Efficient Algorithms for Learning Depth-2 Neural Networks with General ReLU Activations

Abstract: We present polynomial time and sample efficient algorithms for learning an unknown depth-2 feedforward neural network with general ReLU activations, under mild non-degeneracy assumptions. In particular, we consider learning an unknown network of the form $$f(x) = {a}<sup>{\mathsf{T}}\sigma({W}<sup>\mathsf{T}x+b)$$, where $x$ is drawn from the Gaussian distribution, and $\sigma(t) := \max(t,0)$ is the ReLU activation. Prior works for learning networks with ReLU activations assume that the bias $b$ is zero. In order to deal with the presence of the bias terms, our proposed algorithm consists of robustly decomposing multiple higher order tensors arising from the Hermite expansion of the function $f(x)$. Using these ideas we also establish identifiability of the network parameters under minimal assumptions.