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A control system framework for counterfactuals: an optimization based approach

T0 review · 4 major / 7 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Counterfactual explanations recast as minimum-effort endpoints of a control system.

desk verdict A sensible reframing of counterfactuals as minimum-effort optimal control, but the moment-based extraction step relies on an unjustified atomicity assumption that can produce non-reachable barycenters. read the letter →

arxiv 2501.12914 v1 pith:RSLMHTSW submitted 2025-01-22 eess.SY cs.SY

Pierluigi Francesco De Paola , Jared Miller , Alessandro Borri , Alessia Paglialonga , Fabrizio Dabbene This is my paper · ORCID
classification eess.SYcs.SY MSC 49K1590C22
keywords counterfactualexplanationsoptimalcontroloccupationmeasuresmoment-SOShierarchyminimum-effortglucose-insulinregulationuncertainsystemsreachability
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

Counterfactual explanations in machine learning usually perturb feature values without checking whether the resulting state is physically reachable. This paper proposes a control-theoretic alternative: define the factual as an initial state in an unsafe set and the counterfactual as the terminal state reached by the minimum-effort control that steers the system into a safe set. It solves this optimal control problem through occupation measures and the moment-SOS hierarchy, so the answer respects the system's dynamics, and it extends the construction to systems with uncertain parameters. The approach is demonstrated on a standard glucose-insulin regulation model, where the resulting counterfactuals lie near the safe-set boundary for known dynamics and shift deeper into the safe set when parameter uncertainty is taken into account.

What carries the argument

The engine is the occupation-measure formulation of optimal control. The trajectory is represented by three measures, the initial measure $\mu_0$, the occupation measure $\mu$, and the terminal measure $\mu_\tau$, linked by Liouville's equation $\delta_T \otimes \mu_T = \delta_0 \otimes \mu_0 + L^*\mu$. This turns the nonlinear reachability-and-effort problem into an infinite-dimensional linear program, which the moment-SOS hierarchy approximates by a sequence of semidefinite programs in truncated moment space. The terminal measure's first-order moment $y_{1,\tau}$, or in the dual a polynomial subsolution $v$ of the Hamilton-Jacobi-Bellman equation, is what the extraction algorithms use to read off the counterfactual.

What would settle it

For a two-state linear system with two equally cheap safe endpoints, solving the moment relaxation and checking whether the first moment of the terminal measure is a reachable endpoint under any admissible control would settle the extraction claim: if it is not, Algorithm 1's output is a barycenter rather than a counterfactual.

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Extended reading notes

Core claim

The paper's central claim is that a counterfactual can be defined without reference to any classifier, purely from the system dynamics: given a factual $x_f \in X_0$, the CS-counterfactual is $x_{cf} = x(\tau \mid x_f, u^*)$, the endpoint of the trajectory generated by the control $u^*$ that minimizes $\int_0^\tau u^2\,dt$ subject to the dynamics and $x(\tau) \in X_T$. This minimum-effort steering problem is lifted to a linear program over occupation measures, where Liouville's equation encodes the dynamics as a linear constraint. Solving a finite-dimensional moment-SOS relaxation yields a lower bound on the true cost; under the atomicity condition, the first-order moment of the terminal measure gives the counterfactual, and the dual polynomial provides a control law that can be simulated to recover the full trajectory. For uncertain parameters, the same construction is applied to an extended system in which unknown parameters become zero-dynamics states, producing robust counterfactuals that respect the whole range of parameter values.

Load-bearing premise

The extraction step assumes that the optimal terminal measure is concentrated on a single point, so its first-order moment is itself a reachable counterfactual state; finite moment relaxations do not guarantee this atomicity.

Editorial extensions

If this is right

  • Counterfactuals produced this way are actual trajectory endpoints, so they are reachable under the system's dynamics rather than arbitrary feature perturbations.
  • The method applies to general polynomial, control-affine nonlinear systems and does not require a trained classifier, only a safe set and a dynamic model.
  • For uncertain parameter values, the extended-state formulation yields counterfactuals that remain safe for every parameter in the specified set, at the price of lying farther from the boundary.
  • The dual solution gives a feedback law, so one can simulate the full factual-to-counterfactual trajectory, not just the terminal point.
  • The measure relaxation also computes a lower bound on the minimum control effort, which could serve as a quantitative measure of how hard it is to transition from an unsafe to a safe state.

