arXiv subjects

Fred Roosta

Publications and source records attributed to Fred Roosta.

At least 19 recordsLinked to original sources

arXiv2026

Primal-Dual Inexact Newton-MR for Nonconvex Optimization with Equality Constraints

Optimization problems with nonlinear equality constraints arise throughout science, engineering, and increasingly in machine learning. Prominent methods for solving such problems include sequential quadratic programming and, more broadly, primal-dual Newton methods. Classical analyses of these methods typically rely on strong assumptions, perhaps most notably positive definiteness of the Lagrangian Hessian on the null space of the constraint Jacobian. In practice, this assumption often necessitates strong regularization or the use of a positive definite Hessian surrogate. Moreover, in large-scale settings, solving the primal-dual Newton subproblem exactly is often computationally infeasible. To address these issues, we propose an inexact primal-dual Newton method with an inner solver based on the conjugate residual (CR) method. Exploiting recently established properties of CR, including negative-curvature detection, iterate monotonicity, and descent guarantees, our method handles indefiniteness in the subproblem directly as it arises. Our method thereby avoids detrimental regularization of the Lagrangian Hessian while naturally accommodating inexact solves. We establish worst-case global convergence guarantees and demonstrate strong empirical performance on large-scale, nonconvex problems.

math.OC↗

arXiv2026

Is the Last Layer Sufficient for Uncertainty Quantification?

Epistemic uncertainty quantification (UQ) for deep neural networks (DNNs) is a requirement for safe adoption of AI in mission-critical settings. Several leading methods for UQ linearize DNNs to form Bayesian Generalized Linear Models (GLMs), where epistemic uncertainty is modeled via the predictive posterior distribution. Linearizing around the parameters of the final connected layer of a DNN is a commonly used approximation for reducing the computational burden of such GLMs, though it is often believed to come at the cost of degraded performance. In this work, we compare GLMs arising from full-network and last-layer linearization using both theoretical and empirical approaches. We first employ tools from random matrix theory to conduct a theoretical comparison; this analysis reveals no meaningful improvement in the UQ capabilities of full linearization. Coupled with a large-scale empirical evaluation across a range of modern machine learning tasks, we arrive at the following conclusion: a last-layer approximation yields comparable UQ performance while offering substantially improved computational efficiency.

stat.ML↗

arXiv2026

The Geometric Wall: Manifold Structure Predicts Layerwise Sparse Autoencoder Scaling Laws

Sparse autoencoders (SAEs) operationalise the linear representation hypothesis: they reconstruct model activations as sparse linear combinations of interpretable dictionary atoms, on the implicit assumption that activation space is well approximated by a globally linear structure. Their reconstruction error varies sharply across layers in ways that existing scaling laws, fitted at single layers, do not explain. We argue that this variation is the empirical trace of a geometric mismatch: where the activation manifold is curved and its intrinsic dimension varies across layers, no sparse linear dictionary can match it uniformly, and the SAE's width-sparsity scaling becomes a layer-dependent function of manifold structure rather than a single universal law. We conduct the first cross-layer SAE scaling study, fitting and regressing on 844 residual-stream Gemma Scope SAE checkpoints across 68 layers of Gemma 2 2B and 9B. Stage 1 fits a per-layer scaling-law surface; Stage 2 regresses the fitted parameters and the derived per-layer width exponents on four layerwise geometric summaries. We find that manifold geometry predicts the per-layer width exponent in both models, and that the same regression coefficients learnt on one model predict the other model's per-layer exponents under cross-model transfer, indicating a transferable geometric law. At the showcase layers where richer width grids permit identification of the asymptotic floor, we find that the fitted floor tracks the layerwise geometric ordering: higher curvature and intrinsic dimension correspond to higher floor, consistent with the irreducible second-order residual that any sparse linear approximation of a curved manifold must leave behind. SAEs thus encounter not a finite-resource ceiling but a geometry-dependent wall, set by the manifold they are trying to reconstruct.

