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REVIEW 4 major objections 3 minor 29 references

A Convex Optimization Approach to Model-Free Inverse Optimal Control with Provable Convergence

T0 review · 4 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read Model-free inverse optimal control can be reformulated as a convex program, with a provable O(1/k) convergence rate for the block-coordinate solver.

desk verdict Clever convex reformulation of model-free IOC, but the O(1/k) convergence claim rests on a false Lagrangian dual derivation and the main theorem does not hold as stated. read the letter →

arxiv 2507.19965 v1 pith:ZJXB5RU4 submitted 2025-07-26 math.OC

Meiling Yu , Lechen Feng , Lei Jiang , Yuan-Hua Ni This is my paper · ORCID
classification math.OC MSC 49N4590C2590C2293B30
keywords inverseoptimalcontrollinearquadraticregulatormodel-freeconvexoptimizationconicprogrammingblockcoordinatedescentconvergenceratesystemidentification
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Model-free inverse optimal control asks for the unknown system matrices and unknown LQR cost weights that explain one observed optimal trajectory; this paper claims that this non-convex joint estimation problem is exactly equivalent to a convex conic feasibility problem. The key is a variable change that turns the bilinear algebraic Riccati equation into linear equalities, so the remaining constraints are positive-semidefinite cones. The paper solves the Lagrange dual with block successive upper-bound minimization, proves that every block update has a closed form, and shows the dual objective gap converges at $\mathcal{O}(1/k)$. If the proof stands, this is the first model-free IOC method with an explicit, non-asymptotic convergence-rate guarantee, and the simulations show it reproducing the expert trajectory with mean-squared error around $10^{-6}$ in well under a second.

What carries the argument

The load-bearing device is the variable change $Z=\hat{A}^\top \hat{P}$ and $G=\hat{P}\hat{A}_K$, which turns the bilinear Riccati equation into linear equalities and makes the joint estimation a convex feasibility problem in the stacked variable $\xi$. The solver works on the Lagrange dual, where the KKT relation $\xi^*(\lambda)=-\tfrac12(\Omega^\top\Omega)^\dagger U^\top\lambda$ produces the quadratic-conic dual objective, and each of the three block updates is a closed-form resolvent/projection step. This structure is what allows the proof of the $\mathcal{O}(1/k)$ convergence rate.

What would settle it

Compute the Hessian $H$ of the dual objective and look for a nonzero direction $d$ inside the semidefinite cone with $H d = 0$ and $W^\top d = 0$; such a direction makes the objective constant along a ray, so the level-set radius $R$ is infinite, $\sigma = 1/(9\|H\|R^2)$ vanishes, and the theorem's $\mathcal{O}(1/k)$ inequality is vacuous. Run the proposed updates on an instance with a rank-deficient $Q$ on the data-supported subspace and check whether the dual gap obeys the claimed bound with the stated constants.

Watch

Extended reading notes

Core claim

The paper's central claim is that jointly recovering $(A,B,Q,R)$ from a single optimal LQR trajectory, usually regarded as a non-convex inverse problem, can be recast without loss as a convex problem. Concretely, Theorem 3.5 proves that optimal solutions of the original parameter optimization are exactly the feasible points of a system of linear equalities plus semidefinite cone constraints. Vectorizing those equalities and minimizing the residual gives a least-squares problem over PSD cones whose Lagrangian dual is a convex quadratic-conic objective; Theorem 3.13 proves that the block-coordinate dual updates reduce the objective gap at rate $\mathcal{O}(1/k)$. The paper further claims this is the first explicit convergence-rate guarantee for model-free IOC and reports an order-of-magnitude speedup over a differentiable-programming benchmark in reconstruction accuracy and robustness.

Load-bearing premise

The $\mathcal{O}(1/k)$ proof assumes the set of dual variables no worse than the starting point stays within a bounded region; the paper does not prove this, and with unbounded semidefinite cones the rate constant can become infinite, making the bound vacuous.