Reading between the lines

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

  • Editorial inference: the same LP-in-measures machinery could certify infeasibility, meaning that if no measure satisfies the constraints, no control of the allowed form can reach the safe set, which amounts to a formal certificate that a counterfactual does not exist within budget.
  • Editorial inference: the dense clusters of counterfactuals seen in the experiments are likely shaped by the system's controllability and the value function's level sets; one could test this by comparing cluster locations against level sets of $v$ from the dual.
  • Editorial inference: when the terminal measure is not atomic, a more honest output would be a set or distribution of safe endpoints; checking the rank of the moment matrix $M_d(y_\tau)$ could flag this before extracting a single point.
  • Editorial inference: the framework could be extended to stochastic dynamics by replacing Liouville's equation with the Fokker-Planck equation, giving counterfactuals that are reachable with high probability rather than deterministically.
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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 / 7 minor

Summary. The paper proposes a control-theoretic definition of counterfactuals: given a factual initial condition in an unsafe set, the counterfactual is the terminal state of a minimum-effort optimal control problem that steers the system to a safe set with free terminal time. The problem is formulated as an infinite-dimensional linear program in occupation measures and solved through the moment-SOS hierarchy. Three extraction algorithms are proposed: reading the counterfactual as the first-order moment of the terminal measure (Algorithms 1 and 3), and simulating a closed-loop trajectory from the approximate dual value function (Algorithm 2). The approach is extended to systems with uncertain parameters by augmenting the state with the uncertain parameters, and numerical experiments are reported on a Bergman glucose-insulin model.

Significance. The idea of grounding counterfactuals in dynamical reachability and minimum effort, rather than in classifier geometry alone, is timely and potentially valuable for safety-critical and biomedical applications. The paper correctly identifies occupation measures and the moment-SOS hierarchy as natural tools for this problem, and the robust extension in Section 4 is a useful addition. However, the central computational claim, that a counterfactual can be read off as the first moment of the terminal measure from a finite relaxation, is not justified and is false in general. Because the numerical results in Sections 3.5 and 4.5 rely on this extraction, the contribution as written is not yet established. If the extraction step is repaired with atomicity certificates or a certified trajectory-recovery procedure, the framework could become a useful addition to the counterfactual-explanation literature.

major comments (4)
  1. [§3.4, Proposition 1 and Algorithm 1] The proposition infers from mass(mu_tau)=1 that mu_tau is a Dirac measure. This inference is invalid: a non-atomic measure can have total mass 1, and a finite-order truncated moment vector need not even have a representing measure. Consequently, xcf = y_1,tau may be a barycenter of several candidate endpoints and need not be a reachable terminal state of any feasible trajectory. The extraction step must be justified by a flat-extension or rank-one certificate on M_d(y_tau), or replaced by a recovery method that reconstructs an atomic terminal measure; the numerical sections should report such certificates.
  2. [§4.2, equations (20b)-(22c)] The claim that constraint (22c) together with Liouville's equation implies that the final measure mu_t* is atomic is incorrect. Choosing the test function v=1 in (20b) only gives conservation of mass, <1,mu_0>=<1,mu_t*>, which says nothing about atomicity. The subsequent statement that if (22) admits s optimal solutions then mu_t* is s-atomic is also false: the feasible set of (22) is convex, so any convex combination of optimal terminal measures is again optimal, and the set of optimal endpoints can be a continuum. These errors invalidate Proposition 3 and Algorithm 3 as stated.
  3. [§3.4, Proposition 2 and Algorithm 2] The dual-based trajectory recovery is not certified. For a finite relaxation order d, the dual solution v_d is only a subsolution of the value function, and the control u*_d(x) obtained from it is not guaranteed to steer the closed-loop system to X_T at time tau=mass(mu). No reachability or optimality proof is provided for Algorithm 2, so this alternative extraction path also does not currently support the paper's claims.
  4. [§3.4, Theorems 1-3] The formal results are either near-tautological or not established. Theorem 1 only asserts boundedness of trajectories, which follows from compactness of X when the setup is well posed. Theorem 2's proof assumes V(x)=0 on X_T and V(x)>0 elsewhere, which is not established by (6a)-(6d), and the inequality L_f V <= -phi with phi=u^2 does not imply asymptotic stability of X_T because u^2 is a control-dependent quantity rather than a positive definite function of the state alone. Theorem 3 is a boundedness statement on a compact set and provides no convergence or reachability guarantee. These results should be corrected, or their statements and proofs removed, so that the advertised guarantees match what is actually shown.
minor comments (7)
  1. [§2.1] The sentence 'For alpha=1 we get the mass of the measure' is incorrect in multi-index notation: the mass is y_0=<1,mu>, not the first-order moment y_1.
  2. [§2.1] The line defining the adjoint operator says 'H*=H', which is not generally true; the correct identity is <Hv,mu>=<v,H*mu>.
  3. [§2.3, equation (2)] The constraint 'p(x)-sum_i a_i(x)v_i >= 0' is written with 'for all i=1,...,I_max', but the positivity condition applies to the single polynomial in the sum, not to each index i separately.
  4. [§3.4, Remark after Proposition 1] The expression 'xcf = spt(mu_tau=delta_x_tau)=sum_i beta_i*x_tau_i' is not well-formed: spt of a measure is a set of atoms, not a weighted sum. The intended meaning should be written with the measure itself, e.g. mu_tau=sum_i beta_i*delta_x_tau_i.
  5. [§3.5] Equation (13) is labeled with two numbers, (13) and (14), for a single displayed polynomial; the duplicate label should be removed.
  6. [§4.5] The text says 'parameters p3 and p3 are related to the dynamics'; the second occurrence should presumably be p2.
  7. [§4.5] The observation that counterfactuals lie near the boundary of the safe set is a direct consequence of the free-terminal-time, no-terminal-cost minimum-effort formulation, not an independent empirical discovery; the paper should present it as such.

