Variational Online Mirror Descent for Robust Learning in Schrödinger Bridge
Authors: Dong-Sig Han
Jaein Kim
HEE BIN YOO
Byoung-Tak Zhang
Venue: Accepted by TMLR ()

Abstract: The Schrödinger bridge (SB) has evolved into a universal class of probabilistic generative models. In practice, however, estimated learning signals are innately uncertain, and the reliability promised by existing methods is often based on speculative optimal case scenarios. Recent studies regarding the Sinkhorn algorithm through mirror descent (MD) have gained attention, revealing geometric insights into solution acquisition of the SB problems. In this paper, we propose a variational online MD (OMD) framework for the SB problems, which provides further stability to SB solvers. We formally prove convergence and a regret bound for the novel OMD formulation of SB acquisition. As a result, we propose a simulation-free SB algorithm called Variational Mirrored Schrödinger Bridge (VMSB) by utilizing the Wasserstein-Fisher-Rao geometry of the Gaussian mixture parameterization for Schrödinger potentials. Based on the Wasserstein gradient flow theory, the algorithm offers tractable learning dynamics that precisely approximate each OMD step. In experiments, we validate the performance of the proposed VMSB algorithm across an extensive suite of benchmarks. VMSB consistently outperforms contemporary SB solvers on a wide range of SB problems, demonstrating the robustness as well as generality predicted by our OMD theory.

Published in Transactions on Machine Learning Research (01/2026)
Variational Online Mirror Descent for Robust Learning in Schrödinger Bridge
Dong-Sig Han,' 1 Jaein Kim,² Hee Bin Yoo,² and Syoung-Tak Zhang²
DDepartment of Computing, Imperial College London Artificial Intelligence Institute, Seoul National University
Reviewed on OpenReview: https: openreview. net/ forum? id= g3SsM9FLpm
Abstract
The Schrödinger bridge (SB) has evolved into a universal class of probabilistic generative models. In practice, however, estimated learning signals are innately uncertain, and the reliability promised by existing methods is often based on speculative optimal case scenarios. Recent studies regarding the Sinkhorn algorithm through mirror descent (MD) have gained attention, revealing geometric insights into solution acquisition of the SB problems. In this paper, we propose a variational online MD (OMD) framework for the SB problems, which provides further stability to SB solvers. We formally prove convergence and regret bound for the novel OMD formulation of SB acquisition. As result, propose simulation- free SB algorithm called Variational Mirrored Schrödinger Bridge (VMSB) by utilizing the Wasssrsttenn-isshere
geometry of the Gaussian mixture parameterization for Schrödinger potentials. Based on the Wasserstein gradient flow theory, the algorithm offers tractable learning dynamics that precisely approximate each OMD step. In experiments, we validate the performance of the proposed VMSB algorithm across an extensive suite of benchmarks. VMSB consistently outperforms contemporary SB solvers wide range of SB problems, demonstrating the robustness well as generality predicted by our OMD theory.
1 Introduction
The Schrödinger bridge (SB; Schrödinger 1932) has emerged a universal class of probabilistic generative models. Nevertheless, the methodologies used in SB algorithms remain somewhat atypical, as each algorithm involves a series of sophisticated approaches to derive a solution. The excessive specialization in existing algorithms arises from the absence of a unified perspective, which has led contemporary studies to rest on overly idcalized or unrcalistic assumptions. To address the issue, a formal discussion of the SB algorithms' robustness, akin to discussions in classical optimization theory (Xu et al., 2008 Duchi & Namkoong. 202T) is beneficial for ensuring the stability and reliability of their solutions against uncertainty, thereby improving overall performance. Considering the problem as an instance of ordinary optimization opens avenues for progress, particularly for improving stability.
We draw inspiration from the rich geometric properties of the Schrödinger bridge problem (SBP; Léonard 2013) induced by entropy regularization (Nutz 2021), which is closely related to the field of information geometry (Amari 2016). In this type of geometry, mirror descent (MD; Nemirovsky & Yudin 1983)
provides natural methodology for discrete updates, which has proven effective various scenarios (Beck
&c Tcboulle, 2003b Shalev-Shwartz et al., 2012 Amid & Warmuth, 2020a b. Ghai et al. 2020 Han ct al.
2022). For example, given a strictly convex function Ω R& IR { [+0)) the Euclidean space, update
of classical MD with respect to cost function F 12 + R U [+oo} is formulated
v v((w)) 7), VF(w),
where the gradient operator V defines an injective mapping from the parameter space to the dual space. Recent studies have shown that the Sinkhorn algorithm for training SB models can be reinterpreted

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Primal Space
Dual Space
A technical overview. Combining chracter- istics of existing methods, VMSB offers simulation- free SB solver that produces iterative OMD solutions. Our VMSB additionally provides a strong theoretical guarantee of convergence based on a regert analysis.
Table 1:
I1,
D
Wi
6,52
M
empirical estimates
Iterative
Simulation-free
Regret analysis
DSB
Figure 1: A schematic illustration. The primal and dual spaces (C,D) retain bidirectional maps (662) II and 1 indicate projection spaces of "1 μ and 92 = v, respectively. The current Te performs an online learning update following "un- reliable" leader oi in a region shaded in gray.
(Dh Bortoli al)
DSRM
/
X
X
(Shi et all)
LightSB
X
(Kurotin et al.
LightSB-M (Gusbchi
X
/
x
VMSB (ours)
mirror descent using log-Schrödinger potentials as dual parameters (Peyré et al. 2019 Léger 2021 Aubin-
Frankowski et all 2022). These prior analyses have predominantly focused on the optimal-case scenario with fixed F for simplicity. However, theoretical improvements for worst-case scenarios remain underexplored, particularly with respect to online convex Icarning methods (Zinkevich, 2003) for unknown sequences of cost
functions ( a selting known as online mirror descent (OMD; Srebro el al. 2011: Lei & Zhou 2020
Since the SBP is an
optimization (Karimi et al. 2024h within optimal transport (OT),
applying general OMD arguments to the SB context requires thcoretical development and a computational
strategy to ensure practical performance improvements. Leveraging the Wasserstein gradient flow discovered by Jordan, Kinderlehrer, and Otto (JKO; Jordan et al.] 1998), this work proposes to solve these iterative optimization problems of OMD updates via gradient dynamics endowed with the Wasserstein metric which has been canonically used in solving OT, replacing standard Euclidean gradients flows (Santambrogio 2017). We further establish a fundamental insight that OMD acts as the informational counterpart of gradient. descent in Wasserstein geometry, thus providing a core theoretical rationale for our algorithm.
In this paper, we present a novel algorithm for SB acquisition through the lens of information geometry to further enhance stability in the learning process. As illustrated in Fig. | we leverage primal and dual geometries (C, D) under the Bregman potential (2-)) KL(-Ile-< @ v), where the transformations between the coupling and the dual potential fo, @ th, are uniquely defined by first variation operators (de, dp) (Aubin- FrankooFraoowkii et al.) 2022). For online learning, we account for the solvers optimization errors due to uncertainty (gray region), and we propose an online mirror descent framework to minimize the regret resulting from these errors. We show that this framework enables a complete form of online learning that tolerates unreliable empirical estimates from arbitrary data-driven SB solvers. To this end, we propose a robust, simulation-free SB algorithm, called variational mirrored Schrödinger bridge (VMSB), with new parametric update rule that we call variational online mirror descent (VOMD). We soek an exact and computationally feasible MD update mechanism by narrowing down the solution space to subset of tractable distributions, which is often referred to as taking a variational form (Paisley et al. 2012 Blei et all. 2017). To solve the SBP in a computationally viable manner, our VOMD method utilizes a variational approach [Lambert et al. 2022) based on the Wasserstein gradient flow with respect to the Wasserstein- Fisher-Rao (WFR) geometry. The proposed VMSB offers closed-form mirror descent updates by formulating the dynamics of itcrative subproblems using Wasscrstoin gradient flows, and our experiments indicate that VMSB generally outperforms existing simulation-free methods in various benchmarks.
Our contributions. This work builds novel algorithm grounded in statistical learning theory, formally derived from the information geometric perspective for SBPs. Recently, a Gaussian mixture parameterization of the Schrödinger potentials has been proposed by Korotin et al. (2024). The simulation-free LightSB solver is simple yet general, and its authors proved a universal approximation property for the parameterization. The expressiveness of the solver aligns with the geometric propertics of Gaussian variational inference and
mixture models (Chen et al. 2018) Daudel et al., [202]] Lambert et al.
2022 Diao et al. 2023). However,
its shortcoming- -nnniinnnnndddgooo that of other simulation-free solvers (Tong et al.
2024b Gushchin et al. 2024a is
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the uncertainty of data-driven learning signals for non-convex objectives. We argue that this presents an opportunity for improvement by leveraging the rich geometric properties of the SBP in variational form. To the best of our knowledge, VMSB is the first SB algorithm based on VOMD and inherits the theoretical essence of OMD. Table I shows that VMSB is a simulation-free solver that admits a rigorous convergence analysis in general learning scenarios for probabilistic generative models. We verify the validity of our core theoretical principles through an extensive suite of SB benchmarks, including real-time online learning, standard benchmarks in optimal transport, and to-ge-to translation tasks. Our main contributions are summarized below:
We develop a robust SB learning algorithm built upon the online mirror descent formulation, whose specific update rules follow Wasserstein-2 dynamics derived from local MD objectives. Under mild assumptions, we formally prove the convergence of the proposed algorithm in general online learning scenarios ($ - We introduce a simulation-free SB method, leveraging the Beassesttin-Fisher---r asymptotic stability within Wasserstein gradient flows. The resulting VMSB algorithm admits closed- form dynamics, enabling an accurate and efficient. implementation via the LightSB parameterization ($ 5 We validate our algorithm across diverse SB problem settings, highlighting the effectiveness of our VOMD- based framework in various scenarios, including online learning, classical optimal transport benchmarks, and saage-to-image translation tasks. Empirical results consistently demonstrate that our proposed meth- ods outperform existing simulation-free solvers, validating our theoretical assumptions and corresponding claims (8 6).
geometry to ensure
2 Related Work
Simulation-free SB. The Schrödinger bridge problems are originated from physical formulation for evolution of a dynamical system between measures [Téonard 2012 Pavon & Wakklbinger 1991). The study of SB has gained popularity due to its connection to cntropy-regularing
optimal transport (EOT; Peyré et al. 2019 Nutz 2021). Its association with optimal transport suggests various applications across various fields related to machine learning, such as image processing, natural language processing, and control systems CCaon et al. 2020: Liu et al., 2023 Alvarez-Melis & Jaakkola,
2018 Chen et al. 2022 Historically, the
most representative algorithm for SBP is Sinkhorn (Kullback 1968). and there has been progress in training
simulation- SB with nonlinear networks (Vargas et al., 2021 De Bortoli et al., (021) by "matching"
with simulation data, of half-bridge of forward and backward diffusion at each time. An SB solver is called simulation-free [Tong al. 2024a b) if the solver is trained without samples from the simulation of SB diffusion processes. LightSB (Korotin et al., 2024 is special lype of simulation-free solver using the maximum likelihood method of Gaussian mixture models (GMMs). Building upon these advancements, our approach focuses on enhancing simulation-free SB solvers by leveraging gcometric insights derived from the generalized dual geometry inherent to the SBP.
MD and Sinkhorn. The Bregman divergence (Bregman 1967 is a family of statistical divergence that is particularly useful when analyzing constrained convex problems (Beck & Teboulle 2003a; [oyd & Van- denberghe, 2004 Hiriart- Urruty & Lemaréchal 2004). Notably, Léger (2021) and
et al. (2022) adopted the Bregman divergence into entropic optimal transport and SB problems with probability measures, and the studies revealed that Sinkhorn can be considered to be mirror descent with a constant step size 17 = In statistical geometries, the Bregman divergence is a first-order approximation of a Hessian structure (Shima & Yagil 1997 Butnariu & Resmerita, 2006), which interprets MD as natural discretization [De et al. (2023) introduced Wasscrstein mirror flow, and the results include a geometric interpretation of Sinkhorn for unconstrained OT, i.e., when & + 0 from the entropic regularization setup. Karimi et al. (2024) formulated a half-iteration of the Sinkhorn algorithm into mirror flow, i.e., 7)t 0.
on a gradient flow. [eb
Wasserstein gradient flows have attracted considerable attention in machine learning where their intrinsic geometry is governed by the Wasserstein-2 metric. (Ambrosio et al. 2005a Villani 2009: Santambrogio 2017). Otto (200I) introduced a formal Riemannian structure to interpret various evolutionary equations as gradient flows with the Wasserstein space, which is closely related to our variational approach. The mirror
Langevin dynamics is an early work describing the evolution of the Langevin diffusion (Hsich al.
2018
and was later incorporated in the geometry of the Bregman Wasserstein divergence (Rankin & Wong
2023
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We relate methodology with recent approaches of variational inference the Sarrs-Wassserteein space
[Lambert et al.
2022
Diao et al. 2023) Utilizing Bures Wasserstein geometry, the Wasserstein-Fishher-
geometry (Liero et al. 2016; Chizat et al. 2018 Liero et al. 2018 Lambert et al. 2022) additionally provides
"liftings" which yield an interaction among measures.
Learning theory and Robustness. Suppose we have time-varying costs { {Fr) We generally referred to learning through these signals as online learning (Fiat & Woeginger 1998p. Our interest lies in temporal costs defined in a probability space, where following the ordinary gradient may not be the best choice due to the geometric constraints (Amari 2016 Amari & Nagaoka, 2000). In this sense, primarily relate our work
to OMD, an online learning form of mirror descent (Srebro et al. 2011 Raskutti & Mukherjec, 2015 Lei &
Zhou 2020). The OMD amgorithm provides generalization of robust learning by seeking solutions that are optimal in a worst case sense, ensuring performance guarantees under adversarial or uncertain conditions
(Xu al. 2008 Zinkevich| 2003 Madry al., 2017). Another relevant design of the online algorithm the (FTRL; McMMahan 2011). OMD focuses on scheduling proximity of updates through ) whereas FTRL minimizes historical losses with a fixed proximity term. Meanwhile, there have been several notions of robustness. Distributionally robust optimization (DRO; Kuhn et al. 2025 addresses robustness by solving minimax objective under adversarial distributions, a concept originating from Knightian uncertainty. Our work addresses this via regret minimization, which is aligned with the core argument of Wasserstein distributionally robust regret optimization (DRRO; Fiechtner & Blanchet 2025) The main difference of our work from DRRO is the premise of the ambiguity sets: whereas DRRO relies on Wasserstein balls, we model uncertainty in the dual geometry under statistical assumptions.
3 General Properties of Schrödinger Bridge Problems
Notation. Let P(S) (P2(S)) denote the set of (absolutely continuous) Borel probability measures on S C Rd (with a finite second moment). For marginals (s, E PS)),, I(,,, u) denotes the set of couplings (Peyré et al.) 2019). For a coupling n, we often use a shorthand notation ग" (#²) to denotes conditional distribution for a sample data #(-[x) (or #-ly)); see Fig. 2 We use KT.(-II-) to denote the KT. functional (e.g-, formally defined at Eq. 22) and assume +oo if an argument is not absolutely continuous.
Noise (W*)
(x
Y1
X1
H.
We employ notation P([0, S) to denote set of path measures from the time interval [0, 1].
Figure The SB problem.
The static SB problem and Sinkhorn. We first introduce a problem definition of a static variant of Schrödinger bridge problems. For a regularization coefficient IR+, the static SB problem, or the rnpic optimal transport problem with quadratic cost function c(x, y) = - defined as optimization of finding unique minimizer of optimal coupling x* with repect to the following objective
Jf cxx,y) dn (x, v) (KL(=
OTe(u,)) = inf rell(w.)) r
(2)
where μ &u& @ denotes, the product of marginals and KL(-II-) in the regularization term denotes the Kullback- Leibler (KL) functional, or relative entropy, defined as
KL(T|/u = Mosss log d(u 8 v)
dR
du, π HKHθ
x μ Kn.
It is well studied that the EOT problem 2 can be translated to a singular form of divergence minimization problem (Nutz) 2021), d.e., [L(]||R) where the reference mcasure is of Gibbs parameterization, satisfying π'оw: = - with C₂ c/s. We are particularly interested in its dual aspect, which contains a pair of log-Schrüdinger potentials as an alternative representation for EOT solutions.
the Nikodyy derivative of dR
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Lemma 1 (Duality; Nutz 2021p. Suppose existence of optimal coupling '² E I(p,v) and log-Schrödinger potentials (f,", )² E I. (1) (v). Then, the dual prohlem of 22 is found without gop by
inf KL(T]|R) xell(u.))
sup VEIA())EEN)))
(d(p) + v(b) Jssxs eF dR+1. 1,
(3)
where pl(p) ( v(v) = = 1 dv; the operator @ indicates the direct sum of two potentials; the symbol R denoles the reference of Gibbs measure: dR dl @ v) with ce(x,, v) = zे || y yl|².
By the duality, the Lebesgue integrable functions (0* E (H) x LL'(')) represent dual solution of (3) associated with n² by dx ex "e" = v), (n @ allmostt surely. Apparently, the result is a general form of Legendre duality from convex optimization, and the realization that every element of the dual space can be split into two separate potential functions has inspired halfway projection method known as the Sinkhorn algorithm (Cuturi 2013). The Sinkhorn algorithm is defined as following alternating updates:
(2442(2) = - -lg / uz (
V2++1(1) = - llog ( -al(()))a(a.))) p(dx),
(4)
where each update called a iterative proportional fitting (IPF; Kullback 1968) procedure. Based on the geometric understanding of Bregman gradient descent and flow (Leger 2021: Karimi al. 2024), the static SB problem and Sinkhorn reveal a profound connection the information geometry incurred by the KL functional. Of course, this implies that that the Sinkhorn algorithm not the only way of finding the solution space. Akin to differential geometry, JKO discovered that each density be analyzed using the standard geometry of the 2 distance, defined as (Jordan et al. 1998
Villani 2009)
W2(1v)= = inf (Rreswiwa ) TEIl(H.))
(5)
where is a joint distribution that has μ and v its marginals. In essence, the distance measures the minimum cost to transform the distribution into v, and provides an LL-like metric that quantifies the discrepancy between two marginal distributions by considering the distance that mass must travel. Our method utilizes the theory of Wasserstein gradient flows whose technical machinery is noted in Appendix B
The dynamic SB problem. An alternative perspective on the Schrödinger bridge is the interpretation in terms of fluid dynamics or diffusion. In this setup, one considers a Wiener process W with volatility e R to be the reference for SB. The dynamic SBP (Vargas et al. 2021 aims to find process T such that
T aigmin KL (T WW"), TeQ(u.)
(6)
where Q(H, v) C P((0, 1), is the set of processes with marginals μ and The SB process T is uniquely described by stochastic differential equation (SDE): dX, (1, X1) + dWF in E [0, 1), governed by a drift function g" along with noises generated by Wi. Addressing T involves continuous-time Jog-Schröd potentials {prheejo.1]- {wxheeoo.n and the time-dependent density Pt, governed by the following differential 2020
equations for Po = and p1 = = Gigli Tamanini
0:p: 1 p 2 A
dept + V. (Vpip)) Appr 0.p. + V. (VDp)) S.Pp..
0, = V ot 2 A
where the operators V., A) denote gradient, divergence, and Laplacian, respectively. The pairs on the left and the right are commonly known as Hamilton-Jacobi -acl----ma-- equations and Fokker-Planck equations, respectively. The fundamental equivalence between static and dynamic SBPs (Pavon & Wakolbinger, 1991 Lóonard 2012) allows us to consider the optimal coupling x¹ when finding the SB process T*, vice versa.
V.
The Bregman divergence. To advance the discussion, one needs alternative notion of gradients in order to generalize the classical Bregman divergence, defined as Da(x]lv) = 2(x) [2(y) (Vi(v), v)- However, since the domain of Ω considered in this paper (the measure space of SB solutions) has
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Gâteaux differentiability is not ensured. Consequently, to discuss OMD here, we adopt the following definitions of directional derivatives (Aliprantis & Border (006) and first variations PArboo-rrankkwwki et al., 2022) in order to discuss MD. We also refer the readers to Definition 7.12 of Santambrogio (2015) for alternative description on this topic.
empty interior [Aubin-Frankower et al. 2022
Definition 1 (Directional derivative). Given a locally convex topologiaal vector space M, the directional derivative of F in the direction& & is defined as F(;; = lim Flo+h()-F(x)
Definition 2 (First, variation). Given a topological vector space M and a convex constraint C S M and a function F : -M- + R U {oo); we define the first variation of F over C evaluated at x E C dom(F), as ((x) E M", where M is the topological dual of M, such that it holds for all y E C dom(F) and V リ= --- E M: (f.F())) = d+ d=;(z; (:--) denotes the duality product of M and M
Following Karimi al. 2024)), this work considers the following generalized divergence defined with weak notion of the directional derivative. We explicitly set the Bregman potential (2(-) = KT[[lee=Ss @ v) in the SB problems for the rest of the paper, which enforces the Gibbs parameterization for the OT couplings. Definition 3 (Gencralized Bregman divergence). Let a convex functional Q M - RU{++0} be a Brogman potential. Define the Bregman divergence associated with as
Da(<ly)) = Mxx) - P(y)- d dy; - -y),
(8)
for every pair of elements in the domain y MM.
The above definition preserves the essential role of the Bregman divergence as the first-order Taylor expansion of Ω allowing us to utilize properties in optimization similar to those used in the classical setting. To account for smoothness- and convexity- properties of a functional defined on metric spaces, we employ the notions of relative smoothness and convexity with respect to the generalized Bregman divergence. These notions were first introduced by BBinbbam al. (011)) and later popularized by Aubin--Prankowsk et al. (2022), who applied them to spaces of measures to provide theoretical foundation for the Sinkhorn and EM algorithms.
The formal definitions are given in Definition G
Asymptotically log-concave distributions. Lastly, our theoretical analysis works with a certain form of measure concentration property, and we formally address this property with asymptotically strong log- concave (ale) distributions, This section rigorously states the resulting assumption which is later used in address the desired properties of OMD, analougous to strong convexity assumption in the classical literature. Let us consider the following informal definition of asymptotically strong log-coneave distributions
Pale = { ((dx) = exp(-U(z))dr= : U (2R), U is asymptotically strongly convex }
(9)
where Appendix A contains formal version on asymptotical convexity. Note that asymptotically log-concave functions satisfy a certain form of logarithmic Sobolev inequality (LSI; Gross 1975). The condition can be extension of Sobolev space (Adams & Fournier 2003) for informational geometric problems. The simplest case of such condition for the Gaussian measure is represented as follows.
Remark 1 (LSI for the standard Gaussian). Suppose that f is a nonnegative function, integrable with respect to a mcasure 7, and that the entropy is defined as Ent, (f) = S2a log fdy - (Jge fdy) log(Sge fdy). the log Sobolev inequality when y is the standard Gaussian measure reads Ent, < 2 Jg. df.
Historically, the log Sobolev inequality condition arises from the implication of satisfying the Talagrand's inequality for bounding the Wasserstein-2 distance, and is closely related to measure concentration (Otto & Villanj 2000 The important extension of asymptotically strong log-concave distributions for Schrödinger bridge dm eP @ v), (u & v)-a.s. is that induced SB model also satisfies asymptotically strongly log-conoe log.concaveness and the LSI condition (Conforti, 2024). For a representative model related to our work, the Gaussian mixture parameterization (Korotin et al. 2024 is a representative model that our theoretical analysis holds, bocause Gaussian mixture weights docs not alter the asymptotic charactoristic.
Remark 2 (Conforti 2024). Let J4, v E (RRe) with finite entropy on Lebesgue measures and a C be a coupling in the static Schrödinger bridge problem. Then, for quadratic cost function, the coupling distribution is also asymptotically log-concave and satisfies form of logarithmic Sobolev inequality.
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C
IPF
V.P(.))
V F(()))
77
[m Tt-1
7T
774
172
Figure 3: Learning for an SB model {mr in the primal space (see Fig. ! for the details), Left: Sinkhorn (Lemma 2) Middle: Wasserstein gradient descent the distributional space for fixed F (Lemma 3
Right: Variational online mirror descent with sequence of convex costs using uncertain estimates { [t=)"=
- Let us suppose that a parametorized SB model dı, = - v) obeys the following constraints for marginals and potentials:
c= : (1, E P2(R") (RI)(R"),, (p) E (n) x (v),,, and p. E (RR) n Lip
(10)
where Lip(KC denotes a set of functions with K--iipshitz continuity. Using the disintegration theorem for probability measures (Léonard 2014), we assume the boundedness of Bregman divergence between two trans- port plans using derivatives of first variations with a positive constraint >> 0 by the following assumption.
Assumption (LSI for couplings). Let us suppose ?2()) KL((I|R) for reference measure R. Suppose that arbitrary π, E C satisfy type of logarithmic Sobolev inequality for relative entropy (KL divergence) is upper bounded by the relative Fisher information (Gross 1975
for some w E R4 as
1 2w /
//
dr dr (2, 2
[L(T||R) <
V log
T(dz, dy),
(11)
dR(x,, y)
By the equivalence of the first variations of Bregman divergences (detailed in Lemma convergence in KL(T|]m) is equivalent to convergence in ()) Introducing the relative logarithm logæ (T) == log(
and adopting the framework of Conforti 2024), we can assume there exists a constant w > 0 such that
Da(=| S V[6.0())) - 6,5(()) (3()))
(12)
2w
holds for Ω KL(-Ile & and the first variation & We call the condition LSI(w).
Since, as illustrated in Fig. Pi e wh = (.((T.)) for arbitrary Te, the assumption geometrically enforces a quadratic upper bound on the Bregman divergence in terms of dual space gradients. In general, the LSI condition also has often been used to analyze the convergence of partial differential equations (Malrieu 2001) To make our analysis on improvement (Lemma 14) and solid regret bound of OMD (Lemına 16 this work finds that Assumption is necessary to ensure certain asymptotical type of measure concentration.
4 Learning Schrödinger Bridge via Online Mirror Descent
The goal in this section is to derive OMD update rule for SB, and analyze its convergence. To accomplish this objective, we postulate the existence of temporal estimates and online learning problem. Our analysis suggests that applying an MD approach can reduce the uncortainty of these estimates.
4.1 Sinkhorn and Wasserstein descent as mirror descent algorithms
We start with a novel characterization of Sinkhorn as hcuristic form of online mirror descent as illustrated in the left side of Fig. 3 which will lead to better understanding of our arguments. OMD updates are determined by the first order approximation of costs F, and proximity of previous iterate with respect to a Bregman divergence (Beck & Teboulle 2003a).. Using the first variation &, in Definition instead of standard gradient V, the proximal form of generalized OMD over C is derived as (Karimi et al. 2024
T++ = arg min (dcFi(T)), ग - -m) + J. D
(13)
EEC

