Safe Time-Varying Optimization based on Gaussian Processes with Spatio-Temporal Kernel (2409.18000v1)

Abstract: Ensuring safety is a key aspect in sequential decision making problems, such as robotics or process control. The complexity of the underlying systems often makes finding the optimal decision challenging, especially when the safety-critical system is time-varying. Overcoming the problem of optimizing an unknown time-varying reward subject to unknown time-varying safety constraints, we propose TVSafeOpt, a new algorithm built on Bayesian optimization with a spatio-temporal kernel. The algorithm is capable of safely tracking a time-varying safe region without the need for explicit change detection. Optimality guarantees are also provided for the algorithm when the optimization problem becomes stationary. We show that TVSafeOpt compares favorably against SafeOpt on synthetic data, both regarding safety and optimality. Evaluation on a realistic case study with gas compressors confirms that TVSafeOpt ensures safety when solving time-varying optimization problems with unknown reward and safety functions.

• The paper proposes TVSafeOpt, a novel algorithm that extends safe Bayesian optimization to time-varying settings with spatio-temporal kernels.
• It employs Gaussian Processes and time Lipschitz constants to update safe sets, reducing unsafe decisions while adapting to dynamic environments.
• Empirical evaluations show significant reductions in unsafe decisions and competitive performance in both synthetic examples and a gas compressor case study.

Safe Time-Varying Optimization based on Gaussian Processes with Spatio-Temporal Kernel

The paper "Safe Time-Varying Optimization based on Gaussian Processes with Spatio-Temporal Kernel" by Li et al. proposes a novel algorithm called TVSafeOpt, which addresses the problem of optimizing unknown, time-varying reward functions under unknown, time-varying safety constraints. This research extends the framework of Bayesian Optimization (BO) to time-varying settings, ensuring safe decision-making without the need for explicit change detection mechanisms.

Overview

TVSafeOpt represents a significant advancement in safe Bayesian optimization by incorporating a spatio-temporal kernel and leveraging time Lipschitz constants to track time-varying safe regions. This enables safe optimization in environments where both the reward and safety constraints change over time without prior knowledge about these changes or the need to explicitly detect them. The algorithm builds upon the SafeOpt framework but includes several crucial modifications to handle dynamic changes safely.

Methodology

Problem Definition

The optimization problem is defined as maximizing a time-varying reward function $f(\mathbf{x}, t)$ over a finite set of decisions $\mathcal{X}$, subject to multiple time-varying safety constraints $c_i(\mathbf{x}, t) \ge 0$. Both the reward function and the constraints are unknown but can be evaluated. The key challenge is to maintain safety over consecutive iterations while the underlying functions vary with time.

TVSafeOpt Algorithm

The TVSafeOpt algorithm distinguishes itself from traditional SBO (Safe Bayesian Optimization) methods through its spatio-temporal kernel. This kernel captures the continuity of the reward and safety functions over time, supporting real-time adaptations. Safety is reinforced by robustly shrinking the safe set with each iteration, subtracting a safety margin instead of relying on previous safe regions, which may become unsafe due to temporal changes.

1. Initialization: The algorithm starts with an initial safe set $S_0$ and computes confidence intervals for the auxiliary function wrapping the reward and safety functions.
2. Posterior Updates: Gaussian Process (GP) models with spatio-temporal kernels are used to update the posterior distributions of the functions based on new observations.
3. Safe Set Updates: The safe set $S_k$ is updated at each iteration by considering the uncertainty and temporal variability, ensuring the set's elements meet safety constraints with high probability.
4. Decision-Making: Decisions are made by balancing exploration and exploitation. Potential maximizers are defined, and the most uncertain safe decision is selected to either improve the reward function or expand the safe set.
5. Safety Guarantees: The paper provides theoretical guarantees for maintaining safety across iterations using Lipschitz constants and bounds on the RKHS norm of the functions.
6. Optimality: Near-optimality guarantees are established for scenarios where the reward function becomes stationary, allowing convergence to near-optimal solutions within a defined number of iterations.

Experimental Evaluation

Synthetic Example

In a synthetic experiment, TVSafeOpt was applied to a two-dimensional, time-varying optimization problem with dynamic reward and safety constraints. The results demonstrated that TVSafeOpt outperformed SafeOpt by maintaining fewer unsafe decisions and adapting more effectively to time variations. The cumulative regret of TVSafeOpt was 77.3% lower than that of SafeOpt, indicating superior performance in terms of reward optimization while ensuring safety.

Gas Compressor Case Study

The algorithm was also evaluated in a practical setting involving the optimization of a gas compressor station's operating parameters, considering time-varying demand and degradation. TVSafeOpt achieved a significant reduction in unsafe decisions compared to SafeOpt and approximate optimization methods, with 73.9% fewer violations than SafeOpt. However, this increase in safety came at the cost of optimality, as the cumulative regret of TVSafeOpt was higher by 74.9%.

Implications and Future Directions

The implications of this work are substantial for fields requiring safe and adaptive decision-making in dynamic environments. By ensuring safety in time-varying settings, TVSafeOpt can be applied to real-time process control in robotics, medical dosage design, and adaptive control in autonomous vehicles and industrial processes. Future research could explore further refinement of the theoretical bounds for non-stationary cases and the integration of more sophisticated temporal models to enhance performance in rapidly changing environments.

Conclusion

Li et al.'s TVSafeOpt algorithm marks a significant step forward in the field of safe Bayesian optimization. By incorporating spatio-temporal kernels and considering time Lipschitz constants, the algorithm provides a robust framework for safely and effectively optimizing time-varying reward functions under uncertain and changing safety constraints. Theoretical guarantees ensure its reliability, and empirical results validate its practical utility, making it a valuable tool for dynamic, safety-critical applications.