Probabilistic Guarantees Research Articles

Machine learning enabled uncertainty set for data-driven robust optimization

The way how the uncertainties are represented by sets plays a vital role in the performance of robust optimization (RO). This paper presents a novel approach leveraging machine learning (ML) techniques to construct data-driven uncertainty sets from historical uncertainty data for RO problems. The proposed method integrates Density-Based Spatial Clustering of Applications with Noise (DBSCAN), Gaussian Mixture Model (GMM), and Principle Component Analysis (PCA) systematically to eliminate the influence of uncertainty scenarios with low occurrence probability and generate a nonconvex uncertainty set that is a union of multiple basic subsets (box or ellipsoid) without sacrificing its computational tractability. In addition to presenting a comprehensive algorithm for uncertainty set development, this paper offers detailed guidelines for parameter tuning and performance analysis. By harnessing the well-established ML packages scikit-learn, a Python-based toolkit for implementing the proposed approach is also provided. Furthermore, a computationally efficient solution for a two-stage linear RO problem with the proposed data-driven uncertainty set is derived, alongside establishing a probabilistic guarantee of constraint satisfaction for out-of-sample uncertainties. Extensive numerical experiments, conducted on both synthetic and real-world datasets as well as an optimization-based control problem, are performed to demonstrate the efficacy of the proposed methodology.

A stochastic Bregman golden ratio algorithm for non-Lipschitz stochastic mixed variational inequalities with application to resource share problems

In the study of stochastic mixed variational inequalities(SMVIs), Lipschitz is an indispensable assumption for the convergence analysis. However, practical applications may not satisfy this assumption. In this paper, we propose a stochastic Bregman golden ratio algorithm for solving non-Lipschitz SMVIs. Since our algorithm only requires to calculate one stochastic approximation of the expected mapping per iteration, the computations can be reduced. Under some moderate conditions, we prove the almost surely convergence of the iteration sequence and the O(1/K) convergence rate, where K denotes the maximum iteration. Furthermore, we derive the probabilities of large deviation results, which provide a high probability guarantee for the convergence of the proposed algorithm. Numerical experiments on Logistic regression problems and modified entropy regularized LP boosting problems show that our algorithm is competitive compared with some existing algorithms. Finally, we apply our algorithm to solve a non-Lipschitz resource sharing problem.

Optical circuit switched three-stage twisted-folded Clos-network design model guaranteeing admissible blocking probability

Some data center networks have already started to use optical circuit switching (OCS) with potential performance benefits, including high capacity, low latency, and energy efficiency. This paper addresses a switching network design to maximize the network radix, i.e., the number of terminals connected to the network under the condition that a specified number of identical switches with the size N×N and the maximum admissible blocking probability are given. Previous work presented a two-stage twisted and folded Clos network (TF-Clos) with a blocking probability guarantee for OCS, which has a larger network radix than TF-Clos with a strict-sense non-blocking condition. Expanding the number of stages allows for enhancing the network radix. This paper proposes a model designing an OCS three-stage TF-Clos structure with a blocking probability guarantee to increase the network radix compared to the two-stage TF-Clos. We formulate the problem of obtaining the network configuration that maximizes the network radix as an optimization problem. We conduct an algorithm based on an exhaustive search to obtain a feasible solution satisfying the constraints of the optimization problem. This algorithm identifies the structure with the largest network radix in non-increasing order to avoid unnecessary searches. Numerical results show that the proposed model achieves a larger network radix than the two-stage model.

Efficient Approximation Algorithms for Minimum Cost Seed Selection with Probabilistic Coverage Guarantee