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where Fe denotes temporal cost function for SB models in C. In Eq. 13 the updates are determined by the first order approximation of Fr and proximity of previous iterate with respect to the Bregman divergence (Beck & Teboulle) 2003a). In contrast to the 'half-bridge" interpretation provided by Kariiii et al. (2024, the online MD iteration (13) involves a temporal cost Ft, which offers more general reinterpretation of the Sinkhorn algorithm. Using the leasible model space C in (10 IPF projections are reformulated as following subproblems of alternating Bregman projections:
arg min ( [T[[=)))) E C, 122h v), =
arg arg (L =E
CC = p):
(14)
where (1)(()) == [ y) dy and (2) (2; y) dæ denotes the marginalization operations, and the symbols (11 112)) denote the Sinkhorn projection spaces that preserve the property of marginals. As a
generalized optimization problem (L2TT [p) with sequence {a f0, 0, I...... J. By construction, we show that an online form of MD for F. with a constant step size 7/, 1 matches the Sinkhorn.
C, one can consider a temporal cost FT( = a a (lx) + (1
Lemma 2 (Sinkhorn). For S(-)) = [T[T]e v); ilerales from Te+1 arg xec ( -m)
[=() are equivalent estimates from (4t,, 4.) of 4), for every update step t No.
The proof is in Appendix A Consoquently, we established that the Sinkhorn algorithm corresponds to an instance OMD; however, its inherent structure limits flexibility on step sizes and other underlying assump- tions, making it challenging to analyze directly using standard OMD theoretical arguments. Instead, one can alternatively consider OMD by recovering "static" objective, namely F(-) == KL(-Il="), where the KL functional is originated from the formal definition of SBP (Vargas et al.) 2021 Chen et al. 2022). The following lemma shows that the MD) updates for the static cost correspond to discretization of a Wasserstein gradient flow for SB models, a Ricmannian steepest descent in the model space of C.
Lemma 3 (MD in the Wasserstein space). Suppose that F(T) == KL(x[[=") for π E C. The MD formulation of F correspords to a discretization of a geodesic flow such that lim 01t1-Ke (Fe), where V. denotes the Wasserstein-2 gradient operator.
According Lomma 3 gradients of F are tangential to the geodesic curve from To to x² (green line in the middle of Fig. 3] in terms of the Wasserstein-2 metric W2. Hence, the geometrie interpretation allows us to consider the static cost F as the ground-truth cost for optimization in our variational OMD framework. However, the ideal case falls short in practice since n² is inherently unknown. Therefore, we postulate on an online learning problem that nonstationary estimates {x] ) are offered instead of X² as learning signals, making an optimization process with F+(+) = KL(-I|=?"). As shown in in the right side of Fig. 3 we focus on the online learning setting where { 10 are fundamentally uncertain with perturbation, since the optimal coupling n" is not accessible during the training time.
4.2 Online mirror descent for Schrödinger bridges: theoretical analysis
In this section, we examine the robustness of our OMD algorithm by analyzing its convergence behavior under statistical uncertainty. From a learning theoretic standpoint, an apparent yet. understated premise is that an SB algorithm does not retain the global target x" in practice. The global objective F (also Fi) are fundamentally unknown, but are instcad inferred, imposing innate uncertainty of optimization. Instcad, we postulate an ouline learning problem that nonstalionary ergodic estimates {m])ss are offered instead of x² (gray region in Fig. [] Let Ω be the Fenchel conjugate of Ω + ic with the convex indicator ic. For
the space D = &C), a directional derivative åp of Ω exists by the Danskin's theorem (Danskin 1967 Bernhard & Rapaport 1995 such that
6 ( v) = arg max (pp + w) - ((=))
(15)
REC
Note that d,.0)) form bidirectional maps; direct sum of potentials yp B &b E D represent an element of the generalized dual space. The key assumption is that the learning target o. is asymptotically mean
Defined as (x() = if E and too otherwise.
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Lignts (19)
LightSB
redbo
Do poctiinnn
POOSET
(m)
Ours
Figure 4: Loss landscapes and gradient dynamics in a 2D problem. Left: In an early stage, parameters of three modalities (m=)2=1 (mean ostimations) for both LightSB (top) and VMSB (bottom) methods approach the optimality with different costs. Right: When magnified the landscapes in the late stages (10 times), while LightSB is vibrant, whereas our method emits strictly convex landscape and stable dynamics.
stationary Grryy & Kiefler 1980) for the dual space, which have been used to analyze stochastic dynamics. Since iterates updated through dual parameters in MD, refer to the process as being dually stationary. Asummtion 2 (Dually stationary process). Suppose that Tι E C exists, which is the primal representation of asymptotic dual average T == 6 (lim;=+oo [=()),((;))), where the notation E, denotes the time-average.
Plugging a temporal c.s = [L(+ to 13 from 4.J we achieve dis- tinct problem setup of online mirror descent. Fig. shows demonstrative ex- periment regarding our online learning hypothesis. OMD decomposes the global problem into local convex problems, and prevented iterates from being vibrant by stopping at single point nh. This verifies that OMD stabilizes learning of even when the reference कद. tends to inherently have some perturbation. Addi- tionally, we state two conditions for the sequence (7 ) which will be justified in Theorem I and Proposition
m/1
WW
[ Fig. 5 shows a plot of harmonic progression a+td for a E R and d R+ with respect to which satisfies both conditions.
Figure 5: A sequence ample of n and 1 7h
Assumption 3 (Step sizes). Assume
conditions for step sizes {m (a) Convergent sequence &
divergent series: lim Th 0 and E co. (b) Convergent series for squares: E < co.
Under the above assumptions, we firstly argue that OMD for the temporal cost KL(-IIT)) with respect to the Bregman potential Ω = KL(- @ v) requires step size scheduling for the sake of convergence. The following theorem states that convergence in the case of π = T is assured when ?e follows Assumption (31). In contrast to the well-known linear convergence guarantees for the Sinkhorn algorithm under bounded costs with fixed marginals (Carlier 2022), our OMD-based analysis establishes sublinear convergence rates, accommodating scenarios involving unbounded and non-stationary costs.
Theorem 1 (Step size considerations). Suppose the idealized case of x² = n.. Then, for ( CC we get limr-po (D [T)] = 0 if and only if Assumption (3a) is satisfied. Furthermore, if the step size is in the form of f) E+i then E.r (Da( = (1/T).
2
Therefore, we can guarantee the ideal convergence in the SB learning when the scheduling of тee follows the step size assumptions. Next, we argue that general convergence toward # is guaranteed under Assump- tion 3D). Given the convex nature of SB cost. functionals, we argue that this convergence toward x is beneficial long as of is trained to approximate T' and remain bounded. Therefore, argue that the convergence of SB is beneficial and address the following statement.
Proposition 1 (Convergence). Suppose that x" + n,, hence xnc E[F((=)] satisfies Assumption 3 then lim;- Eral ( converges to almost surely.
0.0 0. If the step sizes (n))
Lastly, assume that the log Sobolev inequality in Assumption ! holds with continuity of potentials. We establish an online learning regret. bound of ((VT) for certain instance of step sizes, demonstrating that
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a
Ground-truth marginal
b
Trained distribution
Zoom 4X
S-curve
Moons
Swiss Roll
S
Carto
0o000 0a o 0o
oooo 0000 000
00o 0000
Varistional
Diract KL
Figure 6: Variational MD with synthetic datasets. (a) A distribution is accessible by finite batch data. (b) 3D surfaces of (#h; m trained by Monte Carlo method for KT. (top) and variational MD (bottom) show that the MD results in more stable outcomes. (c) The plots show the estimated KL( a*) with different step size schoduling (3 runs), with red dashed basclines (I
imposing specific measure-theorete properties in SBPs generalizes classical OMD results (Nesterov) 2009
Srebro al. 2011 Orabona & Pál 2018 Lei & Zhou 2020). The analysis on our general online learning setup compatible with these results by using the dual norm llāell, which represents a distance ween the OMD iterate Te and the empirical estimate zi in the dual space D, where the generalized notional of duality comes from the transformation 8,5)()) defined by applying the first variation of Ω with respect to C.
Proposition 2 (Regret bound). Assume ip, 1 E C²(RR)) n Lip(K), Assumption | holds, and the given costs (F.)E= are bounded. For arbilrary u E C and a total slep T, define D2 = maxises7 Du((u]TT)), mt)- (a) When the number of time step is known a priori, the regret is bounded to 2DV for a constant step size n V2KT
DVG
(b) For an adoptive scheduling tu = DV// / 2i the regret is bounded to 2D 2D1/20-1 2 |la.11 where 9. = (x.) (z)))
Note that although our analysis establishes a rigorous connection between SB and OMD, it inherits cer- tain limitations from classical regret analyses. For instance, sublinear regrets in Proposition 2 relies on an additional boundedness assumption on costs, and there exist some cases of Assumption that may yield asymptotically lincar regrct (Orabona & Pál, 2018). Addressing these limitations may involve advancod hybrid OMD methodologies which are actively being studied, such dual averaging (Fang al., 2022) or
FTRL (Chen & Orabona 2023). As exploring (as well computing) such extensions for SBPs falls beyond our scope, from now we focus practical implementations of OMD that demonstrate theoretical conver- gence (Proposition [ over a relatively long horizon of several hundred steps. Hence, we adopt Assumption 3 and provide corresponding experimental evidence to substantiate its validity in practical scenarios.
4.3 Online mirror descent updates using Wasserstein gradient flows
The remaining challenge is practicality of our OMD theory, namely the accurate computability of each critical point for (3). To resolve this issue, this work newly presents an approximation method using Wasserstein gradient flows (Jordan et al., 1998) through an equivalence property of first variations. Suppose we expand
a subinterval [t,t+1] 1) for each OMD step [13 into continuous dynamics of p(T) C for a T E oo). By
Otto's calculus on the Wasserstein space, known as the Otto calculus (Otto) 2001 see Appendix B.I. one
can describe the dynamics of pr for minimizing strictly convex functional & C R as the PDE
= =
(16)
where Vw denotes the Wasserstein-2 gradient operator V = (oV In this work, we adopt the Wasserstein gradient flow theory (Jordan et al. 1998) to efficiently perform OMD where the equilibrium indicates the subsequent iterate T++1- Note that Wasserstein gradient flows are asymptotically stable by the LaSalle's invariance principle (Carrillo et al.) 2023). We present a simple and exact closed-form expression for the VOMD update. Note that the cost F.(-) = KL(-I[=?) satisfies the L-relative and 1-strong- convexity relative to f2 (see Definition 0 Aubin-Frankowskk Frankowski et al. 2202)). Then, first variation of the OMD problem can be decomposed into multiple variations of another problem with similar characteristics (c.g., equilibrium, snoothness, and convexity). We present the following theorem for the computation of OMD.
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Theorem 2 (Dynamics equivalence in first variation). Consider the Wasserstein gradient dynamics of (16
which solves local updale of (13). The gradient dynamics of updates are equivalent to that of a linear combination of KL functionals such that for any EC
m.dcE((p)) = - (e) + (LLm))))))))))) Vp, EC,
(17)
and the PDE 17
conweryes to unique crilical point of subsequent OMD iterale 13
0s +
Sketch of Proof. We identify gE, as dynamics that reaches an equilibrium solution for minimize (6.Fi(TT), π m.) (T
xEC
minimize The FEC
Da(T
+ (1- -c) (=(T))))
(18)
empirical estimates
proximity
and then the equivalence of first variation for recursively defined Bregman divergences is applied (Lemma 6 At a glance, Eq. (18) appears analogous to the interpolation search between two points, where the influence of σद is controlled by T)-- We leave the full version of proof in Appendix A.5
Theorem 2 holdd practical importance for OMD computation, since following the argument allows us to per- form gradient-basse updates without directly constructing a desired Bregman divergence. That is, updates be drawn based on linear combination of gradient flows 7) (p that, just like ordinary gradients, Wasserslein gradient operators
|| + (1 VLL Note measure space allow for this direct translation of Eq. (17). where such expression has been extensively studied both theoretically and computa- tionally (Carrillo et al. 2023 Lambert et al. 2022). Therefore, we can utilize interpolation of Wasserstein gradient flows for performing updates and utilize a certain variational class for reducing the computational cost. Fig. shows our actual experiments using GMMs. Let a reference estimation be fitted using a Monte Carlo method, and our model be trained through variational OMD method which is explained in the subsequent section. We initially observed that the VOMD method provides stability improvement when 7 1. In contrast, the condition of 17 > performed worse than the Monte Carlo method and 1) showed almost equivalent performance. Furthermore, the performance of VOMD was greatly improved by choosing a harmonic step size scheduling in the interval [1.0, 0.05). All of these results on variational approximation precisely matches our analysis.
5 Variational Mirrored Schrödinger Bridge
In this section, we propose variational mirrored Schrödinger bridge, a simulation-free method that offers iterative MD updates for parameterized SB models with mixture models, using the Wasserstein-Fist Fisher---o-r geometry. We provide tractable and exact VOMD-based update rule for LightSB models and draw practical VOMD updates algorithm that closely resembles ordinary machine learning methods.
5.1 Gaussian mixture parameterization for the Schrödinger bridge problem
In order translate theoretical arguments VOMD into practical algorithm implementation, this section focuses on a computational implementation of our theory. Recently, Korotin et al. (2024) proposed the GMM parameterization, which provides theoretically and computationally desirable models for our variational OMD approach The parameterization considers the adjusted Schrödinger potential u'(x) == /2e) and v v²(y) == exp((" (y) Iivll /2ll such that we have proportional property π T"(lax) x exp(y"(xx)- - ||=|
exp((x.0)/e)) (v). With finite of paramcters 6 á {ag, my, Ex for weights og > 0, mcans mę E Rd and covariances Σg St,,, Korotin al. (2024) proposed to approximate the adjusted Schrödinger potential
ve and conditional probability density ma
(v)
an N(yllm,,EEn
(v)
o of Nyl(y Ex),
(19)
#
2 "Adapting the GMM parameterization our theory straightffrwwar, achieved by specifying the cast cz( v) 1/2e|lx- vll
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where GMM component for #6 conditioned by an input a: m == my + Exe, og == og exp ( F. ( (m)
2x
"g Σ k= og (see Proposition 3.2 of Korotin et al.p. For this parameterization, the closed-frod from expression of SB process To is given as the following SDE for E [0, 1):
To dX, = go,t, dt + VEdW,
k
(20)
=
t)La)
Σ
a N(mx (0.)) Au)(),
m where mx(t, 22 e(i-1) + -- my and (t) a(1 an Ia 1 Therefore, the LightSB parameterization represent both static and dynamic SB models and arbitrary SB solvers can be applied without restrictions. We utilize the GMM paramoterization for our computational algorithm for three key reasons. Firstly, the parameterization induce the universal approximation property for both Re and To (Korotin et al. 2024). Secondly, GMMs are asymptotically log-concave (see Lemma 1. which is a fundamental assumption for our theory. Lastly, the GMM parameterization makes the computation of Wasserstein gradient flows with respect to the KL divergence tractable, which in turn enables to apply canonical treatment, akin to solving ordinary convex optimization problems endowed with a Riemannian geometry.
5.2 Computation of VOMD in the Wasserstein-Fisher-ee geometry
For tractable computation, we formally derive a particular variant of Wasserstein gradient flow for the GMM parameterization. The space of Gaussian parameters Rd S (d-dimensinoal mean vectors and positive definite, symmetric convariance matrices), endowed with the Wasserstein-2 metric W2, is formally recognized as the Burcs-Waartt Wasserstein (BW) geometry (Bures) 1969 Bhatia et al. 2019 Lambert et al. 2022 BW(R C (R The Wasserstein--isher--Bae (WFR) geometry, equivalently characterized by the spherical Kellinger-Kantoree
distance, extends this setting by considering liftings of positive, complete, and separable measures while preserving total mass (Liero et al. 2018 Chizat et al. 2018 Lu et al., 2019). Building upon the BW space, the Wasserstein Desssertein-Fisher-RRe provides liftings of Gaussian particles satisfying distributional consistency. In this work, we introduce the following proposition, which refines and extends the results from Lambert et al. 20222) $ 6) specifically
geometry of GMMs, namely PB(BB( naturally
enhancing their framework through the introduction of freely trainable GMM weights ai-
Proposition 3 (WFR gradient dynamics). Suppose a time-varying GMM model po. with the porameter 0. (akrr MK. 1k,TH at time т. Let yk.r N(mm Ek.)) denote a sample from the k-th Gaussian particle of po.. Then, the WFR dynamics VarRKL(po, I10²) wrr 0, (drk,r, ing,r Ex,r ) are given as
du,r
IE log
Po- (yh.T)
1
ar
Po- log
(ver)) Jak.n pr
2r
(21)
mag log Po- (Werr) pa
EKT= -E
722
log
Por pa
(ye,.) Enat - Ek, F
p8- V2 V²
(Whrr) pa
for E [0, 00), where 2, k== ORi V and 2 denole gradient and Hessian with respect to yk.r.
Appendices A.6 and contain the complete theory. Proposition 3 argues that the one parameter family 0, predicts a gradient-based algorithm of (oo,LL(), and thus Eq. (21 can be directly used for training GMM models. Recall that GMMs have a closed form expression of likelihoods, which means each log likelihood difference can be calculated without errors. Given that the target has the identical mumber of Gaussian particles, both Eq. (21) and its approximation using finite samples strictly induce zero gradicnts at the equilibrium. Hence, we argue that the simulation-free algorithm VMSB will result in more robust and stable outcomes than standard data-driven SB learning. Our VOMD framework can be implemented in modern deep learning librarics and, when coupled with advanced optimizers that provide inherently more stable gradient estimates (e.g., adaptive learning rate schedules), the proposed method empirically find convergence even faster than the conservative rates predicted by theory.
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Algorithm Variational Mirrored SB (VMSB).
Input: SB models (To, πΦ) parameterized by Gaussian mixtures, step sizes (fhs (r), nyy E EEN.
for t 1 to I' do
Acquire with an external data-driven SB solver.
3: 0,-0 0, 1/t 6- 1/( + ( (--/(--))
4: for 4-1 to N do
5:
{xi))=1 + sample mini batch data from μr.
OC
B Σ 7 WFRgraad 0z, Bty + (1 7)) n:)FFRgrad(; 0, 0,, nyy)
30
7: Update f with the gradient 0c
end for end for
Output: Trained SB model Tg.
5.3 Algorithmic considerations
