Conformal Prediction in The Loop: A Feedback-Based Uncertainty Model for Trajectory Optimization
Authors: Han Wang
Chao Ning
Venue: NeurIPS 2025 poster ()
Keywords: Conformal prediction
Feedback
Trajectory optimization
Uncertainty quantification
Risk allocation
Primary area: social_and_economic_aspects_of_machine_learning
TL;DR: This paper proposes a feedback-based conformal prediction framework for trajectory optimization that dynamically adjusts prediction regions using realized trajectories, improving decision performance while preserving coverage guarantees.

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.

Conformal Prediction in The Loop: A Feedback-Based
Uncertainty Model for Trajectory Optimization
Han Wang
Chao Ning"
School of Automation and Intelligent Sensing Shanghai Jiao Tong University Shanghai 200240, China wang@s tu. edu. cn
School of Automation and Intelligent Sensing
Shanghai Jiao Tong University
Shanghai 200240, China
chao. ning@s jtu. jtreddssttueuuucco edu.
Abstract
Conformal Prediction (CP) is powerful statistical machine learning tool struct uncertainty sets with coverage guarantees, which has fueled its extensive adoption in generating prediction regions for decision-making tasks, e.g., Trajec- tory Optimization (TO) in uncertain environments. However, existing methods predominantly employ sequential scheme, where decisions rely unidirectionally on the prediction regions, and consequently the information from decision-making fails be fed back instruct CP. In this paper, we propose a novel Feedback-Bas -Bssed CP (Fb-CP) framework for barinking-horizon TO with 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 provable online improvement in trajectory performance. Furthermore, we theoretically prove that such adjustments consistently maintain the coverage guar- antees of the prediction regions, thereby ensuring provable safety. Additionally, we develop a -fouuse 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.
Introduction
In recent years, Trajectory Optimization (TO) has recently garnered significant attention the aca- demic community Pan et al. (024) and has achieved significant success in fields such as autonomous driving Zhou et al. [2020). autonomous surface vessels Tsolakis et al.) [2024), and coverage control Davis et al. [2066] However, collision-free TO in uncertain environments is formidable challenge, because the intentions of obstacles are unknown. A crucial aspect of collision avoidance involves predicting obstacle trajectories. Existing trajectory prediction tools are unable to predict fully accurate trajectories. Therefore, a common approach is to generate the (1-a) -ovveraeeee prediction regions of the obstacle trajectories. If these regions contain the true trajectories with probability of at least 1 a, they are considered valid. The key to probabilistic collision-fre TO lies in adjusting the prediction regions while remaining valid to improve the trajectory performance.
Conformal Prediction (CP) is an attractive framework produce prediction regions with finite- sample guarantees of validity Vovk et al. (2005) [Shafer and Vovk [2008. Without imposing any assumptions
"Corresponding author.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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about prediction algorithms and data distributions, CP utilizes a calibration dataset to obtain valid prediction region for test data. Owing to its simplicity and versatility, CP and its variants have
been widely applied in various safety-critical applications, such as probabilistic collision-free TO
Lindemann et al. [2023], reliable estimation of graph neural networks H. Zargarbashi et al. 2023)
and and language modeling Quach et al. [024].
However, there is a disconnect between existing research on CP theory and CP application for decision-making. On the side of CP theory, most existing work primarily focuses on upstream data, developing new CP algorithms to enhance prediction performance, such as addressing distributional shifts Gibbs and Candes] (202T). performing multi-step time forecasting [Snn and Yu [2023), and improving the efficiency of prediction regions Bai al.) [202]] There is a lack of CP algorithms focused on enhancing the performance of downstream decisions. On the side of CP application for decision-maker makinn. most existing work embeds the CP into decision-making pipelines as a method for generating prediction regions, and employs sequential approach, i.e. the prediction region is first computed using CP and then the decision depends unidirectionally on the prediction region without considering the favorable impact of the decision on the prediction region. However, this information channel blockage from the decision -aaiinnn to the CP seriously prevents the CP from leveraging the information of past decisions to boost the performance of future decisions. Therefore, there is a pressing research need to develop a closed-loop framework that seamlessly integrates CP with Decision-making fully exploiting the information of past decisions to adjust prediction regions and thereby remarkably enhance the performance of future decisions.
To fill the aforementioned research gap, we propose Feedback- CP (Fb-CP) framework for shrinking-horiin horizon TO uncertain environments and the collision risk over the total mission time constrained at all times. The proposed framework leverages CP to construct the prediction regions of obstacle positions and adjusts these regions online in a closed-loop fashion while ensuring coverage guarantees, i.e. validity. In particular, we propose novel posterior probability calculation method to obtain the posterior probability of collision conditional on realized trajectories. The posterior collision probability is then used to adjust the allowable collision risk, which is allocated to future times to yield prediction regions. In this manner, information from past trajectories is fed back to the CP through the posterior probability calculation, guiding the feedback-basee adjustments of the prediction regions. Such adjustments in Fb-CP not only offer provable performance improvements but also consistently maintain the validity of the prediction regions. With the adjusted prediction regions, the trajectory is obtained by solving the resulting TO problem. Additionally, we further propose a pecision-focused risk allocation method, i.e. Iterative Risk Allocation (IRA), which aims to optimize the trajectory performance by iteratively allocating the allowable risk to future times while enjoying the convergence guarantee. We highlight the main contributions of our work below.
We propose, for the first time in the literature Fb-CP, a general uncertainty quantification framework closely associated with downstream pocision-making which enables the adjustment of prediction regions using the feedback information embedded in decisions.
We prove that 1) the feedback- adjustments in Fb-CP do not compromise the coverage guar- antees of prediction regions, and 2) Fb-CP offers guarantees for decision becision-making performance improvement. In other words, Fb-CI enjoys both validity and superior performance.
We propose drecision-fouuse risk allocation algorithm with theoretical convergence analysis for Fb-CP, which optimizes the risk allocation to enhance decision decision-making performance.
We extend Fb-CP to handle distribution shift by applying a weighting scheme to the test and calibration data and demonstrate its effectiveness.
2 Related work
Conformal Prediction. Conformal prediction originated in the early work Vovk al. [1999 2005
Shafer and Vovk (2008) to generate the prediction region. The salient advantage of CP lies in its ability to offer coverage guarantees regardless of prediction algorithms and data distributions. Most recently, various variants of CP have been developed to handle upstream data with different characteristics H. Zargarbashi et al. [2024), Liu et al. [0244] or to produce prediction regions in a wide array of forms Angelopoulos et al. [024]] Auer et al. 2023 In response to the distribution shift
in the upstream data, GCbss and Candes [021] Podkopaev et al. [2024). Zaffran et al. [0222]
and EnbPI Xu and Xie [202] 1 2023 developed CP through online learning and sliding window,

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where each of the N joint respectively, and achieved asymptotic validity. In the context of multi-step time series forecasting, Sun and Yu [2023] combined CP with copula to propose the CopulaCPTS, while (CeaveeCen et al. [2024] employed an optimization-baset method. [Zhou et al. [2024b] presented a new conformal method for time series forecasting. addition, numerous studies focused on improving the efficiency of the prediction region by changing the region shape Xu et al. [024). minimizing the region length Kiyani et al. (2024), or optimizing the region construction function Bai et al. (2022). Note that the prediction regions are typically utilized by downstream tasks in sequential manner. However, the above research work primarily aims to enhance the predictive performance of CP rather than directly improving the performance of downstream decision- making..
Decision-making with CP. Many studies have focused on the integration of CP with decision- making. \ovk and Bendtsen (2018) was the first to the method of CP to make it applicable to decision-making. Additionally, Fannjian et al. [2022] proposed CP with coverage guarantee under one-step Feedback Covariate Shift (FCS), in which the test data depend on the training data. Building upon the aforementioned work, |Prinster al. (2024) refined its theoretical framework and extended it to multistep FCS. Furthermore, [Lekeufack et al. (0244 introduced a conformal decision theory, which follows the ACI concept to directly provide provable statistical guarantees of having low risk for decisions made based on uncertainty-awanr predictions.
TO using CP. Lindemann et [2023] and Strawn et al.
2023 applied CP to the safe planning for Muthal et al. [2023] and Kuipers et al. 2024] extended it to multi-robot
single-robot systems, while Muthali systems. Additionally, |Dixtt et al. [2023] and Zhou et al. [2024a] employed the ACI to address the obstacle trajectory distribution shift. Staamouii et al. [2024] proposed novel nonconformity score for shrinking-horizon TO. All the above methods directly employ CP in a sequential way to generate prediction regions. Nevertheless, the performance of realized trajectories has yet to be conveyed to the upstream CP as feedback information to adjust the prediction regions, which has the great
potential to further boost the performance of trajectory.
3 Problem formulation and background
3.1 Problem formulation
Consider discrete-time nonlinear dynamics as follows.
It-1 f(z,, Uz), Io Tinit
(I)
where x, E X C " and u, u C RT are the state and control time f O,... T, respectively,
and T 2 1 is the total mission time. The sets 21 and X' represent the admissible sets of control inputs and system states, respectively. The function f R== R + R represents the system dynamics and Tinst is the initial state. For brevity, let ,, == (It, .... It,) and ,t == (Me, .... Utg) denote the state and control sequences from t to f2, respectively. The system operates in an environment with M dynamic obstacles with priori unknown trajectories. Let Y, = (Y.,,..., Ye,M) represent the joint obstacle state at time I, where Yt, E RR denotes the state of obstacle j at time t. The dynamic of the above joint obstacle system can be expressed as follows.
Y+1 gY Y(-1) We
where the window length, g(-) represents the model learned from the historical data of the obstacle using machine learning tool, i.e., Long Short-Term Memory (LSTM). Naturally, readers may choose to adopt more powerful predictors for improved accuracy, such as [Salzmann et al.) 2020). Yuan et al. [2021] Salzmann et al. [2023]. Yuan and Kitani 2020]. wy captures the modeling error. Additionally, we define Y == (Yo ..., Yr) as the entire trajectories of the obstacles, which is generated by sampling the initial state Yo from an unknown probability distribution D and by evolving it based on the dynamics (2). The system can observe the joint obstacle states Yo..., Ye.
when making the decision at time t. We assume the availability of an offline dataset as follows.
Assumption 3.1 We have calibration dataset Deat ==
obstacle trajectories is generated by independently sampling an initial state ( from D and evolving it based on its ground truth dynamics.
With Assumptions 3.1
can conclude that the real joint obstacle trajectory Y and the N available joint obstacle trajectories (+) are independently and identically distributed (i.i.d.,, and are also
3

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exchangeable. Assumption 3.1 is not restrictive in practice, e.g. the historical trajectories of obstacles. In Appendix G we introduce an extension address the case where a shift exists between the initial state distributions of calibration data and test data, and the error wi represents state-dependent noise rather than being i.i.d..
We focus on the TO problem whose objective is to find the sequences I1.T and u0:T-1 that minimize the cost function (IIITT uO:T-1) subject the dynamics and constraints. The TO is performed in shrinking-horizon fashion, with the optimization problem at time formulated as follows.
min 2++1:7 T1
JI UoTT (TT)
+ Σ y=t Ly (27, Ur)
(3a)
8.t.
I++1 = f(0,, 24);
VT = 1,.., T
(3b)
Jr E X
VT + 1, T
(3c)
Ur E U,
VT= = t....T
(3d)
P P(n T
(d( p Y+) 2 20 21 a
(3e)
where P(X) denotes the probability of event X, the constraint function c := R= x R + R is L-Lipschitz continuous, which can encode various tasks, such collision avoidance. Due to the uncertainty of the joint obstacle state Y,, we impose the joint chance constraint (3e) with failure probability E (0, 1) to ensure that the joint probability of satisfying the constraint over the total mission time is no less than - a. To ensure the initial feasibility of the TO problem, we assume that the initial state satisfies the constraint, i.e. c( Yo) 2 O, with probability 1.
3.2 Conformal prediction
In this subsection, we provide brief introduction to the theoretical results for CP and refer readers to Ao/Aneloopouoss and Bates 2021] for thorough introduction. Given N + exchangeable random variables R!!") R ..... R which is usually referred to as the nonconformity score, CP aims to find a probabilistic upper bound for R based on R("), .... R(N) such that R is less than this upper bound with high probability. In practice, R()) represents the test data point, while R()), ..... R(N)
denote the calibration dataset. Formally, the central idea behind CP is summarized in Lemma A.1 provided in Appendix A.I
4 Feedback-basse conformal prediction
The challenge in solving the TO problem ( lies in the computation of the joint probability (3e). Existing literature predominantly employs sequential way of using CP, i.e. the prediction regions of obstacle positions are first computed based on the failure probability depends mey on the prediction regions. However, it is important to note that the shrinking- horizon TO framework, at time t the past decisions To:t are available and typically contain rich information that can instrumentally assist in refining the prediction regions at subsequent time steps, thereby considerably improving the performance of TO. Therefore, we propose novel Feedback- based Conformal Prediction (Fb-CP). Fb-CP not only exploits the feedback information provided by realized trajectories to perform closed-loop adjustments of the prediction regions but also maintains coverage guarantees. To begin with, the joint chance constraint (3e) can be reformulated as a set of individual chance constraints and total risk constraint by using Boole's 's inequality follows.
and then the decision of TO
P(n [c(xT, Y+) ) 2 03 2 21- Dt <a
P [c( Y,) 2 0} 2 1 - ar; VT = 1,.... T
P
The risk a, at each time can be initially allocated uniformly at time f = 0, i.c. a, a/T. and remains constant throughout the darinking-horizon TO process, as in Lindemann However, at time f, the system states x, for < t are available, which grants us to compute the posterior probability 3-- (c(x,, Y,) > 0|æ,} and the permissible risk for future times, which is then used to adjust the prediction regions. Using the information in the realized trajectories, the feedback-band -ased adaptation of the prediction regions tremendously reduces the conservatism of trajectory online while ensuring coverage guarantees. In Subsection 4.1 we present the individual chance constraint reformulation using the prediction regions derived. In Subsection 4.2 we present a CP-based method for calculating 3,, We reformulate the TO problem in Subsection
al. (2023)

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4.1 Constraint reformulation using conformal prediction region
We randomly divide the calibration dataset Dest into two subsets D and Deat with K and L joint obstacle trajectories, respectively, where K + L = N. Without loss of generality, we reassign indices to the joint obstacle trajectories as D == {Y ...., } and D == (Y(K+1), ..... Y +) At time t, we obtain the prediction of the joint obstacle state Y= for all future time = t t+...... T using s(-) described in (2). Similarly, the prediction y for each trajectory y(i in Deat is derived by using the same method. We define the nonconformity score as follows.
Roje ||,,, - Yill,, R7/ || - 1i=1.... K
(5)
Note that Y,, Y
Y K) are exchangeable and the prediction function g()) is trained from Derein independent of D oet. Therefore, given an allocated risk a, for future time т. the random variables
Rrju, 7/ .... r/e are exchangeable and the prediction region with coverage guarantee is derived according to Lemma
((K)
A. as follows.
P(I|Y, - P S C 2 O+
(6a)
Ch Quantile (R 7|:::... .... [2(k) .00
(6b)
Based on the (1 O.,)) --ov--coeaaage prediction region {y/ lll - Yrlll I S C2r }. the individual chance constraint in can be reformulated as the following lemma proven in Appendix B.1
Lemma 4.1 (chance constraint) If Asssuptiin 3.1 holds, the constraint function c is L--iiscchtz
continuous and c(z,, Yxi) 2 LCC--- is satisfied where T/t Ch--a- is calculated by (bb). then the individual chance constraint P(c(x,, 2 0} 2 1 at is satisfied.
For general collision avoidance chance constraint of the form P[[|I - 2 ay. Lemma 4.1 effectively converts it into deterministic constraint that requires the distance between I, and the predicted location ÝzJe to exceed the inflated radius derived from the prediction error, i.e.,
||z,- Y+1 - C1-a+ 2
4.2 Posterior probability conditional on past decisions
At time t, the states x for all past time 7 = 1, .... & are deterministic and available to the trajectory optimizer. We assume that x; is the true system state at time r. Note that x is feasible solution to the TO problem (3) at time - with the reformulated constraints through Lemma 4.1 Therefore, the individual chance constraint ([(( Y2) 2 0) 2 1 ar, is satisfied at time T - and will be naturally satisfied for all time r' 2 1. However, the constraint violation probability ox is priori probability allocated before time T that tends to overestimate the violation probability and thus leads to conservative results. Fortunately, the determined x; allows to compute the posterior probability of constraint violation By, which, as we theoretically prove, is less than a, with high probability. The risk redundancy between a; and B, can be allocated across future times. In this way, the information embedded x; is fed back from the decision-mi makinn to the CP to readjust the prediction region and to achieve trajectory with notably improved performance. To compute 3, using Lemma A. propose a novel nonconformity score as follows.
S, c(=", --- + w+) c( Y+). S(O cx2", E + (!)) Vi 1, K + (7)
where wi(() is the modeling error of the joint obstacle trajectory (()) at time т, which can be obtained through (f) y!! g( We note that w., ..... W7 LK+ are i...d., and if 2 is
fixed and independent of wr. w ..... wi (K+L) the random variables S,, s ..... s (L++) are also exchangeable. However, as x. is derived through the TO problem B at time 7 1, it depends on DI and the random variables ST, ...., s(K) are no longer exchangeable. Therefore, we only use the subset D2 D2 i.e. s(K+1) ..... LL)+L) to compute 3y. The upper bound of the posterior violation probability 8, is computed via the following lemma, whose proof is given in Appendix B.2
5

