{"id":"55cbcc4d-1978-4fc9-9d1f-0ff53c86e7f2","arxiv_id":"2412.04346","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"The paper defines the distributionally robust performative optimum (DRPO), minimizes worst-case performative risk over a KL uncertainty set of distribution maps, and proves it approximates the true performative optimum under misspecification.","lead":"Performative prediction models how a deployed machine learning model changes the data it later sees. This paper adds a robust safeguard: a distributionally robust performative optimum that stays dependable even when the model's assumptions about that data shift are wrong.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 3.3 is false as stated: the proof drops PR(θPO,true)−PRtrue(θPO,true), so coverage at θDRPO alone cannot yield the advertised local bound.","rationale":"The paper's central claim is that the DRPO localizes excess risk to the true PO under map misspecification. The load-bearing step is Proposition 3.3. On inspection, the proof does not establish the advertised bound: the last equality in Appendix D.3 conflates the KL-DRO sensitivity gap relative to the nominal map with the gap relative to the true map. The difference PR(θPO,true)−PRtrue(θPO,true) is not controlled by coverage at θDRPO, and the finite Bernoulli example shows the stated theorem is actually false. This is an internal inconsistency, not merely a disagreement with the field, and it directly undermines the strongest claim. The reader's weakest assumption correctly pointed at the KL coverage condition, but that condition is necessary yet not sufficient; the missing condition is coverage of the true map at the true PO (or uniformly over Θ), or an explicit extra term in the bound. I would not reject the paper outright: the DRPO formulation, the dual reformulation, and the experiments are valuable, and the proof is repairable by strengthening the assumption or weakening the theorem statement. The verdict should remain conditional, but the condition must include fixing Proposition 3.3 before the local guarantee is accepted.","tokens_in":20894,"tokens_out":17653,"duration_ms":177768,"concrete_test":"Evaluate the finite counterexample: Θ={A,B}, Z={0,1}, D(A)=Bernoulli(0.1), Dtrue(A)=Bernoulli(0.11), D(B)=Bernoulli(0.1), Dtrue(B)=Bernoulli(0.9), ℓ(1;A)=5, ℓ(0;A)=0, ℓ(0;B)=1, ℓ(1;B)=0, ρ=0.001. Compute DRPR(A)=5·q_max and DRPR(B)=1−q_min by solving KL(q||0.1)=0.001, verify θDRPO=A, and compare PRtrue(A)−PRtrue(B) with √(ρ Var_{Dtrue(B)}[ℓ(·;B)]). If LHS≈0.45 and RHS≈0.0095, Proposition 3.3 is falsified.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The last step of Appendix D.3 asserts DRPR(θ*PO)−PRtrue(θ*PO)=√(ρ Var_{D(θ*PO)}[ℓ(·;θ*PO)])+o(√ρ). This does not follow from the KL-DRO sensitivity theorem, which controls DRPR(θ*PO)−PR(θ*PO), the gap to the nominal distribution D(θ*PO). The remaining term PR(θ*PO)−PRtrue(θ*PO) is not controlled by the stated assumption D(Dtrue(θDRPO)||D(θDRPO))≤ρ; it can be O(1). A finite counterexample: Θ={A,B}, Z={0,1}; D(A)=Bernoulli(0.1), Dtrue(A)=Bernoulli(0.11), D(B)=Bernoulli(0.1), Dtrue(B)=Bernoulli(0.9); ℓ(1;A)=5, ℓ(0;A)=0, ℓ(0;B)=1, ℓ(1;B)=0; ρ=0.001. Then KL(Dtrue(A)||D(A))≈0.00054≤ρ, so the hypothesis holds. DRPR(A)=5·q_max≈0.567 and DRPR(B)=1−q_min≈0.913, so θDRPO=A. The true PO is B, with PRtrue(B)=0.1 and PRtrue(A)=0.55, so the excess risk is 0.45. The right-hand side of (3.3) is √(0.001·Var_{Bernoulli(0.9)}[ℓ(·;B)])=√(0.001·0.09)≈0.0095. Thus the central local guarantee is violated by a factor of roughly 47. The fix is to require D(Dtrue(θPO,true)||D(θPO,true))≤ρ (or uniform coverage over Θ), or to add the extra misspecification term to the bound; Proposition C.1 inherits the same flaw at its θ*PO term.