Source Clarifications

Source: Performative Prediction, PMLR 119 and its official supplement. RRM denotes repeated population risk minimization; RERM uses empirical risk, and REGD uses empirical projected gradients.

Proposition 3.6(c): compatible regularity constants

• Arbitrary positive $\beta,\gamma$ → $0<\gamma\leq\beta$ (Proposition 3.6, p. 5). Strong convexity gives $\gamma|u-v|^2\leq(\ell'(u)-\ell'(v))(u-v)$; gradient smoothness bounds the right side by $\beta|u-v|^2$. Thus a nontrivial parameter interval requires $\gamma\leq\beta$.
• The squared-loss example for equal constants → $\ell(z;\theta)=\gamma\theta^2/2-\beta z\theta$ for every compatible pair (Appendix E.2). With $D(\theta)=\delta_{1+\epsilon\theta}$, RRM gives $G(\theta)=\beta/\gamma+(\beta\epsilon/\gamma)\theta$. At the paper’s threshold $\epsilon\geq\gamma/\beta$, the multiplier is at least one and the intercept is positive, so the orbit from zero diverges. At equality, each step adds $\beta/\gamma$. Here $\delta_x$ is point mass at $x$.

Proposition 3.6(a)–(b): concrete counterexample models

• Part (b)’s unspecified large hinge penalty → $C=\gamma(\epsilon^{-2}+2)$, with $a=-1/(2\epsilon)$ and parameter interval $[a,2]$. For every $\epsilon,\gamma>0$, this satisfies $\gamma(1-a)\leq2C\epsilon$ and $2\gamma\leq C$, making the frozen-risk minimizers alternate $2,a,2,a,\ldots$ in the source’s regularized-hinge construction (Appendix E.2). This is a witness choice, not a population assumption.

Theorem 3.10: dimension and adaptive uniformity

Source: Theorem 3.10, p. 6; Appendix E.4.

• One printed all-dimension sample budget → separate checked routes, without the printed big-O formula. The direct RERM route covers every positive data dimension with an explicit selected-shell schedule: inverse-square times one radius logarithm at $m=1$, inverse-square times a squared-radius logarithm at $m=2$, and the supercritical inverse-dimension power at $m>2$. The REGD route remains the explicit supercritical ($m>2$) schedule. Here $m$ is data dimension, $\epsilon$ distribution sensitivity, $\delta$ target distance, and $p$ total failure probability. A Bernoulli mass estimate fluctuates on scale $n^{-1/2}$, exactly its Wasserstein error on $\{0,1\}$, so the printed $r^{-1}$ sample dependence at $m=1$ does not supply accuracy $r. Fournier–Guillin Theorems 1–2 distinguish $m=1$, $m=2$ with a logarithmic correction, and $m>2$.
• RERM regularity on $\Theta$ and $Z=\bigcup_{\theta\in\Theta}\operatorname{supp}D(\theta)$ → the source-domain model carrier. The recovered direct raw-$W_1$ RERM route uses its laws, loss integrability, A1/A2 lower model, sensitivity, and minimization only on the declared feasible domain. It does not retain the historical data-Lipschitz loss bridge. The separate RGD finite-sample route still records its older global analytic regularity boundary explicitly.
• A finite exponential moment separately at each deployment → one common bound over every law reached along every adaptive history: $$\int e^{\lambda\|z\|^\alpha}\,dD(\theta)(z)\leq K, \qquad\lambda>0,\quad\alpha>1+s,\quad0<s\leq1.$$ Pointwise finiteness does not itself give a uniform $K$ for an adaptive sequence. This is a substantive uniformity premise of the current proof; the batch selector splits bounded-region and tail errors and allocates round failure budgets $6p/[\pi^2(t+1)^2]$. Necessity of the uniform bound under all source hypotheses is unresolved.
• Explicit sample-count formula → dimension-regime schedules; logarithmic burn-in → recovered source-style entry threshold. For contraction factor $q$, target radius $R$, sampling perturbation $e$, and outer factor $q\leq q_{\rm out}<1$, retain $$e\leq(q_{\rm out}-q)R,\qquad e\leq(1-q)R.$$ These inequalities keep sampling noise within the contraction slack. The proof instantiates the source-style condition $$t_0\ \geq\ \frac{\log(d_0/R)}{1-q_{\rm out}}$$ (with the RERM outer factor $2\epsilon\beta/\gamma$) and then keeps every later iterate within $R$ with probability at least $1-p$. The printed common all-dimension sample-count formula is not valid in every positive dimension; the proved dimension-regime schedules replace it.
• RERM gradient regularity → direct minimizer perturbation, without a data-Lipschitz loss comparison. The recovered RERM proof uses A1’s data-gradient bound and A2’s strong-convexity lower model to convert its raw $W_1$ event directly into a parameter perturbation. REGD instead uses $e=h\beta W$ for step $h\geq0$. Both branches retain the integrability and numerical tolerances their respective adaptive concentration constructions actually consume.

