Gradient flows and randomised thresholding: sparse inversion and classification

• URL: http://arxiv.org/abs/2203.11555v1
• Date: Tue, 22 Mar 2022 09:21:14 GMT
• Title: Gradient flows and randomised thresholding: sparse inversion and classification
• Authors: Jonas Latz
• Abstract summary: Sparse inversion and classification problems are ubiquitous in modern data science and imaging. In classification, we consider, e.g., the sum of a data fidelity term and a non-smooth Ginzburg--Landau energy. Standard (sub)gradient descent methods have shown to be inefficient when approaching such problems.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Sparse inversion and classification problems are ubiquitous in modern data science and imaging. They are often formulated as non-smooth minimisation problems. In sparse inversion, we minimise, e.g., the sum of a data fidelity term and an L1/LASSO regulariser. In classification, we consider, e.g., the sum of a data fidelity term and a non-smooth Ginzburg--Landau energy. Standard (sub)gradient descent methods have shown to be inefficient when approaching such problems. Splitting techniques are much more useful: here, the target function is partitioned into a sum of two subtarget functions -- each of which can be efficiently optimised. Splitting proceeds by performing optimisation steps alternately with respect to each of the two subtarget functions. In this work, we study splitting from a stochastic continuous-time perspective. Indeed, we define a differential inclusion that follows one of the two subtarget function's negative subgradient at each point in time. The choice of the subtarget function is controlled by a binary continuous-time Markov process. The resulting dynamical system is a stochastic approximation of the underlying subgradient flow. We investigate this stochastic approximation for an L1-regularised sparse inversion flow and for a discrete Allen-Cahn equation minimising a Ginzburg--Landau energy. In both cases, we study the longtime behaviour of the stochastic dynamical system and its ability to approximate the underlying subgradient flow at any accuracy. We illustrate our theoretical findings in a simple sparse estimation problem and also in a low-dimensional classification problem.

Related papers

• A Stochastic Approach to Bi-Level Optimization for Hyperparameter Optimization and Meta Learning [74.80956524812714]
  We tackle the general differentiable meta learning problem that is ubiquitous in modern deep learning. These problems are often formalized as Bi-Level optimizations (BLO) We introduce a novel perspective by turning a given BLO problem into a ii optimization, where the inner loss function becomes a smooth distribution, and the outer loss becomes an expected loss over the inner distribution.
  arXiv  Detail & Related papers  (2024-10-14T12:10:06Z)
• An Accelerated Algorithm for Stochastic Bilevel Optimization under Unbounded Smoothness [15.656614304616006]
  This paper investigates a class of bilevel optimization problems where the upper-level function is non- unbounded smoothness and the lower-level problem is strongly convex. These problems have significant applications in data learning, such as text classification using neural networks.
  arXiv  Detail & Related papers  (2024-09-28T02:30:44Z)
• Efficient Low-rank Identification via Accelerated Iteratively Reweighted Nuclear Norm Minimization [8.879403568685499]
  We introduce an adaptive updating strategy for smoothing parameters. This behavior transforms the algorithm into one that effectively solves problems after a few iterations. We prove the global proposed experiment, guaranteeing that every iteration is a critical one.
  arXiv  Detail & Related papers  (2024-06-22T02:37:13Z)
• A Mean-Field Analysis of Neural Stochastic Gradient Descent-Ascent for Functional Minimax Optimization [90.87444114491116]
  This paper studies minimax optimization problems defined over infinite-dimensional function classes of overparametricized two-layer neural networks. We address (i) the convergence of the gradient descent-ascent algorithm and (ii) the representation learning of the neural networks. Results show that the feature representation induced by the neural networks is allowed to deviate from the initial one by the magnitude of $O(alpha-1)$, measured in terms of the Wasserstein distance.
  arXiv  Detail & Related papers  (2024-04-18T16:46:08Z)
• On Learning Gaussian Multi-index Models with Gradient Flow [57.170617397894404]
  We study gradient flow on the multi-index regression problem for high-dimensional Gaussian data. We consider a two-timescale algorithm, whereby the low-dimensional link function is learnt with a non-parametric model infinitely faster than the subspace parametrizing the low-rank projection.
  arXiv  Detail & Related papers  (2023-10-30T17:55:28Z)
• Improved Convergence Rate of Stochastic Gradient Langevin Dynamics with Variance Reduction and its Application to Optimization [50.83356836818667]
  gradient Langevin Dynamics is one of the most fundamental algorithms to solve non-eps optimization problems. In this paper, we show two variants of this kind, namely the Variance Reduced Langevin Dynamics and the Recursive Gradient Langevin Dynamics.
  arXiv  Detail & Related papers  (2022-03-30T11:39:00Z)
• Subgradient methods near active manifolds: saddle point avoidance, local convergence, and asymptotic normality [4.709588811674973]
  We show that aiming and subgradient approximation fully expose the smooth substructure of the problem. We prove these properties hold for a wide class of problems, including cone reducible/decomposable functions and generic semialgebraic problems. The normality results appear to be new even in the most classical setting.
  arXiv  Detail & Related papers  (2021-08-26T15:02:16Z)
• A Retrospective Approximation Approach for Smooth Stochastic Optimization [0.2867517731896504]
  Gradient (SG) is the defactorandom iterative technique to solve optimization (SO) problems with a smooth (non-fimation) objective $imation.
  arXiv  Detail & Related papers  (2021-03-07T16:29:36Z)
• Exploiting Higher Order Smoothness in Derivative-free Optimization and Continuous Bandits [99.70167985955352]
  We study the problem of zero-order optimization of a strongly convex function. We consider a randomized approximation of the projected gradient descent algorithm. Our results imply that the zero-order algorithm is nearly optimal in terms of sample complexity and the problem parameters.
  arXiv  Detail & Related papers  (2020-06-14T10:42:23Z)
• Explicit Regularization of Stochastic Gradient Methods through Duality [9.131027490864938]
  We propose randomized Dykstra-style algorithms based on randomized dual coordinate ascent. For accelerated coordinate descent, we obtain a new algorithm that has better convergence properties than existing gradient methods in the interpolating regime.
  arXiv  Detail & Related papers  (2020-03-30T20:44:56Z)
• Towards Better Understanding of Adaptive Gradient Algorithms in Generative Adversarial Nets [71.05306664267832]
  Adaptive algorithms perform gradient updates using the history of gradients and are ubiquitous in training deep neural networks. In this paper we analyze a variant of OptimisticOA algorithm for nonconcave minmax problems. Our experiments show that adaptive GAN non-adaptive gradient algorithms can be observed empirically.
  arXiv  Detail & Related papers  (2019-12-26T22:10:10Z)

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.