Data-driven Reachability using Christoffel Functions and Conformal Prediction

• URL: http://arxiv.org/abs/2309.08976v1
• Date: Sat, 16 Sep 2023 12:21:57 GMT
• Title: Data-driven Reachability using Christoffel Functions and Conformal Prediction
• Authors: Abdelmouaiz Tebjou, Goran Frehse, Fa\"icel Chamroukhi
• Abstract summary: An important tool in the analysis of dynamical systems is the approximation of the reach set, i.e., the set of states reachable after a given time from a given initial state. Data-based approaches are promised to avoid difficulties by estimating the reach set based on a sample of states. A recently proposed approach for data-based reach set approximation uses Christoffel functions to approximate the reach set.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: An important mathematical tool in the analysis of dynamical systems is the approximation of the reach set, i.e., the set of states reachable after a given time from a given initial state. This set is difficult to compute for complex systems even if the system dynamics are known and given by a system of ordinary differential equations with known coefficients. In practice, parameters are often unknown and mathematical models difficult to obtain. Data-based approaches are promised to avoid these difficulties by estimating the reach set based on a sample of states. If a model is available, this training set can be obtained through numerical simulation. In the absence of a model, real-life observations can be used instead. A recently proposed approach for data-based reach set approximation uses Christoffel functions to approximate the reach set. Under certain assumptions, the approximation is guaranteed to converge to the true solution. In this paper, we improve upon these results by notably improving the sample efficiency and relaxing some of the assumptions by exploiting statistical guarantees from conformal prediction with training and calibration sets. In addition, we exploit an incremental way to compute the Christoffel function to avoid the calibration set while maintaining the statistical convergence guarantees. Furthermore, our approach is robust to outliers in the training and calibration set.

Related papers

• Accelerated zero-order SGD under high-order smoothness and overparameterized regime [79.85163929026146]
  We present a novel gradient-free algorithm to solve convex optimization problems. Such problems are encountered in medicine, physics, and machine learning. We provide convergence guarantees for the proposed algorithm under both types of noise.
  arXiv  Detail & Related papers  (2024-11-21T10:26:17Z)
• Safe and High-Performance Learning of Model Predicitve Control using Kernel-Based Interpolation [44.99833362998488]
  We present a method, which allows efficient and safe approximation of model predictive controllers using kernel. Since the computational complexity of the approximating function scales linearly with the number of data points, we propose to use a scoring function which chooses the most promising data. In order to guarantee safety and high performance of the designed approximated controller, we use reachability analysis based on Monte Carlo methods.
  arXiv  Detail & Related papers  (2024-10-09T11:04:15Z)
• Auxiliary Functions as Koopman Observables: Data-Driven Analysis of Dynamical Systems via Polynomial Optimization [0.0]
  We present a flexible data-driven method for system analysis that does not require explicit model discovery. The method is rooted in well-established techniques for approxing the Koopman operator from data and is implemented as a semidefinite program that can be solved numerically.
  arXiv  Detail & Related papers  (2023-03-02T18:44:18Z)
• Numerically Stable Sparse Gaussian Processes via Minimum Separation using Cover Trees [57.67528738886731]
  We study the numerical stability of scalable sparse approximations based on inducing points. For low-dimensional tasks such as geospatial modeling, we propose an automated method for computing inducing points satisfying these conditions.
  arXiv  Detail & Related papers  (2022-10-14T15:20:17Z)
• TACTiS: Transformer-Attentional Copulas for Time Series [76.71406465526454]
  estimation of time-varying quantities is a fundamental component of decision making in fields such as healthcare and finance. We propose a versatile method that estimates joint distributions using an attention-based decoder. We show that our model produces state-of-the-art predictions on several real-world datasets.
  arXiv  Detail & Related papers  (2022-02-07T21:37:29Z)
• Data-Driven Reachability analysis and Support set Estimation with Christoffel Functions [8.183446952097528]
  We present algorithms for estimating the forward reachable set of a dynamical system. The produced estimate is the sublevel set of a function called an empirical inverse Christoffel function. In addition to reachability analysis, the same approach can be applied to general problems of estimating the support of a random variable.
  arXiv  Detail & Related papers  (2021-12-18T20:25:34Z)
• Calibrating Over-Parametrized Simulation Models: A Framework via Eligibility Set [3.862247454265944]
  We develop a framework to develop calibration schemes that satisfy rigorous frequentist statistical guarantees. We demonstrate our methodology on several numerical examples, including an application to calibration of a limit order book market simulator.
  arXiv  Detail & Related papers  (2021-05-27T00:59:29Z)
• Scalable Marginal Likelihood Estimation for Model Selection in Deep Learning [78.83598532168256]
  Marginal-likelihood based model-selection is rarely used in deep learning due to estimation difficulties. Our work shows that marginal likelihoods can improve generalization and be useful when validation data is unavailable.
  arXiv  Detail & Related papers  (2021-04-11T09:50:24Z)
• Using Data Assimilation to Train a Hybrid Forecast System that Combines Machine-Learning and Knowledge-Based Components [52.77024349608834]
  We consider the problem of data-assisted forecasting of chaotic dynamical systems when the available data is noisy partial measurements. We show that by using partial measurements of the state of the dynamical system, we can train a machine learning model to improve predictions made by an imperfect knowledge-based model.
  arXiv  Detail & Related papers  (2021-02-15T19:56:48Z)
• The Shooting Regressor; Randomized Gradient-Based Ensembles [0.0]
  An ensemble method is introduced that utilizes randomization and loss function gradients to compute a prediction. Multiple weakly-correlated estimators approximate the gradient at randomly sampled points on the error surface and are aggregated into a final solution.
  arXiv  Detail & Related papers  (2020-09-14T03:20:59Z)
• Learning while Respecting Privacy and Robustness to Distributional Uncertainties and Adversarial Data [66.78671826743884]
  The distributionally robust optimization framework is considered for training a parametric model. The objective is to endow the trained model with robustness against adversarially manipulated input data. Proposed algorithms offer robustness with little overhead.
  arXiv  Detail & Related papers  (2020-07-07T18:25:25Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.