Existing inferential methods for small area data involve a trade-off between maintaining area-level frequentist coverage rates and improving inferential precision via the incorporation of indirect information. In this article, we propose a method to obtain an area-level prediction region for a future observation which mitigates this trade-off. The proposed method takes a conformal prediction approach in which the conformity measure is the posterior predictive density of a working model that incorporates indirect information. The resulting prediction region has guaranteed frequentist coverage regardless of the working model, and, if the working model assumptions are accurate, the region has minimum expected volume compared to other regions with the same coverage rate. When constructed under a normal working model, we prove such a prediction region is an interval and construct an efficient algorithm to obtain the exact interval. We illustrate the performance of our method through simulation studies and an application to EPA radon survey data.
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Functional linear and single-index models are core regression methods in functional data analysis and are widely used methods for performing regression when the covariates are observed random functions coupled with scalar responses in a wide range of applications. In the existing literature, however, the construction of associated estimators and the study of their theoretical properties is invariably carried out on a case-by-case basis for specific models under consideration. In this work, we provide a unified methodological and theoretical framework for estimating the index in functional linear and single-index models; in the later case the proposed approach does not require the specification of the link function. In terms of methodology, we show that the reproducing kernel Hilbert space (RKHS) based functional linear least-squares estimator, when viewed through the lens of an infinite-dimensional Gaussian Stein's identity, also provides an estimator of the index of the single-index model. On the theoretical side, we characterize the convergence rates of the proposed estimators for both linear and single-index models. Our analysis has several key advantages: (i) we do not require restrictive commutativity assumptions for the covariance operator of the random covariates on one hand and the integral operator associated with the reproducing kernel on the other hand; and (ii) we also allow for the true index parameter to lie outside of the chosen RKHS, thereby allowing for index mis-specification as well as for quantifying the degree of such index mis-specification. Several existing results emerge as special cases of our analysis.
We develop an \textit{a posteriori} error analysis for a numerical estimate of the time at which a functional of the solution to a partial differential equation (PDE) first achieves a threshold value on a given time interval. This quantity of interest (QoI) differs from classical QoIs which are modeled as bounded linear (or nonlinear) functionals {of the solution}. Taylor's theorem and an adjoint-based \textit{a posteriori} analysis is used to derive computable and accurate error estimates in the case of semi-linear parabolic and hyperbolic PDEs. The accuracy of the error estimates is demonstrated through numerical solutions of the one-dimensional heat equation and linearized shallow water equations (SWE), representing parabolic and hyperbolic cases, respectively.
Estimating an individual treatment effect (ITE) is essential to personalized decision making. However, existing methods for estimating the ITE often rely on unconfoundedness, an assumption that is fundamentally untestable with observed data. To assess the robustness of individual-level causal conclusion with unconfoundedness, this paper proposes a method for sensitivity analysis of the ITE, a way to estimate a range of the ITE under unobserved confounding. The method we develop quantifies unmeasured confounding through a marginal sensitivity model [Ros2002, Tan2006], and adapts the framework of conformal inference to estimate an ITE interval at a given confounding strength. In particular, we formulate this sensitivity analysis problem as a conformal inference problem under distribution shift, and we extend existing methods of covariate-shifted conformal inference to this more general setting. The result is a predictive interval that has guaranteed nominal coverage of the ITE, a method that provides coverage with distribution-free and nonasymptotic guarantees. We evaluate the method on synthetic data and illustrate its application in an observational study.
We consider a linear stochastic bandit problem involving $M$ agents that can collaborate via a central server to minimize regret. A fraction $\alpha$ of these agents are adversarial and can act arbitrarily, leading to the following tension: while collaboration can potentially reduce regret, it can also disrupt the process of learning due to adversaries. In this work, we provide a fundamental understanding of this tension by designing new algorithms that balance the exploration-exploitation trade-off via carefully constructed robust confidence intervals. We also complement our algorithms with tight analyses. First, we develop a robust collaborative phased elimination algorithm that achieves $\tilde{O}\left(\alpha+ 1/\sqrt{M}\right) \sqrt{dT}$ regret for each good agent; here, $d$ is the model-dimension and $T$ is the horizon. For small $\alpha$, our result thus reveals a clear benefit of collaboration despite adversaries. Using an information-theoretic argument, we then prove a matching lower bound, thereby providing the first set of tight, near-optimal regret bounds for collaborative linear bandits with adversaries. Furthermore, by leveraging recent advances in high-dimensional robust statistics, we significantly extend our algorithmic ideas and results to (i) the generalized linear bandit model that allows for non-linear observation maps; and (ii) the contextual bandit setting that allows for time-varying feature vectors.
Multiple hypothesis testing has been widely applied to problems dealing with high-dimensional data, e.g., selecting significant variables and controlling the selection error rate. The most prevailing measure of error rate used in the multiple hypothesis testing is the false discovery rate (FDR). In recent years, local false discovery rate (fdr) has drawn much attention, due to its advantage of accessing the confidence of individual hypothesis. However, most methods estimate fdr through p-values or statistics with known null distributions, which are sometimes not available or reliable. Adopting the innovative methodology of competition-based procedures, e.g., knockoff filter, this paper proposes a new approach, named TDfdr, to local false discovery rate estimation, which is free of the p-values or known null distributions. Simulation results demonstrate that TDfdr can accurately estimate the fdr with two competition-based procedures. In real data analysis, the power of TDfdr on variable selection is verified on two biological datasets.