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a distributionally robust version of performative prediction: given a nominal distribution map D, it defines the distributionally robust performative risk DRPR(θ) as the worst-case expected loss over a KL ball of distribution maps around D, and studies the resulting solution concept DRPO. The main theoretical contribution is an excess-risk comparison claiming that, when the true map is covered by the KL ball at θDRPO, the DRPO has excess risk no larger than sqrt(ρ Var_{D(θPO,true)}[ℓ]) + o(sqrt ρ), thereby 'localizing' the error to the true performative optimum, whereas the standard PO has an excess-risk bound involving a supremum over the whole parameter space. The paper also gives a strong-duality reformulation, an alternating-minimization algorithm, a tilted-risk interpretation, calibration procedures for ρ, and experiments in strategic classification, partial identification, and fairness settings.","tokens_in":21259,"tokens_out":7128,"duration_ms":68108,"significance":"If the central theorem were correct, this would be a useful and timely contribution: it connects KL-divergence DRO with performative prediction, gives a clean dual reformulation, and provides a concrete robust alternative to the usual PO under distribution-map misspecification. The paper also honestly notes that its comparison of upper bounds is not tight, and the empirical study is informative. However, the central claim of Proposition 3.3 is false as stated, and the flaw propagates to Proposition C.1. Since the advertised 'localization to the true PO' is the paper's main theoretical selling point, the manuscript needs substantive revision before the claims can be accepted.","major_comments":[{"comment":"The proof of Proposition 3.3 is invalid at its last step. The paper claims DRPR(θ*PO) − PRtrue(θ*PO) = sqrt(ρ Var_{D(θ*PO)}[ℓ(Z; θ*PO)]) + o(sqrt ρ), citing the sensitivity property of KL-DRO. That sensitivity result controls DRPR(θ*PO) − PR(θ*PO), i.e., the gap with respect to the nominal distribution D(θ*PO), not with respect to the true distribution Dtrue(θ*PO). The remaining term PR(θ*PO) − PRtrue(θ*PO) is not controlled by the stated assumption D(Dtrue(θDRPO)∥D(θDRPO)) ≤ ρ. The statement is false as written: take Θ = {A,B}, Z = {0,1}, D(A) = Bernoulli(0.1), Dtrue(A) = Bernoulli(0.11), D(B) = Bernoulli(0.1), Dtrue(B) = Bernoulli(0.9), ℓ(1;A)=5, ℓ(0;A)=0, ℓ(0;B)=1, ℓ(1;B)=0, and ρ = 0.001. Then KL(Dtrue(A)∥D(A)) ≈ 0.00054 ≤ ρ, DRPR(A) ≈ 0.567, DRPR(B) ≈ 0.913, so θDRPO = A; the true PO is B, the excess risk is PRtrue(A) − PRtrue(B) = 0.45, while the right-hand side of (3.3) is about 0.0095. Coverage at θDRPO alone is therefore insufficient, and the theorem needs either coverage at θPO,true (or uniformly over Θ) or an added misspecification term.","section":"Proposition 3.3 and Appendix D.3"},{"comment":"Proposition C.1 inherits the same flaw. In the proof in Appendix D.4, the term DRPR(θ*PO) − PRtrue(θ*PO) is again replaced by the KL-DRO sensitivity bound, which only controls the gap to the nominal distribution D(θ*PO). The projection term involving P at θDRPO does not repair the unmodeled misspecification at θ*PO. The stated generalized bound is therefore not established, and the claim that it reduces to Proposition 3.3 when D(Dtrue(θDRPO)∥D(θDRPO)) ≤ ρ does not rescue either statement.","section":"Proposition C.1 and Appendix D.4"},{"comment":"The convergence claim for Algorithm 1 is unsupported. The text states that 'the alternating minimization algorithm in common practice guarantees global convergence (to stationary point)' and that strong convexity gives convergence to a global minimum, but no assumptions are stated under which ψ(θ, µ) is sufficiently well behaved, and θ-updates are only assumed to be solved by an unspecified performative-risk minimization method. Appendix I later says convergence guarantees must be established case-by-case, which is inconsistent with the main-text claim. Please provide a formal convergence theorem with explicit conditions or remove the unqualified claim.","section":"Section 4.1 and Appendix I"}],"minor_comments":[{"comment":"There are numerous typos and small errors, including 'advatanges' in the abstract, 'Pinksker's' in Appendix D.2, 'hitogram' in Appendix B, 'Precicely' and 'estiamte' in Section 5.2, 'misspeicification' and 'indentified' in Section 5.2, 'Algirthm' in Appendix H, and 'reformualtion' in Appendix H.","section":"Throughout"},{"comment":"The main text says each shaded region shows the standard