PaperlayerView exact revision on arXiv

Source-linked AI summary

Uniform Error Bounds for Gaussian Process Regression with Application to Safe Control

Armin Lederer, Jonas Umlauft, Sandra Hirche

arXiv:1906.01376v2cs.LGeess.SYstat.ML

TL;DR

Safety-critical use of data-driven control is hindered by limited, noisy data and restrictive assumptions in existing Gaussian-process uniform error bounds. The paper uses the GP function-space distribution and continuity arguments to derive a weaker-assumption bound, probabilistic Lipschitz constants, and asymptotic conditions. It then derives safety bounds for GP-based tracking control and evaluates them in robotic-manipulator simulations.

  • Problem

    Limited and noisy training data make model-error quantification essential, while existing GP uniform error bounds require restrictive assumptions for safety-critical control.

  • Method

    The paper exploits the GP distribution and continuity arguments to derive a uniform error bound, probabilistic Lipschitz constants, asymptotic conditions, and Lyapunov-based safety guarantees.

  • Results

    The derived bounds provide safety guarantees for unknown dynamical systems and are evaluated in simulation on a robotic manipulator.

  • Takeaways & Limitations

    The GP-based analysis supports considering the simulated robotic manipulator controller safe because the system is guaranteed not to leave a depicted task-space area.

  • Takeaways & Limitations

    The asymptotic guarantee depends on posterior-standard-deviation convergence conditions tied to the training-data distribution and covariance-kernel structure.

Abstract

from arXiv · show

Data-driven models are subject to model errors due to limited and noisy training data. Key to the application of such models in safety-critical domains is the quantification of their model error. Gaussian processes provide such a measure and uniform error bounds have been derived, which allow safe control based on these models. However, existing error bounds require restrictive assumptions. In this paper, we employ the Gaussian process distribution and continuity arguments to derive a novel uniform error bound under weaker assumptions. Furthermore, we demonstrate how this distribution can be used to derive probabilistic Lipschitz constants and analyze the asymptotic behavior of our bound. Finally, we derive safety conditions for the control of unknown dynamical systems based on Gaussian process models and evaluate them in simulations of a robotic manipulator.

1 Introduction

Learning-based control can address complex unknown systems, but safety-critical deployment requires reliable model-error quantification. This paper develops weaker-assumption Gaussian-process bounds and uses them to establish safety for unknown-system control.

  • 1 Introduction: Limited and noisy training data create model imperfections, making uncertainty quantification important for safety-critical learning-based control.Gaussian processes provide a measure of their own model imprecision.
  • 1 Introduction: Existing Gaussian-process uniform error bounds rely on restrictive assumptions that limit their use in control applications.The paper identifies these assumptions as a central barrier to applying GP-based guarantees more broadly.
  • 1 Introduction: The paper derives a novel GP uniform error bound requiring less prior knowledge and fewer assumptions than previous approaches.The intended consequence is applicability to a wider range of problems.
  • 1 Introduction: The authors derive probabilistic Lipschitz constants for GP samples and analyze asymptotic behavior to study arbitrarily small error bounds with sufficient resources and data.These results connect the GP distribution and continuity arguments to the bound’s limiting behavior.
  • 1 Introduction: Lyapunov-based safety analysis shows that a GP-model controller for a robotic manipulator can make the closed-loop system converge to a small fraction of the state space.The paper presents this convergence as a basis for considering the controlled system safe.

2 Background

Gaussian-process regression models functions through Gaussian-distributed observations and supports uniform regression-error analysis on compact domains. Prior bounds draw on interpolation, kernel-regression, RKHS, information-gain, and GP-support perspectives, with noisy-observation settings remaining incompletely analyzed.

  • 2 Background: Gaussian process regression assumes every finite collection of random variables follows a joint Gaussian distribution with a specified mean and covariance kernel.The observations are generated from a GP sample function with additive zero-mean Gaussian noise.
  • 2 Background: Conditioned on training inputs and observations, GP regression produces a posterior mean and standard deviation describing the distribution at a query point.The posterior distribution is represented as a normal distribution whose parameters depend on the kernel matrix and kernel vector.
  • 2 Background: A probabilistic uniform error bound controls GP regression error over a compact set with probability at least 1 −δ.Uniform boundedness is defined through an error-bound function η(x).
  • 2 Background: Related bounds use scattered-data interpolation, regularized kernel regression, empirical covering numbers, effective dimension, kernel approximations, maximal information gain, and RKHS norms.These approaches extend across noise-free and noisy regression settings but introduce different complexity measures and assumptions.
  • 2 Background: Prior work also derives bounds from GP suprema and prior-distribution support, but noisy-observation GP uniform error has not been fully analyzed through that support-based perspective.The related-work discussion distinguishes these approaches from the paper’s distribution-based treatment.

3 Probabilistic Uniform Error Bound

The paper derives a probabilistic uniform error bound for noisy Gaussian process regression using the GP’s function-space distribution and continuity, requiring weaker assumptions than prior approaches. It also obtains probabilistic Lipschitz constants and shows conditions under which the bound can vanish asymptotically.

