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.

• All dimensions with sample budget $n_t=O((\epsilon\delta)^{-m}\log(t/p))$ → $m>2$ and an explicit sufficient batch selector, without the printed big-O budget. 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 displayed $r^{-1}$ 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 $m>2$.
• Regularity on $\Theta$ and $Z=\bigcup_{\theta\in\Theta}\operatorname{supp}D(\theta)$ → regularity on the entire parameter and data spaces. A1/A2 hold for all parameter/data pairs; sensitivity, loss/gradient integrability, and loss differentiation hold for all ambient parameters. RERM’s data-Lipschitz loss bound is global too. This restricts the models covered by the current proof; necessity of the stronger scope is unresolved.
• 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 and burn-in formulas → sufficient error budgets and some finite entry time. 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 derives a finite $t_0$ after which the iterates stay within $R$ with probability at least $1-p$. Matching the paper’s displayed sample-count and logarithmic burn-in bounds remains unproved.
• RERM gradient regularity alone → an additional data-Lipschitz loss comparison: for loss constant $L_z$, Wasserstein tolerance $W$, and risk tolerance $\tau$, require $L_zW\leq\tau$ and use $e=\sqrt{4\tau/\gamma}$. Strong convexity converts risk error to parameter error; gradient smoothness alone does not give this loss-Lipschitz premise. REGD instead uses $e=h\beta W$ for step $h\geq0$. Both branches need integrable sampled quantities and tolerances fitting the concentration selector’s bounded-region and tail budgets. Necessity of these current-proof premises is unresolved.

Proposition 4.1: stable-point existence

• Continuity of the loss and distribution map on the source domain $\Theta$ and support union $Z$ → expected-loss integrability for every ambient deployed/evaluated parameter pair and joint continuity of the decoupled expected risk on the full ambient parameter product. The current target also requires loss convexity on $\Theta$ for every datum in the chosen data carrier. Defining a total model outside $\Theta$ is a representation choice, and taking the data carrier to be $Z$ makes the all-datum quantifier source-relative. The global integrability and continuity conditions still constrain the off-domain extension; they are sufficient for the current fixed-point proof, and their necessity is unresolved.
• 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: population scope

• 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 current targets also require 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 current formalization beyond the paper; no derivation from the source-domain assumptions is proved, and necessity is unresolved.

Corollary 5.1: whole-space population scope

• 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 target has no separate domain argument and starts 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.