A randomized algorithm to solve reduced rank operator regression

• URL: http://arxiv.org/abs/2312.17348v1
• Date: Thu, 28 Dec 2023 20:29:59 GMT
• Title: A randomized algorithm to solve reduced rank operator regression
• Authors: Giacomo Turri, Vladimir Kostic, Pietro Novelli, Massimiliano Pontil
• Abstract summary: We present and analyze an algorithm designed for addressing vector-valued regression problems involving possibly infinite-dimensional input and output spaces. The algorithm is a randomized adaptation of reduced rank regression, a technique to optimally learn a low-rank vector-valued function.
• Score: 27.513149895229837
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We present and analyze an algorithm designed for addressing vector-valued regression problems involving possibly infinite-dimensional input and output spaces. The algorithm is a randomized adaptation of reduced rank regression, a technique to optimally learn a low-rank vector-valued function (i.e. an operator) between sampled data via regularized empirical risk minimization with rank constraints. We propose Gaussian sketching techniques both for the primal and dual optimization objectives, yielding Randomized Reduced Rank Regression (R4) estimators that are efficient and accurate. For each of our R4 algorithms we prove that the resulting regularized empirical risk is, in expectation w.r.t. randomness of a sketch, arbitrarily close to the optimal value when hyper-parameteres are properly tuned. Numerical expreriments illustrate the tightness of our bounds and show advantages in two distinct scenarios: (i) solving a vector-valued regression problem using synthetic and large-scale neuroscience datasets, and (ii) regressing the Koopman operator of a nonlinear stochastic dynamical system.

Related papers

• Accelerated zero-order SGD under high-order smoothness and overparameterized regime [79.85163929026146]
  We present a novel gradient-free algorithm to solve convex optimization problems. Such problems are encountered in medicine, physics, and machine learning. We provide convergence guarantees for the proposed algorithm under both types of noise.
  arXiv  Detail & Related papers  (2024-11-21T10:26:17Z)
• Optimal Rates for Vector-Valued Spectral Regularization Learning Algorithms [28.046728466038022]
  We study theoretical properties of a broad class of regularized algorithms with vector-valued output. We rigorously confirm the so-called saturation effect for ridge regression with vector-valued output. We present the upper bound for the finite sample risk general vector-valued spectral algorithms.
  arXiv  Detail & Related papers  (2024-05-23T16:45:52Z)
• Stochastic Optimization for Non-convex Problem with Inexact Hessian Matrix, Gradient, and Function [99.31457740916815]
  Trust-region (TR) and adaptive regularization using cubics have proven to have some very appealing theoretical properties. We show that TR and ARC methods can simultaneously provide inexact computations of the Hessian, gradient, and function values.
  arXiv  Detail & Related papers  (2023-10-18T10:29:58Z)
• Retire: Robust Expectile Regression in High Dimensions [3.9391041278203978]
  Penalized quantile and expectile regression methods offer useful tools to detect heteroscedasticity in high-dimensional data. We propose and study (penalized) robust expectile regression (retire) We show that the proposed procedure can be efficiently solved by a semismooth Newton coordinate descent algorithm.
  arXiv  Detail & Related papers  (2022-12-11T18:03:12Z)
• 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)
• Alternating minimization for generalized rank one matrix sensing: Sharp predictions from a random initialization [5.900674344455754]
  We show a technique for estimating properties of a rank random matrix with i.i.d. We show sharp convergence guarantees exact recovery in a single step. Our analysis also exposes several other properties of this problem.
  arXiv  Detail & Related papers  (2022-07-20T05:31:05Z)
• Momentum Accelerates the Convergence of Stochastic AUPRC Maximization [80.8226518642952]
  We study optimization of areas under precision-recall curves (AUPRC), which is widely used for imbalanced tasks. We develop novel momentum methods with a better iteration of $O (1/epsilon4)$ for finding an $epsilon$stationary solution. We also design a novel family of adaptive methods with the same complexity of $O (1/epsilon4)$, which enjoy faster convergence in practice.
  arXiv  Detail & Related papers  (2021-07-02T16:21:52Z)
• Sparse Bayesian Learning via Stepwise Regression [1.2691047660244335]
  We propose a coordinate ascent algorithm for SBL termed Relevance Matching Pursuit (RMP) As its noise variance parameter goes to zero, RMP exhibits a surprising connection to Stepwise Regression. We derive novel guarantees for Stepwise Regression algorithms, which also shed light on RMP.
  arXiv  Detail & Related papers  (2021-06-11T00:20:27Z)
• High Probability Complexity Bounds for Non-Smooth Stochastic Optimization with Heavy-Tailed Noise [51.31435087414348]
  It is essential to theoretically guarantee that algorithms provide small objective residual with high probability. Existing methods for non-smooth convex optimization have complexity bounds with dependence on confidence level. We propose novel stepsize rules for two methods with gradient clipping.
  arXiv  Detail & Related papers  (2021-06-10T17:54:21Z)
• Fast OSCAR and OWL Regression via Safe Screening Rules [97.28167655721766]
  Ordered $L_1$ (OWL) regularized regression is a new regression analysis for high-dimensional sparse learning. Proximal gradient methods are used as standard approaches to solve OWL regression. We propose the first safe screening rule for OWL regression by exploring the order of the primal solution with the unknown order structure.
  arXiv  Detail & Related papers  (2020-06-29T23:35:53Z)
• Effective Dimension Adaptive Sketching Methods for Faster Regularized Least-Squares Optimization [56.05635751529922]
  We propose a new randomized algorithm for solving L2-regularized least-squares problems based on sketching. We consider two of the most popular random embeddings, namely, Gaussian embeddings and the Subsampled Randomized Hadamard Transform (SRHT)
  arXiv  Detail & Related papers  (2020-06-10T15:00:09Z)

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.