Flow matching (FM) has gained significant attention as a simulation-free
generative model. Unlike diffusion models, which are based on stochastic
differential equations, FM employs a simpler approach by solving an ordinary
differential equation with an initial condition from a normal distribution,
thus streamlining the sample generation process. This paper discusses the
convergence properties of FM in terms of the $p$-Wasserstein distance, a
measure of distributional discrepancy. We establish that FM can achieve the
minmax optimal convergence rate for $1 \leq p \leq 2$, presenting the first
theoretical evidence that FM can reach convergence rates comparable to those of
diffusion models. Our analysis extends existing frameworks by examining a
broader class of mean and variance functions for the vector fields and
identifies specific conditions necessary to attain these optimal rates.

该论文讨论了流匹配在 $p$-Wasserstein 距离方面的收敛性质，通过研究一类更广泛的向量场的均值和方差函数，确定实现这些最优速率所必需的特定条件，并且证明了流匹配能够达到与扩散模型相当的收敛速率，从而为流匹配作为一种无需模拟的生成模型提供了第一条理论证据。

流匹配实现最小极大均衡收敛

Flow matching achieves minimax optimal convergence

Distributional reinforcement learning (DRL), which cares about the full
distribution of returns instead of just the mean, has achieved empirical
success in various domains. One of the core tasks in the field of DRL is
distributional policy evaluation, which involves estimating the return
distribution $\eta^\pi$ for a given policy $\pi$. A distributional temporal
difference (TD) algorithm has been accordingly proposed, which is an extension
of the temporal difference algorithm in the classic RL literature. In the
tabular case, \citet{rowland2018analysis} and \citet{rowland2023analysis}
proved the asymptotic convergence of two instances of distributional TD, namely
categorical temporal difference algorithm (CTD) and quantile temporal
difference algorithm (QTD), respectively. In this paper, we go a step further
and analyze the finite-sample performance of distributional TD. To facilitate
theoretical analysis, we propose non-parametric distributional TD algorithm
(NTD). For a $\gamma$-discounted infinite-horizon tabular Markov decision
process with state space $S$ and action space $A$, we show that in the case of
NTD we need $\wtilde O\prn{\frac{1}{\varepsilon^{2p}(1-\gamma)^{2p+2}}}$
iterations to achieve an $\varepsilon$-optimal estimator with high probability,
when the estimation error is measured by the $p$-Wasserstein distance. Under
some mild assumptions, $\wtilde O\prn{\frac{1}{\varepsilon^{2}(1-\gamma)^{4}}}$
iterations suffices to ensure the Kolmogorov-Smirnov distance between the NTD
estimator $\hat\eta^\pi$ and $\eta^\pi$ less than $\varepsilon$ with high
probability. And we revisit CTD, showing that the same non-asymptotic
convergence bounds hold for CTD in the case of the $p$-Wasserstein distance.

本文研究分布增强学习中的分布策略评估问题，并提出了非参数分布时序差分算法 (NTD) 来分析其有限样本性能，通过理论分析得出了 NTD 算法的迭代次数和估计误差之间的关系，同时也证明了 Categorical Temporal Difference Algorithm (CTD) 在 $p$-Wasserstein 距离下具有非渐近收敛性。