Given a social network G , a cost associated with each user, and an influence threshold η, the minimum cost seed selection problem (MCSS) aims to find a set of seeds that minimizes the total cost to reach η users. Existing works are mainly devoted to providing an <u>e</u>xpected <u>c</u>overage <u>g</u>uarantee on reaching η, classified as MCSS-ECG, where their solutions either rely on an impractical influence oracle or cannot attain the expected influence threshold. More importantly, due to the expected coverage guarantee, the actual influence in a campaign may drift from the threshold evidently. Thus, the advertisers would like to request for a probability guarantee of reaching η. This motivates us to further solve the MCSS problem with a <u>p</u>robabilistic <u>c</u>overage <u>g</u>uarantee, termed MCSS-PCG. In this paper, we first propose our algorithm CLEAR to solve MCSS-ECG, which reaches the expected influence threshold without any influence oracle or influence shortfall but a practical approximation ratio. However, the ratio involves an unknown term (i.e., the optimal cost). Thus, we further devise the STAR method to derive a lower bound of the optimal cost and then obtain the first explicit approximation ratio for MCSS-ECG. In MCSS-PCG, it is necessary to estimate the probability that the current seeds reach η, to decide when to stop seed selection. To achieve this, we design a new technique named MRR, which provides efficient probability estimation with a theoretical guarantee. With MRR in hand, we propose our algorithm SCORE for MCSS-PCG, whose performance guarantee is derived by measuring the gap between MCSS-ECG and MCSS-PCG, and applying the theoretical results in MCSS-ECG. Finally, extensive experiments demonstrate that our algorithms achieve up to two orders of magnitude speed-up compared to alternatives while meeting the requirement of MCSS-PCG with the smallest cost.

Changes in precipitation phases based on the multi-discrimination method in the Tibetan Plateau

Global warming is rapidly changing the precipitation structure, and the shift in precipitation phases is even more pronounced in the Tibetan Plateau (TP), a region sensitive to global climate change. The accuracy of precipitation phase discrimination is essential for studying precipitation variability. Based on the observed data from 112 meteorological stations in the study area obtained from the China Meteorological Administration (CMA) and the National Climatic Data Center (NCDC) before 1980, we compared and assessed six methods for distinguishing precipitation phases and the optimal method was determined to complete the precipitation phase data after 1980 in the study area. Subsequently, the changes in precipitation amount, intensity, and frequency were analyzed by classifying them into light, moderate, heavy, and extreme precipitation using the percentile threshold discrimination method. It found that the dynamic threshold temperature method (presented by Ding et al., 2014) was the most effective to distinguish precipitation phases in the TP, and its performance was further improved from 80.9% to 86.1% after using frequency intersection and probability guarantee methods. Precipitation was mainly concentrated in July. Meanwhile, the proportion of snowfall to total precipitation (S/P) was approximately 9%. From 1961 to 2020, the amount of total precipitation, rainfall and snowfall in the TP exhibited increasing trends, while the sleet demonstrated a persistent decline. Spatial and temporal heterogeneity existed in the changes of precipitation of different phases. The average single precipitation amount increased. Moreover, the changes in the amount and frequency of extreme precipitation were more pronounced. Increased frequency and amount of both rainfall and snowfall were distributed in the central and eastern parts of the study area with a more significant proportion. In contrast, the peripheral areas witnessed a decrease. The spatial distribution of frequency and amount of different precipitation intensities was comparable to that of the total precipitation. This study offers novel procedure into changes of precipitation phases at high altitudes, which will inform future research on climate change and water resource management.

Part-X: A Family of Stochastic Algorithms for Search-Based Test Generation With Probabilistic Guarantees

Part-X: A Family of Stochastic Algorithms for Search-Based Test Generation With Probabilistic Guarantees

Robust Corrupted Data Recovery and Clustering via Generalized Transformed Tensor Low-Rank Representation.

Tensor analysis has received widespread attention in high-dimensional data learning. Unfortunately, the tensor data are often accompanied by arbitrary signal corruptions, including missing entries and sparse noise. How to recover the characteristics of the corrupted tensor data and make it compatible with the downstream clustering task remains a challenging problem. In this article, we study a generalized transformed tensor low-rank representation (TTLRR) model for simultaneously recovering and clustering the corrupted tensor data. The core idea is to find the latent low-rank tensor structure from the corrupted measurements using the transformed tensor singular value decomposition (SVD). Theoretically, we prove that TTLRR can recover the clean tensor data with a high probability guarantee under mild conditions. Furthermore, by using the transform adaptively learning from the data itself, the proposed TTLRR model can approximately represent and exploit the intrinsic subspace and seek out the cluster structure of the tensor data precisely. An effective algorithm is designed to solve the proposed model under the alternating direction method of multipliers (ADMMs) algorithm framework. The effectiveness and superiority of the proposed method against the compared methods are showcased over different tasks, including video/face data recovery and face/object/scene data clustering.