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Lemma 4.2 (posterior probability calculation) Assume that x; is the true state of the system at time T, then the upper bound of the posterior violation probability at time is as follows.
P(c(e,, < 0) EI+E/ei (stkit Ki)) < <0))/(1+D) (1+
(8)
where I[()) the indicator functionn
Lemma 4.2 essentially computes the posterior probability by evaluating the fraction of calibration set D cal samples whose actual trajectories would collide with the given realized position x; Some might raise the concern that 3, could be higher than or,, which could result in more conservative trajectory when using 3, in subsequent times. However, we can restricts the upper bound of the expectation of Br in Corollary Corollary|A.2ppmmm provided in Appendix A.2 Additionally, we report the empirical observation that our method consistently tends to improve performance in practice, which is further discussed in Remark A.3 provided in Appendix A.2
4.3 Optimization problem reformulation
By making use of the joint chance constraint reformulation 4. the individual constraint reformulation
in Lemma 4.1 and the posterior probability in Lemma 4.2
the TO (3) is transformed as follows.
min 1 Aee 1
J(1411I -1) (xT)) + Σ L (Iiys Uty)
(9a)
8.L.
3b
3d
(9b)
c P111) 2
LC r/P
VT = 1,.., T
(9c)
ar 0,
= I,.., T
(9d)
Σ T
+==++ a, Sa
3
(9c)
where Constraint (9c) ensures the satisfaction of individual chance constraints (4) for future times
+ I..,, T through Lemma 4.1 and C.-a is calculated by (6b). Constraint (9d) is imposed to ensure the non-negativity of at,, Constraint (9c) is the most important part for feedback-bard adjustments of the prediction region and online performance enhancement of the optimized trajectory. It is derived by replacing a, for past time r = I...., t in the total risk constraint 4 with 33, calculated through Lemma 4.2 The information embedded in past decisions zi...., . influences the future values of 0t++1, .... orr through the calculation of ...... 3r thereby reshaping the prediction region of CP in an end-to-end way. Based on Corollary A.2 and Remark [A.3] 3, is highly likely to be less than Or in practice. Consequently, using 3, grants more risk to be reserved for future times, resulting in much compact prediction regions and tremendously improved optimization performance.
However, it is important to note that Cl depends a, and D Consequently, treating Q1+1:T as decision variables alongside 2t+1:T and u;T-1 would make the optimization problem ( computationally demanding to solve for larger values of T and K. Therefore, we will present an allocation method for a++1:T that aligns with the optimization problem (9) in the next section.
5 Shrinking-horizon trajectory optimization using Fb-CP
The shrinking-horizon TO framework using Fb- Db-CPin is illustrated in Figure ! The information in z guides the feedback- based adjustments of the size of the prediction regions C1- through posterior probability calculations. Solving the TO problem (9) is divided into two steps: 1) risk allocation and 2) TO with the fixed 0+41:7 The TO problem with the fixed a411T formalized as follows.
min 02:T-1
[E MTT s.t. 96
(9c
(10)
The problem 100 can be readily solved to obtain o 1:T and uh-T and only the first system input ut implemented the control input. Therefore, as the actual time progresses, the optimization horizon gradually shrinks. For the risk allocation, general approach is the Average-based based Risk Allocation (ARA), i.e. the allocable risk is evenly allocated across future times at time t below.
0, (a Et=o t) Vr t+ 1+,1.... 1,...."
(11)

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Allocable risk
Risk allocation
CP regions generation
TO
Robot system
Can
B
(x,
1+
r20
Maa
lieration
Data
Feedback information
CP-based posterior probability
Figure 1: Shrinking-horrtt horizon trajectory optimization framework using Fb-CP.
Although the ARA method has the advantage of computational efficiency, the fixed proportion allocation significantly diminishes the flexibility in modifying the prediction regions for future times. Therefore, we extend the IRA Ono and Williams 2008 the Fb-CP. We refer the TO problem with the fixed 04+1:2' (TO) and the risk allocation problem as the lower-stage and the upper-stage problem, respectively. The system states 27+1.77 and inputs "7-7-1 1 as well as the risk allocation 01+1::7 are obtained by iteratively solving the lower and upper-stage problems. We denote the feasible region of (10) with fixed 1,41T as (x((++1.T). The upper-stage problem is formally stated below.
millgu1 ('(+11))T
(12a)
s,t.a, 2 0, T
=t+
(12b)
Ei=41 a Sa- a Ei 3r
(12c)
0:41:7 E {a 3 (z, u) E Re(a++17))
(12d)
where (((0++117) is the optimal objective function of (10) given 04+11:7 If a risk allocation 04+1:T7
satisfies (2d) then we refer 0++1:7 feasible risk allocation. However, the lower-stage problem (12) is challenging to solve due to the computational complexity arising from its objective
(Tab and Constraint (1cc). To solve (12) efficiently, we introduce a descent algorithm, i.e. IRA for
Fb-CP which is based on the monotonicity of J' (a++1)) below, which is proven in Appendix B.4
Lemma 5.1 (monotonicity of J*) At time I, the following inequalities always hold.
J.J" (01+1))
<0 =-=t+1,...
(13)
Oo,
The monotonicity of in Lemma 5.1 shows that increasing the allocated risk a any time step will strictly decrease the optimal cost of the trajectory optimizer. This insight enables our risk reallocation strategy: by reducing redundant risk at inactive time steps and reallocating lower the overall optimal cost.
active ones, we can
We assume that 1. represents the feasible risk allocation obtained after the rath iteration time t. n+] in the (n + 1)th iteration such that J (a)+1:7) S
IRA aims to obtain a feasible risk allocation a J (aT++::) the (n 1)th iteration, IRA first solves the lower-stage problem 10p using of 1:7 to obtain the optimal solution 2 1:7 and - 1 Subsequently, based on 1:71- Constraint ( in the lower-stage problem 10 is categorized into active and inactive constraints. The active and mactive t 1,.., T) and 1, ..., 7}, respectively. In summary, IRA consists two steps: 1) tightening the inactive constraints and 2) relaxing the active constraints.
constraint sets are formally defined as Toct {T c(x7 Y=1) LC,
Iima = {T & Tact.
Tightening the inactive constraints is first implemented construct ã 0:+1: from 02+1.-- Specifically, (=1-a,
for T E Iect. set a = a. Based on the definition of C+T a
6bb. Cl-ar is non-increasing with
respect a, for the fixed Dia- Thus for Inna choose a I a" so that
c(x",, Pr11) 2 LC1- T/K 2 LC1-a)
(14)
7/r
7

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Therefore, the optimal solution (rrr, (xr-1) ) for 0++1r is also the optimal solution for a a++1: and thus J (aa++11)) = J (" 1-)) Finally, is straightforward to show that ã 1++1T is a feasible risk
Based on (14. it can be deduced that (2i+1:7, ( E Re(ā(=1T) S Re(a
allocation, because (i) (1bb follows from (14 and the fact that when ae + 0, C+p + oc; (ii)
12c) is satisfied since
T O, S E "² S Ei= Bri (iii) 12d) is satisfied because 1:77 - is feasible for 01+1:7- The specific construction of σ is as follows.
{{
a (1 (1=n)a; + naa
T TE Inct
07
(15)
T Tina
where 17 E (0, is the step size and a. is the lower bound of ā² calculated as in Lemma 5.2
Lemma 5.2 (constraint tightening) Assume that 1it1; is feasible for the problem 10) with of and oHtr < For J E Ii the lower bound of 0²2 while satisfying (14) is as follows.
a² = (1+ E,-, (c(x%, (r1)) < (R) /(L+K) K)
(16)
Furthermore, it is deterministic that o² S <aa.
Lemma 5.2 is proven in Appendix constraints as follows.
B.5 Then n+1 is constructed from 0ž+1:r to relax the active a:41:7
(a - Ea
Br
Ef / T E Inct
(17)
T E Tina
where Nact represents the number of elements in the set Toct. It can be easily verified that 1 n+1 is a feasible risk allocation. Note that a;+1 > a since o 12d). and thus 1:77
satisfies
12b
satisfies
T2c Therefore, the following inequality is obtained by implying Lemma 5.1
Ja n+1 S J (a++1.) = (a
(18)
By recursively constructing 0f 0'+1:T,,.... 1:+1: in this manner, J monotonically decreases. The algorithm of Fb-CP using IRA at time isis delineated in Algorithm Note that at time = 0, the input parameter ag:r is initialized and is given small tolerance. At time t. the robot first obtains r, and Y (Line 2). Then, based on ... the future joint obstacle states Ye+1JT:..., Y,,,,r are predicted using LSTMss (Line 3). Additionally, based 2; and D2 Deat- the posterior collision risk can be calculated through (8) (Line 4). After initialization (Line 5), IRA IRA jointly optimizes risk allocation and trajectory through iteration (Line 6-1 Specifically, in each iteration, IRA first computes the optimal control - state 1:7 and cost J J'(alta:r) for the current iteration based on the risk allocation o obtained from the previous iteration (Line 7). The active and inactive constraint sets Luct, Tina are determined based on the optimal state +1 (Line 8). And then, by sequentially applying the inactive constraint tightening (15) (Line 9) and the redundant risk reallocation (17) (Line 10), the updated risk allocation a n+1 0++1:7 is obtained. Finally, if the convergence condition is
satisfied, the the optimal control up,T1 and the risk allocation w are output; otherwise, the next iteration begins (Line 12). Finally, the convergence of Algorithm and the overall risk guarantee are established in Theorems [53 and [.4 whose proofs are provided in Appendices B.6 and B.7 respectively.
Theorem 5.3 (convergence guarantee) Assume that 2: 1+ ue-r-1 are are feasible in problem 10 with risk allocation 1.T- If the sets X, U are bounded and the objective function ]IL41:T M:T-1) 1) is continuous, then the sequence of the optimal objective (J' (of+1r) ) Iner converges to a finite limit.
Theorem 5.4 (overall risk guarantee for entire trajectory) Given an overall risk tolerance a, if the posterior risk 31a is calculated through Lemma 4.2 the risk ax+ 1.T is allocated through ARA (17) or IRA (2), the planned state oi+te is a feasible solution of the TO problem (70) with Q+4 1:T, then the entire trajectory time satisfies the overall risk guarantee PI (C(", Y+) 2 0}}
a.
One may notice that the calculation of 3- in Lemma 4.2 is similar to the computation of σ in Lemma 5.2 This observation is correct. The key difference between the two lies in that the former utilizes the dataset D2 independent with D to achieve the coverage guarantee for Br. By contrast, as step in solving (11), the latter does not need to consider the coverage guarantee and thus directly uses Dia The use of different datasets results in the former providing probabilistic guarantee, while the latter achieves deterministic guarantee (a S o.)).
Remark 5.5
8

---

Algorithm Fb-CP using IRA at time
1: Input: a, OuT, 90.t-1, 7/. Deatt D
oat
2: Observe the system state 2, and joint obstacle states Y
3: Yetije...., Vr,e+ Trajectory prediction using LSTMs based on Yo. ..., Yt
4: Be < PPosterior probability calculation
(8) {Using He and D 1:7, 0 (Initialization of IRA]
5: (a(att1:)) &- oc, o 0++1:T
6: repeat
7:
I'( IITI,T " f Solve the lower-stage problem 10 with 1+17
8:
Tact, Tina: Nact f Identification of active and inactive constraints
9:
a1T17 f Transitional risk allocation calculation (15)
10:
0+11: f- New risk allocation calculation
n-11
17
11:
12:
until (a(
J (0 <
n-1 13: Output: Bu.d, " a1+1T a++1T
n-1
6 Experiments
All the experiments are performed on a personal computer with 2.10 GHz Inter Core 17-13700 CPU and 32 GB RAM. We conduct 1,000 Monte Carlo experiments on a kinematic vehicle model
Lekeufack et al. [024]. 3D linear quadrotor model Dixit et al. [2023] dynamic bicycle model
Hakobyan and Yang| [222T] and the Stanford Drone Dataset Robicquet al. (2016) to compare our
method with the state-of-art art methods as follows
ii Conformal Control (CC) proposed in Lekeufack et
2024
(ii) ACI for Motion Planning (ACI-MP) proposed in Dixit et al.
2023
(iii) Recursively Feasible MPC using CP (RF-CP) proposed Stamouli et al. [2024
(iv) Sequential CP (S-CP) proposed in Lindemann et al. [2023).
(v) Fb-CP with ARA (Fb-CP-ARA) The method based on Fb-CP using average risk allocation.
(vi) Fb-CP with IRA (Fb-CP-IRA): The method based on Fb-CP using iterative risk allocation.
In this section, we present the main experimental results, while the full set of results can be found in Appendix C Figure shows the simulation results from of the 1,000 simulations using the 2D vehicle model. At t = 0, the vehicle performs the first TO. For Fb-CP-IRA IRA allows for flexible allocation of the risks across future times. Therefore, by assigning more risk to 9, which leads to a compact prediction region, a trajectory passing between Obstacles 2 and 3 is obtained. However, with the fixed risk allocation at = S-CP and Fb-CP-ARA can only optimize the trajectory based on fixed prediction regions. Consequently, they can only navigate around to pass between Obstacles I and Note that at t O, no deterministic vehicle position is available for posterior probability calculation. Thus Fb-CP-ARA degenerates into S-CP, resulting in both methods obtaining essentially the same trajectory. As time progresses, more and more vehicle positions become available. For Fb-CP-ARA, 31:3 can be computed at I = 3 and is with high probability less than 01:33 as outlined in Corollary A.2 The reduction from 01:3 31:3 leads to an increased allowable risk for future times, corresponding to narrowing in the prediction regions,Theee Therefore, compared with S-CP, Fb-CP-ARA generates less conservative trajectory. Similarly, Fb-CP-IRA also leverages $1:3 increase the total allocable risk, thereby further enhancing the flexibility in allocating risks. As shown in Figure 2 the trajectory obtained by Fb-CP-IRA at 3 exhibits reduced conservativeness compared with the trajectory obtained at & = 0.
Table summarizes the average cost, computation time, and collision avoidance rate of 1,000 simulations using the quadrotor model with different methods. As shown in Table the Fb-CP-ARA reduces the cost by an average of 1 11.34% compared with S-CP thanks to the feedback information of posterior probabilities, with a negligible additional computational burden. Furthermore, by flexibly allocating the additional allowable risk provided by posterior probabilities, Fb-CP-IRA achieves an 58.50% reduction in average cost compared with S-CP. However, since IRA needs to solve the TO problem 10) iteratively, the average computation time increases significantly. Additionally, since CC and ACI-MP fail to fully utilize the information in the calibration dataset, they incur higher costs, which are 184% and 296% higher than those of Fb-CP-IRA, respectively. Particularly for CC, directly controls the collision avoidance rate by adjusting the weight of the collision penalty term

