{"id":8512,"date":"2025-11-20T07:02:29","date_gmt":"2025-11-20T07:02:29","guid":{"rendered":"https:\/\/mailitics.com\/index.php\/2025\/11\/20\/2511-14827\/"},"modified":"2025-11-20T07:02:29","modified_gmt":"2025-11-20T07:02:29","slug":"2511-14827","status":"publish","type":"post","link":"https:\/\/mailitics.com\/index.php\/2025\/11\/20\/2511-14827\/","title":{"rendered":"Implicit Bias of the JKO Scheme"},"content":{"rendered":"

Implicit Bias of the JKO Scheme
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arXiv:2511.14827v1 Announce Type: new
\nAbstract: Wasserstein gradient flow provides a general framework for minimizing an energy functional $J$ over the space of probability measures on a Riemannian manifold $(M,g)$. Its canonical time-discretization, the Jordan-Kinderlehrer-Otto (JKO) scheme, produces for any step size $eta>0$ a sequence of probability distributions $rho_k^eta$ that approximate to first order in $eta$ Wasserstein gradient flow on $J$. But the JKO scheme also has many other remarkable properties not shared by other first order integrators, e.g. it preserves energy dissipation and exhibits unconditional stability for $lambda$-geodesically convex functionals $J$. To better understand the JKO scheme we characterize its implicit bias at second order in $eta$. We show that $rho_k^eta$ are approximated to order $eta^2$ by Wasserstein gradient flow on a emph{modified} energy [ J^{eta}(rho) = J(rho) – frac{eta}{4}int_M BiglVert nabla_g frac{delta J}{delta rho} (rho) BigrVert_{2}^{2} ,rho(dx), ] obtained by subtracting from $J$ the squared metric curvature of $J$ times $eta\/4$. The JKO scheme therefore adds at second order in $eta$ a textit{deceleration} in directions where the metric curvature of $J$ is rapidly changing. This corresponds to canonical implicit biases for common functionals: for entropy the implicit bias is the Fisher information, for KL-divergence it is the Fisher-Hyv{“a}rinen divergence, and for Riemannian gradient descent it is the kinetic energy in the metric $g$. To understand the differences between minimizing $J$ and $J^eta$ we study emph{JKO-Flow}, Wasserstein gradient flow on $J^eta$, in several simple numerical examples. These include exactly solvable Langevin dynamics on the Bures-Wasserstein space and Langevin sampling from a quartic potential in 1D.<\/div>\n

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\n Peter Halmos, Boris Hanin
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Implicit Bias of the JKO Scheme arXiv:2511.14827v1 Announce Type: new Abstract: Wasserstein gradient flow provides a general framework for minimizing an energy functional $J$ over the space of probability measures on a Riemannian manifold $(M,g)$. Its canonical time-discretization, the Jordan-Kinderlehrer-Otto (JKO) scheme, produces for any step size $eta>0$ a sequence of probability distributions $rho_k^eta$ that […]<\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[62,187,113,2846,112],"tags":[4085,1297,2036],"class_list":["post-8512","post","type-post","status-publish","format-standard","hentry","category-aimldsaimlds","category-cs-ai","category-cs-lg","category-math-ap","category-stat-ml","tag-eta","tag-jko","tag-rho"],"_links":{"self":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/8512"}],"collection":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/comments?post=8512"}],"version-history":[{"count":0,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/posts\/8512\/revisions"}],"wp:attachment":[{"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/media?parent=8512"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/categories?post=8512"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mailitics.com\/index.php\/wp-json\/wp\/v2\/tags?post=8512"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}