Optimality of Sub-network Laplace Approximations: New Results and Methods

• URL: http://arxiv.org/abs/2605.09075v1
• Date: Sat, 09 May 2026 17:37:06 GMT
• Title: Optimality of Sub-network Laplace Approximations: New Results and Methods
• Abstract summary: Sub-network Laplace approximations construct surrogates by restricting attention to a small subset of parameters.<n>We show that sub-network Laplace methods systematically underestimate the predictive variance of the full Laplace posterior.<n>We propose two principled, analytically grounded sub-network Hessian approximations.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Although the Laplace approximation offers a simple route to uncertainty quantification in deep neural networks, its reliance on inverting large Hessian matrices has motivated a range of computationally feasible low-dimensional or sparse approximations. A prominent class of such methods - sub-network Laplace approximations, constructs surrogates by restricting attention to a small subset of parameters. Existing approaches in this family typically rely on diagonal, layer-wise, or other architectural heuristics for subset selection, which ignore cross-parameter interactions and lack formal optimality guarantees. In this paper, we provide a rigorous theoretical analysis of the sub-network Laplace paradigm. We prove that all sub-network Laplace methods systematically underestimate the predictive variance of the full Laplace posterior, and that this bias decreases monotonically as the retained sub-matrix expands. Leveraging this insight, we propose two principled, analytically grounded sub-network Hessian approximations: \textit{Gradient-Laplace} selects parameters with the largest average squared gradients of the model output with respect to the parameters over a reference dataset; while \textit{Greedy-Laplace} iteratively refines this selection by accounting for off-diagonal interactions in the precision matrix. We establish theoretical guarantees characterizing their optimality properties and show that Gradient-Laplace provably outperforms existing heuristic approaches. Extensive numerical studies across diverse settings indicate that these methods perform strongly relative to existing benchmarks.

Related papers

• Inversion-Free Natural Gradient Descent on Riemannian Manifolds [5.556612138252781]
  This paper proposes an inversion-free natural gradient method for probability distributions with convex parameters.<n>The manifold setting offers several advantages: one can implicitly enforce parameter constraints.<n>A limited-memory variant of the algorithm with sub-quadratic storage complexity is proposed.
  arXiv  Detail & Related papers  (2026-04-03T11:08:59Z)
• Revisiting Zeroth-Order Optimization: Minimum-Variance Two-Point Estimators and Directionally Aligned Perturbations [57.179679246370114]
  We identify the distribution of random perturbations that minimizes the estimator's variance as the perturbation stepsize tends to zero.<n>Our findings reveal that such desired perturbations can align directionally with the true gradient, instead of maintaining a fixed length.
  arXiv  Detail & Related papers  (2025-10-22T19:06:39Z)
• On the Optimal Construction of Unbiased Gradient Estimators for Zeroth-Order Optimization [57.179679246370114]
  A potential limitation of existing methods is the bias inherent in most perturbation estimators unless a stepsize is proposed.<n>We propose a novel family of unbiased gradient scaling estimators that eliminate bias while maintaining favorable construction.
  arXiv  Detail & Related papers  (2025-10-22T18:25:43Z)
• Beyond Coordinates: Meta-Equivariance in Statistical Inference [0.0]
  Optimal statistical decisions should transcend the language used to describe them.<n>How do we guarantee that the choice of coordinates does not subtly dictate the solution?<n>We first analyse the optimal combination of twoally normal estimators under a strictly convex trace-AMSE risk.
  arXiv  Detail & Related papers  (2025-04-14T19:40:39Z)
• Regularized Projection Matrix Approximation with Applications to Community Detection [1.3761665705201904]
  This paper introduces a regularized projection matrix approximation framework designed to recover cluster information from the affinity matrix. We investigate three distinct penalty functions, each specifically tailored to address bounded, positive, and sparse scenarios. Numerical experiments conducted on both synthetic and real-world datasets reveal that our regularized projection matrix approximation approach significantly outperforms state-of-the-art methods in clustering performance.
  arXiv  Detail & Related papers  (2024-05-26T15:18:22Z)
• A Unified Theory of Stochastic Proximal Point Methods without Smoothness [52.30944052987393]
  Proximal point methods have attracted considerable interest owing to their numerical stability and robustness against imperfect tuning. This paper presents a comprehensive analysis of a broad range of variations of the proximal point method (SPPM)
  arXiv  Detail & Related papers  (2024-05-24T21:09:19Z)
• A unified error analysis for randomized low-rank approximation with application to data assimilation [39.47298454012977]
  We propose a unified framework for the analysis of the low-rank approximation error in Frobenius norm for centered and non-standard matrices.<n>Under minimal assumptions, we derive accurate bounds in expectation and probability.<n>Our bounds have clear interpretations that enable us to derive properties and motivate practical choices.
  arXiv  Detail & Related papers  (2024-05-08T04:51:56Z)
• Riemannian stochastic optimization methods avoid strict saddle points [68.80251170757647]
  We show that policies under study avoid strict saddle points / submanifolds with probability 1. This result provides an important sanity check as it shows that, almost always, the limit state of an algorithm can only be a local minimizer.
  arXiv  Detail & Related papers  (2023-11-04T11:12:24Z)
• Stochastic Mirror Descent for Large-Scale Sparse Recovery [13.500750042707407]
  We discuss an application of quadratic Approximation to statistical estimation of high-dimensional sparse parameters. We show that the proposed algorithm attains the optimal convergence of the estimation error under weak assumptions on the regressor distribution.
  arXiv  Detail & Related papers  (2022-10-23T23:23:23Z)
• Pathwise Conditioning of Gaussian Processes [72.61885354624604]
  Conventional approaches for simulating Gaussian process posteriors view samples as draws from marginal distributions of process values at finite sets of input locations. This distribution-centric characterization leads to generative strategies that scale cubically in the size of the desired random vector. We show how this pathwise interpretation of conditioning gives rise to a general family of approximations that lend themselves to efficiently sampling Gaussian process posteriors.
  arXiv  Detail & Related papers  (2020-11-08T17:09:37Z)
• Understanding Implicit Regularization in Over-Parameterized Single Index Model [55.41685740015095]
  We design regularization-free algorithms for the high-dimensional single index model. We provide theoretical guarantees for the induced implicit regularization phenomenon.
  arXiv  Detail & Related papers  (2020-07-16T13:27:47Z)
• Support recovery and sup-norm convergence rates for sparse pivotal estimation [79.13844065776928]
  In high dimensional sparse regression, pivotal estimators are estimators for which the optimal regularization parameter is independent of the noise level. We show minimax sup-norm convergence rates for non smoothed and smoothed, single task and multitask square-root Lasso-type estimators.
  arXiv  Detail & Related papers  (2020-01-15T16:11:04Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.