We consider the problem of learning function classes computed by neural networks with various activations (e.g. ReLU or Sigmoid), a task believed to be computationally intractable in the worst-case. A major open problem is to understand the minimal assumptions under which these classes admit provably efficient algorithms. In this work we show that a natural distributional assumption corresponding to {\em eigenvalue decay} of the Gram matrix yields polynomial-time algorithms in the non-realizable setting for expressive classes of networks (e.g.

Wealso compare CoPrivwith SOTA network optimization methods, including SNL, MetaPruning, etc. CoPriv achieves 9.98 and 3.88 online and total communication reduction with a higher accuracy compared to SNL,respectively.

Neural Information Processing Systems http://nips.cc/

We study the parameterized complexity of training two-layer neural networks with respect to the dimension of the input data and the number of hidden neurons, considering ReLU and linear threshold activation functions. Albeit the computational complexity of these problems has been studied numerous times in recent years, several questions are still open. We answer questions by Arora et al. (ICLR 2018) and Khalife and Basu (IPCO 2022) showing that both problems are NP-hard for two dimensions, which excludes any polynomial-time algorithm for constant dimension. We also answer a question by Froese et al. (JAIR 2022) proving W[1]-hardness for four ReLUs (or two linear threshold neurons) with zero training error. Finally, in the ReLU case, we show fixed-parameter tractability for the combined parameter number of dimensions and number of ReLUs if the network is assumed to compute a convex map. Our results settle the complexity status regarding these parameters almost completely.

Deep learning relies on good initialization schemes and hyperparameter choices prior to training a neural network. Random weight initializations induce random network ensembles, which give rise to the trainability, training speed, and sometimes also generalization ability of an instance. In addition, such ensembles provide theoretical insights into the space of candidate models of which one is selected during training. The results obtained so far rely on mean field approximations that assume infinite layer width and that study average squared signals. We derive the joint signal output distribution exactly, without mean field assumptions, for fully-connected networks with Gaussian weights and biases, and analyze deviations from the mean field results. For rectified linear units, we further discuss limitations of the standard initialization scheme, such as its lack of dynamical isometry, and propose a simple alternative that overcomes these by initial parameter sharing.