Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSharp Bounds on the Approximation Rates, Metric Entropy, and $n$-widths of Shallow Neural Networks
116
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
In this article, we study approximation properties of the variation spaces corresponding to shallow neural networks with a variety of activation functions. We introduce two main tools for estimating the metric entropy, approximation rates, and $n$-widths of these spaces. First, we introduce the notion of a smoothly parameterized dictionary and give upper bounds on the non-linear approximation rates, metric entropy and $n$-widths of their absolute convex hull. The upper bounds depend upon the order of smoothness of the parameterization. This result is applied to dictionaries of ridge functions corresponding to shallow neural networks, and they improve upon existing results in many cases. Next, we provide a method for lower bounding the metric entropy and $n$-widths of variation spaces which contain certain classes of ridge functions. This result gives sharp lower bounds on the $L^2$-approximation rates, metric entropy, and $n$-widths for variation spaces corresponding to neural networks with a range of important activation functions, including ReLU$^k$, sigmoidal activation functions with bounded variation, and the B-spline activation functions.
rate research
Read More
We consider the approximation rates of shallow neural networks with respect to the variation norm. Upper bounds on these rates have been established for sigmoidal and ReLU activation functions, but it has remained an important open problem whether these rates are sharp. In this article, we provide a solution to this problem by proving sharp lower bounds on the approximation rates for shallow neural networks, which are obtained by lower bounding the $L^2$-metric entropy of the convex hull of the neural network basis functions. In addition, our methods also give sharp lower bounds on the Kolmogorov $n$-widths of this convex hull, which show that the variation spaces corresponding to shallow neural networks cannot be efficiently approximated by linear methods. These lower bounds apply to both sigmoidal activation functions with bounded variation and to activation functions which are a power of the ReLU. Our results also quantify how much stronger the Barron spectral norm is than the variation norm and, combined with previous results, give the asymptotics of the $L^infty$-metric entropy up to logarithmic factors in the case of the ReLU activation function.
We consider shallow (single hidden layer) neural networks and characterize their performance when trained with stochastic gradient descent as the number of hidden units $N$ and gradient descent steps grow to infinity. In particular, we investigate the effect of different scaling schemes, which lead to different normalizations of the neural network, on the networks statistical output, closing the gap between the $1/sqrt{N}$ and the mean-field $1/N$ normalization. We develop an asymptotic expansion for the neural networks statistical output pointwise with respect to the scaling parameter as the number of hidden units grows to infinity. Based on this expansion, we demonstrate mathematically that to leading order in $N$, there is no bias-variance trade off, in that both bias and variance (both explicitly characterized) decrease as the number of hidden units increases and time grows. In addition, we show that to leading order in $N$, the variance of the neural networks statistical output decays as the implied normalization by the scaling parameter approaches the mean field normalization. Numerical studies on the MNIST and CIFAR10 datasets show that test and train accuracy monotonically improve as the neural networks normalization gets closer to the mean field normalization.
We consider the variation space corresponding to a dictionary of functions in $L^2(Omega)$ and present the basic theory of approximation in these spaces. Specifically, we compare the definition based on integral representations with the definition in terms of convex hulls. We show that in many cases, including the dictionaries corresponding to shallow ReLU$^k$ networks and a dictionary of decaying Fourier modes, that the two definitions coincide. We also give a partial characterization of the variation space for shallow ReLU$^k$ networks and show that the variation space with respect to the dictionary of decaying Fourier modes corresponds to the Barron spectral space.
In this paper, we extend the recently proposed multivariate rank energy distance, based on the theory of optimal transport, for statistical testing of distributional similarity, to soft rank energy distance. Being differentiable, this in turn allows us to extend the rank energy to a subspace robust rank energy distance, dubbed Projected soft-Rank Energy distance, which can be computed via optimization over the Stiefel manifold. We show via experiments that using projected soft rank energy one can trade-off the detection power vs the false alarm via projections onto an appropriately selected low dimensional subspace. We also show the utility of the proposed tests on unsupervised change point detection in multivariate time series data. All codes are publicly available at the link provided in the experiment section.
We consider learning two layer neural networks using stochastic gradient descent. The mean-field description of this learning dynamics approximates the evolution of the network weights by an evolution in the space of probability distributions in $R^D$ (where $D$ is the number of parameters associated to each neuron). This evolution can be defined through a partial differential equation or, equivalently, as the gradient flow in the Wasserstein space of probability distributions. Earlier work shows that (under some regularity assumptions), the mean field description is accurate as soon as the number of hidden units is much larger than the dimension $D$. In this paper we establish stronger and more general approximation guarantees. First of all, we show that the number of hidden units only needs to be larger than a quantity dependent on the regularity properties of the data, and independent of the dimensions. Next, we generalize this analysis to the case of unbounded activation functions, which was not covered by earlier bounds. We extend our results to noisy stochastic gradient descent. Finally, we show that kernel ridge regression can be recovered as a special limit of the mean field analysis.
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
