We investigate random feature models in which neural networks sampled from a prescribed initialization ensemble are frozen and used as random features, with only the readout weights optimized. Adopting a statistical-physics viewpoint, we study the training, test, and generalization errors beyond the mean-kernel approximation. Since the predictor is a nonlinear functional of the induced random kernel, the ensemble-averaged errors depend not only on the mean kernel but also on higher-order fluctuation statistics. Within an effective field-theoretic framework, these finite-width contributions naturally appear as loop corrections. We derive the loop corrections to the training, test, and generalization errors, obtain their scaling laws, and support the theory with experimental verification.

翻译：我们研究随机特征模型，其中从预设初始化系综中采样的神经网络被冻结并用作随机特征，仅优化读出权重。采用统计物理视角，我们研究超越平均核近似的训练、测试和泛化误差。由于预测变量是诱导随机核的非线性泛函，系综平均误差不仅依赖于平均核，还依赖于高阶涨落统计量。在有效场论框架内，这些有限宽度贡献自然表现为圈修正。我们推导了训练、测试和泛化误差的圈修正，获得了其标度律，并通过实验验证支持该理论。