arXiv subjects

Armin Eftekhari

Publications and source records attributed to Armin Eftekhari.

At least 19 recordsLinked to original sources

arXiv2025

A Unified Model for High-Resolution ODEs: New Insights on Accelerated Methods

Recent work on high-resolution ordinary differential equations (HR-ODEs) captures fine nuances among different momentum-based optimization methods, leading to accurate theoretical insights. However, these HR-ODEs often appear disconnected, each targeting a specific algorithm and derived with different assumptions and techniques. We present a unifying framework by showing that these diverse HR-ODEs emerge as special cases of a general HR-ODE derived using the Forced Euler-Lagrange equation. Discretizing this model recovers a wide range of optimization algorithms through different parameter choices. Using integral quadratic constraints, we also introduce a general Lyapunov function to analyze the convergence of the proposed HR-ODE and its discretizations, achieving significant improvements across various cases, including new guarantees for the triple momentum method$'$s HR-ODE and the quasi-hyperbolic momentum method, as well as faster gradient norm minimization rates for Nesterov$'$s accelerated gradient algorithm, among other advances.

math.OC↗

arXiv2024

Computing second-order points under equality constraints: revisiting Fletcher's augmented Lagrangian

We address the problem of minimizing a smooth function under smooth equality constraints. Under regularity assumptions on these constraints, we propose a notion of approximate first- and second-order critical point which relies on the geometric formalism of Riemannian optimization. Using a smooth exact penalty function known as Fletcher's augmented Lagrangian, we propose an algorithm to minimize the penalized cost function which reaches $\varepsilon$-approximate second-order critical points of the original optimization problem in at most $\mathcal{O}(\varepsilon^{-3})$ iterations. This improves on current best theoretical bounds. Along the way, we show new properties of Fletcher's augmented Lagrangian, which may be of independent interest.

math.OC↗

arXiv2022

Nonlinear matrix recovery using optimization on the Grassmann manifold

We investigate the problem of recovering a partially observed high-rank matrix whose columns obey a nonlinear structure such as a union of subspaces, an algebraic variety or grouped in clusters. The recovery problem is formulated as the rank minimization of a nonlinear feature map applied to the original matrix, which is then further approximated by a constrained non-convex optimization problem involving the Grassmann manifold. We propose two sets of algorithms, one arising from Riemannian optimization and the other as an alternating minimization scheme, both of which include first- and second-order variants. Both sets of algorithms have theoretical guarantees. In particular, for the alternating minimization, we establish global convergence and worst-case complexity bounds. Additionally, using the Kurdyka-Lojasiewicz property, we show that the alternating minimization converges to a unique limit point. We provide extensive numerical results for the recovery of union of subspaces and clustering under entry sampling and dense Gaussian sampling. Our methods are competitive with existing approaches and, in particular, high accuracy is achieved in the recovery using Riemannian second-order methods.

stat.ML↗

arXiv2022

An Inexact Augmented Lagrangian Framework for Nonconvex Optimization with Nonlinear Constraints

We propose a practical inexact augmented Lagrangian method (iALM) for nonconvex problems with nonlinear constraints. We characterize the total computational complexity of our method subject to a verifiable geometric condition, which is closely related to the Polyak-Lojasiewicz and Mangasarian-Fromowitz conditions. In particular, when a first-order solver is used for the inner iterates, we prove that iALM finds a first-order stationary point with $\tilde{\mathcal{O}}(1/ε^4)$ calls to the first-order oracle. If, in addition, the problem is smooth and a second-order solver is used for the inner iterates, iALM finds a second-order stationary point with $\tilde{\mathcal{O}}(1/ε^5)$ calls to the second-order oracle, which matches the known theoretical complexity result in the literature. We also provide strong numerical evidence on large-scale machine learning problems, including the Burer-Monteiro factorization of semidefinite programs, and a novel nonconvex relaxation of the standard basis pursuit template. For these examples, we also show how to verify our geometric condition.

