Net-negative effect of error-correction on the learning coefficient for Turing machines

Determine whether there exist Turing machines augmented with run-time error-correction (that correct all error syndromes of weight at most C) for which the overall introduction of error-correction reduces the local learning coefficient λ([M], q) of the average negative log-likelihood L around the code [M], despite the increase in complexity from using more transition tuples. Concretely, ascertain whether error-correction can be engineered so that the net effect—combining the tendency for added tuples to raise λ with the upper bound λ([M], q) ≤ d/(2(C+1)) for machines that correct weight-≤C errors—results in a strictly smaller λ([M], q) compared to the original machine without error-correction.

Background

The paper links the internal structure of Turing machines to the geometry of singularities in the average negative log-likelihood via singular learning theory, with the local learning coefficient λ([M], q) governing asymptotic Bayesian model comparison. In Section 6.1, the authors formalize error-correction for Turing machines as the ability to correct all error syndromes up to a specified weight C when simulated by their staged pseudo-UTM.

They prove an upper bound λ([M], q) ≤ d/(2(C+1)) on the local learning coefficient when a machine can correct errors of weight ≤ C, where d is the dimension of the parameter space of noisy codes. They also note that implementing error-correction typically increases the number of transition tuples, which tends to increase λ. The open question asks whether the trade-off can be made net negative—i.e., whether the decrease due to error-correction can outweigh the increase due to added structure—thereby reducing λ and making such machines more favored by the Bayesian posterior among classical solutions.