Deep neural networks are widely used prediction algorithms whose performance often improves as the number of weights increases, leading to over-parametrization. We consider a two-layered neural network whose first layer is frozen while the last layer is trainable, known as the random feature model. We study over-parametrization in the context of a student-teacher framework by deriving a set of differential equations for the learning dynamics. For any finite ratio of hidden layer size and input dimension, the student cannot generalize perfectly, and we compute the non-zero asymptotic generalization error. Only when the student's hidden layer size is exponentially larger than the input dimension, an approach to perfect generalization is possible.
相關內容
In this paper, we study the following nonlinear matrix decomposition (NMD) problem: given a sparse nonnegative matrix $X$, find a low-rank matrix $\Theta$ such that $X \approx f(\Theta)$, where $f$ is an element-wise nonlinear function. We focus on the case where $f(\cdot) = \max(0, \cdot)$, the rectified unit (ReLU) non-linear activation. We refer to the corresponding problem as ReLU-NMD. We first provide a brief overview of the existing approaches that were developed to tackle ReLU-NMD. Then we introduce two new algorithms: (1) aggressive accelerated NMD (A-NMD) which uses an adaptive Nesterov extrapolation to accelerate an existing algorithm, and (2) three-block NMD (3B-NMD) which parametrizes $\Theta = WH$ and leads to a significant reduction in the computational cost. We also propose an effective initialization strategy based on the nuclear norm as a proxy for the rank function. We illustrate the effectiveness of the proposed algorithms (available on gitlab) on synthetic and real-world data sets.
Dynamical mean-field theory is a powerful physics tool used to analyze the typical behavior of neural networks, where neurons can be recurrently connected, or multiple layers of neurons can be stacked. However, it is not easy for beginners to access the essence of this tool and the underlying physics. Here, we give a pedagogical introduction of this method in a particular example of generic random neural networks, where neurons are randomly and fully connected by correlated synapses and therefore the network exhibits rich emergent collective dynamics. We also review related past and recent important works applying this tool. In addition, a physically transparent and alternative method, namely the dynamical cavity method, is also introduced to derive exactly the same results. The numerical implementation of solving the integro-differential mean-field equations is also detailed, with an illustration of exploring the fluctuation dissipation theorem.
With deep learning deployed in many security-sensitive areas, machine learning security is becoming progressively important. Recent studies demonstrate attackers can exploit system-level techniques exploiting the RowHammer vulnerability of DRAM to deterministically and precisely flip bits in Deep Neural Networks (DNN) model weights to affect inference accuracy. The existing defense mechanisms are software-based, such as weight reconstruction requiring expensive training overhead or performance degradation. On the other hand, generic hardware-based victim-/aggressor-focused mechanisms impose expensive hardware overheads and preserve the spatial connection between victim and aggressor rows. In this paper, we present the first DRAM-based victim-focused defense mechanism tailored for quantized DNNs, named DNN-Defender that leverages the potential of in-DRAM swapping to withstand the targeted bit-flip attacks. Our results indicate that DNN-Defender can deliver a high level of protection downgrading the performance of targeted RowHammer attacks to a random attack level. In addition, the proposed defense has no accuracy drop on CIFAR-10 and ImageNet datasets without requiring any software training or incurring additional hardware overhead.
This research considers the ranking and selection with input uncertainty. The objective is to maximize the posterior probability of correctly selecting the best alternative under a fixed simulation budget, where each alternative is measured by its worst-case performance. We formulate the dynamic simulation budget allocation decision problem as a stochastic control problem under a Bayesian framework. Following the approximate dynamic programming theory, we derive a one-step-ahead dynamic optimal budget allocation policy and prove that this policy achieves consistency and asymptotic optimality. Numerical experiments demonstrate that the proposed procedure can significantly improve performance.
We propose an online learning algorithm for a class of machine learning models under a separable stochastic approximation framework. The essence of our idea lies in the observation that certain parameters in the models are easier to optimize than others. In this paper, we focus on models where some parameters have a linear nature, which is common in machine learning. In one routine of the proposed algorithm, the linear parameters are updated by the recursive least squares (RLS) algorithm, which is equivalent to a stochastic Newton method; then, based on the updated linear parameters, the nonlinear parameters are updated by the stochastic gradient method (SGD). The proposed algorithm can be understood as a stochastic approximation version of block coordinate gradient descent approach in which one part of the parameters is updated by a second-order SGD method while the other part is updated by a first-order SGD. Global convergence of the proposed online algorithm for non-convex cases is established in terms of the expected violation of a first-order optimality condition. Numerical experiments have shown that the proposed method accelerates convergence significantly and produces more robust training and test performance when compared to other popular learning algorithms. Moreover, our algorithm is less sensitive to the learning rate and outperforms the recently proposed slimTrain algorithm. The code has been uploaded to GitHub for validation.
