Title: Conformal Prediction in The Loop: A Feedback-Based Uncertainty Model for Trajectory Optimization
Venue: NeurIPS 2025
Accepted: yes
Keywords: Conformal prediction

Feedback

Trajectory optimization

Uncertainty quantification

Risk allocation
Abstract: Conformal Prediction (CP) is a powerful statistical machine learning tool to construct uncertainty sets with coverage guarantees, which has fueled its extensive adoption in generating prediction regions for decision-making tasks, e.g., Trajectory Optimization (TO) in uncertain environments. However, existing methods predominantly employ a sequential scheme, where decisions rely unidirectionally on the prediction regions, and consequently the information from decision-making fails to be fed back to instruct CP. In this paper, we propose a novel Feedback-Based CP (Fb-CP) framework for shrinking-horizon TO with a joint risk constraint over the entire mission time. Specifically, a CP-based posterior risk calculation method is developed by fully leveraging the realized trajectories to adjust the posterior allowable risk, which is then allocated to future times to update prediction regions. In this way, the information in the realized trajectories is continuously fed back to the CP, enabling attractive feedback-based adjustments of the prediction regions and a provable online improvement in trajectory performance. Furthermore, we theoretically prove that such adjustments consistently maintain the coverage guarantees of the prediction regions, thereby ensuring provable safety. Additionally, we develop a decision-focused iterative risk allocation algorithm with theoretical convergence analysis for allocating the posterior allowable risk which closely aligns with Fb-CP. Furthermore, we extend the proposed method to handle distribution shift. The effectiveness and superiority of the proposed method are demonstrated through benchmark experiments.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper addresses the problem of complementing time-series prediction with reliable prediction intervals using efficient conformal prediction sets. To improve the efficiency of these sets, the authors propose re-estimating the miscoverage rate at past time steps in order to reallocate the budget for future steps. Across a range of experiments, the proposed method outperforms baseline approaches that fix the miscoverage requirement beforehand and do not adapt it over time.\n\nSTRENGTHS AND WEAKNESSES:\nThe idea of \"stacking\" CP predictions to reliably estimate the conservativeness of the policy is interesting and novel. In practice, this is shown to greatly improve the performance of time-series CP without compromising coverage.\n\nMy main concerns about the paper are as follows:\n- The assumption of i.i.d. prediction errors is quite restrictive, as one would expect the errors to depend on past quantities. The main text mentions that this assumption can be removed, and Appendix D provides further discussion. However, based on the material in Appendix D, it is unclear how one would extend Lemma 4.2 and Corollary 4.3 once this assumption is dropped.\n- The guarantees of the algorithm are not clearly stated. The original \"uncorrected\" coefficients are guaranteed to produce policies that satisfy the constraint requirements. Do the same guarantees hold for the adjusted coefficients?”\n\nQUESTIONS:\n- CP sets are known to have two-sided guarantees and to be asymptotically optimal. That is, one can show that the coverage probability is also lower bounded, and that the bound approaches the target level $\\alpha$ as the sample size $n$ increases. Given this, it seems that ‘recalibrating’ the thresholds is not necessary for large $n$, as these sets are not conservative in that regime. Where does the conservativeness of CP arise in the scenario considered in the paper? Why would one need to use the proposed method for large $n$?\n\n-I believe Remark 4.4 should be toned down. Corollary 4.3 shows that the expected value of the adjusted reliability level is lower than a quantity that depends on the original value. However, it is not straightforward to conclude that $\\beta < \\alpha$, especially for small $N$ and $L$. Claiming that the proposed approach performs no worse than the non-adjusted one seems to be a stretch.\n\nSome minor mistakes:\n- line 200 \"an feasible...\"\n- line 230 \"c(.)\"\n- In eq.9  should not $x_\\tau^*$ depend on (i) since it is the policy for the specific trajectory?\n\nLIMITATIONS:\n- The main limitation of this work is the i.i.d. residual error assumption, which is quite restrictive in time-series forecasting. This is especially true given that the paper deals with CP, whose main strength lies in being a distribution-free framework.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 2, "DATE": "2025-06-23"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper discuss the application of conformal prediction (CP) for trajectory optimization. The authors identify a gap between the theoretical and application side of CP research and proposed a feedback-based CP frame work for srhinking horizon TO in uncertain environments with dynamic obstacles. The author provide theoretical analysis and numerical experiment in a simulation environment to show case the model is able to achieve good collision avoidance rate.\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths\n- The paper is well motivated. The authors were able to provide ample amount intuition pointing out the gap in current reseach for comformal prediction for trajectory optimization problem\n- The author provides comprehensive theoratical analysis on the effect of incorporating feeback information from realization of agent decision as time progresses.\n\n### Weaknesses\n- Despite the comprehensive theoratical analysis, the overall paper are quite dense and unfriendly to readers outside of the comformal prediction research domain. The author should consider reorganize the paper and put some of the lemma or corollary in the appendex.\n- Consider the main contribution of the paper lies in providing a feedback-based CP for trajectory optimization, the author should consider put the algorithm in the appendix in the main paper. The author should also put more numerical results in the main paper to demonstrate the theoratical analysis.\n\nQUESTIONS:\n- Why not using more powerful trajectory predictors for $g()$ in eqution 2?\n\nLIMITATIONS:\nThe author mentioned limitation of the Fb-CP lies in the size of the calibaration data.\nConsider the ample amount of trajectory prediciton works, the author should consider adopt a more sophisiticated algorithm to proper gate obstacle trajectory such as [1] and [2]. And consider apply this in a real world dataset such as [3].\n\n[1].Salzmann, Tim, et al. \"Trajectron++: Dynamically-feasible trajectory forecasting with heterogeneous data.\" Computer Vision–ECCV 2020: 16th European Conference, Glasgow, UK, August 23–28, 2020, Proceedings, Part XVIII 16. Springer International Publishing, 2020.