Minimax Rates for Conditional Generative Learning under Wasserstein Loss

Sungwon Park; Kyoung-Sook Moon; Hongjoong Kim

math.STOct 26, 2026

Minimax Rates for Conditional Generative Learning under Wasserstein Loss

Authors: Sungwon Park, Kyoung-Sook Moon, Hongjoong Kim

Organizations: Department of Mathematics, Korea University · Department of Finance and Big Data, Gachon University

Abstract

We study minimax rates for conditional generative learning under integrated Wasserstein loss, with Hölder continuous conditional laws and an unrestricted covariate distribution. We evaluate unrestricted generators through their conditional output distributions. For expected Wasserstein loss, the minimax rate agrees with the rate under uniform covariates and follows from balancing localization bias against unconditional distribution estimation error. Higher moments of the loss can instead be dominated by unobserved covariate regions, which force an additional risk contribution of order $n^{-1}$ , where $n$ is the sample size. An adaptive localization argument gives matching upper bounds. The results cover unrestricted response laws and possibly vanishing Besov densities for $W_1$ , and extend to higher Wasserstein orders under uniform response density bounds and integrability conditions. A multiscale lower bound closes the logarithmic gap in unconditional $W_1$ minimax rates in dimension two for both unrestricted distributions and Besov density classes. We also establish minimax rates for conditional Brenier maps under density, integrability, and uniform curvature conditions.

Figures & tables

Regime	Unconditional rate $a_{d_{y},\tau}(m)$	Conditional rate $A_{n}(\tau)$
$d_{y}=1$	$m^{-1/2}$	$n^{-\beta/(2\beta+d_{x})}$
$d_{y}=2$	$\{\log(em)/m\}^{1/2}$	$\{\log(en)/n\}^{\beta/(2\beta+d_{x})}$
$d_{y}>2$ , $\tau=0$	$m^{-1/d_{y}}$	$n^{-\beta/(\beta d_{y}+d_{x})}$
$d_{y}>2$ , $\tau>0$	$m^{-(\tau+1)/(2\tau+d_{y})}$	$n^{-\beta(\tau+1)/[\beta(2\tau+d_{y})+d_{x}(\tau+1)]}$

Table 1: Unconditional and conditional rates. The response dimension determines the critical behavior, while the covariate dimension enters through localization.

Figure 1: Conditional distribution estimation for

q_{x}(y)=1+0.8\sin(2\pi x)(2y_{1}-1)

with

X

uniform on

[0,1]

. Panel (a) shows mean integrated

W_{1}

error of the grid empirical estimator in response dimensions

d=d_{y}\in\{1,2,4\}

. Panel (b) shows the mean sampling loss

W_{1}(\widehat{Q}_{I},Q_{I})

for

d=1,2

, where

Q_{I}

is the population law pooled over the query cell. In panel (b), each curve is multiplied by

\sqrt{m_{n,d}}

, with

m_{n,d}=n/K_{d}(n)

, and normalized at

n=512

. The dotted reference is

\{\log(em_{n,2})/\log(em_{512,2})\}^{1/2}

, and the horizontal reference is one. Shaded bands are pointwise

95\%

bootstrap intervals incorporating numerical transport bounds. Appendix G.1 gives the bandwidths, repetition counts, and evaluation procedures.

Figure 2: Conditional distribution learning for ( 28 ). The left and right columns use uniform and rare covariates, respectively. Panels (a) and (b) show mean integrated

W_{1}

loss, and panels (c) and (d) show mean integrated

W_{1}^{4}

loss, averaged over the two signs. All axes are logarithmic, with common vertical limits within each row. Gray dashed segments indicate reference powers with arbitrary vertical scaling. Shaded bands are pointwise

95\%

bootstrap intervals over independent training datasets, with stratified resampling for rare covariates. Appendix G.2 gives the repetition counts and integration procedures.

