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Depth separation results propose a possible theoretical explanation for the benefits of deep neural networks over shallower architectures, establishing that the former possess superior approximation capabilities. However, there are no known results in which the deeper architecture leverages this advantage into a provable optimization guarantee. We prove that when the data are generated by a distribution with radial symmetry which satisfies some mild assumptions, gradient descent can efficiently learn ball indicator functions using a depth 2 neural network with two layers of sigmoidal activations, and where the hidden layer is held fixed throughout training. By building on and refining existing techniques for approximation lower bounds of neural networks with a single layer of non-linearities, we show that there are $d$-dimensional radial distributions on the data such that ball indicators cannot be learned efficiently by any algorithm to accuracy better than $\Omega(d^{-4})$, nor by a standard gradient descent implementation to accuracy better than a constant. These results establish what is to the best of our knowledge, the first optimization-based separations where the approximation benefits of the stronger architecture provably manifest in practice. Our proof technique introduces new tools and ideas that may be of independent interest in the theoretical study of both the approximation and optimization of neural networks.

Owing to their superior modeling capabilities, gated Recurrent Neural Networks, such as Gated Recurrent Units (GRUs) and Long Short-Term Memory networks (LSTMs), have become popular tools for learning dynamical systems. This paper aims to discuss how these networks can be adopted for the synthesis of Internal Model Control (IMC) architectures. To this end, first a gated recurrent network is used to learn a model of the unknown input-output stable plant. Then, a controller gated recurrent network is trained to approximate the model inverse. The stability of these networks, ensured by means of a suitable training procedure, allows to guarantee the input-output closed-loop stability. The proposed scheme is able to cope with the saturation of the control variables, and can be deployed on low-power embedded controllers, as it requires limited online computations. The approach is then tested on the Quadruple Tank benchmark system and compared to alternative control laws, resulting in remarkable closed-loop performances.

Physically-inspired latent force models offer an interpretable alternative to purely data driven tools for inference in dynamical systems. They carry the structure of differential equations and the flexibility of Gaussian processes, yielding interpretable parameters and dynamics-imposed latent functions. However, the existing inference techniques associated with these models rely on the exact computation of posterior kernel terms which are seldom available in analytical form. Most applications relevant to practitioners, such as Hill equations or diffusion equations, are hence intractable. In this paper, we overcome these computational problems by proposing a variational solution to a general class of non-linear and parabolic partial differential equation latent force models. Further, we show that a neural operator approach can scale our model to thousands of instances, enabling fast, distributed computation. We demonstrate the efficacy and flexibility of our framework by achieving competitive performance on several tasks where the kernels are of varying degrees of tractability.

We introduce Stochastic Asymptotical Regularization (SAR) methods for the uncertainty quantification of the stable approximate solution of ill-posed linear-operator equations, which are deterministic models for numerous inverse problems in science and engineering. We prove the regularizing properties of SAR with regard to mean-square convergence. We also show that SAR is an optimal-order regularization method for linear ill-posed problems provided that the terminating time of SAR is chosen according to the smoothness of the solution. This result is proven for both a priori and a posteriori stopping rules under general range-type source conditions. Furthermore, some converse results of SAR are verified. Two iterative schemes are developed for the numerical realization of SAR, and the convergence analyses of these two numerical schemes are also provided. A toy example and a real-world problem of biosensor tomography are studied to show the accuracy and the advantages of SAR: compared with the conventional deterministic regularization approaches for deterministic inverse problems, SAR can provide the uncertainty quantification of the quantity of interest, which can in turn be used to reveal and explicate the hidden information about real-world problems, usually obscured by the incomplete mathematical modeling and the ascendence of complex-structured noise.