---

All trajectories
Squence CP
Fb-CP with ARA
Fb-CP with IRA
Figure 2: Trajectories the vehicle with different TO methods. (Numbers on the circles denote indices of obstacles. The diamond and pentagon symbols represent the initial and target points of the vehicle, respectively. The translucent circles represent the planned positions of the vehicle and the prediction regions for further time. In particular, the colored and transparent circles with black edges denote the planned positions and the prediction regions for T = 9, respectively.)
in the objective function, which results in higher average cost. However, it should be noted that, in practice, CC and ACI-MP are better suited for scenarios where the test data exhibit distribution shift, rather than the setup considered in our work. For RF-CP, thanks to the proposed normalized nonconformity score, its average cost is comparable to that of Fb-CP-ARA, but it remains 85.8% higher than Fb-CP-IRA. However, the normalized nonconformity score introduces mixed-integer variables into the TO problem, significantly increasing the computation time. Specifically, the average computation time of RF-CP is more than an order of magnitude higher than that of Fb-CP-IRA.
Additionally, we have investigated the impact of using prior versus posterior probabilities on the prediction regions by analyzing the prediction region radius at different time f and in Appendix H Furthermore, we have also extended Fb-CP to handle distribution shift and empirically demonstrate its effectiveness. Detailed methodology and corresponding experiments can be found in Appendix
Table 1: Average cost, computation time, and collision avoidance rate using the quadrotor model with different methods (1) the learning rate of CC).
Fb-CP
CC
ACI-MP
RF-CP
S-CP
ARA
IRA
1000
59.25
0.05
17.970
15.794
17.321
15.356
7.189
Average cost
7 500
47.50
0.10
17.263
14.378
16.17
14.228
6.798
100
22.46
= 0.15
16.096
11.922
14.83
12.354
6.191
7 50
21.34
0.20
15.310
10.032
13.217
10.22
5,398
1000
0.019
a = 0.05
0.022
0.487
0.022
0.027
0.038
Average
7 500
0.019
0.10
0.026
0.494
0.020
0.021
0.039
computation time
100
0.021
0.15
0.021
0.545
0.021
0.020
0.037
7 50
0.022
0.20
0.022
0.500
0.020
0.019
0.036
1000
97.0%
0.05
98.6%
98.7%
98.8%
98.2%
96.3%
Collision
500
92.8%
0.10
93.3%
96.9%
93.5%
94.6%
94.1%
avoidance
100
82.5%
0.15
91.5%
92.4%
92.0%
90.2%
91.9%
rate
7 50
79.1%
0.20
87.9%
90.0%
88.2%
86.7%
88.2%
7 Conclusion and Limitations
In this paper, we proposed a Fb-CP framework for thinking-hhorioon TO with a joint risk constraint uncertain environments. This method enables the feedback the information the realized trajectory from the decision-making to the CP, guiding the closed-loop adjustments of the prediction regions. The proposed adjustment rule balances both performance and safety, offering provable performance and coverage guarantees. The limitations are discussed in detail in Appendix L
10

---

Acknowledgements
This work was supported in part by the National Natural Science Foundation of China under Grants 62473256 and 62103264.
References
Alexandre Alahi, Kratarth Goel, Vignesh Ramanathan, Alexandre Robicquet, Li Fei-Fei, and Silvio Savarese. Social Istm: Human trajectory prediction in crowded spaces. In Proceedings of the IEEE conference on computer vision and pattern recognition, pages 961-971 2016.
Anastasios Angelopoulos, Emmanuel Candes, and Ryan J Tibshirani. Conformal pid control for time series prediction. Advances in neural information processing systems, 36, 2024.
Anastasios Angelopoulos and Stephen Bates. A gentle introduction to conformal prediction and distribution-free uncertainty quantification. arXiv preprint arXiv:2/07.07511, 2021..
Andreas Auer, Martin Gauch, Daniel Klotz, and Sepp Hochreiter. Conformal prediction for time series with modern hopfield networks. Advances in Neural Information Processing Systems, 36: 56027-56074, 2023.
Yu Bai, Song Mei, Huan Wang, Yingbo Zhou, and Caiming Xiong. Efficient and differentiable conformal prediction with general function classes. arXiv preprint (xXxi:2202.1109) 2022.
Matthew Cleaveland, Insup Lee, George Pappas, and Lars Lindemann. Conformal prediction regions for time series using linear complementarity programming. In Proceedings of the AAAI Conference on Artificial Intelligence, volume 38, pages 20984-20992, 2024.
Bobby Davis, Ioannis Karamouzas, and Stephen J Guy. C-opt: Coverage-aware trajectory optimiza- tion under uncertainty. IEEE Robotics and Automation Letters, 12)::1020-1027, 2016.
Anushri Dixit, Lars Lindemann, Skylar X Wei, Matthew Cleaveland, George J Pappas, and Joel W Burdick. Adaptive conformal prediction for motion planning among dynamic agents. Learning for Dynamics and Control Conference, pages 300-314. PMLR, 2023.
Clara Fannjiang. Stephen Bates, Anastasios N Angelopoulos, Jennifer Listgarten, and Michael I Jordan. Conformal prediction under feedback covariate shift for biomolecular design. Proceedings of the National Academy of Sciences, 119(43):c2244561111 2022.
Isaac Gibbs and Emmanuel Candes. Adaptive conformal inference under distribution shift. Advances in Neural Information Processing Systems, 34: 1e6166601672. 2021.
Soroush H. Zargarbashi, Simone Antonelli, and Aleksandar Bojchevski. Conformal prediction sets for graph neural networks. In Proceedings of the 40th International Conference on Machine Learning, volume 202, pages 12292- 12318. PMLR, 2023.
Soroush H. Zargarbashi, Mohammad Sadegh Akhondzadeh, and Aleksandar Bojchevski. Robust yet efficient conformal prediction sets. In Proceedings of the 41st International Conference on Machine Learning, volume 235, pages 17123-17147, PMLR, 2024.
Astghik Hakobyan and Insoon Yang. Wasserstein distributionally robust motion control for collision avoidance using conditional value-at-risk.. IEEE Transactions Robotics, 38(2):939-957, 2021.
Shayan Kiyani, George Pappas, and Hamed Hassani. Length optimization in conformal prediction. arXiv preprint arXiv:2406. / 188/4, 2024.
Parth Kothari, Sven Kreiss, and Alexandre Alahi, Human trajectory forecasting in crowds: A deep learning perspective. IEEE Transactions on Intelligent Transportation Systems, 2307):738--7400 2021.
Tom Kuipers. Renukanandan Tumu, Shuo Yang, Milad Kazemi, Rahul Mangharam, and Nicola Pao- letti. Conformal off-policy prediction for multi-agent systems. arXiv preprint arXiv:2403, 16871, 2024.

---

Jordan Lekeufack, Anastasios N Angelopoulos, Andrea Bajesy, Michael I Jordan, and Jitendra Malik. Conformal decision theory: Safe autonomous decisions from imperfect predictions. In 2024 IEEE International Conference Robotics and Automation (ICRA), pages 16668-1 11668-16775.. IEEE, 2024.
Lars Lindemann, Matthew Cleaveland, Gihyun Shim, and George J Pappas. Safe planning in dynamic environments using conformal prediction. IEEE Robotics and Automation Letters, 2023.
Yi Liu, Alexander Levis, Sharon-Lise Normand, and Larry Han, Multi-source conformal inference under distribution shift. In Proceedings of the 41st International Conference on Machine Learning, volume 235, pages 31344-31382. PMLR, 2024.
Anish Muthali, Haotian Shen, Sampada Deglurkar, Michael H Lim, Rebecca Roelofs, Aleksandra Faust, and Claire Tomlin. Multi-agent reachability calibration with conformal prediction. In 2023 62nd IEEE Conference on Decision and Control (CDC), pages 6596-6603 IEEE, 2023.
Masahiro Ono and Brian C Williams. Iterative risk allocation: A new approach to robust model predictive control with a joint chance constraint. In 2008 47th IEEE Conference on Decision and Control, pages 3427-3432. IEEE, 2008
Chaoyi Pan, Zeji Yi, Guanya Shi, and Guannan Qu. Model-based diffusion for trajectory optimization. In The Thirty-eighth Annual Conference on Neural Information Processing Systems, 2024.
Aleksandr Podkopaev, Dong Xu, and Kuang- chih Lee. Adaptive conformal inference by betting, In Forty-first International Conference on Machine Learning, 2024.
Drew Prinster, Samuel Don Stanton, Anqi Liu, and Suchi Saria. Conformal validity guarantees exist for any data distribution (and how to find them). In Forty-firsi International Conference on Machine Learning, 2024.
Victor Quach, Adam Fisch, Tal Schuster, Adam Yala, Jae Ho Sohn, Tommi S. Jaakkola, and Regina Barzilay. Conformal language modeling. In The Twelfth International Conference on Learning Representations, 2024.
Alexandre Robicquet, Amir Sadeghian, Alexandre Alahi, and Silvio Savarese. Learning social etiquette: Human trajectory understanding in crowded scenes. In European conference on computer vision, pages 549-565. Springer, 2016.
Tim Salzmann, Boris Ivanovic, Punarjay Chakravarty, and Marco Pavone. Trajectron+++: Dynamically- feasible trajectory forecasting with heterogeneous data, In European Conference on Computer Vision, pages 683-700. Springer, 2020.
Tim Salzmann, Hao-Tien Lewis Chiang, Markus Ryll, Dorsa Sadigh, Carolina Parada, and Alex Bewley. Robots that can see: Leveraging human pose for trajectory prediction. IEEE Robotics and Automation Letters, 8(1 ((1):7090-7099) 2023.
Glenn Shafer and Vladimir Vovk. A tutorial conformal prediction. Journal of Machine Learning Research, 9(3), 2008
Charis Stamouli, Lars Lindemann, and George Pappas. Recursively feasible shrinking-horizon mpe in dynamic environments with conformal prediction guarantees. In 6th Annual Learning for Dynamics & Control Conference, pages 1330-1342, PMLR, 2024.
Kegan J Strawn, Nora Ayanian, and Lars Lindemann. Conformal predictive safety filter for rl controllers in dynamic environments. IEEE Robotics and Automation Letters, 2023.
Sophia Huiwen Sun and Rose Yu. Copula conformal prediction for multi-step time series prediction. In The Twelfth International Conference on Learning Representations, 2023.
Ryan J Tibshirani, Rina Foygel Barber, Emmanuel Candes, and Aaditya Ramdas. Conformal prediction under covariate shift. Advances in neural information processing systems, 32, 2019.
Anastasios Tsolakis, Rudy R Negenborn, Vasso Reppa, and Laura Ferranti. Model predictive trajectory optimization and control for autonomous surface vessels considering traffic rules. IEEE Transactions Intelligent Transportation Systems, 2024.
12

---

Vladimir Vovk. Conditional validity of inductive conformal predictors. In Asian conference on machine learning, pages 475-490. PMLR, 2012.
Vladimir Vovk and Claus Bendtsen. Conformal predictive decision making. In Conformal and Probabilistic Prediction and Applications, pages 52-62. PMLR, 2018.
Vladimir Vovk, Alexander Gammerman, and Glenn Shafer. Algorithmic learning in random world, volume 29. Springer, 2005.
Volodya Vovk, Alexander Gammerman, and Craig Saunders. algorithmic randomness, 1999.
Machine-learning applications of
Chen Xu and Yao Xie. Conformal prediction interval for dynamic time-series. In International Conference on Machine Learning, pages 11559-11569, PMLR, 2021.
Chen Xu and Yao Xie. Sequential predictive conformal inference for time series. In International Conference on Machine Learning, pages 38707-38727. PMLR, 2023.
Chen Xu, Hanyang Jiang, and Yao Xie. Conformal prediction for multi-dimensional time series by ellipsoidal sets. In Proceedings of the 41st International Conference Machine Learning, volume 235, pages (006995 PMLR, 2024.
Ye Yuan and Kris M. Kitani. Diverse trajectory forecasting with determinantal point processes. In International Conference on Learning Representations, 2020.
Ye Yuan, Xinshuo Weng, Yanglan Ou, and Kris M Kitani. Agentformer: Agent-aware transformers for socio-temporal multi-agent forecasting. In Proceedings of the IEEF/CVF international conference on computer vision, pages 9813-9823, 2021.
Margaux Zaffran, Olivier Féron, Yannig Goude, Julie Josse, and Aymeric Dieuleveut. Adaptive conformal predictions for time series. In International Conference Machine Learning, pages 25834-25866. PMLR, 2022.
Hao Zhou, Yanze Zhang, and Wenhao Luo. Safety-critical control with uncertainty quantification using adaptive conformal prediction. In 2024 American Control Conference (ACC), pages 574-580. IEEE, 2024a.
Jinyun Zhou, Runxin He, Yu Wang, Shu Jiang. Zhenguang Zhu, Jiangtao Hu, Jinghao Miao, and Qi Luo. Autonomous driving trajectory optimization with dual-loop iterative anchoring path smoothing and piecewise-jerk speed optimization. IEEE Robotics and Automation Letters, 6(2):
439 439 446, 2020.
Yanfei Zhou, Lars Lindemann, and Matteo Sesia. Conformalized adaptive forecasting heteroge- neous trajectories. In Proceedings of the 41st International Conference on Machine Learning, volume 235, pages 22002-62056. PMLR, 21-27 Jul 2024b.
Steve Zymler, Daniel Kuhn, and Berç Rustem. Distributionally robust joint chance constraints with second-order moment information. Mathematical Programming, 137:167-198, 2013.
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A Auxiliary Results
A.1 Conformal Prediction Lemma
Lemma A.1 [Lemma in Tibshirani et al. (20/9]/ (coverage guarantee) If R( R(), ...., R(M) are
N exchangeable random variables, then for failure probability a E (0, 1), holds that
PRR & Quantilel-a
R(M, .00} R0,) 2 a
(19)
where the function - -...y R(N), 00) denotes the level 1 a quantile of the empirical distribution of the values R(()) .... R(N), oc follows.
Quantiler-a((tt) R(N, oc) = inf PPZ S =) a-a
(20a)
[E RR + 6xx) /(N +
(20b)
where 8pc) and & denote the Dirac delta function R and co, respectively.
A.2 Upper bound of the expectation of posterior risk
Corollary A.2 (upper bound of (3)) Suppose that 5 E (0,1) and K > (- (-In5)/(2a?), we have
P P( - (1 L (a+ + V- [1+ 2 1
(21)
Furthermore if K,L oc, then E(8,) S a, holds with probability one.
Remark A.3 Corollary A.2 provides a performance guarantee for Fb-CP, i.e. Fb-CP performs at least as well as the sequential method Lindemann et al. (2023) with high probability. Furthermore, the experiments in Section demonstrate that the proposed method performs significantly better in practice. This is attributed to the conservatism of the inequality (38) in the proof of Corollary A.2 Since c( Yx) contains the information provided by c(.) (e.g. the shape of the robot and obstacles) and x;,, it rypically occurs that P(c[=,, Y+) <0 P(Ily, -Yiell C)-a,)
B Proofs
B.1 Proof of Lemma 4.1
According to Assumptions 3.1 as well as the calculation of Cl-,,
(6b), the (1 crvv-reeeaage
guarantee of the prediction (6a) is obtained through Lemma A.1 Note that the function is L- Lipschitz continuous, the following inequality obtained.
||([,, Yx) c(Ir. |1 & L||Y, Y-ull = c(æ,, 2 c(z,, 811) LIY, Y-jell (22)
at- is satisfied, we have the following inequality. the constraint c(r,, Pr1r) 2 7/8
c( Y,) 2 L(C 7/e
||Y, Y=lell)
(23)
According to the (1 a, coveraae guarantee (6a) P(C 1| lemma is proven.
Yziell 2 0) 2 ar, the
7/t
B.2 Proof of Lemma 4.2
According Assumption 3.1 know that all trajectories ( generated by sampling their
initial state (i) from the same distribution ID and then evolving using common dynamics (the ground truth dynamics). As result, the random variables wy
(1) ,.... w. (K-L) are i.i.d. Note that x
the true state of the system time т, thus x; is fixed and independent of wn, w7 (K+1) (K+L) s(K+1)
W
Therefore, the random variables S, ..... s!. <LK+L) are exchangeable.
21