cs.LG↗

arXiv2026

Counterfactual Explanations on Robust Perceptual Geodesics

Latent-space optimization methods for counterfactual explanations - framed as minimal semantic perturbations that change model predictions - inherit the ambiguity of Wachter et al.'s objective: the choice of distance metric dictates whether perturbations are meaningful or adversarial. Existing approaches adopt flat or misaligned geometries, leading to off-manifold artifacts, semantic drift, or adversarial collapse. We introduce Perceptual Counterfactual Geodesics (PCG), a method that constructs counterfactuals by tracing geodesics under a perceptually Riemannian metric induced from robust vision features. This geometry aligns with human perception and penalizes brittle directions, enabling smooth, on-manifold, semantically valid transitions. Experiments on three vision datasets show that PCG outperforms baselines and reveals failure modes hidden under standard metrics.

cs.LG↗

arXiv2025

Latent Refinement via Flow Matching for Training-free Linear Inverse Problem Solving

Recent advances in inverse problem solving have increasingly adopted flow priors over diffusion models due to their ability to construct straight probability paths from noise to data, thereby enhancing efficiency in both training and inference. However, current flow-based inverse solvers face two primary limitations: (i) they operate directly in pixel space, which demands heavy computational resources for training and restricts scalability to high-resolution images, and (ii) they employ guidance strategies with prior-agnostic posterior covariances, which can weaken alignment with the generative trajectory and degrade posterior coverage. In this paper, we propose LFlow (Latent Refinement via Flows), a training-free framework for solving linear inverse problems via pretrained latent flow priors. LFlow leverages the efficiency of flow matching to perform ODE sampling in latent space along an optimal path. This latent formulation further allows us to introduce a theoretically grounded posterior covariance, derived from the optimal vector field, enabling effective flow guidance. Experimental results demonstrate that our proposed method outperforms state-of-the-art latent diffusion solvers in reconstruction quality across most tasks. The code will be publicly available at https://github.com/hosseinaskari-cs/LFlow .

cs.CV↗

arXiv2025

Faithful-Newton Framework: Bridging Inner and Outer Solvers for Enhanced Optimization

Newton-type methods enjoy fast local convergence and strong empirical performance, but achieving global guarantees comparable to first-order methods remains challenging. Even for simple strongly convex problems, no straightforward variant of Newton's method matches the global complexity of gradient descent. While more sophisticated variants can improve iteration complexity, they typically require solving difficult subproblems with high per-iteration costs, leading to worse overall complexity. These limitations stem from treating the subproblem as an afterthought, either as a black box, yielding overly complex and impractical formulations, or in isolation, without regard to its role in advancing the optimization of the main objective. By tightening the integration between the inner iterations of the subproblem solvers and the outer iterations of the optimization algorithm, we introduce simple Newton-type variants, called Faithful-Newton framework, which, in a sense, remain faithful to the overall simplicity of classical Newton's method by retaining simple linear system subproblems. The key conceptual difference, however, is that the quality of the subproblem solution is directly assessed based on its effectiveness in reducing optimality, which in turn enables desirable convergence complexities across a variety of settings. Under standard assumptions, we show that our variants, depending on parameter choices, achieve global superlinear convergence, condition-number-independent linear convergence, and/or local quadratic convergence, even when using inexact Newton steps, for strongly convex problems; and competitive iteration complexity for general convex problems. Numerical experiments further demonstrate that our proposed methods perform competitively compared with several alternative Newton-type approaches.

math.NA↗

arXiv2025

Importance Sampling for Nonlinear Models

While norm-based and leverage-score-based methods have been extensively studied for identifying "important" data points in linear models, analogous tools for nonlinear models remain significantly underdeveloped. By introducing the concept of the adjoint operator of a nonlinear map, we address this gap and generalize norm-based and leverage-score-based importance sampling to nonlinear settings. We demonstrate that sampling based on these generalized notions of norm and leverage scores provides approximation guarantees for the underlying nonlinear mapping, similar to linear subspace embeddings. As direct applications, these nonlinear scores not only reduce the computational complexity of training nonlinear models by enabling efficient sampling over large datasets but also offer a novel mechanism for model explainability and outlier detection. Our contributions are supported by both theoretical analyses and experimental results across a variety of supervised learning scenarios.

