arXiv subjects

Jonathan W. Siegel

Publications and source records attributed to Jonathan W. Siegel.

35 records · Page 2Linked to original sources

arXiv2024

Efficient Structure-Informed Featurization and Property Prediction of Ordered, Dilute, and Random Atomic Structures

Structure-informed materials informatics is a rapidly evolving discipline of materials science relying on the featurization of atomic structures or configurations to construct vector, voxel, graph, graphlet, and other representations useful for machine learning prediction of properties, fingerprinting, and generative design. This work discusses how current featurizers typically perform redundant calculations and how their efficiency could be improved by considering (1) fundamentals of crystallographic (orbits) equivalency to optimize ordered cases and (2) representation-dependent equivalency to optimize cases of dilute, doped, and defect structures with broken symmetry. It also discusses and contrasts ways of (3) approximating random solid solutions occupying arbitrary lattices under such representations. Efficiency improvements discussed in this work were implemented within pySIPFENN or python toolset for Structure-Informed Property and Feature Engineering with Neural Networks developed by authors since 2019 and shown to increase performance from 2 to 10 times for typical inputs. Throughout this work, the authors explicitly discuss how these advances can be applied to different kinds of similar tools in the community.

cond-mat.mtrl-sci↗

arXiv2024

Optimal Approximation Rates for Deep ReLU Neural Networks on Sobolev and Besov Spaces

Let $Ω= [0,1]^d$ be the unit cube in $\mathbb{R}^d$. We study the problem of how efficiently, in terms of the number of parameters, deep neural networks with the ReLU activation function can approximate functions in the Sobolev spaces $W^s(L_q(Ω))$ and Besov spaces $B^s_r(L_q(Ω))$, with error measured in the $L_p(Ω)$ norm. This problem is important when studying the application of neural networks in a variety of fields, including scientific computing and signal processing, and has previously been solved only when $p=q=\infty$. Our contribution is to provide a complete solution for all $1\leq p,q\leq \infty$ and $s > 0$ for which the corresponding Sobolev or Besov space compactly embeds into $L_p$. The key technical tool is a novel bit-extraction technique which gives an optimal encoding of sparse vectors. This enables us to obtain sharp upper bounds in the non-linear regime where $p > q$. We also provide a novel method for deriving $L_p$-approximation lower bounds based upon VC-dimension when $p < \infty$. Our results show that very deep ReLU networks significantly outperform classical methods of approximation in terms of the number of parameters, but that this comes at the cost of parameters which are not encodable.

stat.ML↗

arXiv2024

Sharp Bounds on the Approximation Rates, Metric Entropy, and $n$-widths of Shallow Neural Networks

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$^k$ activation functions and sigmoidal activation functions with bounded variation.

stat.ML↗

arXiv2023

A qualitative difference between gradient flows of convex functions in finite- and infinite-dimensional Hilbert spaces

We consider gradient flow/gradient descent and heavy ball/accelerated gradient descent optimization for convex objective functions. In the gradient flow case, we prove the following: 1. If $f$ does not have a minimizer, the convergence $f(x_t)\to \inf f$ can be arbitrarily slow. 2. If $f$ does have a minimizer, the excess energy $f(x_t) - \inf f$ is integrable/summable in time. In particular, $f(x_t) - \inf f = o(1/t)$ as $t\to\infty$. 3. In Hilbert spaces, this is optimal: $f(x_t) - \inf f$ can decay to $0$ as slowly as any given function which is monotone decreasing and integrable at $\infty$, even for a fixed quadratic objective. 4. In finite dimension (or more generally, for all gradient flow curves of finite length), this is not optimal: We prove that there are convex monotone decreasing integrable functions $g(t)$ which decrease to zero slower than $f(x_t)-\inf f$ for the gradient flow of any convex function on $\mathbb R^d$. For instance, we show that any gradient flow $x_t$ of a convex function $f$ in finite dimension satisfies $\liminf_{t\to\infty} \big(t\cdot \log^2(t)\cdot \big\{f(x_t) -\inf f\big\}\big)=0$. This improves on the commonly reported $O(1/t)$ rate and provides a sharp characterization of the energy decay law. We also note that it is impossible to establish a rate $O(1/(tϕ(t))$ for any function $ϕ$ which satisfies $\lim_{t\to\infty}ϕ(t) = \infty$, even asymptotically. Similar results are obtained in related settings for (1) discrete time gradient descent, (2) stochastic gradient descent with multiplicative noise and (3) the heavy ball ODE. In the case of stochastic gradient descent, the summability of $\mathbb E[f(x_n) - \inf f]$ is used to prove that $f(x_n)\to \inf f$ almost surely - an improvement on the convergence almost surely up to a subsequence which follows from the $O(1/n)$ decay estimate.