Algorithm ] outlines the overall procedure. The proposed VMSB algorithm requires SB parameters and o, which represents āे and #. from the theoretical framework in $ 4.2 The target model 7s is independently fitted using an arbitrary SB solver. By the results of analysis, one can schedule of the step size d with a harmonic progression satisfying Assumption 3 thus, we propose to schedule by the series for 2 m 2 tr 0 in Line 3 of the algorithm. In our settings, the hyperparameters which varies depending each length of training. The algorithm can also put "warm up" steps leveraged by a existing solver, and start from A = oi enforcing = 1 for a certain period of the carly stage.
set 7h 1 and 7r {0.05, 0.01]
The VMSB algorithm is essentially designed to perform the following approximation the WFR gradient operation 21 in Proposition 3 approximated by the following equation with finite data samples {- ~ p
B DE WFRgrad di Ni, ny) R VuER (T.(Fo
where each expectation is estimated using ny samples from each Gaussian particle. Following Theorem we propose to update the SB model 7e with WFRgrad dr, vi, nyy) + (1 (1-1) WFFgradd( 0,- 1, 2i, fly) each VOMD iteration (see Line 6). When μ is pero-centered distribution, we set B = and 0 for the fast training time. This trick is equivalent to training the adjusted Schrödinger potential (Korotin et al.) 2024)
Vo = > N Im,, <x)) x na(+/æ 0) directly, which enables the VMSB algorithm run efficiently for certain tasks. We argue that our design provides a simple yet faithful realization of OMD updates, yielding procedure that closcly resembles classical gradient descent in machine learning. Although our methodology and computational strategy build on well-established ideas [Lambert et al. 2022 2022 Karimi et al. 2024), we deliberately integrate them into a unified framework to verify our novel online learning theory for the SB problem. As a result, VMSB emerges as robust solver that embeds OMD within variational formulation, offering both rigorous theoretical guarantees and clear computational advantages.
et al.
a
b
Moone Srrrree
S cure
0102
0.03
vIDE
filter (45")
Ereagy Ustance
0.07
0.1L
0/01
Vibble from Dudon
Lok mtem parii
sirs partic
Gaoste pericce
Geloli perides
Figure 7: Online SBPs for synthetic datasct streams. (a) We designed an online learning problem with a rotating filter where an algorithm is allowed to observe the data in y ~ v only 12.5% at time. (b) The plots show that our VMSB and VMSB-M show consistent improvements from their references regarding the ED metric with 95% confidence intervals for 5 runs with different seeds.
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Table 2: EOT benchmark scores with cBW3-uvp + (%) between the optimal coupling x² and the learned model Th (5 runs). Results of classical EOT solvers marked with f are taken from (Korotin et al. 2024), and $ from (Gushchin et al., 2024a). Additionally, LightSB- EMBA indicates a hybrid approach using the exponential moving average techniques (EMA; Morales- Brotons ct al., 2024) for LightSB parameters (decay = 0.99). Our VMSB and VMSB-M results are highlighted in bold when VMSB methods exceed their reference algorithm.
0.1
Type
Solver
d
2
d
16
d
64
d
128
d
2
d m 16
d
64
d
128
d=
2
d
16
d 64
128
Classical solvers (best:
Korotin et aL
1.94
13.67
11.71
11.1
1.01
9.08
18.05
15.23
1.40
1.27
2.36
1.31
Bridg-M
DSBM
Shi et al
5.2
10.8
37.3
35
0.3
1.1
9.7
31
105
3357
15000
Bridge-M
[Tong of al.
0.54
3.7
9.5
10.9
0.2
1.1
23
0.31
4.9
319
819
rev. KL
LightSU
Korotin et al.
0.007
0.040
0.100
0.140
0.014
0.026
0.060
0.140
0.019
0.027
0.062
0.092
Bridge-i M
LightBBMM Gustrctan et al
0.017
0.088
0.201
0.316
0.020
0.069
0.131
0.294
0.011
0.029
0.207
0.747
EMA
LightSIEFMA
0.005
0.040
0.078
0.149
0.012
0.022
0.051
0.127
0.017
0.021
0.025
0.042
MD
VMSB (ours) VMSB-M (ours)
0.004
0.012
0.038
0.101
0.010
0.018
0.044
0.114
0.013
0.019
0.021
0.040
Var-MD
0.015
0.067
0.108
0.253
0.010
0.019
0.094
0.222
0.013
0.029
0.193
0.748
6 Experimental Results
Experiment goals. We aimed to test our online learning hypothesis and verify that the VMSB effectively induces OMD updates. Since our theoretical claims are intended to be highly versatile, consistent perfor- mance improvements for each setting coincides with the generality of the proposed VOMD method. We delineate our objectives as follows: ! We aimed to affirm our online learning hypothesis by demonstrating consistent improvements. 2 We sought to corroborate our theoretical results, aiming for stable performance that consistently exceeds that of benchmarks.
@ We aimed to verify that our algorithm effectively induces OMD by the Wasserstein gradient flow. To achieve these goals, we validate our algorithm across diverse SB problem settings, including online learning scenarios, classical OT benchmarks, and image translation tasks.
Baselines and VMSB variants. Korotin et al. (2024) proposed a streamlined, simulation-free solver referred to LightSB that optimizes Φ through Monte Carlo approximation of KL(**I| As an alter- native, LightSB-M (Gushchin et al., (2024a) reformulated the reciprocal projection from DSBM (Shi et al., 2023) to a projection method termed optimal projection, establishing approximated bridge matching for the path measure Ts. Applying Algorithm D we derived two distinct methods called VMSB and VMSB-M (Ma), trained upon LightSB and LightSB-M solvers (Āø), respectively. Since the theoretical arguments imply that the algorithm is agnostic to targets, the performance benefits of VMSB variants from their references support the generality of our claims. Additionally, we adopted VMSB on hybrid settings, leveraging networks embeddings for complex problems. We refer readers to Appendix D for additional experimental setups.
6.1 Online SB learning for synthetic data streams
To validate our online learning hypothosis, we first considered two-dimensional synthetic SB problems for data streams depicted in Fig. 7 (a). We applied an angle-based rotating filter, making the marginal data stream where only 12.5% (or 45-degree angle) of the total data is accessible for each step t. We trained conditional models 7e with VMSB and To with other baseline SB solvers for the 2D problem, respectively. Fig. (b) shows the plots of squared energy distance (ED), which is a special instance of squared maximum mean discrepancy (MMD). approximating the 1.2 distance between distributions: ED(P, Q) = [(P(z) - (zx ((()))) (Rizzo & Székely 2016). In our ED evaluation, the VMSB algorithm achieved a strictly lower divergence than the LightSB and LightSB-M solvers for various numbers of Gaussian particles K. Therefore, we conclude that these results aligned with our hypothesis and theory of online mirror descent.
6.2 Quantitative Evaluation
EOT benchmark. Next, we considered the EOT benchmark proposed by Gushchin et al. (0024b), which contains 12 entropic OT problems with different volatility and dimensionality sottings. Table 2 shows that LightSB and VMSB methods outperforms other EOT methods in terms of the cBW2-UVP metric, conditional Bures Wasserstein unexplained variance percentage, soldifying previous reports by Korotin
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VMSB-add (MNIST)
LighSB (EMNIST)
VMSB (EMNISTT)
Table 4: FID and MSD scores in EMNIST-to-MMNIST translation lasks. Hyperparamelers between LightSB and VMSB are shared. We examined the scores with five runs for the ALAE case.
b a
Pixel Space
2/0000
4
5 1.
007335300 8
#
1-
b
c.d.
Method
FID
MSD
VMSR (MNNSST)
Light5n (EMNNIST)
VMSn (EMNIST)
Lateut Space
2
4 J Jeoe
/ 1
MF*M-Sink
23.215
0.456
S
F A 8 R
J I
U-net
DSBM-IPF
15.211
0.352
Q 2
He4 T 2 c
b 6 4
J JTLTII
DSBM-IMF
11.429
0.373
J N43J51SCC
4866DI
5 1 1.A1a2b66
Ja f
ACTU
Pixel
LightSB-ady
20.017
0.362
4 64 1
VMSB-adv (ours)
15.471
0.356
// LEFLTEJA
LightSB
9.183 g0.600
0.371 0.0014 #0.018
Figure 8: Generated MINST/EMMNH samples. Top:
ALAE
VMSB (ours)
2.774
0.365
Raw pixel SB results. Bottom: Latent SB results.
et al. 2024] and Gushchin et al. (2024a). We also observed that a hybrid approach combining LightSB and the exponential moving average (EMA: MMrales- Brotons et al. 2024) named as LightSB-EMA was effective for improving stability. Among 24 different settings, our MD approach exceeded the reference model and the EMA method in 23 settings in terms of the cBW2-UVP metric (Gushchin et al.] 2024b). Our replication of LightSB/ BightSB/Ligh LightSB-M achieved better performance than originally reported results, and performance in this benchmark with stability, which
our method accordingly reached the state-of-the-a represents strong evidence of Proposition]] [] Among all cases, the only exception was LightSB-M, which had the highest dimension and volatility. We suspected that the drift form Eq. (20), which is proportional to E, may have violated our assumptions Assumption 2 and the boundedness assumption during the training, Thus, we conclude that our variational MD training is effective in various EOT setups. Tables 9 and 10 in the appendix show comprchensive statistics on this benchmark with wider range of SB solvers.
SB on single cell dynamics. We evaluated VMSB on unpaired single-cell data problems in the high-dimensional single cell dynamics experiment (Tong et al., (0244)). The dataset provided single cell data from four donors on days 2, 3, 4, and 7, describing the gene ex- pression levels of distinct cells. Given samples collected on two different dates, the task in- volves performing inference on temporal evo- lution, such as interpolation and extrapola- tion of PCA projections with (50, 100, 1000)
Table 3: Energy distance on the MSCT dataset (95% confi- dence interval, ten trials with instances of two setups). Re-
sults marked with $ are from (Gushchin ct al. 2024a
Type
Solver
d= 50
d= 100
d 1000
Sinkhorn Bridge-M
Vargas
2021
2.34
1.864
DSBM
2.46 =
2.35 + 0.1
1.00
Bridge-M
MF-M-
Tong et al.
2.66 4: 0.18
2.52 ::2:::00 0.17
1.38 0.05
rov. KL
LightSB
2.31 1 0.08
2.15 0.00
1.264 1 0.06
Bridge-M M
LightBB-M
2,30 0.08
2.13 0.08
1.267 = 0.06
MD Var-MD
VMSB (ours)
2.28 1. 0.09
2.13 0.09
1.260 1. 0.06
VMSB-M (ous)
2.26 0.10
2.12
1.265 = 0.05
dimensions. We evaluated the energy distance over ten trials, which were divided into two distinct scttings with five runs each (Tong al.) 2024a). The first setting spanned from Day 2 to Day 4 (evaluated al Day 3), while the second setting considers duration from Day 3 to Day 7 (evaluated at Day 4). The quantitative results in Table ssoow that our VMSB method exhibited competitive results even for large dimensionalitics. demonstrating its competitiveness as practical SB solver for real-world problems.
6.3 Unpaired image-to-image transfer
We applied VMSB to unpaired image translation tasks for MNIST and EMNIST datasets. In these tasks, LightSB methods struggled to generate raw pixels due to the limited scalability of the loss function. To solve this issuc, we opted to find a viable alternative to LightSB the raw pixel space, and we discovered that the capabilities of GMM parameterization can be extended by incorporating the adversarial learning technique (Goodfellow et al.) 2014 see Appendix D.5) was effective in providing rich lcarning signals for zg. Therefore, we named the adversarial method and the VMSB adaptation LightSB- adv and VMSB-adv. Also, we pretrained encoder networks using the Adversarial Latent AutoEncoder (ALAE; Pidhorskyi et al. 2020) technique, and applied the LightSB and VMSB algorithms on the 128-
MNIST-EMNIST.
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Embedding energy distance
LigRibsSM VWGB-M
Chili - Advit
VMSB
VMS/--M
Female Maie
VMSR
VMSS--M
01
0.5
Figure Imago-to-Image translation on latent space. Left: Generation results for the FFHQ dataset (1024 1024) using our two SB variants. Right: Quantitative results of ED metrics for ALAE embeddings.
dimensional latent space that represent the both of data. Fig. Sishows that VMSB/VAL /VSSB-adv outperformed hight/LighttB-ad (with identical architecture) in the fidelity of samples and semantics of letters for latent and pixel spaces. In Table 4 the VMSB method on the ALAE embedding space was able to surpass deep SB models with a fewer number of parameters of K = 256. Even for raw pixels, our algorithm also achieved competitive FID and uuput/outptt MSD similarity scores for K 4096. The consistent performance gains from the LightSB and LightSB-ady algorithms strongly supports our theoretical claims on online learning.
FFHQ. Following the latent SB setting of Korotin el al. (2024, we assessed our method by utilizing pretrained ALAE model for generating 1024 1024 images of the FFHQ dataset (Karras et al. 2019). With the predefined 512-dimensional embedding space, we trained our SB models on the latent space to solve four distinct tasks: Adult-+ Child, Child Adult, Female- Male, and Male Female. Fig. g illustrates that our method delivered high-quality translation results. We also conducted a quantitative analysis using the ED on the ALAE embedding as a metric for evaluation, and the corresponding quantitative results are reported in Table 13 The result also verifies that our VMSB and VMSB-M M algorithms consistently achieved lower ED scores than other baselines, demonstrating its applicability for the high dimensional embedding space. Consoquently, the image-to-image transfer results showed that the generality of our online lcarning hypothesis and that the proposed algorithm is highly capable of interacting with neural networks of complex learning dynamics. Considering the significantly higher dimensionality of image domains relative to the batch sizes used in VOMD, the consistent and stable performance improvements demonstrated in our experiments strongly validate our theoretical claims regarding the robustness of our approach in online learning scenarios.
7 Conclusion
In this paper, we introduced VMSB, practical simulat ion-free Schrödinger bridge algorithm based on the online learning theory, designed for effectively solving SB problems encountered in real-world scenarios. We proposed a robust theoretical learning framework applicable to general SB solvers, leveraging a dual geometric interpretation of convex optimization to construct a robust OMD algorithm with rigorous guarantees convergence and regret. bounds. Furthermore, we proposed the computational algorithm for our OMD geometry. Through extensive empirical evaluation,
framework by employing the Sontersstin--Fsher---e we validated the effectiveness of VMSB across diverse settings, including high-dimensionall spaces, limited- sample regimes, and online learning environments. The experimental results consistently demonstrated stable and superior benchmark performance, highlighting the enhanced robustness of our approach. Consoquently, we argue that the proposed VMSB algorithm and our theoretical arguments regarding VOMD provide promising and robust methodology for probabilistic generative modeling within learning- theoretic contexts.
Limitations. In this work, we significantly reduced the computational complexity inherent in the MD framework by adopting the Wasserstein--isher--e geometry. GMM-based models, due to the lack of deep structural processing, tend to focus on instance-level associations of images in EOT couplings rather than the subinstance- or feature-level associations that are intrinsic to deep generative models. As a result, while VMSB produces statistically valid representations of optimal transportation within the given architectural constraints, these outcomes may be perceived somewhat, synthetic compared to large neural networks.
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Nevertheless, GMM-based models still hold irreplaceable role in numerous problems such as latent diffu- sion and variational methods, due to their simplicity and distinctive properties Korotin et al. (2024). As we successfully demonstrated in two distinet ways of interacting with neural networks for solving unpaired image transfor, we hope our thcoretical and empirical findings help novel neural architecture studics. While VMSB strictly outperforms existing SB solvers across standard numerical benchmarks, the performance gains are marginal some scenarios. For instance, the single-cell dynamics experiment utilizes a PCA-based prepro- cessing pipeline (Tong et al. 2024ah, and the transformation into this lower-dimensional space compromises the biological relevance of the SB objective, consequently narrowing the discernible performance gap.
Future rescarch. One line of future studies in SB is a general understanding of learning in diffusion models with various regularizations. This includes diffusion models in various problem-specific constraints, and geometric constraints from manifolds. Another direction is the extension the theoretical results into network architecture design. From $ 4.2 a pair of Schrödinger potentials represent dual representation of SB in a statistical manifold. In Gigli & Tamanini (2020), such potentials satisfy the HamiltonJacobi- Dattttnn--acobb-Bellaa Bellman (HJB) equations and, this can be trained with forward-backward SDE (SB-FBSDE) as presented by Liu et al. (2022) However, this requires many simulation samples from SDEs, and the requirements for applying VMSB contain a tractable way of estimating gradient flows, and a guarantce of mcasure concentration. Therefore, we expect there will be a new studies of mergy--assed neural architecture for efficiently representing SB, which will advance various subfields of machine learning. Lastly, a theoretical generalization of our work can be done by considering the Orlicz space for EOT studied by Lorenz & Mahler (0022). Since we essentially devised our theoretical framework to be compatible with arbitrary Bregman potentials, we believe controlling regularity of Young functionals can find more generalized learning algorithms for a wider range of OT problems.
References
Robert A Adams and John JF Fournier. Soboleu spaces, volume 140. Elsevier, 2003.
Charalambos D. Aliprantis and Kim C. Border. Infinite Dimensional Analysis: a Hitchhiker's Guide, Springer, 2006.
Jason Altschuler, Sinho Chewi, Patrik Gerber, and Austin Stromme. Averaging on the Bures- Wasserstein man- ifold: dimension-free convergence of gradient descent. Advonces in Neural Information Processing Systems, 34: 22132-22143 2021.
David Alvarez-Melis and Tommi Jaakkola. Gromov Wassenstein alignment of word embedding spaces. arXio preprint arXiu:1809.00 18190080000001 2018.
Shun-ichi Amari. Information geometry and its applications, volume 194. Springer, 2016.
Shun-ichi Amari and Hiroshi Nagaoka. Methods of information geometry, volume 191. American Mathematical Soc., 2000.
Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré. Gradient flows: in metric speces and in the space of probability measures. Springer Science Business Media, 2005a.
Luigi Ambrosio, Nicola Gigli, and Giuseppe Savaré. Gradient flows: in metric spaces and in the space of probability measures. Springer Science & Business Media, 2005b.
Ehsan Amid and Manfred K. Warmuth, Reparameterizing mirror descent as gradient descent. Advances in Newral Information Processing Systems, 33:8430 8439, 2020a.
Ehsan Amid and Manfred K Warmuth. Winnowing with gradient descent. In Conference on Learning Theory, pp. 163-182. PMIRR, 2020h.
Pierre-Cyril Sabbin-Frankwskki, Anna Korba, and Flavien Léger. Mirror descent with relative smoothness measure spaces, with application to Sinkhorn and EM. Advances in Neural Information Processing Syslems, 35:17263 17275, 2022.
Peter Auer, Nicolo Cesa-Biande Bianchi, and Claudio Gentile. Adaptive and self-confident on-line learning algorithms. Journal of Computer and System Sciences, 64(1):48-75, 2002.
Amir Beck and Mare Teboulle. Mirror descent and nonlinear projected subgradient methods for convex optimization. Operations Research Letters, (3):1667717 2003a.
17