We develop a stable finite difference method for the elastic wave equation in bounded media, where the material properties can be discontinuous at curved interfaces. The governing equation is discretized in second order form by a fourth or sixth order accurate summation-by-parts operator. The mesh size is determined by the velocity structure of the material, resulting in nonconforming grid interfaces with hanging nodes. We use order-preserving interpolation and the ghost point technique to couple adjacent mesh blocks in an energy-conserving manner, which is supported by a fully discrete stability analysis. In our previous work for the wave equation, two pairs of order-preserving interpolation operators are needed when imposing the interface conditions weakly by a penalty technique. Here, we only use one pair in the ghost point method. In numerical experiments, we demonstrate that the convergence rate is optimal, and is the same as when a globally uniform mesh is used in a single domain. In addition, with a predictor-corrector time integration method, we obtain time stepping stability with stepsize almost the same as given by the usual Courant-Friedrichs-Lewy condition.

We study sparse linear regression over a network of agents, modeled as an undirected graph and no server node. The estimation of the $s$-sparse parameter is formulated as a constrained LASSO problem wherein each agent owns a subset of the $N$ total observations. We analyze the convergence rate and statistical guarantees of a distributed projected gradient tracking-based algorithm under high-dimensional scaling, allowing the ambient dimension $d$ to grow with (and possibly exceed) the sample size $N$. Our theory shows that, under standard notions of restricted strong convexity and smoothness of the loss functions, suitable conditions on the network connectivity and algorithm tuning, the distributed algorithm converges globally at a {\it linear} rate to an estimate that is within the centralized {\it statistical precision} of the model, $O(s\log d/N)$. When $s\log d/N=o(1)$, a condition necessary for statistical consistency, an $\varepsilon$-optimal solution is attained after $\mathcal{O}(\kappa \log (1/\varepsilon))$ gradient computations and $O (\kappa/(1-\rho) \log (1/\varepsilon))$ communication rounds, where $\kappa$ is the restricted condition number of the loss function and $\rho$ measures the network connectivity. The computation cost matches that of the centralized projected gradient algorithm despite having data distributed; whereas the communication rounds reduce as the network connectivity improves. Overall, our study reveals interesting connections between statistical efficiency, network connectivity \& topology, and convergence rate in high dimensions.

Graph neural networks (GNNs) are a popular class of machine learning models whose major advantage is their ability to incorporate a sparse and discrete dependency structure between data points. Unfortunately, GNNs can only be used when such a graph-structure is available. In practice, however, real-world graphs are often noisy and incomplete or might not be available at all. With this work, we propose to jointly learn the graph structure and the parameters of graph convolutional networks (GCNs) by approximately solving a bilevel program that learns a discrete probability distribution on the edges of the graph. This allows one to apply GCNs not only in scenarios where the given graph is incomplete or corrupted but also in those where a graph is not available. We conduct a series of experiments that analyze the behavior of the proposed method and demonstrate that it outperforms related methods by a significant margin.

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.

Dynamic programming (DP) solves a variety of structured combinatorial problems by iteratively breaking them down into smaller subproblems. In spite of their versatility, DP algorithms are usually non-differentiable, which hampers their use as a layer in neural networks trained by backpropagation. To address this issue, we propose to smooth the max operator in the dynamic programming recursion, using a strongly convex regularizer. This allows to relax both the optimal value and solution of the original combinatorial problem, and turns a broad class of DP algorithms into differentiable operators. Theoretically, we provide a new probabilistic perspective on backpropagating through these DP operators, and relate them to inference in graphical models. We derive two particular instantiations of our framework, a smoothed Viterbi algorithm for sequence prediction and a smoothed DTW algorithm for time-series alignment. We showcase these instantiations on two structured prediction tasks and on structured and sparse attention for neural machine translation.

Methods that align distributions by minimizing an adversarial distance between them have recently achieved impressive results. However, these approaches are difficult to optimize with gradient descent and they often do not converge well without careful hyperparameter tuning and proper initialization. We investigate whether turning the adversarial min-max problem into an optimization problem by replacing the maximization part with its dual improves the quality of the resulting alignment and explore its connections to Maximum Mean Discrepancy. Our empirical results suggest that using the dual formulation for the restricted family of linear discriminators results in a more stable convergence to a desirable solution when compared with the performance of a primal min-max GAN-like objective and an MMD objective under the same restrictions. We test our hypothesis on the problem of aligning two synthetic point clouds on a plane and on a real-image domain adaptation problem on digits. In both cases, the dual formulation yields an iterative procedure that gives more stable and monotonic improvement over time.