We prove several hardness results for training depth-2 neural networks with the ReLU activation function; these networks are simply weighted sums (that may include negative coefficients) of ReLUs. Our goal is to output a depth-2 neural network that minimizes the square loss with respect to a given training set. We prove that this problem is NP-hard already for a network with a single ReLU. We also prove NP-hardness for outputting a weighted sum of k ReLUs minimizing the squared error (for k>1) even in the realizable setting (i.e., when the labels are consistent with an unknown depth-2 ReLU network). We are also able to obtain lower bounds on the running time in terms of the desired additive error ฯต. To obtain our lower bounds, we use the Gap Exponential Time Hypothesis (Gap-ETH) as well as a new hypothesis regarding the hardness of approximating the well known Densest k-Subgraph problem in subexponential time (these hypotheses are used separately in proving different lower bounds). For example, we prove that under reasonable hardness assumptions, any proper learning algorithm for finding the best fitting ReLU must run in time exponential in (1/epsilon)^2. Together with a previous work regarding improperly learning a ReLU (Goel et al., COLT'17), this implies the first separation between proper and improper algorithms for learning a ReLU. We also study the problem of properly learning a depth-2 network of ReLUs with bounded weights giving new (worst-case) upper bounds on the running time needed to learn such networks both in the realizable and agnostic settings. Our upper bounds on the running time essentially matches our lower bounds in terms of the dependency on epsilon.
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Time/Accuracy Tradeoffs for learning a ReLU with respect to Gaussian Marginals
We consider the problem of computing the best-fitting ReLU with respect to square-loss on a training set when the examples have been drawn according to a spherical Gaussian distribution (the labels can be arbitrary). Let ๐๐๐<1 be the population loss of the best-fitting ReLU. We prove:
1. Finding a ReLU with square-loss ๐๐๐+ฯต is as hard as the problem of learning sparse parities with noise, widely thought to be computationally intractable. This is the first hardness result for learning a ReLU with respect to Gaussian marginals, and our results imply -{\emph unconditionally}- that gradient descent cannot converge to the global minimum in polynomial time.
2. There exists an efficient approximation algorithm for finding the best-fitting ReLU that achieves error O(๐๐๐^{2/3}). The algorithm uses a novel reduction to noisy halfspace learning with respect to 0/1 loss.
Prior work due to Soltanolkotabi [Sol17] showed that gradient descent can find the best-fitting ReLU with respect to Gaussian marginals, if the training set is exactly labeled by a ReLU.
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- Award ID(s):
- 1717896
- NSF-PAR ID:
- 10190451
- Date Published:
- Journal Name:
- Advances in neural information processing systems
- ISSN:
- 1049-5258
- Page Range / eLocation ID:
- 8582-8591
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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