Safety-Informed Costing

• Safety-Informed Costing is an approach that translates safety risk data into explicit cost metrics for decision-making, using CMDP and economic frameworks.
• It integrates learned or physics-derived safety signals with cost optimization methods in reinforcement learning and engineering applications.
• The methodology bridges safety engineering and economics by embedding hazard quantification into lifecycle valuations and risk-budgeting strategies.

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to=arxiv_search 下载彩神争霸 Searching arXiv for papers on safety-informed costing and related safe RL cost-learning methods. Safety-informed costing denotes a family of methodologies in which safety-relevant information is translated into explicit cost terms, constraints, or standardized cost accounts that can be optimized, budgeted, or propagated through financial analysis. In recent arXiv literature, the term covers constrained Markov decision process formulations in which unsafe behavior is encoded as a learned or constructed cost signal, and engineering-economic frameworks that enumerate hazards, map mitigating scope into formal accounts, and carry the resulting safety burden into lifecycle and valuation metrics such as LCOE, NPV, IRR, MIRR, and WACC (Chirra et al., 2024, Woodruff et al., 22 Feb 2026, Wortman et al., 2020).

1. Formal problem classes

In safe reinforcement learning, safety-informed costing is formulated on a CMDP with state space $\mathcal S$, action space $\mathcal A$, transition kernel $P(s' \mid s,a)$, reward $r$, cost $c$, discount $\gamma$, and a safety budget. The canonical objective is to maximize expected cumulative reward while satisfying an expected cumulative cost constraint, for example $\mathcal J^c(\pi)\le c_{\max}$ or $\mathbb E_{\tau\sim\pi}[C(\tau)]\le \kappa$. Offline variants additionally constrain policy shift by imposing $D_{KL}(\pi\|\pi_\beta)\le \delta$, reflecting the need to remain near the behavior policy and avoid large OOD shifts when only a static dataset is available (Chirra et al., 2024, Koirala et al., 2024).

In risk-based protective-system economics, the primitive object is not a policy but a candidate design $\alpha$, characterized by a lifetime probability of catastrophic failure $\mathcal A$0, a non-catastrophe social cost $\mathcal A$1, and a catastrophe premium $\mathcal A$2. The expected lifecycle social cost is

$\mathcal A$3

and the regulated, risk-averse enterprise selects

$\mathcal A$4

This makes safety information economically operative through failure probability, catastrophe premium, and regulatory context rather than through an RL constraint set (Wortman et al., 2020).

A plausible implication is that the same conceptual move recurs across otherwise different domains: safety knowledge becomes decision-relevant only after it is embedded into an additive object that can be optimized, constrained, or rolled up into an aggregate objective.

2. Learning safety costs from sparse feedback

One line of work treats the safety cost itself as unknown and infers it from sparse supervision. In "Safety through feedback in Constrained RL" (Chirra et al., 2024), Safety-Informed Costing is presented as RLSF, a CMDP framework with unknown binary cost $\mathcal A$5. Instead of labeling every state-action pair, an external evaluator provides binary labels on trajectory segments: $\mathcal A$6 means the segment was entirely safe, and $\mathcal A$7 means it was unsafe at least once. The method learns a parametric safety model $\mathcal A$8, replaces the unstable exact segment likelihood with a surrogate per-state cross-entropy loss, and thresholds the converged model as $\mathcal A$9. Its guarantee

$P(s' \mid s,a)$0

implies that any policy safe under the inferred cost is guaranteed safe under the ground-truth cost. To reduce evaluator burden, novelty-based sampling hashes states with SimHash, maintains counts over evaluated hashes, and queries feedback only for trajectories containing at least $P(s' \mid s,a)$1 distinct unseen states. The reported ablations state that novelty sampling requires $P(s' \mid s,a)$2 fewer trajectory queries to reach the same return/cost-violation performance and that the surrogate loss tolerates up to $P(s' \mid s,a)$3 segment-label noise without degradation (Chirra et al., 2024).

