Source
Felekis et al. (2024), LIT-1806, §4 (pp. 5–8), §6 (Fig. 2 and Table 1, p. 10) and Appendices A–B (pp. 14–16) of arXiv 2312.08107v1; read in NOTE-1342.
What was actually shown
The problem as written. For each intervention ι in a maximal chain, a plan P^ι in the transport polytope U(π_ι): its column sums equal the base model’s empirical distribution under ι and its row sums the abstracted model’s under ω(ι) (eq. 6, p. 6). COTA minimises Σ_ι κ⟨C_ω, P^ι⟩ + λᵀD(P^ι, P^{ι+1}) − μH(P^ι) over these plans, with κ, λ, λ′, μ ≥ 0 summing to 1 (eq. 12, p. 7).
The term. D = [δ, δ′]. The base part is δ_{ι,η}(P^ι, P^η) = d(Σ_i P^η_{ij}, (1/Z^η_j) Σ_i P^ι_{ij}) over columns compatible with η (eq. 9), the abstracted part the same on rows (eq. 10). The normaliser Z^η is a ratio of sums of entries of P^ι over index sets fixed by conditions on the base state x_j (eq. 17, p. 14), and Z^{ω(η)} the same on rows.
The derivation. Every quantity inside δ is a sum of entries of P^ι or P^η over whole columns, so it is a function of the column marginals; δ′ is a function of the row marginals. On U(π) those marginals are the data. So D(P^ι, P^η) has one value on the feasible set. The minimiser of eq. (12) is then the minimiser of Σ κ⟨C_ω, P^ι⟩ − μH(P^ι), which depends on (κ, λ, μ) only through μ/κ, and λ enters that ratio only through the constraint κ + λ + λ′ + μ = 1. The proof of Theorem 1 states the term in this form, δ(P_n, P_{n+1}) = d(P_n 1, P_{n+1} 1), with P_n 1 “the marginals of P_n” (p. 15).
What the paper offers as evidence. Fig. 2 (p. 10) sweeps (κ, λ, μ) over the simplex and finds high error near λ = 0: “demonstrating the benefit of the do-calculus constraints in the OT problem”. Moving along that simplex at fixed μ/κ would not change the plans; moving λ at fixed μ changes μ/κ. The figure does not hold μ/κ fixed. In Table 1 (p. 10) the best COTA setting and the baseline with independent plans and no causal term have the same Wasserstein error, 0.011.
What could have failed. The normalisers could have depended on entries of a plan other than through whole-column sums, or the marginal constraints could have been relaxed; either would have made the term act. Eq. (17)‘s index sets are defined by conditions on x_j (or x′_i) only, and eq. (12) keeps the plans in U(π).
What this does not say
- That COTA’s results are wrong. The errors reported are what the method produced. They are explained by the ω-cost and the entropy weight, not by the causal term (THEORY-1137).
- That the released code solves eq. (12). Read without running, it puts every chain into one problem and gives a plan marginal constraints only the first time its intervention appears; later appearances are new variables without them, coupled by the δ term (NOTE-1342). There the term can act. That variant is not described in the paper.
- Anything about Approximate COTA (Appendix F), whose element-wise term does depend on the plans’ entries.
- That do-calculus constraints cannot help abstraction learning. A formulation in which they couple the entries of the plans, or a joint plan over interventions, is untested here.