A novel surrogate polytope method for day-ahead virtual power plant scheduling with joint probabilistic constraints

Virtual Power Plant (VPP) is an emerging concept that can effectively manage a large number of distributed energy resources (DERs). However, the inherent uncertainty of load and renewable generation poses a challenge to reliable VPP power scheduling. This manuscript proposes a novel method for day-ahead VPP scheduling with a joint probabilistic guarantee on its power availability without violating the DER constraints and network constraints. A surrogate polytope is first used to find the inner approximation of the VPP power, implicitly including the low-level DER power, DER constraints, and network constraints. Then, a multivariate Gaussian distribution is used to fit the random parameters of the surrogate polytope, after which the iterative supporting hyperplane algorithm is used to solve the VPP scheduling problem. Extensive case studies based on real-world renewable generation scenarios demonstrate the superior performance of the proposed method in out-of-sample cost and reliability, with a manageable computing complexity.

Set-valued rigid-body dynamics for simultaneous, inelastic, frictional impacts

Robotic manipulation and locomotion often entail nearly-simultaneous collisions—such as heel and toe strikes during a foot step—with outcomes that are extremely sensitive to the order in which impacts occur. Robotic simulators and state estimation commonly lack the fidelity and accuracy to predict this ordering, and instead pick one with a heuristic. This discrepancy degrades performance when model-based controllers and policies learned in simulation are placed on a real robot. We reconcile this issue with a set-valued rigid-body model which generates a broad set of outcomes to simultaneous frictional impacts with any impact ordering. We first extend Routh’s impact model to multiple impacts by reformulating it as a differential inclusion (DI), and show that any solution will resolve all impacts in finite time. By considering time as a state, we embed this model into another DI which captures the continuous-time evolution of rigid-body dynamics, and guarantee existence of solutions. We finally cast simulation of simultaneous impacts as a linear complementarity problem (LCP), and develop an algorithm for tight approximation of the post-impact velocity set with probabilistic guarantees. We demonstrate our approach on several examples drawn from manipulation and legged locomotion, and compare the predictions to other models of rigid and compliant collisions.

DET-LSH: A Locality-Sensitive Hashing Scheme with Dynamic Encoding Tree for Approximate Nearest Neighbor Search

Locality-sensitive hashing (LSH) is a well-known solution for approximate nearest neighbor (ANN) search in high-dimensional spaces due to its robust theoretical guarantee on query accuracy. Traditional LSH-based methods mainly focus on improving the efficiency and accuracy of the query phase by designing different query strategies, but pay little attention to improving the efficiency of the indexing phase. They typically fine-tune existing data-oriented partitioning trees to index data points and support their query strategies. However, their strategy to directly partition the multi-dimensional space is time-consuming, and performance degrades as the space dimensionality increases. In this paper, we design an encoding-based tree called Dynamic Encoding Tree (DE-Tree) to improve the indexing efficiency and support efficient range queries based on Euclidean distance. Based on DE-Tree, we propose a novel LSH scheme called DET-LSH. DET-LSH adopts a novel query strategy, which performs range queries in multiple independent index DE-Trees to reduce the probability of missing exact NN points, thereby improving the query accuracy. Our theoretical studies show that DET-LSH enjoys probabilistic guarantees on query accuracy. Extensive experiments on real-world datasets demonstrate the superiority of DET-LSH over the state-of-the-art LSH-based methods on both efficiency and accuracy. While achieving better query accuracy than competitors, DET-LSH achieves up to 6x speedup in indexing time and 2x speedup in query time over the state-of-the-art LSH-based methods.

Statistical Verification using Surrogate Models and Conformal Inference and a Comparison with Risk-Aware Verification

Uncertainty in safety-critical cyber-physical systems can be modeled using a finite number of parameters or parameterized input signals. Given a system specification in Signal Temporal Logic (STL), we would like to verify that for all (infinite) values of the model parameters/input signals, the system satisfies its specification. Unfortunately, this problem is undecidable in general. Statistical model checking (SMC) offers a solution by providing guarantees on the correctness of CPS models by statistically reasoning on model simulations. We propose a new approach for statistical verification of CPS models for user-provided distribution on the model parameters. Our technique uses model simulations to learn surrogate models , and uses conformal inference to provide probabilistic guarantees on the satisfaction of a given STL property. Additionally, we can provide prediction intervals containing the quantitative satisfaction values of the given STL property for any user-specified confidence level. We compare this prediction interval with the interval we get using risk estimation procedures. We also propose a refinement procedure based on Gaussian Process (GP)-based surrogate models for obtaining fine-grained probabilistic guarantees over sub-regions in the parameter space. This in turn enables the CPS designer to choose assured validity domains in the parameter space for safety-critical applications. Finally, we demonstrate the efficacy of our technique on several CPS models.