---

Without loss of generality, we assume that the dataset {-St :i= 1,..,, L) are sorted in non-
decreasing order. We first assume that -t S 0, and then we define the maximum index & that makes- -S, (() I 0 hold as follows.
max 1=1...
(24)
s.t -(( 00
Then the posterior satisfaction probability can be computed below.
P(c(#,, Y+) 2 0} P(-S, 03 2 P(-S, S -
(25)
It is assumed that there are terms in (-St (()+ 1, ..... [.] identical - (((+t) i.e.
S(K+e-0 < < S(((KtE-)+)) - S -S(K+(+1)
(26)
St (+) can be equivalently reformulated as follows.
Then
-
Quantileg(-S(k+)) .... SkK+t), -,KK, AB
L-t
(27)
1+ 1+L
Combining
25 and 27
we have
Pfc( Y+)
0) 2 P(-S+ S Quantileg(-S!ktt (K+))) ..... - -((t))) x0))
(28)
Note that the random variables S,, S, k++)) (0, 1), and thus we can apply Lemma A.1 and obtain
o ((++)) exchangeable and B
(- f
I+L
8-1
P(c( Y+) 2 0) 8, ABE
(29)
I+L'I++
Therefore, the upper bound of the posterior violation probability can be computed by
(1-
P(c( Y.) < 51 B, vB E
IIL'I+
(30)
To minimize this upper bound, take the maximum value of B and
30
becomes
31
f Pfe(,, Y7) < 03 1 I+I
(31)
According to the definition of l (24), we can compute & as follows.
((K+)) 20) I. Σ
I (((K+)) <0)
ST
(32)
1=1
Combining 31 and 32
we have
1 D4 1
(s(K+)) <0)
(33)
1
Finally, we consider the scenario in which S k +1) Then the inequality (33) is simplified as follows.
0, which means S (K+))
< 0 1, 1,... L.
P((((;,, < <0) <
(34)
which is always true. Thus, the Lemma is proven.
22

---

B.3 Proof of Corollary A.2
Taking expectations on both sides of Equation 8 we can obtain
B(3-) = 1 LP
0} 1+ LP < <0} 1+ LF((E, 1,) < 0}
(35)
1+L
1+L
1
The second equality holds because S,, 5 SK-1 ; SK+L are exchangeable. Note that the function c() is L-Lipschitz continuous and x is a feasible solution of problem with the reformulated
constraint through Lemma [.11, and the following incquality can be derived in the same manner as inequalities and (3 in the Proof of Lemma 4.1 (Appendix(B...
d,,, I(C 11, Pr1(1) 7/6
(36)
Based on 336 we can obtain
d <2)))=000))) Y Y-jel Chi
(37)
And the following inequality is derived.
P(e(at, <0 S P(|Y, Yrlell > Chrar
(38)
Combining 35 and 388
we have
1+ LF - Yriell > Cl
E(3,)
(39)
For ar, d E 10 and K > (- 11 () we can apply Vovk 2012), Proposition 2a] so that
Frrll PP -II,
- -Yill s ChIe } 2 a+ + In ( 2
(40)
which can be equivalently transformed into the following expression.
Y+ull I(P((Ix-Y+ul1>>> -
PfP
- S a, + - lm)/((2 21-8
(41)
Combining 33 and (4)), we can finally obtain the inequality 21
When K, L co, we can further assume that L 1/5 and K 2 max (n 1/6). Note that for fixed a.,, we can always find small enough positive such that a+ + v (- In (n < Orr + (n <1 1. Therefore for a small enough positive ō we have
8 + a V 0/2
B(B,) 5
6+1
1 (ar / In 1
(42)
>P
B(B,) <
21-8
1
Let + 0+, we finally obtain that (E(3,) I a, holds with probability one.
B.4 Proof of Lemma 5.1
Let aftir and aftr. be two risk allocations at time t. Based on the definition of C, (6b).
Cl- is non-increasing with respect to a, for fixed D Therefore, if af S a² VT = 1+ 1, I.... T,
then cl-a: 2 c--a which further leads to Re(a/+1r) S R2(a4+17) Since J (*(4+11:T) is
the minimum of the objective problem (10) with the feasible region R Re(oo-1:), J (*(a(1T)) 2 J (aa+11r) can be obtained and the lemma is proven.
23

---

B.5 Proof of Lemma 5.2
The computation of the lower bound is analogous to the calculation of 3, in Lemma 4.1 except that Lemma[A.) is not required to obtain coverage guarantees. Therefore, the computation is based on Diat. Without loss of generality, we assume that the dataset { r/t i I,.., K} is sorted in non-decreasing order. Note that 4241:T is feasible for the problem (TOp with of and T E Tinas the inequality c(z,,
LC LQuantiléi-a; (
(R
7/t R(k oo) holds true. Since
o < 1, it follows that c(a?, Fr1e) 7/t
> Re Therefore, we define the maximum index K. that makes
c(z", (+) 2 L hold follows.
K
max k k= k=l....K K
(43)
s.t. c( Y 2 LR
It is assumed that there are / terms in [R : = 1,.., "...,,K K} identical to R and thus we can obtain
R(C-1) ((K-++1) =.. = R// S c( y f/pp ((++1)
(44)
We aim to determine the maximum value of "² (the minimum value of a;) while satisfying
Ch1 S ((=,, which is equivalent to C1 S &R because Chr can only take
values finite number of discrete points R(() T|e R co. Furthermore, R can be equivalently reformulated as follows.
(K- (
R = Quantile (R)
Rejer... my R/ 0o) c.
vB E
[
(45)
i I+K
Therefore, the constraint C² S R is equivalent to the following expression.
-l- < 7/p 38 E (K-t_ K.
(46)
Note that CH,, non-decreasing with respect to 8 for fixed D Constraint 46) is further reformu- lated as follows.
K.
2
(47)
1+ K
According to the definition of K.
43
compute K. in (48).
K
LR..) K
Σ
T (elo Ieeoo F71e) IR
(48)
i=1
Combining 44 and 48 the lower bound of ā is computed as follows.
1 1+ I (c(27) Erje) < LR(
(49)
1 K
We note that of is the lower bound of a that ensures the constraint c(27, V,1)) 2 IC,, 2. Furthermore, since o+tr: is feasible for the problem 110 with 0.41:7- the constraint c(x"), V, 2 LC1-a, is satisfied. Therefore, a S at is naturally obtained. Thus the Lemma is proven.
r/t
B.6 Proof of Theorem 5.3
The proof adapts elements of the proof from Zymler et al. 2013). If 00 #1+1rr "ir-1 is feasible solution for the risk allocation a 1.-1. the update law a:+1:7 guarantees that the sequence of the optimal objective values (J* (a)+1x))men monotonically decreasing, as previously mentioned. Since the sets X' and ll are bounded, 1:71.7 and u:r-1 are bounded. Because the objective function J (I++1:,, u:T-1) is continuous, the boundedness of 1441:T, W: and the monotonicity of the optimal objective value sequence imply that {J* (or))r)))e converges to finite limit.
24

---

B.7 Proof of Theorem 5.4
At each time the realized state i available. According Lemma 4.2
we know that
P(c(=t, 0} S By,
VT 1, L...,
(50)
For the planned state P.+1:7 the corresponding risk allocations 1:7 are obtained either via ARA or IRA (2). In both cases, they satisty
Σ ar a -
(51)
We note that if the planned state o 1:7 is a feasible solution of the TO problem (10) with 01441::7 the following inequality can be obtained through Lemma 4.
P(c( Y, 2 0} 2 - ary
V7
1 1,...,
T
(52)
Finally, by combining the above three inequalities 50 inequality, we obtain
51
and
52
together with Boole's
P
(o 2 20) 1
(53)
r=1
which means that the overall risk of the entire trajectory at time f, .ier is below the risk tolerance Thus, Theorem 5.4jis proven.
C Experiment details and additional results
C.1 Simulation for kinematic vehicle model
We examine the kinematic vehicle model Lekeufack et al. 20244 with the following nonlincar
dynamics.
Postl Py,t-1
Pn, + Au, cos 0g
Py.t Au sin O,
(54)
0.11
O. A % tan o;
0t+1
D2 + Aa,
where p: := (Pe.e. Py.e), 9s v2 are the position, orientation, and velocity of the vehicle, respectively. I : 0.2 is the length. and A = 0.125 is the sampling time. The system inputs are the steering angle The objective
de [-x/6, /6, / /66 and the acceleration o, E [-5, 5). The total time is set to 7 is to reach the vicinity of the target point while avoiding collisions with obstacles. Formally, the objective function is defined as J Σ 100 a? to minimize energy consumption and the constraint |ppr- - Ptarl1e S 0.2 is incorporated into 9 to ensure the vehicle reaches the target point, where Ptor is the target point. The constraint function for collision avoidance is follows.
c(pT, Y+) minj=11....MM [|p-, Yrulla
Ts
(55)
where r, and ro are the inflation radius of the vehicle and obstacle, respectively. Ts is the safety margin. The interior- interior-point method-based solver IPOPT (v3.12.9) was used to solve the TO problem (9). Similar to Lindemann et al. [2023], we consider M = obstacles, with their trajectories 2021 which publicly available htttp::/github.com/// ppttaraanetplussllusaseeiiee lnd is licensed under the MIT License. We generate 13,000 joint obstacle trajectories and randomly divide them into training Dtrains calibration Deats and test. Drest datasets with the set sizes 2,000, 10,000, and 1,000, respectively. We train an LSTM Alahi et al. (2016) using Dtroin the trajectory predictor. For the proposed Fb-CP, Deat is further divided into D and D with sizes D 2, 000 and DD2 = 000. We conduct 1,000 Monte Carlo simulations using Drest- As we discussed in Section 6. the methods S-CP, Fb-CP-ARA, and Fb-CP-IRA are analyzed.
generated by TrajNet+ Kothari
Table 2 shows the average cost, average computation time, and collision avoidance rate of 1,000 simulations using the kinematic vehicle model with different methods. We collect the simulation data under different total risk tolerances 0.05, 0.10, 0.15, 0.20. On one hand, with increase in
25

---

total risk tolerance, the average cost of all methods decreases. On the other hand, benefiting from the feedback information of posterior probabilities, the average cost Fb-CP-ARA shows reduction of 7.21 % to 11.26 1.266% compared to S-CP. Furthermore, by flexibly allocating the allowable risk provided by posterior probabilities, the average cost of Fb-CP with IRA exhibits a significant reduction compared with S-CP. Additionally, the increase in total risk tolerance provides greater flexibility in the risk allocation of Fb-CP-IRA resulting in a significant reduction in its average cost. As mentioned in Section 6 the calculation of posterior probabilities does not incur additional computational burden. Therefore, the average computation time of Fb-CP-ARA essentially comparable to that S-CP. The collision rates of all methods do not exceed their corresponding total risk tolerances.
Table 2: Average cost, computation time, and collision avoidance rate using the kinematic vehicle model with different methods.
Fb-CP
Sequential CP
with ARA
with IRA
a' 0.05
22.20
20.46
4.77
a 0.10
20.24
18.78
3.52
Average cost
a 0.15
19.22
17.35
3.18
0.20
17.05
15.13
2.89
Q 0.05
0.111
0.100
0.128
α 0.10
0.093
0.087
0.126
Average computation time
a 0.15
0.078
0.085
0.130
a 0.20
0.076
0.078
0.131
Q 0.05
95.4%
95.7%
98.4%
Q 0.10
93.9%
93.1%
98.3%
Collision avoidance rate
a 0.15
90.8%
89.8%
97.6%
a 0.20
88.4%
89.1%
91.2%
C.2 Simulation for linear quadrotor model
We examine the quadrotor model Dixit et al.
2023
with the following linear dynamics.
J g0
g
U1
lq u4
(56)
Tax U2
Tyo U3
T..
where 9.81 represents the gravitational acceleration, me 0.65 denotes the mass, and lo 0.23 is the distance between the quadrotor and the rotor. Ia- 0.0075, Lyy 0.0075, and I:x 0.013 correspond to the area moments of inertia about the principle axes in the body frame. The states are the position and orientation with corresponding velocities and rates
(z, y, v, 2; J. 2, o, 0, th, ó, Ó, d) E 122 The control inputs U/ U22 u3, 4 correspond to the thrust force in the body frame and three moments. The system (66 is discretized using the sampling time = 0.125, and the total time also set to T = 20.
Similar to the experiments based on the kinematic vehicle model in Appendix C.I the objective is to control the quadrotor to reach the target point Ptar while navigating around M moving obstacles. The target point constraint and obstacle avoidance constraints are consistent with those used in the simulation using the kinematic vehicle model. We randomly generate 13,000 obstacle trajectories and assign them as in Appendix C.I The following state-of-art methods recently proposed in the
literature are analyzed through 1,000 Monte Carlo simulations.
(i) Conformal Control (CC) proposed in Lekeufack et al. (2024]
(ii) ACI for Motion Planning (ACI-MP) proposed in Dixit el al.
2023
(iii) Recursively Feasible MPC using CP (RF-CPP proposed in Stamouli et al. [024
(iv) Sequential CP (S-CP) proposed in Lindemann et al. 2023
Computation of the CP region and
TO is performed sequentially.
(v) Fb-CP with ARA (Fb-Cl (1b-CP-ARA); The method based on Fb-CP using average risk allocation.
(vi) Fb-CP with IRA (Fb-CP-IRA): The method based Fb-CP using iterative risk allocation.
Table shows the average cost, average computation time, and collision avoidance rate of 1,000 simulations using the quadrotor model with different methods. For the methods S-CP, Fb-CP-ARA.
26

---

Table 3: Average cost, computation time, and collision avoidance rate using the quadrotor model with
different methods (1) the learning rate of CC).
Fb-CP
CC
ACI-MP
RF-CP
S-CP
ARA
IRA
1000
59.25
a = 0.05
17.970
15.794
17.321
15.356
7.189
Average
500
47.50
0.10
17.263
14.378
16.17
14.228
6.798
cost
100
22.46
a = 0.15
16.096
11.922
14.83
12.354
6.191
7 50
21.34
0.20
15.310
10.032
13.217
10.22
5.398
1000
0.019
a = 0.05
0.022
0.487
0.022
0.027
0.038
Average
500
0.019
0.10
0.026
0.494
0.020
0.021
0.039
computation time
= 100
0.021
a = 0.15
0.021
0.545
0.021
0.020
0.037
7 50
0.022
0.20
0.022
0.500
0.020
0.019
0.036
1000
97.0%
a = 0.05
98.6%
98.7%
98.8%
98.2%
96.3%
Collision
500
92.8%
0.10
93.3%
96.9%
93.5%
94.6%
94.1%
avoidance
= 100
82.5%
a = 0,15
91.5%
92.4%
92.0%
90.2%
91.9%
rate
50
79.1%
0.20
87.9%
90.0%
88.2%
86.7%
88.2%
and Fb- Fb-CP-IRA, the experimental results using the quadrotor model are fundamentally consistent with those derived from the experiments using the kinematic vehicle model. Specifically, compared with S-CP, Fb-CP-ARA benefits from the posterior probabilities calculation, leading to a moderate improvement in performance. Fb-CP-IRA, leveraging the combined use of posterior probabilities and more flexible risk allocation, exhibits a significant enhancement in performance. Additionally, since CC and ACI-MP fail to fully utilize the information in the calibration dataset, they incur higher costs, which are 184% and 296% higher than those of Fb-CP-IRA respectively. Particularly for CC, directly controls the collision avoidance rate by adjusting the weight of the collision penalty in the objective function, which results in higher average cost. However, it should be noted that, in practice, CC and ACI-MP are better suited for scenarios where the test data exhibit distribution shift, rather than the setup considered in our work. For RF-CP, thanks to the proposed normalized nonconformity score, its average cost is comparable that of Fb-CP-ARA, but it remains 85.8% higher than Fb-CP-IRA However, the normalized nonconformity score introduces mixed-integer variables into the TO problem, significantly increasing the computation time. As shown in Table 3, the average computation time of RF-CP is more than an order of magnitude higher than that of Fb-CP-IRA.
C.3 Simulation for dynamic bicycle model
We examine vehicle with the following dynamic bicycle model Hakobyan and Yang [021
= Up cos 0 - "y sin 0
(57)
12 sin 0 My cos 0
(58)
(59)
2(C, + C+) CC))
-(1,,, (21,0,-2,,, + +1x
20's 8s r
vi
(60)
my
mVVE
my
+11,Cy+h0a.
12Ux Uy 212C, 212C
21,0181
(61)
12Un
12
where v, y are the vehicle's central of mass, uy. and are lateral velocity, orientation, and yaw rate, respectively. Furthermore, Vg is the constant longitudinal velocity. my denotes the mass of the vehicle, Cy and C, represent the cornering stiffness coefficients of the front and rear tires respectively, Ly and L, denote the distances from the center of mass to the front and rear wheels, and I2 corresponds to the moment inertia around the z-axis. The input variable is the front wheel steering angle 5y. The system (57) is discretized using the sampling time A 0.125, and the total time is also set = 20. According to Hakobyan and Yang [2021), the parameters the dynamic bicycle model used in this simulation are listed in Table
27

