Papers › Spurious Local Minima are Common in Two-Layer ReLU Neural Networks
Spurious Local Minima are Common in Two-Layer ReLU Neural Networks
Itay Safran, Ohad Shamir
We consider the optimization problem associated with training simple ReLU neural networks of the form 𝐱↦∑ᵢ₌₁ᵏmax{0,𝐰ᵢ^⊤ 𝐱} with respect to the squared loss. We provide a computer-assisted proof that even if the input distribution is standard Gaussian, even if the dimension is arbitrarily large, and even if the target values are generated by such a network, with orthonormal parameter vectors, the problem can still have spurious local minima once 6≤k≤20. By a concentration of measure argument, this implies that in high input dimensions, \emph{nearly all} target networks of the relevant sizes lead to spurious local minima. Moreover, we conduct experiments which show that the probability of hitting such local minima is quite high, and increasing with the network size. On the positive side, mild over-parameterization appears to drastically reduce such local minima, indicating that an over-parameterization assumption is necessary to get a positive result in this setting.
In Syntology Open this paper in Syntology's Atlas, the map of the papers in Syntology's graph and their citations.
Code
Repository list and official/mentioned flags are the archive's, frozen 2025-07-28. Reachability, where shown, is from one Syntology probe window (2026-09-16 to 2026-09-18); repositories not probed show nothing. GitHub stars are not tracked.
Code Syntology ran Syntology
Not run by Syntology. Nothing on this page verifies that the listed code works.
Tasks
Results from the paper archive 2025-07-28
No leaderboard rows for this paper in the archive.
Report a problem or propose a change · a person checks every report against the paper or source before anything changes; decisions are listed on /corrections