---

Published in Transactions on Machine Learning Research (01/2026)
Amir Beck and Marc Teboulle. Mirror descent and nonlinear projected subgradient methods for convex optimization. Operalions Research Letters, 31(3):167-117 175, 2003b. ISSN 0167-6377.
Jean-David Benamou and Yann Brenier. A computational fluid mechanics solution to the Monge-Kantorovich mass transfer problem. Numerische Mathematik,, 84(3):375 393, 2000.
Pierre Bernhard and Alain Rapaport. On a theorem of Danskin with an application to a theorem of von neumann-sion. Nonlinear Analysis: Theory, Methods & Applications, 24(8):1163 1181, 1995.
Rajendra Bhatia, Tanvi Jain, and Yongdo Lim. On the Bures Wasserstein distance between positive definite matrices. Expositiones Mathematicae, (7(2):165 191, 2019.
Benjamin Birnbaum, Nikhil R. Devanur, and Lin Xiao. Distributed algorithms via gradient descent for fisher markets. In Proceedings of the 12th ACM conference on Electronic commence, pp. 127-136, 2011.
David M Blei, Alp Kucukelbir, and Jon D McAuliffe. Variational inference: A review for statisticians. Journal of the American statistical Association, 112(518):859 877, 2017.
Stephen Boyd and Lieven Vandenberghe. Conven optimization. Cambridge university press, 2004.
James Bradbury, Roy Frostig, Peter Hawkins, Matthew James Johnson, Chris Leary, Dougal Maclaurin, George Nec- ula, Adam Paszke, Jake VanderPlas, Skye Wanderman- Milne, and Qiao Zhang. JAX: composable transformations of Nummy programs. 2018. URL http://github.. http:///gtthuuu.oo//ggooo///
Lev M. Bregman. The relaxation method of finding the common point of convex sets and application the solution of problems in convex programming. USSR computational mathematics and mathematical physics, 7(3): 200 217, 1967.
Donald Bures. An extension Kakutanis theorem on infinite product measures to the tensor product semifinite w² algeebars.s. Transactions of the American Mathematical Sociely, 135:199 212, 1969.
Dan Butnariu and Elena Resmerita. Bregman distances, totally convex functions, and a method for solving operator equations in Banach spaces. In Abstract and Applied Analysis, volume 2006, 2006.
Guillaume Carlier. On the linear convergence of the multimarginal sinkhorn algorithm. STAM Journal on Optimiza- tion, 32(2):786-77) 794, 2022.
Mathilde Caron, Ishan Misra, Julien Mairal, Priya Goyal, Piotr Bojanowski, and Armand Joulin. Unsupervised learning of visual features by contrasting cluster assignments. Advances in neural information processing systems, 33:9912 9924, 2020.
José Carrillo, Rishabh S Gvalani, and Jeremy S-H Wu. An invariance principle for gradient flows in the space of probability measures. Journal of Differential Equations, 345:233-28 2244, 2023.
Gong Chen and Mare Teboulle. Convergence analysis of proximal-like minimization algorithm using bregman functions. SIAM Journal on Optimization, 3(3)5388 543, 1993.
Keyi Chen and Francesco Orabona. Generalized implicit follov--heerrgglaiiieddleddd In International Conference on Muchine Learningg, pp. 4826 4838. PMLR, 2023.
Tianrong Chen, CnannHorng Liu, and Evangelos A. Theodorou. Likelihood training of Schrödinger bridge using (orward-backward sdes theory. In 10th International Conference Learning Representations, 2022.
Yongxin Chen, Tryphon T Georgiou, and Allen Tannenbaum. Optimal transport for Gaussian mixture models. JEEE Access, 7:6269 6278, 2018.
Lenaic Chizat, Gabriel Peyré, Bernhard Schmitzer, and François- Xavier Vialard. An interpolating distance between optimal transport and Fisher Rao metrics. Foundations of Computational Mathematics, 18:1 44, 2018.
Giovanni Conforti. Weak semiconvexity estimates for schrödinger potentials and logarithmic sobolev inequality for schrödinger bridges. Probability Theory and Related Fields, pp. 1 1-27, 2024.
Giovanni Conforti, Alain Durmuss, and Giacomo Greco. Quantitative contraction rates for sinkhorn algorithm: beyond bounded costs and compact marginals. arXio preprint ar arXiv:2304.0442 04451,, 2023.
18

---

Published in Transactions on Machine Learning Research (01/2026)
Isaac P. Cornfeld, Sergej V Fomin, and Yakov Grigorevich Sinai. Ergodic theory, volume 245. Springer Science & Business Media, 2012.
Marco Cuturi. Sinkhorn distances: lightspeed computation of optimal transport. In Advances in Newral Information Processing Systems, volume 26, 2013.
John M Danskin. The theory of mar-min and its application to weapons allocation problems, volume 5. Springer, 1967.
Kamélia Daudel et al. Mixture weights optimisation for alpha-divergence variational inference. Advances in Neural Information Processing Systems, 34:3977 1408, 2021.
Valentin De Bortoli, James Thornton, Jeremy Heng, and Arnand Doucet. Diffusion Schrödinger bridge with appli- cations to score-based generative modeling. Adnances in Neural Information Processing Systems, 34:1 14:17695-17709, 2021.
Nabarun Deb, Young-Heon Kim, Soumik Pal, and Geoffrey Schiebinger. Wasserstein mirror gradient flow as the limit of the Sinkhorn algorithm. arXie preprint or orXie:2377114221 2023.
Michael Ziyang Diao, Krishna Balasubramanian, Sinho Chewi, and Adil Salim. tional inference via JKO in the Bures- Wasserstein space. In International Conference on Machine Learning, pp. 7960-7991. PMLR, 2023.
Gaussian varia-
John C Duchi and Hongseok Namkoong, Learning models with uniform performance via distributionally robust optimization. The Annals of Statistics, 49(3):1378-14066 2021.
Kwok Sau Fa. Solution of fokker-planck equation for a broad class of drift and diffusion coefficients. Physical Review E, (1110122, 2011.
Huang Fang, Nicholas JA Harvey, Victor S Portella, and Michael P Friedlander. Ouline mirror descent and dual averaging: keeping pace in the dynamic case. Journal of Machine Learning Research, 23(121):1 38, 2022.
Amos Fiat and Gerhard J Woeginger. Online algorithms: The state of the art, volume 1442. Springer, 1998.
Lakas-Benediikt Fiechtner and Jose Blanchet. Wasserstein distributionally robust regret optimization. arXio preprint 2rX:::2504.1009 10796, 2025.
Udaya Ghai, Flad Hazan, and Yoram Singer. Exponentiated gradient meets gradient descent. In Algorithmic learning theory, pp. 186-407. PMLR, 2020.
Nicola Gigli and Luca Tamanini. Samamon-Brenias and duality formulas for the entropic cost on RCD" RCD*(K, spaces. Probability Theory and Related Fields, 176(1-2):1-34 2020.
lan Goodfellow, Jean Pouget- Mehdi Mirza, Bing Xu, David Warde-Farley, Sherjil Ozair, Aaron Courville, and Yoshua Bengio. Generative adversarial nets. In Advances in Newral Information Processing Systems, pp. 2672-2680, 2014.
Robert M Gray and John C Kieffer. Asymptotically mean stationary measures. The Annols of Probability, pp. 962 973, 1980.
Leonard Gross. Logarithmic Sobolev inequalities. American Journal of Mathematics, 97(4):1061-1083, 1975.
Suriya Gunasekar, Blake E. Woodworth, and Nathan Srebro. Mirrorless mirror descent: A more natural discretization of riemannian gradient flow. CoRR, 0oss/204.01100 2020.
Nikita Gushchin, Sergei Kholkin, Evgeny Burnaev, and Alexander Korotin. Light and optimal schrödinger bridge matching. In International Conference on Machine Learnin. PMLR, 2024a.
Nikita Gushchin, Alexander Kolesov, Petr Mokrov, Polina Karpikova, Audrei Spiridonov, Evgeny Burnaev, and Alexander Korotin. Building the bridge of Schrödinger: A continuous entropic optimal transport benchmark. Advances Neural Information Processing Systems, 36, 2024b.
Dong-Sig Han, Hyunseo Kim, Hyundo Lee, JeHwan Ryu, and Byoung- Tak Zhang, Robust imitation via mirror descent inverse reinforcement learning. Advonces in Neural Information Processing Systems, 35:30031-300 -30043, 2022.
19

---

Published in Transactions on Machine Learning Research (01/2026)
Jean-Baptiste Hiriart-Urruly and Claude Lemaréchal. Fandomentals of convex analysis. Springer Science & Business Media, 2004.
Ya-Ping Hsieh, Ali Kavis, Paul Rolland, and Volkan Cevher. Mirrored Langevin dynamics. Information Processing Systems, 31, 2018.
Advances in Neural
Eric Jang, Shixiang Gu, and Ben Poole. Categorical reparameterization with gumbel-softmax. 5th International Conference on Leorning Representations, 2016.
Sadeep Jayasumana, Srikumar Ramalingam, Andreas Veit, Daniel Glasner, Ayan Chakrabarti, and Sanjiv Kumar. Rethinking Towards a better evaluation metric for image generation. Proceedings of the TEEE/CVP Con- ference on Computer Vision and Pattern Recognition, pp- 9307-9315, 2024.
Richard Jordan, David Kinderlehrer, and Felix Otto. The variational formulation of the Fokker Planck equation. SIAM journal on mathematical analysis, 29(1):1-17, 1998.
Mohammad Reza Karimi, Ya-Ping Hsieh, and Andreas Krause. Sinkhorn flow as mirror flow: a continuous-time framework for generalizing the sinkhorn algorithm. In International Conference Artificial Intelligence and Statistics, 4186-4194. PMLR, 2024.
Tero Karras, Samuli Laine, and Timo Aila. A style-based generator architecture for generative adversarial networks. In Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, pp. 4401-4410, 2019.
Alexander Korotin, Nikita Gushchin, and Evgeny Burnaev. Light Schrödinger bridge. In 18th International Confer- ence Learning Representations, 2024.
Daniel Kuhn, Soroosh Shafiee, and Wolfram Wiesemann. Distributionally robust optimization. Acta Numerica, 34: 579-804, 2025.
Solomon Kullback. Probability densities with given marginals. The Annals of Mathematical Statistics, 39(4): 1236 1243, 1968.
Mare Lambert, Sinho Chewi, Francis Bach, Silv ère Bonnabel, and Philippe Rigollet. Variational inference via Wasserstein gradient flows. Advances in Newral Information Processing Systems, 35:14134 14447, 2022.
Flavien Léger. A gradient descent perspective on Sinkhorn. Applied Mathematics 8 Optimization, 84(2):18443 1855, 2021.
Yunwen and Ding- Xuan Zhou. Convergence of ouline mirror descent. Applied and Computational Harmonic Analysis, 48(1):343-311 2020.
Christian Léonard. From the Schrödinger problem to the Monge Kantorovich problem. Journal of Functional Analysis, 262(4))1879-197)))) 2012.
Christian Léonard. A survey the Schrödinger problem and some of its connections with optimal transport. arXiv preprint Xi0:1103 0215, 2013.
Christian Léonard. Some properties of path measures. Séminaire de Probabilités XLVI, pp. 207-230, 2014.
Matthias Liero, Alexander Mielke, and Giuseppe Savaré. Optimal transport in competition with reaction: The Hellinger Hellinger-Kantooorie Kantorovich distance and geodesic curves. SIAM Journal on Mathematical Analysis, 48(4):2869-290 2016.
Matthias Liero, Alexander Mielke, and Giuseppe Savaré. Optimal tranpoot problems and a new Hellinger- Kantorovich distance between positive measures. Inventiones mathemalicae, 211(3):969 1117, 2018
Guan-Horng Liu, Tianrong Chen, Oswin So, and Evangelos Theodorou. Deep generalized Schrödinger bridge. In Advances Newral Information Processing Systems, pp. 9374 9388, 2022.
Guan-Horng Liu, Arash Vahdat, De-An Huang, Evangelos A Theodorou, Weili Nie, and Anima Anandkumar. I-sb: Image-to-image Schrödinger bridge. In International Conference on Machine Learning, pp. 22042-22062. PMLR, 2023.
Dirk Loreuz and Hinrich Mahler. Orlicz space regularization of continuous optimal transport problems. Applied Mathematics 8 Optimization, 85(2):14, 2022.
20

---

Published in Transactions on Machine Learning Research (01/2026)
John Lott. Some geometric calculations Wasserstein space. Commaaiicatioss in Malhemalical Physics, (2) 423 423-437, 2008.
Haihao Lu, Robert M Freund, and Yurii Nesterov. Relatively smooth convex optimization by first-order methods, and applications. SIAM Journal on Optimization, 28(1):333-335 354, 2018.
Yulong Lu, Jianfeng Lu, James Nolen. Accelerating langevin sampling with birth-death. arXiv preprint arX5v:1905. 098663, 2019.
Aleksander Madry, Aleksandar Makelov, Ludwig Schmidt, Dimitris Tsipras, and Adrian Vladu. Towards deep learning models resistant to adversarial attacks. arXiv preprint arXio:1706 06083, 2017.
Florient Malrieu. Logarithmic sobolev inequalities for some nonlinear pde's. Stochastic processes and their applica- lions, 95(1):109- 132, 2001.
Brendan McMahan. Follow-the www.ttee--aaaiied-eeeeeeeee and mirror descent: Equivalence theorems and 11 regularization. In Proceedings of the Fourleenth International Conference Artificial Intelligence and Slatistics, pp. 525-533. JMLR Workshop and Conference Proceedings, 2011.
John Milnor. Microbundles: Part i. Topology, 3:53-80, 1964.
Daniel Morales-Brotones Thijs Vogels, and Hadrien Hendrikx. Exponential moving average of weights in deep learning:
Dynamics and benefits. Transactions on Machine Learning Research, 2024.
Edward Nelson. Dynamical theories of Brownian motion. Princeton University Press, 2001.
Arkadií Semenovich Nemirovsky and David Borisovich Yudin. Problem complenity and method efficiency in optimiza- Gion. Wiley-Interscience publication. Wiley, 1983.
Yurii Nesterov. Primal-dual subgradient methods for convex problems. Mathematical programmingg 120(1):221 259, 2009.
Marcel Nutz. Introduction to entropic optimal transport. Lecture notes, Columbia University, 2021.
Marcel Nutz and Johannes Wiesel. Stability of Schrödinger potentials and convergence of sinkhorns algorithm. The Annals of Probebility, 51(2):699-722, 2023.
Bernt Øksendal. Stochastic Differential Equations: An Introduction with Applications. Springer, 2003.
Francesco Orabona and Dávid Pál. Scale-free online learning. Theoretical Computer Science, 716:50 69, 2018.
Felix Otto. The geometry of dissipative evolution equations: the porous medium equation. Communications in Partial Differential Equations, 26(1-2):101-174, 2001.
Felix Otto and Cédric Villani. Generalization of an inequality by talagrand and links with the logarithmic sobolev inequality. Journal of Functional Analysis, 173(2):361-400, 2000.
John Paisley, David Blei, and Michael Jordan. Variational Bayesian inference with stochastic search. or preprinl arXin: 1206.1200..00 64300,66..666 2012.
Michele Pavon Anton Wakolbinger. free energy, stochastic control, and Schrödinger processes. Model- ing, Estimation and Control of Systems with Uncertainly: Proceedings of Conference held in Sopron, Hungory, September 1990, pp. 334-348. Springer, 1991.
Gabriel Peyré, Marco Cuturi, et al. Computational optimal transport: With applications to data science. Foundations and TrendsG in Machine Learning, 11(5-6):355 607, 2019.
Stanislav Pidhorskyi, Donald A Adjeroh, and Gianfranco Doretto. Adversarial latent autoencoders. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition, pp. 14104-14113, 2020.
Alec Radford. Unsupervised representation learning with deep convolutional generative adversarial networks. arXi preprint #xXi::511.06453 2015.
Cale Rankin and Ting- Kam Leonard Wong. Bregman-Vassserstinn Wasserstein divergence: geometry and applications. ar preprint w.Xi:2332.00883 2023.
21

