Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSharp Lower Bounds on the Approximation Rate of Shallow Neural Networks
158
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
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.
rate research
Read More
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.
Consider the problem: given the data pair $(mathbf{x}, mathbf{y})$ drawn from a population with $f_*(x) = mathbf{E}[mathbf{y} | mathbf{x} = x]$, specify a neural network model and run gradient flow on the weights over time until reaching any stationarity. How does $f_t$, the function computed by the neural network at time $t$, relate to $f_*$, in terms of approximation and representation? What are the provable benefits of the adaptive representation by neural networks compared to the pre-specified fixed basis representation in the classical nonparametric literature? We answer the above questions via a dynamic reproducing kernel Hilbert space (RKHS) approach indexed by the training process of neural networks. Firstly, we show that when reaching any local stationarity, gradient flow learns an adaptive RKHS representation and performs the global least-squares projection onto the adaptive RKHS, simultaneously. Secondly, we prove that as the RKHS is data-adaptive and task-specific, the residual for $f_*$ lies in a subspace that is potentially much smaller than the orthogonal complement of the RKHS. The result formalizes the representation and approximation benefits of neural networks. Lastly, we show that the neural network function computed by gradient flow converges to the kernel ridgeless regression with an adaptive kernel, in the limit of vanishing regularization. The adaptive kernel viewpoint provides new angles of studying the approximation, representation, generalization, and optimization advantages of neural networks.
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.
This paper considers the growth in the length of one-dimensional trajectories as they are passed through deep ReLU neural networks, which, among other things, is one measure of the expressivity of deep networks. We generalise existing results, providing an alternative, simpler method for lower bounding expected trajectory growth through random networks, for a more general class of weights distributions, including sparsely connected networks. We illustrate this approach by deriving bounds for sparse-Gaussian, sparse-uniform, and sparse-discrete-valued random nets. We prove that trajectory growth can remain exponential in depth with these new distributions, including their sparse variants, with the sparsity parameter appearing in the base of the exponent.
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