cs.LG↗

arXiv2025

Determinant Estimation under Memory Constraints and Neural Scaling Laws

Calculating or accurately estimating log-determinants of large positive definite matrices is of fundamental importance in many machine learning tasks. While its cubic computational complexity can already be prohibitive, in modern applications, even storing the matrices themselves can pose a memory bottleneck. To address this, we derive a novel hierarchical algorithm based on block-wise computation of the LDL decomposition for large-scale log-determinant calculation in memory-constrained settings. In extreme cases where matrices are highly ill-conditioned, accurately computing the full matrix itself may be infeasible. This is particularly relevant when considering kernel matrices at scale, including the empirical Neural Tangent Kernel (NTK) of neural networks trained on large datasets. Under the assumption of neural scaling laws in the test error, we show that the ratio of pseudo-determinants satisfies a power-law relationship, allowing us to derive corresponding scaling laws. This enables accurate estimation of NTK log-determinants from a tiny fraction of the full dataset; in our experiments, this results in a $\sim$100,000$\times$ speedup with improved accuracy over competing approximations. Using these techniques, we successfully estimate log-determinants for dense matrices of extreme sizes, which were previously deemed intractable and inaccessible due to their enormous scale and computational demands.

stat.ML↗

arXiv2025

First-ish Order Methods: Hessian-aware Scalings of Gradient Descent

Gradient descent is the primary workhorse for optimizing large-scale problems in machine learning. However, its performance is highly sensitive to the choice of the learning rate. A key limitation of gradient descent is its lack of natural scaling, which often necessitates expensive line searches or heuristic tuning to determine an appropriate step size. In this paper, we address this limitation by incorporating Hessian information to scale the gradient direction. By accounting for the curvature of the function along the gradient, our adaptive, Hessian-aware scaling method ensures a local unit step size guarantee, even in nonconvex settings. Near a local minimum that satisfies the second-order sufficient conditions, our approach achieves linear convergence with a unit step size. We show that our method converges globally under a significantly weaker version of the standard Lipschitz gradient smoothness assumption. Even when Hessian information is inexact, the local unit step size guarantee and global convergence properties remain valid under mild conditions. Finally, we validate our theoretical results empirically on a range of convex and nonconvex machine learning tasks, showcasing the effectiveness of the approach.

math.OC↗

arXiv2025

Uncertainty Quantification with the Empirical Neural Tangent Kernel

While neural networks have demonstrated impressive performance across various tasks, accurately quantifying uncertainty in their predictions is essential to ensure their trustworthiness and enable widespread adoption in critical systems. Several Bayesian uncertainty quantification (UQ) methods exist that are either cheap or reliable, but not both. We propose a post-hoc, sampling-based UQ method for over-parameterized networks at the end of training. Our approach constructs efficient and meaningful deep ensembles by employing a (stochastic) gradient-descent sampling process on appropriately linearized networks. We demonstrate that our method effectively approximates the posterior of a Gaussian process using the empirical Neural Tangent Kernel. Through a series of numerical experiments, we show that our method not only outperforms competing approaches in computational efficiency-often reducing costs by multiple factors-but also maintains state-of-the-art performance across a variety of UQ metrics for both regression and classification tasks.