---

Table 4: Dynamic bicycle model parameters.
my
C5
C
1g
Lg
Ly
1700kg 50kN/rad
50kN/rod
6000kg m²
1.2m
1.3m
5m/
The task is to steer the vehicle to its target point p while avoiding M 2 moving obstacles. Similar to the experiments in Appendix C.1 the target point constraint and obstacle avoidance constraints incorporated into the optimization problem ensure the vehicle reaches the target point while avoiding collisions with obstacles. We collect 13,000 joint obstacle trajectories and assign them as in Appendix C.1 The methods S-CP, Fb-CP-ARA, and Fb-CP-IRA analyzed through 1,000 Monte Carlo simulations.
Table 5 shows the average cost, average computation time, and collision avoidance rate of 1,000 simulatrons using the dynamic bicycle model with different methods. The experimental results are generally consistent with those obtained from the experiments using the kinematic vehicle mode and the quadrotor model. The performance Fb-CP shows a certain degree of improvement over S-CP based posterior probability calculations. Based posterior probability calculations, Fb-CP-ARA demonstrates certain level of performance improvement compared to S-CP, while Fb-CP-IRA further attains significant performance by leveraging the combined of posterior probabilities and a more flexible risk allocation. should be noted that, due to the simulation of a relatively complex nonlinear model in this experiment, the average computation time inevitably increases. Furthermore, may be observed that the reduction in average cost achieved by Fb-CP-IRA compared to S-CP decreases in this experiment (47.2% reduction) compared with the experiment using the kinematic vehicle model in Appendix [C.1] (81.9% reduction). This because, compared to relatively simple scenarios (2 obstacles, dynamic bicycle model experiment). more complex scenarios obstacles, kinematic vehicle model experiment) better highlight the performance improvements enabled by the flexibility in risk allocation.
Table 5: Average cost, computation time, and collision avoidance rate using the dynamic bicycle model with different methods.
Fb-CP
S-CP
ARA
IRA
a 0.05
23.05
20.91
13.77
0,10
22.38
18.39
11.35
Average cost
Q = 0.15
20.71
16.99
10.17
0.20
16.55
14.78
8.58
a 0.05
0.365
0.361
0.884
0.10
0.339
0.335
0.817
Average computation time
Q 0.15
0.494
0.506
1,292
0.20
0.309
0.407
1.003
a 0.05
96.8%
96.5%
97.0%
0.10
94.8%
94.0%
94.3%
Collision avoidance rate
Q 0.15
91.5%
90.0%
91.5%
= 0.20
89.5%
87.8%
89.5%
In summary, the three simulation experiments demonstrate the general applicability of the proposed method, achieving significant performance improvements across various system models while satisfy- ing probabilistic collision avoidance requirements. In fact, the complexity of different system models only affects the average computation time. addition, simulations demonstrate that Fb-CP-IRA achieves more significant performance improvements in relatively complex scenarios.
C.4 Stanford Drone Dataset
We perform comparative evaluation of different methods on the Stanford Drone Dataset. Specifically, the task is steer the vehicle its target point while avoiding moving obstacles (humans). Similar to the experiments in Appendix C.I the target point constraint and obstacle avoidance constraints
28

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are incorporated into the optimization problem to ensure the vehicle reaches the target point while avoiding collisions with obstacles. The average cost of different methods aresummarized Table 6
As shown Table 6 compared with S-CP, Fb-CP-ARA benefits from the posterior probabilities average reduction of at least 20.7%. Fb-CP-IRA reduces the cost by least 46% compared to S-CP. Additionally, since CC and ACI-MP fail to fully utilize the information in the calibration dataset, they incur higher costs, which are 176% and 113 higher than those of Fb-CP-IRA, respectively. Note that RF-CP is not included as a baseline because its formulation introduces mixed-intens -exeed-nttegerreeee variables through nonconformity score definitions, which-when when combined with the nonlinear vehicle dynamics- leads to highly complex trajectory optimization problems with prohibitively long computation times.
calculation, resulting
Table Average cost of different methods on the Real-World Stanford Drone Dataset with different methods.
Fb-CP
CC
ACI-MP
S-CP
ARA
IRA
1) 1000
95.51
a: 0.05
40.02
38.58
30.58
20.81
Average
500
77.68
Q 0.10
35.94
34.75
27.52
18.52
cost
77 100
41.27
a 0.15
32.57
30.48
23.75
15.58
50
38.73
0.20
29.89
28.59
21.25
14.02
D Experiment with different trajectory predictor
As noted in Subsection 3.1 employing the more advanced trajectory predictors can enhance control performance. Accordingly. we perform experiments on the kinematic vehicle model (54) to compare the average Trajectron+++ [alzmann et al. [2020] AgentFormer Yuan et al. [0211) with the results summarized in Table 7 As shown in Table 7 when switching to more advanced predictors Trajectron++ and AgentFormer, both Fb-CP and S- see improved performance, but the proposed method (Fb-CP)
of S-CP and Fb-CP when using different predictors Social LSTTM|laa et al. 2016).
continues to outperform the S-CP under all tested predictors.
Table 7: Average Cost of Different Methods using Different Trajectory Predictors (a 0.2).
S-CP
Fb-CP-ARA
Fb-CP-IRA
Social LSTM
17.30
15.49
2.97
Trajectron++
13.58
10.58
1.81
AgentFormer 12.83
9.71
1.72
E Experiment with small calibration size
As discussed in the Limitation (Appendix D), the proposed method requires a larger calibration dataset because the calibration dataset must be divided into two parts. To rigorously evaluate its performance under data-scarce conditions, we conduct experiments on the kinematic vehicle model (54), where both S-CP and Fb-CP are tested using the same small calibration dataset (N 400). The corresponding average costs, computation time, and collision avoidance rate are reported in Table g As shown in Table although all methods see decrease in performance under limited data, Fb-CP still achieves substantially lower cost than S-CP and remains safety. This highlights the robustness and practical effectiveness of the proposed method, even when the calibration dataset is small.
Experiment with higher-dimenssonal settings
In this appendix, we validate the performance of the proposed method in high-dimensional settings. Specifically, conduct an additional case study controlling the trajectory of high-dimensionaal

---

Table 8: Average Cost, computation time, and collision avoidance rate of Different methods with calibration data size of N 400.
S-CP
Fb-CP-ARA
Fb-CP-IRA
Average cost
18.03
16.82
3.52
Average computation time
0.078
0.081
0.122
Collision avoidance rate
88.9%
89.5%
91.7%
double integrator model, where shrinking-horizon trajectory optimization is performed and 10-dimensional spaces. The average costs and collision probabilities of different methods under proposed Fb-CP framework
various dimensions are summarized Table 2 As shown Table
consistently outperforms S-CP while maintaining safety, in these higher-dimension scenarios. These results validate the applicability and effectiveness of the proposed approach in high-dimensional problems.
Table 9: Average cost and collision avoidance rate of different methods (a = 0.2) with varying dimensions.
Dimension
S-CP
Fb-CP-ARA
Fb-CP-IRA
5
44.52
39.67
30.25
Average cost
10
72.66
65.33
60.18
5
92.5%
92.1% 93.5%
93.5%
Collision avoidance rate
10
91.2%
91.4%
G Extension and experiments on distribution shift
The individual chance constraint reformulation in Lemma 4.1 and the posterior probability calculation Lemma 4.2 rely on Assumption 3.1 and the i.i.d. property of w, which imply that real joint obstacle trajectory and those in the training and calibration datasets follow the same distribution. To enhance the generalizability of the proposed method across different scenarios, we extend our method to address the case where a shift exists between the initial state distributions of calibration data and test data, and the error w/ represents state-dependent noise rather than being i.i.d.
Specifically, there exists shift between the initial obstacle state distribution Diest of the test data and the initial obstacle state distribution Deal of the calibration data. Moreover, the error w/ constitutes state-de dete-dependent noise, i.e., its distribution Dw, is conditioned on the current state Y. Clearly, the difference between Drest and Deats along with the state-dependent nature wi, implies that the test
trajectories Y, and the calibration trajectories y are not exchangeable. Fortunately. the distribution shift between Y, and y can be characterized as a covariate shift, defined in Tibshirani al. [201)) This allows us to adapt the approach proposed in Tibshirani et al. [2019] to extend our method accordingly. The extended method is described in detail below.
Intuitively, we reweight the calibration data by computing the likelihood ratio between the test and calibration distributions. The reweighted data then used compute the prediction regions and the posterior probabilities, thereby enabling robustness to covariate shift. Specifically, at each time t, we begin by applying an uncertainty propagation technique to derive the distributions Dy, and Dy, of the test obstacle state Y, and calibration obstacle state (z(i) respectively. As a result, given a data Y. we can calculate the likelihood ratio as follows.
vYY dP Dr, (d)/ (Y)
(62)
In the forward phase of confidence region computation, the weights for the test data and the calibration data in Dim are computed using the likelihood ratios 622) as follows.
o(Y)) = o(Xx())) + o(y))
"(Y,()))
(Y))
pi()(())
Vi = 1,.., (63)
ΣK (XX + oXY)'
30

---

Following the approach in Tibshirani et al. 2019 we replace the (1 - a)- quantile in equation 6b with the following weighted form.
Cl-ar Quantilei-ar T/t
Σ
pi (X;)6 + P 2
(64)
In the process of backward posterior probability computation, we calculate the weights for the test data and the calibration data in D as follows.
(Yi) Zj=K U( +o())'
v(y(ktt) Lj=K v(Y(5)) + v(y))
(Yt)
P2(Y
Vi 1,..., L (65)
KKL
For the posterior probability computation, we replace equation weighted form.
8 in Lemma 4.2 with the following
L P(c(,t, <0} S 3, = P2(()) + Σ 1=1
p2 (Yx 1 (s(Kti) ( <0)
(66)
It is evident that when P2(Yi) p2
1/ 1). equation 666 degenerates equation (8). By applying the weighting schemes in equations (65) and (66) to the test and calibration data, the extended method capable of handling the covariate shift between them. We refer to this extended approach as Weighted Fb-CP.
To demonstrate the Weighted Fb-CP exhibits robustness the covariate shift, we design experiments to compare the effects of covariate shift between test trajectories and calibration trajectories on the performance and safety of the proposed method. Apart from the method of generating obstacle trajectories, the experimental setup is identical to that of the kinematic vehicle model experiment in Appendix C.1 To obtain obstacle trajectories with different distributions, the obstacles are modeled using the tollowing double integrator model.
Ps.t Aus,t + A²
Pr,t+1 Py,t+1
2 ar,t
Pu.t + Auy + 2 aye
+ wie
(67)
0g,t+1
Un, + Aaz,
0y,t+1
0y. + Aay.
where (Pr,Py, Uz Byy) is the state of an obstacle, consisting of its center of mass and velocity vector. The control input (az, ay) is the acceleration vector. Similarly, the sampling time A is selected as 0. 125. The obstacle trajectories from a given start point to the target point are obtained by solving an optimization problem. wt sampled from zero-mean Gaussian distribution, with its variance determined by the current state (Pa, Py: Ux, Uyy) The initial states of the trajectories in the test dataset are sampled from Gaussian distribution with mean (-5, O, O, 0)T while the initial states of the trajectories in the calibration dataset are sampled from Gaussian distribution with mean (0, -5, 0, 0)² As result, there exists distribution shift between the trajectories in the test dataset and the calibration dataset. We conducted 1,000 Monte Carlo experiments in this scenario to compare the Weighted Fb-CP, Fb-CP, S-CP Lindemann et al. 2023 and ACI-MP Dixit et al. [2023.
Table 10: Average cost, computation time, and collision avoidance rate using the kinematic vehicle model with different methods (a 0.2).
Fb-CP
Weighted Fb-CP
ACI-MP
S-CP
ARA
IRA
ARA
IRA
Average cost Collision avoidance rate
19.27
17.56
14.35
11.37
16.27
13.72
84.8%
78.7%
78.6%
76.8%
84.6%
83.4%
Table 10 shows the average cost and collision avoidance rate of 1,000 simulations using the kinematic vehicle model with different methods. It can be observed that due to the covariate shift, both the S-CP and the proposed Fb-CP methods exceed the specified risk tolerance (a 0.2). Owing to the reweighting of the test and calibration data, the Weighted Fb-CP method is able to maintain the
31

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collision probability within the prescribed risk tolerance, accompanied by justifiable and expected increase average cost. For ACI-MP, since it does not rely on the calibration set, naturally does not exceed the risk tolerance. However, precisely because it cannot leverage the information contained in the calibration dataset, its average cost 40.5% higher than that of Weighted Fb-CP-IRA. In summary, we empirically demonstrate that the Weighted Fb-CP is capable of maintaining the risk probability below the specified tolerance under covariate shift, while also achieving a favorable trade-off in terms of average cost.
H Details about the prediction regions
In this Section, we investigate the impact of using prior versus posterior probabilities on the prediction regions. To this end, we collect the prediction region radius for time f, denoted as C2o using the vehicle model with different methods across 1,000 simulations, as illustrated Figure It can be observed that C20Je decreases as t increases, which is reasonable since the error of the trajectory predictor diminishes t approaches т. Note that since S-CP only uses prior probabilities compute C2j2e throughout the entire planning process, which depends solely on Deats C20/: remains constant for fixed By contrast, for Fb-CP-ARA, C20 also depends on the actual obstacle positions and past decisions due to the use of the posterior probabilities, which leads to the variability of C20 across 1,000 simulations. The distribution of C20 shown in the right panel of Figure 3 It can be seen that C20je computed by Fb-CP-ARA is typically smaller than that computed by S-CP. As t increases, more posterior probabilities can be used, leading to a growing gap between the C20je calculated by the two methods, which corroborates Corollary A.2
Prediction region radius for r- 20
Fb-CP-ARA (posterior probability) S-CP (priori probability)
0.67
0.66
Region radius (m))
0.8
Region radius (m)
0.65
0.6
0.63
0.4
0.62
0.2
0.61
200
600
Actual time
Number
Figure 3: Left: prediction region radius for r 20 each time (C2oj)) using the vehicle model with different methods across 1,000 simulations. Right: distributions of C2oja-
Furthermore, Table [] shows the prediction region radius for different time I and T (C)))) At the initial state (1. = 0), realized state available for calculating posterior probabilities. As result, C+1o for all r obtained by S-CP and Fb-CP-ARA are essentially identical, with minor differences arising from the fact that S-CP utilizes the calibration Deat- whereas Fb-CP-ARA only employs D As the system operates, an increasing number of realized states 2; are available for the calculation of posterior probabilities, enabling Fb-CP to yield relatively narrower prediction region, corresponding to smaller Crjt- As shown in Table TT the average ratio the predicted region radius obtained by Fb-CP-ARA to those by S-CP generally exhibits decreasing trend as time / increases. Moreover, when & > 10, based on sufficient posterior probabilities, the prediction region radius obtained by -b-CP-ARA is reduced by more than 50% compared to that of S-CP.
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Table 11: Prediction region radius for different т and & (Cr)e) using the kinematic vehicle model with
different methods (S-CP and Fb-CP-ARA) across 1,000 simulations (a 0.2).
== 3
==
6
т 9
12
7 -= 15
7 == 18
Ratio
t Fb-CP-ARA
0.125
0.258
0.374
0.519
0.698
0.913
1.005
S-CP
0.129
0.247
0.374
0.520
0.691
0.902
Fb-CP-ARA
0.102
0.216
0.351
0.527
0.705
t
0.793
S-CP
0.129
0.263
0.409
0.598
0.738
Fb-CP-ARA
0.141
0.289
0.442
0.604
t = 6
0.956
S-CP
0.152
0.304
0.451
0.625
Fb-CP-ARA
0.134
0.268
0.414
t = 9
0.889
S-CP
0.149
0.305
0.466
Fb-CP-ARA
0.059
0.139
t = 12
0.453
S-CP
0.138
0.291
Fb-CP-ARA
0.054
1 = 15
0.470
S-CP
0.115
I Computation time of IRA and the hybrid method of ARA and IRA
Table 12 shows the average computation time at each time & using the kinematic vehicle model with Fb-CP-IRA (a 0.2). It can be observed that, in practice, the majority of the additional computational burden introduced by IRA arises the initial time t 0, where is used to obtain the initial risk allocation and initial trajectory. At subsequent time steps. by using the optimal solution from the previous time step (or iteration) as the initial value for the next time step (or iteration), the IRA algorithm can converge quickly. It is important to note that the TO problem at the initial time step can solved offline, while the average computation time for TO subsequent time steps is much smaller than the sampling time (0.125s). Therefore, Fb-CP-IRA is well-suited for real-time TO.
Table 12: Average Computation Time (ACT) each time t using the kinematic vehicle model with Fb-CP-IRA (a 0.2).
0 2,302
2 0.033
3
4 0.028
5 0.026
6 0.024
7
8 0.021
9 0.022
ACT
0.035
0.032
0.022
10
11
12
13
14
15
16
17
18 0.005
19 0.002
ACT
0.015
0.011
0.010
0.009
0,008
0.007
0.006
0.005
Furthermore, we explore a hybrid method of ARA and IRA achieve trade-off between average cost and average computation time. Specifically, we define a switching time l., such that when I < the IRA method is used, and when / > l... the ARA method is applied. Table T3 shows the average cost and computation time using the kinematic vehicle model with different switching time l (a 0.2). can be observed that using IRA only at the initial time step results in 66.56% reduction in average cost. This is because the trajectory obtained at the initial time step determines the overall path of the entire trajectory. As t, increases, the average cost naturally decreases. When reaches 11, further increases in f., no longer lead to significant reduction in the average cost. This is because, in our scenario, the interaction between obstacles and the vehicle most intensive at the middle of the mission time. After t > 11, the obstacles and the vehicle have moved apart, significantly reducing collision risk, which results in ARA and IRA optimizing nearly identical trajectory. Although the aforementioned hybrid method balances the trade-off between average cost and computation time, as previously mentioned, executing IRA the initial time step is the primary source of both cost reduction and increased computation time.
33

