Riemannian Stochastic Optimization for Sufficient Dimension Reduction

• URL: http://arxiv.org/abs/2606.00413v1
• Date: Fri, 29 May 2026 23:06:44 GMT
• Title: Riemannian Stochastic Optimization for Sufficient Dimension Reduction
• Abstract summary: Sufficient dimension (SDR) makes high-dimensional regression tractable by projecting the cos onto a low-dimensional subspace.<n>The resulting algorithm, SMAVE, combines the empirical reduction on the Stiefel manifold with closed-form gradient.
• Score: 4.340040784481499
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Sufficient dimension reduction (SDR) makes high-dimensional regression tractable by projecting the covariates onto a low-dimensional subspace that preserves the conditional mean of the response. Existing gradient-based estimators either operate in the ambient space and suffer from the curse of dimensionality, or localize in the reduced space at a per-outer-iteration cost at least quadratic in the sample size. We show that minimizers of the population Minimum Average Variance Estimation (MAVE) risk approximate the same Grassmannian target as the Outer Product of Gradients (OPG), and recast the empirical criterion as a smooth maximization on the Stiefel manifold with closed-form Riemannian gradient. The resulting algorithm, SMAVE, combines sparse projected-space nearest-neighbor localization with Riemannian stochastic gradient ascent. A simplified version comes with almost-sure convergence and a non-asymptotic rate matching the standard non-convex stochastic first-order scaling. Empirically, SMAVE matches or improves on RMAVE's synthetic subspace recovery at moderate-to-high ambient dimension, and on four real datasets it uniformly improves over OPG and is competitive with or outperforms RMAVE at orders of magnitude lower runtime.