Editorial extensions

If this is right

  • If the reformulation is valid, model-free IOC no longer needs non-convex optimization: any data-compatible $(A,B,Q,R)$ can be found by convex feasibility, so local minima are not an issue.
  • The $\mathcal{O}(1/k)$ bound gives an explicit iteration budget, so a safety-critical deployment can precompute how many iterations are needed for a required tolerance on the dual objective gap.
  • Closed-form block updates make each iteration cheap and step-size free; in the paper's experiment the method converges in 19 iterations and 0.3 seconds, versus 2000 iterations for the differentiable-programming baseline.
  • Because all zero-objective solutions form an equivalence class, the algorithm's guarantee is convergence to an equivalent LQR explanation of the expert's behavior, not to a unique ground-truth parameter set.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The paper leaves implicit that the real identifiability bottleneck is the identification of the feedback gain $K^*$ from the trajectory; if the data fail the persistent-excitation condition, no convex reformulation can recover a meaningful cost because $K^*$ itself is not unique.
  • A natural testable extension would be to apply the same variable-change and dual block-coordinate machinery to discrete-time LQR or output-feedback problems, where the cones and block updates change but the convexification strategy may carry over.
  • Because convergence is measured on the dual objective gap rather than on parameter error, practitioners should validate recovered models by trajectory reproduction error, as the simulations do, rather than by closeness of $(A,B,Q,R)$ to an unknown ground truth.
  • A downstream safety implication the paper does not develop: since positive scaling of $(Q,R)$ does not change the optimal policy, the magnitude of the recovered cost is not identifiable, so any guarantee stated in terms of $Q$ or $R$ must be invariant to that scaling.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 3 minor

Summary. The paper studies model-free inverse optimal control for continuous-time LQR systems, where both the dynamics (A,B) and cost weights (Q,R) are unknown. It proposes to identify the optimal feedback gain K* from a single expert trajectory, then reformulates the joint estimation problem as the convex feasibility problem (14) and the equivalent least-squares conic program (16). The authors derive a Lagrangian dual (19), solve it with a block successive upper-bound minimization algorithm, and claim a sublinear O(1/k) convergence rate. Simulation comparisons with Pontryagin Differentiable Programming are reported as showing faster and more accurate reconstruction.

Significance. If the central claims were correct, the paper would make a substantial contribution: a convex reformulation of a nonconvex inverse LQR problem, a closed-form block-coordinate solver, and the first explicit non-asymptotic convergence-rate guarantee for model-free IOC. The paper contains several genuinely useful pieces: Lemma 3.3 and Lemma 3.4 give a clean trajectory-equivalence argument, and the SDP feasibility formulation in (14) is a reasonable convex relaxation of the original problem. However, the load-bearing dual derivation in Lemma 3.7 and Theorem 3.8 is invalid for the stated problem, and the convergence-rate bound in Theorem 3.13 rests on unverified assumptions. Because these flaws directly undermine the algorithm's claimed guarantees, the paper cannot be accepted in its present form.

