LIT-541: Deep Learning is Singular, and That's Good

Murfet, Wei, Gong, Li, Gell-Redman and Quella (2020) — ARXIV-2010.11560. Read as NOTE-286.

Key takeaways

• The distinction the whole school rests on. A model is regular if the map w ↦ p(y|x,w) is one-to-one and the Fisher information matrix is positive definite; otherwise it is strictly singular. Neural networks have been known to be singular since Amari et al. (2003) and Watanabe (2007): the set of weights equivalent to the truth under KL divergence is a real analytic variety that fails to be a manifold.
• BIC is derived from an approximation that does not hold here. BIC = n L_n(w_MLE) + (d/2) log n comes from the Laplace approximation, which applies to regular models. Watanabe (2013)‘s correct criterion, for regular and singular models alike, is n L_n(w_0) + λ log n with λ the RLCT — and since λ may be far below d/2, a network can have high marginal likelihood, which BIC says it cannot.
• λ is a volume codimension, not a curvature. For the volume of almost-true parameters V(t, v_0) = ∫_{K(w)<t} φ(w) dw, one has V(t, v_0) = c t^λ + o(t^λ). In the minimally singular case K(w) = Σᵢ cᵢ wᵢ² over d′ < d coordinates, λ = d′/2: there are d − d′ directions in which the parameter can move without changing the model at all. The constants cᵢ — the curvature — do not enter. For strictly singular models 2λ need not even be an integer.
• The generalisation result. E_n G(n) = λ/n + o(1/n) when q̂_n is the Bayes predictive distribution (Watanabe 2009, Thm 1.2 and 7.2), against E_n G(n) = C/n + o(1/n) for MAP or MLE, where C is the maximum of a Gaussian process. Regular models have λ = C = d/2; singular models generally have C > λ, so the Bayes predictive distribution should be preferred. Experimentally, being Bayesian in just the final layers beats MAP, and the Laplace approximation performs poorly as well as being inappropriate.
• λ depends on the truth, not only the model. In regular models the effective parameter count is a function of the (model, prior) pair; the RLCT is a function of the (model, truth, prior) triplet, and rises as the true distribution grows complex relative to the model. Verified on ReLU and SiLU families.

Standing in the anthology

Active, and it closes a citation the record could not resolve. LIT-476 compares its own local-volume power law against “Local Learning Coefficient results where d/2 is a strict upper bound not seen in practice”, and NOTE-225 names the LLC as “the singular-learning-theory neighbour”. The record was citing this school’s findings while holding none of its papers.

It is a position paper and says so — “an invitation to singular learning theory … and suggest important future work to make singular learning theory directly applicable to how deep learning is performed in practice”. The theorems it relies on are Watanabe’s and are not proved here; what is new is the argument for relevance plus small confirming experiments.

Its bearing on SOTA-012 is direct and is why that practice moves to v3. The sentence “we will explain why this matters more than the curvature of those directions (as measured for example by eigenvalues of the Hessian) laying bare some of the confusion over ‘flat’ minima” is a published argument that the quantity a sharpness visualisation displays is not the quantity that governs generalisation.