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Future Aware Safe Active Learning of Time Varying Systems using Gaussian Processes

Markus Lange-Hegermann, Christoph Zimmer

TL;DR

This work tackles safe active learning for time-varying systems by introducing Time-aware IMSPE ($T$-IMSPE), which minimizes posterior variance across present and future time steps. It establishes a general, closed-form computability of IMSPE/$T$-IMSPE under the class of $P$-elementary covariance marginalizable kernels and finite measures, and extends the method to NX-structured dynamics. A PyTorch-oriented cookbook guides kernel constructions to retain closed-form solvability, enabling practical deployment. Empirical results on seasonal change, drift, and a rail-pressure NX system show significant RMSE improvements and robust safety guarantees over entropy and IMSPE baselines, with substantial reductions in data collection cost. The framework thus offers scalable, theoretically sound guidance for time-varying safe learning with broad applicability across engineering domains.

Abstract

Experimental exploration of high-cost systems with safety constraints, common in engineering applications, is a challenging endeavor. Data-driven models offer a promising solution, but acquiring the requisite data remains expensive and is potentially unsafe. Safe active learning techniques prove essential, enabling the learning of high-quality models with minimal expensive data points and high safety. This paper introduces a safe active learning framework tailored for time-varying systems, addressing drift, seasonal changes, and complexities due to dynamic behavior. The proposed Time-aware Integrated Mean Squared Prediction Error (T-IMSPE) method minimizes posterior variance over current and future states, optimizing information gathering also in the time domain. Empirical results highlight T-IMSPE's advantages in model quality through toy and real-world examples. State of the art Gaussian processes are compatible with T-IMSPE. Our theoretical contributions include a clear delineation which Gaussian process kernels, domains, and weighting measures are suitable for T-IMSPE and even beyond for its non-time aware predecessor IMSPE.

Future Aware Safe Active Learning of Time Varying Systems using Gaussian Processes

TL;DR

This work tackles safe active learning for time-varying systems by introducing Time-aware IMSPE (-IMSPE), which minimizes posterior variance across present and future time steps. It establishes a general, closed-form computability of IMSPE/-IMSPE under the class of -elementary covariance marginalizable kernels and finite measures, and extends the method to NX-structured dynamics. A PyTorch-oriented cookbook guides kernel constructions to retain closed-form solvability, enabling practical deployment. Empirical results on seasonal change, drift, and a rail-pressure NX system show significant RMSE improvements and robust safety guarantees over entropy and IMSPE baselines, with substantial reductions in data collection cost. The framework thus offers scalable, theoretically sound guidance for time-varying safe learning with broad applicability across engineering domains.

Abstract

Experimental exploration of high-cost systems with safety constraints, common in engineering applications, is a challenging endeavor. Data-driven models offer a promising solution, but acquiring the requisite data remains expensive and is potentially unsafe. Safe active learning techniques prove essential, enabling the learning of high-quality models with minimal expensive data points and high safety. This paper introduces a safe active learning framework tailored for time-varying systems, addressing drift, seasonal changes, and complexities due to dynamic behavior. The proposed Time-aware Integrated Mean Squared Prediction Error (T-IMSPE) method minimizes posterior variance over current and future states, optimizing information gathering also in the time domain. Empirical results highlight T-IMSPE's advantages in model quality through toy and real-world examples. State of the art Gaussian processes are compatible with T-IMSPE. Our theoretical contributions include a clear delineation which Gaussian process kernels, domains, and weighting measures are suitable for T-IMSPE and even beyond for its non-time aware predecessor IMSPE.
Paper Structure (42 sections, 7 theorems, 51 equations, 13 figures, 2 tables, 1 algorithm)

This paper contains 42 sections, 7 theorems, 51 equations, 13 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Background
  3. Gaussian processes (GPs)
  4. Time varying systems
  5. Safe active learning
  6. Integrated Mean Squared Prediction Error (IMSPE)
  7. Related work
  8. Acquisition functions for active learning
  9. Active learning in dynamic systems
  10. Safe learning
  11. T-IMSPE - Time-aware Integrated Mean Squared Prediction Error
  12. T-IMSPE for GPs with time amongst its inputs
  13. T-IMSPE for GPs with NX-structure
  14. P-elementary functions for closed form (T-)IMSPE
  15. Computability in closed form
  16. ...and 27 more sections

Key Result

Theorem 4.4

Assume the prior covariance function $k$ of a prior GP $g$ to be $P$-elementary covariance marginalizable w.r.t. a measure $\mu$. Then, IMSPE and T-IMSPE are $P$-elementary, i.e. they can be computed in closed form in the programming framework $P$.

Figures (13)

  • Figure 1: This illustrative sketch showcases the efficacy of our acquisition function T-IMSPE in variance reduction on a domain $D$ in present (dark color) and future (light color). The darker color at the bottom exhibits the transition from the prior blue line to the posterior green dashed line when conditioning on the red data point $x$ at present time. The lighter colored top delineates the induced variance reduction in a future time step. The left subplot (a) highlights the suboptimal variance reduction when employing the entropy acquisition function: it leads to a measurement at the boundary and to a very low reduction of the future variance. The right subplot (b) reveals the capability of T-IMSPE to outperform entropy in reducing the average variance at current and future time steps.
  • Figure 2: The IMSPE criterion in the motivating example. Without the (green) data at $x=1$, the IMSPE criterion would be symmetric around zero. With the (green) data at $x=1$, the optimal choice according to IMSPE for the (red) next measurement point $x_*$ is the red position at $x_*=-0.5479204538$.
  • Figure 3: Plots for the seasonal change experiment (Subsection \ref{['subsection_seasonal']}) for $t=0$ (left) and $t=6$ (right).
  • Figure 4: Box plots of RMSE value in the safe area of the experiments from active learning with seasonal changes (left, Subsection \ref{['subsection_seasonal']}) and under drift (right, Subsection \ref{['subsection_drift']}). The x-axis shows these value on averaged over all time steps and at certain ascending time steps from 10% to 100%. The left diagram compares the results of 450 runs between T-IMSPE (blue), entropy (red), and IMSPE (green), where T-IMSPE is highly significantly ($p<2.2\text{e}{-16}$) superior to entropy in all eleven comparison, whereas T-IMSPE is superior over IMSPE on average ($p<2.1\text{e}{-3}$). The median error reached by T-IMSPE at 30 % of the time is reached by entropy at 50 % of the time, which is a 40 % reduction of measuremt time and cost. The right diagram shows the gain of RMSE values in 500 runs when choosing T-IMSPE over entropy (red) and over IMSPE (green). The results are significantly ($p<1\text{e}{-10}$) better for T-IMSPE over entropy in all 11 comparisons and and significantly better in 10 out of 11 comparisons for T-IMSPE over IMSPE.
  • Figure 5: Plots of the drift experiment (Subsection \ref{['subsection_drift']}) for $t=0$ (left) and $t=100$ (right).
  • ...and 8 more figures

Theorems & Definitions (25)

  • Remark 2.1
  • Definition 4.1
  • Example 4.2
  • Definition 4.3
  • Theorem 4.4
  • Corollary 4.5
  • Corollary 4.6
  • Example A.1
  • Definition A.2
  • Example A.3
  • ...and 15 more