Importance sampling is a popular technique in Bayesian inference: by reweighting samples drawn from a proposal distribution we are able to obtain samples and moment estimates from a Bayesian posterior over some $n$ latent variables. Recent work, however, indicates that importance sampling scales poorly -- in order to accurately approximate the true posterior, the required number of importance samples grows is exponential in the number of latent variables [Chatterjee and Diaconis, 2018]. Massively parallel importance sampling works around this issue by drawing $K$ samples for each of the $n$ latent variables and reasoning about all $K^n$ combinations of latent samples. In principle, we can reason efficiently over $K^n$ combinations of samples by exploiting conditional independencies in the generative model. However, in practice this requires complex algorithms that traverse backwards through the graphical model, and we need separate backward traversals for each computation (posterior expectations, marginals and samples). Our contribution is to exploit the source term trick from physics to entirely avoid the need to hand-write backward traversals. Instead, we demonstrate how to simply and easily compute all the required quantities -- posterior expectations, marginals and samples -- by differentiating through a slightly modified marginal likelihood estimator.
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Robust Markov decision processes (MDPs) are used for applications of dynamic optimization in uncertain environments and have been studied extensively. Many of the main properties and algorithms of MDPs, such as value iteration and policy iteration, extend directly to RMDPs. Surprisingly, there is no known analog of the MDP convex optimization formulation for solving RMDPs. This work describes the first convex optimization formulation of RMDPs under the classical sa-rectangularity and s-rectangularity assumptions. By using entropic regularization and exponential change of variables, we derive a convex formulation with a number of variables and constraints polynomial in the number of states and actions, but with large coefficients in the constraints. We further simplify the formulation for RMDPs with polyhedral, ellipsoidal, or entropy-based uncertainty sets, showing that, in these cases, RMDPs can be reformulated as conic programs based on exponential cones, quadratic cones, and non-negative orthants. Our work opens a new research direction for RMDPs and can serve as a first step toward obtaining a tractable convex formulation of RMDPs.
There are many unsolved problems in vascular image segmentation, including vascular structural connectivity, scarce branches and missing small vessels. Obtaining vessels that preserve their correct topological structures is currently a crucial research issue, as it provides an overall view of one vascular system. In order to preserve the topology and accuracy of vessel segmentation, we proposed a novel Morphology Edge Attention Network (MEA-Net) for the segmentation of vessel-like structures, and an Optimal Geometric Matching Connection (OGMC) model to connect the broken vessel segments. The MEA-Net has an edge attention module that improves the segmentation of edges and small objects by morphology operation extracting boundary voxels on multi-scale. The OGMC model uses the concept of curve touching from differential geometry to filter out fragmented vessel endpoints, and then employs minimal surfaces to determine the optimal connection order between blood vessels. Finally, we calculate the geodesic to repair missing vessels under a given Riemannian metric. Our method achieves superior or competitive results compared to state-of-the-art methods on four datasets of 3D vascular segmentation tasks, both effectively reducing vessel broken and increasing vessel branch richness, yielding blood vessels with a more precise topological structure.
We present a modification to RingCT protocol with stealth addresses that makes it compatible with Delegated Proof of Stake based consensus mechanisms called Delegated RingCT. Our scheme has two building blocks: a customised version of an Integrated Signature and Encryption scheme composed of a public key encryption scheme and two signature schemes (a digital signature and a linkable ring signature); and non-interactive zero knowledge proofs. We give a description of the scheme, security proofs and a prototype implementation whose benchmarking is discussed. Although Delegated RingCT doesn't have the same degree of anonymity as other RingCT constructions, we argue that the benefits that the compatibility with DPoS consensus mechanisms brings constitutes a reasonable trade-off for being able to develop an anonymous decentralised cryptocurrency that is faster and more scalable than existing ones.
Sparse attention as a efficient method can significantly decrease the computation cost, but current sparse attention tend to rely on window self attention which block the global information flow. For this problem, we present Shifted Cross Chunk Attention (SCCA), using different KV shifting strategy to extend respective field in each attention layer. Except, we combine Dilated Attention(DA) and Dilated Neighborhood Attention(DNA) to present Shifted Dilated Attention(SDA). Both SCCA and SDA can accumulate attention results in multi head attention to obtain approximate respective field in full attention. In this paper, we conduct language modeling experiments using different pattern of SCCA and combination of SCCA and SDA. The proposed shifted cross chunk attention (SCCA) can effectively extend large language models (LLMs) to longer context combined with Positional interpolation(PI) and LoRA than current sparse attention. Notably, SCCA adopts LLaMA2 7B from 4k context to 8k in single V100. This attention pattern can provide a Plug-and-play fine-tuning method to extend model context while retaining their original architectures, and is compatible with most existing techniques.
