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We {formalize the} definition of a stable algorithm that is (i) adapted to the use of multiple and variable precision arithmetic, (ii) sufficiently close to the actual practice of computing to be useful, and (iii) sufficiently robust from a mathematical point of view as to allow for the rigorous proof of theorems. This allows us to state some widely satisfied hypotheses, depending only on two functions $f$ and $g$, under which the composition of a stable algorithm for $f$ and a stable algorithm for $g$ is a stable algorithm for the composition $f \circ g$.

The estimation of information measures of continuous distributions based on samples is a fundamental problem in statistics and machine learning. In this paper, we analyze estimates of differential entropy in $K$-dimensional Euclidean space, computed from a finite number of samples, when the probability density function belongs to a predetermined convex family $\mathcal{P}$. First, estimating differential entropy to any accuracy is shown to be infeasible if the differential entropy of densities in $\mathcal{P}$ is unbounded, clearly showing the necessity of additional assumptions. Subsequently, we investigate sufficient conditions that enable confidence bounds for the estimation of differential entropy. In particular, we provide confidence bounds for simple histogram based estimation of differential entropy from a fixed number of samples, assuming that the probability density function is Lipschitz continuous with known Lipschitz constant and known, bounded support. Our focus is on differential entropy, but we provide examples that show that similar results hold for mutual information and relative entropy as well.

In this paper, we characterize the noise of stochastic gradients and analyze the noise-induced dynamics during training deep neural networks by gradient-based optimizers. Specifically, we firstly show that the stochastic gradient noise possesses finite variance, and therefore the classical Central Limit Theorem (CLT) applies; this indicates that the gradient noise is asymptotically Gaussian. Such an asymptotic result validates the wide-accepted assumption of Gaussian noise. We clarify that the recently observed phenomenon of heavy tails within gradient noise may not be intrinsic properties, but the consequence of insufficient mini-batch size; the gradient noise, which is a sum of limited i.i.d. random variables, has not reached the asymptotic regime of CLT, thus deviates from Gaussian. We quantitatively measure the goodness of Gaussian approximation of the noise, which supports our conclusion. Secondly, we analyze the noise-induced dynamics of stochastic gradient descent using the Langevin equation, granting for momentum hyperparameter in the optimizer with a physical interpretation. We then proceed to demonstrate the existence of the steady-state distribution of stochastic gradient descent and approximate the distribution at a small learning rate.

To accurately reproduce measurements from the real world, simulators need to have an adequate model of the physical system and require the parameters of the model be identified. We address the latter problem of estimating parameters through a Bayesian inference approach that approximates a posterior distribution over simulation parameters given real sensor measurements. By extending the commonly used Gaussian likelihood model for trajectories via the multiple-shooting formulation, our chosen particle-based inference algorithm Stein Variational Gradient Descent is able to identify highly nonlinear, underactuated systems. We leverage GPU code generation and differentiable simulation to evaluate the likelihood and its gradient for many particles in parallel. Our algorithm infers non-parametric distributions over simulation parameters more accurately than comparable baselines and handles constraints over parameters efficiently through gradient-based optimization. We evaluate estimation performance on several physical experiments. On an underactuated mechanism where a 7-DOF robot arm excites an object with an unknown mass configuration, we demonstrate how our inference technique can identify symmetries between the parameters and provide highly accurate predictions. Project website: //uscresl.github.io/prob-diff-sim

The present study supposes a single unit and investigates cumulative damage and catastrophic failure models for the unit, in situations where the interarrival times between the shocks, and the magnitudes of the shocks, involve two different stochastic processes. In order to consider two essentially different stochastic processes, integer gamma and Weibull distributions are treated as distributions with two parameters and extensions of exponential distributions. With respect to the cumulative damage models, under the assumption that the interarrival times between shocks follow exponential distributions, the case in which the magnitudes of the shocks follow integer gamma distributions is analyzed. With respect to the catastrophic failure models, the respective cases in which the interarrival times between shocks follow integer gamma and Weibull distributions are discussed. Finally, the study provides some characteristic values for reliability in such models.

