On the non-universality of deep learning: quantifying the cost of symmetry

We prove computational limitations for learning with neural networks trained by noisy gradient descent (GD). Our result applies whenever GD training is equivariant (true for many standard architectures), and quantifies the alignment needed between architectures and data in order for GD to learn. As applications, (i) we characterize the functions that fully-connected networks can weak-learn on the binary hypercube and unit sphere, demonstrating that depth-2 is as powerful as any other depth for this task; (ii) we extend the merged-staircase necessity result for learning with latent low-dimensional structure [ABM22] to beyond the mean-field regime. Our techniques extend to stochastic gradient descent (SGD), for which we show nontrivial hardness results for learning with fully-connected networks, based on cryptographic assumptions.