Complex Dynamics in Simple Neural Networks: Understanding Gradient Flow in Phase Retrieval

• URL: http://arxiv.org/abs/2006.06997v1
• Date: Fri, 12 Jun 2020 08:21:12 GMT
• Title: Complex Dynamics in Simple Neural Networks: Understanding Gradient Flow in Phase Retrieval
• Abstract summary: We show that gradient-based algorithms can find good minima instead of being trapped in spurious ones. By numerical experiments we show that in this regime the gradient flow algorithm is not trapped; it drifts away from the critical points along the unstable direction and succeeds in finding the global minimum.
• Score: 31.993312235435916
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Despite the widespread use of gradient-based algorithms for optimizing high-dimensional non-convex functions, understanding their ability of finding good minima instead of being trapped in spurious ones remains to a large extent an open problem. Here we focus on gradient flow dynamics for phase retrieval from random measurements. When the ratio of the number of measurements over the input dimension is small the dynamics remains trapped in spurious minima with large basins of attraction. We find analytically that above a critical ratio those critical points become unstable developing a negative direction toward the signal. By numerical experiments we show that in this regime the gradient flow algorithm is not trapped; it drifts away from the spurious critical points along the unstable direction and succeeds in finding the global minimum. Using tools from statistical physics we characterize this phenomenon, which is related to a BBP-type transition in the Hessian of the spurious minima.

Related papers

• Stable Minima of ReLU Neural Networks Suffer from the Curse of Dimensionality: The Neural Shattering Phenomenon [22.29950158991071]
  We study the implicit bias of flatness / low (loss) curvature and its effects on generalization in ReLU networks.<n>We show that while flatness does imply generalization, the resulting rates of convergence necessarily deteriorate exponentially as the input dimension grows.
  arXiv  Detail & Related papers  (2025-06-25T19:10:03Z)
• 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)
• From Zero to Hero: How local curvature at artless initial conditions leads away from bad minima [9.50832466973301]
  We focus on the phase retrieval problem as a case study for complex loss landscapes. We show that a transition in the spectrum takes place: the direction is lost and the system gets trapped in bad minima. Our analysis sheds light on this new mechanism that facilitates gradient descent dynamics in finite dimensions.
  arXiv  Detail & Related papers  (2024-03-04T19:12:13Z)
• On the Convergence of Gradient Descent for Large Learning Rates [55.33626480243135]
  We show that convergence is impossible when a fixed step size is used. We provide a proof of this in the case of linear neural networks with a squared loss. We also prove the impossibility of convergence for more general losses without requiring strong assumptions such as Lipschitz continuity for the gradient.
  arXiv  Detail & Related papers  (2024-02-20T16:01:42Z)
• Law of Balance and Stationary Distribution of Stochastic Gradient Descent [11.937085301750288]
  We prove that the minibatch noise of gradient descent (SGD) regularizes the solution towards a balanced solution whenever the loss function contains a rescaling symmetry. We then derive the stationary distribution of gradient flow for a diagonal linear network with arbitrary depth and width. These phenomena are shown to exist uniquely in deep networks, implying a fundamental difference between deep and shallow models.
  arXiv  Detail & Related papers  (2023-08-13T03:13:03Z)
• The Implicit Bias of Minima Stability in Multivariate Shallow ReLU Networks [53.95175206863992]
  We study the type of solutions to which gradient descent converges when used to train a single hidden-layer multivariate ReLU network with the quadratic loss. We prove that although shallow ReLU networks are universal approximators, stable shallow networks are not.
  arXiv  Detail & Related papers  (2023-06-30T09:17:39Z)
• On the ISS Property of the Gradient Flow for Single Hidden-Layer Neural Networks with Linear Activations [0.0]
  We investigate the effects of overfitting on the robustness of gradient-descent training when subject to uncertainty on the gradient estimation. We show that the general overparametrized formulation introduces a set of spurious equilibria which lay outside the set where the loss function is minimized.
  arXiv  Detail & Related papers  (2023-05-17T02:26:34Z)
• Implicit Bias in Leaky ReLU Networks Trained on High-Dimensional Data [63.34506218832164]
  In this work, we investigate the implicit bias of gradient flow and gradient descent in two-layer fully-connected neural networks with ReLU activations. For gradient flow, we leverage recent work on the implicit bias for homogeneous neural networks to show that leakyally, gradient flow produces a neural network with rank at most two. For gradient descent, provided the random variance is small enough, we show that a single step of gradient descent suffices to drastically reduce the rank of the network, and that the rank remains small throughout training.
  arXiv  Detail & Related papers  (2022-10-13T15:09:54Z)
• Gradient flow dynamics of shallow ReLU networks for square loss and orthogonal inputs [19.401271427657395]
  The training of neural networks by gradient descent methods is a cornerstone of the deep learning revolution. This article presents the gradient flow dynamics of one neural network for the mean squared error at small initialisation.
  arXiv  Detail & Related papers  (2022-06-02T09:01:25Z)
• 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)
• Continuous vs. Discrete Optimization of Deep Neural Networks [15.508460240818575]
  We show that over deep neural networks with homogeneous activations, gradient flow trajectories enjoy favorable curvature. This finding allows us to translate an analysis of gradient flow over deep linear neural networks into a guarantee that gradient descent efficiently converges to global minimum. We hypothesize that the theory of gradient flows will be central to unraveling mysteries behind deep learning.
  arXiv  Detail & Related papers  (2021-07-14T10:59:57Z)
• Analytic Signal Phase in $N-D$ by Linear Symmetry Tensor--fingerprint modeling [69.35569554213679]
  We show that the Analytic Signal phase, and its gradient have a hitherto unstudied discontinuity in $2-D $ and higher dimensions. This shortcoming can result in severe artifacts whereas the problem does not exist in $1-D $ signals. We suggest the use of Linear Symmetry phase, relying on more than one set of Gabor filters, but with a negligible computational add-on.
  arXiv  Detail & Related papers  (2020-05-16T21:17:26Z)

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.