---

Published in Transactions on Machine Learning Research (01/2026)
G. Raskutti and S. Mukherjee. The information geomelry of mirror descent. IEEE' Transactions Information Theory, 61(3):1451-144) 144577, 2015.
Maria L Rizzo and Gábor J Székely. Energy distance. wiley interdisciplinary reviews: Computational statistics, 8(1): 27-38, 2016.
Halsey Lawrence Royden and Patrick Fitzpatrick. Real analysis, volume 32. Macmillan New York, 1988.
Filippo Santambrogio. Optimal transport for applied mathematicians. Birkūuser, NY, 55(58-63):94, 2015.
Filippo Santambrogio. [Euclidean, metric, and Wasserstein] gradient flows: an overview. Bulletin of Mathematical Sciences, 7:87 154, 2017.
Erwin Schrödinger. Sur théorie relativiste de l'électron l'interpéé tation de mécanique quantique. Annoles de l'institut Henri Poincaré, volume 2, pp. 269-310, 1932.
Shai Shalev-Shwartz et al. Online learning and online convex optimization. Foundations and Trends in Machine Learning, 1(2):107-19)) 194, 2012.
Yuyang Shi, Valentin De Bortoli, Andrew Campbell, and Arnaud Doucet. Diffusion Schrödinger bridge matching, In Adeances in Neural Information Processing Systems, volume 36, pp. 62183 62183-62223, 2023.
Hirohiko Shima and Katsumi Yagi. Geometry of Hessian manifolds. Differentiol Geometry and its Applications, (3):277-290, 1997. ISSN 0926-2245.
Nati Srebro, Karthik Sridharan, and Ambuj Tewari. On the universality of online mirror descent. In Advances in Neural Information Processing Systems, pp. 2645 2653, 2011.
Alexander Tong, Nikolay Malkin, Kilian Fatras, Lazar Atanackovic, Yanlei Zhang, Guillaume Huguet, Guy Wolf, and Yoshua Bengio. Simulation-free Schrödinger bridges via score and flow matching, arXiv preprint Xrxi::2377. 03672, 2023.
Alexander Tong, Kilian Fatras, Nikolay Malkin, Guillanme Huguet, Yanlei Zhang, Jarrid Rector-Brooks, Guy Wolf, and Yoshua Bengio. Improving and generalizing flow-based generative models with minibatch optimal transport. Transactions Machine Learning Research, 2024a. ISSN 2835-8856. Expert Certification.
Alexander Y. Tong, Nikolay Malkin, Kilian Fatras, Lazar Atanackovic, Yanlei Zhang, Guillaume Hugnet, Guy Wolf, and Yoshua Bengio. Simulation-fre Schrödinger bridges via score and flow matching. Proceedings of The 27th International Conference Artificial Intelligence and Statistics, volume 238 of Proceedings of Machine Leorning Research, pp. 1279-1287. PMLR, 2024b.
Francisco Vargas, Pierre Thodoroff, Austen Lamacraft, and Neil Lawrence. Solving Schrödinger bridges via maximum likelihood. Entropy, 23(9):1131, 2021.
Cédric Villaui. Optimal tronsport: old and new, volume 338, Springer, 2009.
Cédric Villani. Topics in optimal transportation, volume 58. American Mathematical Soc., 2021.
Huan Xu, Constantine Caramanis, and Shie Mannor. Robust regression and lasso. Advances in neural information processing systems, 21, 2008.
Martin Zinkevich. Online convex programming and generalized infinitesimal gradient ascent. In Proceedings of the 20th international conference on machine learning (ieml-03), pp. 928-93989- 9366, 2003.
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Appendices for
Variational Online Mirror Descent for Robust Learning in Schrödinger Bridge
Abbreviation and Notation
Abbreviation
Expansion
Notation
Usage
SB
Schrödinger Bridge
marginal distributions volatility of reference measure
SBP
Schrödinger Bridge Problem
C
EOT
Entropy-rogularized Optimal Transport
cost c. (x y) = 2 || - y
MD
Mirror Descent
a coupling of and conditional distributions n-th marginal
OMD
Online Mirror Descent
f
KL
Kullback-Leibler
IPF
Iterative Proportional Fitting
log-Schrödinger potential
BW
Bures Wasserstein
0, Dn
Bregman dnfential/divergenee
WFR
Pheserrstein-Fisee Rao
d+
directional derivative
SDE
Stochastic Differential Equation
6e, 6p
First variations
PDE
Partial Differcutial Equation
W
Wasserstein-2 2 gradient operator
FP
FokkerPlanch
dynamic stochastic process SB drift function
GMM
Gaussian mixture model
g
indicator function
S
positive definite and symmetric matrices
A Theoretical Details and Proofs
Background first variation operators. In this paper, utilize the notation of first variation operators & and & to identify the generalized primal and dual spaces in Schrödinger bridge. Since the problems are classified as an infinteediimensioaal optimization (Aliprantis & Border 2006), we introduce the essential background supporting the necessity of these operators. We introduce Gâteaux and Fréchet diffcrentiablility Frabis-Frankowski ct al. 2022 Karimi ct al. 2024).
Definition 4 (Gâteaux & Fréchet differentiabliltty). Let M be topological vector space of measures space X. Define the Gâteaux differentiablity of functional F; M + R, if there exists gradient operator Vea such that for an arbitrary direction E .,, defined as the limit
F(x + hv) F(x) DEM
VGa(F(z)(o)
lim A-50
h
If the limit exists in the unit ball in M, the function F is called Fréchet differentiable with VRrF((r).
The problem of the Gâteaux and Fréchet differentiability in the context of SB is that the limit must be given in all dircctions, implying that every neighboring point must be within the domain of the topological space M. For the case of functionals such as the KL divergence functional F()) KL(-Ix*), the domain of F and has an empty interior [Aubin-Fraane Frankowski et al. 2022). To resolve this issue, we use directional derivatives and first variations, defined in Definitions and
First variations KL. Suppose that we have two distributions p.pl E P2(A), s Rd, Let us consider the log likelihood of p': (a(x) = (ogp'(x). and an element of a (topological) tangent space v E T,P2(X)
(Milnor 1964). Then, can achieve the followings:
kL(elle') log p(z) dp(x) - ((x) dp(z)
(22)
f (())((())= + ho(x)] dr f (()))(()) dr to+hl h (()))(())))
+h
(23)
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Given that log(2 + (2 + =) log(z)2 [log(2) + 1|s + o(E), and fx v(x) dr 0, achieve
J log(p(x) ho(z)) (p(x) hu(xr) dx
x log ((g)(())(a)))) dr + (x) 1]bo(z)))))))))))))))))) + o(h) dx
log log)(z)p(z) dr + h log logp(z))(2)dr + h v(x)) + o(h)
(24)
Combining Eqs.
22 24 we achieve
+(p+ho)= = F()) 1
( log ( / +
(25)
By Eq. 25 and Definition 2 the first variation 8132 E (()) exists for infinitesimal A > 0. Therefore, the first variation of KL is derived as 8KL(plpp') = log e- In machine learning, log likelihoods of probabilistic models are often given in closed-form expression, incentivizing development of computational continuous EOT/SB methods. Generally, identical arguments generally apply to all KL functionals with respect distributions (T, æt and marginals) in our setup.
Asymptotically log-concave distributions. For convergence analysis, we assume each marginal distri- bution is in log-concave distribution, particularly satisfying the log Sobolev inequality of measures, motivated by relevant literature (Otto & Villani 2000 Conforti 2024). This assumption works a wider range of costs
and marginals beyond popular choices with boundedness and compactness [Nutz & Wiesel 2023
Conforti
at al 2023). Suppose that marginals admit densities of the form
ı(dx)= = exp(-Uu(x))dr and
v(dy) = exp(-Uu(y))
(26)
We exploit the following definition from (Conforti et al. 2023 in order to describe asymptotically log- concaveness.
Definition 5 (Asymptotically strongly log-concavity; Conforti et al. 2023 Suppose that marginals p and v admit. a positive density against the Lebesgue measure, which can be written in the form 266 In particular, consider collection of functions g {g C²((0, (oc0,,+)),,A,, B4)lr + r 1/2 "2gg(r" 72 jis non-increasing and concave, lim, rg(r) 0}. Accordingly, define set
I {g E 9 bounded and s.t. lim g(r) O, g' 2 and 2g" + gg' 0) Cg.
r-U+
R4 + IR of a differentiable function U as the following and converity profile ky
Ky (r) = inf (VU(x) - (U()), |x vl =r
x y/
We say a potential is asymptotically strongly convex if there exists ay 4 and gu E & such that
(er()) 2 oy r-- rrr'rrr))
(27)
holds for all > O. We consider the set of asymptotically strongly log-concave probability measures (RR) (C(dz) exp( exp(-U(x))dr: U C2(RR)) U is asymptotically strongly convex).
It is essential to note that a mixture of asymptotically log concave is also asymptotically log concave. Lemma 4. For posilive weighls { with ΣK 3x and asymplolically log concave distributions {px) π= =K 3kpk-
Proof. Let us reformulate the mixture as log log = log Ee Br exp(-Ux(x)) for asymptotically strongly convex functions U kK The gradient is
VUI
V log ग = 3 P, -
p = softmax(log B -U)
[Uk]
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If each Ug of mixture satisfy Eq. (27 with ou, and 9U2 there exist ay minigksk au, and (u(()
-rlog Ek=1 exp(-r- that satisfies the condition (27 for - log n. By direct calculation, one can easily see that soft min-like property of gu from {r 90. Jem does not change the conditions of G. ]
General assumptions and justifications. We additionally need the following general assumptions for our OMD framework. @ (Existence) The sequence of MD from Eq. 13 exists {Te C, and are unique, @ (Relative smoothusss/coneesee
For some 1, T. 2 O, the functional Fe is T-smooth and 1-strongly- convex relative to Q. @ (Existence of first variations) For each t 2 O, the first variation (()(T)))))))))))) exists. O (Boundedness estimations) The asymptotic dual mean a is almost surely bounded Pr(Da(T (T ( ( I R)
for some R O. @ (Ergodicity) The estimation process of {=() ] is governed by a measure-preserve transformation on a mcasure space ()/, E) with s()) 1; for every event E Σ, ((T-((E)A)) (that is, E invariant), either sEE) Fi()) KL(+ is well defined since KL is strong Bregman divergence with lower semicontinuity, where the existence of a primal solution in guaranteed as discussed in Aubin-Frarn Frankowski et al. (2022). For @-@. we can identify = L 1 and close-form expression of the first variation that is shown in Definition 6 and Proposition 3 For the assumptions D-@, we postulate the existence of estimates produced from a
0
or sEE) = (ornfflld et al., 2012) For O, the temporal cost
Monte- Carlo method, using a fixed amount of updates on topological vector space. Hence, it natural to consider that these estimates will be bounded probabilistic sense and yield Markovian transitions, which are aperiodic and irreducible.
A.1 Proofs of Lemmas 2 and
The EOT in Eq. can be reformulated divergence minimization problem with respective to a reference measure. If Gibbs parameterization is enforced with the quadratic cost functional cr(x, v) * || y/l for 0, it is well-known that the problem has the equivalence with the entropy regularized optimal transport problem Nutz 2021
OT~(H,V)= = inf (T TEll(H.))
(28)
Note that the above equation corresponds to the constrained minimization of (L(T][WEs) in Eq. 6 by the disintegration theorem of Schrödinger bridge (Appendix A of Vargas et al. 2021 While the Bregman projection formulation of Sinkhorn Eq. (14p are described by the spaces (IT [T ). it is (equally) natural to think that considering the problem as convex problem with the distributional constraint C (see the primal space in illustrated in Fig. 1. As a problem the constraint C, one can consider a temporal cost functional Fi( = a KL a [|)) + (1 az) (LL)T(()) with sequences {ar] = {0, 1, 0, 1....) } for 71T(x) == J T<x yd and (y) = j (x, y) dr. By construction, we have the following MD update:
(5 n.) + Da(TIllx.).
(29)
The optimization problem 29
equivalent to having the property for subsequent T++1:
d F( m.) + Dn(T (.)) 2 d Fi (m m+41 T.) ) + Do
(30)
(o. - (. m+1) + (I((T) - ((T+1)) 2 0, VT EC
Setting the free parameter π = T+1 + h(π T and taking the limit h + 0+ yields described the time evolution of the log-Schrödinger potentials for ग. = -p. -ce dedepeedppp @ v):
d(H1m)) -a spt 40 + log
eve- u(dv)).
c: log dv.
(31a)
Rd
b. log d(127T)))
8 the - + log
I ePI-p'' Rd
"Pi-P" u(dr))
(31b)
dp.
Here, A denotes the symmetric difference, equivalent exclusive-or with respect to membership.
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2014 we also achieve the first variation of with respect to Published in Transactions on Machine Learning Research (01/2026)
for de and B at Setting discrete approximation of dynamics Eq. 1/++1 the + 4/2 yields the following alternating updates:
31
Pi+1
(e + yo and
12)+ -log
(ald))))allss)))d(( P2++2(2) llg (xwua)))-aatww((())))))
Rd
gd
Therefore, the proof of Lemma
2 is complete.
et al. 2022). the static, idcalized MD cost F()) <L(- yield the following form expression for the first variation:
From the dual itcration of KL stated in Eq. (44
())(()))) 6<(((m++1) = =m (6.s(mm)) 6cf(T")),
where the equation implies that setting ne 1 for MD yields one-step optimality x² in this idealized condition. Utilizing the equivalence of first variation stated in Lemma and the disintegration theorem for the Radon-Nikodym derivatives, get the first variation of F with respect to for all æ
(F(m)) log dx dr
(32)
And by the disintegration theorem # for all x as
[Lonard
( log
(33)
d
where /(#*) = KL(** (*) Since this disintegration theorem always hold for every directional derivative, can expression for म" and interchangeably. It is well-known that MD is discretization of natural gradient descent [Gunasekar et al. 2020), and our setting for Ω generates the geometry governed by the (generalized) Fisher information. In this particular case, one can use Otto's formalization of Riemannian calculus (Otto 2001 $ 3.2), and the probability space equipped with the Wasserstein-2 metric (P*),,), W2). is generally represented as a Wasserstein gradient flow
OET -
VaT C,
(34)
where Vy denotes the Wasserstein-2 gradient operator yiclds
VH = V. (p sn). In particular, plugging Eq. 32
05mg (Vg + Aw,
(35)
where A denotes the Laplace operator. The foundational results concerning Wasserstein gradients were initially established by JKO (Jordan et al.] 1998 who demonstrated that the formulation in Eq. (34] corresponds precisely to the FokkerPlanck equat ion (35 Consequently, it follows that Wasserstein gradients
characterize the tangential direction of flows on a manifold constrained by distributional properties and
endowed with the W2 metric.
A.2 Proof of Theorem 1
We start with introducing basic properties of the Bregman divergence in Definition 3 First, the idempotence property states that Bregman divergence associated with another Bregman divergence Da(-ly) remains the identical divergence with the original. Note that the (global universal) idempotence initially stated by et al. (2022), but apply some changes to the statement and only work with localized version of idempotence for the purposc of this paper.
Lemma 5 (Idempotence). Suppose convez potential M(X) RRUR {+oo),, where M(X) denotes topological vector space for X. Assume that for all dom(?), d.((2) exists. Then, Vx, y C dom()): DDo(ll)) Dn(xly).
More precisely, needs apply Lemma for KT,, and the disintegration theorem get Eq.
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Proof of Lemma Both Bregman divergences and Bregman potentials are convex functionals. By definition, we have D (zly) = (n(ell=) (vll=)= (bgS(v) dz((=)) y) for arbitrary 2, and setting 2 = y completes the proof. Another (informal) point of view is considering the Bregman divergence as a first-order approximation of a Hcssian structure, and DD((-s) converges to Da(-12) by taking a limit, knowing that Da(vly) 0.
We then proceed to the equivalence property of the family recursive Bregman divergences. The property is important for proving the theorem and representing the dual representation of MD. Moreover, it also usod Theorem 2 as a key ingredient which constructs our VOMD framework.
Lemma (Equivalence of first variations). Suppose Ω M(x) + U (+00} Assume that for all E dom( the first voriation 6g5((2) erists, then, for all z, yr 92 E dom(S)), the first variation taken for the first argument of the following Bregman divergences are equivalent: deDa(zly) &e Da 6e Da(1a)) (xlly)
Proof of Lemma 0 First, can be analytically driven daDly)) del((z) - 6, Next, by def- inition, taking the first variation of D (=)())s)((l)) with respect to for arbitrary dom(ft) yields & (=](rl)=) - 5e (P()) - 0(=), v)- Knowing that the second term 6c( - I((2, - y) is linear, we achieve (.Dp((1))(l)) = (x)) - 0,52(z) - (6.5(y) - 6x[((z) = d,l(z)) - 6,S(s), which completes the proof.
By an inductive reasoning, we arrive at the basic property of family of Bregman divergences, that all divergence recursively defined by the Bregman potential Ω, has the (local) idempotence and the (global) equivalence of first variation. To address characteristics for particular Bregman potential Q we apply the notions of relative smoothness and convexity with respect 2 which was first introduced by Birnbaum
et all 2011).
Definition (Relative smoothness and convexity). Let G M(x) + R R {+00} be proper convex functional. Given scalar 1, L 2 0, we define that G L-smooth and (-strongly-conver relative lt over C if for every z, y E dom(G) n dom( nc, we have
De(xlly)= < LDa(=llv,
De(ally) 2 (Da(zly),
respectively, where De and Da are Bregman divergences associated with G defined in Definition 3
Applying the idempotence lemma Lemma 5 we immediately recognize that the Bregman divergence Dg is relatively 1-smooth and l-strongly-convex for Q. To start our analysis, we reintroduce the well-known three-point identily for a Bregman divergence.
Lemma 7 (Three-point identity). For all Tai Tb. T. E C n dom(!), we have the following identity (8.S(TT))= - dGsl(m)) me - mb) = Da( - Da(mellTTa) + Da(TollTT))
when Da is the Bregman divergence defined in Definition
Proof of Lemma 7 By the definition of Bregman divergence, we have
Da( - (a( + Da(mll|=a) = a((m)) 2(Tx) 10000(66660 (6.s((T)) Tee Tbb)
- -(((T)) + P(TT) + (6(l(TT))))))))))))))))))))))() TTe TTa)
+ +[(TT) - ((TTT) (8ol(Ta)) TT
= -(6(((TT.) - {(T))) Te Tb)-
Therefore, the proof is complete.
Utilizing the three-point identity, we present the following useful lemmas for dealing incqualities regarding improvements by Han et al. (2022), which we call left and right Bregman differences.
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Lemma 8 (Left Bregman difference). For all Noz Tb, Tee C dom( the following ?dentity holds.
Da(TOI|Ta)) - DDn(TellTaT) = -66 de)l( te - mb) + Da(TFolFoO.
(36)
Proof of Lemma Using Lemma we have
Da(Foll=a) Da( - - + (6el(TTa) 5,5((T)); mb)- -oT)
Utilizing an identity of two Bregman divergences for arbitrary (p. p):
Dalellō) + Da(āllo) = (peR(p) ),5( p)-
(37)
We separate (T,)( (16) into dal(( (xl((Te) and 8,0((T)) 6,.((TT)) and write the rest of the derivation as follows.
Da((ollTa) - To(
-DD((TelTTO + (6:l(me)) 6,50(mb), Te - Tb) +(6:l((Ta) 6cs((Te), Te -mb)
Eq.
Do(Tol(Te) + ( de52(Te). Tb)
Therefore, we achieve the desired identity.
Lemma 9 (Right. Bregman difference). For all Ta Th, Tor the following identity holds.
Da(me mm) Da( (a(mal + (5.l(mma) 5.52(Tb); Tee na)
(38)
Proof of Lemma
By Lemma we have
Da((all=b) Du(TFll=a)= = - (TT [Ta) + (6.5((m.) 5,5((T)), n. -m>-
We separate T Tb into ae and Taa Tu and write the rest of the derivation follows.
De( ((a
=-Da(To|IT) (66S((Ta)) - 6,50(m)) Ta Tb) +(6Ll(TT)) 6cL((T)), me -ma)
Eq.
37
DD((Tal(T)) + (6. (.(()))) Te ma)
Therefore, achieve the desired identity.
Additionally, we introduce the three-point inequality (Chen & Teboulle 1993), which has been key state- ment for proving MD convergence for static cost functional (Aubin- Frankowski et al. 2022), and OMD improvement for temporal costs. The proof mostly follows Aubin-Frnn Frann-Frannoski et al. (2022) with a slight change of notation.
Lemma 10 (Three-point inequality). Given æ E MM(X) and some proper conver functional W M(x) + [+ if &olt exists, as well as p = arg min pee (V( Dn(p]( then for all E ( ddom()))))) (p) Dn(pl(=) (D)) Da(ll Da(pp)p).
Proof of Lemma 10 The existence of 6.52 implics C dommma( (a(-l))) C n dom( n dom( Set G(-) V(-)) + Dn(-lly). By linearity and idempotence, we have for any p dom(A)) n dV
Da(eloø) = D Dn(pllp) 2 Da(elp))
(39)
By / being the optimality for G, for all x C,
G((1 - h) + hp) G(p) 20, d'G(p, p p) lim h-0+
h
which suggests G(p) G(ō) Do(el(o). Applying
39 to this inequality complete the proof.
7
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The following argument is from the convergence rate of mirror descent for relatively smooth and convex pairs of functionals, and extend to infinite dimensional convergence results of Lu et al. (2018) and Aubin- Frankowski et al. (2022). We aim to reformulate the statements in online learning, addressing one-step improvement OMD.
Lemma 11 (OMD improvement). Suppose temporal cost Ft M(X) + R which is L-smooth and I- strongly-conwee convex relative to Q and nt S -- Then, OMD improves for current cost Fe(T(+1) S Ft(m)))
Proof of Lemma Ii Since F is relatively smooth, we initially have the inequality
F((T+11) F((TT) d'F T141 - -Te) + LD
(40)
Applying the three-point incquality (Lemma 10 to Eq. 40 and sctting a linear functional (()) Fi (T mt), p ne and p p= Tc+1 yields
d' FT m++1 - ne) + ( S d' Fe me) m
(p((pm)) 1 (o(llltet1).
Since Ft is assumed to be I-strongly convex relative to Ω, we also have
d'F(mpp- m.) - F((p) F((T)) - (Da(p)mm)
(41)
Then, by using
41
Eq.
40
becomes
F((me+1) S ((()) G. - ())D((p) - m
Dr((p)et1)) + (L dr D
(42)
By substituting p Te, since Da(pTT+1) 2 0 and L decrcasing at each iteration. This completes the proof. n s 0, this shows Fi(T((1) S Fe(Te)) i.e., Fe is
A fundamental property with the dual space D induced by the first variation de holds in our OMD setting. The existence of such learning eequence-particnlle in Sinkhorn-is well discussed by Nutz (2021) and Aubin- Frankowski et all (2022). Focusing on the dual geometry, explicitly call this relationship with arbitrary step size 7) as "dual iteration."
Lemma 12 (Dual itcration). Suppose that first variations (e(.)) and 6c52(T)) cxists fort 2 0. Then, online mirror descent updales Eg. 13 is equivalend to 8 5.(((m)) -1(.eF((mt)) for all E C,, N.
Proof of Lemma
The optimization
is equivalent to having the property for subsequent T++1:
d F((mm - ne) + 1 (a([T( 2 d Fe(m m++1 - nt) + Da Da ( (T+)
(43)
(dcFe(TTe) 1 v (.))), T + ي/2 (S( ((T+1)) 2 0, VT EC
Setting the free parameter π = ue+1 + h(5 T++1 and taking the limit h + 0+ yields the result.
Remark 3. With applications of Lemma 12 and Lemma we can achieve concise form of iteration in the dual using our temporal cost as:
8,5((T)) (.T((1)))
-
(44)
=ne (
where H denotes the entropy, d.e., the minus KL divergence with the Lebesgue measure.
