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We give a number of approximation metatheorems for monotone maximization problems expressible in the first-order logic, in substantially more general settings than the previously known. We obtain * constant-factor approximation algorithm in any class of graphs with bounded expansion, * a QPTAS in any class with strongly sublinear separators, and * a PTAS in any fractionally treewidth-fragile class (which includes all common classes with strongly sublinear separators. Moreover, our tools also give an exact subexponential-time algorithm in any class with strongly sublinear separators.

We characterize the complexity of the lattice decoding problem from a neural network perspective. The notion of Voronoi-reduced basis is introduced to restrict the space of solutions to a binary set. On the one hand, this problem is shown to be equivalent to computing a continuous piecewise linear (CPWL) function restricted to the fundamental parallelotope. On the other hand, it is known that any function computed by a ReLU feed-forward neural network is CPWL. As a result, we count the number of affine pieces in the CPWL decoding function to characterize the complexity of the decoding problem. It is exponential in the space dimension $n$, which induces shallow neural networks of exponential size. For structured lattices we show that folding, a technique equivalent to using a deep neural network, enables to reduce this complexity from exponential in $n$ to polynomial in $n$. Regarding unstructured MIMO lattices, in contrary to dense lattices many pieces in the CPWL decoding function can be neglected for quasi-optimal decoding on the Gaussian channel. This makes the decoding problem easier and it explains why shallow neural networks of reasonable size are more efficient with this category of lattices (in low to moderate dimensions).

We introduce kernel thinning, a new procedure for compressing a distribution $\mathbb{P}$ more effectively than i.i.d. sampling or standard thinning. Given a suitable reproducing kernel $\mathbf{k}$ and $\mathcal{O}(n^2)$ time, kernel thinning compresses an $n$-point approximation to $\mathbb{P}$ into a $\sqrt{n}$-point approximation with comparable worst-case integration error across the associated reproducing kernel Hilbert space. With high probability, the maximum discrepancy in integration error is $\mathcal{O}_d(n^{-\frac{1}{2}}\sqrt{\log n})$ for compactly supported $\mathbb{P}$ and $\mathcal{O}_d(n^{-\frac{1}{2}} \sqrt{(\log n)^{d+1}\log\log n})$ for sub-exponential $\mathbb{P}$ on $\mathbb{R}^d$. In contrast, an equal-sized i.i.d. sample from $\mathbb{P}$ suffers $\Omega(n^{-\frac14})$ integration error. Our sub-exponential guarantees resemble the classical quasi-Monte Carlo error rates for uniform $\mathbb{P}$ on $[0,1]^d$ but apply to general distributions on $\mathbb{R}^d$ and a wide range of common kernels. We use our results to derive explicit non-asymptotic maximum mean discrepancy bounds for Gaussian, Mat\'ern, and B-spline kernels and present two vignettes illustrating the practical benefits of kernel thinning over i.i.d. sampling and standard Markov chain Monte Carlo thinning, in dimensions $d=2$ through $100$.

Reinforcement learning, mathematically described by Markov Decision Problems, may be approached either through dynamic programming or policy search. Actor-critic algorithms combine the merits of both approaches by alternating between steps to estimate the value function and policy gradient updates. Due to the fact that the updates exhibit correlated noise and biased gradient updates, only the asymptotic behavior of actor-critic is known by connecting its behavior to dynamical systems. This work puts forth a new variant of actor-critic that employs Monte Carlo rollouts during the policy search updates, which results in controllable bias that depends on the number of critic evaluations. As a result, we are able to provide for the first time the convergence rate of actor-critic algorithms when the policy search step employs policy gradient, agnostic to the choice of policy evaluation technique. In particular, we establish conditions under which the sample complexity is comparable to stochastic gradient method for non-convex problems or slower as a result of the critic estimation error, which is the main complexity bottleneck. These results hold in continuous state and action spaces with linear function approximation for the value function. We then specialize these conceptual results to the case where the critic is estimated by Temporal Difference, Gradient Temporal Difference, and Accelerated Gradient Temporal Difference. These learning rates are then corroborated on a navigation problem involving an obstacle, providing insight into the interplay between optimization and generalization in reinforcement learning.

We consider approximating analytic functions on the interval $[-1,1]$ from their values at a set of $m+1$ equispaced nodes. A result of Platte, Trefethen & Kuijlaars states that fast and stable approximation from equispaced samples is generally impossible. In particular, any method that converges exponentially fast must also be exponentially ill-conditioned. We prove a positive counterpart to this `impossibility' theorem. Our `possibility' theorem shows that there is a well-conditioned method that provides exponential decay of the error down to a finite, but user-controlled tolerance $\epsilon > 0$, which in practice can be chosen close to machine epsilon. The method is known as \textit{polynomial frame} approximation or \textit{polynomial extensions}. It uses algebraic polynomials of degree $n$ on an extended interval $[-\gamma,\gamma]$, $\gamma > 1$, to construct an approximation on $[-1,1]$ via a SVD-regularized least-squares fit. A key step in the proof of our possibility theorem is a new result on the maximal behaviour of a polynomial of degree $n$ on $[-1,1]$ that is simultaneously bounded by one at a set of $m+1$ equispaced nodes in $[-1,1]$ and $1/\epsilon$ on the extended interval $[-\gamma,\gamma]$. We show that linear oversampling, i.e., $m = c n \log(1/\epsilon) / \sqrt{\gamma^2-1}$, is sufficient for uniform boundedness of any such polynomial on $[-1,1]$. This result aside, we also prove an extended impossibility theorem, which shows that the possibility theorem (and consequently the method of polynomial frame approximation) is essentially optimal.

