{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T10:56:06Z","timestamp":1772103366989,"version":"3.50.1"},"reference-count":31,"publisher":"Springer Science and Business Media LLC","issue":"3","license":[{"start":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T00:00:00Z","timestamp":1772064000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T00:00:00Z","timestamp":1772064000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100001747","name":"Hong Kong Baptist University","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100001747","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Mach Learn"],"published-print":{"date-parts":[[2026,3]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    Performative prediction refers to scenarios where model predictions influence the underlying data distribution they aim to predict. A desirable property in this context is\n                    <jats:italic>performative stability<\/jats:italic>\n                    , where model predictions are already optimal for the distribution they induce, indicating converged model parameters and no need for further retraining. Achieving performative stability requires characterizing the data distribution map\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\mathcal {D}(\\theta )$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    , i.e., the relationship between predictions and the resulting distribution shifts. Current studies typically quantify distribution differences using metrics like\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\mathcal {W}_1$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    distance or\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\chi ^2$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    divergence, which may not provide isometric embeddings or maintain metric equivalence in practical scenarios, limiting their applicability across various data distribution maps. Moreover, the crucial smoothness parameter\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\beta $$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    in existing work is often unobtainable in performative scenarios, constraining the real-world utility of current theoretical results and methods. To address these challenges, we develop an algorithm that learns a performatively stable model for arbitrary data distribution maps without requiring the joint smoothness parameter\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\beta $$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    . Specifically, we introduce a new\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\hat{\\varepsilon }$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    -sensitivity measure for\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\mathcal {D}(\\theta )$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    , quantified by the gradient of the loss function, which naturally and directly characterizes how distribution shifts affect the optimization of the objective function. Based on this sensitivity, we formulate a\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\gamma $$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    -strongly convex loss function and optimize the deployed model accordingly, where\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\gamma $$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    is derived from the defined\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\hat{\\varepsilon }$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    , eliminating the need for the\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\beta $$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    -joint smoothness assumption. Our theoretical results guarantee the convergence of the deployed model to performative stability. Extensive experiments on synthetic and real-world datasets with diverse data distribution maps demonstrate the superiority of our method over state-of-the-art techniques in two key aspects: prediction accuracy and performative stability.\n                  <\/jats:p>","DOI":"10.1007\/s10994-025-06980-1","type":"journal-article","created":{"date-parts":[[2026,2,26]],"date-time":"2026-02-26T09:56:27Z","timestamp":1772099787000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Performative Prediction in the Wild: Adapting to Arbitrary Data Distribution Maps"],"prefix":"10.1007","volume":"115","author":[{"given":"Guangzheng","family":"Zhong","sequence":"first","affiliation":[]},{"given":"Yang","family":"Liu","sequence":"additional","affiliation":[]},{"given":"Ruichen","family":"Liu","sequence":"additional","affiliation":[]},{"given":"Jiming","family":"Liu","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2026,2,26]]},"reference":[{"key":"6980_CR1","doi-asserted-by":"crossref","unstructured":"Bauer, H. 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