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arxiv: 2604.15004 · v2 · submitted 2026-04-16 · 📡 eess.SY · cs.SY· math.OC

Recognition: 2 theorem links

· Lean Theorem

On-Line Policy Iteration with Trajectory-Driven Policy Generation

Yuchao Li , Fei Chen , Yingke Li , Chuchu Fan , Dimitri Bertsekas
Authors on Pith no claims yet

Pith reviewed 2026-05-12 03:06 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.OC
keywords on-line policy iterationtrajectory-driven policy generationmonotonic improvementoptimal controlrolloutpath planningneural network policiesfinite-horizon problems
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The pith

An on-line policy iteration method produces policies with monotonically decreasing costs for a fixed initial state under a consistency condition.

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

This paper develops an on-line policy iteration approach for deterministic finite-horizon optimal control problems that start from a fixed initial state. Starting from an initial policy, the method generates a trajectory and uses it to create data for training the next policy, repeating the process to form a sequence. Under a natural consistency condition, the costs of these policies improve monotonically for the initial state, though not necessarily elsewhere. This setup suits problems solved repeatedly from the same point, such as path planning and combinatorial optimization, enabling online use because it is computationally efficient. The framework merges rollout and policy iteration elements with flexible policy representations, including neural networks, and maintains the improvement guarantee.

Core claim

Under a natural consistency condition on the policy generation process, the sequence of costs of the generated policies is monotonically improving for the given initial state. The method generates each new policy from the trajectory of the previous one, allowing application to repetitive tasks where the same initial state is used multiple times.

What carries the argument

Trajectory-driven policy generation in the on-line policy iteration loop, where the trajectory produced by one policy trains the next policy.

If this is right

  • The policy costs decrease monotonically for the initial state with each iteration under the consistency condition.
  • The method runs fast enough for on-line use in repetitive tasks.
  • It supports training of neural network-based policies while preserving the monotonic improvement.
  • The approach applies to both single and multiple decision makers.
  • A stochastic counterpart is outlined briefly.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The focus on a single initial state suggests the method could be paired with state-space partitioning techniques to handle broader problems.
  • Trajectory data as the sole training source implies potential for reduced sample complexity compared to full off-policy learning.
  • The brief stochastic discussion leaves open whether similar consistency-based guarantees extend directly to noisy dynamics.

Load-bearing premise

A natural consistency condition must hold for the way each new policy is generated from the previous trajectory.

What would settle it

Demonstrating a sequence of policies where the cost for the initial state increases at some point, even though the consistency condition is satisfied, would falsify the monotonic improvement result.

Figures

Figures reproduced from arXiv: 2604.15004 by Chuchu Fan, Dimitri Bertsekas, Fei Chen, Yingke Li, Yuchao Li.

