This paper considers the problem of offline optimization, where the objective
function is unknown except for a collection of ``offline" data examples. While
recent years have seen a flurry of work on applying various machine learning
techniques to the offline optimization problem, the majority of these work
focused on learning a surrogate of the unknown objective function and then
applying existing optimization algorithms. While the idea of modeling the
unknown objective function is intuitive and appealing, from the learning point
of view it also makes it very difficult to tune the objective of the learner
according to the objective of optimization. Instead of learning and then
optimizing the unknown objective function, in this paper we take on a less
intuitive but more direct view that optimization can be thought of as a process
of sampling from a generative model. To learn an effective generative model
from the offline data examples, we consider the standard technique of
``re-weighting", and our main technical contribution is a probably
approximately correct (PAC) lower bound on the natural optimization objective,
which allows us to jointly learn a weight function and a score-based generative
model. The robustly competitive performance of the proposed approach is
demonstrated via empirical studies using the standard offline optimization
benchmarks.

该研究论文探讨了离线优化的问题，其中目标函数除了一系列 “离线” 数据示例外是未知的。与近年来关于将各种机器学习技术应用于离线优化问题的研究相比，大多数工作集中于学习未知目标函数的替代模型，然后应用现有的优化算法。与学习未知目标函数并优化它的想法相比，本文采用了一种不太直观但更直接的观点，即优化可以被视为从生成模型中进行采样的过程。通过对离线数据示例进行有效的生成模型学习，我们考虑了 “重新加权” 的标准技术，并且我们的主要技术贡献是对自然优化目标的可能近似正确 (PAC) 下界的确定，这使得我们能够共同学习一个加权函数和基于分数的生成模型。通过使用标准离线优化基准进行的实证研究证明了所提方法的稳健竞争性能。

从函数到分布建模：离线优化的 PAC - 生成方法

From Function to Distribution Modeling: A PAC-Generative Approach to  Offline Optimization

Since being proposed, Neural Radiance Fields (NeRF) have achieved great
success in related tasks, mainly adopting the hierarchical volume sampling
(HVS) strategy for volume rendering. However, the HVS of NeRF approximates
distributions using piecewise constant functions, which provides a relatively
rough estimation. Based on the observation that a well-trained weight function
$w(t)$ and the $L_0$ distance between points and the surface have very high
similarity, we propose $L_0$-Sampler by incorporating the $L_0$ model into
$w(t)$ to guide the sampling process. Specifically, we propose to use piecewise
exponential functions rather than piecewise constant functions for
interpolation, which can not only approximate quasi-$L_0$ weight distributions
along rays quite well but also can be easily implemented with few lines of code
without additional computational burden. Stable performance improvements can be
achieved by applying $L_0$-Sampler to NeRF and its related tasks like 3D
reconstruction. Code is available at this https URL .

通过将 $L_0$ 模型应用于权重函数 $w (t)$，我们提出了 $L_0$-Sampler，用于指导采样过程并改进 NeRF 及其相关任务的性能。

L0 采样器：一种 L0 模型引导的 NeRF 体积采样

$L_0$-Sampler: An $L_{0}$ Model Guided Volume Sampling for NeRF

In this paper, we propose a so-called probabilistic non-local means (PNLM)
method for image denoising. Our main contributions are: 1) we point out defects
of the weight function used in the classic NLM; 2) we successfully derive all
theoretical statistics of patch-wise differences for Gaussian noise; and 3) we
employ this prior information and formulate the probabilistic weights truly
reflecting the similarity between two noisy patches. The probabilistic nature
of the new weight function also provides a theoretical basis to choose
thresholds rejecting dissimilar patches for fast computations. Our simulation
results indicate the PNLM outperforms the classic NLM and many NLM recent
variants in terms of peak signal noise ratio (PSNR) and structural similarity
(SSIM) index. Encouraging improvements are also found when we replace the NLM
weights with the probabilistic weights in tested NLM variants.

本文提出了一种称之为概率非局部均值（PNLM）的图像去噪方法，利用新的概率权重函数，成功地改进了 NLM 算法并取得了良好的性能，尤其在峰值信噪比和结构相似性指数方面表现卓越。

概率非局部均值

Probabilistic Non-Local Means

We prove the first Chernoff-Hoeffding bounds for general nonreversible
finite-state Markov chains based on the standard L_1 (variation distance)
mixing-time of the chain. Specifically, consider an ergodic Markov chain M and
a weight function f: [n] -> [0,1] on the state space [n] of M with mean mu =
E_{v <- pi}[f(v)], where pi is the stationary distribution of M. A t-step
random walk (v_1,...,v_t) on M starting from the stationary distribution pi has
expected total weight E[X] = mu t, where X = sum_{i=1}^t f(v_i). Let T be the
L_1 mixing-time of M. We show that the probability of X deviating from its mean
by a multiplicative factor of delta, i.e., Pr [ |X - mu t| >= delta mu t ], is
at most exp(-Omega(delta^2 mu t / T)) for 0 <= delta <= 1, and exp(-Omega(delta
mu t / T)) for delta > 1. In fact, the bounds hold even if the weight functions
f_i's for i in [t] are distinct, provided that all of them have the same mean
mu.
We also obtain a simplified proof for the Chernoff-Hoeffding bounds based on
the spectral expansion lambda of M, which is the square root of the second
largest eigenvalue (in absolute value) of M tilde{M}, where tilde{M} is the
time-reversal Markov chain of M. We show that the probability Pr [ |X - mu t|
>= delta mu t ] is at most exp(-Omega(delta^2 (1-lambda) mu t)) for 0 <= delta
<= 1, and exp(-Omega(delta (1-lambda) mu t)) for delta > 1.
Both of our results extend to continuous-time Markov chains, and to the case
where the walk starts from an arbitrary distribution x, at a price of a
multiplicative factor depending on the distribution x in the concentration
bounds.

该论文基于标准的 L_1 混合时间证明了一般性的非可逆有限状态马尔科夫链的 Chernoff-Hoeffding 边界。