We study generalised linear regression and classification for a synthetically
generated dataset encompassing different problems of interest, such as learning
with random features, neural networks in the lazy training regime, and the
hidden manifold model. We consider the high-dimensional regime and using the
replica method from statistical physics, we provide a closed-form expression
for the asymptotic generalisation performance in these problems, valid in both
the under- and over-parametrised regimes and for a broad choice of generalised
linear model loss functions. In particular, we show how to obtain analytically
the so-called double descent behaviour for logistic regression with a peak at
the interpolation threshold, we illustrate the superiority of orthogonal
against random Gaussian projections in learning with random features, and
discuss the role played by correlations in the data generated by the hidden
manifold model. Beyond the interest in these particular problems, the
theoretical formalism introduced in this manuscript provides a path to further
extensions to more complex tasks.

使用统计物理学中的复制法，我们针对一个综合数据集，研究了广义线性回归和分类问题，在超参数化和不充分参数化的条件下，为这些问题提供了渐近泛化表现的闭式表达式，特别地，我们展示了逻辑回归的双重下降效应，突显了用正交投影相比随机高斯投影在学习随机特征时的优越性，讨论了隐藏流形模型中数据相关性的作用。