Fatty liver classification via risk controlled neural networks trained on grouped ultrasound image data

Ultrasound imaging is a widely used technique for fatty liver diagnosis as it is practically affordable and can be quickly deployed by using suitable devices. When it is applied to a patient, multiple images of the targeted tissues are produced. We propose a machine learning model for fatty liver diagnosis from multiple ultrasound images. The machine learning model extracts features of the ultrasound images by using a pre-trained image encoder. It further produces a summary embedding on these features by using a graph neural network. The summary embedding is used as input for a classifier on fatty liver diagnosis. We train the machine learning model on a ultrasound image dataset collected by Taiwan Biobank. We also carry out risk control on the machine learning model using conformal prediction. Under the risk control procedure, the classifier can improve the results with high probabilistic guarantees.

Simplifying Complex Observation Models in Continuous POMDP Planning with Probabilistic Guarantees and Practice

Solving partially observable Markov decision processes (POMDPs) with high dimensional and continuous observations, such as camera images, is required for many real life robotics and planning problems. Recent researches suggested machine learned probabilistic models as observation models, but their use is currently too computationally expensive for online deployment. We deal with the question of what would be the implication of using simplified observation models for planning, while retaining formal guarantees on the quality of the solution. Our main contribution is a novel probabilistic bound based on a statistical total variation distance of the simplified model. We show that it bounds the theoretical POMDP value w.r.t. original model, from the empirical planned value with the simplified model, by generalizing recent results of particle-belief MDP concentration bounds. Our calculations can be separated into offline and online parts, and we arrive at formal guarantees without having to access the costly model at all during planning, which is also a novel result. Finally, we demonstrate in simulation how to integrate the bound into the routine of an existing continuous online POMDP solver.

Monitoring of Perception Systems: Deterministic, Probabilistic, and Learning-Based Fault Detection and Identification (Abstract Reprint)

This paper investigates runtime monitoring of perception systems. Perception is a critical component of high-integrity applications of robotics and autonomous systems, such as self-driving cars. In these applications, failure of perception systems may put human life at risk, and a broad adoption of these technologies requires the development of methodologies to guarantee and monitor safe operation. Despite the paramount importance of perception, currently there is no formal approach for system-level perception monitoring. In this paper, we formalize the problem of runtime fault detection and identification in perception systems and present a framework to model diagnostic information using a diagnostic graph. We then provide a set of deterministic, probabilistic, and learning-based algorithms that use diagnostic graphs to perform fault detection and identification. Moreover, we investigate fundamental limits and provide deterministic and probabilistic guarantees on the fault detection and identification results. We conclude the paper with an extensive experimental evaluation, which recreates several realistic failure modes in the LGSVL open-source autonomous driving simulator, and applies the proposed system monitors to a state-of-the-art autonomous driving software stack (Baidu's Apollo Auto). The results show that the proposed system monitors outperform baselines, have the potential of preventing accidents in realistic autonomous driving scenarios, and incur a negligible computational overhead.

Enumerating Safe Regions in Deep Neural Networks with Provable Probabilistic Guarantees

Identifying safe areas is a key point to guarantee trust for systems that are based on Deep Neural Networks (DNNs). To this end, we introduce the AllDNN-Verification problem: given a safety property and a DNN, enumerate the set of all the regions of the property input domain which are safe, i.e., where the property does hold. Due to the #P-hardness of the problem, we propose an efficient approximation method called ε-ProVe. Our approach exploits a controllable underestimation of the output reachable sets obtained via statistical prediction of tolerance limits, and can provide a tight —with provable probabilistic guarantees— lower estimate of the safe areas. Our empirical evaluation on different standard benchmarks shows the scalability and effectiveness of our method, offering valuable insights for this new type of verification of DNNs.

Study of post-earthquake train running performance considering earthquake-induced track irregularity

It is necessary to study the dynamic performance of high-speed trains to obtain the safe operation. In this paper, a seismic analysis of a track-bridge system subjected to near-field random earthquakes is carried out, and the geometric distribution of the residual and dynamic irregularities of rails after earthquake is explored. Based on the reliability theory and probability guarantee rate, an equivalent coefficient of earthquake-induced dynamic irregularity is proposed. The dynamic analysis of the train-track-bridge system is carried out, considering earthquake-induced track irregularity, and the speed limit of high-speed trains after the seismic event is proposed, ensuring the running safety. The results show that the amplitude of residual irregularity is much larger than that of the dynamic irregularity. The influence of the earthquake-induced track irregularity on the derailment coefficient and transverse acceleration of high-speed trains is significant increasing with the increase of the train operation speed. The earthquake-induced track irregularity has small effect on the vertical dynamic performance of high-speed trains. Based on probability guarantee rate, considering the random initial track irregularity, the operation speed limit of high-speed trains is set to 72 km/h. The research results can provide certain theoretical basis for the post-earthquake running safety evaluation of high-speed trains.

