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A priori error bounds have been derived for different balancing-related model reduction methods. The most classical result is a bound for balanced truncation and singular perturbation approximation that is applicable for asymptotically stable linear time-invariant systems with homogeneous initial conditions. Recently, there have been a few attempts to generalize the balancing-related reduction methods to the case with inhomogeneous initial conditions, but the existing error bounds for these generalizations are quite restrictive. Particularly, it is required to restrict the initial conditions to a low-dimensional subspace, which has to be chosen before the reduced model is constructed. In this paper, we propose an estimator that circumvents this hard constraint completely. Our estimator is applicable to a large class of reduction methods, whereas the former results were only derived for certain specific methods. Moreover, our approach yields to significantly more effective error estimation, as also will be demonstrated numerically.

A prediction interval covers a future observation from a random process in repeated sampling, and is typically constructed by identifying a pivotal quantity that is also an ancillary statistic. Analogously, a tolerance interval covers a population percentile in repeated sampling and is often based on a pivotal quantity. One approach we consider in non-normal models leverages a link function resulting in a pivotal quantity that is approximately normally distributed. In settings where this normal approximation does not hold we consider a second approach for tolerance and prediction based on a confidence interval for the mean. These methods are intuitive, simple to implement, have proper operating characteristics, and are computationally efficient compared to Bayesian, re-sampling, and machine learning methods. This is demonstrated in the context of multi-site clinical trial recruitment with staggered site initiation, real-world time on treatment, and end-of-study success for a clinical endpoint.

Dyadic data is often encountered when quantities of interest are associated with the edges of a network. As such it plays an important role in statistics, econometrics and many other data science disciplines. We consider the problem of uniformly estimating a dyadic Lebesgue density function, focusing on nonparametric kernel-based estimators which take the form of U-process-like dyadic empirical processes. We provide uniform point estimation and distributional results for the dyadic kernel density estimator, giving valid and feasible procedures for robust uniform inference. Our main contributions include the minimax-optimal uniform convergence rate of the dyadic kernel density estimator, along with strong approximation results for the associated standardized $t$-process. A consistent variance estimator is introduced in order to obtain analogous results for the Studentized $t$-process, enabling the construction of provably valid and feasible uniform confidence bands for the unknown density function. A crucial feature of U-process-like dyadic empirical processes is that they may be "degenerate" at some or possibly all points in the support of the data, a property making our uniform analysis somewhat delicate. Nonetheless we show formally that our proposed methods for uniform inference remain robust to the potential presence of such unknown degenerate points. For the purpose of implementation, we discuss uniform inference procedures based on positive semi-definite covariance estimators, mean squared error optimal bandwidth selectors and robust bias-correction methods. We illustrate the empirical finite-sample performance of our robust inference methods in a simulation study. Our technical results concerning strong approximations and maximal inequalities are of potential independent interest.

This paper is focused on the optimization approach to the solution of inverse problems. We introduce a stochastic dynamical system in which the parameter-to-data map is embedded, with the goal of employing techniques from nonlinear Kalman filtering to estimate the parameter given the data. The extended Kalman filter (which we refer to as ExKI in the context of inverse problems) can be effective for some inverse problems approached this way, but is impractical when the forward map is not readily differentiable and is given as a black box, and also for high dimensional parameter spaces because of the need to propagate large covariance matrices. Application of ensemble Kalman filters, for example use of the ensemble Kalman inversion (EKI) algorithm, has emerged as a useful tool which overcomes both of these issues: it is derivative free and works with a low-rank covariance approximation formed from the ensemble. In this paper, we work with the ExKI, EKI, and a variant on EKI which we term unscented Kalman inversion (UKI). The paper contains two main contributions. Firstly, we identify a novel stochastic dynamical system in which the parameter-to-data map is embedded. We present theory in the linear case to show exponential convergence of the mean of the filtering distribution to the solution of a regularized least squares problem. This is in contrast to previous work in which the EKI has been employed where the dynamical system used leads to algebraic convergence to an unregularized problem. Secondly, we show that the application of the UKI to this novel stochastic dynamical system yields improved inversion results, in comparison with the application of EKI to the same novel stochastic dynamical system.