The impact of outliers and anomalies on model estimation and data processing is of paramount importance, as evidenced by the extensive body of research spanning various fields over several decades: thousands of research papers have been published on the subject. As a consequence, numerous reviews, surveys, and textbooks have sought to summarize the existing literature, encompassing a wide range of methods from both the statistical and data mining communities. While these endeavors to organize and summarize the research are invaluable, they face inherent challenges due to the pervasive nature of outliers and anomalies in all data-intensive applications, irrespective of the specific application field or scientific discipline. As a result, the resulting collection of papers remains voluminous and somewhat heterogeneous. To address the need for knowledge organization in this domain, this paper implements the first systematic meta-survey of general surveys and reviews on outlier and anomaly detection. Employing a classical systematic survey approach, the study collects nearly 500 papers using two specialized scientific search engines. From this comprehensive collection, a subset of 56 papers that claim to be general surveys on outlier detection is selected using a snowball search technique to enhance field coverage. A meticulous quality assessment phase further refines the selection to a subset of 25 high-quality general surveys. Using this curated collection, the paper investigates the evolution of the outlier detection field over a 20-year period, revealing emerging themes and methods. Furthermore, an analysis of the surveys sheds light on the survey writing practices adopted by scholars from different communities who have contributed to this field. Finally, the paper delves into several topics where consensus has emerged from the literature. These include taxonomies of outlier types, challenges posed by high-dimensional data, the importance of anomaly scores, the impact of learning conditions, difficulties in benchmarking, and the significance of neural networks. Non-consensual aspects are also discussed, particularly the distinction between local and global outliers and the challenges in organizing detection methods into meaningful taxonomies.
In survival analysis, complex machine learning algorithms have been increasingly used for predictive modeling. Given a collection of features available for inclusion in a predictive model, it may be of interest to quantify the relative importance of a subset of features for the prediction task at hand. In particular, in HIV vaccine trials, participant baseline characteristics are used to predict the probability of infection over the intended follow-up period, and investigators may wish to understand how much certain types of predictors, such as behavioral factors, contribute toward overall predictiveness. Time-to-event outcomes such as time to infection are often subject to right censoring, and existing methods for assessing variable importance are typically not intended to be used in this setting. We describe a broad class of algorithm-agnostic variable importance measures for prediction in the context of survival data. We propose a nonparametric efficient estimation procedure that incorporates flexible learning of nuisance parameters, yields asymptotically valid inference, and enjoys double-robustness. We assess the performance of our proposed procedure via numerical simulations and analyze data from the HVTN 702 study to inform enrollment strategies for future HIV vaccine trials.
The standard mathematical approach to fourth-down decision making in American football is to make the decision that maximizes estimated win probability. Win probability estimates arise from a statistical model fit from historical data. These machine learning models, however, are overfit high-variance estimators, exacerbated by the highly correlated nature of football play-by-play data. We develop a machine learning framework that accounts for this auto-correlation and knits uncertainty quantification into our decision making. In particular, we recommend a fourth-down decision when we are confident it has higher win probability than all other decisions. Our final product is a major advance in fourth-down strategic decision making: far fewer fourth-down decisions are as obvious as analysts claim.
We adopt the integral definition of the fractional Laplace operator and study an optimal control problem on Lipschitz domains that involves a fractional elliptic partial differential equation (PDE) as state equation and a control variable that enters the state equation as a coefficient; pointwise constraints on the control variable are considered as well. We establish the existence of optimal solutions and analyze first and, necessary and sufficient, second order optimality conditions. Regularity estimates for optimal variables are also analyzed. We develop two finite element discretization strategies: a semidiscrete scheme in which the control variable is not discretized, and a fully discrete scheme in which the control variable is discretized with piecewise constant functions. For both schemes, we analyze the convergence properties of discretizations and derive error estimates.
We present a new stability and error analysis of fully discrete approximation schemes for the transient Stokes equation. For the spatial discretization, we consider a wide class of Galerkin finite element methods which includes both inf-sup stable spaces and symmetric pressure stabilized formulations. We extend the results from Burman and Fern\'andez [\textit{SIAM J. Numer. Anal.}, 47 (2009), pp. 409-439] and provide a unified theoretical analysis of backward difference formulae (BDF methods) of order 1 to 6. The main novelty of our approach lies in the use of Dahlquist's G-stability concept together with multiplier techniques introduced by Nevannlina-Odeh and recently by Akrivis et al. [\textit{SIAM J. Numer. Anal.}, 59 (2021), pp. 2449-2472] to derive optimal stability and error estimates for both the velocity and the pressure. When combined with a method dependent Ritz projection for the initial data, unconditional stability can be shown while for arbitrary interpolation, pressure stability is subordinate to the fulfillment of a mild inverse CFL-type condition between space and time discretizations.
This paper considers the problem of robust iterative Bayesian smoothing in nonlinear state-space models with additive noise using Gaussian approximations. Iterative methods are known to improve smoothed estimates but are not guaranteed to converge, motivating the development of more robust versions of the algorithms. The aim of this article is to present Levenberg-Marquardt (LM) and line-search extensions of the classical iterated extended Kalman smoother (IEKS) as well as the iterated posterior linearisation smoother (IPLS). The IEKS has previously been shown to be equivalent to the Gauss-Newton (GN) method. We derive a similar GN interpretation for the IPLS. Furthermore, we show that an LM extension for both iterative methods can be achieved with a simple modification of the smoothing iterations, enabling algorithms with efficient implementations. Our numerical experiments show the importance of robust methods, in particular for the IEKS-based smoothers. The computationally expensive IPLS-based smoothers are naturally robust but can still benefit from further regularisation.
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