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Table 13: Average cost and computation time using the kinematic vehicle model with different switching time (a 0.2).
0 (ARA)
6
11
16
20 (IRA)
Average cost
13.13
5.06
3.74
2.90
2.89
2.89
Average computation time
0.078
0.118
0.121
0.129
0.129
0.131
J Sensitivity analysis of the calibration set division
Table [4 shows the average cost, average computation time and collision avoidance rate using the kinematic vehicle model and Fb-CP-ARA (a 0.2) with 10 different random calibration set divisions. Specifically, for each experiment, we randomly divide the calibration dataset Deat into Dial and D It can be observed that, for the ten random experiments, the standard deviation of the average cost is 0.643, and the coefficient of variation is 4.2%, indicating relatively low volatility. For the collision avoidance rate, its volatility is only 2.3%, and all values do not exceed the given tolerance (80%). Therefore, the proposed method is insensitive the calibration set division.
Table 14: Average Cost (AC), Average Computation Time (ACT) and Collision Avoidance Rate (CAR) using the kinematic vehicle model and Fb-CP-ARA (a 0.2) with 10 different random calibration set divisions.
Index
2
3
4
7
8
9
10
AC
15.13
14.78
15.11
16.03
15.96
14.37
14.60
15.73
15.64
16.19
ACT
0.078
0.078
0.079
0.077
0.072
0.080
0.065
0.074
0.086
0.084
CAR
89.1%
89.2%
90.5%
88.2%
88.6%
88.4%
90.5%
89.2%
89.3%
89.9%
K
An extension using the normalized nonconformity score
Stamouli et al. 2024 proposed normalized nonconformity score, which can improve performance Lindemaan et al.) [2033. In this section, we incorporate this normalized noncon- formity score into Fb-CP to further enhance its performance follows. First, still follow the content outlined prior to Section 4.1 Instead of using the original nonconformity score as in Section 4.1
compared to S-CP
we can redefine the nonconformity score time follows replace 5
|| Y-Hel
R2 max
t=0 t=0,.
7
(68)
|| y-181
max R(O)
Vi= 1...,, K
I=0....T-
where
T/A
max jETyran
||
--00
Vt, T>t
(69)
where Ierain (j Y E Derain} denotes the of indices of the data in the training set Deroin. We note that, compared to the nonconformity score in Stamouli et al. [2024]. we separate the nonconformity score each time r. which facilitates the reallocation the risk each time step. Similarly, given an allocated risk a, for future time r, the random variables R. R(1) .... R(K) are exchangeable and the prediction region with coverage guarantee is derived follows.
| Y-lell
TP
< C.
Cl-aa
2
Oty
(70a)
t=0 0
Trit
Cl-a. Quantilei-a- (R(1),
R(K) 0c)
(70b)
Based on the (1 0, cr)-ecvee prediction region defined in (7a).. the individual chance constraint P (c(z,, Y,) 2 0} a, can be reformulated the following lemma.
34

---

Lemma K.1 (chance constraint) If Assumption 3.7 holds, the constraint function e is L-Lipschitz
continuous and maxos=ge(c(xs, Yoto) - T 2 0 is satisfied where Crp -yyrt C7--0
then the individual chance constraint P(c(z,, Y,) 2 0}
a, is satisfied.
Proof: According to the (1 - a, a.vo-reeaaage prediction region defined in 70a we can obtain that
||, Yilll
P
21-a,
(71)
Tte
According to that fact l S T and the definition, we have the following inequality.
Ir. & - - 0 P
n [Ix Sulll - cl
2 Or
(72)
Based on (72). we can further obtain the following inequality.
P CcT - I|, - Y- 2 o) 2 1 ar, Vs = 0,..,
(73)
Note that the function is L-Lipschitz continuous, the following inequality is obtained.
||< Yr) - c(Fr, [1) II = L||Y, - Yrlell = c(E,, Y,) 2 c(ær, F11)) L||,, - Yell If the constraint maxo fc(xr, 771() - -C 2 is satisfied, we have the following inequality. O.... t c(x,, Y,) 2 LC - ||Y, Yr1=ll)
(74)
(75)
By combining 73 and 75 we ultimately obtain P(c(2,, Y,) 03 0
Finally, is sufficient replace constraint (Iee) in the TO problem (11) with maxos=si Y+1) "C 2 O. For the posterior probability calculation and risk allocation method, since we have separated the nonconformity score at each time step our proposed framework re- mains fully applicable. It is important to note that, as described in Section C the constraint maxos=5((c(2,, Y=1) - "C } 2 introduces mixed-integer variables into the TO problem. which significantly increases the solution time, especially when using the IRA method and dealing with nonlinear systems.
L Limitations
The main limitation lies in the reliance of the proposed method on the size of the calibration dataset. As previously mentioned, to ensure coverage guarantees within the closed-loop framework, the calibration dataset needs to be split into two parts: one for forward computation prediction regions and the other for backward computation of posterior probabilities. This requirement results in the proposed method needing a larger calibration dataset compared to standard CP methods. However, extensive data can be sourced from advanced high fighliidliity simulators robotic applications like autonomous vehicles, where datasets are increasingly accessible. Thus we believe that the reliance on data quantity will not present substantial challenge.
M Broader Impacts
This work proposes a novel Fb-CP framework for trajectory optimization under uncertainty, with provable safety guarantees and adaptive risk control. The method has potential positive societal impacts on safety-critice applications such as autonomous vehicles, robotics, and disaster response, by improving the reliability and efficiency of decision-makinn under uncertainty. It also contributes to the development of trustworthy Al through its theoretical guarantees and feedback beedback-based adaptability.
However, potential negative societal impacts include misuse in high-risk or adversarial settings (e.g., autonomous weapons), privacy concerns from trajectory data collection, and overconfident decisions if the system fails under distribution shift. To mitigate these risks, future deployments should incorporate rigorous validation, privacy safeguards, and oversight mechanisms to ensure safe and ethical use of the proposed method.
35