Related papers

• The Role of Gradient Modification in Heavy-Tailed Nonconvex Stochastic Min-Max Optimization [50.59470683896735]
  min-max optimization has attracted increasing attention due to its applications in modern machine learning.<n>We provide a comprehensive theoretical of gradient descent (SGDA)<n>We develop new clipping-free algorithms, i.e., Stoc-GDAM and Stoc-TRmax, and they both can achieve the optimal dependence on the target accuracy without using gradient clipping.
  arXiv  Detail & Related papers  (2026-09-05T12:49:28Z)
• 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)
• Euclidean Distance Matrix Completion via Asymmetric Projected Gradient Descent [25.846262685970164]
  This paper proposes and analyzes a gradient-type algorithm based on Burer-Monteiro factorization.<n>It reconstructs the point set configuration from partial Euclidean distance measurements.
  arXiv  Detail & Related papers  (2025-04-28T07:13:23Z)
• Riemannian Federated Learning via Averaging Gradient Streams [10.533809913888591]
  Federated learning (FL) as a distributed learning paradigm has a significant advantage in addressing large-scale machine learning tasks.<n>This paper presents and analyzes a new efficient server aggregation -- averaging gradient streams, called RFedAGS.<n>We show that the proposed RFedAGS has global convergence and sublinear convergence rate under decaying step sizes cases.
  arXiv  Detail & Related papers  (2024-09-11T12:28:42Z)
• Last-Iterate Convergence of Adaptive Riemannian Gradient Descent for Equilibrium Computation [52.73824786627612]
  This paper establishes new convergence results for textitgeodesic strongly monotone games.<n>Our key result shows that RGD attains last-iterate linear convergence in a textitgeometry-agnostic fashion.<n>Overall, this paper presents the first geometry-agnostic last-iterate convergence analysis for games beyond the Euclidean settings.
  arXiv  Detail & Related papers  (2023-06-29T01:20:44Z)
• Sampling from Gaussian Process Posteriors using Stochastic Gradient Descent [43.097493761380186]
  gradient algorithms are an efficient method of approximately solving linear systems. We show that gradient descent produces accurate predictions, even in cases where it does not converge quickly to the optimum. Experimentally, gradient descent achieves state-of-the-art performance on sufficiently large-scale or ill-conditioned regression tasks.
  arXiv  Detail & Related papers  (2023-06-20T15:07:37Z)
• Parameter-free projected gradient descent [0.0]
  We consider the problem of minimizing a convex function over a closed convex set, with Projected Gradient Descent (PGD) We propose a fully parameter-free version of AdaGrad, which is adaptive to the distance between the initialization and the optimum, and to the sum of the square norm of the subgradients. Our algorithm is able to handle projection steps, does not involve restarts, reweighing along the trajectory or additional evaluations compared to the classical PGD.
  arXiv  Detail & Related papers  (2023-05-31T07:22:44Z)
• Proximal Subgradient Norm Minimization of ISTA and FISTA [8.261388753972234]
  We show that the squared proximal subgradient norm for the class of iterative shrinkage-thresholding algorithms converges at an inverse square rate. We also show that the squared proximal subgradient norm for the class of faster iterative shrinkage-thresholding algorithms (FISTA) is accelerated to convergence at an inverse cubic rate.
  arXiv  Detail & Related papers  (2022-11-03T06:50:19Z)
• Adaptive Zeroth-Order Optimisation of Nonconvex Composite Objectives [1.7640556247739623]
  We analyze algorithms for zeroth-order entropy composite objectives, focusing on dependence on dimensionality. This is achieved by exploiting low dimensional structure of the decision set using the mirror descent method with an estimation alike function. To improve the gradient, we replace the classic sampling method based on Rademacher and show that the mini-batch method copes with non-Eucli geometry.
  arXiv  Detail & Related papers  (2022-08-09T07:36:25Z)
• First-Order Algorithms for Min-Max Optimization in Geodesic Metric Spaces [93.35384756718868]
  min-max algorithms have been analyzed in the Euclidean setting. We prove that the extraiteient (RCEG) method corrected lastrate convergence at a linear rate.
  arXiv  Detail & Related papers  (2022-06-04T18:53:44Z)
• Faster One-Sample Stochastic Conditional Gradient Method for Composite Convex Minimization [61.26619639722804]
  We propose a conditional gradient method (CGM) for minimizing convex finite-sum objectives formed as a sum of smooth and non-smooth terms. The proposed method, equipped with an average gradient (SAG) estimator, requires only one sample per iteration. Nevertheless, it guarantees fast convergence rates on par with more sophisticated variance reduction techniques.
  arXiv  Detail & Related papers  (2022-02-26T19:10:48Z)
• A Dimensionality Reduction Method for Finding Least Favorable Priors with a Focus on Bregman Divergence [108.28566246421742]
  This paper develops a dimensionality reduction method that allows us to move the optimization to a finite-dimensional setting with an explicit bound on the dimension. In order to make progress on the problem, we restrict ourselves to Bayesian risks induced by a relatively large class of loss functions, namely Bregman divergences.
  arXiv  Detail & Related papers  (2022-02-23T16:22:28Z)
• Nonconvex Stochastic Scaled-Gradient Descent and Generalized Eigenvector Problems [98.34292831923335]
  Motivated by the problem of online correlation analysis, we propose the emphStochastic Scaled-Gradient Descent (SSD) algorithm. We bring these ideas together in an application to online correlation analysis, deriving for the first time an optimal one-time-scale algorithm with an explicit rate of local convergence to normality.
  arXiv  Detail & Related papers  (2021-12-29T18:46:52Z)
• Unified Convergence Analysis for Adaptive Optimization with Moving Average Estimator [75.05106948314956]
  We show that an increasing large momentum parameter for the first-order moment is sufficient for adaptive scaling.<n>We also give insights for increasing the momentum in a stagewise manner in accordance with stagewise decreasing step size.
  arXiv  Detail & Related papers  (2021-04-30T08:50:24Z)
• Sparse Representations of Positive Functions via First and Second-Order Pseudo-Mirror Descent [15.340540198612823]
  We consider expected risk problems when the range of the estimator is required to be nonnegative. We develop first and second-order variants of approximation mirror descent employing emphpseudo-gradients. Experiments demonstrate favorable performance on ingeneous Process intensity estimation in practice.
  arXiv  Detail & Related papers  (2020-11-13T21:54:28Z)
• Zeroth-Order Regularized Optimization (ZORO): Approximately Sparse Gradients and Adaptive Sampling [29.600975900977343]
  We propose a new $textbfZ$eroth-$textbfO$rder $textbfR$egularized $textbfO$ptimization method, dubbed ZORO. When the underlying gradient is approximately sparse at an iterate, ZORO needs very few objective function evaluations to obtain a new iterate that decreases the objective function.
  arXiv  Detail & Related papers  (2020-03-29T11:01:17Z)

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.