We introduce a causal regularisation extension to anchor regression (AR) for
improved out-of-distribution (OOD) generalisation. We present anchor-compatible
losses, aligning with the anchor framework to ensure robustness against
distribution shifts. Various multivariate analysis (MVA) algorithms, such as
(Orthonormalized) PLS, RRR, and MLR, fall within the anchor framework. We
observe that simple regularisation enhances robustness in OOD settings.
Estimators for selected algorithms are provided, showcasing consistency and
efficacy in synthetic and real-world climate science problems. The empirical
validation highlights the versatility of anchor regularisation, emphasizing its
compatibility with MVA approaches and its role in enhancing replicability while
guarding against distribution shifts. The extended AR framework advances causal
inference methodologies, addressing the need for reliable OOD generalisation.

我们通过引入因果正则化扩展到锚定回归（AR）来改善超出分布的广义。我们提出与锚定框架一致的锚定兼容损失，以确保对分布变化的鲁棒性。各种多变量分析（MVA）算法，如（正交化）PLS，RRR 和 MLR，都适用于锚定框架。我们观察到简单的正则化能增强在超出分布设置下的鲁棒性。我们提供了选择算法的估计器，展示了它们在合成和真实的气候科学问题中的一致性和功效。经验证实，锚定正则化的通用性凸显了其与 MVA 方法的兼容性，并在增强可复制性的同时防范了分布变化。扩展的 AR 框架推进了因果推断方法，解决了可靠的超出分布广义化的需求。

通过锚定多元分析提高泛化能力

Improving generalisation via anchor multivariate analysis

We propose a novel algorithm for data augmentation in nonlinear
over-parametrized regression. Our data augmentation algorithm borrows from the
literature on causality and extends the recently proposed Anchor regression
(AR) method for data augmentation, which is in contrast to the current
state-of-the-art domain-agnostic solutions that rely on the Mixup literature.
Our Anchor Data Augmentation (ADA) uses several replicas of the modified
samples in AR to provide more training examples, leading to more robust
regression predictions. We apply ADA to linear and nonlinear regression
problems using neural networks. ADA is competitive with state-of-the-art
C-Mixup solutions.

我们提出了一种用于非线性过参数化回归的数据增强算法，该算法借鉴了因果关系的文献，并扩展了最近提出的锚点回归 (AR) 方法，用于数据增强，与目前依赖于 Mixup 文献的领域不可知解决方案形成鲜明对比。我们的锚点数据增强 (ADA) 使用 AR 中修改后样本的多个副本，提供更多的训练样本，从而得到更强壮的回归预测。我们将 ADA 应用于使用神经网络的线性和非线性回归问题。ADA 与最先进的 C-Mixup 解决方案相竞争。

锚点数据增强

Anchor Data Augmentation

We consider the problem of predicting a response variable from a set of
covariates on a data set that differs in distribution from the training data.
Causal parameters are optimal in terms of predictive accuracy if in the new
distribution either many variables are affected by interventions or only some
variables are affected, but the perturbations are strong. If the training and
test distributions differ by a shift, causal parameters might be too
conservative to perform well on the above task. This motivates anchor
regression, a method that makes use of exogeneous variables to solve a
relaxation of the causal minimax problem by considering a modification of the
least-squares loss. The procedure naturally provides an interpolation between
the solutions of ordinary least squares and two-stage least squares. We prove
that the estimator satisfies predictive guarantees in terms of distributional
robustness against shifts in a linear class; these guarantees are valid even if
the instrumental variables assumptions are violated. If anchor regression and
least squares provide the same answer (anchor stability), we establish that OLS
parameters are invariant under certain distributional changes. Anchor
regression is shown empirically to improve replicability and protect against
distributional shifts.

本文研究了在训练数据与测试数据分布不同的情况下，利用 anchor regression 方法通过使用外生变量来预测响应变量，并提高其可复现性和分布鲁棒性。