Reviews:
[{"review_id": "rkaGMdwRCT", "reviewer": "Reviewer_62Tj", "summary": "This 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.", "questions": "## 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?", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "## 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.", "was_revised": true, "final_justification": "I think that this is technically solid work introducing an uncertainty-aware algorithm for trajectory optimization that utilizes conformal prediction. For this reason, I would argue for the acceptance of the paper.", "rebuttal": "Thank you very much for your valuable comments, which have helped us to improve the quality of this manuscript.\n\n>**The 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>\n>**The 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\nThank you very much for your kind recognition of the proposed method. First, we would like to clarify that our method is specifically designed for Trajectory Optimization (TO) tasks, where the problem dimension is inherently low (typically 2D or 3D) due to the physical configuration of robots and vehicles. In such settings, Conformal Prediction (CP) is practical and effective.\n\nSecond, while our problems are in low-dimensional spaces (by the nature of TO), the experimental evaluation is actually not limited. In the original paper (Appendix C), we conducted extensive experiments on three distinct dynamical systems—a kinematic vehicle [1], a 3D linear quadrotor [2], and a dynamic bicycle model [3]—and compared against four state-of-the-art baselines: Conformal Control (CC) [1], ACI for Motion Planning (ACI-MP) [2], Recursively Feasible CP (RF-CP) [4], and Sequential CP (S-CP) [5]. In all cases, our proposed Fb-CP framework consistently demonstrates superior performance.\n\nMoreover, to further showcase the potential of our method in real-world problems, we have additionally evaluated it on the real-world Stanford Drone Dataset [6]. As shown in Table R1 below, Fb-CP with IRA outperforms all baselines and achieves the lowest average cost. In particular, it reduces the cost by at least 46% compared to S-CP. Note that RF-CP is not included as a baseline because its formulation introduces mixed-integer variables through nonconformity score definitions, which—when combined with the nonlinear vehicle dynamics—leads to highly complex trajectory optimization problems with prohibitively long computation times. These results support the applicability of our method to real-world problems.\n\n*Table R1: Average Cost of Different Methods on the Real-World Stanford Drone Dataset*\n| $\\alpha$ / $\\eta$ | CC | ACI-MP | S-CP | Fb-CP-ARA | Fb-CP-IRA |\n| :---- | :----: | :----: | :----: | :----: | :----: |\n| 0.05/1000 | 95.51 | 40.02 | 38.58 | 30.58 | 20.81 |\n| 0.1/500 | 77.68 | 35.94 | 34.75 | 27.52 | 18.52 |\n| 0.15/100 | 41.27 | 32.57 | 30.48 | 23.75 | 15.58 |\n| 0.2/50 | 38.73 | 29.89 | 28.59 | 21.25 | 14.02 |\n\nFurthermore, to directly evaluate the effectiveness of our framework in higher-dimensional settings, we conducted an additional numerical case study involving a generic dynamical system under a shrinking-horizon setup with higher-dimensional state spaces. As shown in Table R2 below, our feedback-based CP framework consistently outperforms baseline methods while maintaining safety, even in these higher-dimensional scenarios. These results validate the applicability and effectiveness of our approach in high-dimensional problems.\n\n*Table R2: Average Cost and Collision Avoidance Rate of Different Methods ($\\\\alpha=0.2$) under Varying Dimensions*\n|  | Dimension | S-CP | Fb-CP-ARA | Fb-CP-IRA |\n| :- | :-: | :-: | :-: | :-: |\n| **Average cost** | 5 | 44.52 | 39.67 | 30.25 |\n|  | 10| 72.66 | 65.33 | 60.18 |\n| **Collision avoidance rate** | 5 | 92.5% | 92.1% | 93.5% |\n|  | 10 | 91.2% | 93.5% | 91.4% |\n\n>**l.116 - dynamica - dynamics l.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\nWe sincerely thank the reviewer for pointing out the typo on line 116 (\"dynamica\")—we have corrected it to \"dynamics\" in the revised manuscript.\n\nRegarding the comment on line 133, the reviewer is absolutely correct. In Assumption 3.1, the calibration dataset $D_{cal}$ is generated using the true underlying obstacle dynamics, not the learned model $g(\\cdot)$. As clarified, $g(\\cdot)$ is trained from a separate dataset and only used at test time to generate predictions $\\hat{Y}_{\\tau|t}$. We have updated the statement in Assumption 3.1 accordingly to avoid confusion.\n\n>**l.eq3, please fix the formatting here.**\n\nThank you for your comment. We have revised the formatting of eq. (3) to improve clarity and readability. In particular, the constraints and their applicable time indices are now properly aligned and consistently presented.\n\n>**l.152 Shouldn't the superscript start from 0?**\n\nThank you for your comment. In our manuscript, the superscript $(i)$ is used to index the calibration samples, while the symbol without a superscript (i.e., $R$) refers to the test data point. Using superscript $(0)$ for the test point is certainly a valid and alternative convention. Following the reviewer’s suggestion, we have updated the notation in the revised manuscript to use $R^{(0)}$ for the test point, and $R^{(1)},…,R^{(N)}$ for the calibration set, for improved clarity and consistency.\n\n>**l.168 - Explain the abbreviation Fb-CP.**\n\nThank you for pointing this out. We have updated the manuscript to clarify the meaning of “Fb-CP” as “**F**eed**b**ack-based **C**onformal **P**rediction” at line 168.\n\n>**eq. 15a - 15d, fix alignment**\n\nThank you for your comment. Similar to our revision of eq. (3), we have fixed the formatting of eq. (15) in the revised manuscript to improve alignment.\n\n>**For 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\nWe fully agree with the reviewer that full observability of obstacle dynamics is unrealistic in real-world scenarios, especially in human-in-the-loop systems such as traffic. Our framework is in fact designed to accommodate such partial observability.\n\nFirst, in our formulation, $Y_{t}$ does not need to represent the complete internal dynamics of obstacles. It can simply represent any observable states relevant to safety—such as position and velocity—which are commonly available from sensors. The learned prediction model $g(\\cdot)$ can be interpreted more precisely as a function of past observations within a time window:\n\n$Y_{t} = g(Y_{t-h},...,Y_{t-1}) + \\omega_{t}$\n\nwhere $h$ is the window length. This allows for the use of flexible, data-driven time-series forecasting models such as Social-LSTM [7], Trajectron++ [8] or AgentFormer [9]. Therefore, our method can naturally operate under partial observability of the obstacle state.\n\nSecond, while partial observability does not invalidate the safety guarantees provided by conformal prediction, it does influence performance. Specifically, limited observability may increase the prediction error $\\omega_{t}$, leading to wider conformal prediction regions. This increased conservatism in the confidence region can result in less efficient trajectories, as the planner needs to avoid these larger obstacle position confidence regions\n\nIn summary, our framework remains applicable and safe under partial observability, though its performance may degrade gracefully depending on the quality of the available observations and the predictive model used. We have also validated the effectiveness of our approach on the real-world Stanford Drone Dataset [6], demonstrating its practical applicability under realistic observation conditions.\n\n[1] *Jordan Lekeufack, Anastasios N Angelopoulos, Andrea Bajcsy, Michael I Jordan, and Jitendra Malik. Conformal decision theory: Safe autonomous decisions from imperfect predictions. IEEE International Conference on Robotics and Automation (ICRA), 2024.*\n\n[2] *Anushri Dixit, Lars Lindemann, Skylar X Wei, Matthew Cleaveland, George J Pappas, and Joel W Burdick. Adaptive conformal prediction for motion planning among dynamic agents. Learning for Dynamics and Control Conference, 2023.*\n\n[3] *Astghik Hakobyan and Insoon Yang. Wasserstein distributionally robust motion control for collision avoidance using conditional value-at-risk. IEEE Transactions on Robotics, 2021.*\n\n[4] *Charis Stamouli, Lars Lindemann, and George Pappas. Recursively feasible shrinking-horizon mpc in dynamic environments with conformal prediction guarantees. Learning for Dynamics and Control Conference, 2024.*\n\n[5] *Lars Lindemann, Matthew Cleaveland, Gihyun Shim, and George J Pappas. Safe planning in dynamic environments using conformal prediction. IEEE Robotics and Automation Letters, 2023.*\n\n[6] *Alexandre Robicquet, Amir Sadeghian, Alexandre Alahi, and Silvio Savarese. Learning social etiquette: Human trajectory understanding in crowded scenes. European Conference on Computer Vision (ECCV), 2016.*\n\n[7] *Alexandre Alahi, Kratarth Goel, Vignesh Ramanathan, Alexandre Robicquet, Li Fei-Fei, and Silvio Savarese. Social lstm: Human trajectory prediction in crowded spaces. IEEE Conference on Computer Vision and Pattern Recognition, 2016.*\n\n[8] *Tim Salzmann, Boris Ivanovic, Punarjay Chakravarty, and Marco Pavone. Trajectron++: Dynamically-feasible trajectory forecasting with heterogeneous data. European Conference on Computer Vision (ECCV), 2020.*\n\n[9] *Ye Yuan, Xinshuo Weng, Yanglan Ou, and Kris M Kitani. Agentformer: Agent-aware transformers for socio-temporal multi-agent forecasting. Proceedings of the IEEE/CVF international conference on computer vision. 2021.*"}, {"review_id": "9ssdTSJZu3", "reviewer": "Reviewer_PHGF", "summary": "The 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.", "questions": "Please check the weaknesses for detail and address the points there.", "limitations": "Yes", "rating": 4, "confidence": 5, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 2, "clarity": 3, "significance": 2, "originality": 3, "strengths_and_weaknesses": "Strengths:\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.", "was_revised": true, "final_justification": "This paper proposes a Feedback-Based Conformal Prediction (Fb-CP) framework to improve trajectory optimization (TO) in uncertain environments. The rebuttal has answered my questions, so I keep my relatively positive rating.", "rebuttal": "Thank you very much for your valuable comments, which have helped us to improve the quality of this manuscript.\n\n>**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).**\n\nWe thank the reviewer for raising this important point regarding computation time. We would like to emphasize that our contribution is a general Fb-CP framework for trajectory optimization, which supports **multiple risk allocation strategies**, including both **IRA** and **ARA**.\n\nWhile Fb-CP-IRA indeed achieves significant performance improvement by iteratively optimizing risk allocations, we fully agree that its computational demand may not be suitable for very high-frequency control. **For such settings, our framework also supports the non-iterative Fb-CP-ARA, which is extremely lightweight.** As shown in Table 2 of the original manuscript, Fb-CP-ARA incurs virtually no additional computation time compared to S-CP, yet still achieves at least a 7.21% reduction in average cost.\n\nFurthermore, to balance performance and computation efficiency, we propose a hybrid scheme (Appendix F) in which IRA is used only at the initial time step (offline), and ARA is used for all online steps. This approach eliminates online overhead of IRA, yet achieves a 66.56% cost reduction over pure ARA. The user can therefore flexibly select between IRA, ARA, or the hybrid variant, depending on their computational budget and real-time requirements.\n\nIn summary, our framework allows users to flexibly choose between IRA, ARA, or a hybrid approach based on the computational constraints and real-time requirements of their specific application. In particular, ARA provides a fast and practical solution, incurring almost no extra runtime compared to baselines, while still benefiting from feedback-based prediction adjustment.\n\n>**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.**\n\nWe thank the reviewer for highlighting this important concern. We agree that integrating feedback-based conformal prediction with trajectory optimization introduces additional system complexity compared to sequential CP methods. However, we believe the significant and consistent performance improvements justify this complexity.\n\nAs shown in Appendix C of the original manuscript, **we conducted extensive experiments across three representative dynamical systems**—a kinematic vehicle [1], a 3D linear quadrotor [2], and a dynamic bicycle model [3]—and **compared against four state-of-the-art baselines**: CC [1], ACI-MP [2], RF-CP [4], and S-CP [5]. The proposed Fb-CP-ARA consistently achieved substantial cost reductions over the S-CP baseline, including:\n- 78.5% reduction on the kinematic vehicle model,\n- 58.5% reduction on the 3D quadrotor model,\n- 40.2% reduction on the dynamic bicycle model.\n\nTo show performance improvement in real-world problems, **we evaluated our method on the Stanford Drone Dataset [6]**. As shown in Table R1 below, Fb-CP-IRA outperforms all baselines and achieves the lowest average cost. In particular, it reduces the cost by at least 46% compared to S-CP. Note that RF-CP is not included as a baseline due to its reliance on mixed-integer variables, which combined with nonlinear dynamics, results in intractable trajectory optimization.\n\n*Table R1: Average Cost of Different Methods on the Stanford Drone Dataset*\n| $\\alpha$ / $\\eta$ | CC | ACI-MP | S-CP | Fb-CP-ARA | Fb-CP-IRA |\n| :- | :-: | :-: | :-: | :-: | :-: |\n| 0.05/1000 | 95.51 | 40.02 | 38.58 | 30.58 | 20.81 |\n| 0.1/500 | 77.68 | 35.94 | 34.75 | 27.52 | 18.52 |\n| 0.15/100 | 41.27 | 32.57 | 30.48 | 23.75 | 15.58 |\n| 0.2/50 | 38.73 | 29.89 | 28.59 | 21.25 | 14.02 |\n\nFinally, to reduce the engineering effort and support adoption, we will release the full implementation of our method upon paper acceptance. We hope this will make our framework easier to use and build upon for both researchers and practitioners.\n\n>**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.**\n\nWe thank the reviewer for the insightful comment. We would like to clarify that the core novelty of our work lies in the Feedback-based Conformal Prediction (Fb-CP) framework, rather than in the design of a specific trajectory predictor. **Our method can work with any trajectory prediction module, including classical models or modern deep learning-based predictors.**\n\nWe fully agree that the quality of the trajectory predictor can affect the performance of the resulting trajectory optimization, as it influences the prediction error and thus the conservativeness of the conformal prediction region. **This sensitivity, however, is inherent to all CP-based decision-making methods, not unique to our framework.** As shown in Table 3 of the original manuscript, when using the same predictor, our method consistently outperforms existing baselines across, which highlights the strength of our feedback mechanism regardless of the predictor accuracy.\n\nTo further validate that our performance gains are independent of the choice of predictor, we conducted additional experiments comparing our method and S-CP under different predictors. As shown in Table R2 below, when switching to more advanced predictors Trajectron++ [7] and AgentFormer [8], both our method and the baseline see improved performance, but our framework continues to outperform the baseline under all tested predictors.\n\n*Table R2: Average Cost of Different Methods using Different Trajectory Predictors*\n|  | S-CP | Fb-CP-ARA | Fb-CP-IRA |\n| :- | :-: | :-: | :-: |\n| Social LSTM [9] | 17.30 | 15.49 | 2.973 |\n| Trajectron++ [7] | 13.58 | 10.58 | 1.809 |\n| AgentFormer [8] | 12.83 | 9.712 | 1.722 |\n\nThis confirms that while the choice of predictor affects the absolute performance—as expected—our Fb-CP framework consistently maintains higher performance compared to the baseline method regardless of predictor quality. Additionally, in practice, **our framework can be directly integrated with more sophisticated predictors** as needed in real-world applications, enabling further performance gains without changing the core structure of our algorithm.\n\n>**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\nThank you very much for your valuable comment. Frist, we would like to clarify that in all our experiments, the total amount of calibration data used by Fb-CP and the baselines is exactly the same. Although the split reduces the number of samples used in each stage (forward prediction or backward feedback), the performance gains brought by feedback-based adjustment significantly outweigh the minor statistical loss from halving the calibration size.\n\nTo directly address the reviewer’s concern, we conducted additional experiments under a data-scarce regime with only $N=400$ calibration samples. As shown in Table R3 below, although all methods see a decrease in performance under limited data, Fb-CP still achieves substantially lower cost than S-CP and remains safety. This highlights the robustness and practical effectiveness of our method, even when the calibration dataset is small.\n\n*Table R3: Average Cost of Different Methods with the Calibration Data Size of $N=400$*\n|  | S-CP | Fb-CP-ARA | Fb-CP-IRA |\n| :- | :-: | :-: | :-: |\n| Average cost | 18.03 | 16.82 | 3.521 |\n| Average computation time | 0.078 | 0.081 | 0.122 |\n| Collision avoidance rate | 88.9% | 89.5% | 91.7% |\n\n[1] *Conformal decision theory: Safe autonomous decisions from imperfect predictions. ICRA, 2024.*\n\n[2] *Adaptive conformal prediction for motion planning among dynamic agents. L4DC, 2023.*\n\n[3] *Wasserstein distributionally robust motion control for collision avoidance using conditional value-at-risk. IEEE T-RO, 2021.*\n\n[4] *Recursively feasible shrinking-horizon mpc in dynamic environments with conformal prediction guarantees. L4DC, 2024.*\n\n[5] *Safe planning in dynamic environments using conformal prediction. IEEE R-AL, 2023.*\n\n[6] *Learning social etiquette: Human trajectory understanding in crowded scenes. ECCV, 2016.*\n\n[7] *Trajectron++: Dynamically-feasible trajectory forecasting with heterogeneous data. ECCV, 2020.*\n\n[8] *Agentformer: Agent-aware transformers for socio-temporal multi-agent forecasting. ICCV. 2021.*\n\n[9] *Social lstm: Human trajectory prediction in crowded spaces. CVPR, 2016.*"}, {"review_id": "kjjqOoMVgm", "reviewer": "Reviewer_n484", "summary": "The 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.", "questions": "- Why not using more powerful trajectory predictors for $g()$ in eqution 2?", "limitations": "The 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.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 2, "originality": 3, "strengths_and_weaknesses": "### 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.", "was_revised": true, "final_justification": "Overall the paper has provide ample amount of theoratical groundings, the author provided more follow up experiments that address the reviewer's concern, hence raised the score.", "rebuttal": "Thank you very much for your valuable comments, which have helped us to improve the quality of this manuscript.\n\n>**Weakness 1**\n\nWe thank the reviewer for this valuable suggestion. To address the concern regarding accessibility and clarity, especially for readers who may not have a background in conformal prediction, we have made the following two major revisions.\n\nFirst, **we reorganized the structure of the main text and added informal, intuitive explanations after presenting mathematically dense definitions, theorems, or derivations**. This is intended to guide the reader through the logic of the proposed framework more smoothly, and to convey the key ideas in a more approachable way before introducing formal notation. Specifically:\n- After Lemma 4.1, we added an explanatory sentence directly to provide readers with an intuitive understanding of the lemma. The inserted text reads \n>“For a general collision avoidance chance constraint of the form $\\\\mathbb{P} \\\\{ \\\\Vert x_{\\\\tau} - Y_{\\\\tau} \\\\Vert - r \\\\geq 0 \\\\} \\\\geq 1 - \\\\alpha_{\\\\tau}$, Lemma 4.1 effectively converts it into a deterministic constraint that requires the distance between $x\\_{\\\\tau}$ and the predicted location $\\\\hat{Y}\\_{\\\\tau|t}$ to exceed the inflated radius derived from the prediction error, i.e., $\\\\Vert x\\_{\\\\tau} - \\\\hat{Y}\\_{\\\\tau|t} \\\\Vert - r - C\\_{\\\\tau|t}^{1-\\\\alpha\\_{\\\\tau}}  \\\\geq 0$.\"\n- After Lemma 4.2, we inserted a clarification sentence to convey the core idea behind the lemma in more intuitive terms. The newly added explanation states:\n>“Lemma 4.2 essentially computes the posterior probability by evaluating the fraction of calibration set $D\\_{cal}^{2}$ samples whose actual trajectories would collide with the given realized position $x\\_{\\\\tau}^{∗}$.”\n- After Lemma 5.1, we added an explanatory sentence to clarify the implication of the lemma's monotonicity result. Specifically, the inserted text reads:\n>“The monotonicity of $J^{∗}$ in Lemma 5.1 shows that increasing the allocated risk $\\\\alpha\\_{\\\\tau}$ at any time step will strictly decrease the optimal cost of the trajectory optimizer. This insight enables our risk reallocation strategy: by reducing redundant risk at inactive time steps and reallocating it to active ones, we can lower the overall optimal cost.”\n- After Lemma 5.2, we added Remark 5.4 to provide an insightful explanation of the relationship between Lemma 5.2 and Lemma 4.2.\n>“**Remark 5.4.** One may notice that the calculation of $\\\\beta\\_{\\\\tau}$ in Lemma 4.2 is similar to the computation of $\\\\widetilde{\\\\alpha}^{n}_{\\\\tau}$ in Lemma 5.2. This observation is correct. The key difference between the two lies in that the former utilizes the dataset $D\\_{cal}^{2}$ independent with $D\\_{cal}^{1}$ to achieve the coverage guarantee for $\\\\beta\\_{\\\\tau}$. By contrast, as a step in solving (11), the latter does not need to consider the coverage guarantee and thus directly uses $D\\_{cal}^{1}$. The use of different datasets results in the former providing probabilistic guarantee, while the latter achieves deterministic guarantee $(\\\\underline{\\\\alpha}\\_{\\\\tau}^{n} \\\\leq \\\\alpha\\_{\\\\tau}^{n})$.”\n\nWe hope that these additions help readers outside the conformal prediction community understand both the motivation and the mechanism of our method more easily.\n\nSecond, to further improve readability, **we moved some non-core theoretical results to the appendix**:\n- Lemma 3.2, which is a direct application of a lemma from [7], has been relocated to the appendix. While useful for completeness, it is not essential for understanding or implementing our proposed method.\n- Corollary 4.3, which provides a theoretical performance guarantee but is not directly involved in the operation of the framework, has also been moved to the appendix.\n\nWe believe that these changes collectively improve the clarity, accessibility, and structure of the paper, and we hope they address the reviewer’s concerns.\n\n>**Weakness 2**\n\nWe thank the reviewer for this insightful suggestion. We fully agree that the feedback-based CP framework is the core contribution of our work, and that the main paper would benefit from a more complete presentation of both the algorithm and supporting numerical results.\n\nDue to the strict 9-page limit of the NeurIPS main paper submission format, in the original manuscript, we placed the pseudocode and explanation of the proposed algorithm in Appendix B, and included only abbreviated experimental results in the main text, with full results provided in Appendix C. Fortunately, the final camera-ready version permitted by NeurIPS allows extending the main paper to 10 pages. As a result, following the reviewer’s suggestion, we revise the manuscript by\n- Move the pseudocode and its explanation from Appendix B into the main paper;\n- Move Table 3 from Appendix C.2 into the main text to provide more detailed numerical results and better support the theoretical analysis.\n\nWe hope these updates will enhance the completeness and clarity of the main paper and better highlight the theoretical and practical contributions of our framework.\n\n>**Questions**\n\nWe thank the reviewer for the helpful comment. We would like to emphasize that the proposed Fb-CP framework is general and modular, and is not restricted to the Social LSTM [6] used in our original experiments. **The trajectory predictor $g(\\\\cdot)$ can be replaced with any more powerful or complex predictor.** Indeed, using stronger predictors such as [1], [2], [4], or [5] should lead to improved prediction accuracy and therefore better downstream performance in trajectory optimization.\n\n**In the revised manuscript, we made this flexibility explicit by updating discussion and citing [1], [2], [4], and [5] around learning of $g(\\\\cdot)$ to include**\n> “*Naturally, readers may choose to adopt more powerful predictors for improved accuracy, such as Salzmann et al. [2020], Yuan et al. [2021], Salzmann et al. [2023], Yuan and Kitani [2020].*”\n\nFinally, to further validate this point, we conducted additional experiments using different predictors within our framework. These results are presented in a later section of our response and demonstrate that Fb-CP maintains superior performance regardless of the specific predictor used, and benefits even more from stronger predictors.\n\n>**Limitations**\n\nWe sincerely thank the reviewer for this helpful suggestion. In response, we have conducted two additional sets of experiments to further validate the generality and real-world applicability of the proposed Fb-CP framework: (1) We evaluated our framework using more powerful trajectory predictors; (2) We applied our method to a real-world dataset, the Stanford Drone Dataset [3]. These experiments are detailed below.\n\n**(1) Using more advanced trajectory predictors.**\n\nWe evaluated the performance of both our method (Fb-CP with ARA) and the baseline (S-CP) under three different prediction models:\n- The original Social LSTM predictor used in the original manuscript;\n- Two more sophisiticated algorithm, Trajectron++ [1] and AgentFormer[2].\n\nAs shown in Table R1 below, the results demonstrate that more powerful trajectory predictors can enhance the performance of all methods by providing more accurate obstacle forecasts, and that Fb-CP consistently maintains superior performance over the baseline under both simple and advanced predictors.\n\n*Table R1: Average Cost of Different Methods using Different Trajectory Predictors*\n|  | S-CP | Fb-CP-ARA | Fb-CP-IRA |\n| :- | :-: | :-: | :-: |\n| Social LSTM (Original) | 17.30 | 15.49 | 2.973 |\n| Trajectron++ (Added) | 13.58 | 10.58 | 1.809 |\n| AgentFormer (Added) | 12.83 | 9.712 | 1.722 |\n\n**(2) Applying to a real-world dataset.**\n\nWe further tested our method on the Stanford Drone Dataset. As shown in Table R2 below, Fb-CP-IRA outperforms all baselines and achieves the lowest average cost. In particular, it reduces the cost by at least 46% compared to S-CP. These results support the applicability of our method to practical, real-world environments with complex agent interactions.\n\n*Table R2: Average Cost of Different Methods on the Stanford Drone Dataset*\n| $\\alpha$ / $\\eta$ | CC | ACI-MP | S-CP | Fb-CP-ARA | Fb-CP-IRA |\n| :- | :-: | :-: | :-: | :-: | :-: |\n| 0.05/1000 | 95.51 | 40.02 | 38.58 | 30.58 | 20.81 |\n| 0.1/500 | 77.68 | 35.94 | 34.75 | 27.52 | 18.52 |\n| 0.15/100 | 41.27 | 32.57 | 30.48 | 23.75 | 15.58 |\n| 0.2/50 | 38.73 | 29.89 | 28.59 | 21.25 | 14.02 |\n\nWe hope that these additional experiments effectively address the reviewer’s suggestions and demonstrate the flexibility and robustness of our proposed framework. If the reviewer has any further questions or concerns, we would be happy to clarify them.\n\n[1] *Tim Salzmann, Boris Ivanovic, Punarjay Chakravarty, and Marco Pavone. Trajectron++: Dynamically-feasible trajectory forecasting with heterogeneous data. European Conference on Computer Vision (ECCV), 2020.*\n\n[2] *Ye Yuan, Xinshuo Weng, Yanglan Ou, and Kris M Kitani. Agentformer: Agent-aware transformers for socio-temporal multi-agent forecasting. Proceedings of the IEEE/CVF international conference on computer vision. 2021.*\n\n[3] *Alexandre Robicquet, et al. Learning social etiquette: Human trajectory understanding in crowded scenes. European Conference on Computer Vision (ECCV), 2016.*\n\n[4] *Tim Salzmann, Hao-Tien Lewis Chiang, Markus Ryll, Dorsa Sadigh, Carolina Parada, and Alex Bewley. Robots that can see: Leveraging human pose for trajectory prediction. IEEE Robotics and Automation Letters, 2023.*\n\n[5] *Ye Yuan and Kris M. Kitani. Diverse trajectory forecasting with determinantal point processes. International Conference on Learning Representations, 2020.*\n\n[6] *Alexandre Alahi, et al. Social lstm: Human trajectory prediction in crowded spaces. IEEE Conference on Computer Vision and Pattern Recognition, 2016.*\n\n[7] *Ryan J Tibshirani, et al. Conformal prediction under covariate shift. Advances in neural information processing systems, 2019.*"}, {"review_id": "5g8wRBtoBw", "reviewer": "Reviewer_4FW9", "summary": "The 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.", "questions": "- 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?", "limitations": "- 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.", "rating": 4, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 2, "significance": 3, "originality": 2, "strengths_and_weaknesses": "The 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?”", "was_revised": true, "final_justification": "The authors have addressed my concerns. I have updated my score.", "rebuttal": "Thank you very much for your valuable comments, which have helped us to improve the quality of this manuscript.\n\n>**Weakness 1.**\n\nWe thank the reviewer for this important and insightful comment. Following your suggestion, we carefully revisited the theoretical assumptions and proofs throughout the paper, and found that **the i.i.d. assumption on prediction errors $\\\\omega\\_{t}$ is indeed unnecessarily strong and can be significantly relaxed, as we now explain.**\n\nSpecifically, it is not necessary to assume that the prediction errors at different time steps within the same trajectory are i.i.d.. Instead, we only require that the prediction errors at the same time step $\\tau$ across different trajectories (i.e., test and calibration) are i.i.d.. This weaker condition holds under the following setup: all trajectories are generated by sampling their initial state $Y\\_{0}^{(i)}$ from the same distribution $\\\\mathcal{D}$ and then evolving using a common dynamics $Y\\_{t+1}=g(Y\\_{t})+\\\\omega\\_{t}$. Importantly, this places no restriction on the temporal dependence within the prediction errors $\\\\omega\\_{1} ,…, \\\\omega\\_{T−1}$ along a single trajectory. Therefore, errors that depend on past quantities are fully allowed within each trajectory, addressing the reviewer's concern.\n\nTo further clarify, the reason this weaker assumption is sufficient lies in the requirement of exchangeability when applying conformal prediction. In the proof of Lemma 4.2, what we need is for the nonconformity scores\n$$S\\_{\\\\tau}  = c(x\\_{\\\\tau}^{\\*},\\\\hat{Y}\\_{\\\\tau|\\\\tau-1} + \\\\omega_{\\\\tau}), \\\\quad S\\_{\\\\tau}^{(i)} = c(x\\_{\\\\tau}^{\\*},\\\\hat{Y}\\_{\\\\tau|\\\\tau-1} + \\\\omega\\_{\\\\tau}^{(i)}) \\\\quad \\\\forall i=K+1,...,K+L$$\nto be exchangeable. This is satisfied as long as the prediction errors at time $\\\\tau$ across trajectories are i.i.d.—**no assumption is needed on how $\\\\omega$ evolves over time within the same trajectory**.\n\nAs for Appendix D, we acknowledge that it presents only a practical approach to handling scenarios where test and calibration trajectories may come from different dynamics models. This is meant to be an empirical workaround when the required exchangeability is violated. However, we agree that this is not a substitute for a rigorous theoretical extension, and have clarified this point more explicitly in the revised version.\n\nWe also emphasize that the obstacle trajectories used in our experiments were either generated using the trajectory simulator TrajNet++ [1] or taken from the real-world Stanford Drone Dataset [2], both of which are temporally correlated time-series data. Our framework achieved strong performance while maintaining safety, further validating its practical applicability under realistic conditions.\n\nWe sincerely thank the reviewer again for raising this question. It helped us **identify and remove an unnecessarily strong assumption**, thus improving both the clarity and generality of the theoretical framework.\n\n>**Weakness 2.**\n\nWe thank the reviewer for pointing out this important concern. We confirm that **our framework always guarantees that the entire planned trajectory satisfies the overall collision risk tolerance**, even when using adjusted risk allocations based on posterior probability. Below, we explain the guarantee in more detail.\n\nAt each time step $t$, the realized state up to that point, $x\\_{1}^{∗},…,x\\_{t}^{∗}$, is already available. Using Lemma 4.1, we can compute the posterior probabilities $\\\\beta\\_{1},…,\\\\beta\\_{t}$ for these realized positions. When optimizing the remainder of the trajectory for future steps $\\\\tau=t+1,…,T$, we allow risk reallocation over these future time steps by adjusting $\\\\alpha\\_{t+1},…,\\\\alpha\\_{T}$. It is important to note that the total risk allocation always satisfies $\\\\sum_{\\\\tau=1}^{t} \\\\beta\\_{\\\\tau} + \\\\sum_{\\\\tau=t+1}^{T} \\\\alpha\\_{\\\\tau} \\\\leq \\alpha$, where $\\\\alpha$ is the overall risk tolerance. This ensures that the cumulative collision risk over the full trajectory remains within $\\\\alpha$, regardless of how the remaining risk is redistributed. For $\\\\tau>t$, we again apply Lemma 4.1 to convert the allocated risk $\\\\alpha\\_{\\\\tau}$ into a deterministic constraint. These constraints are enforced during optimization, ensuring that each planned state $x\\_{\\\\tau}^{∗}$ has a collision probability no greater than $\\\\alpha\\_{\\\\tau}$.\n\nTherefore, even with dynamically adjusted coefficients, the entire trajectory remains safe under the risk threshold. We acknowledge that this guarantee was not made sufficiently explicit in the original paper. In the revised manuscript, we have now clearly stated this theoretical guarantee to improve clarity.\n\n>**Question 1.**\n\nWe thank the reviewer for these thoughtful questions. We believe there may be a misunderstanding regarding our Fb-CP framework. **The primary function of our posterior recalibration is not to correct for potential over-conservativeness of CP sets due to small $n$, but rather to exploit information that becomes available during execution—specifically, the realized state $x^{*}$—to tighten the constraint violation probability estimate.**\n\nTo clarify this with an example, we consider a specific chance constraint $\\\\mathbb{P} \\\\{ \\\\Vert x\\_{\\\\tau}-Y\\_{\\\\tau}\\Vert -r\\\\}\\\\geq 1-\\\\alpha_{\\\\tau}$ . During the forward conformal step, Lemma 4.1 converts this chance constraint into a deterministic constraint of the form $\\\\Vert x\\_{\\\\tau}-\\\\hat{Y}\\_{\\\\tau|t}\\Vert -r-C\\_{\\\\tau|t}^{1-\\\\alpha_{\\\\tau}} \\\\geq 0$, where $C\\_{\\\\tau|t}^{1-\\\\alpha_{\\\\tau}}$ is the conformal correction derived from the calibration set $D\\_{cal}^{1}$. This constraint is then enforced in the trajectory optimizer, resulting in a realized state $x\\_{\\\\tau}^{∗}$ that satisfies the above chance constraint. According to standard CP theory, this guarantee is essentially achieved by ensuring that at least a $1-\\\\alpha\\_{\\\\tau}$ fraction of samples in $D\\_{cal}^{1}$ fall within the region $\\\\{p : \\Vert p-\\\\hat{Y}\\_{\\\\tau|t} \\Vert - r - C\\_{\\\\tau|t}^{1-\\\\alpha\\_{\\\\tau}} < 0 \\\\}$. However, it is important to note that not all samples outside this region would actually result in a collision with the realized state $x\\_{t}^{∗}$. In other words, the original constraint enforces safety without knowledge of the actual realization $x\\_{t}^{∗}$, leading to inherent conservativeness.\n\nOur feedback mechanism (Lemma 4.2) directly addresses this: after observing $x\\_{t}^{∗}$, we evaluate the true violation rate on $D\\_{cal}^{2}$, i.e., the fraction of samples that truly violate $\\\\Vert x\\_{t}^{*} - Y_{t}^{(i)}\\\\Vert-r \\geq 0$, which yields a more precise posterior estimate of the actual risk.\n\nTherefore, the conservativeness of the prior CP constraint arises not from limited sample size $n$, but rather from its inability to incorporate the realized decision $x\\_{\\tau}^{*}$ at planning time. This structural conservativeness remains even when $n$ is large, and our method addresses it directly. The benefit of posterior adjustment is thus present regardless of the dataset size, as it leverages new information unavailable in the original conformal calibration step. We thank the reviewer again for prompting this clarification.\n\n>**Question 2.**\n\nWe thank the reviewer for the constructive feedback. We fully agree that drawing a rigorous performance guarantee solely from Corollary 4.3 is too strong, particularly for small calibration sizes $N$ and $L$. Indeed, the improved performance of our posterior-adjusted method primarily stems from leveraging the realized state, which is not directly captured in Corollary 4.3. Therefore, following the reviewer’s suggestion, we have:\n- Removed the claim that the proposed approach is theoretically no worse than the unadjusted baseline;\n- Retained only the empirical observation that our method tends to improve performance in practice;\n- Moved Remark 4.4 to the appendix, where it serves as a supplemental empirical insight.\n\nWe thank the reviewer again for helping us clarify the limitations of this theoretical interpretation and improve the precision of our manuscript.\n\n>**Minor mistakes.**\n\nThank you for pointing out the typos and raising the insightful questions. We have made the following clarifications and revisions:\n- We have corrected “an feasible” to “a feasible”.\n- We have corrected the notation from \"$c(\\dot)$\" to the correct form \"$c(\\cdot)$\".\n- In our formulation, $x\\_{\\\\tau}^{*}$ denotes the policy at time $\\\\tau$ for the test trajectory $Y$, not for any calibration trajectory $Y^{(i)}$. It is a fixed quantity when computing $S\\_{\\\\tau},S\\_{\\\\tau}^{(1)},...,S\\_{\\\\tau}^{(K+L)}$, and thus does not vary with $(i)$. Therefore, the notation $x\\_{\\\\tau}^{\\*}$ without a superscript $(i)$ is correct in this context.\n\n>**Limitations.**\n\nThank you very much for your valuable comment. We would like to reiterate that the i.i.d. assumption on residual errors $\\omega_{t}$ is stronger than necessary and can in fact be relaxed in our framework.\n\nSpecifically, we do not require the prediction errors $\\\\omega\\_{1},…,\\\\omega\\_{T−1}$ within the same trajectory to be i.i.d.. Instead, we only require that across trajectories, the prediction error at a given time $t$ is i.i.d. between the test and calibration set. This means the residuals can exhibit arbitrary temporal dependence within a single trajectory, making our method broadly applicable to general time-series forecasting settings, including those with strong temporal correlation.\n\nWe have clarified this point in the revised manuscript to avoid confusion. Once again, we sincerely thank the reviewer for highlighting this issue—it helped us further improve the rigor and generality of our assumptions. Should the reviewer have further questions, we would be happy to provide more detail or discussion.\n\n[1] *Human trajectory forecasting in crowds: A deep learning perspective. IEEE T-ITS, 2021.*\n\n[2] *Learning social etiquette: Human trajectory understanding in crowded scenes. ECCV, 2016.*"}]