In a standard regression problem, we have a set of explanatory variables whose effect on some response vector is modeled. For wide binary data, such as genetic marker data, we often have two limitations. First, we have more parameters than observations. Second, main effects are not the main focus; instead the primary aim is to uncover interactions between the binary variables that effect the response. Methods such as logic regression are able to find combinations of the explanatory variables that capture higher-order relationships in the response. However, the number of explanatory variables these methods can handle is highly limited. To address these two limitations we need to reduce the number of variables prior to computationally demanding analyses. In this paper, we demonstrate the usefulness of using so-called cross-leverage scores as a means of sampling subsets of explanatory variables while retaining the valuable interactions.

Branch-and-bound is the workhorse of all state-of-the-art mixed integer linear programming (MILP) solvers. These implementations of branch-and-bound typically use variable branching, that is, the child nodes are obtained by fixing some variable to an integer value $v$ in one node and to $v + 1$ in the other node. Even though modern MILP solvers are able to solve very large-scale instances efficiently, relatively little attention has been given to understanding why the underlying branch-and-bound algorithm performs so well. In this paper our goal is to theoretically analyze the performance of the standard variable branching based branch-and-bound algorithm. In order to avoid the exponential worst-case lower bounds, we follow the common idea of considering random instances. More precisely, we consider random integer programs where the entries of the coefficient matrix and the objective function are randomly sampled. Our main result is that with good probability branch-and-bound with variable branching explores only a polynomial number of nodes to solve these instances, for a fixed number of constraints. To the best of our knowledge this is the first known such result for a standard version of branch-and-bound. We believe that this result provides a compelling indication of why branch-and-bound with variable branching works so well in practice.

Graph Neural Networks (GNNs) have been studied from the lens of expressive power and generalization. However, their optimization properties are less well understood. We take the first step towards analyzing GNN training by studying the gradient dynamics of GNNs. First, we analyze linearized GNNs and prove that despite the non-convexity of training, convergence to a global minimum at a linear rate is guaranteed under mild assumptions that we validate on real-world graphs. Second, we study what may affect the GNNs' training speed. Our results show that the training of GNNs is implicitly accelerated by skip connections, more depth, and/or a good label distribution. Empirical results confirm that our theoretical results for linearized GNNs align with the training behavior of nonlinear GNNs. Our results provide the first theoretical support for the success of GNNs with skip connections in terms of optimization, and suggest that deep GNNs with skip connections would be promising in practice.

Sampling methods (e.g., node-wise, layer-wise, or subgraph) has become an indispensable strategy to speed up training large-scale Graph Neural Networks (GNNs). However, existing sampling methods are mostly based on the graph structural information and ignore the dynamicity of optimization, which leads to high variance in estimating the stochastic gradients. The high variance issue can be very pronounced in extremely large graphs, where it results in slow convergence and poor generalization. In this paper, we theoretically analyze the variance of sampling methods and show that, due to the composite structure of empirical risk, the variance of any sampling method can be decomposed into \textit{embedding approximation variance} in the forward stage and \textit{stochastic gradient variance} in the backward stage that necessities mitigating both types of variance to obtain faster convergence rate. We propose a decoupled variance reduction strategy that employs (approximate) gradient information to adaptively sample nodes with minimal variance, and explicitly reduces the variance introduced by embedding approximation. We show theoretically and empirically that the proposed method, even with smaller mini-batch sizes, enjoys a faster convergence rate and entails a better generalization compared to the existing methods.

We develop an approach to risk minimization and stochastic optimization that provides a convex surrogate for variance, allowing near-optimal and computationally efficient trading between approximation and estimation error. Our approach builds off of techniques for distributionally robust optimization and Owen's empirical likelihood, and we provide a number of finite-sample and asymptotic results characterizing the theoretical performance of the estimator. In particular, we show that our procedure comes with certificates of optimality, achieving (in some scenarios) faster rates of convergence than empirical risk minimization by virtue of automatically balancing bias and variance. We give corroborating empirical evidence showing that in practice, the estimator indeed trades between variance and absolute performance on a training sample, improving out-of-sample (test) performance over standard empirical risk minimization for a number of classification problems.