We study the problem of learning multi-index models in high-dimensions using a two-layer neural network trained with the mean-field Langevin algorithm. Under mild distributional assumptions on the data, we characterize the effective dimension $d_{\mathrm{eff}}$ that controls both sample and computational complexity by utilizing the adaptivity of neural networks to latent low-dimensional structures. When the data exhibit such a structure, $d_{\mathrm{eff}}$ can be significantly smaller than the ambient dimension. We prove that the sample complexity grows almost linearly with $d_{\mathrm{eff}}$, bypassing the limitations of the information and generative exponents that appeared in recent analyses of gradient-based feature learning. On the other hand, the computational complexity may inevitably grow exponentially with $d_{\mathrm{eff}}$ in the worst-case scenario. Motivated by improving computational complexity, we take the first steps towards polynomial time convergence of the mean-field Langevin algorithm by investigating a setting where the weights are constrained to be on a compact manifold with positive Ricci curvature, such as the hypersphere. There, we study assumptions under which polynomial time convergence is achievable, whereas similar assumptions in the Euclidean setting lead to exponential time complexity.

本研究解决了高维环境中多索引模型学习的问题，提出了一种通过均场朗动算法训练的双层神经网络新方法。结果表明，当数据具有低维结构时，有效维度$d_{\mathrm{eff}}$可以显著小于环境维度，从而使样本复杂度几乎线性增长，潜在地提高了计算效率。

通过均场朗动动力学学习多索引模型的神经网络