"TraCeS: Trajectory Based Credit Assignment From Sparse Safety Feedback" (Low et al., 17 Apr 2025) addresses the same basic problem through a trajectory-probability model that decomposes safety into multiplicative timestep credits. Given a binary trajectory label $P(s' \mid s,a)$4, the model estimates

$P(s' \mid s,a)$5

where each $P(s' \mid s,a)$6 captures the marginal effect of the observation at time $P(s' \mid s,a)$7 on the probability that the whole trajectory is safe. By defining the learned per-step cost as

$P(s' \mid s,a)$8

the problem is rewritten as a standard constrained RL objective $P(s' \mid s,a)$9. The safety model is then coupled to PPO-Lagrangian through reward-advantage and cost-advantage terms. This replaces binary trajectory labels with a dense learned cost signal suitable for continuous-control optimization (Low et al., 17 Apr 2025).

These approaches make explicit that, in many safety-critical domains, the central costing problem is epistemic before it is algorithmic: the safety signal must first be inferred, then enforced.

3. Cost shaping, dual penalties, and persistent safety in policy optimization

When a safety cost is available or can be estimated, safety-informed costing also includes the construction of optimization objectives that use that cost without collapsing reward-seeking behavior. "FAWAC: Feasibility Informed Advantage Weighted Regression for Persistent Safety in Offline Reinforcement Learning" (Koirala et al., 2024) is a representative offline formulation. FAWAC defines reward advantage and cost-advantage,

$r$0

and solves a per-state trust-region update under both feasibility and KL constraints. The non-parametric optimal policy takes Boltzmann form,

$r$1

so the actor update becomes weighted maximum likelihood with

$r$2

The method includes FAWAC-M, which learns a state-wise multiplier $r$3, and FAWAC-P, which uses a fixed finite penalty $r$4 together with an indicator of whether $r$5. Proposition 3 gives the persistent-safety bound

$r$6

so the learned policy exceeds the cost budget by at most an $r$7 slack term induced by the KL trust region (Koirala et al., 2024).

"AutoCost: Evolving Intrinsic Cost for Zero-violation Reinforcement Learning" (He et al., 2023) addresses a different failure mode: standard constrained RL can oscillate around the constraint boundary and still incur nonzero violations even when the cost limit is zero. AutoCost augments the extrinsic cost $r$8 with a learned intrinsic cost $r$9, forming $c$0, and performs a bi-level evolutionary search over the intrinsic-cost network. The objective is to choose $c$1 such that the downstream safe-RL solver trained under $c$2 yields a policy with zero true violations under $c$3. The reported result is that converged policies with intrinsic costs in all environments achieve zero constraint violation and comparable performance with baselines (He et al., 2023).

"Safe In-Context Reinforcement Learning" (Moeini et al., 29 Sep 2025) extends the costing idea to parameter-update-free adaptation. Over $c$4 episodes on a novel test MDP, it seeks

$c$5

To avoid episode-specific multipliers $c$6, it introduces a single exact-penalty surrogate

$c$7

with updates

$c$8

During pretraining, each context is paired with a Cost-To-Go token $c$9, and the actor loss adds a penalty $\gamma$0 whenever an episode exceeds the budget. The reported empirical effect is budget-reactive behavior: higher CTG produces more aggressive behavior, and lower CTG produces more conservative behavior (Moeini et al., 29 Sep 2025).

Taken together, these methods show that safety-informed costing is not restricted to static penalties. It also includes dual variables, cost-advantages, intrinsic costs, and context-conditioned budget tokens.