\n[2].Yuan, Ye, et al. \"Agentformer: Agent-aware transformers for socio-temporal multi-agent forecasting.\" Proceedings of the IEEE/CVF international conference on computer vision. 2021.\n[3].A. Robicquet, A. Sadeghian, A. Alahi, S. Savarese, Learning Social Etiquette: Human Trajectory Prediction In Crowded Scenes in European Conference on Computer Vision (ECCV), 2016.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 4, "CLARITY": 2, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes a Feedback-Based Conformal Prediction (Fb-CP) framework to improve trajectory optimization (TO) in uncertain environments. The main technical contribution is to create a closed-loop system where information from the realized past time step's trajectory is used to dynamically adjust the uncertainty prediction regions for future time steps.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. The main technical novelty of the paper is the proposed feedback loop from the decision-making process back to the uncertainty quantification model (the CP prediction regions). It addresses a clear gap in existing literature where CP is used as a static, one-way \"black box\" in decision pipelines. This closed-loop approach is intuitive and makes practical sense for shrinking-horizon problems.\n\n2. The theoretical coverage guarantee highlights the technical contribution. The paper proves that using the posterior risk calculation to adjust future prediction regions does not violate the fundamental coverage guarantees of Conformal Prediction (Lemma 4.2). This is crucial for safety-critical systemss such as autonomous driving.\n\nWeaknesses:\n1.However, the computational cost of the proposed Iterative Risk Allocation (IRA) algorithm may affect the application of the framework. The most performant version of the proposed method, Fb-CP-IRA, involves an iterative optimization process at each time step to allocate risk. As shown in the results Table 1 and Table 8, this process significantly increases computation time compared to the non-iterative methods. While the authors show that much of this cost is at the initial step and the subsequent steps are faster, it could still be a bottleneck for applications that require fast decision-making cycles (very high-frequency control).\n2. The overall framework is very complex to implement, since it needs to combine a shrinking-horizon trajectory optimizer with a two-part conformal prediction system and an iterative risk allocation scheme. Implementing it correctly from scratch would be a non-trivial engineering effort compared to simpler, sequential CP methods in the literature. It needs to be justified with sufficient performance improvement for the system to apply this complex algorithm.\n3. The proposed method and its theoretical guarantees relies on certain assumptions, such as the availability of a reasonably accurate dynamics model for obstacles (even if it has i.i.d. errors) and the ability to define a Lipschitz continuous constraint function. While common in the field, the performance of the method would still be sensitive to the quality of the underlying trajectory predictor. \n4.The primary limitation, which the authors acknowledge, is the need to split the calibration dataset into two subsets (Dcal1 for forward prediction region generation and Dcal2 for backward posterior probability calculation). This means the method requires a larger calibration set than standard CP to achieve the same statistical power in each step. While the authors argue that large datasets are increasingly available, this could be a practical constraint in data-scarce domains.\n\nQUESTIONS:\nPlease check the weaknesses for detail and address the points there.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 5, "CLARITY": 3, "IMPACT": 2, "ORIGINALITY": 3, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes a feedback-based conformal prediction framework for trajectory optimization with safety constraints that dynamically adjusts prediction regions using realised trajectories, improving decision performance while preserving coverage guarantees. It proves that the proposed algorithm (Fb-CP) offers guarantees for performance improvement and provide a convergence analysis of the algorithm.\n\nSTRENGTHS AND WEAKNESSES:\n## Strengths\n\nSafe trajectory optimization algorithms are an active area of research, much work has been seen in combining machine learning methods and OC methods to arrive to better algorithms, especially when considering uncertainty quantification where research has been done in using generative models such as flow, gan's etc. but as well to use zero-order sampling techniques for evaluating safety constraint violation. Conformal prediction offers a training-free alternative to quantifying uncertainty in the models prediction with strong guarantees. \n\nWhile the introduction of conformal prediction has been explored in the trajectory optimization setting, the paper offers a thorough analysis with convergence guarantees that improves the usage of information based on the past encountered states. This will enable principled usage of conformal prediction in a feedback loop, which is very useful for application.\n\nThe paper also explores different risk allocation approaches,  average risk allocation and iterative risk allocation.\n\nThe paper provides bounds on the posterior violation probability as well as performance guarantees showing that the proposed method is at  least as good as the sequential method from Lindemann [2023], but in practice the proposed method performs even better experimentally. \n\n\n## Weaknesses\n\nThe issue with conformal prediction is that one needs good coverage for good guarantees, which is a condition that is hard to satisfy in high-dimensional spaces. \n\nThe experimental setting for verifying the algorithm is very limited, as the environments are low-dimensional, the application to real-world problems is questionable.  Nevertheless, I believe that the proposed algorithm and theoretical insights are useful for the community.\n\nQUESTIONS:\n## Questions \n\nl.116 - dynamica - dynamics\nl.133 - I assume that it's actually not according to eq.2 based on how you explain that g() is modelled by an LLM. Do you mean the ground truth dynamics here in assump. 2? You should clarify this.\n\neq3, please fix the formatting here.\n\nl.152 Shouldn't the superscript start from 0?\n\nl.168 -  Explain the abbreviation Fb-CP. \n\neq. 15a - 15d, fix alignment\n\n\n## General questions\n\nFor the real-world, we cannot expect that we have full-observability of obstacles. An example of this is a car in traffic, we see the cars moving but they follow very complex dynamics with a human involved that are difficult to predict. How does your algorithm deal with those situations / how does this influence the uncertainty quantification via CP?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-04"}