major comments (4)
  1. [§3.1, Lemma 3.7 and Theorem 3.8] The derivation of the dual (19) is invalid because the paper replaces the stationary condition 2Ω^TΩξ + U^Tλ = 0 by the pseudoinverse formula (18) without checking that U^Tλ ∈ Ran(Ω^TΩ). If this range condition fails, the Lagrangian L(ξ;λ) is unbounded below and the true dual function is −∞, not the finite quadratic in (19). The condition fails in the present problem: the equality constraints in (14) are scale-invariant, so for any feasible v = (vec(Z), vec(R), vec(Q), vec(P), vec(G))^T built from ground-truth parameters, Ωv = 0; taking λ = (0, I, 0) with the identity multiplier for P gives U^Tλ · v = trace(P*) > 0, so U^Tλ ∉ Nul(Ω)^⊥ = Ran(Ω^TΩ), and along ξ = tv with t < 0, L(ξ;λ) → −∞. Thus Theorem 3.8 is false as stated, and Algorithm 1 in Section 3.2 minimizes a pseudo-objective rather than the actual dual of (16).
  2. [§3.2, Theorem 3.13] The stated O(1/k) bound depends on R = sup_{λ∈F, λ*∈Γ*_+} ∥λ−λ*∥, but the paper never proves that the level set F = {λ : Jdual(λ) ≤ Jdual(λ0)} is bounded. Since H is only positive semidefinite and the PSD cones are unbounded, directions inside the feasible cone along which the quadratic part vanishes can make F unbounded or the constant σ = 1/(9∥H∥R²) degenerate; no argument rules this out. The proof merely says the problem satisfies all assumptions of Theorem 2 in [26], but boundedness and related regularity hypotheses are not verified. Consequently the advertised convergence-rate guarantee is not established.
  3. [§3.2, Algorithm 1 and Eq. (18)] Even if the dual sequence converged, the algorithm's output ξ*(λ^{k+1}) = −(1/2)(Ω^TΩ)^†U^Tλ^{k+1} is not shown to be primal feasible. The paper proves only a dual objective gap (under the contested Theorem 3.13); it does not show that ∥Ωξ*(λ^{k+1})∥ → 0 or that the cone constraints are satisfied, and the pseudoinverse can return a point with Ωξ ≠ 0 when U^Tλ is not in Ran(Ω^TΩ). The stopping criterion ∥λ^{k+1}−λ^k∥ < ε is a dual-iterate gap and has no proven relation to reconstruction accuracy. Hence the claim that Algorithm 1 converges to a point in the equivalence set Φ* is unsupported.
  4. [Remark 3.9 and problem (16)] Strong duality is asserted via Slater's condition, but strict feasibility of (16) is not proved. A strictly feasible point would require Ωξ = 0 with Q ≻ 0, P ≻ εI, and R ≻ εI; the true parameters only guarantee Q ≽ 0, and Q may be singular, so no argument shows that a positive definite Q can satisfy the equality constraints. Without Slater's condition, the KKT-based recovery of a primal solution from λ* via (18) lacks a rigorous justification. This is an additional unverified assumption in the chain leading to the convergence-rate claim.
minor comments (3)
  1. [Equation (8)] The notation e^AK t should be written as exp((Â − B̂K̂)t) or e^{(Â−B̂K̂)t}; as typeset it is ambiguous between e^A K t and e^{AK}t.
  2. [Lemma 3.3 proof] The characteristic polynomial p(λ) = λ^n + a_{n−1}λ^{n−1} + ... + a_0 is followed by equations (9)–(10) that use coefficients a_1, ..., a_n inconsistently; the induction step should be re-indexed to match a_0, ..., a_{n−1}.
  3. [Section 4.1 and Table 1] The runtime comparison (0.3 s vs. 98.4 s) is reported without hardware/software details, implementation effort, or multiple runs; as a single trial it is anecdotal. Reporting distributions over the Monte Carlo runs would better support the claimed order-of-magnitude improvement.

Circularity Check

1 steps flagged · score 6.0 of 10

Reconstruction accuracy is built into the constraints: every feasible point has K̂ = K* and x̂ = x* by construction, so the Section 4 metrics are forced residuals; the convex reformulation and the externally sourced O(1/k) rate are otherwise not circular.

  1. fitted input called prediction [Theorem 3.5 (Ψfeas ⊆ Φ* direction); Lemma 3.4; Section 4.1 (simulation validation)]
    ""For the gain constraint, we compute K̂ = R̂−1B̂⊤P̂ = R̂−1(R̂K∗P̂−1)P̂ = K∗, which shows that K̂ = K∗." "If the data matrix Λ0 is constructed to include the initial state x∗(0) as one of its column vectors, then the trajectory ˆx(·) generated by the closed-loop dynamics ˙x = ˆAKx with the initial condition ˆx(0) = x∗(0) is identical to the expert trajectory x∗(·)." "The Frobenius norm of the error, ||K̂MFIOC − K∗||F, is as low as 1.424 × 10−4. The high accuracy in policy recovery validates the correctness of our theoretical framework.""