math.OC↗

arXiv2023

Greedy Training Algorithms for Neural Networks and Applications to PDEs

Recently, neural networks have been widely applied for solving partial differential equations (PDEs). Although such methods have been proven remarkably successful on practical engineering problems, they have not been shown, theoretically or empirically, to converge to the underlying PDE solution with arbitrarily high accuracy. The primary difficulty lies in solving the highly non-convex optimization problems resulting from the neural network discretization, which are difficult to treat both theoretically and practically. It is our goal in this work to take a step toward remedying this. For this purpose, we develop a novel greedy training algorithm for shallow neural networks. Our method is applicable to both the variational formulation of the PDE and also to the residual minimization formulation pioneered by physics informed neural networks (PINNs). We analyze the method and obtain a priori error bounds when solving PDEs from the function class defined by shallow networks, which rigorously establishes the convergence of the method as the network size increases. Finally, we test the algorithm on several benchmark examples, including high dimensional PDEs, to confirm the theoretical convergence rate. Although the method is expensive relative to traditional approaches such as finite element methods, we view this work as a proof of concept for neural network-based methods, which shows that numerical methods based upon neural networks can be shown to rigorously converge.

math.NA↗

arXiv2022

On the Activation Function Dependence of the Spectral Bias of Neural Networks

Neural networks are universal function approximators which are known to generalize well despite being dramatically overparameterized. We study this phenomenon from the point of view of the spectral bias of neural networks. Our contributions are two-fold. First, we provide a theoretical explanation for the spectral bias of ReLU neural networks by leveraging connections with the theory of finite element methods. Second, based upon this theory we predict that switching the activation function to a piecewise linear B-spline, namely the Hat function, will remove this spectral bias, which we verify empirically in a variety of settings. Our empirical studies also show that neural networks with the Hat activation function are trained significantly faster using stochastic gradient descent and ADAM. Combined with previous work showing that the Hat activation function also improves generalization accuracy on image classification tasks, this indicates that using the Hat activation provides significant advantages over the ReLU on certain problems.

cs.LG↗

arXiv2022

A Priori Analysis of Stable Neural Network Solutions to Numerical PDEs

Methods for solving PDEs using neural networks have recently become a very important topic. We provide an a priori error analysis for such methods which is based on the $\mathcal{K}_1(\mathbb{D})$-norm of the solution. We show that the resulting constrained optimization problem can be efficiently solved using a greedy algorithm, which replaces stochastic gradient descent. Following this, we show that the error arising from discretizing the energy integrals is bounded both in the deterministic case, i.e. when using numerical quadrature, and also in the stochastic case, i.e. when sampling points to approximate the integrals. In the later case, we use a Rademacher complexity analysis, and in the former we use standard numerical quadrature bounds. This extends existing results to methods which use a general dictionary of functions to learn solutions to PDEs and importantly gives a consistent analysis which incorporates the optimization, approximation, and generalization aspects of the problem. In addition, the Rademacher complexity analysis is simplified and generalized, which enables application to a wide range of problems.

