Gradient-Based Non-Linear Inverse Learning

• URL: http://arxiv.org/abs/2412.16794v1
• Date: Sat, 21 Dec 2024 22:38:17 GMT
• Title: Gradient-Based Non-Linear Inverse Learning
• Authors: Abhishake, Nicole Mücke, Tapio Helin,
• Abstract summary: We study statistical inverse learning in the context of nonlinear inverse problems under random design. We employ gradient descent (GD) and descent gradient (SGD) with mini-batching, both using constant step sizes. Our analysis derives convergence rates for both algorithms under classical a priori assumptions on the smoothness of the target function.
• Score: 2.6149030745627644
• License:
• Abstract: We study statistical inverse learning in the context of nonlinear inverse problems under random design. Specifically, we address a class of nonlinear problems by employing gradient descent (GD) and stochastic gradient descent (SGD) with mini-batching, both using constant step sizes. Our analysis derives convergence rates for both algorithms under classical a priori assumptions on the smoothness of the target function. These assumptions are expressed in terms of the integral operator associated with the tangent kernel, as well as through a bound on the effective dimension. Additionally, we establish stopping times that yield minimax-optimal convergence rates within the classical reproducing kernel Hilbert space (RKHS) framework. These results demonstrate the efficacy of GD and SGD in achieving optimal rates for nonlinear inverse problems in random design.

Related papers

• Directional Smoothness and Gradient Methods: Convergence and Adaptivity [16.779513676120096]
  We develop new sub-optimality bounds for gradient descent that depend on the conditioning of the objective along the path of optimization. Key to our proofs is directional smoothness, a measure of gradient variation that we use to develop upper-bounds on the objective. We prove that the Polyak step-size and normalized GD obtain fast, path-dependent rates despite using no knowledge of the directional smoothness.
  arXiv  Detail & Related papers  (2024-03-06T22:24:05Z)
• Implicit regularization in AI meets generalized hardness of approximation in optimization -- Sharp results for diagonal linear networks [0.0]
  We show sharp results for the implicit regularization imposed by the gradient flow of Diagonal Linear Networks. We link this to the phenomenon of phase transitions in generalized hardness of approximation. Non-sharpness of our results would imply that the GHA phenomenon would not occur for the basis pursuit optimization problem.
  arXiv  Detail & Related papers  (2023-07-13T13:27:51Z)
• 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)
• Experimental Design for Linear Functionals in Reproducing Kernel Hilbert Spaces [102.08678737900541]
  We provide algorithms for constructing bias-aware designs for linear functionals. We derive non-asymptotic confidence sets for fixed and adaptive designs under sub-Gaussian noise.
  arXiv  Detail & Related papers  (2022-05-26T20:56:25Z)
• 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)
• Differentiable Annealed Importance Sampling and the Perils of Gradient Noise [68.44523807580438]
  Annealed importance sampling (AIS) and related algorithms are highly effective tools for marginal likelihood estimation. Differentiability is a desirable property as it would admit the possibility of optimizing marginal likelihood as an objective. We propose a differentiable algorithm by abandoning Metropolis-Hastings steps, which further unlocks mini-batch computation.
  arXiv  Detail & Related papers  (2021-07-21T17:10:14Z)
• Benign Overfitting of Constant-Stepsize SGD for Linear Regression [122.70478935214128]
  inductive biases are central in preventing overfitting empirically. This work considers this issue in arguably the most basic setting: constant-stepsize SGD for linear regression. We reflect on a number of notable differences between the algorithmic regularization afforded by (unregularized) SGD in comparison to ordinary least squares.
  arXiv  Detail & Related papers  (2021-03-23T17:15:53Z)
• On Learning Rates and Schr\"odinger Operators [105.32118775014015]
  We present a general theoretical analysis of the effect of the learning rate. We find that the learning rate tends to zero for a broad non- neural class functions.
  arXiv  Detail & Related papers  (2020-04-15T09:52:37Z)
• Non-asymptotic bounds for stochastic optimization with biased noisy gradient oracles [8.655294504286635]
  We introduce biased gradient oracles to capture a setting where the function measurements have an estimation error. Our proposed oracles are in practical contexts, for instance, risk measure estimation from a batch of independent and identically distributed simulation.
  arXiv  Detail & Related papers  (2020-02-26T12:53:04Z)
• GradientDICE: Rethinking Generalized Offline Estimation of Stationary Values [75.17074235764757]
  We present GradientDICE for estimating the density ratio between the state distribution of the target policy and the sampling distribution. GenDICE is the state-of-the-art for estimating such density ratios.
  arXiv  Detail & Related papers  (2020-01-29T22:10:11Z)

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.