Probabilistic Guarantees Research Articles

The scenario approach for data-driven prognostics

Prognostics is the process of forecasting the time-to-failure or the time-to-alarm of an industrial item using degradation models. Data-driven approaches to prognostics employ regression models ft on condition indicators computed from raw run-to-failure data to extrapolate the degradation behaviour of the item. The development of a reliable data-driven degradation model typically requires many run-to-failure acquisitions to understand the degrading behavior. Such experimental tests are destructive and expensive for items manufacturers. Thus, decreasing the number of run-to-failure experiments is key in reducing predictive maintenance costs. In this work, focusing on time-to-alarm prediction to anticipate items breakdown, we propose a data-driven method based on the scenario approach to characterise the degradation behaviour of an industrial item in certain operative conditions using only one run-to-failure experiment, updating the time-to-alarm prediction only when needed. The scenario approach gives probabilistic guarantees on the time-to-alarm predictions.

On principle of optimality for safety-constrained Markov Decision Process and p-Safe Reinforcement Learning ⋆

We study optimality for the safety-constrained Markov decision process which is the underlying framework for safe reinforcement learning. Specifically, we consider an undiscounted safety-constrained Markov decision process subject to random stopping times. The decision maker's goal is to reach a goal state while avoiding unsafe states with certain probabilistic guarantees. Therefore the underlying Markov chain for any control policy will be Multichain or non-ergodic since by definition there exists a goal set and an unsafe set. Bellman's principle of optimality does not hold for such a safety-constrained Markov decision process in a Multichain setting as highlighted by a counterexample. We resolve the aforementioned counterexample by considering a zero-sum game setting between the policy and the Lagrange multiplier vector. Under suitable assumptions regarding the existence of admissible policy, we propose an off-policy RL algorithm for learning an optimal policy that satisfies the probabilistic safety guarantees. After that, we present the finite time error bound of the proposed RL algorithm. Lastly, we present simulation results of the aforementioned RL algorithm on a robot in a grid world setting.

Aggregated Feasible Active Power Region for Distributed Energy Resources with a Distributionally Robust Joint Probabilistic Guarantee

Aggregated Feasible Active Power Region for Distributed Energy Resources with a Distributionally Robust Joint Probabilistic Guarantee

Stochastic approximations of hybrid systems with continuous flow maps

We consider the behavior of stochastic simulators of non-stochastic hybrid systems. We focus on the case where the simulator uses randomness to approximate a (non-stochastic) flow map. The latter is assumed to be a function that is continuous, or globally Lipschitz, for simplicity. sufficient conditions guaranteeing that the sample paths of solutions to the simulator converge graphically, almost surely as the step size shrinks to zero, to solutions of a corresponding non-stochastic “average” hybrid system are provided. Probabilistic guarantees of closeness on compact time domains are also provided.

Data-Driven Stability Analysis of Switched Linear Systems Using Adaptive Sampling

In the realm of system analysis, data-driven methods have gained a lot of attention in recent years. We introduce a new innovative approach for the data-driven stability analysis of switched linear systems which is adaptive sampling. Our aim is to address limitations of existing approaches, in particular, the fact that these methods suffer from ill-conditioning of the optimal Lyapunov function, which is a direct consequence of the way the data is collected by sampling uniformly the state space. Our adaptive-sampling approach consists in a two-step procedure, in which an optimal sampling distribution is estimated in the first step from data collected with a non-optimal distribution, and then, in the second step, new data points are sampled according to the identified distribution to establish the final probabilistic guarantee for the convergence rate of the system. Numerical experiments show the efficiency of our approach, namely, in terms of the total number of data points needed to guarantee stability of the system with given confidence.

