The remarkable practical success of deep learning has revealed some major surprises from a theoretical perspective. In particular, simple gradient methods easily find near-optimal solutions to non-convex optimization problems, and despite giving a near-perfect fit to training data without any explicit effort to control model complexity, these methods exhibit excellent predictive accuracy. We conjecture that specific principles underlie these phenomena: that overparametrization allows gradient methods to find interpolating solutions, that these methods implicitly impose regularization, and that overparametrization leads to benign overfitting, that is, accurate predictions despite overfitting training data. In this article, we survey recent progress in statistical learning theory that provides examples illustrating these principles in simpler settings. We first review classical uniform convergence results and why they fall short of explaining aspects of the behaviour of deep learning methods. We give examples of implicit regularization in simple settings, where gradient methods lead to minimal norm functions that perfectly fit the training data. Then we review prediction methods that exhibit benign overfitting, focusing on regression problems with quadratic loss. For these methods, we can decompose the prediction rule into a simple component that is useful for prediction and a spiky component that is useful for overfitting but, in a favourable setting, does not harm prediction accuracy. We focus specifically on the linear regime for neural networks, where the network can be approximated by a linear model. In this regime, we demonstrate the success of gradient flow, and we consider benign overfitting with two-layer networks, giving an exact asymptotic analysis that precisely demonstrates the impact of overparametrization. We conclude by highlighting the key challenges that arise in extending these insights to realistic deep learning settings.
more »
« less
Bad Global Minima Exist and SGD Can Reach Them
Several works have aimed to explain why overparameterized neural networks generalize well when trained by Stochastic Gradient Descent (SGD). The consensus explanation that has emerged credits the randomized nature of SGD for the bias of the training process towards low-complexity models and, thus, for implicit regularization. We take a careful look at this explanation in the context of image classification with common deep neural network architectures. We find that if we do not regularize explicitly, then SGD can be easily made to converge to poorly-generalizing, high-complexity models: all it takes is to first train on a random labeling on the data, before switching to properly training with the correct labels. In contrast, we find that in the presence of explicit regularization, pretraining with random labels has no detrimental effect on SGD. We believe that our results give evidence that explicit regularization plays a far more important role in the success of overparameterized neural networks than what has been understood until now. Specifically, by penalizing complicated models independently of their fit to the data, regularization affects training dynamics also far away from optima, making simple models that fit the data well discoverable by local methods, such as SGD.
more »
« less
- Award ID(s):
- 1844951
- NSF-PAR ID:
- 10227030
- Date Published:
- Journal Name:
- Advances in neural information processing systems
- ISSN:
- 1049-5258
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Modern neural networks are typically trained in an over-parameterized regime where the parameters of the model far exceed the size of the training data. Due to over-parameterization these neural networks in principle have the capacity to (over)fit any set of labels including pure noise. Despite this high fitting capacity, somewhat paradoxically, neural network models trained via first-order methods continue to predict well on yet unseen test data. In this paper we take a step towards demystifying this phenomena. In particular we show that first order methods such as gradient descent are provably robust to noise/corruption on a constant fraction of the labels despite over-parametrization under a rich dataset model. In particular: i) First, we show that in the first few iterations where the updates are still in the vicinity of the initialization these algorithms only fit to the correct labels essentially ignoring the noisy labels. ii) Secondly, we prove that to start to overfit to the noisy labels these algorithms must stray rather far from from the initial model which can only occur after many more iterations. Together, these show that gradient descent with early stopping is provably robust to label noise and shed light on empirical robustness of deep networks as well as commonly adopted heuristics to prevent overfitting.more » « less
-
more » « less
While deep learning is successful in a number of applications, it is not yet well understood theoretically. A theoretical characterization of deep learning should answer questions about their approximation power, the dynamics of optimization, and good out-of-sample performance, despite overparameterization and the absence of explicit regularization. We review our recent results toward this goal. In approximation theory both shallow and deep networks are known to approximate any continuous functions at an exponential cost. However, we proved that for certain types of compositional functions, deep networks of the convolutional type (even without weight sharing) can avoid the curse of dimensionality. In characterizing minimization of the empirical exponential loss we consider the gradient flow of the weight directions rather than the weights themselves, since the relevant function underlying classification corresponds to normalized networks. The dynamics of normalized weights turn out to be equivalent to those of the constrained problem of minimizing the loss subject to a unit norm constraint. In particular, the dynamics of typical gradient descent have the same critical points as the constrained problem. Thus there is implicit regularization in training deep networks under exponential-type loss functions during gradient flow. As a consequence, the critical points correspond to minimum norm infima of the loss. This result is especially relevant because it has been recently shown that, for overparameterized models, selection of a minimum norm solution optimizes cross-validation leave-one-out stability and thereby the expected error. Thus our results imply that gradient descent in deep networks minimize the expected error.
-
The risk of overparameterized models, in particular deep neural networks, is often double- descent shaped as a function of the model size. Recently, it was shown that the risk as a function of the early-stopping time can also be double-descent shaped, and this behavior can be explained as a super-position of bias-variance tradeoffs. In this paper, we show that the risk of explicit L2-regularized models can exhibit double descent behavior as a function of the regularization strength, both in theory and practice. We find that for linear regression, a double descent shaped risk is caused by a superposition of bias-variance tradeoffs corresponding to different parts of the model and can be mitigated by scaling the regularization strength of each part appropriately. Motivated by this result, we study a two-layer neural network and show that double descent can be eliminated by adjusting the regularization strengths for the first and second layer. Lastly, we study a 5-layer CNN and ResNet-18 trained on CIFAR-10 with label noise, and CIFAR-100 without label noise, and demonstrate that all exhibit double descent behavior as a function of the regularization strength.more » « less
-
Continuing advances in neural interfaces have enabled simultaneous monitoring of spiking activity from hundreds to thousands of neurons. To interpret these large-scale data, several methods have been proposed to infer latent dynamic structure from high-dimensional datasets. One recent line of work uses recurrent neural networks in a sequential autoencoder (SAE) framework to uncover dynamics. SAEs are an appealing option for modeling nonlinear dynamical systems, and enable a precise link between neural activity and behavior on a single-trial basis. However, the very large parameter count and complexity of SAEs relative to other models has caused concern that SAEs may only perform well on very large training sets. We hypothesized that with a method to systematically optimize hyperparameters (HPs), SAEs might perform well even in cases of limited training data. Such a breakthrough would greatly extend their applicability. However, we find that SAEs applied to spiking neural data are prone to a particular form of overfitting that cannot be detected using standard validation metrics, which prevents standard HP searches. We develop and test two potential solutions: an alternate validation method (“sample validation”) and a novel regularization method (“coordinated dropout”). These innovations prevent overfitting quite effectively, and allow us to test whether SAEs can achieve good performance on limited data through large-scale HP optimization. When applied to data from motor cortex recorded while monkeys made reaches in various directions, large-scale HP optimization allowed SAEs to better maintain performance for small dataset sizes. Our results should greatly extend the applicability of SAEs in extracting latent dynamics from sparse, multidimensional data, such as neural population spiking activity.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Conference Paper:
- The DOI is not currently available.