Leveraging the aforementioned lemmas, we have systematically introduced and rigorously formalized the essential concepts necessary to progress with our analysis within the OMD framework. Finally, we are ready to describe a suitable step size scheduling by the following arguments.
Lemma 13 (Step size I). Suppose that F, KL(T [ and Ω <Laa If 1 lime Me 0+
and 2 Eee ?t ④ n S 1, the OMD algorithm converges to certain z
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Proof of Lemma 13
From Lemma
11
have
(o(F((Te+)) - Fi( - - -Do(Fel( + (np 1)
(45)
Taking lim Tt. ensures improvements; this means for any 0 there exists some 0 < 0 < such that Da(Te] + ( < whenever πh 8. Since convexity and the lower semicontimuity of the Bregman divergence Dn induced by KL, we conclude that OMD to certain point upon the assumed step size schoduling.
Lemma 14 (Step size II. Assume that inf B. (T mi)] 0 for all E [1) Suppose that 0 and limt- Σ D=(m=||=f)] = if and only if Σt=1 ? = +00.
Proof of Lemma 14 We firstly argue that due to dual itcration 44 any improvements on KL in Lemma
are also indicates corresponding improvements in the Bregman divergence, i.e. Dn( ( I D et | and if ?/ then the process { is convergent. By the dominated convergence theorem (Royden & Fitzpatrick, 1988) with the ergodicity assumtion of nonstationary solutions { (Corufeld et al. 2012 Appendix there exists a constant that satisfies
E (Da(#*+ (Po(==+1|1$+))))) 2 E1s+1 (De [1 +
(46)
for t > n for some dp - 0, where expectation subseripted by the range of "1 itt indicates notation of time-averaging from the time step 1 to t. Consequently, we achieve the following inequality
E1 (x<TTPa(T=1ll
E1441 ( +
Eq. (46
2 Eia ( ||]] (6-5((T++1) -z) E14+1 (asiiIDDDooooa ( [[T]]]]]]]]]]]]]]] +
Lemma
F1, (T [[]] 1/ Po (+| + Dn 1 P (F te
Eq. 44
(1 7h 11:: Do [T]] + BI=-1 (Da + Dn te
(1 n: E1: Do [r)] +e'
(47)
for some and 0 < < E where Eq.
46
Lemma and Eq. 44 are used in order.
Necessily. For big enough & 2 n where n E N, we can achieve the inequality in Eq. 47 as
Er.+1 1I 2 (D
(48)
Since we have assumed that Wi converges to 0, consider a step size sequence 0 < 1/t S
for k > 0. Denote
2+E
constant 2+ log x
2+ and apply the elementary inequality (Lei & Zhou 2020
2
1- x 2 exp(-ax) such that 0< x <
2 + k
From Eq. 48
we achieve
E
exp(-anp B1 (D [Da(TT((
for all t 2 n. Iteratively applying this inequality iterative for t = n, n + 1, ... , gives
T-1
EgT (Do(Tr|[=)) Ei.. DD [T))
exp(-ani)
(49)
T-1 Σ Tt Eim (a
exp -a
From the assumption n² * Thy Da( > 0 by the property of divergence. Therefore, by Eq. 149 the convergence lim Er.{ ( [ 0 implics the series E ne diverges too so that exp( Σi m converges to 0.
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Sufficiency. Consider static Schrödinger bridge problem with couplings I1 v), which is in constraint set
C= ( v) E P2(RR)) n Pato (p) E (*) x (1),, and E (² n )
For p. E C, we can see
Dalplpp) = O(p) 2(()) (6.2(p), 20 = -(6.0(p) p) 2(p) 2(p).
By adding (6c9()), p- p); we achieve property:
(6.((p) 6.0(p), 2 Da(elle).
(50)
Then, suppose that we have the asymptotic dual mean TE Using the right Bregman difference Lemma the one-step progress from the perspective of dual mean writes as
9
Da(T (=(+) = (d)l(TT) - 6.((T(+1) - me) + (T(1)
= (((o)((TT)) - &d((T)) - - m)) + Da(Tell=e+1) ((((s((TT))) - &cS((T)), ah - m.) + (e - 5,5((m)), o. S DmB (m))) no (o (6.S((m))) & - me) Da((T (T411)
mm)) + Da((allTeta)
By applying the definition of 2b and ergodicity of { we can bound the expectation by finding some > such that
where the inequality is from Eq. 50
Best1 [De(TT (++1) E1: [11 Dn) (m
B1s41 [DT(Fe|(TTt))]
s En.((1 m ( [|:]] + E1s+1 [(6+ 6.00(= 2
1
2w
En.((11 - (n +
" E1 [(d.((((m)
6.0(m))))
2w
1 2
(51)
where is the Lipschitz constant for each log Schrödinger potential in C. For the first inequality, we use Assumption and we use the log Sobolev inequality LSI(w) from Assumption D in the second inequality, Let {A, 1 denote sequence of A, Enr((D((Tà) m.)]. As a result, we have
A411 < n)) 2ni
44 n,
(22)
where 2 = 1C. For a constant h 0, we argue that A,, h for some t n'. Suppose that this statement is not truc; find some t > t such that A, h, Vt 2 t2. Since lim MJ O, there > la > l2 that 1/ < à
z. However, Eq. 52 tells us that for 2 la, for 12 la.
A41 S1 m)4. + 2m E At
Σ T + 0o (as + 20). k=ts
This results to contradiction, which verifies A, < h for t n'. Since lim; 7tt = 0, can find some t which makes A, monotonically decreasing. Therefore, we conclude the nonnegative sequence {A finds convergence by iteratively applying the upper bound in Eq. 52
2. We now prove Theorem 1 under consideration of the particular case of m) F+7
Then, Eq.
52 becomes
2 (1-
2
42
A41 <
A
Wt n
1+1
112
1)22
It follows that recursive relation writes as
1(1 1) S 1)LA, + 42,
1A 2
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Iterative applying the relation, we achieve the following inequality:
(T 1)TAA I (n 1) + 4=(TT - -n),
VT 2
Therefore, we finally achieve inequality follows:
E.r(Pa((T)lTT)) (n 1)nE1:n ( [[))] (T - 1IT
4z 2
(53)
T
Since we assumed x² nh. E1. Ti.r(Dn(='Ilaa -01 (11TT). Combining this with Lemmas and 14 the proof
of Theorem is complete.
A.3 Proof of Proposition
The proof is based on the Doob's forward convergence theorem.
Theorem 3 (Doob's forward convergence theorem). Let (Xihee be sequence of nonnegaläve random variables and let (Febe be random variable and let (Felee be filtration with F, C F141 for every t E N. Assume that [FF < Xt almost surely for every I E N. Then, the sequence {Xt) converges to nonnegative random variable X almost surely.
We follow the derivation of Eq. 51 there exists E N which satisfies
E, DD ([ s Da(msl|m,) - Vt 2
and since the step size is scheduled as lim4-00 Nt 0, the condition E ni < co enables us to define stochastic process (Xthen:
8 X, = (15 [T+)) 2 K
np.
(54)
It straightforward that the defined random variable satisfies E, [X+11] I X, for 2 n. Since X; 2 0, the process is a sub martingale. By Theorem 3 the sequence heN converges to nonnegative random variable X almost surely. Therefore D me) converges to almost surely.
A.4 Proof of Proposition
To achieve meaningful regret bound for our problem setup, we first demonstrate the following. Lemma 15. For all w arg arg min, (ô. y) + Pn((ll=)} with 0, the following equation. Vu.(ng w u) s Da(ull=) Dn(ullw) (n(wll=)
(55)
Proof of Lemma 15 By the first order optimality of (19.v) Da(v]l=}} function of w. have
(g 5 Dn(w [|2), w) 2 0 (g. w u) - (- Da(w]l=), to u) ) (Dn(wl)=) - (n(ullw) - (o(w(l=))
where used Lemma in the derivation. This completes the proof.
Next, we derive the one-step relationship for OMD. a quadratic expression of ftt, which is a key aspect of sublinear total regret. From a technical standpoint, we can see that the assumption for log Sobolev inequality generally works as a premise for Lipschitz continuity of gradient, i.e., VΩ in classical MD analyses.
The result entails that the regret at each step is related to
Lemma 16 (Single step regret). Suppose static Schrödinger bridge problem with the aforementioned con- straint C. Let Dn be the Bregman divergence wrt Ω: P(X) + R + (+oo). Then,
mi 2w
m.(F((TT)) Fi(u)) & Dn(ullm)) (n(ul|T(41)
C
(56)
holds, where g. = & tn (86S((m)))) ( an MD iteration for the dual space for step size Th, and w > O is drawn from type of log Soboleo inequality in Assumption
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Proof of Lemma 16
Consider single step regrets by the adversary plays of lincarization for 9:: (T(T+) - (r) & (0t, क - uu).
Therefore, we derive a incquality for (gc. Ti u) as follows.
(m-se, - u) (na.r, Tx+1 u) + (n: Ti T++1)
I De( - Da(uII - (r( (T)) + (nir, Tt Te+1)
Da(ullTT) - Da(uu| DR(TT+1l [m)) + (6.S((T(+1) 6,(((T), T -T++1)
Da(ullT)) - (++1)++ + DeT(T [x41).
Since we assumed that 9. (6g2((T)) (c5((=++)) by the dual iteration and that Assumption can achieve the upperbound ( mo+1)
holds, we
zf 2w | (()))) by direct calculation.
We now show our upper bound of total regret by utilizing Lemma T6
Lemma 17. Assume M++1 S Th. Then, u E C,,, the following regret bounds for fixred u E hold
Σ (m)))) Fl(u)= - Da(uI(m)
E (
(57)
131ST Fr
2w
where 9. nu
(6cs((m)) de((m))))).
Proof of Lemma
17
Define D = maxigisT De)(ulle)) We get
T Σ (F)(T)) - F((
Regret(u)
1=1
T
T
<
Drt(ul(T)) - Da(ul(Tet1)
+ Σ
1tt
11 (8
2w
t=1
1=1
1 Da(ul[<1)-
_ -Do(ullTr
1
Da(ullTTT1)
+
Mt
11 (172(
1
Thr
Me++
2w
1
D2 +
2
T-1
D2
+
Mt
11 11:2
71
M++1
Th
2w
2w
Therefore, the proof is complete.
Following Lemma 17 and Assumption ! we can have the inequality
D m
D2 KKT.
Σ F((T)) F((u) S
S + 27/ KT.
Tr
2w
1pr
where 2 maxigier Dn(ull=:) Setting a constant step size Na
DVW yields an upper bound of V2KT
2Dv2w -KTT which proves the regret bound of O(VT). Also, recall that the following lemma.
Lemma 18 (Lemma 3.5 of Aucr et al. 2002). Let a sequence a1, a2....,, ar be non-negative real numbers. af > 0, then
T
I Σ
a
2
04
(58)
Σi=1a
ai
Setting a adaptive scheduling ?). D yields 2D 2w 220 - 2
|/@ which has possibility to be
12 Σ ||
lower than ((VT) depending on {mp]7 |T
Therefore, have formally expanded the convergence results of OMT Lei & Zhou 2020: Orabona & PAl 2018 Srebro et al.] 2011) to SBPs.
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A.5 Proof of Theorem 2
Since Dn(-ll-) == (RL,(IR)((I)))) for reference measure R C, apply Lemma 6 and achieve Eq. 17 We write the following equivalent convex problems, using the equivalence of first variation for recursively defined Bregman divergences.
(6.Fi(m)) x πe) ma(((( (6.Da(Tella)), nι) (6.l(m))) 60((m]) - m²) + n
n DD(
Da(T))))
Dn(TI|=i) Da((F(=T) (
= 1 (a(T)I=T)++ + (1 (7 (a(==(
We refer to Appendix B for the stability of Wasscrstein gradient flows according to the LaSalle's invariance principle. We can now interpret deEt as a dynamics that reaches an equilibrium solution
minimize (f.)) me) n (a( #EC
minimize me #EC
Da(a
+(1 1/) De((T)m)))
empirical
proximity
At glance, the above equation appears analogous to the interpolation search between two points, where the influence of oi is controlled by n).
A.6 Proof of Proposition
The proof is closely related to the work of Lambert et al. (2022) where the difference lies in we correct the Wasserstein gradient term Óū,r for suitable for generally unbalanced weight. Suppose take parameterization 0 E (P2())(R))), WFR), the space of Gaussian mixtures equipped with the Siieesstiin-Fshher-e over the measure space of Gaussian particles. Following the arguments from Appendix B.2 and the studies for this particular GMM problem (Lu et al. 2019 the KL functional is derived as
metric,
Lambert et al. 2022) of the Wasserstein- Fisher--hhhreee of
Vw = (VevōKL(plle") _/2 (SKL(oollD)) (sKLLol)))) -otoollo)ooo
(59)
where we can consider the WFR gradient is taken with respect to 0 of its first argument. By Eq. (59 we separately consider Wasserstein gradient the Seres-Wassorstiin space and the space of lighting that controls the amount of each Gaussian particle.
Given a functional F: P2(x) R [U the Wasserstein gradient G.FnT,Pa such that all {ptHER+ satisfy the continuity eqatuion starting from Po (Jordan et al. 1998 Villani 2021 If the functional is the
KL divergence (L we compute the Bures Wasserstein gradient for the Gaussian distribution with respect to (m, Σ) using Eq.
777
V By F (m, Σ) (V Fm,, 2), E))
( (/mmPra mmp log
Pm. π 2 1z log Pmn. π
with some abuse notation for p. Using the following dlosed-form identities for the Gaussian distributions
Yr. V VmPm,x(x)) -VPm,, and Vs.pm,x(x) -
and the equivalence between the Hessian and Fisher information, we achieve the following form:
E, 2 log L
V F(m, Σ)
E,
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Define Tk.r Vak,r- Since follows the Fisher Rao metric in Definition 7 by the Proposition A.1 from Lu st all (219)) and specialization of Lambert et al. (2022) we can think of dynamics of K Gaussian particles (ak,7 MA,T. 2 Jk=l such that
1K
K Σ
Pa. (Wke.)
1
Pa. p
(OE.T) Mk.T
Kk,7
E log
ael
log
p
2T
p=1
mukk -E V log
pa=
(un.r) SK,T= = -B p²
72
2
log
PB- (sme.)) EkiT DR, V2 llg Por
(Ve.T) pr
Since OkKr = TkiT by previous definition, it is straightforward that
OuNTT - E log
(1E
pa= (U1.T)
1
Σ
K
a
log
Per p (veer)
)
p²
2T t=1
For 0i: > 0. This completes the proof.
A Riemannian Perspective on Wasserstein Geometries
B.1 An introduction to Otto calculus and the LaSalle invariance principle
In this appendix, we introduce a basic notion of Wasserstein gradient flows in the space of continuous
probability measures. We focus on describing the particular example, the KL cost, initially studied by JKO
[Jordan et al. 1998) and formally generalized by Otto (2001) in the context of Riemannian geometry. For
more details and mathematical rigor, refer the reader to (Ambrosio et al., 2005b Carrillo et al. 2023). For
X C Rd, and functions U IRz0 + IR; V.W : -
IR. We first consider energy function &: (2) R:
E(O) / (((()) dr +
/ V(s) dp(x)
1 / (W* ()()) dp(x), PE P2(X).
+
(60)
internal potential 2
external potential By
Interaction energy W
For this function, we refer to the solution of the following PDE:
0.p: = V-= + +V+ W p)].
t t2
(61)
as the Wasserstein gradient flow of E. Following Otto's formalization of Riemannian calculus on the contin-
wous probability space equipped with the Wasserstein metric (P2(X), W2), the PDE (61) can be interpreted close to an ODE of Riemannian gradient flow:
0.p: = -
(62)
where V denotes the Wasserstein-2 gradient operator V. == VV Considering the Otto's Wasserstein- 2 Riemannian metric OOto
2001
Lott
2008).
under the absolute continuity, see that
0 Ep:) = -
0p
0p 0t
0.
= +UU p) dp(x) 50,
(63)
at
at
which closely related to the strict Lyapunov condition. As result, dynamical systems following the PDE are guaranteed to reach equilibrium solution, under the LaSalle invariance principle for probability measures (Carrillo et al. 2023).
For representative example, we identify Eq. 60 for the relative entropy (the KL functional) for target density /" E P2(x') writes
/ V(x) dp(x) C,
= KL(p)p') U (p(x)) dx
xx
Ev
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where U(s) soggs, s, V(x) (z) and C U(p*) Ev(p"). Recall that 8E(p) log p(2) have
p then we
V.E(p) 5, G, (F{() = - [.VVVVE(()) = V ,V log P- _
(64)
where denotes the metric tensor in matrix form. We can derive the the FokkerPlanck equation
dep: - (pV logp") Apt,
describing the time evolution of the probability density. Combining the convexity of KL and the LaSalle invariance principle Wasserstein gradient flows, the PDE reaches a unique stationary solution of V(a)
Jx - vv
B.2
Background on Wasserstein- Rao and other related geometries
The Wasserstein- Fishhseei-----se---aa geometry is also known as Hellinger Kantoroeich in some of papers (Liero al. 2016 2018). In this section, we provide an overview of the geomelry tailored to meet our technical needs. Along the way, also briefly describe various metrics and geometries related to the Wasserstein space.
The Wasserstein space. Let n, E (2(R)) be marginal probability densitics with respect to the Lebesgue measure. We define the squared Wasserstein distance by a problem of couplings (Villani 2009)
inf / rELl(M.)) S 2 (2(
1 -11== dz
(65)
The Wasserstein distance offers a principled metric for quantifying the discrepancy between the probability distributions of random variables X and Y. Morcover, the Wasserstein space admits a fluid-dynamical formulation, where optimal transport represented by space-time velocity fields that satisfy the continuity equation. The Brenier theorem (Villani (021) states that there exists an optimal mapping function that pushes forward a to V, i.e. == V where Ç Ed gd U {+oo} is a convex and lower semicontinuous
function. The property is formally referred to instance of the Monge-Ampère equation (Villani 2009)
9 (V((x)) det det (VC(=)) = f(x) ERR.
(66)
after identifying the source and target densities with μ(x) = f(x)da and v(y) = g respectively. In terms of fluid dynamics, the Brenier map T = V internally yields a constant-speed geodesic (pthejo.1)) time-dependent density evolving from po to Pl, described by the following differential equation
Pc = (VG))
t)id +tVC,
(67)
where Po y and P1 = Assuming the existence of such geodesic, can also understand finding the optimality of (Pthejo.1) with the Benamou Brenier formulation (Benanou & Brenier 2000), which involves velocity field for minimizing the total I cost of transportation
(1e W3(,, min 3
dt Po=Aa -= dep: = - -v. (orp))
(68)
The equation dictates how the fluid should be transported (which shall be controlled by speed v.) while satisfying the continuity equation of path measure on the right hand side. In the Otto calculus (Otto 2001), we can understand the Benamou Brenier formula (68) as a Riemannian geometry with respect the W2 metric. In this geometric interpretation, the tangent space at p P2(X)) are measures of the form 6p - (up) with a velocity field L2,pp Rdd) and the metric is given by
{Llbo] do -- -V--))
|lell3 = inf (eLR(p
(69)
The Sanamou-Breeice formula exhibits dynamics in the Wasserstein space of probability densitics where the metric generally governed by dynamical mass transportation costs with the continuity equation, implying the mass of probability is preserved.
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Fisher-Rao metric. The Fisher-Ras metric is metric on the space of positive measures P- with possibly different total masses. We are interested in the simple case where such measure are represented with fininte number of parameters such as exponential families. We use the following definition throughout the paper.
Definition 7 (Fisher Rao metric). The Fisher Rao distance between measures po, E EM+ is given by
inf L've 1
dpo Yx
dpi
(po. PI)
=
dx
p.w.Alpo.nil
3d
where is an admissible set for a scalar field on positive measures; any reference measure such that p and p are both absolutely continuous with respect to X, with Radon-Nikoddym derivatives de dx
The equivalence between the square Fisher-Rao distance and squared Hellinger distance (Liero et al. 2016
2018 quantifies the similarity between two probability distributions ranging from 0 to 1. The total varia- tion bounds the squared form and is well-studied in the information geomelry (Amari 2016). The partial differential equations of the form dep: axPt are called reaction equations of at, which describes dynamics regarding concentration.
Wasserstein-Fisher-aaas The Wasserstein-Fisher-Baa geometry, or equivalently, spherical Hellinger Kantorovich distance, considers liftings of positive, complete, and separable measures while preserving the total mass. This can be expresses as combining the Fisher-Rao Rao and Wasserstein geometrics characterized
by PDE such as (Liero et al. 2016):
We 2 pt
Oepr + V. (vxp))
(70)
One problem, is that the PDE 70 In order to stay the dynamics on the space of probability measures, which is interest, we adopt the definition from [Lu et al. 2019 Lambert al. 2022) the equation becomes
1 {(B B. / Be
O.P: + (prts) = 2
ded dpe
(71)
which satisfies mass conservation. For the geometry, the norm tangent space is given by
11 ()1..))11== 1WWw-/o Br dp) + ||011 fap.
(72)
and we define the WFR distance as
(ea p1) = inf (L [1 1i) di (pr, 3r, o hejo0.1) satisfies
71
(73)
p.3c.v
Lu et al. (2019) demonstrated that Wasserstein-Fiste space can be analytically derived with closed form expressions. In this work, we were able design a computational method for OMD iterates in the WFR geometry. Using Proposition 3 this geometry allowed the VMSB algorithm to perform tractable gradient computation within Wasserstein space.
Rao gradient dynamics over the Bures Wasserstein
B.3 The Bures wassersesnn space and mixture of Gaussians
The space of Gaussian distribution in the Wasserstein space is known Bures Wasserstein space, denoted (W").. Given 0,, 0 BW((Z),, we can identify the space with the manifold gd x S where Sd, denotes the space of symmetric positive definite matrices. For 6g = (mo. Ev) and 0 = (m1, Σ1) an affine map from Po, to po, is given as a closed-form expression:
V((x) mi + NS 1/
1/2
(2 > / 1/2 1/2 (m))
Note that the constant-speed geodesic also lies in (Rd), pushforward of a Gaussian with an affine map is also a Gaussian. Therefore, it can be said that BW(Rd) is a geodesically convex subset of Pa (RR)). For the Brenier map, a constant-speed geodesic in (WR)., for the tangent vector to the geodesic (r. S)
po. exppo (r, S)) Nmmo br, (LS + T2) Ta)),
(74)
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and the dynamics at its current position at time 0 is represented as
nng = r, Σ. = SEo + EnS.
(75)
(76)
Generalizing this geodesic dynamics, the Bures Wasserstein gradient VEu f of a function f for tangent vector S) at time Altschuler et al. 2021
Rd f:R² S + R
(VunS(mo: Eo), (r, 8)) Öef Σx)
Identifying each component, we achieve the following result of Wasserstein gradient flow in Bures Wasserstein space as
Vau = (V WV (Vm.,2Vx.)),
(77)
where Vm and Vx denote Euclidean gradient. We refer readers Appendix A of Altschuler et al. (2021)
and Appendix B of Lambert al. (2022) for further useful geometric properties of Wasserstein spaces and dedicated discussions for the Bures Wasserstein space.
C Background on the Dynamic Schrödinger Bridge Problem
The equivalence between static and dynamic SBPs Pavon & Wakolbinger 1991 Léonard 2012 has been
studied, allowing us to consider the both problems interchangeably. This appendix introduces general control dynamic formulation for describing SB and SB-FBSDE theory (Gigli & Tamanini 2020 Chen et al. 2022). These formulations establish fundamental links to optimal control theory and diffusion models.
For variable drift and diffusion coefficient, a stochastic process, and its time reversal respectively follow the forward and backward Kolmogorov (or FokkerPlanck) equations (Fa 2011 Gigli & Tamanini 2020
3p V. [(f+ + Vppe). 2 (9:95
(78a)
0t
Pp. + + V 2 22 (9:9) pt). Ōt
(78b)
where ft and 9r are time-varying base drift and diffusion cocfficients which together determine reference
measure R Vi denotes divergence; V denotes squared divergence. By subtracting Eq. (78b) from Eq. 78a
the continuity equation
ap + V. (up) =
(79)
01
is achieved by u = f V - Veb.. Adding and scaling of Eq.
78
also derives another identity
(V + V+)p= V (ggTp)
(80)
where we can derive an explicit form of the score function
V V gg V log p + V. g.g.
(81)
heat flux
The rightmost term of [81 heat flux, indicating the amount of diffusion per unit time.
The time-symmetry relation can be considered as multivariate derivation of the the stochastic mechanics. Generalizing the Nelson's 's notation [Nelson 2001), let us define the SB drifts and current drifts for continuous- time path measures P and Q receptively satisfying Eq.