4. Physics-informed and barrier-derived safety signals

A second line of work derives safety costs from physical structure rather than from labels alone. "Physics-informed RL for Maximal Safety Probability Estimation" (Hoshino et al., 2024) starts from the multiplicative long-term safety probability

$\gamma$1

Because a naive logarithmic reduction does not fit standard MDP value-function theory, the paper introduces an augmented state $\gamma$2, declares unsafe or expired states absorbing, and uses a per-step reward

$\gamma$3

Proposition 1 shows that the resulting additive value $\gamma$4 equals the original safety probability $\gamma$5. In continuous time, the optimal action-value satisfies an HJB-type PDE, and PIRL trains a Q-network with a combined loss

$\gamma$6

where $\gamma$7 is the TD loss, $\gamma$8 enforces the PDE residual, and $\gamma$9 enforces terminal and boundary conditions. The paper states that physics constraints can serve as an alternative to reward shaping, and reports that PIRL can recover long-horizon safety probability from short-horizon samples with low mean-squared error while pure DQN degrades sharply (Hoshino et al., 2024).

"Beyond Safety Filtering: Control Barrier Function-Informed Reinforcement Learning for Connected and Automated Vehicles" (Xu et al., 16 May 2026) derives a safety penalty directly from CBF constraint values. For a TTCBF constraint value $\mathcal J^c(\pi)\le c_{\max}$0, safety requires $\mathcal J^c(\pi)\le c_{\max}$1, and violations are mapped to a clipped normalized penalty

$\mathcal J^c(\pi)\le c_{\max}$2

Road-boundary and inter-vehicle penalties are combined as

$\mathcal J^c(\pi)\le c_{\max}$3

and total reward is $\mathcal J^c(\pi)\le c_{\max}$4. In the four-way multi-lane intersection benchmark, the mean total reward across all tested hyperparameters was $\mathcal J^c(\pi)\le c_{\max}$5 (std $\mathcal J^c(\pi)\le c_{\max}$6) for the CBF-informed reward, compared with $\mathcal J^c(\pi)\le c_{\max}$7 (std $\mathcal J^c(\pi)\le c_{\max}$8) for TTC-based and $\mathcal J^c(\pi)\le c_{\max}$9 (std $\mathbb E_{\tau\sim\pi}[C(\tau)]\le \kappa$0) for distance-based baselines; under best hyperparameters, the CBF activation degree was $\mathbb E_{\tau\sim\pi}[C(\tau)]\le \kappa$1 intervention versus $\mathbb E_{\tau\sim\pi}[C(\tau)]\le \kappa$2 for distance and $\mathbb E_{\tau\sim\pi}[C(\tau)]\le \kappa$3 for TTC (Xu et al., 16 May 2026).

These formulations distinguish safety-informed costing from purely heuristic reward design. In one case, physics constraints transmit risk information through a PDE residual; in the other, exact CBF-constraint values are converted into a scalar penalty.

5. Margins of safety, negligence, and economically optimal protection

In protective-system economics, safety-informed costing is centered on the relation among failure probability, margin of safety, regulatory authority, and liability. The protection efficacy of design $\mathbb E_{\tau\sim\pi}[C(\tau)]\le \kappa$4 is

$\mathbb E_{\tau\sim\pi}[C(\tau)]\le \kappa$5

and, relative to a reference efficacy $\mathbb E_{\tau\sim\pi}[C(\tau)]\le \kappa$6, the margin of safety is

$\mathbb E_{\tau\sim\pi}[C(\tau)]\le \kappa$7

Substituting $\mathbb E_{\tau\sim\pi}[C(\tau)]\le \kappa$8 into the lifecycle cost expression gives

$\mathbb E_{\tau\sim\pi}[C(\tau)]\le \kappa$9

so each additional unit of safety margin is priced through the catastrophe premium $D_{KL}(\pi\|\pi_\beta)\le \delta$0. The paper then links this to the calculus of negligence, or Hand Rule, $D_{KL}(\pi\|\pi_\beta)\le \delta$1, and derives the marginal condition $D_{KL}(\pi\|\pi_\beta)\le \delta$2 as the economically optimal point at which the marginal cost of further reducing failure probability equals the marginal benefit measured by loss avoidance. Prescriptive regulation fixes a minimum margin of safety, whereas performance-based regulation imposes an expected-loss cap or an equivalent Lagrangian constraint and allows the enterprise to select the economically optimal margin (Wortman et al., 2020).

