arXiv subjects

Elisa Riccietti

Publications and source records attributed to Elisa Riccietti.

At least 19 recordsLinked to original sources

arXiv2026

Mixed precision Newton's method for optimization

Second-order optimization methods, such as Newton's algorithm, achieve fast local convergence and high accuracy, but their practical use is often limited by high computational costs. To mitigate this issue, variants such as inexact and quasi-Newton methods are widely used. A complementary and promising approach to improve the efficiency of the method is to employ mixed precision arithmetic, using different floating-point precisions for different operations, based on their impact on the convergence and accuracy of the method. In this work, we perform an error analysis of Newton's method accounting for different sources of inexactness, including approximations and rounding errors. We present a convergence analysis for the generated sequence, establishing bounds on the convergence rate and attainable accuracy. This theoretical framework covers quasi-Newton and inexact Newton methods, and is leveraged to propose mixed precision algorithms. We present a wide set of numerical experiments to illustrate our theoretical results and the behavior of Newton's method and its approximate variants in mixed precision floating-point arithmetic.

math.OC↗

arXiv2026

Multiresolution Adaptive Block-Coordinate Forward-Backward for Image Reconstruction

Classical first-order optimization methods for imaging inverse problems scale poorly with image resolution. Wavelet based multilevel strategies can accelerate convergence under strong blur, but their fixed coarse-to-fine schedules lose effectiveness in moderate-blur or noise-dominated regimes. In this work, we propose an adaptive multiresolution block coordinate Forward-Backward algorithm for image restoration. Multiresolution block selection is driven by the local magnitude of the proximal update via a stochastic non-smooth Gauss-Southwell rule applied to the wavelet decomposition of the image. This adaptive selection strategy dynamically balances updates across scales, emphasizing coarse or fine blocks according to the degradation regime. As a result, the proposed method automatically adapts to varying blur and noise levels without relying on a predefined hierarchical update scheme.

eess.SP↗

arXiv2026

Introduction to optimization methods for training SciML models

Optimization is central to both modern machine learning (ML) and scientific machine learning (SciML), yet the structure of the underlying optimization problems differs substantially across these domains. Classical ML typically relies on stochastic, sample-separable objectives that favor first-order and adaptive gradient methods. In contrast, SciML often involves physics-informed or operator-constrained formulations in which differential operators induce global coupling, stiffness, and strong anisotropy in the loss landscape. As a result, optimization behavior in SciML is governed by the spectral properties of the underlying physical models rather than by data statistics, frequently limiting the effectiveness of standard stochastic methods and motivating deterministic or curvature-aware approaches. This document provides a unified introduction to optimization methods in ML and SciML, emphasizing how problem structure shapes algorithmic choices. We review first- and second-order optimization techniques in both deterministic and stochastic settings, discuss their adaptation to physics-constrained and data-driven SciML models, and illustrate practical strategies through tutorial examples, while highlighting open research directions at the interface of scientific computing and scientific machine learning.

math.NA↗

arXiv2025

Mixed precision accumulation for neural network inference guided by componentwise forward error analysis

This work proposes a mathematically founded mixed precision accumulation strategy for the inference of neural networks. Our strategy is based on a new componentwise forward error analysis that explains the propagation of errors in the forward pass of neural networks. Specifically, our analysis shows that the error in each component of the output of a linear layer is proportional to the condition number of the inner product between the weights and the input, multiplied by the condition number of the activation function. These condition numbers can vary widely from one component to the other, thus creating a significant opportunity to introduce mixed precision: each component should be accumulated in a precision inversely proportional to the product of these condition numbers. We propose a numerical algorithm that exploits this observation: it first computes all components in low precision, uses this output to estimate the condition numbers, and recomputes in higher precision only the components associated with large condition numbers. We test our algorithm on various networks and datasets and confirm experimentally that it can significantly improve the cost--accuracy tradeoff compared with uniform precision accumulation baselines.

cs.LG↗

arXiv2025

A multilevel stochastic regularized first-order method with application to finite sum minimization

