We derive a concentration bound of the type `for all $n \geq n_0$ for some
$n_0$' for TD(0) with linear function approximation. We work with online TD
learning with samples from a single sample path of the underlying Markov chain.
This makes our analysis significantly different from offline TD learning or TD
learning with access to independent samples from the stationary distribution of
the Markov chain. We treat TD(0) as a contractive stochastic approximation
algorithm, with both martingale and Markov noises. Markov noise is handled
using the Poisson equation and the lack of almost sure guarantees on
boundedness of iterates is handled using the concept of relaxed concentration
inequalities.

我们得到了一种关于使用线性函数逼近进行 TD (0) 的集中界限。我们通过从底层马尔科夫链的单个样本路径中获取样本来处理在线 TD 学习，这使得我们的分析与离线 TD 学习或使用来自马尔科夫链的稳态分布的独立样本进行 TD 学习的情况显着不同。我们将 TD (0) 视为一种具有收敛性的随机逼近算法，其中同时存在鞅和马尔科夫噪声。使用泊松方程处理马尔科夫噪声，而使用松弛的集中不等式概念处理迭代的有界性的几乎确定性保证的缺失。

TD (0) 函数逼近的集中界

A Concentration Bound for TD(0) with Function Approximation

This paper proves that robustness implies generalization via data-dependent
generalization bounds. As a result, robustness and generalization are shown to
be connected closely in a data-dependent manner. Our bounds improve previous
bounds in two directions, to solve an open problem that has seen little
development since 2010. The first is to reduce the dependence on the covering
number. The second is to remove the dependence on the hypothesis space. We
present several examples, including ones for lasso and deep learning, in which
our bounds are provably preferable. The experiments on real-world data and
theoretical models demonstrate near-exponential improvements in various
situations. To achieve these improvements, we do not require additional
assumptions on the unknown distribution; instead, we only incorporate an
observable and computable property of the training samples. A key technical
innovation is an improved concentration bound for multinomial random variables
that is of independent interest beyond robustness and generalization.

通过数据相关的广义界限，证明了健壮性意味着泛化。我们提出了几个例子，包括 Lasso 和深度学习，证明了我们的界限是明显优于以前的界限的。技术创新包括针对多项式随机变量的改进的浓缩界限。

鲁棒性意味着数据相关的泛化界限推广

Robustness Implies Generalization via Data-Dependent Generalization Bounds

In several real-world applications involving decision making under
uncertainty, the traditional expected value objective may not be suitable, as
it may be necessary to control losses in the case of a rare but extreme event.
Conditional Value-at-Risk (CVaR) is a popular risk measure for modeling the
aforementioned objective. We consider the problem of estimating CVaR from
i.i.d. samples of an unbounded random variable, which is either sub-Gaussian or
sub-exponential. We derive a novel one-sided concentration bound for a natural
sample-based CVaR estimator in this setting. Our bound relies on a
concentration result for a quantile-based estimator for Value-at-Risk (VaR),
which may be of independent interest.

本文提出了一种基于样本计算条件风险价值 (CVaR) 的方法，通过统计方法得到了一个新的置信边界来估计 CVaR。该方法和得到的置信边界是基于量化反应的风险价值 (VaR) 估计的经验。

经验条件风险价值的集中度界限：无界情形

Concentration bounds for empirical conditional value-at-risk: The  unbounded case

Let $W_n= \frac{1}{\sqrt n} M_n$ be a Wigner matrix whose entries have
vanishing third moment, normalized so that the spectrum is concentrated in the
interval $[-2,2]$. We prove a concentration bound for $N_I = N_I(W_n)$, the
number of eigenvalues of $W_n$ in an interval $I$.
Our result shows that $N_I$ decays exponentially with standard deviation at
most $O(\log^{O(1)} n)$. This is best possible up to the constant exponent in
the logarithmic term. As a corollary, the bulk eigenvalues are localized to an
interval of width $O(\log^{O(1)} n/n)$; again, this is optimal up to the
exponent. These results strengthen recent results of Erdos, Yau and Yin (under
the extra assumption of vanishing third

该研究使用 Wigner 矩阵并证明了一个关于其特征值数量分布的浓度界限，结果表明本身分布的性质导致了其具有固定的解析性质，包括指数衰减和区间局部化。