  • The proposed bound exploits the GP’s inherent probability distribution rather than restricting analysis to smaller RKHS subspaces.This targets noisy observations while avoiding constants that are difficult to determine in prior approaches.
  • Lipschitz continuity of the covariance kernel and unknown function supports continuity of the posterior mean and standard deviation used in the uniform bound.Common squared exponential and Matérn covariance kernels satisfy the kernel-continuity restriction.
  • The bound is constructed from grid-based Gaussian concentration, covering numbers, posterior quantities, and continuity terms.The grid constant τ controls discretization, while β(τ) depends on the covering number and γ(τ) captures continuity contributions.
  • The asymptotic analysis shows that a vanishing uniform error bound can be proven under weak assumptions despite growth of intermediate terms with the number of observations.The analysis depends on the convergence rate of the posterior standard deviation, which is tied to the distribution of training data and kernel structure.
  • When prior knowledge of the unknown function’s Lipschitz constant is unavailable, the GP distribution yields a probabilistic Lipschitz constant.The result holds with probability at least 1 −δL for sample functions under the stated kernel conditions.

4 Safety Guarantees for Control of Unknown Dynamical Systems

The paper applies GP error bounds to tracking control of unknown dynamical systems, using Lyapunov analysis to establish safety through ultimate tracking-error bounds.

  • 4 Safety Guarantees for Control of Unknown Dynamical Systems: GP-based safety guarantees target unknown control-affine systems whose dynamics are modeled from noisy observations.The unknown function is assumed to be a GP sample, while the state is measured without noise and derivative observations may be noisy.
  • 4.1 Tracking Control Design: The controller uses feedback linearization with the GP posterior mean to compensate for the unknown nonlinear dynamics.The policy tracks the desired output x1 using the tracking error and filtered state r.
  • 4.2 Stability Analysis: An upper tracking-error bound enables verification that controller parameters and the learned model satisfy safety constraints.Ultimate boundedness requires trajectories to enter and remain within a fixed bound after a transient period.
  • 4.2 Stability Analysis: Lyapunov analysis shows tracking error convergence when the feedback term dominates the GP model error.The GP uniform error bound yields a computable set B in which the ultimate bound holds.
  • 4.2 Stability Analysis: The ultimate bound can be reduced by increasing controller gains or adding training points that decrease posterior uncertainty.The paper states that the relevant bound is computable and can be made arbitrarily small under these changes.

5 Numerical Evaluation

Simulations evaluate the theoretical bounds on a synthetic system and a two-degree-of-freedom robotic manipulator, showing conservative safety regions and safe tracking behavior.

  • 5 Numerical Evaluation: The evaluation studies probabilistic Lipschitz estimation, uneven training-sample effects, and GP-based feedback-linearizing control.The experiments cover an unknown synthetic system and a robotic manipulator tracking sinusoidal joint trajectories.
  • 5.1 Synthetic System with Unknown Lipschitz Constant Lf: In Figure 1, the red set B is smaller in low-uncertainty regions than in high-uncertainty regions as the state approaches the desired trajectory.The trajectory is blue, the desired trajectory is green, and backgrounds indicate uncertainty levels.
  • 5.1 Synthetic System with Unknown Lipschitz Constant Lf: In Figure 2, larger ultimate bounds coincide with increased tracking error because the model is less precise.The ultimate bound is shown in red and tracking error in blue.
  • 5.1 Synthetic System with Unknown Lipschitz Constant Lf: 81 training points with noise variance σ_n^2 = 0.01 are used for the synthetic-system experiment.The experiment uses a squared exponential kernel, δ = 0.01, δL = 0.01, and τ = 10^-8.
  • 5.1 Synthetic System with Unknown Lipschitz Constant Lf: The estimated Lipschitz constant is conservative by a factor of 10–100, but the resulting effect is mitigated by choosing τ arbitrarily small.The experiment reports that γ(τ) is dominated by β(τ)ωσN(τ).
  • 5.1 Synthetic System with Unknown Lipschitz Constant Lf: The synthetic-system safety bound is conservative because the violation probability is set to 1%.The reported results are shown in Figures 1 and 2.
  • 5.2 Robotic Manipulator with 2 Degrees of Freedom: In Figure 3, the manipulator remains within B after a transient phase while joint angles and velocities converge toward desired trajectories.The task-space safe region is X \ B, and the right panel shows convergence relative to dashed desired trajectories.
  • 5.2 Robotic Manipulator with 2 Degrees of Freedom: For the manipulator, the learned dynamics yield a joint-space error bound that transforms into a task-space region the robot is guaranteed not to leave.The manipulator has two degrees of freedom, four-dimensional state space, and 81 training points distributed in [−1,1]^4.

6 Conclusion

The paper concludes that GP-distribution-based uniform bounds support probabilistic Lipschitz estimation, arbitrarily small bounds under sufficient resources, and safety analysis for GP-controlled tracking.