(78) with the shared diffusion coefficient:
== fr.t + Verras Sp = fp. + Vira
f4.- = Sq.t + ViQ...
fq. = V.lo..
(82)
$ fo
fo - 5e
up
rq 2
(83)
2
TTn
base drift. is and diffusion is Ve.
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Then, consider the Girsanov theorem Oksendal 2003 for the drifts. The theorem formulate Radon- Nikodym derivatives between path measures of stochastic processes by the fact that potential gradients reconstruct the score function Vy: + Vel = = log Pt.
Lemma 19 (Girsanov theorem; Theorem 8.6.3 of Oksendal 2003 For adapted processes with a given time interval [0, T), let Bg be an Ilō process solving SDE
dB, = ao(w, 8) dB'
(84)
for E X, 0 S and Bg = 0, where a salisfies the Novikous condilion. Then, Bg is a Brownian motion with respect to the path measure Q, satisfying the Radon- Madon-Nikodymm derivative
dP dQ (w) = exp alw, - -f"
2 laaws s)|{?ss)
(85)
Next, present the disintegration theorem in the context of probability measures (Lóonard 2014 Vargas et al. 202T), which exlends the product rule to measures that do not admit the traditional producl rule. Similar to the product rule, these theorems are essential for decomposing and manipulating path measures for dynamic SBPs, eventually connecting various formulation of Schrödinger bridge problems.
Lemma 20 (Disintegration for continuous probability measures). For a probability space (2,F,P) where Z is a product space: 2 == = y and ① x, Rd and de : + Z, is a measurable function known as the canonical projection operator (i.e., (1( and ' {(x, ( = x}), There exists a measure ()))()))),, such that
Hkw ((.N))P(O) - 1/ Jxxxy (a, (ulz)dP(( (x))
(86)
where = = ((O is probability measure, referred to as push-forward measure, and corresponds to the marginal distribution.
The theorem suggests that one way to achieve KL projection between path measures by matching drifts with the time reversal drifts of (x',7 y>) ). Under the Girsanov theorem, we observe that
KI(PIIQ) = KL((,IQQ.) Ep [/' 11 fo..13dd
(87)
= =))(),1(0) + Ep
2 [1ls
Let us consider a reference measure Q = R with based drift and diffusion i.e. f/= =
Knowing the boundary conditions Po Ro- = μ and Rx = QT = Eq. S7 can be reduce to the following problems
(L(P)) = inf (ut ) 2 |1] (2)) (())) dxdt,
(88a)
inf _0 2 1oo, (x) (x) dædt,
[/]
(88b)
and Eqs. 6 and 7 in the main lext represent a simplified case when == We, Furthermore, by combining Eq. we get a symmetrized version of relative entropy (KL functional)
KL 1 kL())))))) + (L 1 Ep [["
1 115 155-5111aa fall 4 115E f5112 +
(89)
Finally, using Eq. 80 we get the constrained problem:
( oit 1 x2 v logū(t, 1) + g v gg T| pla, t) dt dæ,
inf (0,0)
0p + v. [f + +) O. p(0.-) = μ. p(T,) = M.
such that
0t
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To solve this problem, we convert the problem to the Lagrangian function:
((p,v)= [[ 001, (111,,))
V log p(4, x)) po0 ((, + g! gll 99" (1,2)) ((,,))
8
Op 01
+ *(tt, x)
+ v. ((f 0) dr dt
where À is Cag-Lagranginn multiplier. After integration by part, assuming that limits for + are zero, and observing that the boundary values are constant over T( v), we resort to the following problem:
inf / (0.))PPVV Jz [ x) +)1 1 iv log x) )) + 1 IV gg (t,x))1²+
T
(-oe ax vA. 8t
p(t, p(t,z)
(90)
Pointwise minimization with respect to 7 for each fixed flow of probability densities 7 gives v² (x, -V(xx Plugging this form of the optimal control into Eq. 190), we get the functional of E P:
J(p) - 3z -
0t 1 ax +f VA |VAi/2 + log ((t VX+
1 8
1 V gg (t, 12 p(t, x) 8
+
Utilizing the existence and uniqueness of SDE solutions [0ssenda] 2003 the optimality of the problem is uniquely identified by VÁ being the mean current of SB-FBSDE which is introduced in (Chen et al 2022 Liu gt al. 2022 Therefore, both static and dynamic Schrödinger bridge problems solve identical optimization problem of KI. minimization, with different appearence by the problen-specifie origin of objectives, entropies, and geometries.
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Table Hyperparameters.
2D
EOT
MSCI
MNIST (Pixel)
MNIST (Latent)
FFHQ
Dimension
(2, 16, 64, 128)
{50, 100, 1000)
784
Modality K
{8, 20, 50}
[5, 100]
(256, 1024, 4096)
{256, 10243
Volatility e
0.1
(0.1, 1, 10)
0.1
10
10
{0.1, 0.5, 1.0, 10.0)
Total steps (T)
20,000
30,000
10,000
100,000
30,000
20,000
OMD steps (t)
400
600
200
1000
375
400
Experimental Details
D.1 Rationales of the GMM parameterization for VMSB
Our parameterization choice follows LightSB (Korotin al. 2024) because of the following two key reasons. concentration, which is suitable for
First, GMMs ensure that the model space satisfies certain analyzing theoretical properties of SB models (Conforti al. 2023). Firstly, analyzed the regret under the log Sobolev incquality in Proposition 2 Enforcing the LightSB paramcterization will automatically satisfy Assumption [ Secondly, VMSB requires tractable gradient computation of Wasserstein gradient flow in 4.3 As shown in Proposition 3 we can perform VMSB using the variational inference in the WFR
geometry of the GMM parameterization.
D.2 Hyperparameters.
The hyperparameters are displayed in Table 5 For step size scheduling, we followed the theoretical result in Theorem 1 and Proposition]] [ and chose 7) and W/ E {0.05, 0.1} with harmonic sequences, as illustrated -ENIST (784d), and latent FFHQ Image- to Image transfer tasks (512d), the initial warm up steps for 10% of the total learning helped starting a
in Fig. 5 For high dimensional tasks in MSCI (1000d),
training sequence from a reasonable starting point as this set The = 1 as verified in Fig. 6 (c).
p ~
0.25
10
S
Irput diarrbttion Torget dear bubin
Hartc-p THATTE
Figure 10: SB in 2D synthetic datasots. SB processes To with different volatility e.
D.3 2D Synthetic datasets and the online learning setup
Fig. demonstrates that our method achieved the SB model for the various volatility For various configurations, most of baseline SB algorithms capable of learning in the 2D space (10). In order to align our theoretical arguments for online learning, we selectively offered with rotating filter that only 12.5% of the samples to the SB solvers basod on the angles measured from the origin. For instance, we provided data for angle of [0, r/4] for first. / E [o, 25) steps, and so on. This partial observability is periodically rotated through the data stream, thereby testing the algorithms ability to learn robustly under sparse and shifting information. Since this requires 200 batches for the full rotation of the filter, the problem became substantially more challenging, and LightSB and LightSB-M algorithms oftentimes failed on this online learning setting.
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D.4 Entropic optimal transport benchmark
Our hyperparameter for the EOT benchmarks choices mostly follow the official repositories of the Li and LightSB-M Since it is known that imitial distribution is the standard Gaussian distribution (uushchin gt all, 2024b), we only trained 0g using the variational MD algorithm. Due to the huge number of configura- tions, some hyperparameter settings were not clearly reported. Thus, we conducted our own examination on these cases; we replicated better performance than the reported numbers by carefully dealing each benchmark configuration.
D.5 SB learning with adversarial networks
Suppose a discriminator network, denoted as D, is equipped with useful architectural properties for discrim- inaingg images. The discriminator outputs a binary classification regarding authenticity through sigmoidal outputs, Le., ((x) [0, 1] Vo 222x22x11 For image samples {a), x..,,,") ,4, we trained the discriminator D with the logistic regression:
Elog log D(v") 1 Σ log(1 D)
maximize
(91)
where Dh in the right-hand side denotes sample from SB model parameterized by é, generated using an input rm From our experiment setting, we use the SB distribution Ps which is generated by मे from samples of the marginal This makes the objective of adversarial learming of training the law of SB process at time = For a completely separable metric space, is well known that the discriminator converges at
D(x) = v(z)+po(*)
v(#)
Goodfellow et al. 2014
Table 6: Model hyperparameters for D.
In the adversarial learning technique, retaining fully differen- tiable computation path from the inpul pixels to the discriminator outputs is essential. Therefore, we implemented a differentiable inference function using the categorical reparamctorization trick with Gumbel-softmax (Jang et al. 2016), as well as the Gaussian reparaneterization trick. These reparameterization tricks enabled learning with samples generated through LightSB-adv-K, directly by maximizing
Layer Type
Shape
Input Layer
(-1, 28, 28, I)
Cony Layer 1
(-1, 14, 14, 64)
Cony Layer 2
(-1, 7, 7, 128)
Batch Norm
(-1, 7, 7, 128)
Flatten
(-1, 6272)
Dense
(-1, 1024)
Dense
(-1, 1)
M Σ
log D(v log(1 D(v))),
M
where the term essentially represents the logit function logit(D(v)) = log D-D(g) equilibrium, we can approximate the following KL learning
D(v)
When approaches the
I(0) / log
v(v) pa(v) Poly)dy- (L(p))P))
where the KL functional directly corresponds to the divergence minimization of the SB problems ( and 28 under the disintegration theorem of Schrödinger bridge (Léonard 2014
In the MNIST-EMNISS image transfer tasks, we set one of the bascline as the aforementioned adversarial learning as the baseline for training the SB model for the pixel space. Among our attempts, while the LightSB-ady method successfully generated learning signals to train GMM-based models, the losses proposed by LightSB (Korotin et al. 2024) and LightSB-M (Gushchin et al. 2024a failed to generate relevant images with high fidelity. For the discriminator, we used the DCGAN (Radford 2015) architecture shown in Table 6 and this can be replaced with more complex architecture for more realistic images with high fidelity. We fixed the covariance after warm-ups 10,000 steps, and we used the entropy coefficient E = 10-4 based on our hyperparameter search.
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Table 7: Training time for the 100-dimension single-cell data problem.
Sinkhorn (IPF) LightSB
VMSB
8m (GPU)
668 (CPU) 32s (GPU) / 22m (CPU)
Table 8: Generation time for the 784-dimension MNIST pixel data.
61
256
K 1024
1096
NN (SDE)
GPU
us
726
739ps
740μs
1.372s
CPU
60.140ms
133.333ms
128.433ms
1.527s
D.6 Latent diffusion experiments
For the latent space, we pretrained ALAE (Pidhorskyi et al., 2020) model using the both MNIST and EMNIST (first ten letters) datasets. The ALAE is a high-fidelity autoencoder internally use an adversarial learning to generate high-fidelity images. For the encoder network, as well as decoder network, we mostly adopt the DCGAN architecture. Therefore, the encoder is mostly identical to Table G except the point the final layer is 128 dimension instead of 1, and the decoder is convolutional neural network with four convolutional layers.
Following the latent SB setting (Korolin et al. 2024 we assessed our method by utilizing the ALAE model Pidhhorsky et al., 2020) for generating 1024 x 1024 images of the FFHQ dataset (Krrras et al. 2019 The
base generative model has a latent embedding layer which represent 512-dimensional embedding space. The goal is to transport a point latent space to another, performing unpaired imagetto-image translation tasks for four distinet cases: Adult-> Child, Child Adult, Female Male, and Male Female. We conducted a quantitative analysis using the ED on the predefined ALAE embedding as metric for evaluation.
E Discussion on Implementation of VMSB
Limitations. GMM-based SB models, due to the lack of deep structural processing, tend to focus on instance-level associations of images in EOT couplings rather than the subinstance- or feature-level associ- ations that are intrinsic to deep generative models. As result, while VMSB produces statistically valid representations of optimal transportation within the given architectural constraints, these outcomes may be perceived as somewhat "synthetic." Nevertheless, GMM-based models still hold an irreplaceable role in numerous problems such as latent diffusion and variational methods, due to their simplicity and distinctive properties (Korotin et al.] (024)). As we successfully demonstrated two distinet ways of interacting with neural networks for solving unpaired image transfer, we hope our theoretical and empirical findings help novel neural architecture studies.
Computation. For fast computation, we utilized the JAX automatic differentiation library (Bradbury et all, 2018) for computing gradients and Hessians in Proposition 3 For each input, the computational of VMSB requires quadratic time for computing the Wasserstein gradient flow (asymptotically O(K"ny)) and the memory footprint for estimating with internal Gaussian particles is linear (asymptotically ((Kny)). There are inherent trade-offs between accuracy and computational efficiency when choosing between LightSB and VMSB; nevertheless, VMSB remains significantly more manageable and computationally tractable com- pared to deep learning methods for moderate settings. For instance, we have presented performance regarding efficiency and scalability up 1,000 dimensions in the experiments. Driven by parallel nature of Gaussian particles, we observed that the computation of Proposition B favors vectorized instructions, and the ex- pected speed enhancement from using GPUs is much more evident in neural network cases. In Table 7 we report the wall-clock time for a 100-dimensional single-cell data problem Vargas et al. (2021): Korotin et al. (2024), where the performance reported in Table 3] Additionally, training time in the MNIST-EMNIST translation is reported in Table II in the ablation study. This property also holds for generation, allowing practitioners to deploy the model much faster on GPUs. In Table g we also report that generating 100
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Published in Transactions on Machine Learning Research (01/2026)
MNIST samples from 4096 Gaussian particles, equipped with competitive performance, can be done 1,854 times faster under the same hardware. Since VMSB a simulation-free, the GMM generation process does not suffer from discretization errors of SDE.
Reproducibility statement. Comprehensive justification and theoretical background are presented in Appendices A and B Since the primary contributions of this paper pertain to the learning methodology, we ensured that all architectures and hyperparameters remained consistent across the Light variants. All datasets utilized in this study are available for download alongside the training scripts. Please refer to Appendix D for more information on the experimental setups.
Table 9: EOT Benchmark scores of BW3-UvP & (%).
61
1- 10
Type
Sulvxe
Clamkaal wiver
11006
2015
0:55
0.21
1L305
0101
0.56
0.32
u10
112
0.35
1423
Bedggok
DSBS
DG3
5.15
2,30
0.84
0.00
7.3
0.56
100
3553
15000
Belgek
518
177
4.7
S19
KL.
Kontie
0/04 6.84
5991 0:304
3/25 0031
0101 3003
0 3305
10000
0000
0:002004
0.052000
D666 20000
M
0.035 0.003
24112 0.001
01011 8.33
01831 1. 30012
0:451 1-911
0167 1.18T
8157 Lim0
0116 1:09
0161 1080
6507 1006
0:33 1100
1:40 11092
EMA
00011
3014 = 0.301
3/221 =: 0.003
0 =: 30011
0,00 3,00
0.000
00000
00005000000
0 00000
0.00
0.066 0:000
0.029 00.00:00015
06:0:20.000
VarMD
VMRS (suas)
0.006
0.004
VMSSI-M (per)
0000 0.000
0.000 0007
0.056 0.055
0.015
0.005
0.0022
6.002 0.066
0.007 0.001
0.003 0.003
0.011 000001
0.317 00.012
0:09 1..115
Table 10: EOT scores of cBW3-uvP + (%), the fully extended version of Table 2
Type
Sohvee
16
4- 51
d= 179
4-9 9
di 61
A- 176
x= 9
d= 16
4- 65
Jin 138
1367 35
11.
15:33
1.77
Bidlge3
s0 Sp
1341
1.3
23
0.33
4/1
NDD
no. KL,
Konsia 21
0:00 0.065
0140 = 0:223
3,00 8013
0.11= 2/03
0:06 3/033
0 2.02
0000
0:140:200000
00000200000
0,07 0.06
0.52 (L000
bntaww
0017 00004
=
3.34 6034
=0007
20000
0.014:0000
0.000
0.747 0.000
Yar
VMSR (sens)
0,04 0.001
0.012
- 0.002
0:038 0.00
0010 0.002
0.000
0.001
0.01%
-000
6.011
0.00
0:111:-6:001
0.000
0.019
0.:19-00.00
0.021
0.005
0.040 - 0.000
Van-ID
13888-M (sun)
0.015 0.018
0.057 = 0.030
0.106
1:006:0000 0.020
0.255 =: 0.107
0.010 0.001
0.0L9
0.000
6.004
0,223
0.085
0.013:00.01
0.023 0,003
0.153 2 0.013
0.745 (L000
F Additional Experimental Results
F.1 Additional results on the EOT benchmark
We present the full results of EOT benchmark experiments. Tables and 10 show comprehensive statistics on the EOT benchmark with more SB solvers. As mentioned in $ 6.2 the VMSB and VMSB-M solvers consistently brought. better performance with low standard deviations of scores for cBW3-uvP and BW3- UVP measures. We note that the experiment was conducted in a highly controlled setting with identical model configurations; with all other aspects controlled and outcomes differing only by learning mothods, the consistent performance gains of our work were a well-anticipated result from our theoretical analysis.
F.2 Additional image generation results
In the unpaired EMNIST-to-MNIST translation task for the raw 784 pixel, we measured FID scores for various K for the SB parameterization. We considered K- E [64, 256, 1024, 4096] with e = 10-4 for our VMSB algo- rithm. Our observations, both qualitative and quantitative, indicate that higher modalities yield higher-quality samples. In every case of K, VMSB-ady outperformed its counterpart. For instance, Fig. [I demonstrates that VMSB generates more diverse samples with high fidelity. Notably, we achieved the competitive FID score of 15.471 using a standard neural net work discrimina- tor with relatively low MSD similarity scores. As the latent VMSB model for 128-dimensional embeddings also achieved the considcrably low FID score of 9.558 (Table 4), we concluded that VMSB showed promising quality im- provements for the both case, and this supports the generality of our theory.
100
FID
DSBM
64
256
1024
4006
Number of Gausiian modaltins K
Figure 11: FID vs. modality.
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Published in Transactions on Machine Learning Research (01/2026)
LightSB-adee adv
Ours- 256
Ligh
Ours- 1024
LightSB-adve 4dv-
Ours-4096
MINIST-1-EMMS EMIN1ST-to-MMN Mo-MMNSSSTT
5 0
202
A
5 2
3
3
3 1
2 /
4
0
0
5
1 3 3 3
6
0 0
1
9
0
0 2
/ 3
0 0
1 4
1
1
3
1
5 5
05
0
6
0
9 2
0
0
4
3
00000000
1000
3
8
0
2000
1
6
1
7
1
00000000
3
0
0
D 6
00033
0 8
6.66
C/003
00000-
3 4 5
5
1 2
C
O
3 0.000
O 2
LightSB-adee adv 256
Ours- 256
LightSB-ade adv-1024
Ours- Ours-1024
LightSB B0ynSB--dy-Baa 4dv-4096
Ours-4096
5
7
A
B-T
2
F
f
1
d
d
L
9
/
- 1
1
W 1
[
/-
C
d
/
J
4
C
H d
w.bbb
1
FFF44
1/4H
9
//- A
5
1
b 1
5
/
/ 7
WEBC66
/
4 1
C 6
Figure 12:
Generation results of unpaired image-to-image translation in the raw pixel space. We considered image data from MNIST and EMNIST (containing the first ten letters), sized as 28x28 pixels. For comparison, we trained GMM-based models with adversarial learning using a simple logistic discriminator (Table This was used as both a benchmark and tractable target SB model (LightSB-adv-K), VMSB in the raw pixel domain remensttrat qualitalive improvements in terms of diversity and clarity of image samples.
Adul
Clvid
VMSB
VMSB-M
Made
Fomale
VMSBB
VISB-.M
VNSR-M
Figure 13: Image-to- Image translation on a latent space for the VMSB and VMSB-M algorithms.
Fig. 12 demonstrates that VMSB generated more diverse samples with high fidelity. Note that the pro- posed mothod suffers less from mode collapse than LightSB method (especially on the transfer MNIST- to-EMNIST), with the same Gaussian mixture setting. This result especially a good point where the difference only lies the learning methodology, which aligns with our theory. Tables TT and 12] effectively show the statistics and FID scores on both the train and the test datasets. The quantitative results highlight that the VMSB solver is more performant with less overfitting than its counterpart. Consequently, our claim regarding the stability of SB solution acquisition is verified by additional experiments involving pixel spaces.
We present Embedding-EBD scores Jayasumana et al. 2024 in Table 13 and some qualitative generation results which is visualized in Fig. 9 For quantitative results, we calculated statistics from ED scores on embeddings of the ALAE model Pidhorskky et al. 2020), for the four different unpaired image-to-imane translation tasks. The results show that VMSB is capable of translating an arbitrary representation, which
Table 11: MNIST transfer statistics.
Table 12: FID scores and differences for generated MNIST.
Time
Parameters
FID (Train)
FID (Test)
Diff. (test train).
T6ghnSB-256
61.257
30m
0.4M
LightBB-102
287
53m
L.M
Lddv256
60.746
21.604
0.858
LightBB4499
20.017
135m
6.4M
Ligh
25.934
26.569
0.635
Ligh:SB-ady-4096
19.960
20.196
0.237
VMSB-256
52.634
76m
0.4M
VMSB-1024
24.022
203m
1.6M
VMSB-adv-256
51.684
52.283
0.599
VMSD-4096
15.471
44h
6.4M
VMSB-adv-1024
23.853
24.053
0.200
VMSB-adv-4096
15.508
15.496
0.012
DSBM-IMP
11.429
42h
6.6M
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Published in Transactions on Machine Learning Research (01/2026)
is closer to target domain than baselines. In Fig. 13 as well as Fig. 9 M algorithms generate FFHQ data with given translation task. To qualitatively verify these generation results, we generated images using LightSB and VMSB in Figures 14 and 15 Since these improvements are purely based on information geometry and learning theory, we anticipate that following works on the variational principle application across various fields such as image processing, natural language processing,
that VMSB and VMSB-
and control systems Caron al. 2020: Liu al. 2023
Alvarez-Melis & Jaakkola,
2018 Chen et al., 2022).
Table 13: ALAE Embeddingg ED scores. To evaluate the performance, we computed averages and standard deviations the ED scores across four different transfer tasks.
== 0.1
== 0.5
== 1.0
= 10.0
MFuk 0.02916 1 0.00145
0.04112 1 0.00191
0.05670 1 0.00249
0.06641 1: 0.00441
DSBM-IMF
0.02275 1 0.00101
0.03358 1 0.00142
0.04866 1 0.00168
0.06474 1: 0.00381
LightSB
0.01086 + 0.00045
0.02382 + 0.00093
0.03462 + 0.00 140
0.05376 + 0.00273
LightSB-M
0.01066 0.00055
0.02366 + 0.00107
0.03519 + 0.00153
0.05973 = 0.00298
VMSB
0.01002 = 0.00055
0.02288 + 0.00101
0.03396 + 0.00174
0.05315 + 0.00307
VMSB-M
O. 0.00997- 1 O. 0.00054
0. 0.02298 1 0.00106
0.03391 1 0.00140
0 0.05351 1 0.00241
VMSB, E 0.1
VMSB. 01.5
VMSB, 1.0
VMSB, E 10.0
Figure 14: Generation results of VMSB (Adult + Child) with different volatility settings
46