math.OC↗

arXiv2022

The Nonconvex Geometry of Linear Inverse Problems

The gauge function, closely related to the atomic norm, measures the complexity of a statistical model, and has found broad applications in machine learning and statistical signal processing. In a high-dimensional learning problem, the gauge function attempts to safeguard against overfitting by promoting a sparse (concise) representation within the learning alphabet. In this work, within the context of linear inverse problems, we pinpoint the source of its success, but also argue that the applicability of the gauge function is inherently limited by its convexity, and showcase several learning problems where the classical gauge function theory fails. We then introduce a new notion of statistical complexity, gauge$_p$ function, which overcomes the limitations of the gauge function. The gauge$_p$ function is a simple generalization of the gauge function that can tightly control the sparsity of a statistical model within the learning alphabet and, perhaps surprisingly, draws further inspiration from the Burer-Monteiro factorization in computational mathematics. We also propose a new learning machine, with the building block of gauge$_p$ function, and arm this machine with a number of statistical guarantees. The potential of the proposed gauge$_p$ function theory is then studied for two stylized applications. Finally, we discuss the computational aspects and, in particular, suggest a tractable numerical algorithm for implementing the new learning machine.

math.OC↗

arXiv2022

The Forward-Backward Envelope for Sampling with the Overdamped Langevin Algorithm

In this paper, we analyse a proximal method based on the idea of forward-backward splitting for sampling from distributions with densities that are not necessarily smooth. In particular, we study the non-asymptotic properties of the Euler-Maruyama discretization of the Langevin equation, where the forward-backward envelope is used to deal with the non-smooth part of the dynamics. An advantage of this envelope, when compared to widely-used Moreu-Yoshida one and the MYULA algorithm, is that it maintains the MAP estimator of the original non-smooth distribution. We also study a number of numerical experiments that corroborate that support our theoretical findings.

math.NA↗

arXiv2021

Over-Parametrized Matrix Factorization in the Presence of Spurious Stationary Points

Motivated by the emerging role of interpolating machines in signal processing and machine learning, this work considers the computational aspects of over-parametrized matrix factorization. In this context, the optimization landscape may contain spurious stationary points (SSPs), which are proved to be full-rank matrices. The presence of these SSPs means that it is impossible to hope for any global guarantees in over-parametrized matrix factorization. For example, when initialized at an SSP, the gradient flow will be trapped there forever. Nevertheless, despite these SSPs, we establish in this work that the gradient flow of the corresponding merit function converges to a global minimizer, provided that its initialization is rank-deficient and sufficiently close to the feasible set of the optimization problem. We numerically observe that a heuristic discretization of the proposed gradient flow, inspired by primal-dual algorithms, is successful when initialized randomly. Our result is in sharp contrast with the local refinement methods which require an initialization close to the optimal set of the optimization problem. More specifically, we successfully avoid the traps set by the SSPs because the gradient flow remains rank-deficient at all times, and not because there are no SSPs nearby. The latter is the case for the local refinement methods. Moreover, the widely-used restricted isometry property plays no role in our main result.

cs.LG↗

arXiv2021

Subquadratic Overparameterization for Shallow Neural Networks

Overparameterization refers to the important phenomenon where the width of a neural network is chosen such that learning algorithms can provably attain zero loss in nonconvex training. The existing theory establishes such global convergence using various initialization strategies, training modifications, and width scalings. In particular, the state-of-the-art results require the width to scale quadratically with the number of training data under standard initialization strategies used in practice for best generalization performance. In contrast, the most recent results obtain linear scaling either with requiring initializations that lead to the "lazy-training", or training only a single layer. In this work, we provide an analytical framework that allows us to adopt standard initialization strategies, possibly avoid lazy training, and train all layers simultaneously in basic shallow neural networks while attaining a desirable subquadratic scaling on the network width. We achieve the desiderata via Polyak-Lojasiewicz condition, smoothness, and standard assumptions on data, and use tools from random matrix theory.

cs.LG↗

arXiv2021

Limitations of Implicit Bias in Matrix Sensing: Initialization Rank Matters

In matrix sensing, we first numerically identify the sensitivity to the initialization rank as a new limitation of the implicit bias of gradient flow. We will partially quantify this phenomenon mathematically, where we establish that the gradient flow of the empirical risk is implicitly biased towards low-rank outcomes and successfully learns the planted low-rank matrix, provided that the initialization is low-rank and within a specific "capture neighborhood". This capture neighborhood is far larger than the corresponding neighborhood in local refinement results; the former contains all models with zero training error whereas the latter is a small neighborhood of a model with zero test error. These new insights enable us to design an alternative algorithm for matrix sensing that complements the high-rank and near-zero initialization scheme which is predominant in the existing literature.

