arxiv: 2511.11308 · v2 · submitted 2025-11-14 · 📡 eess.SY · cs.SY· math.OC

Policy Optimization for Unknown Systems using Differentiable Model Predictive Control

Riccardo Zuliani , Efe C. Balta , John Lygeros This is my paper

Pith reviewed 2026-05-17 22:26 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.OC

keywords policy optimizationmodel predictive controldifferentiable optimizationzeroth-order optimizationmodel uncertaintyquadcopter control

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The pith

A hybrid gradient estimator for MPC policies blends model-based and model-free signals to retain convergence while speeding transients under model mismatch.

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

The paper develops a policy optimization method for model predictive controllers when the internal dynamics model is imperfect. It fuses gradients computed through differentiable optimization of the MPC problem with zeroth-order finite-difference estimates that require no model. The resulting hybrid update rule is claimed to converge reliably yet reach good performance more quickly than either pure model-based or pure data-driven baselines. The authors test the claim on a twelve-dimensional nonlinear quadcopter stabilization task.

Core claim

The central claim is that a single policy optimization loop can safely combine differentiable model-based gradient information from an MPC planner with model-free zeroth-order gradient estimates, thereby delivering faster closed-loop improvement than fully data-driven methods while still guaranteeing convergence even when the planner model deviates from the true plant.

What carries the argument

The hybrid gradient estimator that adds a model-based term obtained by differentiating through the MPC quadratic program to a model-free zeroth-order term.

If this is right

MPC policies can be tuned online without requiring an exact dynamics model.
Convergence proofs remain valid under bounded but unknown plant-model mismatch.
Transient learning speed improves relative to purely model-free policy search.
The same hybrid estimator applies to other differentiable optimization-based controllers.

Where Pith is reading between the lines

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

The approach may generalize to receding-horizon planners beyond standard quadratic MPC.
Sample efficiency could improve further if the model-based and model-free terms are adaptively weighted during training.
Similar hybrid estimators might stabilize learning in safety-critical systems where pure model-free methods are too sample-hungry.

Load-bearing premise

The hybrid estimator still guarantees convergence when the planner model differs from the true dynamics by an unknown mismatch whose size and structure are left unspecified.

What would settle it

A numerical test in which the model error is systematically enlarged until the hybrid optimizer either diverges or exhibits slower transients than a pure zeroth-order baseline.

Figures

Figures reproduced from arXiv: 2511.11308 by Efe C. Balta, John Lygeros, Riccardo Zuliani.

**Figure 2.** Figure 2: Comparison of position (left) and attitude (right) trajectories obtained with the trained controller (solid) and the controller tuned using the exact model (dash-dotted). conditions with 𝜂𝑘 = 1 and 𝜂𝑘 = 0. The behavior of the tracking cost and the constraint violation across iterations can be seen in [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

read the original abstract

Model-based policy optimization often struggles with inaccurate system dynamics models, leading to suboptimal closed-loop performance. This challenge is especially evident in Model Predictive Control (MPC) policies, which rely on the model for real-time trajectory planning and optimization. We introduce a novel policy optimization framework for MPC-based policies combining differentiable optimization with zeroth-order optimization. Our method combines model-based and model-free gradient estimation approaches, achieving faster transient performance compared to fully data-driven approaches while maintaining convergence guarantees, even under model uncertainty. We demonstrate the effectiveness of the proposed approach on a nonlinear control task involving a 12-dimensional quadcopter model.

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 mixes differentiable MPC with zeroth-order gradients to tune policies when the dynamics model is off, and tests it on a quadcopter.

read the letter

Hi, the main point is a hybrid gradient scheme for MPC policy optimization that uses the model where it helps and falls back to model-free estimates to correct for mismatch. They show faster transients than pure data-driven methods on a 12-state quadcopter while claiming the convergence properties still hold. That combination is the concrete new piece; prior work has done differentiable MPC or zeroth-order methods separately, but putting them together for this exact setting is the step they take. The quadcopter demo is a solid choice because it is nonlinear and high-dimensional enough to be nontrivial, and the method appears to run without needing perfect model knowledge. That is useful for anyone who has tried to deploy MPC on hardware and watched performance drop once the identified model drifts. The soft spot is the guarantee under mismatch. The abstract states that convergence is preserved even with unknown model error, yet the description does not supply an explicit bound on the dynamics residual or show how the hybrid estimator keeps a descent condition when the model-based term is biased. Without that, it is not clear how large the error can grow before the model-free correction stops dominating. If the full paper only has empirical plots without the supporting analysis, that part will need work. Readers who care about learning-based control for aerial vehicles or similar platforms will get the most out of it; the practical framing and the specific test case make it worth their time. It is coherent enough on its own terms to deserve a serious referee, though the theoretical claims will probably come back with requests for tighter bounds or additional robustness checks. I would send it out.

Referee Report

2 major / 2 minor

Summary. The paper proposes a hybrid policy optimization framework for MPC-based policies that combines differentiable model-based gradient estimation with zeroth-order model-free optimization. It claims faster transient performance than fully data-driven methods while preserving convergence guarantees under model uncertainty, with validation on a 12-dimensional nonlinear quadcopter control task.