Ensemble Variance Reduction Methods for Stochastic Mixed-Integer Programming and their Application to the Stochastic Facility Location Problem

Sample average approximation (SAA), the standard approach to stochastic mixed-integer programming, does not provide guidance for cases with limited computational budgets. In such settings, variance reduction is critical in identifying good decisions. This paper explores two closely related ensemble methods to determine effective decisions with a probabilistic guarantee. (a) The first approach recommends a decision by coordinating aggregation in the space of decisions as well as aggregation of objective values. This combination of aggregation methods generalizes the bagging method and the “compromise decision” of stochastic linear programming. Combining these concepts, we propose a stopping rule that provides an upper bound on the probability of early termination. (b) The second approach applies efficient computational budget allocation for objective function evaluation and contributes to identifying the best solution with a predicted lower bound on the probability of correct selection. It also reduces the variance of the upper bound estimate at optimality. Furthermore, it adaptively selects the evaluation sample size. Both approaches provide approximately optimal solutions even in cases with a huge number of scenarios, especially when scenarios are generated by using oracles/simulators. Finally, we demonstrate the effectiveness of these methods via extensive computational results for “megascale” (extremely large scale) stochastic facility location problems. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. Funding: This work was supported by The Office of Naval Research [Grant N00014-20-1-2077] and the Air Force Office of Scientific Research [Grant FA9550-20-1-0006]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2021.0324 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2021.0324 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Localization and Estimation of Unknown Forced Inputs: A Group LASSO Approach

This paper studies the problem of locating the sparse set of sources of forcing inputs driving linear systems from noisy measurements when the initial state is unknown. This problem is particularly relevant to detecting forced oscillations in electric power networks. We express measurements as an additive model comprising the initial state and inputs grouped over time, both expanded in terms of the basis functions (i.e., impulse response coefficients). Using this model, with probabilistic guarantees, we recover the locations and simultaneously estimate the initial state and forcing inputs using a variant of the group LASSO (linear absolute shrinkage and selection operator) method. Specifically, we provide upper bounds on: (i) the probability that the group LASSO estimator incorrectly identifies the locations and (ii) the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$\ell _{2}$</tex-math></inline-formula> -norm of the estimation error. Our bounds depend on the number of measurements, inputs, and sensors; the sensor noise variance; and the minimum singular value of the observability and impulse response matrices. Our theoretical analysis is one of the first to provide a complete treatment for the group LASSO estimator for the left invertible linear systems with delay. Finally, we validate the performance of the estimator on synthetic models and the IEEE 68-bus, 16-machine power system.

Iceberg Hashing: Optimizing Many Hash-Table Criteria at Once

Despite being one of the oldest data structures in computer science, hash tables continue to be the focus of a great deal of both theoretical and empirical research. A central reason for this is that many of the fundamental properties that one desires from a hash table are difficult to achieve simultaneously; thus many variants offering different trade-offs have been proposed. This article introduces Iceberg hashing, a hash table that simultaneously offers the strongest known guarantees on a large number of core properties. Iceberg hashing supports constant-time operations while improving on the state of the art for space efficiency, cache efficiency, and low failure probability. Iceberg hashing is also the first hash table to support a load factor of up to 1 - o(1) while being stable, meaning that the position where an element is stored only ever changes when resizes occur. In fact, in the setting where keys are Θ (log n ) bits, the space guarantees that Iceberg hashing offers, namely that it uses at most \(\log \binom{|U|}{n} + O(n \log \ \text{log} n)\) bits to store n items from a universe U , matches a lower bound by Demaine et al. that applies to any stable hash table. Iceberg hashing introduces new general-purpose techniques for some of the most basic aspects of hash-table design. Notably, our indirection-free technique for dynamic resizing, which we call waterfall addressing, and our techniques for achieving stability and very-high probability guarantees, can be applied to any hash table that makes use of the front-yard/backyard paradigm for hash table design.

Investigating earthquake-induced irregularity based on post-earthquake running performance analysis of track-bridge system

In evaluating post-earthquake damage to track-bridge systems, this study conducts nonlinear time history analysis on the CRTS II ballastless track simply-supported beam system subjected to transverse earthquake loading and explores the characteristics of residual displacement and stiffness degradation of the track-bridge system under transverse earthquakes. The effect of the earthquake-induced stiffness degradation of track-bridge systems on the dynamic performance of high-speed trains is investigated, and earthquake-induced dynamic irregularities are constructed to quantify the overall stiffness degradation degree of the track-bridge system. Furthermore, a reconstruction method for the earthquake-induced dynamic irregularity characteristic curve is proposed, considering probability guarantee rates. The research results indicate that the earthquake-induced dynamic irregularity can effectively quantify the running performance degree of high-speed trains under earthquake-induced stiffness degradation conditions. The transverse acceleration and derailment coefficient of high-speed trains increase as the ground motion intensity rises. The amplitude of the characteristic curve grows significantly with the increase in pier height.

