cosmological models. Current surveys, such as Euclid or Rubin, will provide new weak lensing datasets of very high quality. Accurately quantifying uncertainties in mass maps is therefore critical to perform reliable cosmological parameter inference. In this paper, we extend the conformalized quantile regression (CQR) algorithm, initially proposed for scalar regression, to inverse problems. We compare our approach with another distribution-free approach based on risk-controlling prediction sets (RCPS). Both methods are based on a calibration dataset, and offer finite-sample coverage guarantees that are independent of the data distribution. Furthermore, they are applicable to any mass mapping method, including blackbox predictors. In our experiments, we apply UQ on three mass-mapping method: the Kaiser-Squires inversion, iterative Wiener filtering, and the MCALens algorithm. Our experiments reveal that RCPS tends to produce overconservative confidence bounds with small calibration sets, whereas CQR is designed to avoid this issue. Although the expected miscoverage rate is guaranteed to stay below a user-prescribed threshold regardless of the mass mapping method, selecting an appropriate reconstruction algorithm remains crucial for obtaining accurate estimates, especially around peak-like structures, which are particularly important for inferring cosmological parameters. Additionally, the choice of mass mapping method influences the size of the error bars.

astro-ph.CO↗

arXiv2025

Stochastic Monotone Inclusion with Closed Loop Distributions

In this paper, we study in a Hilbertian setting, first and second-order monotone inclusions related to stochastic optimization problems with decision dependent distributions. The studied dynamics are formulated as monotone inclusions governed by Lipschitz perturbations of maximally monotone operators where the concept of equilibrium plays a central role. We discuss the relationship between the $\mathbb{W}_1$-Wasserstein Lipschitz behavior of the distribution and the so-called coarse Ricci curvature. As an application, we consider the monotone inclusions associated with stochastic optimisation problems involving the sum of a smooth function with Lipschitz gradient, a proximable function and a composite term.

math.OC↗

arXiv2024

Global non-asymptotic super-linear convergence rates of regularized proximal quasi-Newton methods on non-smooth composite problems

In this paper, we propose two regularized proximal quasi-Newton methods with symmetric rank-1 update of the metric (SR1 quasi-Newton) to solve non-smooth convex additive composite problems. Both algorithms avoid using line search or other trust region strategies. For each of them, we prove a super-linear convergence rate that is independent of the initialization of the algorithm. The cubic regularized method achieves a rate of order $\left(\frac{C}{N^{1/2}}\right)^{N/2}$, where $N$ is the number of iterations and $C$ is some constant, and the other gradient regularized method shows a rate of the order $\left(\frac{C}{N^{1/4}}\right)^{N/2}$. To the best of our knowledge, these are the first global non-asymptotic super-linear convergence rates for regularized quasi-Newton methods and regularized proximal quasi-Newton methods. The theoretical properties are also demonstrated in two applications from machine learning.

math.OC↗