Although ubiquitous in the sciences, histogram data have not received much
attention by the Deep Learning community. Whilst regression and classification
tasks for scalar and vector data are routinely solved by neural networks, a
principled approach for estimating histogram labels as a function of an input
vector or image is lacking in the literature. We present a dedicated method for
Deep Learning-based histogram regression, which incorporates cross-bin
information and yields distributions over possible histograms, expressed by
$\tau$-quantiles of the cumulative histogram in each bin. The crux of our
approach is a new loss function obtained by applying the pinball loss to the
cumulative histogram, which for 1D histograms reduces to the Earth Mover's
distance (EMD) in the special case of the median ($\tau = 0.5$), and
generalizes it to arbitrary quantiles. We validate our method with an
illustrative toy example, a football-related task, and an astrophysical
computer vision problem. We show that with our loss function, the accuracy of
the predicted median histograms is very similar to the standard EMD case (and
higher than for per-bin loss functions such as cross-entropy), while the
predictions become much more informative at almost no additional computational
cost.

本文提出一种基于深度学习的直方图回归方法，它包含交叉区间信息并产生可能直方图的分布，并将针球损失应用于累积直方图中以获得新的损失函数，该函数在预测中位数直方图的准确度上类似于标准的 Earth Mover's distance (EMD)。（$\tau=0.5$），并将其推广到任意分位数。

移动地球销球器损失：直方图值回归的分位数

The Earth Mover's Pinball Loss: Quantiles for Histogram-Valued  Regression

Among the many ways of quantifying uncertainty in a regression setting,
specifying the full quantile function is attractive, as quantiles are amenable
to interpretation and evaluation. A model that predicts the true conditional
quantiles for each input, at all quantile levels, presents a correct and
efficient representation of the underlying uncertainty. To achieve this, many
current quantile-based methods focus on optimizing the so-called pinball loss.
However, this loss restricts the scope of applicable regression models, limits
the ability to target many desirable properties (e.g. calibration, sharpness,
centered intervals), and may produce poor conditional quantiles. In this work,
we develop new quantile methods that address these shortcomings. In particular,
we propose methods that can apply to any class of regression model, allow for
selecting a trade-off between calibration and sharpness, optimize for
calibration of centered intervals, and produce more accurate conditional
quantiles. We provide a thorough experimental evaluation of our methods, which
includes a high dimensional uncertainty quantification task in nuclear fusion.

本文提出了新的分位数方法，可以适用于任何回归模型，并允许在校准和锐度之间进行权衡，优化中心区间的校准度，并产生更准确的条件分位数。

超越 Pinball Loss：分位数方法用于校准不确定性量化

Beyond Pinball Loss: Quantile Methods for Calibrated Uncertainty  Quantification

We propose a novel framework for fitting additive quantile regression models,
which provides well calibrated inference about the conditional quantiles and
fast automatic estimation of the smoothing parameters, for model structures as
diverse as those usable with distributional GAMs, while maintaining equivalent
numerical efficiency and stability. The proposed methods are at once
statistically rigorous and computationally efficient, because they are based on
the general belief updating framework of Bissiri et al. (2016) to loss based
inference, but compute by adapting the stable fitting methods of Wood et al.
(2016). We show how the pinball loss is statistically suboptimal relative to a
novel smooth generalisation, which also gives access to fast estimation
methods. Further, we provide a novel calibration method for efficiently
selecting the 'learning rate' balancing the loss with the smoothing priors
during inference, thereby obtaining reliable quantile uncertainty estimates.
Our work was motivated by a probabilistic electricity load forecasting
application, used here to demonstrate the proposed approach. The methods
described here are implemented by the qgam R package, available on the
Comprehensive R Archive Network (CRAN).

该研究提出了一种新的框架，用于拟合添加量分位回归模型，提供关于条件分位数的精确推断和平滑参数的快速自动估计，适用于多样化的模型结构，同时保持数值效率和稳定性。

快速校准的加性分位数回归

Fast calibrated additive quantile regression

The so-called pinball loss for estimating conditional quantiles is a
well-known tool in both statistics and machine learning. So far, however, only
little work has been done to quantify the efficiency of this tool for
nonparametric approaches. We fill this gap by establishing inequalities that
describe how close approximate pinball risk minimizers are to the corresponding
conditional quantile. These inequalities, which hold under mild assumptions on
the data-generating distribution, are then used to establish so-called variance
bounds, which recently turned out to play an important role in the statistical
analysis of (regularized) empirical risk minimization approaches. Finally, we
use both types of inequalities to establish an oracle inequality for support
vector machines that use the pinball loss. The resulting learning rates are
min--max optimal under some standard regularity assumptions on the conditional
quantile.

本研究通过建立不等式来描述近似的 pinball 风险最小化器与相应条件分位数之间的接近程度，再利用这些不等式建立了所谓的方差界，并利用两种类型的不等式来证明使用 pinball 损失函数的支持向量机的 Oracle 不等式，从而实现了最小 - 最大下限对某些标准规定的条件分位数的正则性假设。