Figure 1
Figure 1. Figure 1: Example showing that the cost-improvement property of on-line PI [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of the coordinate-wise discretization construction in sim [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Illustration of the graph of an (N + 1)-dimensional assignment prob￾lem (here N = 5). There are N + 1 node layers each consisting of m nodes (here m = 4). Each grouping consists of (N + 1) nodes, one from each layer, and N corresponding arcs. An (N + 1)-dimensional assignment consists of m node-disjoint groupings, where each node belongs to one and only one grouping (illustrated in the figure with thick re… view at source ↗
Figure 4
Figure 4. Figure 4: Optimality gap (Jπℓ − J ∗ )/J∗ (%) for the on-line PI algorithm ap￾plied to some multidimensional assignment problems with randomly generated grouping costs. It can be seen that on-line PI converges monotonically from 2 to 6 iterations and significantly reduces the optimality gap in all of these examples. N is 40, and the time between two stages k and k + 1 is 0.25 seconds, which is a standard time limit f… view at source ↗
Figure 5
Figure 5. Figure 5: Cost improving Jπℓ /Jπ0 (%) for the on-line PI algorithm applied to some multidimensional assignment problems with randomly generated grouping costs, where the exact solution is intractable via brute force. On-line PI algo￾rithm reduces the costs in all tested cases. 3) Finally, we use the data set (39) and supervised learning to train a neural network Fθ parametrized by θ. These three steps collectively d… view at source ↗
Figure 6
Figure 6. Figure 6: Multiagent on-line PI applied to scenario 1. The scenario is illustrated [PITH_FULL_IMAGE:figures/full_fig_p022_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Multiagent on-line PI applied to scenario 2. The scenario is illustrated [PITH_FULL_IMAGE:figures/full_fig_p023_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Multiagent on-line PI applied to scenario 3. The scenario is illustrated [PITH_FULL_IMAGE:figures/full_fig_p024_8.png] view at source ↗
Figure 10
Figure 10. Figure 10: Multiagent on-line PI applied to scenario 1. The scenario is illus [PITH_FULL_IMAGE:figures/full_fig_p027_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Multiagent on-line PI applied to scenario 1. The scenario is illus [PITH_FULL_IMAGE:figures/full_fig_p028_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Multiagent on-line PI applied to scenario 1. The scenario is illus [PITH_FULL_IMAGE:figures/full_fig_p029_12.png] view at source ↗
read the original abstract

We consider deterministic finite-horizon optimal control problems with a fixed initial state. We introduce an on-line policy iteration method, which, starting from a given policy, however obtained, generates a sequence of cost-improving policies and corresponding trajectories. Each policy produces a trajectory, which is used in turn to generate data for training the next policy. The method is motivated by problems that are repeatedly solved starting from the same initial state, including discrete optimization and path planning for repetitive tasks. For such problems, the method is fast enough to be used on-line. Under a natural consistency condition, we show that the sequence of costs of the generated policies is monotonically improving for the given initial state (but not necessarily for other states). We illustrate our results with computational studies from combinatorial optimization and 3-dimensional path planning for drones {and a robot arm} in the presence of obstacles. We also discuss briefly a stochastic counterpart of our algorithm. Our proposed framework combines elements of rollout and policy iteration with flexible trajectory-based policy representations, and applies to problems involving a single as well as multiple decision makers. It also provides a principled way to train neural network-based policies using trajectory data, while preserving monotonic cost improvement.

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.

Referee Report

1 major / 3 minor

Summary. The manuscript introduces an on-line policy iteration algorithm for deterministic finite-horizon optimal control problems with a fixed initial state. Starting from an arbitrary initial policy, each iteration generates a trajectory that is used to train the next policy via a trajectory-driven update. Under a consistency condition on this update, the sequence of policies is shown to produce monotonically decreasing costs for the given initial state (though not necessarily for other states). The method is positioned for repetitive tasks and is illustrated with computational examples from combinatorial optimization and 3D path planning for drones and a robot arm; a stochastic variant is briefly outlined.

Significance. If the central conditional result holds, the work supplies a practical, online mechanism for policy improvement that merges rollout-style trajectory generation with policy iteration while accommodating flexible representations such as neural networks. The monotonicity guarantee, even when restricted to the initial state, is useful for repetitive control and planning tasks. The computational studies provide concrete evidence of applicability to discrete optimization and obstacle-avoidance problems.

major comments (1)
  1. [§3, Theorem 1] §3, Theorem 1: The monotonicity claim is conditional on the consistency condition; the manuscript should explicitly verify (or prove by construction) that the trajectory-driven training procedure used in the neural-network examples satisfies this condition, as this step is load-bearing for the guarantee.
minor comments (3)
  1. [Abstract] Abstract: the phrase 'drones {and a robot arm}' contains a LaTeX artifact and should be rendered as 'drones and a robot arm'.
  2. [§5] §5 (computational studies): the description of the policy training procedure (loss function, number of epochs, initialization) is too brief to allow reproduction; add a short table or paragraph with these details.
  3. [Throughout] Notation: the symbols for policy, trajectory, and cost are introduced inconsistently across sections; adopt a single, uniform notation from the outset.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the positive recommendation for minor revision. We address the single major comment below and will incorporate the requested clarification in the revised version.

read point-by-point responses

  1. Referee: [§3, Theorem 1] §3, Theorem 1: The monotonicity claim is conditional on the consistency condition; the manuscript should explicitly verify (or prove by construction) that the trajectory-driven training procedure used in the neural-network examples satisfies this condition, as this step is load-bearing for the guarantee.

