Failure Probability Estimation for Black-Box Autonomous Systems using State-Dependent Importance Sampling Proposals

Harrison Delecki; Mykel J. Kochenderfer; Sydney M. Katz

REVIEW 4 major objections 4 minor 27 references

Failure Probability Estimation for Black-Box Autonomous Systems using State-Dependent Importance Sampling Proposals

T0 review · 4 major / 4 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read SPAIS estimates rare failure probabilities in black-box sequential systems by training a state-dependent importance sampling proposal with a forward-KL objective, achieving under 10% bias and at least 2x lower variance than baselines on…

desk verdict A practically promising adaptive IS method for black-box failure probability, but the paper's theoretical justification has an unaddressed stale-weight problem in its MCMC step. read the letter →

arxiv 2412.02154 v1 pith:6ZQJCDOY submitted 2024-12-03 cs.RO cs.AIcs.LGstat.ML

Harrison Delecki , Sydney M. Katz , Mykel J. Kochenderfer This is my paper · ORCID

classification cs.ROcs.AIcs.LGstat.ML

keywords failureprobabilityestimationadaptiveimportancesamplingblack-boxsafetyvalidationautonomoussystemsMarkovscoreascentstate-dependentproposalrareevent

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper proposes SPAIS, an adaptive importance sampling algorithm for estimating the probability of failure of black-box sequential autonomous systems. SPAIS parameterizes a proposal over disturbances that depends on the current state, so the importance weight factorizes across timesteps and the high-dimensional proposal-fitting problem reduces to a per-timestep one. The proposal is trained by minimizing the forward KL divergence to a smooth relaxation of the failure distribution, using Markov score ascent with an Independent Metropolis-Hastings kernel to draw near-target trajectories. On four validation systems (an inverted pendulum, a crosswalk scenario, aircraft collision avoidance, and an F-16 ground collision avoidance model), SPAIS reports relative errors within 10% and at least 2x lower empirical variance than Monte Carlo, cross-entropy, and policy-gradient baselines. A reader should care because failure probability estimation is the part of safety validation that says how likely failures are, not merely that they exist, and prior methods do not scale to long horizons with continuous state spaces.

What carries the argument

The central object is the sequential state-dependent proposal $q_\theta(x \mid s)$, a multivariate Gaussian whose mean and covariance are neural networks of the state; it makes importance weights factorize across timesteps, turning the trajectory-fitting problem into conditional density estimation. The mechanism carrying the argument is the forward-KL objective with the relaxed failure distribution $\tilde p(\tau) \propto p(\tau)P_\beta(f(\tau)-\gamma)$, where $P_\beta$ is the logistic CDF with scale $\beta$, optimized via Markov score ascent using an Independent Metropolis-Hastings kernel. The IMH acceptance ratio $\min(1, \tilde w(\tau')/\tilde w(\tau))$ uses unnormalized weights under the relaxed target, avoiding any need for the normalizing constant, and the logistic relaxation keeps acceptance rates from collapsing to zero when proposed trajectories are safe.

What would settle it

Re-run the algorithm with a control variant that recomputes all particle weights under the current proposal $\theta_k$ before every IMH acceptance step and compare the resulting failure probability estimates and acceptance rates on the four benchmark problems; if they differ materially, the stale-weight shortcut is load-bearing rather than harmless.

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Extended reading notes

Core claim

The central claim is that a state-conditioned proposal $q_\theta(x \mid s)$, where $x$ is a disturbance and $s$ is the current state, can be trained to approximate the optimal importance sampling distribution over failure trajectories by minimizing the forward KL divergence to the relaxed target $\tilde p(\tau) \propto p(\tau)P_\beta(f(\tau)-\gamma)$. Because the nominal trajectory density and the proposal density share all transition, observation, and policy factors, the importance weight collapses to $\prod_t d(x_t \mid s_t)/q_\theta(x_t \mid s_t)$, so learning the proposal reduces to fitting a per-timestep conditional density. Markov score ascent supplies low-variance gradients by running a single-step IMH kernel whose acceptance ratio uses unnormalized weights under the relaxed target, and the same MCMC samples are used to form the final importance sampling estimate. The experiments report that SPAIS achieves under 10% empirical bias and at least 2x lower variance than the baselines on all four systems, including two where a state-independent CEM proposal misses failure modes through mode collapse.

Load-bearing premise

The load-bearing premise is that reusing importance weights computed under older versions of the proposal in the acceptance step does not bias the accepted trajectories away from the failure distribution the algorithm is trying to sample.

