UQO Working Paper No. 28· v1.1· November 2026· 95 pages

Beyond One-Step Distributed Maximum Likelihood: Local Bias Cancellation, Global Synchronization, and the Price of Inexact Curvature

A joint order–round frontier, its sharpness in an exact model, and measured communication costs

Finance & Economics

Local jackknife debiasing of order r plus q Newton rounds matches the centralized MLE whenever K = o(N^(1−1/(2^(q+1) r))), but in the measured designs two Newton steps from the plain average beat the half-block jackknife with one step in 173 of 175 cells.

The exponent makes one extra local extrapolation level and one extra synchronization round exchangeable, and it is sharp for this construction in the exponential model. The frontier is a sufficient condition for fixed dimension and i.i.d. data; it does not say which protocol is the practical choice, and the paper reports that the component that is new here, a local order above one, is dominated in its own experiments. No communication lower bound is claimed.

Simon-Pierre Boucher — contact@spboucher.ai

Paper PDF (revised)Read online Code & data repository

1 − 1/(2^(q+1) r)

Exponent of the joint order–round frontier for fixed dimension: the number of workers K may grow as any power of N below it; it depends on (r, q) only through the depth 2^q r

7/8 vs 3/4

Exponent for the half-block jackknife with one Newton step against the plain average with one step; the quarter-block extrapolation with one step gives 11/12

−4θ0c^4

Limiting shift at the boundary for the half-block jackknife with one step in the exponential model, where the sufficient exponent is also necessary for this construction

173 of 175

Cells in which two Newton steps from the plain average have a smaller gap than the half-block jackknife with one step; they also fail less often and are faster in every design without injected latency except, marginally, at K = 256 small blocks

37 of 44 / 0 of 9

Baseline logistic cells, and contaminated cells, in which the analytic Cox–Snell correction with one step beats the half-block jackknife with one step; it wins in every exponential and Poisson cell

≈ 0.39 × 2^(−b)

Floor left by rounding per-worker scores to b bits, which does not fall with N: the Hessians may be quantized, the scores must be sent at full precision

15 ms

Synchronized derivative phase at p = 8 and K = 16 with a BLAS Hessian kernel (35 ms with the einsum kernel, 54 ms in the original runs); at p = 64, 0.058 s against 1.83 s. Every timing is specific to this hardware and implementation

Abstract

Distributed maximum likelihood estimation trades statistical efficiency for communication. We study a two-phase protocol in which each of K workers corrects the bias of its local maximum likelihood estimator by an r-th order split-sample extrapolation over nested sub-blocks, the coordinator averages the corrected estimates, and q globally synchronized Newton corrections follow. For fixed-dimensional regular likelihoods with N observations we prove a joint order–round frontier: first-order equivalence to the centralized estimator holds whenever K = o(N^(1−1/(2^(q+1) r))), so that one extra local extrapolation level and one extra synchronization round are exchangeable in the exponent; the half-block jackknife with one Newton step tolerates K = o(N^(7/8)) partitions, against K = o(N^(3/4)) for the plain average with one step. In the exponential model the exponent is sharp for the construction, with limiting shift −θ0(βr c^r)^(2^q) when K ∼ cN^α(r,q). A perturbation theorem for inexact curvature recovers the known single-machine-Hessian frontiers and gives the number of bits to which the per-worker Hessians may be quantized at each local order; the scores must be sent at full precision. Paired Monte Carlo experiments (N up to 2^22), an exact-model boundary study (N up to 2^32) and timed benchmarks on an Apple-silicon cluster quantify the finite-sample costs: two Newton steps from the plain average have the smaller gap than the half-block jackknife with one step in 173 of 175 cells, fail less often and are faster in every design without injected latency except, marginally, at K = 256 small blocks; the analytic Cox–Snell correction with one step beats it in every exponential and Poisson cell and in 37 of 44 baseline logistic cells, but in none of the 9 contaminated cells, where its bias formula is wrong; the regimes in which the jackknife is cheaper (reply latency of tens to hundreds of milliseconds, many small blocks) never meet the accuracy criterion. The components are known; the joint frontier, its sharpness and the Hessian-channel bit requirement are new. No communication lower bound is claimed.