Figure 3: Transport separation for cosine densities using multiple levels or the finest level alone, with amplitude

\varepsilon_{J}=0.025\,2^{-2J}

. Panels (a) and (b) show

W_{1}/\varepsilon_{J}

and

W_{1}/(\varepsilon_{J}\sqrt{J})

. Markers and lines show means of numerical interval midpoints; black segments give mean lower and upper bounds. Shaded bands incorporate these bounds and pointwise

95\%

bootstrap uncertainty over

32

independent pairs of sign realizations at each resolution. Appendix G.3 gives the transport computations and numerical bounds.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Figure G.1: Fourth-moment risk of the occupancy estimator under rare covariates. The three contributions in ( 88 ) are multiplied by

n

. Both axes are logarithmic. The dotted curve is the lower bound

nB_{n}

from ( 89 ). Shaded bands are pointwise

95\%

bootstrap intervals, using

5{,}632

stratified repetitions at each sample size.

Conditional Generative Modeling

Conditional generative modeling learns to produce data distributions steered by labels, inputs, text, or physical and structural constraints.

See all 393 papers

On the Error Analysis of Triangular Transport Maps for Conditional Generation via MMD-based Objectives
Oct 24, 2026Michele Martino, Bamdad Hosseini, Amir Sagiv +1Conditional Generative Modeling Conditional Distribution Estimation
Department of Applied Mathematics, University of Washington, Seattle, WA, USA · Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ, USA · Department of Aeronautics and Astronautics, University of Washington, Seattle, WA, USA
A Temporal Spatial Minimax Rate for Smoothly-Varying Distributions in Wasserstein Space
Jun 5, 2026Munsik KimMinimax Estimation Spatiotemporal Forecasting
Conditioning Degenerate Diffusion Models
Sep 3, 2026Uğur Aydın, Tamer BaşarDiffusion Model Guidance Stochastic Optimal Control
Department of Electrical and Computer Engineering and Coordinated Science Laboratory University of Illinois Urbana Champaign Illinois, IL 61801, USA

Oct 24, 2026J/K move · Enter open · S save

On the Error Analysis of Triangular Transport Maps for Conditional Generation via MMD-based Objectives

Michele Martino, Bamdad Hosseini, Amir Sagiv, Amirhossein Taghvaei

Department of Applied Mathematics, University of Washington, Seattle, WA, USA · Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ, USA · Department of Aeronautics and Astronautics, University of Washington, Seattle, WA, USA

Conditional Generative Modeling Conditional Distribution Estimation

Open paper AskPDF arXiv

Conditional generative models parameterized by triangular transport maps are widely used for conditional sampling, yet theoretical guarantees on the quality of the conditional measures they produce remain scarce. We address this question for minimum divergence estimators of triangular maps. We use MMD-based objectives as a tractable setting in which the gap between joint and conditional errors can be analyzed precisely, and we measure performance through the average conditional estimation error, defined as the squared maximum mean discrepancy (MMD) between the estimated and the true conditionals, averaged over the conditioning variable. We first show that the common practice of minimizing a joint MMD may fail to recover the conditionals: writing the joint and the conditional criteria as quadratic energies of the same conditional mean embedding residual, we prove that, for parametric classes, joint minimizers are generically inconsistent in the misspecified setting. We then analyze an estimator that directly minimizes the conditional criterion through a surrogate loss computable from joint target samples and reference samples. We prove a non-asymptotic error bound that decomposes the average conditional estimation error into approximation, estimation, and reference sampling errors, with fast rates independent of the ambient dimension in the parametric setting. Finally, for joint MMDs built from concentrating Gaussian kernels on the conditioning variable, we show that the joint criterion approximates a density-reweighted version of the conditional one, and we derive an error bound for the joint minimizer whose rate suffers from a curse of dimensionality in the conditioning variable. Our results suggest that training criteria for conditional generative models should target the conditional rather than the joint error, a principle that extends to other divergences and to dynamic transport.

Abstract

Conditional Generative Modeling

Explore similar work

On the Error Analysis of Triangular Transport Maps for Conditional Generation via MMD-based Objectives

A Temporal Spatial Minimax Rate for Smoothly-Varying Distributions in Wasserstein Space

Conditioning Degenerate Diffusion Models