error of the mean from 30 trials, but Appendix G.1 says the training procedure is repeated for 10 trials. Please reconcile this inconsistency.","section":"Section 5 and Appendix G.1"},{"comment":"The sentence 'we should have θTPO with α equals θDRPO with ρ = (µ⋆)^{-1}(1/α)' is garbled. For fixed µ, argmin_θ ψ(θ, µ) equals argmin_θ E_{Z∼D(θ)}[e^{ℓ(Z;θ)/µ}], so the correspondence should be written explicitly as θTPO(α) = θDRPO(ρ(α)) with α = 1/µ⋆(ρ), making the dependence on ρ and α clear.","section":"Section 4.2"},{"comment":"The sentence 'the excess risks of them are comparable, thus doing no harm' overclaims: comparing two upper bounds that are not tight does not establish that the actual excess risks are comparable, so this conclusion should be softened or removed.","section":"Section 3.2"}],"recommendation":"major_revision","confidential_remarks":"The central theoretical claim of the paper is false as stated, as shown by a small finite counterexample. The good news is that the error is localized to the proof's last step and the framework could be repaired by changing the coverage assumption to θPO,true (or uniform coverage) or by adding an explicit misspecification term. However, the advertised message—that DRPO localizes the excess risk to the true PO under coverage only at θDRPO—would need to be substantially revised. I recommend major revision rather than outright rejection, but the revised paper must contain a correct proof of the main excess-risk statement and a clear discussion of what assumption is actually needed."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nYou should know two things about this paper before deciding whether to trust it. First, the distributionally robust performative optimum (DRPO) is a genuinely new and sensible idea: minimize worst-case performative risk over a KL ball of distribution maps, and the dual reformulation as an augmented/tilted performative problem is clean and connects nicely to standard DRO machinery. Second, the paper's central theoretical guarantee — Proposition 3.3 — is not proved and appears false as stated.\n\nThe issue is in Appendix D.3. The proof bounds PRtrue(θDRPO) − PRtrue(θ*PO) by DRPR(θ*PO) − PRtrue(θ*PO), then asserts this equals √(ρ Var_{D(θ*PO)}[ℓ]) + o(√ρ) via the KL-DRO sensitivity theorem. That theorem controls DRPR(θ) − PR(θ), the gap to the nominal distribution. The additional term PR(θ*PO) − PRtrue(θ*PO) is not controlled by the assumption KL(Dtrue(θDRPO)∥D(θDRPO)) ≤ ρ; it can be O(1). The stress-test example is correct: a two-point model with D(A)=Bernoulli(0.1), Dtrue(A)=Bernoulli(0.11), D(B)=Bernoulli(0.1), Dtrue(B)=Bernoulli(0.9), and losses that make B the true PO while the DRPO is A, gives excess risk 0.45 versus a claimed bound of about 0.0095. Proposition C.1 inherits the same flaw at its θ*PO term.\n\nThe rest of the paper is in decent shape. The duality (Prop 3.1) is standard and correctly cited. The excess-risk bound for the PO (Prop 3.2) is a straightforward Pinsker argument. The experiments are honest and show that robustness helps under moderate-to-large misspecification, which is a useful qualitative observation even if the theory is currently overstating the guarantee. Algorithm 1's convergence claim is hand-wavy, though the appendix walks it back to case-by-case convergence. No code is provided, which is a minor reproducibility setback but not disqualifying.\n\nWhat would fix this? Either strengthen the assumption to uniform coverage (KL(Dtrue(θ)∥D(θ)) ≤ ρ for all θ) — which would make the proof go through with an extra sup term — or add the misspecification gap at θ*PO to the bound, which weakens the localization claim substantially. As written, the central theorem is not salvageable in place.\n\nThis paper is worth a serious referee: the DRPO concept and the tilted reformulation are likely to be reused, and the empirical findings are suggestive. But the authors need to correct the theorem and its proof before the main claim can be trusted.