Circularity Check

2 steps flagged · score 6.0 of 10

Two central 'findings' reduce to definitions: boundary placement is the min-effort terminal-set objective, and Proposition 1's xcf=y1,τ restates Algorithm 1's extraction rule; the mass-one atomicity inference is asserted rather than derived.

  1. self definitional [Section 3.1, Definition 2; Section 3.5, Results under Fig.1]
    "Given a CS-factual xf ∈ X0, we define the CS-counterfactual xcf associated with the factual xf ∈ X0 as xcf := x(τ |xf,u∗), xcf ∈ XT ... Counterfactuals lie in the safe set in the vicinity of the boundary (i.e., the diabetic threshold) separating the safe set from the unsafe set."

    Definition 2 makes the counterfactual the terminal point of the minimum-effort trajectory that is constrained to end in the safe set XT. Under Assumption 5 the cost is only ∫u^2 with no terminal cost, so the optimal endpoint of a free-terminal-time transfer to XT is generically on the boundary of XT. The observed 'vicinity of the boundary' property is therefore a restatement of the optimization objective and constraints, not an independently derived prediction. The claim is forced by the definition of CS-counterfactual plus Assumption 1.

  2. self definitional [Section 3.4, Algorithm 1 and Proposition 1]
    "Algorithm 1: 3: Extract y1,τ from Md(yτ) 4: xcf = y1,τ ... Proposition 1: If mass(µτ) = 1, then the counterfactual xcf = spt(µτ = δxτ) and it holds that xcf = y1,τ."

    The extraction rule of Algorithm 1 is precisely the equality xcf = y1,τ. Proposition 1 then 'proves' that same equality under the condition mass(µτ)=1, so the conclusion is the algorithm's definition restated rather than a consequence of the OCP definition of counterfactual. The only nontrivial ingredient needed for the claimed derivation is a certificate that a unit-mass final measure is a Dirac delta; the paper asserts this without proof, and it is false in general (a uniform measure has mass one and its first moment is a barycenter, not an atom). The same reduction-by-definition appears in Algorithm 3/Proposition 3 for the robust case.

full rationale

The paper's moment-SOS computation itself is not circular: it solves a genuine infinite-dimensional LP for minimum-effort control, and the convergence statements are imported from the independent occupation-measure literature (Lasserre, Henrion, Korda, etc.), not from the authors' own prior work. No self-citation is load-bearing for the core LP relaxation. However, two claimed outputs do reduce to their inputs by construction. First, the empirical observation that counterfactuals lie near the safe-set boundary is a direct consequence of Definition 2 (terminal state in XT) together with the minimum-effort objective (no terminal cost), so it has no independent predictive content. Second, the extraction step defines xcf as the first moment y1,τ in Algorithm 1 and then Proposition 1 purports to derive xcf=y1,τ from unit mass; since unit mass does not imply atomicity, the proposition supplies no independent derivation and the equality is simply the algorithm's construction. The same issue affects the robust Algorithm 3/Proposition 3 and the §4.2 claim that mass conservation makes μt* atomic. Those issues are also correctness risks, but the circular aspect is that the 'predicted' counterfactual equals the output variable by definition. Because the core optimization framework remains independently meaningful, the circularity is partial (score 6) rather than total.