JEL: C13, C15, C55, C63

Key Results

A joint order–round frontier

Under a uniform stochastic expansion with a bias expansion of order r and local Newton regularity, first-order equivalence to the centralized MLE holds whenever K = o(N^α(r,q)). At q = 1 the theorem is the composition of the order-r pilot rate with the classical one-step argument; its content is the joint indexing by (r, q) and the centered random remainder that the proof keeps.

Exact analysis and sharpness in the exponential model

Exact bias and variance of the corrected local estimators, an exact Newton error identity, the basin of attraction of the Newton map, and the boundary behaviour. Simulations confirm the limit for (1, 1) and (2, 0) and verify the exact finite-m identity along the boundary for (2, 1) and (3, 1).

The price of inexact curvature

With stale averaged local Hessians the exponent becomes 1 − 1/(2(r + q)), with a single-machine Hessian 1 − 1/(2r + q), which reproduces known frontiers for r = 1. Relative quantization of the per-worker Hessians costs nothing asymptotically above a bit count that falls with the local order; no fixed number of bits suffices for the score channel.

A negative finding reported as a result

In the designs measured, a local order above one is dominated by two Newton steps from the plain average and by the analytic correction with one step; the regimes in which the jackknife is cheaper never meet the accuracy criterion.

Failure-aware evidence and measured costs

The experiments report the fraction of local fits that fail, interval coverage, transmitted bytes per round and the measured exchange rate between local fits and synchronization rounds; the finite-sample variance cost of higher-order corrections reverses the asymptotic ranking at moderate block sizes.

Quality control

This paper went through a multi-stage quality control: 7 quality-control reports and 86 points checked. The version presented on this page is the revised manuscript. Scores are on a 0–100 scale and are a diagnostic of the manuscript, not an acceptance probability.

Score after revision

72.1/100

63.4 before quality control · +8.7

Quality control dated November 7, 2026

Quality-control decision: major revision
Scientific integrity: PASS
Reproducibility: R1reproduced from documented analysis data
Maturity: Bone revision cycle from submission
Quality-control reports: 7
Points checked: 86

Scores by dimension

Before quality controlAfter revision

Originality & contribution57.0 → 58.0
Scientific rigor66.3 → 76.0
Identification & methodology57.9 → 72.0
Data & reproducibility71.6 → 82.0
Robustness & validation65.1 → 74.0
Writing70.0 → 72.0
Figures & tables55.3 → 72.0
Literature positioning72.9 → 80.0

Revised manuscript95 pages8 figures33 tables37 equations43 references46,433 words

What quality control changed

Quality control kept the frontier theorem and narrowed what surrounds it. The quantization result now covers the Hessian channel only, sharpness is confirmed by simulation for two cases and verified through an exact identity for the others, and failure-aware experiments led the paper to report that its jackknife component is dominated in practice by two plain Newton steps.

Baseline for the plain average with one step: K = O(N^(1/2)) → K = o(N^(3/4)), the exponent already implied by the earlier literature's bound; the 7/8 exponent of the half-block jackknife with one step is unchanged
"Transmitted derivatives may be quantized" → only the per-worker Hessians may be quantized; the scores must be sent at full precision, since per-worker score rounding leaves a floor of about 0.39 × 2^(−b) that does not fall with N
Sharpness "shown" by the boundary study → the limit is confirmed by simulation for (1, 1) and (2, 0); for (2, 1) and (3, 1) the exact finite-m identity is verified along the boundary to N = 2^32
Added as a result: two Newton steps from the plain average have a smaller gap than the half-block jackknife with one step in 173 of 175 cells, and the analytic correction with one step beats it in every exponential and Poisson cell
"One more round is cheaper than one more local level" from a single kernel → benchmarks re-run with a BLAS kernel on 15 hosts: the derivative phase at p = 64 falls from 1.83 s to 0.058 s, and the jackknife is cheaper only with injected latency or many small blocks, regimes that never meet the accuracy criterion
Theorems restated for the truncated local estimator; the Lean count is qualified as algebraic identities only

Limitations stated in the paper

All exponents are for fixed parameter dimension and i.i.d. data; the results do not transfer to growing dimension without a new analysis.
The frontier orders estimators by the exponent of the bias term alone; at moderate block sizes the random remainder makes the order-3 pilot worse than the order-2 pilot, so doubling the local order to save a round is not free statistically.
Nothing is a minimax or communication lower bound, and the sharpness concerns this estimator family, not all distributed procedures; in exponential families the pooled sufficient statistic recovers the centralized estimator exactly in one round.
Measured times are specific to one cluster, two Hessian kernels, one implementation and K ≤ 256 worker processes; the bit count for Hessian quantization should be read as a rate, not as a number of bits for a given N.