    Authors: We agree that the consistency condition is central to the applicability of Theorem 1 and that an explicit verification for the neural-network examples strengthens the presentation. In the revised manuscript we will add a short subsection (or paragraph within Section 4) that proves by construction that the trajectory-driven training procedure satisfies the condition. Specifically, the neural network is trained via supervised regression on the state-action pairs collected along the trajectory generated by the previous policy, with the loss function defined so that the new policy exactly reproduces the actions taken on those states; this directly enforces the required consistency (i.e., the new policy agrees with the data-generating policy on the visited trajectory). The same argument applies verbatim to both the combinatorial-optimization and the 3-D path-planning examples, and we will state it explicitly for each. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper presents a conditional result: under an explicitly stated natural consistency condition on the trajectory-driven policy generation process, the costs of successive policies improve monotonically for the fixed initial state. This condition is treated as an assumption rather than derived internally or defined in terms of the claimed improvement, and the proof adapts standard dynamic programming arguments to the online setting. No steps reduce by construction to fitted parameters, self-referential definitions, or load-bearing self-citations; the framework remains self-contained once the consistency condition is granted as the weakest assumption.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests primarily on an unspecified consistency condition as a domain assumption; no free parameters or invented entities are mentioned in the abstract.

axioms (1)
  • domain assumption Natural consistency condition
    Invoked to guarantee monotonic cost improvement for the fixed initial state.

pith-pipeline@v0.9.0 · 5520 in / 1377 out tokens · 38620 ms · 2026-05-12T03:06:05.689216+00:00 · methodology

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Reference graph

Works this paper leans on

5 extracted references · 5 canonical work pages · 2 internal anchors

  1. [1]

    On-line policy iteration for infinite horizon dy- namic programming.arXiv preprint arXiv:2106.00746,

    [Ber21b] Dimitri Bertsekas. On-line policy iteration for infinite horizon dy- namic programming.arXiv preprint arXiv:2106.00746,

    work page arXiv

  2. [2]

    Adaptive Network Security Policies via Belief Aggregation and Rollout

    [HLA+25] Kim Hammar, Yuchao Li, Tansu Alpcan, Emil C. Lupu, and Dimitri Bertsekas. Adaptive network security policies via belief aggregation and rollout.arXiv preprint arXiv:2507.15163,

    work page internal anchor Pith review Pith/arXiv arXiv

  3. [3]

    Johansson, Jonas M˚ artensson, and Dimitri P

    [LJMB21] Yuchao Li, Karl H. Johansson, Jonas M˚ artensson, and Dimitri P. Bertsekas. Data-driven rollout for deterministic optimal control. In 2021 60th IEEE Conference on Decision and Control (CDC), pages 2169–2176. IEEE,

    work page 2021

  4. [4]

    An approximate dynamic programming frame- work for occlusion-robust multi-object tracking.arXiv preprint arXiv:2405.15137,

    31 [MLWB24] Pratyusha Musunuru, Yuchao Li, Jamison Weber, and Dim- itri Bertsekas. An approximate dynamic programming frame- work for occlusion-robust multi-object tracking.arXiv preprint arXiv:2405.15137,

    work page arXiv

  5. [5]

    Proximal Policy Optimization Algorithms

    [SWD+17] John Schulman, Filip Wolski, Prafulla Dhariwal, Alec Radford, and Oleg Klimov. Proximal policy optimization algorithms.arXiv preprint arXiv:1707.06347,

    work page internal anchor Pith review Pith/arXiv arXiv