Assumptions & free parameters 0 free parameters · 8 assumptions · 0 invented entities

The method introduces no new free parameters fitted to data: all model constants come from the Bergman model and the normalization constants are coordinate scalings. The main axioms are the standard assumptions for the occupation measure framework plus a specific cost choice (L2 control effort). The value function properties used in the stability proof are assumed rather than derived.

assumptions (8)
  • domain assumption Assumption 1: X0 ⊆ Xu and XT ⊆ Xs, where Xu and Xs are unsafe and safe sets.
    This is a modeling choice that defines the counterfactual problem as a transfer from unsafe to safe regions.
  • domain assumption Assumption 2: The vector field is polynomial and Lipschitz, and the sets X, XT are semialgebraic.
    This is required for the moment-SOS hierarchy to apply.
  • domain assumption Assumption 3: The vector field is affine in the control input.
    This simplifies the dual optimal control problem and is common in the occupation measure literature.
  • domain assumption Assumption 4: The control is Lp-measurable with finite norm.
    Standard in optimal control to ensure existence of solutions.
  • ad hoc to paper Assumption 5: The effort is the L2 norm of the control and there is no terminal cost.
    This specific cost choice defines what 'minimum change' means; other costs would yield different counterfactuals.
  • domain assumption Assumption 6: Uncertain parameters are constant over time and lie in a known set.
    Used to formulate robust counterfactuals via an augmented state-space.
  • standard math Convergence of the moment-SOS hierarchy to the true optimal value as d → ∞ (Theorem 4.1 in [12]).
    The paper relies on this known result to justify the finite-dimensional approximations.
  • domain assumption The value function V(x) satisfies V(x)=0 on XT and V(x)>0 elsewhere.
    This is assumed in the proof of Theorem 2 but not proven from the dynamics and cost.

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Pith. "Pith review of A control system framework for counterfactuals: an optimization based approach." pith.science (2026). https://pith.science/paper/RSLMHTSW

@misc{pith2026250112914,
  author       = {Pith},
  title        = {Pith review of: A control system framework for counterfactuals: an optimization based approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RSLMHTSW}},
  note         = {Machine review of arXiv:2501.12914}
}
read the original abstract

Counterfactuals are a concept inherited from the field of logic and in general attain to the existence of causal relations between sentences or events. In particular, this concept has been introduced also in the context of interpretability in artificial intelligence, where counterfactuals refer to the minimum change to the feature values that changes the prediction of a classification model. The artificial intelligence framework of counterfactuals is mostly focused on machine learning approaches, typically neglecting the physics of the variables that determine a change in class. However, a theoretical formulation of counterfactuals in a control system framework - i.e., able to account for the mechanisms underlying a change in class - is lacking. To fill this gap, in this work we propose an original control system, physics-informed, theoretical foundation for counterfactuals, by means of the formulation of an optimal control problem. We apply the proposed methodology to a general glucose-insulin regulation model and results appear promising and pave the way to the possible integration with artificial intelligence techniques, with the aim of feeding machine learning models with the physics knowledge acquired through the system framework.

Figures

Figures reproduced from arXiv: 2501.12914 by the authors.

Figure 1
Figure 1. Factual-counterfactual pairs as obtained by solv [PITH_FULL_IMAGE:figures/full_fig_p011_1.png] view at source ↗
Figure 2
Figure 2. Factuals (red dots) and counterfactuals (blue dot [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Example of counterfactual trajectory as retrieve [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: Factual-counterfactual pairs as obtained by solv [PITH_FULL_IMAGE:figures/full_fig_p019_4.png]
Figure 5
Figure 5. Figure 5: Factuals vs Counterfactuals obtained for [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]

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