Read online (interactive reader)Revised PDF Revised branch on spbgit

Data

Paired Monte Carlo designs

Exponential model with N from 2^14 to 2^22 and block sizes m from 8 to 1024, 2000 replications per cell up to N = 2^20 and 500 at N = 2^22; logistic and Poisson regressions with 1000 replications in most cells, plus a stress design with p = 20 and robustness designs with misspecification, contamination and very small blocks.

Exact-model boundary study and ablations

Exponential-model runs along the boundary with N up to 2^32, and ablations over split patterns, Newton rounds, curvature and per-worker derivative quantization.

Cluster benchmarks

The coordinator–worker protocol run as written over TCP on a wired local network of Apple-silicon Macs: original runs (75 runs over 23 designs, up to 16 hosts) and quality-control runs (329 runs over 42 designs, five repeats each, 15 hosts) with two Hessian kernels, injected reply latencies and an emulation of the near-frontier regime. Each worker simulates its own block of a logistic regression with N = 2^22 observations, so no data move.

Methodology

Generalized split-sample jackknife

Each worker computes the MLE on its block and on nested sub-blocks and combines them with Richardson-type weights that cancel the first r − 1 terms of the bias expansion while preserving the first-order influence function exactly.

Aggregated Newton synchronization

The coordinator averages the corrected estimates, broadcasts the average and applies q Newton corrections from aggregated scores and Hessians; variants use stale, single-machine or quantized curvature.

Theorems for the truncated estimator

The assumptions are stated for the truncated local estimator, so the logistic and Poisson designs are covered on the event that every sub-block fit exists; cells where that event is rare are outside the theorems and are reported with their failure fractions.

Machine-checked algebra

27 Lean 4 / Mathlib theorems formalize the algebraic identities only (weights and constants, exact biases, the Newton error identity, the boundary constant, the exponent algebra, the exact variance). No probabilistic step is formalized; those rest on the written proofs, with moment formulas verified symbolically.

Cost model calibrated on measurements

A wall-clock model of local fits and synchronization rounds is calibrated on the benchmark timings; it shows on which side of the exchange condition the designs fall and does not establish a general speed advantage of any protocol.

Reproducibility

Every Monte Carlo and benchmark number in the tables, figures and prose is regenerated from committed raw outputs by scripts; none is typed by hand.

Every raw file has a manifest with the code commit, the Python and NumPy versions, the host, the wall time and the SHA-256 of the output; a manifest table collects the 261 rows, and a test asserts that the processed summary is reproduced from the raw files.

The prior exploratory note is kept frozen with checksums; its Monte Carlo was re-run and matched to 5 × 10⁻¹³ before any change was made.

Across computing environments the order-1 and order-2 estimators agree to 10⁻¹⁰ relative in every failure-free cell; order-3 estimators on sub-blocks of eight observations differ by up to 10⁻², and diverged replications differ arbitrarily.

A script checks the 27 Lean theorems; the data are simulated from seeds, so nothing is withheld.

Cite this paper

Boucher, S.-P. (2026). Beyond One-Step Distributed Maximum Likelihood: Local Bias Cancellation, Global Synchronization, and the Price of Inexact Curvature. UQO Working Paper No. 28, Université du Québec en Outaouais.

@techreport{boucher2026wp28,
  author      = {Simon-Pierre Boucher},
  title       = {Beyond One-Step Distributed Maximum Likelihood: Local Bias Cancellation, Global Synchronization, and the Price of Inexact Curvature},
  institution = {Université du Québec en Outaouais, Département des sciences administratives},
  number      = {28},
  type        = {UQO Working Paper},
  year        = {2026},
  month       = {nov},
  url         = {https://www.spboucher.ai/research/wp28},
}