stat.ML↗

arXiv2024

Complexity Guarantees for Nonconvex Newton-MR Under Inexact Hessian Information

We consider an extension of the Newton-MR algorithm for nonconvex unconstrained optimization to the settings where Hessian information is approximated. Under a particular noise model on the Hessian matrix, we investigate the iteration and operation complexities of this variant to achieve appropriate sub-optimality criteria in several nonconvex settings. We do this by first considering functions that satisfy the (generalized) Polyak-Łojasiewicz condition, a special sub-class of nonconvex functions. We show that, under certain conditions, our algorithm achieves global linear convergence rate. We then consider more general nonconvex settings where the rate to obtain first order sub-optimality is shown to be sub-linear. In all these settings, we show that our algorithm converges regardless of the degree of approximation of the Hessian as well as the accuracy of the solution to the sub-problem. Finally, we compare the performance of our algorithm with several alternatives on a few machine learning problems.

math.OC↗

arXiv2024

Inexact Newton-type Methods for Optimisation with Nonnegativity Constraints

We consider solving large scale nonconvex optimisation problems with nonnegativity constraints. Such problems arise frequently in machine learning, such as nonnegative least-squares, nonnegative matrix factorisation, as well as problems with sparsity-inducing regularisation. In such settings, first-order methods, despite their simplicity, can be prohibitively slow on ill-conditioned problems or become trapped near saddle regions, while most second-order alternatives involve non-trivially challenging subproblems. The two-metric projection framework, initially proposed by Bertsekas (1982), alleviates these issues and achieves the best of both worlds by combining projected gradient steps at the boundary of the feasible region with Newton steps in the interior in such a way that feasibility can be maintained by simple projection onto the nonnegative orthant. We develop extensions of the two-metric projection framework, which by inexactly solving the subproblems as well as employing non-positive curvature directions, are suitable for large scale and nonconvex settings. We obtain state-of-the-art convergence rates for various classes of non-convex problems and demonstrate competitive practical performance on a variety of problems.

math.OC↗

arXiv2024

Manifold Integrated Gradients: Riemannian Geometry for Feature Attribution

In this paper, we dive into the reliability concerns of Integrated Gradients (IG), a prevalent feature attribution method for black-box deep learning models. We particularly address two predominant challenges associated with IG: the generation of noisy feature visualizations for vision models and the vulnerability to adversarial attributional attacks. Our approach involves an adaptation of path-based feature attribution, aligning the path of attribution more closely to the intrinsic geometry of the data manifold. Our experiments utilise deep generative models applied to several real-world image datasets. They demonstrate that IG along the geodesics conforms to the curved geometry of the Riemannian data manifold, generating more perceptually intuitive explanations and, subsequently, substantially increasing robustness to targeted attributional attacks.

cs.LG↗

arXiv2024

Bi-level Guided Diffusion Models for Zero-Shot Medical Imaging Inverse Problems

In the realm of medical imaging, inverse problems aim to infer high-quality images from incomplete, noisy measurements, with the objective of minimizing expenses and risks to patients in clinical settings. The Diffusion Models have recently emerged as a promising approach to such practical challenges, proving particularly useful for the zero-shot inference of images from partially acquired measurements in Magnetic Resonance Imaging (MRI) and Computed Tomography (CT). A central challenge in this approach, however, is how to guide an unconditional prediction to conform to the measurement information. Existing methods rely on deficient projection or inefficient posterior score approximation guidance, which often leads to suboptimal performance. In this paper, we propose \underline{\textbf{B}}i-level \underline{G}uided \underline{D}iffusion \underline{M}odels ({BGDM}), a zero-shot imaging framework that efficiently steers the initial unconditional prediction through a \emph{bi-level} guidance strategy. Specifically, BGDM first approximates an \emph{inner-level} conditional posterior mean as an initial measurement-consistent reference point and then solves an \emph{outer-level} proximal optimization objective to reinforce the measurement consistency. Our experimental findings, using publicly available MRI and CT medical datasets, reveal that BGDM is more effective and efficient compared to the baselines, faithfully generating high-fidelity medical images and substantially reducing hallucinatory artifacts in cases of severe degradation.