\n\nBest,\n[Your name]","headline":"Good new concept, clean duality, but the main excess-risk theorem (Prop 3.3) is false as stated: the proof drops the misspecification gap at the true PO, so the advertised localization to the true PO is not established.","tokens_in":21815,"tokens_out":4297,"would_cite":false,"duration_ms":36616,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper's core claim is that the distributionally robust performative optimum (DRPO) minimizes the worst-case performative risk over a KL-divergence ball of distribution maps, and when the true map falls inside that ball at the DRPO…","keywords":["performative prediction","distributionally robust optimization","performative optimum","KL divergence","distribution map misspecification","tilted risk minimization","strategic classification","fairness without demographics"],"falsifier":"Take a one-dimensional location family with nominal map $D(\\theta)=\\mathcal{N}(\\theta,\\sigma^2)$, true map $D_{\\mathrm{true}}(\\theta)=\\mathcal{N}((1+\\delta)\\theta,\\sigma^2)$, quadratic loss $\\ell(z;\\theta)=(z-\\theta)^2$, and vary $\\rho$ and $\\delta$. Compute the exact DRPO and true PO; if, for $\\delta$ values that keep $D_{\\mathrm{KL}}(D_{\\mathrm{true}}(\\theta_{\\mathrm{DRPO}})\\|D(\\theta_{\\mathrm{DRPO}}))\\le \\rho$, the ratio $E(\\theta_{\\mathrm{DRPO}})/\\sqrt{\\rho \\operatorname{Var}_{D(\\theta_{\\mathrm{PO,true}})}[\\ell]}$ does not stay bounded as $\\rho\\to 0$, Proposition 3.3 is false; if coverage is violated, compare the empirical excess risk against the two-term bound of Proposition C.1.","tokens_in":20642,"feed_emoji":"🛡️","tokens_out":7316,"duration_ms":67675,"temperature":0.7,"pith_summary":"Performative prediction wants a model that is good after it changes the data distribution it predicts, but the map describing that change is never known exactly. This paper proposes to optimize, instead of the nominal performative risk, the worst-case performative risk over a KL-divergence ball of distribution maps around the nominal map, and calls the minimizer the distributionally robust performative optimum (DRPO). The key claim is that when the true distribution map falls inside that ball at the DRPO, the DRPO's excess risk relative to the true optimum is bounded by the square root of the loss variance at the true optimum, a localized error that can be much smaller than the PO's error bound, which depends on the worst misspecification over the whole parameter space. This makes DRPO a robust approximation of the true PO under map misspecification, and the paper backs this with an efficient tilted-performative-risk reformulation and with experiments in strategic classification, partially identifiable maps, and fairness without demographics.","feed_headline":"DRPO beats the standard optimum when the map is wrong","feed_subtitle":"Its excess risk is governed by noise at the true optimum, not by worst-case misspecification over all models.","key_machinery":"The load-bearing object is the KL uncertainty collection $\\mathcal{U}(D)$ around the nominal distribution map, with radius $\\rho$ measured in KL divergence at each parameter $\\theta$. The argument runs through the strong-duality identity $\\operatorname{DRPR}(\\theta)=\\inf_{\\mu\\ge 0}\\{ \\mu \\log \\mathbb{E}_{Z\\sim D(\\theta)}[e^{\\ell(Z;\\theta)/\\mu}] + \\mu\\rho \\}$, which turns the infinite-dimensional supremum over maps into a one-dimensional dual problem and identifies the worst-case map as an exponential tilt of $D(\\theta)$. This dual form lets the authors recast DRPO as the minimizer of a tilted performative risk $\\mathbb{E}_{Z\\sim D(\\theta)}[e^{\\alpha\\ell(Z;\\theta)}]$, and their excess-risk proof uses the cancellation $\\operatorname{DRPR}(\\theta_{\\mathrm{DRPO}})-\\operatorname{DRPR}(\\theta_{\\mathrm{PO,true}})\\le 0$ plus a sensitivity expansion of the KL-DRO value at the true optimum to localize the error. The same decomposition gives a generalized bound with an extra infimum term when the coverage condition fails (Proposition C.1).","core_discovery":"Define the distributionally robust performative risk $\\operatorname{DRPR}(\\theta)=\\sup_{\\widetilde{D}\\in\\mathcal{U}(D)} \\mathbb{E}_{Z\\sim\\widetilde{D}(\\theta)}[\\ell(Z;\\theta)]$ with uncertainty collection $\\mathcal{U}(D)=\\{\\widetilde{D}: D_{\\mathrm{KL}}(\\widetilde{D}(\\theta)\\|D(\\theta))\\le \\rho \\text{ for all }\\theta\\}$, where $D$ is the nominal distribution map and $\\rho$ is the robustness radius. The distributionally robust performative optimum $\\theta_{\\mathrm{DRPO}}$ minimizes this quantity. The paper proves that if the true map is covered at the robust solution, $D_{\\mathrm{KL}}(D_{\\mathrm{true}}(\\theta_{\\mathrm{DRPO}})\\|D(\\theta_{\\mathrm{DRPO}}))\\le \\rho$, then the excess risk over the true performative optimum satisfies $E(\\theta_{\\mathrm{DRPO}})\\le \\sqrt{\\rho \\operatorname{Var}_{Z\\sim D(\\theta_{\\mathrm{PO,true}})}[\\ell(Z;\\theta_{\\mathrm{PO,true}})]}+o(\\sqrt{\\rho})$, whereas the plain PO obeys $E(\\theta_{\\mathrm{PO}})\\le \\sqrt{2B}\\sup_{\\theta}\\sqrt{D_{\\mathrm{KL}}(D_{\\mathrm{true}}(\\theta)\\|D(\\theta))}$. The DRPO bound is local: it charges only the variance of the loss at the true optimum, while the PO bound is global and can be inflated by one badly misspecified $\\theta$. The paper also shows DRPR minimization is equivalent to an augmented performative risk minimization, so off-the-shelf PO solvers can be used, and that in the Gaussian location example DRPO is exactly an $L^1$-regularized PO.","pith_inferences":["The localized form of the bound suggests a practical selection rule the authors only partially explore: if the loss variance at the true optimum is small, even a conservatively large $\\rho$ leaves the DRPO close to the oracle, so calibrating $\\rho$ against estimated variance rather than worst-case KL might be cheaper than their grid search.","Because the worst-case map in the KL ball is an exponential tilt, DRPO is implicitly regularizing the right tail of the performative loss; one testable corollary is that DRPO should improve tail quantiles such as the value-at-risk of the performative loss, not only its mean, in heavy-tailed settings.","The paper's Wasserstein-extension remark points to a concrete open target: replacing the variance term by a Lipschitz norm would let DRPO handle losses whose exponential moments diverge, which the KL dual requires."],"forward_implications":["Choosing $\\rho$ to cover the true map makes the true performative risk of $\\theta_{\\mathrm{DRPO}}$ bounded by $\\operatorname{DRPR}(\\theta_{\\mathrm{DRPO}})$, a certificate of out-of-sample performance that the ordinary PO does not offer.","Algorithmically, the dual identity reduces DRPO to alternating minimization in which the hard subproblem is exactly a performative risk minimization, so existing PO solvers can be reused with a tuned tilt parameter $\\alpha$.","When misspecification is small, the DRPO bound and the PO bound are comparable, so robustness comes at no substantial cost; when misspecification grows, DRPO can outperform PO in worst-case performative risk, as shown for strategic classification and partially identified location models.","In the fairness experiment, increasing the tilt shrinks the performative-risk gap between majority and minority subpopulations while raising population risk, offering a demographic-free fairness lever.","All excess-risk statements carry over to general $\\varphi$-divergences, so the same localized guarantee holds for other divergence-based uncertainty sets."],"supporting_citations":[{"why":"Establishes the performative prediction framework and the performative optimum/performative stability definitions that this paper builds on.","marker":"[29]"},{"why":"Proposes a plug-in method for finding the performative optimum, serving as a baseline and as a model-based inner solver for the robust formulation.","marker":"[26]"},{"why":"Gives performative gradient descent for parametric distribution maps, which the authors note can be reused inside their alternating minimization algorithm.","marker":"[14]"},{"why":"Shows that a PO computed from a misspecified nominal map can still approximate the true PO, providing the comparison point for the DRPO's advantages.","marker":"[24]"},{"why":"Supplies the KL-divergence distributionally robust optimization strong-duality result that yields the dual form of the