Many representative graph neural networks, $e.g.$, GPR-GNN and ChebyNet, approximate graph convolutions with graph spectral filters. However, existing work either applies predefined filter weights or learns them without necessary constraints, which may lead to oversimplified or ill-posed filters. To overcome these issues, we propose $\textit{BernNet}$, a novel graph neural network with theoretical support that provides a simple but effective scheme for designing and learning arbitrary graph spectral filters. In particular, for any filter over the normalized Laplacian spectrum of a graph, our BernNet estimates it by an order-$K$ Bernstein polynomial approximation and designs its spectral property by setting the coefficients of the Bernstein basis. Moreover, we can learn the coefficients (and the corresponding filter weights) based on observed graphs and their associated signals and thus achieve the BernNet specialized for the data. Our experiments demonstrate that BernNet can learn arbitrary spectral filters, including complicated band-rejection and comb filters, and it achieves superior performance in real-world graph modeling tasks.

In order to avoid the curse of dimensionality, frequently encountered in Big Data analysis, there was a vast development in the field of linear and nonlinear dimension reduction techniques in recent years. These techniques (sometimes referred to as manifold learning) assume that the scattered input data is lying on a lower dimensional manifold, thus the high dimensionality problem can be overcome by learning the lower dimensionality behavior. However, in real life applications, data is often very noisy. In this work, we propose a method to approximate $\mathcal{M}$ a $d$-dimensional $C^{m+1}$ smooth submanifold of $\mathbb{R}^n$ ($d \ll n$) based upon noisy scattered data points (i.e., a data cloud). We assume that the data points are located "near" the lower dimensional manifold and suggest a non-linear moving least-squares projection on an approximating $d$-dimensional manifold. Under some mild assumptions, the resulting approximant is shown to be infinitely smooth and of high approximation order (i.e., $O(h^{m+1})$, where $h$ is the fill distance and $m$ is the degree of the local polynomial approximation). The method presented here assumes no analytic knowledge of the approximated manifold and the approximation algorithm is linear in the large dimension $n$. Furthermore, the approximating manifold can serve as a framework to perform operations directly on the high dimensional data in a computationally efficient manner. This way, the preparatory step of dimension reduction, which induces distortions to the data, can be avoided altogether.

We introduce a new family of deep neural network models. Instead of specifying a discrete sequence of hidden layers, we parameterize the derivative of the hidden state using a neural network. The output of the network is computed using a black-box differential equation solver. These continuous-depth models have constant memory cost, adapt their evaluation strategy to each input, and can explicitly trade numerical precision for speed. We demonstrate these properties in continuous-depth residual networks and continuous-time latent variable models. We also construct continuous normalizing flows, a generative model that can train by maximum likelihood, without partitioning or ordering the data dimensions. For training, we show how to scalably backpropagate through any ODE solver, without access to its internal operations. This allows end-to-end training of ODEs within larger models.

This paper addresses the problem of formally verifying desirable properties of neural networks, i.e., obtaining provable guarantees that neural networks satisfy specifications relating their inputs and outputs (robustness to bounded norm adversarial perturbations, for example). Most previous work on this topic was limited in its applicability by the size of the network, network architecture and the complexity of properties to be verified. In contrast, our framework applies to a general class of activation functions and specifications on neural network inputs and outputs. We formulate verification as an optimization problem (seeking to find the largest violation of the specification) and solve a Lagrangian relaxation of the optimization problem to obtain an upper bound on the worst case violation of the specification being verified. Our approach is anytime i.e. it can be stopped at any time and a valid bound on the maximum violation can be obtained. We develop specialized verification algorithms with provable tightness guarantees under special assumptions and demonstrate the practical significance of our general verification approach on a variety of verification tasks.

This paper describes a suite of algorithms for constructing low-rank approximations of an input matrix from a random linear image of the matrix, called a sketch. These methods can preserve structural properties of the input matrix, such as positive-semidefiniteness, and they can produce approximations with a user-specified rank. The algorithms are simple, accurate, numerically stable, and provably correct. Moreover, each method is accompanied by an informative error bound that allows users to select parameters a priori to achieve a given approximation quality. These claims are supported by numerical experiments with real and synthetic data.