Probabilistic Safety Guarantees

• Probabilistic safety guarantees are formal quantified bounds defining the likelihood that uncertain control systems satisfy prescribed safety requirements.
• They utilize mathematical formulations, sampling-based verification, and temporal logic to rigorously assess and manage residual risks under uncertainty.
• These methods are applied across safe reinforcement learning, model predictive control, and motion planning to balance performance and safety with explicit risk measures.

Probabilistic safety guarantees refer to formal, explicitly quantified bounds on the likelihood that a stochastic or uncertain control system will satisfy specified safety requirements, often expressed as chance constraints, invariance conditions, or probabilistic temporal logic properties. Unlike deterministic safety, which aims for almost-sure satisfaction, probabilistic approaches recognize and formally quantify residual risks induced by noise, modeling error, online learning, or environment uncertainty. Recent research has produced rigorous, scalable methods for safe reinforcement learning, model predictive control, motion planning, and safety filtering in continuous and discrete domains.

1. Mathematical Formulations of Probabilistic Safety

Probabilistic safety guarantees are grounded in explicit mathematical statements regarding the probability of remaining within a safe set or satisfying a temporal logic specification. For a discrete-time stochastic dynamical system

$x_{k+1} = f(x_k, u_k, w_k)$

with state $x_k \in \mathbb{R}^n$, control $u_k \in \mathbb{R}^m$, and disturbance $w_k$ (random, possibly adversarial), the key safety property is typically: $$\mathbb{P}\left[ x_k \in S \;\forall k=0,\dots,N \right] \geq 1-\epsilon$$ for some safe set $S$ and risk threshold $\epsilon \in (0,1)$. When temporal logic specifications $\psi$ (e.g., signal temporal logic, STL) are involved, probabilistic guarantees take the form: $$\mathbb{P}\left[ \text{trajectory } (x_{0:N}) \models \psi \right] \geq 1-\epsilon$$ A key notion is the robustness function $\rho(x, t)$ associated with $\psi$, satisfying $$\rho(x, t) \geq 0 \Longleftrightarrow (x, t) \models \psi$$.

Probabilistic safety can also be specified over finite horizons using reachability or forward invariance probabilities, or using barrier functions $h(x)$: $$\Pr( h(x_{k+1}) \geq \alpha h(x_k) \mid x_k ) \geq 1-\delta$$ for prescribed $\alpha, \delta$ (Mestres et al., 1 Oct 2025), which yields horizon-wise $\epsilon$-safety via $(1-\delta)^H \geq 1-\epsilon$.

2. Verification Methodologies and Sampling-Based Guarantees

Contemporary methods for certifying probabilistic safety employ rigorous sampling and scenario-based approaches to estimate the probability that a controller or policy will satisfy the safety specification under uncertainty. The scenario approach, as exemplified in (Krasowski et al., 2022), considers a verified controller $u(x)$, perturbed by bounded disturbances $\xi_k \in \mathcal{E}$: $$x_{k+1} = f(x_k, u(x_k) + \xi_k), \quad \xi_k \sim \text{Uniform}(\mathcal{E})$$ The key probabilistic guarantee uses sampled trajectories $\{\varphi_{p_i}\}$ (from random initial conditions and perturbation sequences), robustness evaluations $r_i = \rho(\varphi_{p_i})$, and the minimum $c^* = \min_{i} r_i$. Then, for $\epsilon \in [0,1]$: $$\mathbb{P}^N\left[\,\mathbb{P}\left[\,\rho(\varphi_p) \geq c^*_N\,\right] \geq 1-\epsilon\,\right] \geq 1-(1-\epsilon)^N$$ If $c^*_N \geq 0$, at least $1-\epsilon$ fraction of all possible perturbations yield safe executions, with confidence $1-(1-\epsilon)^N$.

Another approach constructs formal abstractions (e.g., box domains in state space), as in model checking frameworks for deep RL (Bacci et al., 2020), obtaining explicit bounds: $$\Pr_s( \text{unsafe visit within } T ) \leq \widehat{\Pr}_{\max}(\hat{s}, \text{unsafe within } T )$$ through sound over-approximations.

Performance and sample complexity trade-offs are governed by Hoeffding-type inequalities and volume-fraction arguments—e.g., $N \geq (1 / 2\epsilon^2)\ln(2/\delta)$ implies that $|\widehat{\mu} - \mu| \leq \epsilon$ with probability $1-\delta$ for sampling-based shielded RL (Goodall et al., 1 Feb 2024).

