arxiv: 2605.07768 · v1 · submitted 2026-05-08 · 📡 eess.SY · cs.LG· cs.SY

Recognition: 1 theorem link

· Lean Theorem

Interactive Trajectory Planning with Learning-based Distributionally Robust Model Predictive Control and Markov Systems

Erik B\"orve, Leo Laine, Morteza Haghir Chehreghani, Nikolce Murgovski

Authors on Pith no claims yet

Pith reviewed 2026-05-11 02:45 UTC · model grok-4.3

classification 📡 eess.SY cs.LGcs.SY

keywords interactive trajectory planningdistributionally robust MPCPAC learningMarkov systemsstochastic model predictive controlagent decision uncertaintyautonomous planning

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The pith

PAC learning combined with distributionally robust optimization lets model predictive control account for errors in learned agent decision distributions during interactive trajectory planning.

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

The paper develops a framework for planning trajectories when other agents' actions are uncertain. It learns a distribution over Markov states from data and then uses distributionally robust optimization to handle the fact that the learned distribution is only approximate. Probably Approximately Correct learning provides bounds on how much the true distribution could differ, which are incorporated into the robust constraints. This creates a tunable method: with few samples the planner is cautious like robust MPC, and with many samples it approaches the performance of stochastic MPC assuming perfect knowledge. A reader would care because it offers a practical way to balance safety and efficiency in uncertain environments like traffic without requiring infinite data.

Core claim

The authors show that PAC learning can be combined with distributionally robust optimization to account for errors induced by learning a decision distribution over Markov states. This yields a DR-MPC framework that interpolates between a fully robust MPC and an SMPC with perfect knowledge depending on the number of available samples.

What carries the argument

The PAC learning-based distributionally robust MPC framework applied to Markov systems for modeling surrounding agents' decisions.

Load-bearing premise

The decisions of surrounding agents can be modeled as a distribution over Markov states that can be learned from samples, with PAC bounds translating directly into distributionally robust constraints without excessive conservatism or loss of real-time feasibility.

What would settle it

A simulation or real-world test in which the realized rate of safety violations exceeds the PAC-derived guarantee when the learned distribution is used inside the distributionally robust constraints would show that the framework does not fully account for learning errors.

Figures

Figures reproduced from arXiv: 2605.07768 by Erik B\"orve, Leo Laine, Morteza Haghir Chehreghani, Nikolce Murgovski.

**Figure 2.** Figure 2: Example of a scenario tree for |Y| = 2. fh : R Nh,x × R Nh,u 7→ R Nh,x . For notational convenience, we concatenate these attributes as x¯ = [x ⊤ e , x ⊤ h ] ⊤, u¯ = [u ⊤ e , u ⊤ h ] ⊤, f = [f ⊤ e , f ⊤ h ] ⊤. Indeed, the aim of the MPC problem is to obtain the egoagent control actions ue. Following the approach in, e.g Schuurmans and Patrinos (2023); Chen et al. (2022), the human control actions are char… view at source ↗

**Figure 3.** Figure 3: Road crossing case. Each color indicates a path [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

read the original abstract

We investigate interactive trajectory planning subject to uncertainty in the decisions of surrounding agents. To control the ego-agent, we aim to first learn the decision distribution and solve a Stochastic Model Predictive Control (SMPC) problem. To account for errors in the learned distribution, we show that it is possible to utilize Probably Approximately Correct (PAC) learning in combination with Distributionally Robust (DR) optimization to obtain a solution which accounts for the errors induced by the learning model. The results indicate that our PAC learning-based DR-MPC framework provides a method to interpolate between a robust MPC and an omnipotent SMPC, based on the available number of samples.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper combines PAC learning with DR-MPC to give a sample-count knob for robustness in interactive trajectory planning, but the technical steps and sampling assumptions need checking.