Editorial extensions

If this is right

Failure probability estimates with under 10% bias and at least 2x lower variance than baselines become practical for sequential systems with continuous state spaces and horizons up to 200 timesteps, where naive Monte Carlo would need billions of samples.
Because the proposal is state-dependent, SPAIS can cover multiple failure modes (e.g., the pendulum falling left and right) rather than collapsing onto a single mode as CEM and PG-AIS do in the reported experiments.
Since the systems are treated as black boxes, the method applies to non-differentiable simulators and controllers whose internal logic is unknown, provided disturbances can be injected and trajectories scored.
The IS buffer formed from all training proposals provides the final estimate, so no separate evaluation phase is required after training completes.
The paper states that the implementation is open sourced, allowing others to reproduce the experiments and apply the method to new systems.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The algorithm's correctness rests on the IMH step reusing weights computed under earlier proposal parameters; a version that recomputes all stored weights under the current proposal at each iteration would reveal whether the reported accuracy depends on this stale-weight shortcut, and the paper does not demonstrate that the MCMC chain actually mixes to the relaxed failure distribution.
Pooling samples from multiple proposals into a single IS buffer is a valid multiple importance sampling estimator only when every sample is weighted by its own proposal density, a property the paper does not state explicitly or analyze for variance.
The same state-dependent proposal and forward-KL objective could be extended to proposals conditioned on observations or actions, or to non-Gaussian families such as normalizing flows, which would likely help on highly multimodal failure distributions.
A testable extension is to vary the relaxation scale $\beta$ and the number of MCMC steps per iteration; the paper fixes $\beta = 10^{-2}$ and one step, so the sensitivity of the bias and variance claims to these choices remains unknown.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: SPAIS is a sensible and well-engineered adaptive importance sampling recipe for rare-event estimation in sequential black-box systems, and the experiments back it up. The main caveat is that the paper's stated objective—minimizing forward KL via Markov score ascent—is not actually implemented as written, because the IMH acceptance uses weights from an outdated proposal. The final estimator is still unbiased, so the empirical results probably survive, but the explanation of why the optimization works does not.

What's new: the combination of a state-dependent sequential proposal, a logistic relaxation of the failure indicator, and Markov score ascent for the forward-KL objective is not in the cited literature. The decomposition in eq. (9) is clean and the importance weight is correct. The four test problems are well chosen and span different state dimensions and horizons. The results in Table 2 are consistent: SPAIS has lower relative and absolute error than MC, CEM, and PG-AIS on all four systems, with about 10% empirical bias and lower variance. That is a genuine contribution to the safety-validation toolbox, especially for practitioners who need an off-the-shelf estimator. The code is open source, which makes the method testable.

The soft spots: First, the stale MH weight issue. In Algorithm 1, the acceptance ratio min(1, w̃'_n/w̃_n) uses a stored weight w̃_n computed under the proposal that last accepted that trajectory, not under the current q_θ_k. The correct IMH ratio requires p̃(τ')/q_θ_k(τ') over p̃(τ)/q_θ_k(τ). Because θ updates every iteration, the chain is not ergodic for the target p̃, and the samples feeding eq. (13) are not from p̃. So the gradient is biased and the claim that the proposal minimizes the forward KL is not supported as written. This is a real issue, not a nitpick. It does not kill the final IS estimate because each buffer sample carries its own p/q weight, but it means the paper's central explanation of why the algorithm works is currently wrong. The fix is straightforward: recompute the weight of the current trajectory under the current q_θ_k before forming the ratio, or use a proper adaptive MCMC framework. Second, the ground-truth failure probabilities came from 10^7 MC samples, but the paper does not report confidence intervals for those ground truths. At µ around 1e-5, 10^7 MC samples gives only tens of failures, so the "true" µ itself has notable uncertainty. That should be propagated into the reported errors. Third, the analysis is entirely empirical; a consistency or convergence argument for the adaptive chain would strengthen the paper, but for a methods paper this is not surprising.

Who this is for: people working on black-box safety validation and rare-event estimation, especially in robotics and autonomous systems. A serious referee should engage with it, but should ask for the MH weight fix and the ground-truth uncertainty. The paper is worth a revision rather than a rejection.

Referee Report

4 major / 4 minor

Summary. The paper proposes SPAIS, an adaptive importance sampling method for estimating failure probabilities of sequential black-box systems. The method uses a state-dependent Gaussian proposal q_theta(x|s), optimized by minimizing the forward KL divergence to a relaxed failure distribution via a Markov score ascent procedure that employs Independent Metropolis-Hastings steps. The final estimate is an importance sampling average over a buffer of all proposed trajectories, weighted by p(τ)/q_theta(τ). The authors evaluate on four systems (inverted pendulum, crosswalk, collision avoidance, F-16 GCAS) and report lower relative and absolute errors than Monte Carlo, the cross-entropy method, and a policy-gradient-based AIS baseline.

