For well-behaved real-valued multi-index models, the paper gives PAC learners with d^{O(m)} complexity and nearly matching SQ lower bounds, plus a network-size-independent learner for homogeneous Lipschitz ReLU networks.
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Algorithms and SQ Lower Bounds for Robustly Learning Real-valued Multi-index Models
For well-behaved real-valued multi-index models, the paper gives PAC learners with d^{O(m)} complexity and nearly matching SQ lower bounds, plus a network-size-independent learner for homogeneous Lipschitz ReLU networks.