The Monte Carlo simulation (MCS) is a statistical methodology used in a large number of applications. It uses repeated random sampling to solve problems with a probability interpretation to obtain high-quality numerical results. The MCS is simple and easy to develop, implement, and apply. However, its computational cost and total runtime can be quite high as it requires many samples to obtain an accurate approximation with low variance. In this paper, a novel MCS, called the self-adaptive BAT-MCS, based on the binary-adaption-tree algorithm (BAT) and our proposed self-adaptive simulation-number algorithm is proposed to simply and effectively reduce the run time and variance of the MCS. The proposed self-adaptive BAT-MCS was applied to a simple benchmark problem to demonstrate its application in network reliability. The statistical characteristics, including the expectation, variance, and simulation number, and the time complexity of the proposed self-adaptive BAT-MCS are discussed. Furthermore, its performance is compared to that of the traditional MCS extensively on a large-scale problem.

Double Q-learning is a classical method for reducing overestimation bias, which is caused by taking maximum estimated values in the Bellman operation. Its variants in the deep Q-learning paradigm have shown great promise in producing reliable value prediction and improving learning performance. However, as shown by prior work, double Q-learning is not fully unbiased and suffers from underestimation bias. In this paper, we show that such underestimation bias may lead to multiple non-optimal fixed points under an approximate Bellman operator. To address the concerns of converging to non-optimal stationary solutions, we propose a simple but effective approach as a partial fix for the underestimation bias in double Q-learning. This approach leverages an approximate dynamic programming to bound the target value. We extensively evaluate our proposed method in the Atari benchmark tasks and demonstrate its significant improvement over baseline algorithms.

Meta-reinforcement learning (meta-RL) aims to learn from multiple training tasks the ability to adapt efficiently to unseen test tasks. Despite the success, existing meta-RL algorithms are known to be sensitive to the task distribution shift. When the test task distribution is different from the training task distribution, the performance may degrade significantly. To address this issue, this paper proposes Model-based Adversarial Meta-Reinforcement Learning (AdMRL), where we aim to minimize the worst-case sub-optimality gap -- the difference between the optimal return and the return that the algorithm achieves after adaptation -- across all tasks in a family of tasks, with a model-based approach. We propose a minimax objective and optimize it by alternating between learning the dynamics model on a fixed task and finding the adversarial task for the current model -- the task for which the policy induced by the model is maximally suboptimal. Assuming the family of tasks is parameterized, we derive a formula for the gradient of the suboptimality with respect to the task parameters via the implicit function theorem, and show how the gradient estimator can be efficiently implemented by the conjugate gradient method and a novel use of the REINFORCE estimator. We evaluate our approach on several continuous control benchmarks and demonstrate its efficacy in the worst-case performance over all tasks, the generalization power to out-of-distribution tasks, and in training and test time sample efficiency, over existing state-of-the-art meta-RL algorithms.

It is known that the current graph neural networks (GNNs) are difficult to make themselves deep due to the problem known as \textit{over-smoothing}. Multi-scale GNNs are a promising approach for mitigating the over-smoothing problem. However, there is little explanation of why it works empirically from the viewpoint of learning theory. In this study, we derive the optimization and generalization guarantees of transductive learning algorithms that include multi-scale GNNs. Using the boosting theory, we prove the convergence of the training error under weak learning-type conditions. By combining it with generalization gap bounds in terms of transductive Rademacher complexity, we show that a test error bound of a specific type of multi-scale GNNs that decreases corresponding to the depth under the conditions. Our results offer theoretical explanations for the effectiveness of the multi-scale structure against the over-smoothing problem. We apply boosting algorithms to the training of multi-scale GNNs for real-world node prediction tasks. We confirm that its performance is comparable to existing GNNs, and the practical behaviors are consistent with theoretical observations. Code is available at //github.com/delta2323/GB-GNN

Modern neural network training relies heavily on data augmentation for improved generalization. After the initial success of label-preserving augmentations, there has been a recent surge of interest in label-perturbing approaches, which combine features and labels across training samples to smooth the learned decision surface. In this paper, we propose a new augmentation method that leverages the first and second moments extracted and re-injected by feature normalization. We replace the moments of the learned features of one training image by those of another, and also interpolate the target labels. As our approach is fast, operates entirely in feature space, and mixes different signals than prior methods, one can effectively combine it with existing augmentation methods. We demonstrate its efficacy across benchmark data sets in computer vision, speech, and natural language processing, where it consistently improves the generalization performance of highly competitive baseline networks.

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.