Why depth yields a genuine computational advantage over shallow methods remains a central open question in learning theory. We study this question in a controlled high-dimensional Gaussian setting, focusing on compositional target functions. We analyze their learnability using an explicit three-layer fitting model trained via layer-wise spectral estimators. Although the target is globally a high-degree polynomial, its compositional structure allows learning to proceed in stages: an intermediate representation reveals structure that is inaccessible at the input level. This reduces learning to simpler spectral estimation problems, well studied in the context of multi-index models, whereas any shallow estimator must resolve all components simultaneously. Our analysis relies on Gaussian universality, leading to sharp separations in sample complexity between two and three-layer learning strategies.

The problem is challenging because the loss distribution and threshold value of each arm are unknown. We study this novel setting by establishing its'equivalence' to Multiple-Play Multi-Armed Bandits (MP-MAB) andCombinatorial Semi-Bandits.

This work focuses on the active learning in video captioning.

Consider F = {f , w(x) = c hx, i) | w 2 Sd 1}. Let = (d) > 0 beasequence 2 (0,1) a constant, and{Fd}d2N beasequence Rd.

These limitations, both theoretical and practical, underscore the need for novel sketching algorithms.