A number calculated exactly
An arXiv preprint dated Aug. 20, 2026, presents a deterministic algorithm intended to calculate local real log canonical thresholds, or RLCTs, exactly for any two-dimensional model whose Kullback–Leibler distance is contact equivalent to a polynomial. The RLCT is the local learning coefficient the paper tracks at a specified point. In the theorem stated by the authors, Algorithm 1 computes the RLCT at the origin for every rational bivariate polynomial—a polynomial in two variables with rational-number coefficients.
At the center of the method is a geometric and algebraic Newton-polygon procedure for the local RLCT at the origin. In practical terms, it works from the structure of a polynomial rather than producing an estimate from a sampling run. The input class is written f in Q[x,y], meaning bivariate polynomials with rational coefficients, and the paper states that the procedure returns the exact RLCT for that class.
The authors report an upper bound on the number of algorithmic steps that is quadratic in the polynomial’s degree. They leave open whether that bound is tight, so the paper gives a worst-case ceiling without settling the procedure’s best possible scaling.
Turning a neural-network distance into a polynomial
The neural-network example uses a polynomial neural-network regression model with two parameters, θ=(θ1,θ2), and a repeated 2×2 polynomial weight matrix. The setup is described at activation degree r=2; the benchmark table compares it with r=3 and r=4 while varying depth L from 1 to 5.
Because the PNN’s Kullback–Leibler distance is not a polynomial, the application replaces K(θ) with a contact-equivalent polynomial H(θ). Under the stated equivalence, both have the same local RLCT at the origin. That puts the application into the polynomial form required by Algorithm 1 while preserving the local quantity being calculated.
Exact calculations used SageMath on an Intel Core Ultra 7. The comparator was SGLD, the sampling-based method, run with γ=1, ε=10^-5, five chains and T=10,000 in Google Colab with two virtual CPUs. In the comparison, SGLD supplies approximate estimates, while the exact procedure supplies symbolic values.
Exact fractions, unfinished runs
For r=2, exact RLCT values declined at every listed depth: 3/4 at L=1, 1/4 at L=2, 3/28 at L=3, 1/20 at L=4 and 3/124 at L=5. These are exact symbolic outputs, not estimates with statistical intervals.
At higher activation degrees, the calculation reached fewer entries. The r=3 row gives 3/4, 3/16, 3/52 and 3/160 for L=1 through L=4, then records >1 hr at L=5. The r=4 row gives 3/4, 3/20 and 1/28 through L=3, then records >1 hr at both L=4 and L=5. The >1 hr labels are noncompletion markers, not RLCT values.
Across the supplied grid, the reported local learning coefficients decrease with both depth and activation degree. At the same time, polynomial expansion becomes the bottleneck as settings grow, and the table contains both >1 hr and NaN entries. The pattern is therefore a benchmark result for this specified construction, not a universal scaling result.
A direct comparison with sampling
SGLD returned approximate values for many of the same cells, with an absolute error in parentheses. For r=2, the estimates from L=1 to L=5 were 0.677 (0.072), 0.154 (0.095), 0.038 (0.068), 0.014 (0.035) and 0.006 (0.017). For r=3, they were 0.677 (0.072), 0.081 (0.106), 0.025 (0.033), 0.007 (0.011) and NaN; for r=4, they were 0.677 (0.072), 0.093 (0.142), 0.019 (0.017), NaN and NaN. No convergence guarantee or statistical confidence interval was reported for these runs.
At the two r=2 endpoints with timings shown, the exact calculation was faster. At L=1, its total time was 0.047 seconds versus 461 seconds for SGLD; at L=5, it was 220.2 seconds versus 581 seconds. The comparison is tied to the reported hardware and settings, so it does not establish universal runtime superiority.
The contrast is also incomplete: SGLD has missing deeper entries, while the exact table has runs that exceeded one hour. The study compares numerical estimates, exact values and wall-clock time in a symbolic benchmark; it does not provide predictive-performance evidence or show that a lower RLCT causes better prediction, generalization or model selection.
Where the result stops
The method’s reach is deliberately narrow. The authors limit Algorithm 1 to biparametric models, meaning models with two parameters, and say suitable coordinate systems for its Newton-diagram analysis remain unknown in dimensions above two. The method also assumes the relevant singular point is already known; if it is not, broader model selection would require stratifying the singular locus—working out where the different singular cases lie before applying a local calculation.
That boundary matters for interpreting the neural-network example. The exact theorem is stated for rational bivariate polynomials, and the benchmark uses a repeated-weight, two-parameter construction. Nothing in the supplied results validates the algorithm for models with more than two parameters, and the paper leaves open whether the quadratic step bound is tight.
The authors present the calculation as a possible way to calibrate sampling estimators and to support model selection in smaller models. But the reported decline in RLCT is an observed pattern in the listed configurations, not evidence that a lower coefficient causes better generalization or prediction. Nor does the work identify a global singular locus when the relevant point is unknown.
For readers assessing reproducibility, the paper says its SGLD experiments used Timaeus’s DevInterp library. The supplied text identifies no author-maintained code repository or separate supplementary file, and it contains no funding statement. Replicating the comparison would therefore require working from the stated benchmark settings and the implementation details available with the preprint.
The result is best read as a precise answer to a narrow computational problem: a deterministic route to exact local RLCTs for the stated two-dimensional polynomial class, with a symbolic neural-network demonstration that was faster than SGLD at the two reported r=2 timing points. The unanswered questions are whether the method can move beyond two parameters, whether its complexity bound can be tightened, and whether the timing and sampling comparison survives broader hardware and stronger convergence checks.
Paper data and sources
Original title: Exact Algebraic Computation of Learning Coefficients for Two-Dimensional Singular Models
Authors: Grégoire Sergeant-Perthuis, Elias Tsigaridas, Jules Tsukahara
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-20
DOI: Not available
Original paper · Full text