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
- 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) - The Butterfly Effect: Neural Network Training Trajectories Are Highly Sensitive to Initial Conditions [51.68215326304272]
We show that even small perturbations reliably cause otherwise identical training trajectories to diverge-an effect that diminishes rapidly over training time.<n>Our findings provide insights into neural network training stability, with practical implications for fine-tuning, model merging, and diversity of model ensembles.
arXiv Detail & Related papers (2025-06-16T08:35:16Z) - 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) - Unraveling the Hessian: A Key to Smooth Convergence in Loss Function Landscapes [0.0]
We theoretically analyze the convergence of the loss landscape in a fully connected neural network and derive upper bounds for the difference in loss function values when adding a new object to the sample.
Our empirical study confirms these results on various datasets, demonstrating the convergence of the loss function surface for image classification tasks.
arXiv Detail & Related papers (2024-09-18T14:04:15Z) - Semantic Ensemble Loss and Latent Refinement for High-Fidelity Neural Image Compression [58.618625678054826]
This study presents an enhanced neural compression method designed for optimal visual fidelity.
We have trained our model with a sophisticated semantic ensemble loss, integrating Charbonnier loss, perceptual loss, style loss, and a non-binary adversarial loss.
Our empirical findings demonstrate that this approach significantly improves the statistical fidelity of neural image compression.
arXiv Detail & Related papers (2024-01-25T08:11:27Z) - 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) - 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) - Compressed Regression over Adaptive Networks [58.79251288443156]
We derive the performance achievable by a network of distributed agents that solve, adaptively and in the presence of communication constraints, a regression problem.
We devise an optimized allocation strategy where the parameters necessary for the optimization can be learned online by the agents.
arXiv Detail & Related papers (2023-04-07T13:41:08Z) - Linear Classification of Neural Manifolds with Correlated Variability [3.3946853660795893]
We show how correlations between object representations affect the capacity, a measure of linear separability.
We then apply our results to accurately estimate the capacity of deep network data.
arXiv Detail & Related papers (2022-11-27T23:01:43Z) - 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) - Fundamental Limits and Tradeoffs in Invariant Representation Learning [99.2368462915979]
Many machine learning applications involve learning representations that achieve two competing goals.
Minimax game-theoretic formulation represents a fundamental tradeoff between accuracy and invariance.
We provide an information-theoretic analysis of this general and important problem under both classification and regression settings.
arXiv Detail & Related papers (2020-12-19T15:24:04Z) - Optimizing Mode Connectivity via Neuron Alignment [84.26606622400423]
Empirically, the local minima of loss functions can be connected by a learned curve in model space along which the loss remains nearly constant.
We propose a more general framework to investigate effect of symmetry on landscape connectivity by accounting for the weight permutations of networks being connected.
arXiv Detail & Related papers (2020-09-05T02:25:23Z) - Tangent Space Sensitivity and Distribution of Linear Regions in ReLU
Networks [0.0]
We consider adversarial stability in the tangent space and suggest tangent sensitivity in order to characterize stability.
We derive several easily computable bounds and empirical measures for feed-forward fully connected ReLU networks.
Our experiments suggest that even simple bounds and measures are associated with the empirical generalization gap.
arXiv Detail & Related papers (2020-06-11T20:02:51Z) - 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.