Maximum Safe Dynamics Learning

• Maximum Safe Dynamics Learning is a framework for modeling unknown nonlinear dynamics while strictly enforcing continuous safety constraints.
• It uses Gaussian Processes with probabilistic confidence bounds to update model accuracy online and guarantee safe policy exploration.
• The approach is validated in autonomous vehicles and drones, achieving finite-time learning with no episodic resets and near-optimal reward maximization.

Maximum Safe Dynamics Learning refers to the process of learning unknown – typically nonlinear – system dynamics from sequential data while strictly enforcing safety constraints at every stage, ensuring that the system never leaves a certified safe operating region and that the accuracy of the learned model improves up to a user-specified tolerance. This setting is central in real-world domains where unsafe exploration is catastrophic and episodic resets are infeasible, such as autonomous vehicles, high-performance robotics, and aerial navigation. The framework introduced in (Prajapat et al., 20 Sep 2025) provides the first online, non-episodic guarantee of sufficient dynamics learning, high-probability persistent safety, and near-optimal reward maximization.

1. Problem Formulation and Safety Concepts

The framework considers an unknown discrete-time system

$x(k+1) = f(x(k), u(k)) + \eta(k)$

where $x$ is the state, $u$ the control, $f$ is an unknown dynamics function, and $\eta(k)$ is noise (assumed sub-Gaussian and bounded). The agent has no direct model of $f$ but must operate in continuous time, gathering samples through a single trajectory, and must ensure safety throughout (i.e., the system must remain in a constraint set $\mathcal{X}$ and ultimately return to a robustly invariant “safe set” $\mathcal{X}_0$).

Notions of Safety

• Pessimistically Safe Policy (Editor’s term): Ensures all closed-loop trajectories under any model in the current confidence set will remain safe and will be returnable to the initial safe region.
• Optimistic Exploration: While acting safely for all plausible models, the agent plans informative actions that would reach high-uncertainty areas under some model, even if model uncertainty precludes actually reaching these points.
• The key safety requirement: At every time step, for any plausible dynamics $f$, the planned finite-horizon action sequence never leaves constraints and ends in the safe set.

2. Learning Dynamics via Probabilistic Confidence Bounds

The unknown dynamics $f$ is modeled using probabilistic confidence sets constructed from sequential measurements, typically using independent vector-valued Gaussian Processes (GPs). At each state-input pair $z=(x,u)$, the GP yields a mean $\mu_n(z)$ and standard deviation $\sigma_n(z)$ so that with high probability

$$|f_i(z) - \mu_{n,i}(z)| \leq \sqrt{\beta_{n,i}}\, \sigma_{n,i}(z), \ \forall i$$

where $\beta_{n,i}$ is a scaling parameter chosen via concentration inequalities to ensure the coverage.

The set of plausible dynamics is defined by

$$\mathcal{F}_n = \left\{ f : |f_i(z) - \mu_{n,i}(z)| \leq \sqrt{\beta_{n,i}}\ \sigma_{n,i}(z), \ \forall z, i \right\}$$

These confidence sets update online as new measurements are collected, continuously shrinking as model certainty improves.

3. Online Safe Policy Space Exploration

At each time step:

• The agent plans finite-horizon policies over horizon $H$.
• It restricts candidate policies to those that are safe for all $f \in \mathcal{F}_n$, i.e., robust to worst-case plausible dynamics (pessimistically safe).
• Simultaneously, the agent searches for policies that maximize the minimum GP confidence width (optimistically explores).

Key Algorithms

A. Maximum Safe Dynamics Exploration:

• Given the current state, find a policy from the pessimistic safe set that steers the system to a region where the minimum of the GP’s confidence width is above threshold $\epsilon$.
• Execute this policy; update the dataset with observations, thus refining the model.
• Repeat until no further optimistically informative action is safely reachable; then, dynamics have been learned sufficiently to the desired tolerance.

B. Reward Maximization Overlay:

• After sufficiently learning the dynamics, maximize the reward (or minimize cost) over the class of policies guaranteed to be safe for all $f \in \mathcal{F}_n$.
• Both the optimistic ("best plausible model") and pessimistic ("worst-case plausible") performance bounds are computed; the process repeats until the performance gap is within a desired bound.

Policy Set Definitions

• The true $\epsilon$-safe policy set contains all policies that are safe up to tolerance $\epsilon$.
• The pessimistic safe policy set enforces safety for every $f \in \mathcal{F}_n$.
• The optimistic safe policy set only requires one $f \in \mathcal{F}_n$ for a safe rollout, enabling selection of informative trajectories.

4. Theoretical Guarantees

The main theoretical results are:

• Finite-Time Completeness: The agent is guaranteed to learn the dynamics up to any arbitrary small tolerance (subject to noise) in a finite number of steps, while always remaining safe with high probability.
• No Resets Required: The framework operates in a non-episodic, fully online fashion. The system never needs to be reset; each action is computed relative to all past data and the current state.
• Provable Safe Operation: At all times, for all possible candidate models, the executed policies maintain safety and returnability.
• Near-Optimal Performance: Once model uncertainty is sufficiently reduced, the agent’s reward can be brought arbitrarily close to optimal over the robustly invariant safe set.

5. Mathematical Formulations

The planning and exploration processes are formalized as constrained finite-horizon optimal control problems: $\begin{aligned} & \text{Find } \pi^p \in \mathcal{P}_n(x(k); H) \ & \text{s.t.~ at some step %%%%21%%%%:}~ w_n(x_h, \pi_h(x_h)) = 2 \sqrt{\beta_n} \sigma_n(x_h, \pi_h(x_h)) \geq \epsilon \end{aligned}$ where $\mathcal{P}_n(x(k);H)$ denotes the pessimistically safe policy set, and $\sigma_n$ is the GP’s posterior standard deviation.

During reward maximization: $$J^p(x(k), \mu_n; \pi) = J(x(k), \mu_n; \pi) - L_r \sum_{h=0}^{H-1} \sum_{i=0}^H L^i w_n(x_i, \pi_i(x_i))$$ where $J^p$ penalizes the optimistic policy reward by uncertainty terms, ensuring that the performance bound is valid for all $f \in \mathcal{F}_n$.

6. Empirical Results and Applications

The framework is evaluated on nonlinear, high-dimensional domains:

• Autonomous Car Racing: The controller learns the vehicle’s nonlinear dynamics online, safely explores near the limits of the track, and achieves near-optimal performance without ever violating state or input constraints.
• Drone Navigation: In the presence of aerodynamic challenges and state uncertainty, the agent safely learns the true model and closely tracks complex reference trajectories.

Performance metrics include cumulative regret (rate of online reward improvement), safety violations (zero in all cases), and model learning efficiency (measured by confidence width reductions).

7. Implications, Contrast with Existing RL Methods, and Significance

This framework provides a new paradigm for safe reinforcement learning and online model-based control:

• Contrast to standard RL: Unlike classical RL, which typically relies on episodic learning and only expects safety after convergence, this method enforces safety at every time step throughout learning.
• Contrast to prior safe learning: Earlier algorithms either required resets, operated only in constrained parts of the state space, or lacked guarantees on finite-time sufficient model learning.
• Suitability: The approach is robust to noise, does not assume resets, enables online deployment, and provides rigorous finite-time guarantees for both identification and control.
• Real-world impact: The approach is applicable in autonomous driving, UAV navigation, and other settings where unsafe exploration is not tolerable and where learning must be accomplished in continuous operation.

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