We provide in this work an algorithm for approximating a very broad class of symmetric Toeplitz matrices to machine precision in $\mathcal{O}(n \log n)$ time with applications to fitting time series models. In particular, for a symmetric Toeplitz matrix $\mathbf{\Sigma}$ with values $\mathbf{\Sigma}_{j,k} = h_{|j-k|} = \int_{-1/2}^{1/2} e^{2 \pi i |j-k| \omega} S(\omega) \mathrm{d} \omega$ where $S(\omega)$ is piecewise smooth, we give an approximation $\mathbf{\mathcal{F}} \mathbf{\Sigma} \mathbf{\mathcal{F}}^H \approx \mathbf{D} + \mathbf{U} \mathbf{V}^H$, where $\mathbf{\mathcal{F}}$ is the DFT matrix, $\mathbf{D}$ is diagonal, and the matrices $\mathbf{U}$ and $\mathbf{V}$ are in $\mathbb{C}^{n \times r}$ with $r \ll n$. Studying these matrices in the context of time series, we offer a theoretical explanation of this structure and connect it to existing spectral-domain approximation frameworks. We then give a complete discussion of the numerical method for assembling the approximation and demonstrate its efficiency for improving Whittle-type likelihood approximations, including dramatic examples where a correction of rank $r = 2$ to the standard Whittle approximation increases the accuracy from $3$ to $14$ digits for a matrix $\mathbf{\Sigma} \in \mathbb{R}^{10^5 \times 10^5}$. The method and analysis of this work applies well beyond time series analysis, providing an algorithm for extremely accurate direct solves with a wide variety of symmetric Toeplitz matrices. The analysis employed here largely depends on asymptotic expansions of oscillatory integrals, and also provides a new perspective on when existing spectral-domain approximation methods for Gaussian log-likelihoods can be particularly problematic.
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In the realm of statistical learning, the increasing volume of accessible data and increasing model complexity necessitate robust methodologies. This paper explores two branches of robust Bayesian methods in response to this trend. The first is generalized Bayesian inference, which introduces a learning rate parameter to enhance robustness against model misspecifications. The second is Gibbs posterior inference, which formulates inferential problems using generic loss functions rather than probabilistic models. In such approaches, it is necessary to calibrate the spread of the posterior distribution by selecting a learning rate parameter. The study aims to enhance the generalized posterior calibration (GPC) algorithm proposed by [1]. Their algorithm chooses the learning rate to achieve the nominal frequentist coverage probability, but it is computationally intensive because it requires repeated posterior simulations for bootstrap samples. We propose a more efficient version of the GPC inspired by sequential Monte Carlo (SMC) samplers. A target distribution with a different learning rate is evaluated without posterior simulation as in the reweighting step in SMC sampling. Thus, the proposed algorithm can reach the desirable value within a few iterations. This improvement substantially reduces the computational cost of the GPC. Its efficacy is demonstrated through synthetic and real data applications.
For a control problem with multiple conflicting objectives, there exists a set of Pareto-optimal policies called the Pareto set instead of a single optimal policy. When a multi-objective control problem is continuous and complex, traditional multi-objective reinforcement learning (MORL) algorithms search for many Pareto-optimal deep policies to approximate the Pareto set, which is quite resource-consuming. In this paper, we propose a simple and resource-efficient MORL algorithm that learns a continuous representation of the Pareto set in a high-dimensional policy parameter space using a single hypernet. The learned hypernet can directly generate various well-trained policy networks for different user preferences. We compare our method with two state-of-the-art MORL algorithms on seven multi-objective continuous robot control problems. Experimental results show that our method achieves the best overall performance with the least training parameters. An interesting observation is that the Pareto set is well approximated by a curved line or surface in a high-dimensional parameter space. This observation will provide insight for researchers to design new MORL algorithms.
In this work, we revisit some combinatorial and information-theoretic extension techniques for detecting non-algebraic matroids. These are the Dress-Lov\'asz and Ahlswede-K\"orner extension properties. We provide optimizations of these techniques to reduce their computational complexity, finding new non-algebraic matroids on 9 and 10 points. In addition, we use the Ahlswede-K\"orner extension property to find better lower bounds on the information ratio of secret sharing schemes for ports of non-algebraic matroids.
We illustrate a time and memory efficient application of Runge-Kutta discontinuous Galerkin (RKDG) methods for the simulation of the ultrasounds advection in moving fluids. In particular, this study addresses to the analysis of transit-time ultrasonic meters which rely on the propagation of acoustic waves to measure fluids flow rate. Accurate and efficient simulations of the physics related to the transport of ultrasounds are therefore crucial for studying and enhancing these devices. Starting from the description of the linearized Euler equations (LEE) model and presenting the general theory of explicit-time DG methods for hyperbolic systems, we then motivate the use of a spectral basis and introduce a novel high-accuracy method for the imposition of absorbing and resistive walls which analyses the incident wave direction across the boundary surface. The proposed implementation is both accurate and efficient, making it suitable for industrial applications of acoustic wave propagation.