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Published in Transactions on Machine Learning Research (01/2026)
1.0
LightSB, =
VMSB, 1.0
LightSB ==
VMSB, == 1.0
Figure 15: Qualitative comparison between LightSB and VMSB for relatively high volatility, = 1.0. Top (Male + Female): We find that VSBM has preserved more facial details, such as wearing glasses, than LightBB. Bottom (Adull + Child); VSBM was stable at retaining facial position even with high e.
47


Reviews:
[{"review_id": "07Yd6ccJHq", "reviewer": "Reviewer_Rqoo", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper introduces a variational online mirror descent (VOMD) framework for learning Schrödinger Bridge (SB) models and instantiates it in a simulation-free algorithm, Variational Mirrored Schrödinger Bridge (VMSB). The theory analyzes OMD dynamics under a KL-based Bregman geometry appropriate for entropic SB, establishes step-size schedules and convergence (Theorem 1; Proposition 1), and proves an O(√T) regret bound under log-Sobolev/regularity assumptions (Proposition 2). Practically, the method leverages a Wasserstein–Fisher–Rao (WFR) gradient flow over a Gaussian mixture (GMM) parameterization of Schrödinger potentials, yielding tractable updates that approximate OMD steps. Experiments span synthetic online settings, an extensive EOT benchmark, and additional generative/image tasks; VMSB variants generally outperform strong baselines (e.g., LightSB, LightSB-M, LightSB-EMA) in cBW2^2-UVP and other metrics.", "claims_and_evidence_yn": "Yes", "claims_explanation": "Here is a list of the paper's contributions: \n1) A formal online mirror-descent formulation of SB learning with KL-based Bregman geometry;  \n2) Step-size conditions guaranteeing convergence in an “idealized” case (π* = π_D) and convergence toward π_D more generally under assumptions on {η_t};  \n3) A regret analysis yielding O(√T) bounds with constants linked to the dual norm defined via first variations;  \n4) A simulation-free algorithm (VMSB) built on WFR gradient dynamics of GMMs, with practical scheduling (harmonic progression, warm-up) and implementation guidance;  \n5) Empirical evidence that VMSB variants are robust and competitive across diverse SB problems.", "audience_yn": "Yes", "audience_explanation": "Bringing an OMD lens to SB with an explicit KL-Bregman geometry and a tractable WFR/GMM instantiation is novel and timely. The combination of theory (step-size, convergence, regret) with a simulation-free algorithm that consistently improves over strong SB baselines suggests the work is both original and impactful for the SB community and, more broadly, for entropic OT-based generative modeling.", "requested_changes": "Questions for the Authors\n1) Can the regret analysis accommodate nonstationary targets explicitly (e.g., with a drift budget), quantifying the price of nonstationarity?  \n2) How sensitive is performance to (i) mixture count K, (ii) warm-up length, (iii) η_1 and η_T? Please include ablations.  \n3) Could the WFR/GMM gradients be combined with adaptive optimizers (Adam, RMSProp) without violating theory? Any empirical observations?  \n4) Beyond GMMs: what breaks if we parameterize potentials via normalizing flows or neural fields? Are there tractable approximations to preserve the OMD interpretation?  \n5) Can you report per-task wall-clock/throughput and memory usage for VMSB vs. LightSB(-EMA) to contextualize the gains?  \n6) In the EOT outlier case (high d/volatility), can you stabilize training by adapting ε or the drift form? Any empirical fix?\n\nTypos / Presentation Nits (non-exhaustive)\n• “showes” → “shows”; “affective” → “effective”.  \n• Ensure consistent formatting of cBW2^2-UVP and BW2^2-UVP across text/tables.  \n• Consider a boxed “Assumptions” list summarizing Assumptions 1–3 and where each is used.  \n• Briefly restate key definitions in the main text (e.g., δ_C, dual norm, Bregman divergence) to improve readability.", "broader_impact_concerns": "None.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "g3QrQGFJNW", "reviewer": "Reviewer_H1Sz", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "Yes", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "The paper tackles the Schrödinger Bridge Problem and proposes a variational online mirror descent model to do so. In fact, as stated by the authors, there are three main contributions in this paper:\n1) the authors provide a learning algorithm for SBP based on online mirror descent together with theoretical guarantees, such as convergence and regret bound;\n2) the authors propose an efficient simulation-free algorithm, Variational Mirrored Schrödinger Bridge, leveraging the Wasserstein-Fisher-Rao geometry;\n3) the authors provide experiments on different synthetic and real-world SBP settings to corroborate their theoretical claim (mainly the online learning hypothesis) and the superiority of their method over existing simulation-free solvers.\n\nStrengths:\n- The related works are discussed and the paper is clearly motivated.\n- The claims are supported by theoretical and empirical evidences.\n\nWeaknesses:\n- Even though the paper is overall well presented and written, supported by relevant tables and illustrations, its reading still is a little tedious since it grasps many different topics.", "claims_and_evidence_yn": "Yes", "claims_explanation": "The authors support their learning framework based on a dual geometric interpretation of convex optimization by providing some guarantees on the convergence and regret bound of the corresponding OMD algorithm. Moreover, they provide experiments to support the relevance of their approach on various scenarios, such as Online SB learning, Optimal Transport or image-to-image translation tasks.", "audience_yn": "Yes", "audience_explanation": "I am not familiar with this literature, but the related works seem thoroughly discussed and the authors identified a hole in the literature regarding SBP. In fact, they propose this well-suited OMD learning framework and propose an efficient algorithm benefiting from Wasserstein-Fisher-Rao geometry. Hence, in my opinion, the TMLR's audience should be interested in this paper, especially since the authors discuss the limitations of their work and some future work directions.", "requested_changes": "N/A", "broader_impact_concerns": "N/A", "additional_comments": "", "extra_scores": {}, "extra_text": {}}, {"review_id": "CA8gbhbWfe", "reviewer": "Reviewer_wQU9", "summary": "", "questions": "", "limitations": "", "rating": null, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "No", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "paper_topic_and_main_contributions": "", "excitement": null, "reproducibility": null, "ethical_concerns": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "summary_of_contributions": "This paper proposes Variational Online Mirror Descent for Robust Learning in Schrödinger Bridge. It shows the convergence of the proposed robust SB learning algorithm under mild conditions. It further introduces a simulation-free SB method. This method further leverages the  Wasserstein-Fisher-Rao geometry to improve the stability. The proposed algorithms are validated using multiple experiments.", "claims_and_evidence_yn": "No", "claims_explanation": "1. The contribution 1 (develop a robust SB learning algorithm built upon the online mirror descent formulation) is not clearly supported. In Section 4, the author introduces the  mirror descent algorithms. Both  Sinkhorn and Wasserstein descent  can be viewed as the special case of the mirror descent (Section 4.1); theoretical analysis indicates that the mirror desent (Eq.(13)) converges under some mild conditions (Section 4.2). In Section 4.3, it describes how to solve the argmin in Eq.(13).\n\n    However, all of these do not support that the proposed algorithm is a robust SB learning algorithm. The author doesn't clearly define what kinds of the robustness is, neither clarifying what is the difference between the robust SB learning and non-robust SB learning. In the distributionally robust optimization, the optimization problem should be formulated into a min-max problem. The max part would be the role of \"robustness\". \n\n2. The contribution 2 (introduce a simulation-free SB method, leveraging the Wasserstein-Fisher-Rao geometry) is not clearly presented. It is not clear what is the difference between the simulation-free SB method and non-simulation-free SB method. It doesn't state how the Wasserstein-Fisher-Rao geometry helps make it simulation-free; it doesn't justify that it is necessary to use this geometry to make it simulation-free.", "audience_yn": "Yes", "audience_explanation": "I believe those who have sufficient backgrounds in this topic may be interested in knowing the findings of this paper. Currently, I may say this paper is too technical for most of TMLR's audiences.", "requested_changes": "### Contributions:\n \nI suggest the author to revise the contribution summarization (those three bullets on page 3) to better describe the contributions of this work. The current summarization doesn't match the content of this paper. \n\n---\n### Missing definitions:\n\nThis paper should include more preliminaries for those who are not familiar with this topic. Many defnitions are missing, which makes it really hard to understand (at least for me). Therefore, I may suggest the author to include the following changes:\n\n1. In Equation (1), the author should specify the domain and the function $\\Omega$ (and the function $F$). If the domain is a metric space of something different from the Euclidean space, it should specify the definition of the *convexity*. Apparantly, the author is discussing the topological vector space and the *dual* represents the topological dual. It should be clearly stated here rather than omit all of them. ---Because for the topological vector space, the convexity can be defined as you want; but for a general topological space, there is no \"convex\" function and there is no \"dual\" space. \n\n2. In **Notations** paragraph, the author should define *KL functionals*. I notice that there is a definition of KL functional below. Can you just add a label to that and put a braket here? For example, add this (see Equation.(2) for its definition).\n\n3. \"Lebesgueintegrable\" . I don't know if it is mis-spelled. It could be \"Lebesgue integrable\". \n\n4. Definition 2 (First variation). $x$ belongs to $C \\cup dom F$ (the union of $C$ and $dom F$) instead of $C \\cap dom F$.  Does it mean that $F(x)$ may be undefined? Moreover, I suggest the author to explicitly state the domain and image space of each mapping; e.g. write $F:X\\to Y$ instead of just $F$. \n\n5. In **The Bregman divergence.** paragraph, it mentions \"topological space of SB\". I don't understand what the topologinal space of SB is. The author may need to include a separate paragraph to define the space of SB, then define the topology on it. From the static SB problem, the space of SB seems to be the space of all random variables with the mariginals $\\mu$ and $\\nu$; it may have multiple topologies  e.g. the one induced by the Wasserstein distance. It should be more clear and I also hope the author include the reason why it \"does not ensure Gâteaux  differentiability\". \n\n6. I may need additional clarification. The space of SB doesn't sound like a vector space. As the linear combination of two random variables may not have the same marginals. \n\n7. Assumption 1 (LSI for couplings). I am not clear about the definition of $\\nabla$ here. It seems to be the gradient of something. In (11), it represents the gradient with respect to $(x,y)$, and how about in (12)? How is Holder's ineuqality applied here?\n\n8. Section 4.1 seems to be unrelated to Section 3. $\\delta_C$ is the gradient and $D_\\Omega$ is the Bregman divergence; so, Eq.(13) seems to be the standard (or generalized) mirror descent formula. Do I understand it correctly? Then Lemma 2 and Lemma 3 put the standard  Sinkhorn algorithm and Wasserstein descent in this mirror descent formula. \n\n9. In (16), the author is using the Wasserstein-2 gradient operator; what is the gradient operator used in previous sections? E.g. Eq.(11) and Eq.(12). I would request the author to include the clear definition of these notations. \n\n10. What is the definition of $\\pi^\\circ$? Has it been defined somewhere? \n\n11. In Section 5, what is $S^d_{++}$? \n\n12. The  Gaussian mixture parameterization is hard for me to understand. I suggest the author to include an example to illustrate it. For example, how does Eq.(2) look like in this parameterization?  It can be much more helpful if the author could re-write the objective function, the parameter space, the algorithm update rule, and recap the goal here using this simple optimal transport problem. \n\nI am not an expert in this topic; some suggestions may not fit the scoop. For example, some commonly known notations in this fields are not necessary to be defined in the paper, but I may not know that. The author doesn't need to take all of these suggestions into account.", "broader_impact_concerns": "This is a purely theoretical work so there are no any ethical concerns.", "additional_comments": "", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)