cs.IT↗

arXiv2021

Principal Component Hierarchy for Sparse Quadratic Programs

We propose a novel approximation hierarchy for cardinality-constrained, convex quadratic programs that exploits the rank-dominating eigenvectors of the quadratic matrix. Each level of approximation admits a min-max characterization whose objective function can be optimized over the binary variables analytically, while preserving convexity in the continuous variables. Exploiting this property, we propose two scalable optimization algorithms, coined as the "best response" and the "dual program", that can efficiently screen the potential indices of the nonzero elements of the original program. We show that the proposed methods are competitive with the existing screening methods in the current sparse regression literature, and it is particularly fast on instances with high number of measurements in experiments with both synthetic and real datasets.

math.OC↗

arXiv2020

Stable Super-Resolution of Images: A Theoretical Study

We study the ubiquitous super-resolution problem, in which one aims at localizing positive point sources in an image, blurred by the point spread function of the imaging device. To recover the point sources, we propose to solve a convex feasibility program, which simply finds a nonnegative Borel measure that agrees with the observations collected by the imaging device. In the absence of imaging noise, we show that solving this convex program uniquely retrieves the point sources, provided that the imaging device collects enough observations. This result holds true if the point spread function of the imaging device can be decomposed into horizontal and vertical components, and if the translations of these components form a Chebyshev system, i.e., a system of continuous functions that loosely behave like algebraic polynomials. Building upon recent results for one-dimensional signals [1], we prove that this super-resolution algorithm is stable, in the generalized Wasserstein metric, to model mismatch (i.e., when the image is not sparse) and to additive imaging noise. In particular, the recovery error depends on the noise level and how well the image can be approximated with well-separated point sources. As an example, we verify these claims for the important case of a Gaussian point spread function. The proofs rely on the construction of novel interpolating polynomials---which are the main technical contribution of this paper---and partially resolve the question raised in [2] about the extension of the standard machinery to higher dimensions.

cs.IT↗

arXiv2020

Double-Loop Unadjusted Langevin Algorithm

A well-known first-order method for sampling from log-concave probability distributions is the Unadjusted Langevin Algorithm (ULA). This work proposes a new annealing step-size schedule for ULA, which allows to prove new convergence guarantees for sampling from a smooth log-concave distribution, which are not covered by existing state-of-the-art convergence guarantees. To establish this result, we derive a new theoretical bound that relates the Wasserstein distance to total variation distance between any two log-concave distributions that complements the reach of Talagrand T2 inequality. Moreover, applying this new step size schedule to an existing constrained sampling algorithm, we show state-of-the-art convergence rates for sampling from a constrained log-concave distribution, as well as improved dimension dependence.

math.ST↗

arXiv2020

Training Linear Neural Networks: Non-Local Convergence and Complexity Results

Linear networks provide valuable insights into the workings of neural networks in general. This paper identifies conditions under which the gradient flow provably trains a linear network, in spite of the non-strict saddle points present in the optimization landscape. This paper also provides the computational complexity of training linear networks with gradient flow. To achieve these results, this work develops a machinery to provably identify the stable set of gradient flow, which then enables us to improve over the state of the art in the literature of linear networks (Bah et al., 2019;Arora et al., 2018a). Crucially, our results appear to be the first to break away from the lazy training regime which has dominated the literature of neural networks. This work requires the network to have a layer with one neuron, which subsumes the networks with a scalar output, but extending the results of this theoretical work to all linear networks remains a challenging open problem.

math.OC↗

arXiv2020

Explicit Stabilised Gradient Descent for Faster Strongly Convex Optimisation

This paper introduces the Runge-Kutta Chebyshev descent method (RKCD) for strongly convex optimisation problems. This new algorithm is based on explicit stabilised integrators for stiff differential equations, a powerful class of numerical schemes that avoid the severe step size restriction faced by standard explicit integrators. For optimising quadratic and strongly convex functions, this paper proves that RKCD nearly achieves the optimal convergence rate of the conjugate gradient algorithm, and the suboptimality of RKCD diminishes as the condition number of the quadratic function worsens. It is established that this optimal rate is obtained also for a partitioned variant of RKCD applied to perturbations of quadratic functions. In addition, numerical experiments on general strongly convex problems show that RKCD outperforms Nesterov's accelerated gradient descent.

