Why neural networks are singular

By the textbook, a billion-parameter network trained on far less data than that should memorise its training set and fail on everything else. In practice these models generalise well, and often better as they grow, against the textbook's direct prediction. That gap is the puzzle that singular learning theory, developed by Sumio Watanabe1, was built to explain.

The resolution starts from a simple observation: the map from parameters to the function a network computes is massively many-to-one. Part of this is exact symmetry, where swapping two matched units or rescaling a ReLU's weights in against out leaves the function untouched. More of it builds up as the network trains and whole directions in weight space go slack, their pull on the output dwindling toward nothing (that slackening is itself a late phase of training, with its own explainer: nothing to read yet#20). Either way, the best-fitting parameters fill a high-dimensional surface in weight space, one that can fold and cross through itself. The two-weight network from earlier is the smallest version of this, where the best fits already trace out a whole curve.

Models whose optimal parameters live on such a degenerate surface are called singular. The mental picture to keep is that the loss landscape near a good solution behaves less like a round bowl with one well-defined bottom and more like a system of flat-floored valleys and creases, where the floor itself can be a ridge or a sheet that extends for a long way.

Here is the same contrast as two loss surfaces over a pair of weights:

This is what breaks the classical toolkit. Standard asymptotics assume the Fisher information matrix is invertible at the optimum, which is the geometric statement that the bowl curves by a comparable amount in every direction. On a singular model the Fisher information is degenerate: along the flat directions of the valley floor it loses rank and its smallest eigenvalues, the curvature in its flattest directions, run to zero. The trough in the demo above is exactly this case: along its flat floor the loss has no curvature at all. Once that happens, the raw parameter count $d$ badly overstates how complex the model really is.

Singular learning theory replaces the classical complexity $d/2$ with a smaller quantity, the real log canonical threshold $\lambda$ (also called the learning coefficient). It counts the model's effective parameters, the ones that actually change the loss near the solution, and it is governed by the geometry of the valley floor: by how quickly the loss climbs as you step off the floor. The flatter the floor, the smaller $\lambda$, and the simpler the model behaves regardless of how many raw parameters it has.

Everything that follows builds on this single picture.

  1. Sumio Watanabe, Algebraic Geometry and Statistical Learning Theory (Cambridge University Press, 2009), the foundational text; and Mathematical Theory of Bayesian Statistics (CRC Press, 2018) for a more accessible treatment. ↩︎