This paper is concerned with the multi-frequency factorization method for imaging the support of a wave-number-dependent source function. It is supposed that the source function is given by the Fourier transform of some time-dependent source with a priori given radiating period. Using the multi-frequency far-field data at a fixed observation direction, we provide a computational criterion for characterizing the smallest strip containing the support and perpendicular to the observation direction. The far-field data from sparse observation directions can be used to recover a $\Theta$-convex polygon of the support. The inversion algorithm is proven valid even with multi-frequency near-field data in three dimensions. The connections to time-dependent inverse source problems are discussed in the near-field case. We also comment on possible extensions to source functions with two disconnected supports. Numerical tests in both two and three dimensions are implemented to show effectiveness and feasibility of the approach. This paper provides numerical analysis for a frequency-domain approach to recover the support of an admissible class of time-dependent sources.
Inverse problems are in many cases solved with optimization techniques. When the underlying model is linear, first-order gradient methods are usually sufficient. With nonlinear models, due to nonconvexity, one must often resort to second-order methods that are computationally more expensive. In this work we aim to approximate a nonlinear model with a linear one and correct the resulting approximation error. We develop a sequential method that iteratively solves a linear inverse problem and updates the approximation error by evaluating it at the new solution. This treatment convexifies the problem and allows us to benefit from established convex optimization methods. We separately consider cases where the approximation is fixed over iterations and where the approximation is adaptive. In the fixed case we show theoretically under what assumptions the sequence converges. In the adaptive case, particularly considering the special case of approximation by first-order Taylor expansion, we show that with certain assumptions the sequence converges to a critical point of the original nonconvex functional. Furthermore, we show that with quadratic objective functions the sequence corresponds to the Gauss-Newton method. Finally, we showcase numerical results superior to the conventional model correction method. We also show, that a fixed approximation can provide competitive results with considerable computational speed-up.
Random smoothing data augmentation is a unique form of regularization that can prevent overfitting by introducing noise to the input data, encouraging the model to learn more generalized features. Despite its success in various applications, there has been a lack of systematic study on the regularization ability of random smoothing. In this paper, we aim to bridge this gap by presenting a framework for random smoothing regularization that can adaptively and effectively learn a wide range of ground truth functions belonging to the classical Sobolev spaces. Specifically, we investigate two underlying function spaces: the Sobolev space of low intrinsic dimension, which includes the Sobolev space in $D$-dimensional Euclidean space or low-dimensional sub-manifolds as special cases, and the mixed smooth Sobolev space with a tensor structure. By using random smoothing regularization as novel convolution-based smoothing kernels, we can attain optimal convergence rates in these cases using a kernel gradient descent algorithm, either with early stopping or weight decay. It is noteworthy that our estimator can adapt to the structural assumptions of the underlying data and avoid the curse of dimensionality. This is achieved through various choices of injected noise distributions such as Gaussian, Laplace, or general polynomial noises, allowing for broad adaptation to the aforementioned structural assumptions of the underlying data. The convergence rate depends only on the effective dimension, which may be significantly smaller than the actual data dimension. We conduct numerical experiments on simulated data to validate our theoretical results.
We study the problem of overcoming exponential sample complexity in differential entropy estimation under Gaussian convolutions. Specifically, we consider the estimation of the differential entropy $h(X+Z)$ via $n$ independently and identically distributed samples of $X$, where $X$ and $Z$ are independent $D$-dimensional random variables with $X$ sub-Gaussian with bounded second moment and $Z\sim\mathcal{N}(0,\sigma^2I_D)$. Under the absolute-error loss, the above problem has a parametric estimation rate of $\frac{c^D}{\sqrt{n}}$, which is exponential in data dimension $D$ and often problematic for applications. We overcome this exponential sample complexity by projecting $X$ to a low-dimensional space via principal component analysis (PCA) before the entropy estimation, and show that the asymptotic error overhead vanishes as the unexplained variance of the PCA vanishes. This implies near-optimal performance for inherently low-dimensional structures embedded in high-dimensional spaces, including hidden-layer outputs of deep neural networks (DNN), which can be used to estimate mutual information (MI) in DNNs. We provide numerical results verifying the performance of our PCA approach on Gaussian and spiral data. We also apply our method to analysis of information flow through neural network layers (c.f. information bottleneck), with results measuring mutual information in a noisy fully connected network and a noisy convolutional neural network (CNN) for MNIST classification.
Knowledge distillation is the technique of compressing a larger neural network, known as the teacher, into a smaller neural network, known as the student, while still trying to maintain the performance of the larger neural network as much as possible. Existing methods of knowledge distillation are mostly applicable for classification tasks. Many of them also require access to the data used to train the teacher model. To address the problem of knowledge distillation for regression tasks under the absence of original training data, previous work has proposed a data-free knowledge distillation method where synthetic data are generated using a generator model trained adversarially against the student model. These synthetic data and their labels predicted by the teacher model are then used to train the student model. In this study, we investigate the behavior of various synthetic data generation methods and propose a new synthetic data generation strategy that directly optimizes for a large but bounded difference between the student and teacher model. Our results on benchmark and case study experiments demonstrate that the proposed strategy allows the student model to learn better and emulate the performance of the teacher model more closely.
注冊地址： 北京市海淀區羊坊店路18號2幢3層301-191