It is known that fiber nonlinearities induce crosstalk in a wavelength division multiplexed (WDM) system, which limits the capacity of such systems as the transmitted signal power is increased. A network user in a WDM system is an entity that operates around a given optical wavelength. Traditionally, the channel capacity of a WDM system has been analyzed under different assumptions for the transmitted signals of the other users, while treating the interference arising from these users as noise. In this paper, we instead take a multiuser information theoretic view and treat the optical WDM system impaired by cross-phase modulation and dispersion as an interference channel. We characterize an outer bound on the capacity region of simultaneously achievable rate pairs, assuming a simplified K-user perturbative channel model using genie-aided techniques. Furthermore, an achievable rate region is obtained by time-sharing between certain single-user strategies. It is shown that such time-sharing can achieve better rate tuples compared to treating nonlinear interference as noise. For the single-polarization single-span system under consideration and a power 4.4 dB above the optimum launch power, treating nonlinear interference as noise results in a rate of 1.67 bit/sym, while time-sharing gives a rate of 6.33 bit/sym.
Reconfigurable intelligent surface (RIS) can effectively control the wavefront of the impinging signals and has emerged as a cost-effective promising solution to improve the spectrum and energy efficiency of wireless systems. Most existing researches on RIS assume that the hardware operations are perfect. However, both physical transceiver and RIS suffer from inevitable hardware impairments in practice, which can lead to severe system performance degradation and increase the complexity of beamforming optimization. Consequently, the existing researches on RIS, including channel estimation, beamforming optimization, spectrum and energy efficiency analysis, etc., cannot directly apply to the case of hardware impairments. In this paper, by taking hardware impairments into consideration, we conduct the joint transmit and reflect beamforming optimization, and reevaluate the system performance. First, we characterize the closed-form estimators of direct and cascaded channels in both single-user and multi-user cases, and analyze the impact of hardware impairments on channel estimation accuracy. Then, the optimal transmit beamforming solution is derived, and a gradient descent method-based algorithm is also proposed to optimize the reflect beamforming. Moreover, we analyze the three types of asymptotic channel capacities with respect to the transmit power, the antenna number, and the reflecting element number. Finally, in terms of the system energy consumption, we analyze the power scaling law and the energy efficiency. Our experimental results also reveal an encouraging phenomenon that the RIS-assisted wireless system with massive reflecting elements can achieve both high spectrum and energy efficiency without the need for massive antennas and without allocating too many resources to optimize the reflect beamforming.
How accurate models of human behavior are needed for human-robot interaction? For automated driving?
There are many examples of cases where access to improved models of human behavior and cognition has allowed creation of robots which can better interact with humans, and not least in road vehicle automation this is a rapidly growing area of research. Human-robot interaction (HRI) therefore provides an important applied setting for human behavior modeling - but given the vast complexity of human behavior, how complete and accurate do these models need to be? Here, we outline some possible ways of thinking about this problem, starting from the suggestion that modelers need to keep the right end goal in sight: A successful human-robot interaction, in terms of safety, performance, and human satisfaction. Efforts toward model completeness and accuracy should be focused on those aspects of human behavior to which interaction success is most sensitive. We emphasise that identifying which those aspects are is a difficult scientific objective in its own right, distinct for each given HRI context. We propose and exemplify an approach to formulating a priori hypotheses on this matter, in cases where robots are to be involved in interactions which currently take place between humans, such as in automated driving. Our perspective also highlights some possible risks of overreliance on machine-learned models of human behavior in HRI, and how to mitigate against those risks.
Label Propagation (LPA) and Graph Convolutional Neural Networks (GCN) are both message passing algorithms on graphs. Both solve the task of node classification but LPA propagates node label information across the edges of the graph, while GCN propagates and transforms node feature information. However, while conceptually similar, theoretical relation between LPA and GCN has not yet been investigated. Here we study the relationship between LPA and GCN in terms of two aspects: (1) feature/label smoothing where we analyze how the feature/label of one node is spread over its neighbors; And, (2) feature/label influence of how much the initial feature/label of one node influences the final feature/label of another node. Based on our theoretical analysis, we propose an end-to-end model that unifies GCN and LPA for node classification. In our unified model, edge weights are learnable, and the LPA serves as regularization to assist the GCN in learning proper edge weights that lead to improved classification performance. Our model can also be seen as learning attention weights based on node labels, which is more task-oriented than existing feature-based attention models. In a number of experiments on real-world graphs, our model shows superiority over state-of-the-art GCN-based methods in terms of node classification accuracy.
Graph convolutional networks (GCNs) have been successfully applied in node classification tasks of network mining. However, most of these models based on neighborhood aggregation are usually shallow and lack the "graph pooling" mechanism, which prevents the model from obtaining adequate global information. In order to increase the receptive field, we propose a novel deep Hierarchical Graph Convolutional Network (H-GCN) for semi-supervised node classification. H-GCN first repeatedly aggregates structurally similar nodes to hyper-nodes and then refines the coarsened graph to the original to restore the representation for each node. Instead of merely aggregating one- or two-hop neighborhood information, the proposed coarsening procedure enlarges the receptive field for each node, hence more global information can be learned. Comprehensive experiments conducted on public datasets demonstrate the effectiveness of the proposed method over the state-of-art methods. Notably, our model gains substantial improvements when only a few labeled samples are provided.
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