eess.IV↗

arXiv2024

Conjugate Direction Methods Under Inconsistent Systems

Since the development of the conjugate gradient (CG) method in 1952 by Hestenes and Stiefel, CG, has become an indispensable tool in computational mathematics for solving positive definite linear systems. On the other hand, the conjugate residual (CR) method, closely related CG and introduced by Stiefel in 1955 for the same settings, remains relatively less known outside the numerical linear algebra community. Since their inception, these methods -- henceforth collectively referred to as conjugate direction methods -- have been extended beyond positive definite to indefinite, albeit consistent, settings. Going one step further, in this paper, we investigate the theoretical and empirical properties of these methods under inconsistent systems. Among other things, we show that small modifications to the original algorithms allow for the pseudo-inverse solution. Furthermore, we show that CR is essentially equivalent to the minimum residual method, proposed by Paige and Saunders in 1975, in such contexts. Lastly, we conduct a series of numerical experiments to shed lights on their numerical stability (or lack thereof) and their performance for inconsistent systems. Surprisingly, we will demonstrate that, unlike CR and contrary to popular belief, CG can exhibit significant numerical instability, bordering on catastrophe in some instances.

math.NA↗

arXiv2023

SALSA: Sequential Approximate Leverage-Score Algorithm with Application in Analyzing Big Time Series Data

We develop a new efficient sequential approximate leverage score algorithm, SALSA, using methods from randomized numerical linear algebra (RandNLA) for large matrices. We demonstrate that, with high probability, the accuracy of SALSA's approximations is within $(1 + O({\varepsilon}))$ of the true leverage scores. In addition, we show that the theoretical computational complexity and numerical accuracy of SALSA surpass existing approximations. These theoretical results are subsequently utilized to develop an efficient algorithm, named LSARMA, for fitting an appropriate ARMA model to large-scale time series data. Our proposed algorithm is, with high probability, guaranteed to find the maximum likelihood estimates of the parameters for the true underlying ARMA model. Furthermore, it has a worst-case running time that significantly improves those of the state-of-the-art alternatives in big data regimes. Empirical results on large-scale data strongly support these theoretical results and underscore the efficacy of our new approach.

stat.ML↗

arXiv2023

Non-Uniform Smoothness for Gradient Descent

The analysis of gradient descent-type methods typically relies on the Lipschitz continuity of the objective gradient. This generally requires an expensive hyperparameter tuning process to appropriately calibrate a stepsize for a given problem. In this work we introduce a local first-order smoothness oracle (LFSO) which generalizes the Lipschitz continuous gradients smoothness condition and is applicable to any twice-differentiable function. We show that this oracle can encode all relevant problem information for tuning stepsizes for a suitably modified gradient descent method and give global and local convergence results. We also show that LFSOs in this modified first-order method can yield global linear convergence rates for non-strongly convex problems with extremely flat minima, and thus improve over the lower bound on rates achievable by general (accelerated) first-order methods.

math.OC↗

arXiv2023

A PAC-Bayesian Perspective on the Interpolating Information Criterion

Deep learning is renowned for its theory-practice gap, whereby principled theory typically fails to provide much beneficial guidance for implementation in practice. This has been highlighted recently by the benign overfitting phenomenon: when neural networks become sufficiently large to interpolate the dataset perfectly, model performance appears to improve with increasing model size, in apparent contradiction with the well-known bias-variance tradeoff. While such phenomena have proven challenging to theoretically study for general models, the recently proposed Interpolating Information Criterion (IIC) provides a valuable theoretical framework to examine performance for overparameterized models. Using the IIC, a PAC-Bayes bound is obtained for a general class of models, characterizing factors which influence generalization performance in the interpolating regime. From the provided bound, we quantify how the test error for overparameterized models achieving effectively zero training error depends on the quality of the implicit regularization imposed by e.g. the combination of model, optimizer, and parameter-initialization scheme; the spectrum of the empirical neural tangent kernel; curvature of the loss landscape; and noise present in the data.

stat.ML↗