Leveraging the computing and sensing capabilities of vehicles, vehicular federated learning (VFL) has been applied to edge training for connected vehicles. The dynamic and interconnected nature of vehicular networks presents unique opportunities to harness direct vehicle-to-vehicle (V2V) communications, enhancing VFL training efficiency. In this paper, we formulate a stochastic optimization problem to optimize the VFL training performance, considering the energy constraints and mobility of vehicles, and propose a V2V-enhanced dynamic scheduling (VEDS) algorithm to solve it. The model aggregation requirements of VFL and the limited transmission time due to mobility result in a stepwise objective function, which presents challenges in solving the problem. We thus propose a derivative-based drift-plus-penalty method to convert the long-term stochastic optimization problem to an online mixed integer nonlinear programming (MINLP) problem, and provide a theoretical analysis to bound the performance gap between the online solution and the offline optimal solution. Further analysis of the scheduling priority reduces the original problem into a set of convex optimization problems, which are efficiently solved using the interior-point method. Experimental results demonstrate that compared with the state-of-the-art benchmarks, the proposed algorithm enhances the image classification accuracy on the CIFAR-10 dataset by 3.18% and reduces the average displacement errors on the Argoverse trajectory prediction dataset by 10.21%.
Federated Learning (FL) is a decentralized machine-learning paradigm, in which a global server iteratively averages the model parameters of local users without accessing their data. User heterogeneity has imposed significant challenges to FL, which can incur drifted global models that are slow to converge. Knowledge Distillation has recently emerged to tackle this issue, by refining the server model using aggregated knowledge from heterogeneous users, other than directly averaging their model parameters. This approach, however, depends on a proxy dataset, making it impractical unless such a prerequisite is satisfied. Moreover, the ensemble knowledge is not fully utilized to guide local model learning, which may in turn affect the quality of the aggregated model. Inspired by the prior art, we propose a data-free knowledge distillation} approach to address heterogeneous FL, where the server learns a lightweight generator to ensemble user information in a data-free manner, which is then broadcasted to users, regulating local training using the learned knowledge as an inductive bias. Empirical studies powered by theoretical implications show that, our approach facilitates FL with better generalization performance using fewer communication rounds, compared with the state-of-the-art.
External knowledge is often useful for natural language understanding tasks. We introduce a contextual text representation model called Conceptual-Contextual (CC) embeddings, which incorporates structured knowledge into text representations. Unlike entity embedding methods, our approach encodes a knowledge graph into a context model. CC embeddings can be easily reused for a wide range of tasks just like pre-trained language models. Our model effectively encodes the huge UMLS database by leveraging semantic generalizability. Experiments on electronic health records (EHRs) and medical text processing benchmarks showed our model gives a major boost to the performance of supervised medical NLP tasks.
External knowledge is often useful for natural language understanding tasks. We introduce a contextual text representation model called Conceptual-Contextual (CC) embeddings, which incorporates structured knowledge into text representations. Unlike entity embedding methods, our approach encodes a knowledge graph into a context model. CC embeddings can be easily reused for a wide range of tasks just like pre-trained language models. Our model effectively encodes the huge UMLS database by leveraging semantic generalizability. Experiments on electronic health records (EHRs) and medical text processing benchmarks showed our model gives a major boost to the performance of supervised medical NLP tasks.
It is important to detect anomalous inputs when deploying machine learning systems. The use of larger and more complex inputs in deep learning magnifies the difficulty of distinguishing between anomalous and in-distribution examples. At the same time, diverse image and text data are available in enormous quantities. We propose leveraging these data to improve deep anomaly detection by training anomaly detectors against an auxiliary dataset of outliers, an approach we call Outlier Exposure (OE). This enables anomaly detectors to generalize and detect unseen anomalies. In extensive experiments on natural language processing and small- and large-scale vision tasks, we find that Outlier Exposure significantly improves detection performance. We also observe that cutting-edge generative models trained on CIFAR-10 may assign higher likelihoods to SVHN images than to CIFAR-10 images; we use OE to mitigate this issue. We also analyze the flexibility and robustness of Outlier Exposure, and identify characteristics of the auxiliary dataset that improve performance.
Multi-relation Question Answering is a challenging task, due to the requirement of elaborated analysis on questions and reasoning over multiple fact triples in knowledge base. In this paper, we present a novel model called Interpretable Reasoning Network that employs an interpretable, hop-by-hop reasoning process for question answering. The model dynamically decides which part of an input question should be analyzed at each hop; predicts a relation that corresponds to the current parsed results; utilizes the predicted relation to update the question representation and the state of the reasoning process; and then drives the next-hop reasoning. Experiments show that our model yields state-of-the-art results on two datasets. More interestingly, the model can offer traceable and observable intermediate predictions for reasoning analysis and failure diagnosis, thereby allowing manual manipulation in predicting the final answer.
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