Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userThe Geometry of Over-parameterized Regression and Adversarial Perturbations
58
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Classical regression has a simple geometric description in terms of a projection of the training labels onto the column space of the design matrix. However, for over-parameterized models -- where the number of fit parameters is large enough to perfectly fit the training data -- this picture becomes uninformative. Here, we present an alternative geometric interpretation of regression that applies to both under- and over-parameterized models. Unlike the classical picture which takes place in the space of training labels, our new picture resides in the space of input features. This new feature-based perspective provides a natural geometric interpretation of the double-descent phenomenon in the context of bias and variance, explaining why it can occur even in the absence of label noise. Furthermore, we show that adversarial perturbations -- small perturbations to the input features that result in large changes in label values -- are a generic feature of biased models, arising from the underlying geometry. We demonstrate these ideas by analyzing three minimal models for over-parameterized linear least squares regression: without basis functions (input features equal model features) and with linear or nonlinear basis functions (two-layer neural networks with linear or nonlinear activation functions, respectively).
rate research
Read More
The bias-variance trade-off is a central concept in supervised learning. In classical statistics, increasing the complexity of a model (e.g., number of parameters) reduces bias but also increases variance. Until recently, it was commonly believed that optimal performance is achieved at intermediate model complexities which strike a balance between bias and variance. Modern Deep Learning methods flout this dogma, achieving state-of-the-art performance using over-parameterized models where the number of fit parameters is large enough to perfectly fit the training data. As a result, understanding bias and variance in over-parameterized models has emerged as a fundamental problem in machine learning. Here, we use methods from statistical physics to derive analytic expressions for bias and variance in two minimal models of over-parameterization (linear regression and two-layer neural networks with nonlinear data distributions), allowing us to disentangle properties stemming from the model architecture and random sampling of data. In both models, increasing the number of fit parameters leads to a phase transition where the training error goes to zero and the test error diverges as a result of the variance (while the bias remains finite). Beyond this threshold in the interpolation regime, the training error remains zero while the test error decreases. We also show that in contrast with classical intuition, over-parameterized models can overfit even in the absence of noise and exhibit bias even if the student and teacher models match. We synthesize these results to construct a holistic understanding of generalization error and the bias-variance trade-off in over-parameterized models and relate our results to random matrix theory.
Over-parameterization and adaptive methods have played a crucial role in the success of deep learning in the last decade. The widespread use of over-parameterization has forced us to rethink generalization by bringing forth new phenomena, such as implicit regularization of optimization algorithms and double descent with training progression. A series of recent works have started to shed light on these areas in the quest to understand -- why do neural networks generalize well? The setting of over-parameterized linear regression has provided key insights into understanding this mysterious behavior of neural networks. In this paper, we aim to characterize the performance of adaptive methods in the over-parameterized linear regression setting. First, we focus on two sub-classes of adaptive methods depending on their generalization performance. For the first class of adaptive methods, the parameter vector remains in the span of the data and converges to the minimum norm solution like gradient descent (GD). On the other hand, for the second class of adaptive methods, the gradient rotation caused by the pre-conditioner matrix results in an in-span component of the parameter vector that converges to the minimum norm solution and the out-of-span component that saturates. Our experiments on over-parameterized linear regression and deep neural networks support this theory.
In this manuscript we consider Kernel Ridge Regression (KRR) under the Gaussian design. Exponents for the decay of the excess generalization error of KRR have been reported in various works under the assumption of power-law decay of eigenvalues of the features co-variance. These decays were, however, provided for sizeably different setups, namely in the noiseless case with constant regularization and in the noisy optimally regularized case. Intermediary settings have been left substantially uncharted. In this work, we unify and extend this line of work, providing characterization of all regimes and excess error decay rates that can be observed in terms of the interplay of noise and regularization. In particular, we show the existence of a transition in the noisy setting between the noiseless exponents to its noisy values as the sample complexity is increased. Finally, we illustrate how this crossover can also be observed on real data sets.
We consider whether algorithmic choices in over-parameterized linear matrix factorization introduce implicit regularization. We focus on noiseless matrix sensing over rank-$r$ positive semi-definite (PSD) matrices in $mathbb{R}^{n times n}$, with a sensing mechanism that satisfies restricted isometry properties (RIP). The algorithm we study is emph{factored gradient descent}, where we model the low-rankness and PSD constraints with the factorization $UU^top$, for $U in mathbb{R}^{n times r}$. Surprisingly, recent work argues that the choice of $r leq n$ is not pivotal: even setting $U in mathbb{R}^{n times n}$ is sufficient for factored gradient descent to find the rank-$r$ solution, which suggests that operating over the factors leads to an implicit regularization. In this contribution, we provide a different perspective to the problem of implicit regularization. We show that under certain conditions, the PSD constraint by itself is sufficient to lead to a unique rank-$r$ matrix recovery, without implicit or explicit low-rank regularization. emph{I.e.}, under assumptions, the set of PSD matrices, that are consistent with the observed data, is a singleton, regardless of the algorithm used.
Over-parametrization is an important technique in training neural networks. In both theory and practice, training a larger network allows the optimization algorithm to avoid bad local optimal solutions. In this paper we study a closely related tensor decomposition problem: given an $l$-th order tensor in $(R^d)^{otimes l}$ of rank $r$ (where $rll d$), can variants of gradient descent find a rank $m$ decomposition where $m > r$? We show that in a lazy training regime (similar to the NTK regime for neural networks) one needs at least $m = Omega(d^{l-1})$, while a variant of gradient descent can find an approximate tensor when $m = O^*(r^{2.5l}log d)$. Our results show that gradient descent on over-parametrized objective could go beyond the lazy training regime and utilize certain low-rank structure in the data.
suggested questions
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
