Learning with rejection is an important framework that can refrain from
making predictions to avoid critical mispredictions by balancing between
prediction and rejection. Previous studies on cost-based rejection only focused
on the classification setting, which cannot handle the continuous and infinite
target space in the regression setting. In this paper, we investigate a novel
regression problem called regression with cost-based rejection, where the model
can reject to make predictions on some examples given certain rejection costs.
To solve this problem, we first formulate the expected risk for this problem
and then derive the Bayes optimal solution, which shows that the optimal model
should reject to make predictions on the examples whose variance is larger than
the rejection cost when the mean squared error is used as the evaluation
metric. Furthermore, we propose to train the model by a surrogate loss function
that considers rejection as binary classification and we provide conditions for
the model consistency, which implies that the Bayes optimal solution can be
recovered by our proposed surrogate loss. Extensive experiments demonstrate the
effectiveness of our proposed method.

针对具有连续和无限目标空间的回归问题，提出了一种新颖的基于成本拒绝的回归模型，其可以通过考虑拒绝成本来拒绝对某些示例进行预测。该研究首先建立了该问题的期望风险模型，然后导出了贝叶斯最优解，表明当使用均方误差作为评估指标时，最优模型应拒绝对方差大于拒绝成本的示例进行预测。此外，提出使用考虑拒绝作为二分类的替代损失函数进行模型训练，并提供了模型一致性的条件，表明我们提出的替代损失函数可以恢复贝叶斯最优解。大量实验证明了我们提出方法的有效性。

基于成本拒绝的回归

Regression with Cost-based Rejection

Learning with rejection is a prototypical model for studying the interaction
between humans and AI on prediction tasks. The model has two components, a
predictor and a rejector. Upon the arrival of a sample, the rejector first
decides whether to accept it; if accepted, the predictor fulfills the
prediction task, and if rejected, the prediction will be deferred to humans.
The learning problem requires learning a predictor and a rejector
simultaneously. This changes the structure of the conventional loss function
and often results in non-convexity and inconsistency issues. For the
classification with rejection problem, several works develop surrogate losses
for the jointly learning with provable consistency guarantees; in parallel,
there has been less work for the regression counterpart. We study the
regression with rejection (RwR) problem and investigate the no-rejection
learning strategy which treats the RwR problem as a standard regression task to
learn the predictor. We establish that the suboptimality of the no-rejection
learning strategy observed in the literature can be mitigated by enlarging the
function class of the predictor. Then we introduce the truncated loss to single
out the learning for the predictor and we show that a consistent surrogate
property can be established for the predictor individually in an easier way
than for the predictor and the rejector jointly. Our findings advocate for a
two-step learning procedure that first uses all the data to learn the predictor
and then calibrates the prediction loss for the rejector. It is better aligned
with the common intuition that more data samples will lead to a better
predictor and it calls for more efforts on a better design of calibration
algorithms for learning the rejector. While our discussions mainly focus on the
regression problem, the theoretical results and insights generalize to the
classification problem as well.

学习与拒绝是研究人类与人工智能在预测任务中相互作用的典型模型，其包含预测者和拒绝者两个组件。我们探讨了回归与拒绝问题，并研究了将该问题视为标准回归任务学习预测者的无拒绝学习策略。通过扩大预测者的函数类，我们发现无拒绝学习策略的次最优性可以得到缓解。在研究中，我们引入了截断损失来单独针对预测者进行学习，并展示了比联合学习更容易建立预测者的一致性替代性。我们的研究结果支持先使用所有数据来学习预测者，然后根据拒绝者的预测损失进行标定的两步学习过程。这更符合常识，即更多的数据样本将会获得更好的预测者，并需要更多的努力设计用于学习拒绝者的标定算法。虽然我们主要讨论了回归问题，但理论结果和观点同样适用于分类问题。

无拒绝学习在具有拒绝的回归问题中的最优性

When No-Rejection Learning is Optimal for Regression with Rejection

Long-tail recognition tackles the natural non-uniformly distributed data in
real-world scenarios. While modern classifiers perform well on populated
classes, its performance degrades significantly on tail classes. Humans,
however, are less affected by this since, when confronted with uncertain
examples, they simply opt to provide coarser predictions. Motivated by this, a
deep realistic taxonomic classifier (Deep-RTC) is proposed as a new solution to
the long-tail problem, combining realism with hierarchical predictions. The
model has the option to reject classifying samples at different levels of the
taxonomy, once it cannot guarantee the desired performance. Deep-RTC is
implemented with a stochastic tree sampling during training to simulate all
possible classification conditions at finer or coarser levels and a rejection
mechanism at inference time. Experiments on the long-tailed version of four
datasets, CIFAR100, AWA2, Imagenet, and iNaturalist, demonstrate that the
proposed approach preserves more information on all classes with different
popularity levels. Deep-RTC also outperforms the state-of-the-art methods in
longtailed recognition, hierarchical classification, and learning with
rejection literature using the proposed correctly predicted bits (CPB) metric.

提出了一种新的解决长尾问题的方法 —— 深度逼真的分类器（Deep-RTC），该模型将现实主义与分层预测相结合，并具有在分类不确定性时拒绝样本的机制。