math.NA↗

arXiv2022

Extended Regularized Dual Averaging Methods for Stochastic Optimization

We introduce a new algorithm, extended regularized dual averaging (XRDA), for solving regularized stochastic optimization problems, which generalizes the regularized dual averaging (RDA) method. The main novelty of the method is that it allows a flexible control of the backward step size. For instance, the backward step size used in RDA grows without bound, while for XRDA the backward step size can be kept bounded. We demonstrate experimentally that additional control over the backward step size can significantly improve the convergence rate of the algorithm while preserving desired properties of the iterates, such as sparsity. Theoretically, we show that the XRDA method achieves the same convergence rate as RDA for general convex objectives.

math.OC↗

arXiv2022

Characterization of the Variation Spaces Corresponding to Shallow Neural Networks

We study the variation space corresponding to a dictionary of functions in $L^2(Ω)$ for a bounded domain $Ω\subset \mathbb{R}^d$. Specifically, we compare the variation space, which is defined in terms of a convex hull with related notions based on integral representations. This allows us to show that three important notions relating to the approximation theory of shallow neural networks, the Barron space, the spectral Barron space, and the Radon BV space, are actually variation spaces with respect to certain natural dictionaries.

stat.ML↗

arXiv2022

Optimal Convergence Rates for the Orthogonal Greedy Algorithm

We analyze the orthogonal greedy algorithm when applied to dictionaries $\mathbb{D}$ whose convex hull has small entropy. We show that if the metric entropy of the convex hull of $\mathbb{D}$ decays at a rate of $O(n^{-\frac{1}{2}-α})$ for $α> 0$, then the orthogonal greedy algorithm converges at the same rate on the variation space of $\mathbb{D}$. This improves upon the well-known $O(n^{-\frac{1}{2}})$ convergence rate of the orthogonal greedy algorithm in many cases, most notably for dictionaries corresponding to shallow neural networks. These results hold under no additional assumptions on the dictionary beyond the decay rate of the entropy of its convex hull. In addition, they are robust to noise in the target function and can be extended to convergence rates on the interpolation spaces of the variation norm. We show empirically that the predicted rates are obtained for the dictionary corresponding to shallow neural networks with Heaviside activation function in two dimensions. Finally, we show that these improved rates are sharp and prove a negative result showing that the iterates generated by the orthogonal greedy algorithm cannot in general be bounded in the variation norm of $\mathbb{D}$.

math.ST↗

arXiv2021

Extensible Structure-Informed Prediction of Formation Energy with Improved Accuracy and Usability employing Neural Networks

In the present paper, we introduce a new neural network-based tool for the prediction of formation energies of atomic structures based on elemental and structural features of Voronoi-tessellated materials. We provide a concise overview of the connection between the machine learning and the true material-property relationship, how to improve the generalization accuracy by reducing overfitting, how new data can be incorporated into the model to tune it to a specific material system, and preliminary results on using models to preform local structure relaxations. The present work resulted in three final models optimized for (1) highest test accuracy on the Open Quantum Materials Database (OQMD), (2) performance in the discovery of new materials, and (3) performance at a low computational cost. On a test set of 21,800 compounds randomly selected from OQMD, they achieve a mean absolute error (MAE) of 28, 40, and 42 meV/atom, respectively. The second model provides better predictions in a test case of interest not present in the OQMD, while the third reduces the computational cost by a factor of 8. We collect our results in a new open-source tool called SIPFENN (Structure-Informed Prediction of Formation Energy using Neural Networks). SIPFENN not only improves the accuracy beyond existing models but also ships in a ready-to-use form with pre-trained neural networks and a GUI interface. By virtue of this, it can be included in DFT calculations routines at nearly no cost.

cond-mat.mtrl-sci↗

arXiv2021

High-Order Approximation Rates for Shallow Neural Networks with Cosine and ReLU$^k$ Activation Functions

