As a natural extension to the standard conformal prediction method, several
conformal risk control methods have been recently developed and applied to
various learning problems. In this work, we seek to control the conformal risk
in expectation for ordinal classification tasks, which have broad applications
to many real problems. For this purpose, we firstly formulated the ordinal
classification task in the conformal risk control framework, and provided
theoretic risk bounds of the risk control method. Then we proposed two types of
loss functions specially designed for ordinal classification tasks, and
developed corresponding algorithms to determine the prediction set for each
case to control their risks at a desired level. We demonstrated the
effectiveness of our proposed methods, and analyzed the difference between the
two types of risks on three different datasets, including a simulated dataset,
the UTKFace dataset and the diabetic retinopathy detection dataset.

我们扩展了标准的符合预测方法，开发了几种符合风险控制方法，并应用于不同的学习问题。本文中，我们致力于控制序列分类任务中的符合风险，该任务在许多实际问题中具有广泛应用。为此，我们首先在符合风险控制框架中对序列分类任务进行了规范，并提供了理论风险边界控制方法。然后，我们提出了两种特别设计用于序列分类任务的损失函数，并开发了相应的算法来确定每种情况的预测集，以便将其风险控制在所需的水平。我们展示了我们提出的方法的有效性，并分析了两种风险在三个不同数据集上的差异，包括模拟数据集、UTKFace 数据集和糖尿病视网膜病变检测数据集。

序数分类的一致风险控制

Conformal Risk Control for Ordinal Classification

Given a sequence of observable variables $\{(x_1, y_1), \ldots, (x_n,
y_n)\}$, the conformal prediction method estimates a confidence set for
$y_{n+1}$ given $x_{n+1}$ that is valid for any finite sample size by merely
assuming that the joint distribution of the data is permutation invariant.
Although attractive, computing such a set is computationally infeasible in most
regression problems. Indeed, in these cases, the unknown variable $y_{n+1}$ can
take an infinite number of possible candidate values, and generating conformal
sets requires retraining a predictive model for each candidate. In this paper,
we focus on a sparse linear model with only a subset of variables for
prediction and use numerical continuation techniques to approximate the
solution path efficiently. The critical property we exploit is that the set of
selected variables is invariant under a small perturbation of the input data.
Therefore, it is sufficient to enumerate and refit the model only at the change
points of the set of active features and smoothly interpolate the rest of the
solution via a Predictor-Corrector mechanism. We show how our path-following
algorithm accurately approximates conformal prediction sets and illustrate its
performance using synthetic and real data examples.

本文使用数值连续技术设计了一种路径跟踪算法，以高效地逼近稀疏线性模型的共形预测集，通过枚举和重新拟合的方式，在选定的活动特征的变化点上交替运作并平滑插值.