Distributed runtime verification of metric temporal properties

Distributed Systems are often composed of geographically separated components, where the clocks may not be perfectly synchronized. As such verifying the correctness of such system properties are a major challenge and are of utmost importance. In this paper, we describe a centralized runtime monitoring technique for distributed system. First, we propose a generalized runtime verification technique for verifying partially synchronous distributed computations for the metric temporal logic (MTL) by exploiting bounded-skew clock synchronization. Second, we introduce a progression-based formula rewriting scheme for monitoring MTL specifications which employs SMT solving techniques and report experimental results. Third, we also quantify each event according to the possible time of occurrence and calculate the probabilistic guarantee for generating the verification verdict. Lastly, we have implemented the entire procedure and report on extensive synthetic experimental results using UPPAAL, a set of cross-chain transactions implemented on Ethereum and an Industrial Control System of a water treatment plant.

OmniSketch: Efficient Multi-Dimensional High-Velocity Stream Analytics with Arbitrary Predicates

A key need in different disciplines is to perform analytics over fast-paced data streams, similar in nature to the traditional OLAP analytics in relational databases - i.e., with filters and aggregates. Storing unbounded streams, however, is not a realistic, or desired approach due to the high storage requirements, and the delays introduced when storing massive data. Accordingly, many synopses/sketches have been proposed that can summarize the stream in small memory (usually sufficiently small to be stored in RAM), such that aggregate queries can be efficiently approximated, without storing the full stream. However, past synopses predominantly focus on summarizing single-attribute streams, and cannot handle filters and constraints on arbitrary subsets of multiple attributes efficiently. In this work, we propose OmniSketch, the first sketch that scales to fast-paced and complex data streams (with many attributes), and supports count aggregates with filters on multiple attributes, dynamically chosen at query time. The sketch offers probabilistic guarantees, a favorable space-accuracy tradeoff, and a worst-case logarithmic complexity for updating and for query execution. We demonstrate experimentally with both real and synthetic data that the sketch outperforms the state-of-the-art, and that it can approximate complex ad-hoc queries within the configured accuracy guarantees, with small memory requirements.

Quality-Guaranteed and Cost-Effective Population Health Profiling: A Deep Active Learning Approach

Reliability and cost are two primary considerations for profiling population-scale prevalence ( PPP ) of multiple non-communicable diseases ( NCDs ). In this paper, we exploit intra-disease and inter-disease correlation in different traditionally-sensed-areas ( TS-A ) to reduce the number of profiling tasks required without compromising data reliability. Specifically, we propose a novel approach called Compressive Population Health TS-A Selection ( CPH-TS ), which blends the state-of-the-art profile inference, data augmentation and active learning in a unified deep learning framework. It can actively select the minimum number of TS-A regions for profiling task allocation in each profiling cycle, while deducing the missing data on the unprofiled regions with a probabilistic guarantee of reliability. We evaluate our approach on real-world prevalence datasets of London, which shows the effectiveness of CPH-TS . In general, CPH-TS assigned 11.1-27.3% fewer tasks than baselines, assigning tasks to only 34.7% of the sub-regions while the profiling error was below 5% for 95% of the cycles.

