Learning sum of diverse features: computational hardness and efficient gradient-based training for ridge combinations
<|reference_start|>Learning sum of diverse features: computational hardness and efficient gradient-based training for ridge combinations: We study the computational and sample complexity of learning a target function $f_*:\mathbb{R}^d\to\mathbb{R}$ with additive structure, that is, $f_*(x) = \frac{1}{\sqrt{M}}\sum_{m=1}^M f_m(\langle x, v_m\rangle)$, where $f_1,f_2,...,f_M:\mathbb{R}\to\mathbb{R}$ are nonlinear link functions of single-index models (ridge functions) with diverse and near-orthogonal index features $\{v_m\}_{m=1}^M$, and the number of additive tasks $M$ grows with the dimensionality $M\asymp d^\gamma$ for $\gamma\ge 0$. This problem setting is motivated by the classical additive model literature, the recent representation learning theory of two-layer neural network, and large-scale pretraining where the model simultaneously acquires a large number of "skills" that are often localized in distinct parts of the trained network. We prove that a large subset of polynomial $f_*$ can be efficiently learned by gradient descent training of a two-layer neural network, with a polynomial statistical and computational complexity that depends on the number of tasks $M$ and the information exponent of $f_m$, despite the unknown link function and $M$ growing with the dimensionality. We complement this learnability guarantee with computational hardness result by establishing statistical query (SQ) lower bounds for both the correlational SQ and full SQ algorithms.<|reference_end|>
@article{oko2024learning,
    title={Learning sum of diverse features: computational hardness and efficient
  gradient-based training for ridge combinations},
    author={Kazusato Oko, Yujin Song, Taiji Suzuki, Denny Wu},
    journal={arXiv preprint arXiv:2406.11828},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.11828},
    primaryClass={cs.LG stat.ML}
}