A simple connection from loss flatness to compressed representations in neural networks

• URL: http://arxiv.org/abs/2310.01770v3
• Date: Tue, 11 Jun 2024 21:11:28 GMT
• Title: A simple connection from loss flatness to compressed representations in neural networks
• Authors: Shirui Chen, Stefano Recanatesi, Eric Shea-Brown,
• Abstract summary: We show that in the final phase of learning in deep neural networks, the compression of the manifold of neural representations correlates with the flatness of the loss around the minima explored by SGD. Our work builds upon the linear stability insight by Ma and Ying, deriving inequalities between various compression metrics and quantities involving sharpness.
• Score: 3.5502600490147196
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: The generalization capacity of deep neural networks has been studied in a variety of ways, including at least two distinct categories of approaches: one based on the shape of the loss landscape in parameter space, and the other based on the structure of the representation manifold in feature space (that is, in the space of unit activities). Although these two approaches are related, they are rarely studied together explicitly. Here, we present an analysis that bridges this gap. We show that in the final phase of learning in deep neural networks, the compression of the manifold of neural representations correlates with the flatness of the loss around the minima explored by SGD. This correlation is predicted by a relatively simple mathematical relationship: a flatter loss corresponds to a lower upper bound on the compression metrics of neural representations. Our work builds upon the linear stability insight by Ma and Ying, deriving inequalities between various compression metrics and quantities involving sharpness. Empirically, our derived inequality predicts a consistently positive correlation between representation compression and loss sharpness in multiple experimental settings. Overall, we advance a dual perspective on generalization in neural networks in both parameter and feature space.

Related papers

• Topological obstruction to the training of shallow ReLU neural networks [0.0]
  We study the interplay between the geometry of the loss landscape and the optimization trajectories of simple neural networks. This paper reveals the presence of topological obstruction in the loss landscape of shallow ReLU neural networks trained using gradient flow.
  arXiv  Detail & Related papers  (2024-10-18T19:17:48Z)
• A U-turn on Double Descent: Rethinking Parameter Counting in Statistical Learning [68.76846801719095]
  We show that double descent appears exactly when and where it occurs, and that its location is not inherently tied to the threshold p=n. This provides a resolution to tensions between double descent and statistical intuition.
  arXiv  Detail & Related papers  (2023-10-29T12:05:39Z)
• Addressing caveats of neural persistence with deep graph persistence [54.424983583720675]
  We find that the variance of network weights and spatial concentration of large weights are the main factors that impact neural persistence. We propose an extension of the filtration underlying neural persistence to the whole neural network instead of single layers. This yields our deep graph persistence measure, which implicitly incorporates persistent paths through the network and alleviates variance-related issues.
  arXiv  Detail & Related papers  (2023-07-20T13:34:11Z)
• The Multiscale Structure of Neural Network Loss Functions: The Effect on Optimization and Origin [12.092361450994318]
  We study the structure of neural network loss functions and its implication on optimization in a region beyond the reach of good quadratic approximation. We show that training data with different magnitudes give rise to different scales of the loss function, producing subquadratic growth or multiple separate scales.
  arXiv  Detail & Related papers  (2022-04-24T17:34:12Z)
• Phenomenology of Double Descent in Finite-Width Neural Networks [29.119232922018732]
  Double descent delineates the behaviour of models depending on the regime they belong to. We use influence functions to derive suitable expressions of the population loss and its lower bound. Building on our analysis, we investigate how the loss function affects double descent.
  arXiv  Detail & Related papers  (2022-03-14T17:39:49Z)
• Deep Networks on Toroids: Removing Symmetries Reveals the Structure of Flat Regions in the Landscape Geometry [3.712728573432119]
  We develop a standardized parameterization in which all symmetries are removed, resulting in a toroidal topology. We derive a meaningful notion of the flatness of minimizers and of the geodesic paths connecting them. We also find that minimizers found by variants of gradient descent can be connected by zero-error paths with a single bend.
  arXiv  Detail & Related papers  (2022-02-07T09:57:54Z)
• Mean-field Analysis of Piecewise Linear Solutions for Wide ReLU Networks [83.58049517083138]
  We consider a two-layer ReLU network trained via gradient descent. We show that SGD is biased towards a simple solution. We also provide empirical evidence that knots at locations distinct from the data points might occur.
  arXiv  Detail & Related papers  (2021-11-03T15:14:20Z)
• The Interplay Between Implicit Bias and Benign Overfitting in Two-Layer Linear Networks [51.1848572349154]
  neural network models that perfectly fit noisy data can generalize well to unseen test data. We consider interpolating two-layer linear neural networks trained with gradient flow on the squared loss and derive bounds on the excess risk.
  arXiv  Detail & Related papers  (2021-08-25T22:01:01Z)
• A neural anisotropic view of underspecification in deep learning [60.119023683371736]
  We show that the way neural networks handle the underspecification of problems is highly dependent on the data representation. Our results highlight that understanding the architectural inductive bias in deep learning is fundamental to address the fairness, robustness, and generalization of these systems.
  arXiv  Detail & Related papers  (2021-04-29T14:31:09Z)
• Understanding Generalization in Deep Learning via Tensor Methods [53.808840694241]
  We advance the understanding of the relations between the network's architecture and its generalizability from the compression perspective. We propose a series of intuitive, data-dependent and easily-measurable properties that tightly characterize the compressibility and generalizability of neural networks.
  arXiv  Detail & Related papers  (2020-01-14T22:26:57Z)
• Avoiding Spurious Local Minima in Deep Quadratic Networks [0.0]
  We characterize the landscape of the mean squared nonlinear error for networks with neural activation functions. We prove that deepized neural networks with quadratic activations benefit from similar landscape properties.
  arXiv  Detail & Related papers  (2019-12-31T22:31:11Z)

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.