read the letter

The main point is a framework that learns a distribution over surrounding agents' Markov decisions, then uses PAC bounds inside a distributionally robust MPC to interpolate between a conservative robust controller and a stochastic one that trusts the learned model completely. More samples tighten the ambiguity set and move the solution toward the learned SMPC; fewer samples keep it safer. That tunable property is the cleanest part of the work and directly targets a deployment headache in robotics where you rarely have enough data for perfect models. Modeling agent behavior as Markov chains is a reasonable simplification that keeps the state space manageable for planning. The authors correctly flag that learning error must be handled rather than ignored, and they reach for established PAC and DR tools to do it without inventing new theory from scratch. The combination for this specific interpolation in trajectory planning is not a routine extension, so there is some novelty in the application. The soft spots are in the execution details that are missing from the abstract. No derivations show how the PAC guarantee is turned into a concrete ambiguity set for the MPC problem, so it is unclear whether the resulting optimization stays real-time feasible or just becomes overly conservative. The stress-test concern about non-i.i.d. samples lands as a real issue: because the ego trajectory affects what the other agents do, any collected data is generated under a policy-dependent measure, which standard PAC statements do not cover. If the full paper does not supply a conditional PAC argument or an online learning fix, the claimed interpolation property weakens. Simulations and baseline comparisons are also not described, leaving the indicative results hard to judge. This is for researchers already working on learning-based MPC for autonomous vehicles or robots who want a practical robustness dial. A reader who knows PAC and DR-MPC will see the application value quickly, but practitioners will need the missing proofs and experiments before trying it. The paper deserves a serious referee because the core idea is relevant and grounded in existing theory, even though the authors will have to add the technical steps and address the sampling dependence to make the contribution solid.

Referee Report

3 major / 1 minor

Summary. The manuscript proposes an interactive trajectory planning framework for an ego-agent under uncertainty in surrounding agents' decisions. Surrounding agents are modeled as Markov systems whose decision distributions are learned from data via PAC learning; the learned distribution is then used to construct an ambiguity set for a distributionally robust MPC (DR-MPC) problem. The central claim is that this PAC-DR combination yields a controller that interpolates between a fully robust MPC and an ideal stochastic MPC as a function of the number of available samples, while explicitly accounting for learning error.

Significance. If the technical steps are sound, the work would supply a concrete, sample-size-dependent mechanism for trading off conservatism and performance in interactive settings without requiring an omnipotent model of other agents. No machine-checked proofs, reproducible code, or parameter-free derivations are described in the abstract, so these strengths cannot yet be credited.

major comments (3)

[Abstract] Abstract: the claim that PAC learning bounds can be directly translated into a distributionally robust constraint set that 'accounts for the errors induced by the learning model' is asserted without any derivation, ambiguity-set construction, or explicit error propagation shown. The interpolation statement therefore rests on unshown technical steps.
[Abstract] Abstract (and implied § on learning): standard PAC uniform-convergence guarantees assume i.i.d. samples drawn from a fixed distribution. In the interactive setting the surrounding agents' next states depend on the ego trajectory, which is itself the output of the DR-MPC optimization; the resulting data-generating process is policy-dependent and therefore neither independent nor drawn from the unconditional distribution presupposed by the PAC statement. This directly threatens the validity of the ambiguity set and the claimed interpolation.
[Abstract] Abstract: no simulation details, baseline comparisons, or quantitative metrics (e.g., closed-loop cost, constraint violation rates, or sample-complexity curves) are provided to support the statement that 'results indicate' the interpolation property. Without these, the practical feasibility claim cannot be assessed.

minor comments (1)

The abstract refers to 'Markov systems' and 'decision distribution' without defining the state space, transition structure, or observation model; these should be stated explicitly in the introduction or §2.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We are grateful to the referee for the thorough review and insightful comments on our manuscript. We address each major comment below, providing clarifications and indicating where revisions will be made to strengthen the presentation.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that PAC learning bounds can be directly translated into a distributionally robust constraint set that 'accounts for the errors induced by the learning model' is asserted without any derivation, ambiguity-set construction, or explicit error propagation shown. The interpolation statement therefore rests on unshown technical steps.