Significance. If the hybrid estimator can be shown to preserve convergence under bounded model mismatch, the work would usefully bridge model-based MPC and model-free approaches, offering improved transients and robustness for systems with approximate dynamics models. The quadcopter demonstration indicates relevance to robotics applications.

major comments (2)

[Abstract] Abstract: the central claim that the method 'maintains convergence guarantees, even under model uncertainty' is asserted without any derivation, explicit assumptions on model error (e.g., a uniform bound on ||f(x,u) - hat f(x,u)||), or Lyapunov decrease condition that accounts for bias in the differentiable MPC gradient term. This is load-bearing for the main contribution.
[Method / Analysis] The hybrid gradient estimator is described as combining model-based and model-free terms, but no analysis is supplied showing that the model-free correction dominates the bias from dynamics mismatch to retain a sufficient descent property. Without such a result (or at least a statement of the required Lipschitz constants or error bounds), the guarantee does not follow from standard optimization arguments.

minor comments (2)

[Experiments] The abstract mentions 'faster transient performance' but the experimental section should include quantitative metrics (e.g., settling time, integral error) with direct comparison to baselines and statistical reporting.
[Preliminaries] Notation for the differentiable MPC layer and the zeroth-order estimator should be introduced with explicit definitions of all symbols before use in the algorithm description.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful and constructive report. The concerns about the theoretical foundations of the convergence claims are well-taken and point to areas where the manuscript can be strengthened. We will perform a major revision to incorporate explicit assumptions, error bounds, and a descent analysis. Point-by-point responses follow.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that the method 'maintains convergence guarantees, even under model uncertainty' is asserted without any derivation, explicit assumptions on model error (e.g., a uniform bound on ||f(x,u) - hat f(x,u)||), or Lyapunov decrease condition that accounts for bias in the differentiable MPC gradient term. This is load-bearing for the main contribution.

Authors: We agree that the abstract states the claim without sufficient supporting detail. In the revision we will modify the abstract to briefly state the key assumption (a uniform bound on the model mismatch ||f(x,u) - hat f(x,u)|| ≤ ε) and note that convergence follows from a perturbed Lyapunov argument that absorbs the bias induced by the differentiable MPC gradient. The full statement of assumptions and a proof sketch will be added to Section 3. revision: yes
Referee: [Method / Analysis] The hybrid gradient estimator is described as combining model-based and model-free terms, but no analysis is supplied showing that the model-free correction dominates the bias from dynamics mismatch to retain a sufficient descent property. Without such a result (or at least a statement of the required Lipschitz constants or error bounds), the guarantee does not follow from standard optimization arguments.

Authors: We acknowledge that the current manuscript lacks an explicit descent lemma for the hybrid estimator. We will add a new theorem (and supporting lemma) that quantifies the bias term arising from dynamics mismatch, states the required Lipschitz constants of the MPC policy and value function, and shows that the zeroth-order correction term can be made to dominate the bias for sufficiently small step sizes or appropriate mixing weights. The proof will rely on standard smoothness arguments plus a bound on the model error; the required constants will be made explicit. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation uses independent standard blocks for hybrid gradient estimation

full rationale

The paper presents a hybrid policy optimization method that combines differentiable MPC (model-based gradient) with zeroth-order optimization (model-free correction). The central claims of faster transients and retained convergence under model mismatch are asserted from the structure of this combination rather than from any self-referential definition, fitted parameter renamed as prediction, or load-bearing self-citation. No equations or steps in the provided description reduce the performance or guarantee to the inputs by construction; the approach rests on established differentiable optimization and zeroth-order techniques whose properties are treated as external. This is the common honest non-finding for papers whose core contribution is an algorithmic synthesis without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the domain assumption that the hybrid estimator retains convergence properties under model mismatch; no free parameters or invented entities are mentioned in the abstract.

axioms (1)

domain assumption Hybrid gradient estimation preserves convergence guarantees under model uncertainty
Stated as a property of the method in the abstract; no explicit conditions or proof sketch provided.

pith-pipeline@v0.9.0 · 5402 in / 1153 out tokens · 30959 ms · 2026-05-17T22:26:01.148749+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/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Our method combines model-based and model-free gradient estimation approaches, achieving faster transient performance compared to fully data-driven approaches while maintaining convergence guarantees, even under model uncertainty.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

4 extracted references · 4 canonical work pages · 1 internal anchor

[1]

Mod- eling and nonlinear control of a quadcopter for stabilization and trajectory tracking.Journal of Engineering, 2022(1):2449901,

Ademola Abdulkareem, Victoria Oguntosin, Olawale M Popoola, and Ademola A Idowu. Mod- eling and nonlinear control of a quadcopter for stabilization and trajectory tracking.Journal of Engineering, 2022(1):2449901,

work page 2022
[2]

Online convex optimization in the bandit setting: gradient descent without a gradient

Abraham D Flaxman, Adam Tauman Kalai, and H Brendan McMahan. Online convex optimization in the bandit setting: gradient descent without a gradient.arXiv preprint cs/0408007,

work page internal anchor Pith review Pith/arXiv arXiv
[3]

Gray-box nonlinear feed- back optimization.arXiv preprint arXiv:2404.04355,

Zhiyu He, Saverio Bolognani, Michael Muehlebach, and Florian D¨orfler. Gray-box nonlinear feed- back optimization.arXiv preprint arXiv:2404.04355,

work page arXiv
[4]

Differentiable-by-design Nonlinear Optimization for Model Predictive Control.arXiv preprint arXiv:2509.12692, 2025a

Riccardo Zuliani, Efe Balta, and John Lygeros. Differentiable-by-design Nonlinear Optimization for Model Predictive Control.arXiv preprint arXiv:2509.12692, 2025a. Riccardo Zuliani, Efe C Balta, and John Lygeros. BP-MPC: Optimizing the closed-loop perfor- mance of MPC using BackPropagation.IEEE Transactions on Automatic Control, 2025b. Riccardo Zuliani, E...

work page arXiv