Is Bitcoin Future as Secure as We Think? Analysis of Bitcoin Vulnerability to Bribery Attacks Launched through Large Transactions

Bitcoin uses blockchain technology to maintain transactions order and provides probabilistic guarantees to prevent double-spending, assuming that an attacker’s computational power does not exceed 50% of the network power. In this article, we design a novel bribery attack and show that this guarantee can be hugely undermined. Miners are assumed to be rational in this setup, and they are given incentives that are dynamically calculated. In this attack, the adversary misuses the Bitcoin protocol to bribe miners and maximize their gained advantage. We will reformulate the bribery attack to propose a general mathematical foundation upon which we build multiple strategies. We show that, unlike Whale Attack, these strategies are practical, especially in the future when halvings lower the mining rewards. In the so-called “guaranteed variable-rate bribing with commitment” strategy, through optimization by Differential Evolution (DE), we show how double-spending is possible in the Bitcoin ecosystem for any transaction whose value is above 218.9BTC, and this comes with 100% success rate. A slight reduction in the success probability, e.g., by 10%, brings the threshold down to 165BTC. If the rationality assumption holds, then this shows how vulnerable blockchain-based systems like Bitcoin are. We suggest a soft fork on Bitcoin to fix this issue at the end.

Impact of vertical girder deformations on the high-speed train operation performance

The deformation of bridge structures will cause deterioration of geometry on the rail surface during the service period, which will threaten the normal operation of train. A train–track–bridge coupling dynamics model was established. In order to simulate real operational conditions, initial geometric irregularity was taken into account, which was combined with rail irregularity caused by different vertical deformation of girder and was superimposed as external excitation. The probabilistic and statistical characteristics of operational performance indexes were analysed. The sensitivity of operational performance indexes to different deformation modes was studied. The vertical deformation thresholds of girder based on probability guarantee rate were obtained. The results show that unloading rate of wheel and wheel/track vertical force are more sensitive to single vertical rotation deformation mode. Vertical acceleration of the train and sperling index are more sensitive to bilateral vertical rotation deformation mode. Unloading rate of wheel is more sensitive to single vertical translation deformation mode. In the process of enhancing daily operation, single vertical rotation and multiple vertical translation should receive significant attention.

Twisted and Folded Clos-Network Design Model With Two-Step Blocking Probability Guarantee

Twisted and Folded Clos-Network Design Model With Two-Step Blocking Probability Guarantee

Analysis of Key Influencing Factors of Track Dynamic Irregularity Induced by Earthquakes in the High-Speed Railway Track–Bridge System

Stiffness degradation of track–bridge systems under earthquake excitations can lead to track dynamic irregularity, impacting the safety of train operations post-earthquake. In this paper, the nonlinear time-history analysis of CRTS II track–bridge system model established by ANSYS finite element analysis software is carried out under the action of transverse random earthquake. The train–track–bridge system model considering the stiffness degradation caused by earthquake is established by MATLAB, and the earthquake-induced track dynamic irregularity sample library is constructed. The probability distribution mode of the power spectral density sample of the earthquake-induced track dynamic irregularity is explored and the calculation method based on the probability guarantee rate is proposed. The influence of structural damping ratio, train running speed and train type on the power spectral density curve of the earthquake-induced track dynamic irregularity is analyzed. The results show that the power spectrum samples of track dynamic irregularity conform to the hypothesis test of a normal distribution. The power spectral density of earthquake-induced dynamic irregularity primarily consists of medium and low-frequency components. When the structural damping ratio increases from 0.03 to 0.04, 0.04 to 0.05, 0.05 to 0.06, and 0.06 to 0.07, the variation gradients are 0.1701, 0.1240, 0.1034 and 0.0999, respectively, which indicates that the structural damping ratio has a significant effect on the power spectral density of earthquake-induced irregularity, and the impacts of train speed and train type on the power spectral density of near-fault earthquake-induced irregularity are minimal.