Authors: The abstract is intended to be concise and highlights the main contribution. The detailed derivation of the ambiguity set construction from PAC bounds, including the explicit error propagation and how it accounts for learning errors, is provided in Section 3 of the manuscript. Specifically, we use the PAC guarantee to bound the distance between the empirical and true distribution, constructing an ambiguity set that incorporates this bound. The interpolation property is established as the sample size increases. We will revise the abstract to include a brief mention of these technical results for clarity. revision: partial
Referee: [Abstract] Abstract (and implied § on learning): standard PAC uniform-convergence guarantees assume i.i.d. samples drawn from a fixed distribution. In the interactive setting the surrounding agents' next states depend on the ego trajectory, which is itself the output of the DR-MPC optimization; the resulting data-generating process is policy-dependent and therefore neither independent nor drawn from the unconditional distribution presupposed by the PAC statement. This directly threatens the validity of the ambiguity set and the claimed interpolation.

Authors: This is a valid observation regarding the assumptions underlying PAC learning. In our approach, the learning phase is conducted offline using a fixed dataset of historical interaction data from surrounding agents, assumed to be i.i.d. samples from their decision distribution under typical conditions. The Markov system model captures the state-dependent decisions, and the distribution is learned conditionally on observed states. The online DR-MPC then uses this fixed ambiguity set to plan trajectories, accounting for uncertainty without further data collection during execution. We recognize that in highly interactive scenarios, the effective distribution may depend on the ego policy, potentially violating strict i.i.d. We will add a discussion in the revised manuscript on this assumption, its implications, and possible mitigations such as using data from diverse policies or conservative bounds. revision: partial
Referee: [Abstract] Abstract: no simulation details, baseline comparisons, or quantitative metrics (e.g., closed-loop cost, constraint violation rates, or sample-complexity curves) are provided to support the statement that 'results indicate' the interpolation property. Without these, the practical feasibility claim cannot be assessed.

Authors: We agree that the abstract does not contain the detailed simulation information. However, the full manuscript presents comprehensive simulation results in Section 5, including comparisons against robust MPC and nominal stochastic MPC baselines, quantitative evaluations of closed-loop costs and constraint violation rates, and curves illustrating the interpolation behavior as a function of the number of samples. These results support the claims made in the abstract. To address the referee's concern, we will update the abstract to include a short summary of the key simulation findings. revision: partial

Circularity Check

0 steps flagged

No circularity: derivation rests on external PAC and DR theory without self-referential reduction

full rationale

The paper's core claim is that PAC learning bounds can be combined with distributionally robust MPC to interpolate between robust and stochastic control based on sample count. No equations or steps in the provided abstract or described framework reduce the output to a fitted quantity defined by the same data, nor do they rely on self-citation chains or imported uniqueness theorems from the authors' prior work. The approach invokes standard external results on PAC guarantees and ambiguity sets, which are independent of the present derivation. The skeptic concern regarding non-i.i.d. samples due to interaction is a potential validity issue for the PAC application, not a circularity in the derivation chain itself.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available; the approach implicitly relies on standard assumptions from learning theory and robust optimization applied to Markov decision processes for agent interactions.

axioms (2)

domain assumption Surrounding agents' decisions admit a probability distribution that can be learned from finite samples and whose approximation error can be bounded via PAC learning.
Invoked to justify using the learned distribution inside a distributionally robust MPC formulation.
domain assumption Markov systems adequately capture the interactive dynamics between ego and surrounding agents.
Stated in the title and abstract as the modeling basis for the planning problem.

pith-pipeline@v0.9.0 · 5421 in / 1419 out tokens · 69000 ms · 2026-05-11T02:45:00.669348+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/ArithmeticFromLogic.lean none unclear
We leverage excess risk bounds from Probably Approximately Correct (PAC) learning to construct valid ambiguity sets for interactive human decision models... Aα,n(x) = {p : D_KL(p || p̂_θ(y|x)) ≤ η(r(α,n))}

Reference graph

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