3. Safety-Constrained Policy Optimization and Action Filtering

To leverage the certified probabilistic safety in closed-loop control and RL, controllers are restricted to act within the verified safety tube or margin: $$\mathcal{A}(x) = u(x) \oplus \mathcal{E} = \{ u(x) + \delta: \delta \in \mathcal{E} \}$$ RL agents are trained to optimize performance purely within $\mathcal{A}(x)$, inheriting the original safety guarantee by design (Krasowski et al., 2022). No further composition is necessary—the probabilistic property is re-applied after RL convergence. Policy-gradient methods are augmented with safety penalties, probabilistic logic returns, and counter-example weighting to ensure RL agents do not exploit shield weaknesses (Goodall et al., 1 Feb 2024).

Safety filters (e.g., QP-based barrier filters) enforce action constraints at each step in real time, transforming probabilistic CBF conditions: $$u_t = \arg\min_{u \in U} \|u - u_{\text{nom}}(x_t)\|^2 \;\text{ s.t.} \;\Pr( \Delta h(x_t, u, w_t) \geq 0 ) \geq 1-\delta$$ using one of several tractable surrogates: Markov/Cantelli mean-variance bounds, empirical quantiles (Hoeffding), scenario optimization, or conformal prediction (Mestres et al., 1 Oct 2025).

4. Temporal Logic and Long-Horizon Safety Specifications

Signal temporal logic (STL) is widely employed to express complex safety specifications over trajectories. These specifications are translated into robustness functions $\rho(x, t)$, and safe RL is tasked with maximizing $P(\psi)$ under disturbance. Probabilistic certification is performed over the STL formula, either via sampling (scenario-based), barrier function approaches, or stochastic reachability.

Recent work introduces probabilistic invariance conditions in probability space, enforcing single-step affine constraints: $$D_h(Z_t, U_t) \geq -\alpha\bigl( h(Z_t) - (1-\epsilon) \bigr)$$ on an augmented state $Z_t$ encoding remaining horizon, margin, barrier value, and state (Wang et al., 23 Apr 2024, Wang et al., 2021). This technique provably maintains long-term safe probability $1-\epsilon$ in expectation, outperforming classic infinitesimal methods.

5. Implementation, Algorithmic Considerations, and Real-World Deployment

Implementation entails iterating between probabilistic verification, policy improvement, and re-verification. The process is efficiently scalable to continuous state/action spaces, compatible with black-box systems. Sampling-based certification requires careful selection of batch size $N$ and risk threshold $\epsilon$; these directly control the confidence and conservatism of the guarantee.

Practical code instances include safe RL with PPO restricted to the certified tube, shielded RL using Dreamer + AMBS, and QP-based safe control using probabilistic CBFs under learned uncertainty models. Deterministic MPC can be rendered probabilistically safe by enforcing state constraints on an eroded safe set $S_\varepsilon = S \ominus B^n(\varepsilon)$, where

$$\varepsilon = \sigma \sqrt{ \frac{L^{2N}-1}{L^2-1} \left(n + 2\ln\frac{1}{\delta} \right) }$$

controls the safety margin (Liu et al., 15 Sep 2025).

Empirical results show that such methods can maintain safety probabilities $>99\%$, reduce safety violations by factors of $2$–$5$ compared to unconstrained RL, and generalize to real robot hardware. Tracking the minimum robustness across verification trials quantifies the preservation and improvement of the safety property through learning.

6. Scalability, Limitations, and Open Challenges

While probabilistic safety guarantees scale to high-dimensional continuous domains and can be integrally compatible with RL and MPC, substantial challenges persist. Achieving ultra-low failure rates ($\delta < 10^{-8}$) in systems interacting with humans is infeasible with present data-driven uncertainty models due to massive sample complexity requirements (typically $N \propto 1/\delta$) (Cheng et al., 2021). Unreliable uncertainty bounds at extreme confidence levels undermine downstream safety proofs.

Suggested mitigations include combining learning-based models with deterministic rules or formal assume-guarantee contracts, using hierarchical fallback strategies, and fusing redundant prediction modules to drive joint $\delta$ ever lower. Practitioners are advised to audit tail behavior rigorously and expose model uncertainty throughout the pipeline.

7. Summary and Forward Directions

Probabilistic safety guarantees provide a rigorous framework for safe control and learning under uncertainty, blending formal verification, randomized sampling, and robust optimization. They admit explicit trade-offs between conservatism and performance, are readily implementable across RL, MPC, and filtering architectures, and fundamentally advance the quantification and certification of safety in stochastic, data-driven environments. Research continues toward higher-confidence guarantees, scalable compositional methods, tighter risk bounds for human-in-the-loop systems, and the integration of probabilistic certificates with neural policy verification, compositional barrier certificates, and scenario-based MPC.