A related economic model appears in cybersecurity planning. SECAdvisor represents annualized security cost as

$D_{KL}(\pi\|\pi_\beta)\le \delta$3

where $D_{KL}(\pi\|\pi_\beta)\le \delta$4 is explicit control cost, $D_{KL}(\pi\|\pi_\beta)\le \delta$5 is the residual breach probability after investment, and $D_{KL}(\pi\|\pi_\beta)\le \delta$6 is the loss magnitude. The expected net benefit of investment is

$D_{KL}(\pi\|\pi_\beta)\le \delta$7

and the first-order condition for an interior optimum is

$D_{KL}(\pi\|\pi_\beta)\le \delta$8

In the segmented formulation, investments $D_{KL}(\pi\|\pi_\beta)\le \delta$9 are allocated across assets or business segments subject to $\alpha$0, and candidate controls can then be selected by a knapsack-style step. Although presented for cybersecurity planning, this is structurally the same optimization problem: choose safety expenditure by balancing direct mitigation cost against residual expected loss (Franco et al., 2023).

This economic literature makes clear that safety-informed costing is not merely a compliance surcharge. It is an optimization framework in which safety margin, residual risk, and regulatory form jointly determine what counts as cost-effective protection.

6. Hazard-to-account mapping, uncertainty propagation, and finance in fusion power plants

In fusion power-plant analysis, safety-informed costing is an explicit methodology for translating hazards into plant scope and then into standardized accounts. The hazard taxonomy in the CATF/pyFECONs framework includes plasma disruption or off-normal plasma behavior, rapid structural oxidation, direct radiation exposure, cooling disruption accidents (LOCA/LOFA/LOHR), corrosion or activated transport, cryogenic hazards, tritium and activated material release, supplementary heating beam hazards, vacuum system hazards, radwaste hazards and handling, and fuel handling hazards. For each hazard, representative mitigations are mapped into the chart of accounts: direct radiation exposure maps concrete bioshield and local shield inserts into Account 21.03 and Account 22.1.07; cooling disruption accidents map additional decay-heat removal systems and thermal-shock-resistant provisions into Account 22.1.11 and Account 22.02; tritium and activated material release map detritiation equipment and site buffer provisions into Account 22.03 and Account 20.01; radwaste handling maps into Account 22.08 and Account 54; fuel handling maps into Account 22.09; licensing and compliance map into Account 62.00; and insurance implications map into Account 53.00 (Woodruff et al., 22 Feb 2026).

The same framework then adds explicit regulatory and financial proxies. Licensing cost is modeled as

$\alpha$1

insurance as

$\alpha$2

and discretionary safety contingency as

$\alpha$3

Safety scope is then propagated through a probabilistic costing layer. The baseline equation is

$\alpha$4

and safety-specific uncertainty can be added either as separate terms

$\alpha$5

or uniformly as a product over $\alpha$6. Monte Carlo sampling produces distributions for total capital and O&M cost, which are passed onward into LCOE and value metrics (Woodruff et al., 22 Feb 2026).

Those value metrics are computed directly from safety-inclusive cash-flow objects. With total overnight CAPEX $\alpha$7, annualized capital cost $\alpha$8, and annual operating cost $\alpha$9 including insurance, licensing, and safety operating provisions, LCOE is

$\mathcal A$00

The same safety-inclusive cash flows enter

$\mathcal A$01

the IRR root condition, the MIRR construction from $\mathcal A$02 and $\mathcal A$03, and the WACC formula

$\mathcal A$04

The illustrative FOAK example yields $\mathcal A$05 without it, while the NOAK example yields $\mathcal A$06 without safety scope. The methodological point is broader than those specific numbers: safety costs are treated as first-class accounting objects, and the stochastic layer does not replace the account taxonomy (Woodruff et al., 22 Feb 2026).