We study the approximation properties of shallow neural networks with an activation function which is a power of the rectified linear unit. Specifically, we consider the dependence of the approximation rate on the dimension and the smoothness in the spectral Barron space of the underlying function $f$ to be approximated. We show that as the smoothness index $s$ of $f$ increases, shallow neural networks with ReLU$^k$ activation function obtain an improved approximation rate up to a best possible rate of $O(n^{-(k+1)}\log(n))$ in $L^2$, independent of the dimension $d$. The significance of this result is that the activation function ReLU$^k$ is fixed independent of the dimension, while for classical methods the degree of polynomial approximation or the smoothness of the wavelets used would have to increase in order to take advantage of the dimension dependent smoothness of $f$. In addition, we derive improved approximation rates for shallow neural networks with cosine activation function on the spectral Barron space. Finally, we prove lower bounds showing that the approximation rates attained are optimal under the given assumptions.

math.NA↗

arXiv2021

Sharp Lower Bounds on the Approximation Rate of Shallow Neural Networks

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.

stat.ML↗

arXiv2021

Training Sparse Neural Networks using Compressed Sensing

Pruning the weights of neural networks is an effective and widely-used technique for reducing model size and inference complexity. We develop and test a novel method based on compressed sensing which combines the pruning and training into a single step. Specifically, we utilize an adaptively weighted $\ell^1$ penalty on the weights during training, which we combine with a generalization of the regularized dual averaging (RDA) algorithm in order to train sparse neural networks. The adaptive weighting we introduce corresponds to a novel regularizer based on the logarithm of the absolute value of the weights. We perform a series of ablation studies demonstrating the improvement provided by the adaptive weighting and generalized RDA algorithm. Furthermore, numerical experiments on the CIFAR-10, CIFAR-100, and ImageNet datasets demonstrate that our method 1) trains sparser, more accurate networks than existing state-of-the-art methods; 2) can be used to train sparse networks from scratch, i.e. from a random initialization, as opposed to initializing with a well-trained base model; 3) acts as an effective regularizer, improving generalization accuracy.

cs.CV↗

arXiv2021

Accelerated First-Order Methods: Differential Equations and Lyapunov Functions

We develop a theory of accelerated first-order optimization from the viewpoint of differential equations and Lyapunov functions. Building upon the previous work of many researchers, we consider differential equations which model the behavior of accelerated gradient descent. Our main contributions are to provide a general framework for discretizating the differential equations to produce accelerated methods, and to provide physical intuition which helps explain the optimal damping rate. An important novelty is the generality of our approach, which leads to a unified derivation of a wide variety of methods, including versions of Nesterov's accelerated gradient descent, FISTA, and accelerated coordinate descent.

math.OC↗

arXiv2021

Accelerated Optimization With Orthogonality Constraints

We develop a generalization of Nesterov's accelerated gradient descent method which is designed to deal with orthogonality constraints. To demonstrate the effectiveness of our method, we perform numerical experiments which demonstrate that the number of iterations scales with the square root of the condition number, and also compare with existing state-of-the-art quasi-Newton methods on the Stiefel manifold. Our experiments show that our method outperforms existing state-of-the-art quasi-Newton methods on some large, ill-conditioned problems.

math.OC↗

arXiv2021

Approximation Rates for Neural Networks with General Activation Functions

We prove some new results concerning the approximation rate of neural networks with general activation functions. Our first result concerns the rate of approximation of a two layer neural network with a polynomially-decaying non-sigmoidal activation function. We extend the dimension independent approximation rates previously obtained to this new class of activation functions. Our second result gives a weaker, but still dimension independent, approximation rate for a larger class of activation functions, removing the polynomial decay assumption. This result applies to any bounded, integrable activation function. Finally, we show that a stratified sampling approach can be used to improve the approximation rate for polynomially decaying activation functions under mild additional assumptions.

math.CA↗