Decentralized and asynchronous communications are two popular techniques to
speedup communication complexity of distributed machine learning, by
respectively removing the dependency over a central orchestrator and the need
for synchronization. Yet, combining these two techniques together still remains
a challenge. In this paper, we take a step in this direction and introduce
Asynchronous SGD on Graphs (AGRAF SGD) -- a general algorithmic framework that
covers asynchronous versions of many popular algorithms including SGD,
Decentralized SGD, Local SGD, FedBuff, thanks to its relaxed communication and
computation assumptions. We provide rates of convergence under much milder
assumptions than previous decentralized asynchronous works, while still
recovering or even improving over the best know results for all the algorithms
covered.

分布机器学习中的去中心化和异步通信是提高通信复杂性的两种流行技术，而将这两种技术结合起来仍然是一个挑战。本文引入了一种名为 AGRAF SGD 的异步 SGD 算法框架，它包括了许多流行算法的异步版本，如 SGD、去中心化 SGD、本地 SGD 和 FedBuff，通过放松通信和计算假设，我们提供了比以前的去中心化异步工作更温和的收敛性速度，同时仍然恢复或甚至改善了所有涵盖算法的最佳结果。

图上异步 SGD：异步分布式和联邦优化的统一框架

Asynchronous SGD on Graphs: a Unified Framework for Asynchronous  Decentralized and Federated Optimization

Optimal transport maps between two probability distributions $\mu$ and $\nu$
on $\mathbb{R}^d$ have found extensive applications in both machine learning
and statistics. In practice, these maps need to be estimated from data sampled
according to $\mu$ and $\nu$. Plug-in estimators are perhaps most popular in
estimating transport maps in the field of computational optimal transport. In
this paper, we provide a comprehensive analysis of the rates of convergences
for general plug-in estimators defined via barycentric projections. Our main
contribution is a new stability estimate for barycentric projections which
proceeds under minimal smoothness assumptions and can be used to analyze
general plug-in estimators. We illustrate the usefulness of this stability
estimate by first providing rates of convergence for the natural
discrete-discrete and semi-discrete estimators of optimal transport maps. We
then use the same stability estimate to show that, under additional smoothness
assumptions of Besov type or Sobolev type, wavelet based or kernel smoothed
plug-in estimators respectively speed up the rates of convergence and
significantly mitigate the curse of dimensionality suffered by the natural
discrete-discrete/semi-discrete estimators. As a by-product of our analysis, we
also obtain faster rates of convergence for plug-in estimators of
$W_2(\mu,\nu)$, the Wasserstein distance between $\mu$ and $\nu$, under the
aforementioned smoothness assumptions, thereby complementing recent results in
Chizat et al. (2020). Finally, we illustrate the applicability of our results
in obtaining rates of convergence for Wasserstein barycenters between two
probability distributions and obtaining asymptotic detection thresholds for
some recent optimal-transport based tests of independence.

本文对通过重心投影定义的常规插值估计器的收敛速度进行了全面的分析，利用这种稳定度量来提高优化运输地图的估计器的速度，减轻维数诅咒，并加速 Wasserstein 距离的收敛速度。

使用插值估算器通过重心投影计算最优传输映射的速率

Rates of Estimation of Optimal Transport Maps using Plug-in Estimators  via Barycentric Projections

We consider a problem of multiclass classification, where the training sample
$S_n = \{(X_i, Y_i)\}_{i=1}^n$ is generated from the model $\mathbb P(Y = m | X
= x) = \eta_m(x)$, $1 \leq m \leq M$, and $\eta_1(x), \dots, \eta_M(x)$ are
unknown $\alpha$-Holder continuous functions.Given a test point $X$, our goal
is to predict its label. A widely used $\mathsf k$-nearest-neighbors classifier
constructs estimates of $\eta_1(X), \dots, \eta_M(X)$ and uses a plug-in rule
for the prediction. However, it requires a proper choice of the smoothing
parameter $\mathsf k$, which may become tricky in some situations. In our
solution, we fix several integers $n_1, \dots, n_K$, compute corresponding
$n_k$-nearest-neighbor estimates for each $m$ and each $n_k$ and apply an
aggregation procedure. We study an algorithm, which constructs a convex
combination of these estimates such that the aggregated estimate behaves
approximately as well as an oracle choice. We also provide a non-asymptotic
analysis of the procedure, prove its adaptation to the unknown smoothness
parameter $\alpha$ and to the margin and establish rates of convergence under
mild assumptions.

研究多类分类问题中 $k$- 近邻分类器的平滑参数选择方法，提出一种基于凸组合的聚合方法并证明其适应未知平滑参数和边际，以及在温和条件下的收敛速率。

自适应多类最近邻分类器

An adaptive multiclass nearest neighbor classifier

Collaborative recommendation is an information-filtering technique that
attempts to present information items that are likely of interest to an
Internet user. Traditionally, collaborative systems deal with situations with
two types of variables, users and items. In its most common form, the problem
is framed as trying to estimate ratings for items that have not yet been
consumed by a user. Despite wide-ranging literature, little is known about the
statistical properties of recommendation systems. In fact, no clear
probabilistic model even exists which would allow us to precisely describe the
mathematical forces driving collaborative filtering. To provide an initial
contribution to this, we propose to set out a general sequential stochastic
model for collaborative recommendation. We offer an in-depth analysis of the
so-called cosine-type nearest neighbor collaborative method, which is one of
the most widely used algorithms in collaborative filtering, and analyze its
asymptotic performance as the number of users grows. We establish consistency
of the procedure under mild assumptions on the model. Rates of convergence and
examples are also provided.

提出一种用于协作推荐的随机模型，深入分析了余弦型最近邻协作过滤算法的渐近性能，建立该程序在模型的温和假设下的一致性，并提供了收敛速度和例子。