اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدOver-parameterized Adversarial Training: An Analysis Overcoming the Curse of Dimensionality
84
0
0.0
(
0
)
اسأل ChatGPT حول البحث
ﻻ يوجد ملخص باللغة العربية
Adversarial training is a popular method to give neural nets robustness against adversarial perturbations. In practice adversarial training leads to low robust training loss. However, a rigorous explanation for why this happens under natural conditions is still missing. Recently a convergence theory for standard (non-adversarial) supervised training was developed by various groups for {em very overparametrized} nets. It is unclear how to extend these results to adversarial training because of the min-max objective. Recently, a first step towards this direction was made by Gao et al. using tools from online learning, but they require the width of the net to be emph{exponential} in input dimension $d$, and with an unnatural activation function. Our work proves convergence to low robust training loss for emph{polynomial} width instead of exponential, under natural assumptions and with the ReLU activation. Key element of our proof is showing that ReLU networks near initialization can approximate the step function, which may be of independent interest.
قيم البحث
اقرأ أيضاً
We propose a novel GAN framework for non-parametric density estimation with high-dimensional data. This framework is based on a novel density estimator, called the hyperbolic cross density estimator, which enjoys nice convergence properties in the mi
xed Sobolev spaces. As modifications of the usual Sobolev spaces, the mixed Sobolev spaces are more suitable for describing high-dimensional density functions. We prove that, unlike other existing approaches, the proposed GAN framework does not suffer the curse of dimensionality and can achieve the optimal convergence rate of $O_p(n^{-1/2})$, with $n$ data points in an arbitrary fixed dimension. We also study the universality of GANs in terms of the existence of ReLU networks which can approximate the density functions in the mixed Sobolev spaces up to any accuracy level.
The inductive bias of a neural network is largely determined by the architecture and the training algorithm. To achieve good generalization, how to effectively train a neural network is of great importance. We propose a novel orthogonal over-paramete
rized training (OPT) framework that can provably minimize the hyperspherical energy which characterizes the diversity of neurons on a hypersphere. By maintaining the minimum hyperspherical energy during training, OPT can greatly improve the empirical generalization. Specifically, OPT fixes the randomly initialized weights of the neurons and learns an orthogonal transformation that applies to these neurons. We consider multiple ways to learn such an orthogonal transformation, including unrolling orthogonalization algorithms, applying orthogonal parameterization, and designing orthogonality-preserving gradient descent. For better scalability, we propose the stochastic OPT which performs orthogonal transformation stochastically for partial dimensions of neurons. Interestingly, OPT reveals that learning a proper coordinate system for neurons is crucial to generalization. We provide some insights on why OPT yields better generalization. Extensive experiments validate the superiority of OPT over the standard training.
Many modern machine learning applications come with complex and nuanced design goals such as minimizing the worst-case error, satisfying a given precision or recall target, or enforcing group-fairness constraints. Popular techniques for optimizing su
ch non-decomposable objectives reduce the problem into a sequence of cost-sensitive learning tasks, each of which is then solved by re-weighting the training loss with example-specific costs. We point out that the standard approach of re-weighting the loss to incorporate label costs can produce unsatisfactory results when used to train over-parameterized models. As a remedy, we propose new cost-sensitive losses that extend the classical idea of logit adjustment to handle more general cost matrices. Our losses are calibrated, and can be further improved with distilled labels from a teacher model. Through experiments on benchmark image datasets, we showcase the effectiveness of our approach in training ResNet models with common robust and constrained optimization objectives.
The movement of large quantities of data during the training of a Deep Neural Network presents immense challenges for machine learning workloads. To minimize this overhead, especially on the movement and calculation of gradient information, we introd
uce streaming batch principal component analysis as an update algorithm. Streaming batch principal component analysis uses stochastic power iterations to generate a stochastic k-rank approximation of the network gradient. We demonstrate that the low rank updates produced by streaming batch principal component analysis can effectively train convolutional neural networks on a variety of common datasets, with performance comparable to standard mini batch gradient descent. These results can lead to both improvements in the design of application specific integrated circuits for deep learning and in the speed of synchronization of machine learning models trained with data parallelism.
We consider solving the low rank matrix sensing problem with Factorized Gradient Descend (FGD) method when the true rank is unknown and over-specified, which we refer to as over-parameterized matrix sensing. If the ground truth signal $mathbf{X}^* in
mathbb{R}^{d*d}$ is of rank $r$, but we try to recover it using $mathbf{F} mathbf{F}^top$ where $mathbf{F} in mathbb{R}^{d*k}$ and $k>r$, the existing statistical analysis falls short, due to a flat local curvature of the loss function around the global maxima. By decomposing the factorized matrix $mathbf{F}$ into separate column spaces to capture the effect of extra ranks, we show that $|mathbf{F}_t mathbf{F}_t - mathbf{X}^*|_{F}^2$ converges to a statistical error of $tilde{mathcal{O}} ({k d sigma^2/n})$ after $tilde{mathcal{O}}(frac{sigma_{r}}{sigma}sqrt{frac{n}{d}})$ number of iterations where $mathbf{F}_t$ is the output of FGD after $t$ iterations, $sigma^2$ is the variance of the observation noise, $sigma_{r}$ is the $r$-th largest eigenvalue of $mathbf{X}^*$, and $n$ is the number of sample. Our results, therefore, offer a comprehensive picture of the statistical and computational complexity of FGD for the over-parameterized matrix sensing problem.
الأسئلة المقترحة
سجل دخول لتتمكن من نشر تعليقات
التعليقات
جاري جلب التعليقات
سجل دخول لتتمكن من متابعة معايير البحث التي قمت باختيارها