Significance. If the method is sound, it addresses a practically important problem: rare-event probability estimation for high-dimensional sequential systems with black-box dynamics. The paper's strengths include an open-source implementation, a clear decomposition of the proposal into state-dependent components, and empirical results on four diverse benchmarks that show substantial improvements over baselines. The final importance-sampling estimator is unbiased if the proposals have full support, which is a valuable property. However, the central theoretical justification is undermined by a flaw in the MCMC update (detailed below), so the paper's claimed link between the optimized proposal and the forward-KL objective is not established as written.

major comments (4)

[Algorithm 1 and §4.3, eq. (14)] The same issue appears in §4.3 where the text says the acceptance probability can be computed using w̃(τ) = p̃(τ)/q_θ(τ); this is only true if the q_θ used to compute both numerator and denominator is the same. The algorithm violates this condition.
[Algorithm 1 in general] The theoretical claims in Section 4.2 should be rewritten to accurately describe what the algorithm guarantees, given this issue.
[Section 5.1 and Table 2] The evaluation metric labeled 'absolute error' is actually absolute relative error (ϵ_abs = |μ̂ − μ|/μ). While the definition is given correctly, the text and table caption call it 'absolute error,' which is misleading. This is a presentation issue, but it affects the interpretation of Table 2 in the main results.
[Section 4.4, hyperparameters] The paper states that 'any small value of β in the range [10^{-4}, 10^{-2}] gives good performance' but provides no sensitivity analysis. Since the relaxed target p̃ is used both for the MCMC kernel and for the loss, the choice of β may affect both the optimization trajectory and the quality of the final IS estimate. A brief sensitivity study or at least a plot of performance vs. β would strengthen the empirical claims.

minor comments (4)

[Abstract and Section 1] The abstract says the method is 'more accurate with fewer samples than MC,' but all methods in Table 2 use the same number of samples (50,000 per trial). The claim of 'fewer samples' appears to refer to the left panel of Figure 1, which shows training progress, but this is not a direct sample-efficiency comparison. Consider clarifying the statement.
[Section 4.2, eq. (13)] The expansion of the loss to eq. (13) is correct, but it would be helpful to explicitly state that the term log p(s1) and the transition probabilities cancel or do not depend on θ, to avoid confusion for readers unfamiliar with the sequential factorization.
[Section 6, Figure 4] The figure captions describe the CEM and SPAIS proposals, but the color bars are not labeled in the main text; adding explicit colorbar labels (e.g., 'disturbance magnitude' in each subplot) would improve readability.
[General] There are a few grammatical slips, such as 'We find using any small value of β in the range [10^{-4}, 10^{-2}] gives good performance' (missing 'that') and 'the algorithm samples N new proposal trajectories τ′' (the symbols should be τ'_n). These do not affect correctness but should be cleaned up.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: SPAIS's failure-probability estimate is an unbiased IS average, and proposal training targets an externally defined relaxed failure distribution; self-citations are not load-bearing.

full rationale

SPAIS's estimator is the standard unbiased IS average of eq. (6) with weights w(τ)=p(τ)/qθ(τ), where qθ is the sequential proposal defined in eqs. (7)-(8). The proposal is trained by minimizing the forward-KL/cross-entropy objective eq. (16) against p̃ ∝ p(τ)Pβ(f(τ)-γ), a relaxation of the failure distribution that is defined externally in terms of p, f, and γ, not in terms of the output estimate. No fitted parameter is renamed as a prediction: β is a hand-set smoothing constant, and θ is optimized before the IS estimate is computed, with the estimate itself being an unbiased Monte Carlo average over buffered samples. The only self-referential aspect is the standard adaptive-IS/Markov-score-ascent loop in which the current proposal generates MCMC samples used to improve the proposal; this is a training dynamics, not a definitional reduction. Self-citations (e.g., prior work from the same lab on adaptive stress testing and baselines) appear in related work and baseline comparisons, but they are not load-bearing for the derivation. A possible technical concern about stale IMH weights in Algorithm 1 is a correctness/detailed-balance issue, not a circularity, and does not make the estimator definitionally dependent on its own output.

Assumptions & free parameters 3 free parameters · 5 assumptions · 0 invented entities

The method rests on standard IS support conditions and on a known disturbance model, plus a hand-chosen relaxation parameter β and the correctness of the IMH acceptance step as implemented. The main addition beyond standard assumptions is the stale-weight validity assumption in Algorithm 1, which is not stated or justified.

free parameters (3)

Logistic scale beta = 1e-2
Chosen by hand in Section 4.4 to relax the failure indicator; the paper states any value in [1e-4, 1e-2] works, but no sensitivity analysis is reported. It controls MCMC acceptance and proposal coverage.
Proposal network architecture = two hidden layers of size 64 and 32
Chosen without ablations in Section 4.4; the expressiveness of the Gaussian mean/covariance networks is central to representing multimodal failure modes.
Particle count N and iterations Niter = not reported, total 5e4 samples per trial
Section 5.1 fixes 50,000 samples per trial but the split between N and Niter is not given, and SGD hyperparameters (learning rate, batch size) are not specified.