    The data-identified gain K* (least-squares fit u* = −K*x*) and trajectory samples (Λ̄1, Λ̄2) are the inputs from which constraints (12)–(15) are built. Theorem 3.5 then constructs  = P̂⁻¹Zᵀ and B̂ = P̂⁻¹K*ᵀR̂ and concludes K̂ = K* for every feasible point; Lemma 3.4 concludes any point satisfying Â_KΛ̄1 = Λ̄2 reproduces x*(·) exactly. Thus the Section 4.1 metrics — ||K̂MFIOC − K*||F = 1.4×10⁻⁴ and trajectory MSE = 8.7×10⁻⁷ — presented as 'reconstruction accuracy' that 'validates the correctness of our theoretical framework' are residuals of constraints whose satisfaction the design forces: K̂ = K* and x̂ = x* are algebraic consequences of feasibility, not empirical findings. The genuinely unknown (Q, R) are never compared with ground truth.

full rationale

The main derivation chain is: (i) Assumptions 2.1/2.2 posit a true LQR expert and K* is identified from the trajectory; (ii) problem (8) is reformulated into feasibility systems (12)–(15) via Z = ÂᵀP̂, ÂK, and G = P̂ÂK (Theorem 3.5); (iii) the feasibility system becomes least-squares problem (16), whose Lagrangian dual (19)/(20) is solved by BSUM; (iv) Theorem 3.13 imports the O(1/k) rate from Theorem 2 of Hong et al. [26], an independent external result with no author overlap. Links (ii) and (iii) are genuine algebra: the constraints are derived from the ARE and data-consistency conditions, not from the target convergence statement, so the reformulation itself is not circular. The O(1/k) claim rests on an independent citation, not on self-citation; however, its assumptions are asserted but never verified — the bounded level set F defining R, Slater/strong duality (Remark 3.9), and the range condition needed for the pseudoinverse expression ξ*(λ) = −(1/2)(ΩᵀΩ)†Uᵀλ in Lemma 3.7/Theorem 3.8. The scaling symmetry the paper itself notes implies Ω is rank-deficient; if the skeptic's argument holds, the true dual is −∞ on directions off Ran(ΩᵀΩ), making Theorem 3.8 false as stated and the O(1/k) result a statement about a pseudo-objective. These are correctness risks, not circularity under the rules, and do not by themselves raise the score. The one genuine circularity is at the validation level: any feasible point of (12)–(15) has K̂ = K* and x̂ = x* by construction (Theorem 3.5 proof and Lemma 3.4), so the Section 4 'reconstruction accuracy' metrics measure constraint satisfaction of a system built from the very quantities being 'recovered'. Presenting high policy-recovery accuracy as validation of the framework, and comparing against PDP on these metrics, treats enforced constraint satisfaction as an empirical success. Self-citation [27] (Feng, Ni) is present but only as an auxiliary BSUM reference; the load-bearing convergence reference [26] is independent, so the score reflects the construction-forced predictions rather than self-citation. Overall: partial circularity in the validation claims, with the central convex-reformulation mathematics retaining independent content.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The optimization framework introduces auxiliary variables Z and G to linearize the ARE, but no new physical entities. The free parameters are algorithmic. The axioms include two control-theory domain assumptions from Section 2 and several unproved regularity conditions (derivative data, Slater, bounded level set) that the reformulation and convergence proof silently rely on.