  • 6 Conclusion: The paper presents a novel GP uniform error bound based on the GP probability distribution rather than the covariance kernel’s RKHS.This permits a wider class of functions and applies under weaker assumptions.
  • 6 Conclusion: Probabilistic Lipschitz constants can be estimated from the GP distribution, and sufficient conditions for arbitrarily small uniform bounds are derived.The conclusion connects these results to the paper’s asymptotic analysis.
  • 6 Conclusion: The derived bounds are used to obtain safety bounds for a tracking controller and evaluated in simulation with a robotic manipulator.The conclusion identifies the control application and simulation setting.

A Proof of Theorem 3.1

The proof constructs the uniform GP error bound by establishing continuity properties for the posterior mean and standard deviation, then extending grid-point bounds across the state space.

  • A Proof of Theorem 3.1: The proof first establishes a Lipschitz constant for the posterior mean and a modulus of continuity for the posterior standard deviation.These ingredients precede the regression-error bound.
  • A Proof of Theorem 3.1: The posterior-mean difference is bounded using kernel differences, Cauchy–Schwarz, and kernel Lipschitz continuity.This proves Lipschitz continuity of νN(x).
  • A Proof of Theorem 3.1: The variance difference is bounded first, after which taking its square root yields a modulus of continuity for the posterior standard deviation.The argument uses positivity, Cauchy–Schwarz, and kernel Lipschitz continuity.
  • A Proof of Theorem 3.1: A grid-based probabilistic bound is extended to the full domain using continuity of f(x), νN(x), and σN(x).The grid resolution is related to the covering number M(τ, X).

B Proof of Theorem 3.2

The proof derives a high-probability Lipschitz bound for Gaussian-process sample functions by bounding their supremum and the suprema of derivative processes.

  • Expected supremum: A metric-entropy argument bounds the expected supremum of a Gaussian-process sample function on a compact domain.The proof bounds covering numbers through the covariance pseudo-metric and relates them to coverings in the original metric.
  • Expected supremum: A covering grid in the original metric yields an upper bound on the covariance-metric covering number used in the supremum estimate.Continuity of the covariance kernel provides the inverse-modulus relationship needed to construct the covering net.
  • High-probability bound: The expected-supremum bound is converted into a high-probability supremum bound using concentration inequalities, specifically the Borell-TIS inequality.The resulting sample-function supremum is bounded with probability at least 1 − δL.
  • Lipschitz constant: Derivatives of a Gaussian-process sample function form samples from derivative Gaussian processes, allowing the supremum bound to be applied to each partial derivative.Applying a union bound over all d derivative processes produces the high-probability Lipschitz constant in Theorem 3.2.

C Proof of Theorem 3.3

The proof establishes asymptotic behavior by combining Theorem 3.1 with bounds on posterior-mean and posterior-standard-deviation regularity, then selecting a sufficiently fast-decaying τ(N).

  • Regularity bounds: The proof bounds posterior-mean regularity using a uniform grid, Gaussian-noise concentration, and a union bound over training-sample sizes.The posterior mean has a high-probability Lipschitz bound, while the posterior standard deviation is controlled through its modulus of continuity.
  • Regularity bounds: The posterior-mean Lipschitz constant satisfies LνN ∈ O(N) with probability at least 1 − δ/2, while the posterior standard deviation is bounded through its modulus of continuity.The logarithmic growth of ηN with N is used in the posterior-mean estimate.
  • Asymptotic behavior: τ(N) must decrease faster than O((N log(N))^-1) for the regression error to vanish as N → ∞.The proof therefore chooses τ(N) ∈ O(N^-2).
  • Asymptotic behavior: Choosing τ(N) ∈ O(N^-2) implies βN(τ(N)) ∈ O(log(N)).This relationship follows from the definition of βN and the selected decay rate for τ(N).

D Proof of Theorem 4.1

The proof uses Lyapunov theory and the high-probability GP model-error bound to establish global ultimate boundedness of the closed-loop system.

  • Lyapunov criterion: Lyapunov theory characterizes global ultimate boundedness through a positive-definite function whose derivative is negative outside a target set.This provides the stability criterion used for the closed-loop analysis.
  • Closed-loop analysis: The proof applies the model-error bound from Theorem 3.1 within a Lyapunov-function analysis of the feedback controller.The controller parameters and GP approximation enter the resulting bound on closed-loop behavior.
  • Conclusion: The closed-loop system is shown to be globally ultimately bounded according to the Lyapunov criterion.This establishes the stability property required by Theorem 4.1.

E Report on Computational Complexity of the Numerical Evaluation

The numerical evaluation reports MATLAB simulations with different runtimes and memory usage for Sections 5.1 and 5.2.

  • Simulation resources: The Section 5.1 simulation took 77s and used 1 MB of workspace memory.It was run in MATLAB 2019a on an i5-6200U CPU with 2.3GHz and 8GB RAM.
  • Simulation resources: The Section 5.2 simulation took 39s and used 134 MB of workspace memory.The code is available as supplementary material.
Loading 1906.01376v2…