Author rebuttal:

Meta-review: (a) Summary

This paper proposes a feedback-based conformal prediction (Fb-CP) framework for trajectory optimization under safety constraints in uncertain, dynamic environments. The core idea is to dynamically adjust the conformal prediction regions for future trajectory steps by incorporating information from realized past trajectories. In doing so, the method could improve decision performance while maintaining statistical coverage guarantees. The authors provide theoretical analysis, including convergence guarantees and bounds on posterior violation probability, and compare their method to existing sequential conformal prediction approaches. Experimental results are limited to low-dimensional simulated environments, but they demonstrate improved empirical performance and safety (collision avoidance) over baselines.

(b) Strengths

1. The paper addresses a clear gap in the literature by introducing a closed-loop feedback mechanism into conformal prediction for trajectory optimization, moving beyond the standard static application of CP. The authors provide rigorous theoretical analysis, including coverage guarantees and convergence proofs, which are crucial for safety-critical applications.
2. The method is well-motivated for real-world problems where uncertainty quantification and safety are paramount, such as autonomous driving.
3.  The paper explores different risk allocation strategies (average and iterative) and provides bounds on violation probabilities, showing both theoretical and empirical improvements over prior work.

(c) Weaknesses

- The experiments are restricted to low-dimensional, simulated environments. The applicability to real-world, high-dimensional problems remains untested, and the scalability of the approach is unclear.
- The iterative risk allocation (IRA) variant, while performant, incurs significant computational overhead, which may limit its use in real-time or high-frequency control settings.
- The proposed framework is complex, requiring integration of trajectory optimization, conformal prediction, and risk allocation, which may hinder adoption without clear evidence of substantial performance gains.
- The method assumes access to accurate dynamics models and requires splitting the calibration dataset, increasing data demands. This could be problematic in data-scarce domains.
- The trajectory prediction component could leverage more sophisticated models from recent literature, and experiments on real-world datasets would strengthen the empirical case.

The paper receives positive scores from four reviewers: One reviewer recommends an accept, three reviewers recommend a borderline accept. The reviewers reach a consensus that the paper is a technically solid and original contribution to the intersection of conformal prediction and trajectory optimization. While the experimental validation is limited and the method is complex (complicated), the theoretical insights and the potential for impact in safety-critical applications outweigh these concerns.
Decision: Accept (poster)