Joint IRS Location and Size Optimization in Multi-IRS Aided Two-Way Full-Duplex Communication Systems

Intelligent reflecting surfaces (IRSs) have emerged as a promising wireless technology for the dynamic configuration and control of electromagnetic waves, thus creating a smart (programmable) radio environment. In this context, we study a multi-IRS assisted two-way communication system consisting of two users that employ full-duplex (FD) technology. More specifically, we deal with the joint IRS location and size (i.e., the number of reflecting elements) optimization in order to minimize an upper bound of system outage probability under various constraints: minimum and maximum number of reflecting elements per IRS, maximum number of installed IRSs, maximum total number of reflecting elements (implicit bound on the signaling overhead) as well as maximum total IRS installation cost. First, the problem is formulated as a discrete optimization problem and, then, a theoretical proof of its NP-hardness is given. Moreover, we provide a lower bound on the optimum value by solving a linear-programming relaxation (LPR) problem. Subsequently, we design two polynomial-time algorithms, a deterministic greedy algorithm and a randomized approximation algorithm, based on the LPR solution. The former is a heuristic method that always computes a feasible solution for which (a posteriori) performance guarantee can be provided. The latter achieves an approximate solution, using randomized rounding, with provable (a priori) probabilistic guarantees on the performance. Furthermore, extensive numerical simulations demonstrate the superiority of the proposed algorithms compared to the baseline schemes. Finally, useful conclusions regarding the comparison between FD and conventional half-duplex (HD) systems are also drawn.

Compositional synthesis of control barrier certificates for networks of stochastic systems against [formula omitted]-regular specifications

This paper is concerned with a compositional scheme for the construction of control barrier certificates for interconnected discrete-time stochastic systems. The main objective is to synthesize switching controllers against ω-regular properties that can be described by accepting languages of deterministic Streett automata (DSA) along with providing probabilistic guarantees for the satisfaction of such specifications. The proposed framework leverages the interconnection topology and a notion of so-called control sub-barrier certificates of subsystems, which are used to compositionally construct control barrier certificates of interconnected systems by imposing some dissipativity-type compositionality conditions. We propose a systematic approach to decompose high-level ω-regular specifications into simpler tasks by utilizing the automata corresponding to the specifications. In addition, we formulate an alternating direction method of multipliers (ADMM) optimization problem in order to obtain suitable control sub-barrier certificates of subsystems while satisfying compositionality conditions. For systems with polynomial dynamics, we provide a sum-of-squares (SOS) optimization problem for the computation of control sub-barrier certificates and local controllers of subsystems. Finally, we demonstrate the effectiveness of our proposed approaches by applying them to a physical case study.

Evolutionary power spectral density study of the earthquake-induced dynamic irregularity based on short-time Fourier transform

The stiffness degradation of the track-bridge system under earthquake is the main reason for the deterioration of track dynamic smoothness. In order to investigate the dynamic smoothness performance of the track surface of the track-bridge system after earthquake. In this paper, a nonlinear time analysis of ballastless track-simply supported bridge system under transverse random earthquakes is carried out, and the degradation mode and degradation degree of each key component of the track-bridge system after transverse earthquakes are investigated. Based on the post-earthquake stiffness of each component, a train-track-bridge system considering earthquake-induced stiffness degradation is established. The sample library of earthquake-induced dynamic irregularity is constructed by additional deformation of rails, and the power spectral density of the earthquake-induced dynamic irregularity is studied. Based on the short-time Fourier transform irregularity inversion technique, the earthquake-induced dynamic irregularity characteristic curve based on the probability guarantee rate is proposed, and the effects of train operating speed and ground vibration intensity on the earthquake-induced dynamic irregularity characteristic curve are investigated. The research results show that: The transverse stiffness degradation of the bearings and piers of the track-bridge system is the most obvious during an earthquake, and the stiffness degradation degree of the middle piers is greater than that of the side piers. The stiffness degradation degree of the track-bridge system rises significantly with increasing PGA, while the variability of stiffness degradation of the middle and side piers drops. The most significant earthquake-induced dynamic irregularity sample is the track alignment irregularity. Low-frequency components dominate the evolutionary power spectral density corresponding to track dynamic irregularity, and the amplitude of the power spectral density reduces considerably with increasing spatial frequency. The vehicle speed has little effect on the earthquake-induced track dynamic irregularity. The irregularity amplitude grows significantly with increasing damage degree. The results indicated that the stiffness degradation damage degree of the track-bridge system under earthquake can be quantified by constructive earthquake-induced track dynamic irregularity.

