In this work we investigate the generalization performance of random feature
ridge regression (RFRR). Our main contribution is a general deterministic
equivalent for the test error of RFRR. Specifically, under a certain
concentration property, we show that the test error is well approximated by a
closed-form expression that only depends on the feature map eigenvalues.
Notably, our approximation guarantee is non-asymptotic, multiplicative, and
independent of the feature map dimension -- allowing for infinite-dimensional
features. We expect this deterministic equivalent to hold broadly beyond our
theoretical analysis, and we empirically validate its predictions on various
real and synthetic datasets. As an application, we derive sharp excess error
rates under standard power-law assumptions of the spectrum and target decay. In
particular, we provide a tight result for the smallest number of features
achieving optimal minimax error rate.

我们研究了随机特征岭回归（RFRR）的泛化性能，并提供了其测试误差的一般确定性等价物。具体而言，在一定的集中性质下，我们表明测试误差可以用一个闭式表达式来良好近似，该表达式仅依赖于特征映射的特征值。值得注意的是，我们的近似保证是非渐近的、乘性的，并且与特征映射的维度无关，允许无限维特征。我们预期这个确定性等价物在我们的理论分析之外广泛适用，并从各种真实和合成数据集上对其预测进行了实证验证。作为一个应用，我们根据谱和目标衰减的标准幂律假设导出了尖锐的超额误差率。特别地，我们提供了实现最优极小极大误差率所需特征数量的紧密结果。