DRPR.","marker":"[13]"},{"why":"Provides the sensitivity expansion of KL-based DRO used to derive the localized variance-scaling excess-risk bound at the true optimum.","marker":"[9]"},{"why":"Introduces repeated distributionally robust optimization for performative stability, the closest prior solution concept and the comparative benchmark in the fairness experiment.","marker":"[28]"}],"fun_headline_variants":["DRPO's risk bound depends only on local variance, not worst-case map error","Robust performative optimum: only pays for misspecification at the optimum","When the distribution map is off, DRPO still gets close to ideal","Distributionally robust PO: local error beats global worst-case"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The clean localized guarantee rests on the true distribution map being close to the nominal map at the robust optimum, in the sense that the true distribution induced at $\\theta_{\\mathrm{DRPO}}$ lies inside the KL ball of radius $\\rho$ around the nominal distribution at $\\theta_{\\mathrm{DRPO}}$; if the true map falls outside that ball, the excess-risk bound gains an extra misspecification term and the localization weakens.","fun_headline_variants_meta":{"raw":{"variants":["DRPO's risk bound depends only on local variance, not worst-case map error","Robust performative optimum: only pays for misspecification at the optimum","When the distribution map is off, DRPO still gets close to ideal","Distributionally robust PO: local error beats global worst-case"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000456,"raw_usage":{"total_tokens":2353,"prompt_tokens":1074,"completion_tokens":1279,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":690,"completion_tokens_details":{"reasoning_tokens":1200}},"tokens_in":690,"tokens_out":1279,"duration_ms":11383,"temperature":1.0,"reasoning_tokens":1200,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T21:30:37.524651+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a one-dimensional location family with nominal map $D(\\theta)=\\mathcal{N}(\\theta,\\sigma^2)$, true map $D_{\\mathrm{true}}(\\theta)=\\mathcal{N}((1+\\delta)\\theta,\\sigma^2)$, quadratic loss $\\ell(z;\\theta)=(z-\\theta)^2$, and vary $\\rho$ and $\\delta$. Compute the exact DRPO and true PO; if, for $\\delta$ values that keep $D_{\\mathrm{KL}}(D_{\\mathrm{true}}(\\theta_{\\mathrm{DRPO}})\\|D(\\theta_{\\mathrm{DRPO}}))\\le \\rho$, the ratio $E(\\theta_{\\mathrm{DRPO}})/\\sqrt{\\rho \\operatorname{Var}_{D(\\theta_{\\mathrm{PO,true}})}[\\ell]}$ does not stay bounded as $\\rho\\to 0$, Proposition 3.3 is false; if coverage is violated, compare the empirical excess risk against the two-term bound of Proposition C.1.","supporting_citations":[{"cited_title":"Outside the echo chamber: Optimizing the performative risk","cited_arxiv_id":null,"evidence_quote":"Proposes a plug-in method for finding the performative optimum, serving as a baseline and as a model-based inner solver for the robust formulation."},{"cited_title":"How to learn when data reacts to your model: performative gradient descent","cited_arxiv_id":null,"evidence_quote":"Gives performative gradient descent for parametric distribution maps, which the authors note can be reused inside their alternating minimization algorithm."},{"cited_title":"Plug-in Performative Optimization","cited_arxiv_id":"2305.18728","evidence_quote":"Shows that a PO computed from a misspecified nominal map can still approximate the true PO, providing the comparison point for the DRPO's advantages."},{"cited_title":"Kullback-leibler divergence constrained distributionally robust optimization","cited_arxiv_id":null,"evidence_quote":"Supplies the KL-divergence distributionally robust optimization strong-duality result that yields the dual form of the DRPR."},{"cited_title":"Long Term Fairness for Minority Groups via Performative Distributionally Robust Optimization","cited_arxiv_id":"2207.05777","evidence_quote":"Introduces repeated distributionally robust optimization for performative stability, the closest prior solution concept and the comparative benchmark in the fairness experiment."}],"review_version":1}