In this paper, we propose a multilevel stochastic framework for the solution of nonconvex unconstrained optimization problems. The proposed approach uses random regularized first-order models that exploit an available hierarchical description of the problem, being either in the classical variable space or in the function space, meaning that different levels of accuracy for the objective function are available. We propose a convergence analysis showing an almost sure global convergence of the method to a first order stationary point. The numerical behavior is tested on the solution of finite sum minimization problems. Differently from classical deterministic multilevel schemes, our stochastic method does not require the finest approximation to coincide with the original objective function along all the optimization process. This allows for significantly decreasing their cost, for instance in data-fitting problems, where considering all the data at each iteration can be avoided.

math.OC↗

arXiv2025

Frugality in second-order optimization: floating-point approximations for Newton's method

Minimizing loss functions is central to machine-learning training. Although first-order methods dominate practical applications, higher-order techniques such as Newton's method can deliver greater accuracy and faster convergence, yet are often avoided due to their computational cost. This work analyzes the impact of finite-precision arithmetic on Newton steps and establishes a convergence theorem for mixed-precision Newton optimizers, including "quasi" and "inexact" variants. The theorem provides not only convergence guarantees but also a priori estimates of the achievable solution accuracy. Empirical evaluations on standard regression benchmarks demonstrate that the proposed methods outperform Adam on the Australian and MUSH datasets. The second part of the manuscript introduces GN_k, a generalized Gauss-Newton method that enables partial computation of second-order derivatives. GN_k attains performance comparable to full Newton's method on regression tasks while requiring significantly fewer derivative evaluations.

cs.LG↗

arXiv2025

A flexible block-coordinate forward-backward algorithm for non-smooth and non-convex optimization

Block coordinate descent (BCD) methods are prevalent in large scale optimization problems due to the low memory and computational costs per iteration, the predisposition to parallelization, and the ability to exploit the structure of the problem. The theoretical and practical performance of BCD relies heavily on the rules defining the choice of the blocks to be updated at each iteration. We propose a new deterministic BCD framework that allows for very flexible updates, while guaranteeing state-of-the-art convergence guarantees on non-smooth nonconvex optimization problems. While encompassing several update rules from the literature, this framework allows for priority on updates of particular blocks and correlations in the block selection between iterations, which is not permitted under the classical convergent stochastic framework. This flexibility is leveraged in the context of multilevel optimization algorithms and, in particular, in multilevel image restoration problems, where the efficiency of the approach is illustrated.

math.OC↗

arXiv2025

A Rescaling-Invariant Lipschitz Bound Based on Path-Metrics for Modern ReLU Network Parameterizations

Robustness with respect to weight perturbations underpins guarantees for generalization, pruning and quantization. Existing guarantees rely on Lipschitz bounds in parameter space, cover only plain feed-forward MLPs, and break under the ubiquitous neuron-wise rescaling symmetry of ReLU networks. We prove a new Lipschitz inequality expressed through the $\ell^1$-path-metric of the weights. The bound is (i) rescaling-invariant by construction and (ii) applies to any ReLU-DAG architecture with any combination of convolutions, skip connections, pooling, and frozen (inference-time) batch-normalization -- thus encompassing ResNets, U-Nets, VGG-style CNNs, and more. By respecting the network's natural symmetries, the new bound strictly sharpens prior parameter-space bounds and can be computed in two forward passes. To illustrate its utility, we derive from it a symmetry-aware pruning criterion and show -- through a proof-of-concept experiment on a ResNet-18 trained on ImageNet -- that its pruning performance matches that of classical magnitude pruning, while becoming totally immune to arbitrary neuron-wise rescalings.

cs.LG↗

arXiv2025

Butterfly factorization with error guarantees

In this paper, we investigate the butterfly factorization problem, i.e., the problem of approximating a matrix by a product of sparse and structured factors. We propose a new formal mathematical description of such factors, that encompasses many different variations of butterfly factorization with different choices of the prescribed sparsity patterns. Among these supports, we identify those that ensure that the factorization problem admits an optimum, thanks to a new property called ``chainability''. For those supports we propose a new butterfly algorithm that yields an approximate solution to the butterfly factorization problem and that is supported by stronger theoretical guarantees than existing factorization methods. Specifically, we show that the ratio of the approximation error by the minimum value is bounded by a constant, independent of the target matrix.

