The study of universal approximation properties (UAP) for neural networks
(NN) has a long history. When the network width is unlimited, only a single
hidden layer is sufficient for UAP. In contrast, when the depth is unlimited,
the width for UAP needs to be not less than the critical width
$w^*_{\min}=\max(d_x,d_y)$, where $d_x$ and $d_y$ are the dimensions of the
input and output, respectively. Recently, \cite{cai2022achieve} shows that a
leaky-ReLU NN with this critical width can achieve UAP for $L^p$ functions on a
compact domain $K$, \emph{i.e.,} the UAP for $L^p(K,\mathbb{R}^{d_y})$. This
paper examines a uniform UAP for the function class $C(K,\mathbb{R}^{d_y})$ and
gives the exact minimum width of the leaky-ReLU NN as
$w_{\min}=\max(d_x+1,d_y)+1_{d_y=d_x+1}$, which involves the effects of the
output dimensions. To obtain this result, we propose a novel
lift-flow-discretization approach that shows that the uniform UAP has a deep
connection with topological theory.

本文研究神经网络的通用逼近性质，发现当网络的深度无限大时，其宽度需要不小于临界宽度以实现通用逼近性质；同时，我们提出了一种新颖的提升 - 流 - 离散化方法，探讨了通用逼近性质与拓扑理论之间的深层联系。

Leaky-ReLU 神经网络均匀全局逼近的最小宽度

Minimum Width of Leaky-ReLU Neural Networks for Uniform Universal  Approximation

We make progress in a long-standing problem of batch reinforcement learning
(RL): learning $Q^\star$ from an exploratory and polynomial-sized dataset,
using a realizable and otherwise arbitrary function class. In fact, all
existing algorithms demand function-approximation assumptions stronger than
realizability, and the mounting negative evidence has led to a conjecture that
sample-efficient learning is impossible in this setting (Chen and Jiang, 2019).
Our algorithm, BVFT, breaks the hardness conjecture (albeit under a stronger
notion of exploratory data) via a tournament procedure that reduces the
learning problem to pairwise comparison, and solves the latter with the help of
a state-action partition constructed from the compared functions. We also
discuss how BVFT can be applied to model selection among other extensions and
open problems.

该研究提出了一种 batch reinforcement learning 的学习算法 BVFT，通过一种基于比较和分区的机制使得学习效率更高并且适用于其他问题和扩展。

仅具可实现性的批次值函数逼近

Batch Value-function Approximation with Only Realizability

This paper establishes minimax rates for online regression with arbitrary
classes of functions and general losses. We show that below a certain threshold
for the complexity of the function class, the minimax rates depend on both the
curvature of the loss function and the sequential complexities of the class.
Above this threshold, the curvature of the loss does not affect the rates.
Furthermore, for the case of square loss, our results point to the interesting
phenomenon: whenever sequential and i.i.d. empirical entropies match, the rates
for statistical and online learning are the same.
In addition to the study of minimax regret, we derive a generic forecaster
that enjoys the established optimal rates. We also provide a recipe for
designing online prediction algorithms that can be computationally efficient
for certain problems. We illustrate the techniques by deriving existing and new
forecasters for the case of finite experts and for online linear regression.

本文研究任意函数类和一般损失的在线回归问题，并确定了最小极大率。结果表明，当函数类的复杂度低于某一阈值时，最小极大率取决于损失函数的曲率和序列复杂度。而当复杂度高于该阈值时，损失的曲率不再影响率。此外，对于平方损失，当顺序和独立同分布的经验熵匹配时，统计和在线学习的速率相同。我们还提供了一种通用的预测器，它具有确定的最优率，并提供了一种设计在线预测算法的方法，用于某些问题可以是计算量有效的。

具有通用损失函数的在线非参数回归

Online Nonparametric Regression with General Loss Functions

We construct data dependent bounds on the risk in function learning problems.
The bounds are based on the local norms of the Rademacher process indexed by
the underlying function class and they do not require prior knowledge about the
distribution of the training examples or any specific properties of the
function class.

本篇论文构建了基于 Rademacher 过程的局部范数的数据相关边界，用以解决函数学习问题的风险问题。这些边界不需要关于训练集分布或函数类的任何先验知识。