math.OC↗

arXiv2020

Scalable Learning-Based Sampling Optimization for Compressive Dynamic MRI

Compressed sensing applied to magnetic resonance imaging (MRI) allows to reduce the scanning time by enabling images to be reconstructed from highly undersampled data. In this paper, we tackle the problem of designing a sampling mask for an arbitrary reconstruction method and a limited acquisition budget. Namely, we look for an optimal probability distribution from which a mask with a fixed cardinality is drawn. We demonstrate that this problem admits a compactly supported solution, which leads to a deterministic optimal sampling mask. We then propose a stochastic greedy algorithm that (i) provides an approximate solution to this problem, and (ii) resolves the scaling issues of [1,2]. We validate its performance on in vivo dynamic MRI with retrospective undersampling, showing that our method preserves the performance of [1,2] while reducing the computational burden by a factor close to 200.

eess.IV↗

arXiv2020

MOSES: A Streaming Algorithm for Linear Dimensionality Reduction

This paper introduces Memory-limited Online Subspace Estimation Scheme (MOSES) for both estimating the principal components of streaming data and reducing its dimension. More specifically, in various applications such as sensor networks, the data vectors are presented sequentially to a user who has limited storage and processing time available. Applied to such problems, MOSES can provide a running estimate of leading principal components of the data that has arrived so far and also reduce its dimension. MOSES generalises the popular incremental Singular Vale Decomposition (iSVD) to handle thin blocks of data, rather than just vectors. This minor generalisation in part allows us to complement MOSES with a comprehensive statistical analysis, thus providing the first theoretically-sound variant of iSVD, which has been lacking despite the empirical success of this method. This generalisation also enables us to concretely interpret MOSES as an approximate solver for the underlying non-convex optimisation program. We find that MOSES consistently surpasses the state of the art in our numerical experiments with both synthetic and real-world datasets, while being computationally inexpensive.

cs.IT↗

arXiv2019

PCA by Optimisation of Symmetric Functions has no Spurious Local Optima

Principal Component Analysis (PCA) finds the best linear representation of data, and is an indispensable tool in many learning and inference tasks. Classically, principal components of a dataset are interpreted as the directions that preserve most of its "energy", an interpretation that is theoretically underpinned by the celebrated Eckart-Young-Mirsky Theorem. This paper introduces many other ways of performing PCA, with various geometric interpretations, and proves that the corresponding family of non-convex programs have no spurious local optima, while possessing only strict saddle points. These programs therefore loosely behave like convex problems and can be efficiently solved to global optimality, for example, with certain variants of the stochastic gradient descent. Beyond providing new geometric interpretations and enhancing our theoretical understanding of PCA, our findings might pave the way for entirely new approaches to structured dimensionality reduction, such as sparse PCA and nonnegative matrix factorisation. More specifically, we study an unconstrained formulation of PCA using determinant optimisation that might provide an elegant alternative to the deflating scheme commonly used in sparse PCA.

math.OC↗

arXiv2019

Sparse non-negative super-resolution -- simplified and stabilised

The convolution of a discrete measure, $x=\sum_{i=1}^ka_iδ_{t_i}$, with a local window function, $ϕ(s-t)$, is a common model for a measurement device whose resolution is substantially lower than that of the objects being observed. Super-resolution concerns localising the point sources $\{a_i,t_i\}_{i=1}^k$ with an accuracy beyond the essential support of $ϕ(s-t)$, typically from $m$ samples $y(s_j)=\sum_{i=1}^k a_iϕ(s_j-t_i)+η_j$, where $η_j$ indicates an inexactness in the sample value. We consider the setting of $x$ being non-negative and seek to characterise all non-negative measures approximately consistent with the samples. We first show that $x$ is the unique non-negative measure consistent with the samples provided the samples are exact, i.e. $η_j=0$, $m\ge 2k+1$ samples are available, and $ϕ(s-t)$ generates a Chebyshev system. This is independent of how close the sample locations are and {\em does not rely on any regulariser beyond non-negativity}; as such, it extends and clarifies the work by Schiebinger et al. and De Castro et al., who achieve the same results but require a total variation regulariser, which we show is unnecessary. Moreover, we characterise non-negative solutions $\hat{x}$ consistent with the samples within the bound $\sum_{j=1}^mη_j^2\le δ^2$. Any such non-negative measure is within ${\mathcal O}(δ^{1/7})$ of the discrete measure $x$ generating the samples in the generalised Wasserstein distance, converging to one another as $δ$ approaches zero. We also show how to make these general results, for windows that form a Chebyshev system, precise for the case of $ϕ(s-t)$ being a Gaussian window. The main innovation of these results is that non-negativity alone is sufficient to localise point sources beyond the essential sensor resolution.

math.OC↗