Towards understanding epoch-wise double descent in two-layer linear neural networks
- URL: http://arxiv.org/abs/2407.09845v2
- Date: Thu, 12 Sep 2024 08:45:41 GMT
- Title: Towards understanding epoch-wise double descent in two-layer linear neural networks
- Authors: Amanda Olmin, Fredrik Lindsten,
- Abstract summary: We study epoch-wise double descent in two-layer linear neural networks.
We identify additional factors of epoch-wise double descent emerging with the extra model layer.
This opens up for further questions regarding unidentified factors of epoch-wise double descent for truly deep models.
- Score: 11.210628847081097
- License: http://creativecommons.org/licenses/by/4.0/
- Abstract: Epoch-wise double descent is the phenomenon where generalisation performance improves beyond the point of overfitting, resulting in a generalisation curve exhibiting two descents under the course of learning. Understanding the mechanisms driving this behaviour is crucial not only for understanding the generalisation behaviour of machine learning models in general, but also for employing conventional selection methods, such as the use of early stopping to mitigate overfitting. While we ultimately want to draw conclusions of more complex models, such as deep neural networks, a majority of theoretical results regarding the underlying cause of epoch-wise double descent are based on simple models, such as standard linear regression. In this paper, to take a step towards more complex models in theoretical analysis, we study epoch-wise double descent in two-layer linear neural networks. First, we derive a gradient flow for the linear two-layer model, that bridges the learning dynamics of the standard linear regression model, and the linear two-layer diagonal network with quadratic weights. Second, we identify additional factors of epoch-wise double descent emerging with the extra model layer, by deriving necessary conditions for the generalisation error to follow a double descent pattern. While epoch-wise double descent in linear regression has been attributed to differences in input variance, in the two-layer model, also the singular values of the input-output covariance matrix play an important role. This opens up for further questions regarding unidentified factors of epoch-wise double descent for truly deep models.
Related papers
- Understanding the Double Descent Phenomenon in Deep Learning [49.1574468325115]
This tutorial sets the classical statistical learning framework and introduces the double descent phenomenon.
By looking at a number of examples, section 2 introduces inductive biases that appear to have a key role in double descent by selecting.
section 3 explores the double descent with two linear models, and gives other points of view from recent related works.
arXiv Detail & Related papers (2024-03-15T16:51:24Z) - 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) - Analysis of Interpolating Regression Models and the Double Descent
Phenomenon [3.883460584034765]
It is commonly assumed that models which interpolate noisy training data are poor to generalize.
The best models obtained are overparametrized and the testing error exhibits the double descent behavior as the model order increases.
We derive a result based on the behavior of the smallest singular value of the regression matrix that explains the peak location and the double descent shape of the testing error as a function of model order.
arXiv Detail & Related papers (2023-04-17T09:44:33Z) - Multi-scale Feature Learning Dynamics: Insights for Double Descent [71.91871020059857]
We study the phenomenon of "double descent" of the generalization error.
We find that double descent can be attributed to distinct features being learned at different scales.
arXiv Detail & Related papers (2021-12-06T18:17:08Z) - 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 Bayesian Perspective on Training Speed and Model Selection [51.15664724311443]
We show that a measure of a model's training speed can be used to estimate its marginal likelihood.
We verify our results in model selection tasks for linear models and for the infinite-width limit of deep neural networks.
Our results suggest a promising new direction towards explaining why neural networks trained with gradient descent are biased towards functions that generalize well.
arXiv Detail & Related papers (2020-10-27T17:56:14Z) - The Neural Tangent Kernel in High Dimensions: Triple Descent and a
Multi-Scale Theory of Generalization [34.235007566913396]
Modern deep learning models employ considerably more parameters than required to fit the training data. Whereas conventional statistical wisdom suggests such models should drastically overfit, in practice these models generalize remarkably well.
An emerging paradigm for describing this unexpected behavior is in terms of a emphdouble descent curve.
We provide a precise high-dimensional analysis of generalization with the Neural Tangent Kernel, which characterizes the behavior of wide neural networks with gradient descent.
arXiv Detail & Related papers (2020-08-15T20:55:40Z) - Early Stopping in Deep Networks: Double Descent and How to Eliminate it [30.61588337557343]
We show that epoch-wise double descent arises because different parts of the network are learned at different epochs.
We study two standard convolutional networks empirically and show that eliminating epoch-wise double descent through adjusting stepsizes of different layers improves the early stopping performance significantly.
arXiv Detail & Related papers (2020-07-20T13:43:33Z) - Kernel and Rich Regimes in Overparametrized Models [69.40899443842443]
We show that gradient descent on overparametrized multilayer networks can induce rich implicit biases that are not RKHS norms.
We also demonstrate this transition empirically for more complex matrix factorization models and multilayer non-linear networks.
arXiv Detail & Related papers (2020-02-20T15:43:02Z)
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.