Proposition 4.1: stable-point existence

• Continuity of the loss and distribution map on the source domain $\Theta$ and support union $Z$ → joint continuity of the decoupled expected risk on $\Theta\times\Theta$. This remains the explicit continuity bridge needed by the compact best-response proof. The current model is defined on $\Theta$, so it does not require a total off-domain law or loss extension; pointwise loss convexity is likewise imposed only on the feasible domain.
• Compact $\Theta$ in the displayed proposition → nonempty compact convex $\Theta$. Convexity is already part of the paper’s global parameter-space setup, so making it explicit here adds no source-model restriction. A compact nonconvex example shows only that the displayed proposition would be false if read without that global setup.

Proposition 4.2: valid Bernoulli probabilities

• Only $|\mu+\epsilon|\leq1/2$ → also $|\mu|\leq1/2$ for the Appendix E.5 witness $Y\mid X\sim\mathrm{Bernoulli}(1/2+(\mu+\epsilon\theta)X)$, uniform $X\in\{-1,1\}$ and $\theta\in[0,1]$. Both endpoint bounds are needed for valid intermediate probabilities.
• Valid parameters: $\mu=-3/8$, $\epsilon=3/4$ give probabilities in $[1/8,7/8]$. With $f_\phi(x)=\phi x+1/2$ and squared loss, $$\operatorname{PR}(\theta)=\tfrac14+\tfrac34\theta-\tfrac12\theta^2.$$ This is strictly concave although the loss is strongly convex and jointly smooth, giving the intended counterexample with a valid data law.

Theorems 3.5 and 3.8: recovered source-domain scope

• The closed convex source parameter space $\Theta$ → the actual Lean parameter carrier. MeasurePerformativeModelOn defines laws, losses, and expected risks only for admissible parameters. Theorem 3.5 now proves its transport contraction directly from on-domain A1/A2, loss integrability, and Wasserstein sensitivity; it has no off-domain or population-gradient integrability premise. Theorem 3.8 records expected-gradient integrability on $\Theta$ because that is the population update displayed in the source, then proves its projected contraction and rate on the same domain. This is a clarification of the paper’s model domain, not an additional economic assumption or a change to either conclusion.
• The source’s finite-dimensional $Theta\subseteq\mathbb R^d$ → explicit finite-dimensional Lean parameter spaces. Theorem 3.8 does not separately assume $\gamma\leq\beta$: A1/A2 derive it on every non-singleton domain; the singleton case satisfies the displayed conclusion directly.

Historical all-ambient route for Theorems 3.5 and 3.8

• A1, A2, Wasserstein sensitivity, and analytic conditions on the source domain $\Theta$ and support union $Z$ → the same conditions over every ambient parameter and every datum in the chosen Lean data carrier. The prior targets also required expected-loss and gradient integrability for every ambient deployed/evaluated parameter pair, together with local loss measurability and pointwise parameter differentiation there. Their contraction, stability, and convergence conclusions remain restricted to $\Theta$. Making the model total outside $\Theta$ is a representation choice, and choosing the data carrier as $Z$ removes the apparent off-support strengthening. The parameter-global hypotheses still restrict the prior formalization beyond the paper; no derivation from the source-domain assumptions is proved, and necessity is unresolved.

Corollary 5.1: recovered strategic source-domain scope

• The strategic-classification corollary → its domain-relative strategic profile and Stackelberg benchmark. The current endpoint keeps the source’s closed convex parameter domain, induced distribution, selected response, and Stackelberg comparison. It obtains RRM convergence from the recovered domain-relative theorem and proves the displayed objective-gap upper bound. No whole-space extension or performative-optimum substitution is required.
• The source’s Euclidean parameter setting → an explicit finite-dimensional parameter-space binder. This is a source specialization of a previously more-general Lean endpoint, not an added condition.

Historical whole-space route for Corollary 5.1

• RRM, performative optimality, stability, and regularity on the source’s closed convex domain $\Theta$ → RRM, optimality, stability, A1, A2, Wasserstein sensitivity, loss Lipschitzness, integrability, measurability, and differentiation on the entire ambient parameter carrier. The current prior target had no separate domain argument and started from any ambient initial point. Choosing the Lean data carrier as $Z$ can make its all-datum quantifiers source-relative; the whole-parameter-space restriction remains. No reduction from the source-domain corollary is proved, and necessity of this stronger scope is unresolved.

• Strategic-classification Stackelberg conclusion → performative-optimum comparison. The target does not identify the performative optimum with a Stackelberg equilibrium; that interpretation bridge remains unproved.