free parameters (2)
  • epsilon (positive-definite relaxation) = epsilon = 1e-6 (suggested in Remark 3.1)
    Replaces strict positivity of R and P with PSD constraints R >= eps I, P >= eps I to make projections well-defined; the value is user-chosen and affects the feasible set and the recovered solution.
  • stopping tolerance = epsilon (same symbol) in Algorithm 1
    Termination criterion ||lambda^{k+1} - lambda^k|| < epsilon; user-chosen, affects reported iteration counts and solution precision.
assumptions (6)
  • domain assumption The true system pair (A,B) is stabilizable and (A, sqrt(Q)) is detectable (Assumption 2.1).
    Required for LQR well-posedness and for the ARE to have a unique stabilizing solution; standard in LQR theory.
  • domain assumption The observed trajectory is generated by an optimal LQR controller and satisfies persistent excitation, so K* is uniquely identified by least squares from u* = -K* x* (Assumption 2.2).
    Load-bearing: if the trajectory is not exactly optimal or not persistently exciting, the identified K* is inaccurate and the feasibility reformulation fails.
  • ad hoc to paper Exact samples of the trajectory derivatives up to order n are available to build the data matrices Lambda_i in Eq. (11).
    Never stated as an assumption; required for the constraint A_K \bar{Lambda}_1 = \bar{Lambda}_2 and hence for the equivalence in Theorem 3.5.
  • ad hoc to paper Slater's condition holds for the primal problem (16), so strong duality with the dual (19) holds (Remark 3.9).
    Asserted without proof; requires a strictly feasible point with Q ≻ 0 and R, P ≻ eps I satisfying the linear equalities, which is not guaranteed when the true Q is singular.
  • ad hoc to paper The dual level set F is bounded, giving a finite R in Theorem 3.13.
    Needed for the O(1/k) rate; not proven and likely false for PSD H on unbounded cones.
  • standard math The BSUM convergence theorem of Hong et al. [26] applies to the indicator-regularized problem (20).
    The paper cites Theorem 2 in [26] as a direct corollary without verifying its assumptions (e.g., bounded level sets, Lipschitz continuity of the smooth part).

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Cite this review

Pith. "Pith review of A Convex Optimization Approach to Model-Free Inverse Optimal Control with Provable Convergence." pith.science (2026). https://pith.science/paper/ZJXB5RU4

@misc{pith2026250719965,
  author       = {Pith},
  title        = {Pith review of: A Convex Optimization Approach to Model-Free Inverse Optimal Control with Provable Convergence},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZJXB5RU4}},
  note         = {Machine review of arXiv:2507.19965}
}
abstract

Inverse Optimal Control (IOC) aims to infer the underlying cost functional of an agent from observations of its expert behavior. This paper focuses on the IOC problem within the continuous-time linear quadratic regulator framework, specifically addressing the challenging scenario where both the system dynamics and the cost functional weighting matrices are unknown. A significant limitation of existing methods for this joint estimation problem is the lack of rigorous theoretical guarantees on the convergence and convergence rate of their optimization algorithms, which restricts their application in safety-critical systems. To bridge this theoretical gap, we propose an analytical framework for IOC that provides such guarantees. The core contribution lies in the equivalent reformulation of this non-convex problem of jointly estimating system and cost parameters into a convex second-order cone programming problem. Building on this transformation, we design an efficient iterative solver based on the block successive upper-bound minimization algorithm. We rigorously prove that the proposed algorithm achieves a sublinear convergence rate of $\mathcal{O}(1/k)$. To the best of our knowledge, this is the first solution for the model-free IOC problem that comes with an explicit convergence rate guarantee. Finally, comparative simulation experiments against a state-of-the-art benchmark algorithm validate the superiority of our proposed method. The results demonstrate that our algorithm achieves an order-of-magnitude improvement in convergence speed while also exhibiting significant advantages in reconstruction accuracy and robustness.

Figures

Figures reproduced from arXiv: 2507.19965 by the authors.

Figure 1
Figure 1. Convergence curve of the dual variable in our proposed algorithm. [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. Comparison of the reconstructed trajectory (MFIOC) and the expert trajectory. [PITH_FULL_IMAGE:figures/full_fig_p015_2.png] view at source ↗
Figure 3
Figure 3. Convergence curve of the loss function for the PDP algorithm. [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Comparison of the reconstructed trajectory (PDP) and the expert trajectory. [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]
Figure 5
Figure 5. Figure 5: Performance comparison over 100 Monte Carlo experiments. [PITH_FULL_IMAGE:figures/full_fig_p017_5.png]

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