math.OC↗

arXiv2024

A path-norm toolkit for modern networks: consequences, promises and challenges

This work introduces the first toolkit around path-norms that fully encompasses general DAG ReLU networks with biases, skip connections and any operation based on the extraction of order statistics: max pooling, GroupSort etc. This toolkit notably allows us to establish generalization bounds for modern neural networks that are not only the most widely applicable path-norm based ones, but also recover or beat the sharpest known bounds of this type. These extended path-norms further enjoy the usual benefits of path-norms: ease of computation, invariance under the symmetries of the network, and improved sharpness on layered fully-connected networks compared to the product of operator norms, another complexity measure most commonly used. The versatility of the toolkit and its ease of implementation allow us to challenge the concrete promises of path-norm-based generalization bounds, by numerically evaluating the sharpest known bounds for ResNets on ImageNet.

stat.ML↗

arXiv2024

IML FISTA: A Multilevel Framework for Inexact and Inertial Forward-Backward. Application to Image Restoration

This paper presents a multilevel framework for inertial and inexact proximal algorithms, that encompasses multilevel versions of classical algorithms such as forward-backward and FISTA. The methods are supported by strong theoretical guarantees: we prove both the rate of convergence and the convergence of the iterates to a minimum in the convex case, an important result for ill-posed problems. We propose a particular instance of IML (Inexact MultiLevel) FISTA, based on the use of the Moreau envelope to build efficient and useful coarse corrections, fully adapted to solve problems in image restoration. Such a construction is derived for a broad class of composite optimization problems with proximable functions. We evaluate our approach on several image reconstruction problems and we show that it considerably accelerates the convergence of the corresponding one-level (i.e. standard) version of the methods, for large-scale images.

math.OC↗

arXiv2024

A multilevel framework for accelerating uSARA in radio-interferometric imaging

This paper presents a multilevel algorithm specifically designed for radio-interferometric imaging in astronomy. The proposed algorithm is used to solve the uSARA (unconstrained Sparsity Averaging Reweighting Analysis) formulation of this image restoration problem. Multilevel algorithms rely on a hierarchy of approximations of the objective function to accelerate its optimization. In contrast to the usual multilevel approaches where this hierarchy is derived in the parameter space, here we construct the hierarchy of approximations in the observation space. The proposed approach is compared to a reweighted forward-backward procedure, which is the backbone iteration scheme for solving the uSARA problem.

math.OC↗

arXiv2023

Does a sparse ReLU network training problem always admit an optimum?

Given a training set, a loss function, and a neural network architecture, it is often taken for granted that optimal network parameters exist, and a common practice is to apply available optimization algorithms to search for them. In this work, we show that the existence of an optimal solution is not always guaranteed, especially in the context of {\em sparse} ReLU neural networks. In particular, we first show that optimization problems involving deep networks with certain sparsity patterns do not always have optimal parameters, and that optimization algorithms may then diverge. Via a new topological relation between sparse ReLU neural networks and their linear counterparts, we derive -- using existing tools from real algebraic geometry -- an algorithm to verify that a given sparsity pattern suffers from this issue. Then, the existence of a global optimum is proved for every concrete optimization problem involving a shallow sparse ReLU neural network of output dimension one. Overall, the analysis is based on the investigation of two topological properties of the space of functions implementable as sparse ReLU neural networks: a best approximation property, and a closedness property, both in the uniform norm. This is studied both for (finite) domains corresponding to practical training on finite training sets, and for more general domains such as the unit cube. This allows us to provide conditions for the guaranteed existence of an optimum given a sparsity pattern. The results apply not only to several sparsity patterns proposed in recent works on network pruning/sparsification, but also to classical dense neural networks, including architectures not covered by existing results.

cs.NE↗

arXiv2023

Butterfly factorization by algorithmic identification of rank-one blocks

Many matrices associated with fast transforms posess a certain low-rank property characterized by the existence of several block partitionings of the matrix, where each block is of low rank. Provided that these partitionings are known, there exist algorithms, called butterfly factorization algorithms, that approximate the matrix into a product of sparse factors, thus enabling a rapid evaluation of the associated linear operator. This paper proposes a new method to identify algebraically these block partitionings for a matrix admitting a butterfly factorization, without any analytical assumption on its entries.