Probabilistic Reaction Time Analysis

In many embedded systems, for instance, in the automotive, avionic, or robotics domain, critical functionalities are implemented via chains of communicating recurrent tasks. To ensure safety and correctness of such systems, guarantees on the reaction time, that is, the delay between a cause (e.g., an external activity or reading of a sensor) and the corresponding effect, must be provided. Current approaches focus on the maximum reaction time, considering the worst-case system behavior. However, in many scenarios, probabilistic guarantees on the reaction time are sufficient. That is, it is sufficient to provide a guarantee that the reaction does not exceed a certain threshold with (at least) a certain probability. This work provides such probabilistic guarantees on the reaction time, considering two types of randomness: response time randomness and failure probabilities. To the best of our knowledge, this is the first work that defines and analyzes probabilistic reaction time for cause-effect chains based on sporadic tasks.

Monitoring of perception systems: Deterministic, probabilistic, and learning-based fault detection and identification

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.

Distributionally Robust Optimal Power Flow via Ellipsoidal Approximation

This paper proposes a distributionally robust joint chance-constrained AC optimal power flow to manage the risk of operational limits violations which are caused by uncertain renewable generation. By determining the dispatch schedule, this approach can help to decrease the risk of renewable curtailment, load shedding, or emergency redispatch in real-time. To model the proposed approach, the renewable uncertainty is first modeled as a distributionally robust ellipsoidal bound based on the Wasserstein metric. This bound is built upon a limited historical renewable forecast errors dataset without any assumption on the probability distribution of uncertainty. Then, this uncertainty bound is adopted within a semidefinite relaxation of the optimal power flow. The system responses to the renewable generation forecast errors are modeled as linear sensitivities, where all conventional generators are responsible for compensating the impacts of renewable forecast errors. Numerical experiments on IEEE 14-bus and 118-bus systems show the validity and scalability of the proposed method. Furthermore, the effectiveness of the proposed approach in terms of meeting probabilistic guarantees, cost-effectiveness and computational time is also demonstrated in the experiments.

Distributed event-triggered unadjusted Langevin algorithm for Bayesian learning

This paper presents a distributed event-triggered unadjusted Langevin algorithm (DETULA) to address the Bayesian learning problem. We consider a set of networked learning agents who have access to their own independently distributed data sets. The objective of each agent is to reconstruct the global posterior of the unknown model parameters through local learning along with interaction with neighboring agents. We propose an event-triggered communication mechanism for a distributed Langevin algorithm to limit the inter-agent interactions and thus reduce the communication overhead. We provide conditions on the algorithm step sizes and the triggering threshold to ensure mean-square consensus of the agents’ parameter estimates and convergence of the estimates to the global posterior as if the data sets were aggregated at a central location. A major improvement of our result over previous studies is the establishment of said consensus without imposing any bounded restriction of the gradient of the objective function. Additionally, we establish probabilistic guarantees to prevent consecutive triggering by any agent while maintaining the same rate of convergence as in the case without event-triggering We demonstrate the DETULA using distributed supervised-learning problems. Our results indicate that the agents successfully recover the global posterior by periodically sharing their samples with neighbors.

Federated Edge Intelligence and Edge Caching Mechanisms

Federated learning (FL) has emerged as a promising technique for preserving user privacy and ensuring data security in distributed machine learning contexts, particularly in edge intelligence and edge caching applications. Recognizing the prevalent challenges of imbalanced and noisy data impacting scalability and resilience, our study introduces two innovative algorithms crafted for FL within a peer-to-peer framework. These algorithms aim to enhance performance, especially in decentralized and resource-limited settings. Furthermore, we propose a client-balancing Dirichlet sampling algorithm with probabilistic guarantees to mitigate oversampling issues, optimizing data distribution among clients to achieve more accurate and reliable model training. Within the specifics of our study, we employed 10, 20, and 40 Raspberry Pi devices as clients in a practical FL scenario, simulating real-world conditions. The well-known FedAvg algorithm was implemented, enabling multi-epoch client training before weight integration. Additionally, we examined the influence of real-world dataset noise, culminating in a performance analysis that underscores how our novel methods and research significantly advance robust and efficient FL techniques, thereby enhancing the overall effectiveness of decentralized machine learning applications, including edge intelligence and edge caching.