assumptions (5)

standard math Importance sampling weights require qθ(x|s)>0 wherever d(x|s)1{failure}>0
Unbiasedness of eq (6) and eq (9) relies on support condition; Gaussian proposals guarantee this for continuous disturbances.
domain assumption The disturbance distribution d(x|s) is known
Eq (9) uses d(x|s)/qθ(x|s); in black-box settings d is assumed known from design, which may not hold for fielded systems.
ad hoc to paper The logistic relaxation Pβ(f(τ)-γ) with small β is a faithful surrogate for the true indicator
Section 4.3 replaces the failure indicator with a logistic CDF; if β is too large the proposal covers safe trajectories, if too small MCMC acceptance collapses. The paper asserts a range works without proof.
ad hoc to paper Stored MH weights remain valid after proposal updates
Algorithm 1 reuses ~w_n from earlier iterations in the IMH acceptance ratio; this is only correct if the proposal parameters have not changed since the weight was computed. The paper does not state or justify this assumption.
standard math Markov score ascent theory applies to this IMH-based estimator
The paper invokes Markov score ascent for low-variance gradient estimates without proving the conditions hold for the relaxed failure target and adaptive proposal.

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Cite this review

Pith. "Pith review of Failure Probability Estimation for Black-Box Autonomous Systems using State-Dependent Importance Sampling Proposals." pith.science (2026). https://pith.science/paper/6ZQJCDOY

@misc{pith2026241202154,
  author       = {Pith},
  title        = {Pith review of: Failure Probability Estimation for Black-Box Autonomous Systems using State-Dependent Importance Sampling Proposals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6ZQJCDOY}},
  note         = {Machine review of arXiv:2412.02154}
}

read the original abstract

Estimating the probability of failure is a critical step in developing safety-critical autonomous systems. Direct estimation methods such as Monte Carlo sampling are often impractical due to the rarity of failures in these systems. Existing importance sampling approaches do not scale to sequential decision-making systems with large state spaces and long horizons. We propose an adaptive importance sampling algorithm to address these limitations. Our method minimizes the forward Kullback-Leibler divergence between a state-dependent proposal distribution and a relaxed form of the optimal importance sampling distribution. Our method uses Markov score ascent methods to estimate this objective. We evaluate our approach on four sequential systems and show that it provides more accurate failure probability estimates than baseline Monte Carlo and importance sampling techniques. This work is open sourced.

Figures

Figures reproduced from arXiv: 2412.02154 by the authors.

**Figure 2.** Illustration of p˜β(τ ). To address the issue of low acceptance rates, we use a smooth approximation of the discontinuous failure distribution. Specifically, we approximate the indicator function in the target by Pβ(f(τ ) − γ) where Pβ is the cumulative distribution function of a logistic distribution with zero mean and scale β. This approximation replaces the discontinuous indicator with a smooth logistic curve … view at source ↗

**Figure 3.** The environments used in failure probability estimation experiments. [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Visited states and applied disturbances by CEM (above) and SPAIS (below) after training. [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.

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Falsification of cyber-physical systems using deep reinforcement learning

Takumi Akazaki, Shuang Liu, Yoriyuki Yamagata, Yihai Duan, and Jianye Hao. Falsification of cyber-physical systems using deep reinforcement learning. In International Symposium on Formal Methods (FM), pages 456--465. Springer, 2018

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Monica F Bugallo, Victor Elvira, Luca Martino, David Luengo, Joaquin Miguez, and Petar M Djuric. Adaptive importance sampling: The past, the present, and the future. IEEE Signal Processing Magazine, 34 0 (4): 0 60--79, 2017

work page 2017

[4] [4]

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Olivier Capp \'e , Arnaud Guillin, Jean-Michel Marin, and Christian P Robert. Population M onte C arlo. Journal of Computational and Graphical Statistics, 13 0 (4): 0 907--929, 2004

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James P. Chryssanthacopoulos, Mykel J. Kochenderfer, and Richard E. Williams. Improved Monte Carlo sampling for conflict probability estimation. In AIAA Non-Deterministic Approaches Conference, 2010. doi:10.2514/6.2010-3012

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A Deep Reinforcement Learning Approach to Rare Event Estimation

Anthony Corso, Kyu-Young Kim, Shubh Gupta, Grace Gao, and Mykel J Kochenderfer. A deep reinforcement learning approach to rare event estimation. arXiv preprint arXiv:2211.12470, 2022

work page Pith review arXiv 2022