math.NA↗

arXiv2023

A Block-Coordinate Approach of Multi-level Optimization with an Application to Physics-Informed Neural Networks

Multi-level methods are widely used for the solution of large-scale problems, because of their computational advantages and exploitation of the complementarity between the involved sub-problems. After a re-interpretation of multi-level methods from a block-coordinate point of view, we propose a multi-level algorithm for the solution of nonlinear optimization problems and analyze its evaluation complexity. We apply it to the solution of partial differential equations using physics-informed neural networks (PINNs) and show on a few test problems that the approach results in better solutions and significant computational savings

cs.LG↗

arXiv2023

Self-supervised learning with rotation-invariant kernels

We introduce a regularization loss based on kernel mean embeddings with rotation-invariant kernels on the hypersphere (also known as dot-product kernels) for self-supervised learning of image representations. Besides being fully competitive with the state of the art, our method significantly reduces time and memory complexity for self-supervised training, making it implementable for very large embedding dimensions on existing devices and more easily adjustable than previous methods to settings with limited resources. Our work follows the major paradigm where the model learns to be invariant to some predefined image transformations (cropping, blurring, color jittering, etc.), while avoiding a degenerate solution by regularizing the embedding distribution. Our particular contribution is to propose a loss family promoting the embedding distribution to be close to the uniform distribution on the hypersphere, with respect to the maximum mean discrepancy pseudometric. We demonstrate that this family encompasses several regularizers of former methods, including uniformity-based and information-maximization methods, which are variants of our flexible regularization loss with different kernels. Beyond its practical consequences for state-of-the-art self-supervised learning with limited resources, the proposed generic regularization approach opens perspectives to leverage more widely the literature on kernel methods in order to improve self-supervised learning methods.

cs.CV↗

arXiv2022

Spurious Valleys, NP-hardness, and Tractability of Sparse Matrix Factorization With Fixed Support

The problem of approximating a dense matrix by a product of sparse factors is a fundamental problem for many signal processing and machine learning tasks. It can be decomposed into two subproblems: finding the position of the non-zero coefficients in the sparse factors, and determining their values. While the first step is usually seen as the most challenging one due to its combinatorial nature, this paper focuses on the second step, referred to as sparse matrix approximation with fixed support. First, we show its NP-hardness, while also presenting a nontrivial family of supports making the problem practically tractable with a dedicated algorithm. Then, we investigate the landscape of its natural optimization formulation, proving the absence of spurious local valleys and spurious local minima, whose presence could prevent local optimization methods to achieve global optimality. The advantages of the proposed algorithm over state-of-the-art first-order optimization methods are discussed.

cs.CC↗

arXiv2022

Efficient Identification of Butterfly Sparse Matrix Factorizations

Fast transforms correspond to factorizations of the form $\mathbf{Z} = \mathbf{X}^{(1)} \ldots \mathbf{X}^{(J)}$, where each factor $ \mathbf{X}^{(\ell)}$ is sparse and possibly structured. This paper investigates essential uniqueness of such factorizations, i.e., uniqueness up to unavoidable scaling ambiguities. Our main contribution is to prove that any $N \times N$ matrix having the so-called butterfly structure admits an essentially unique factorization into $J$ butterfly factors (where $N = 2^{J}$), and that the factors can be recovered by a hierarchical factorization method, which consists in recursively factorizing the considered matrix into two factors. This hierarchical identifiability property relies on a simple identifiability condition in the two-layer and fixed-support setting. This approach contrasts with existing ones that fit the product of butterfly factors to a given matrix via gradient descent. The proposed method can be applied in particular to retrieve the factorization of the Hadamard or the discrete Fourier transform matrices of size $N=2^J$. Computing such factorizations costs $\mathcal{O}(N^{2})$, which is of the order of dense matrix-vector multiplication, while the obtained factorizations enable fast $\mathcal{O}(N \log N)$ matrix-vector multiplications and have the potential to be applied to compress deep neural networks.

cs.LG↗