Source
Perdomo, Zrnic, Mendler-Dünner and Hardt (2020), ICML 2020, LIT-1336, read in full in NOTE-1044 from arXiv v4: Example 2.2, Proposition 4.2 (Appendix D.6) and Theorem 4.3 (Appendix D.7). The re-derivations below are the record’s, following the paper.
What was actually shown
Write DPR(θ, θ′) = E_{Z∼D(θ)} ℓ(Z; θ′). θ_PO minimizes DPR(θ, θ); θ_PS minimizes DPR(θ_PS, ·).
They differ. Example 2.2: X uniform on {±1}, Y | X ~ Bernoulli(½ + µX + εθX), predictor θx + ½, squared loss. Since E[Y − ½ | X] = (µ + εθ)X, X² = 1 and (Y − ½)² = ¼,
DPR(θ, θ′) = ¼ − 2θ′(µ + εθ) + θ′².
Optimizing θ′ for fixed θ gives θ′ = µ + εθ, whose fixed point is θ_PS = µ/(1 − ε). Setting θ′ = θ gives PR(θ) = ¼ − 2µθ + (1 − 2ε)θ², so θ_PO = µ/(1 − 2ε) for ε < ½. They coincide iff ε = 0. At µ = 0.2, ε = 0.3, retraining converged to 0.2857 = θ_PS, while θ_PO = 0.5, with performative risks 0.168 and 0.150. For ε ∈ [½, 1), PR is concave in θ, although ε < γ/β = 1, so retraining converges and direct minimization of PR is non-convex (Proposition 4.2).
The gap is bounded. Let d = ‖θ_PO − θ_PS‖.
- Strong convexity and the first-order condition at θ_PS (Lemma C.1) give DPR(θ_PS, θ_PO) − DPR(θ_PS, θ_PS) ≥ (γ/2)d².
- ℓ(·; θ_PO) is L_z-Lipschitz in z, so by Kantorovich–Rubinstein and ε-sensitivity, DPR(θ_PS, θ_PO) − DPR(θ_PO, θ_PO) ≤ L_zεd.
- Optimality of θ_PO gives DPR(θ_PO, θ_PO) ≤ DPR(θ_PS, θ_PS).
So (γ/2)d² ≤ DPR(θ_PS, θ_PO) − DPR(θ_PS, θ_PS) ≤ DPR(θ_PS, θ_PO) − DPR(θ_PO, θ_PO) ≤ L_zεd, and d ≤ 2L_zε/γ. It holds for every stable point and every optimum, so neither need be unique.
What could have failed: stability is a condition at one point, optimality a condition over all deployments, and nothing forces them near each other in general. Strong convexity in θ and Lipschitz dependence on z are what tie them.
What this does not say
- Not that stable points exist. Existence needs Theorem 3.5’s conditions (THEORY-861) or Proposition 4.1’s compactness and convexity.
- Not how to find the optimum. θ_PO depends on how D(θ) varies away from the deployed model, which retraining never observes.
- Not a bound in risk. Corollary 5.1 converts it to risk when ℓ is also L_θ-Lipschitz in θ, giving 2L_zε(L_θ + L_zε)/γ.
- Not that a stable model is wrong. It is calibrated for the world it makes. A26: “A performatively stable model isn’t necessarily false, unjust, or captured.”
Connections
- CLAIM-402. A category confirmed by the outcomes its own use produces is the stable point. This shows the stable point can be a worse choice than an alternative that would have produced different evidence. That is the claim’s reason to model the effect of use, made exact in one family of models.
- CLAIM-407. Closure under ordinary dynamics does not preserve intervention responses. In the same spirit, optimality under the deployed distribution does not give optimality under the map D(·).