WP28: supplementary benchmarks (p=32, K=256), final build

Co-Authored-By: Claude Fable 5.1 <noreply@anthropic.com>

Simon-Pierre Boucher committed 5 days ago (Oct 29, 2026) parent 8a14819 Browse files

18 changed files +63 −23

modified FINAL_DELIVERY.md +1 −1

				`@@ -16,7 +16,7 @@ Known: generalized jackknife (Quenouille; Schucany–Gray–Owen; Dhaene–Jochm`
16		16
17	## Simulations actually run	17	## Simulations actually run
18	- 39 exponential cells (2000 reps; 500 at N = 2^22), 136 logistic/Poisson/robustness cells on 18 cluster nodes (1000/500 reps), 32 failure-affected cells rerun with valid-worker-only variants (500 reps for the ten N = 2^20 cells), boundary study (60 cells), ablations (8 + 6 cells), unequal sizes (10 cells); all completed; seeds = 20261028 + hash(cell); manifests in experiments/EXPERIMENT_MANIFEST.csv; standard errors in results/processed/*.csv.	18	- 39 exponential cells (2000 reps; 500 at N = 2^22), 136 logistic/Poisson/robustness cells on 18 cluster nodes (1000/500 reps), 32 failure-affected cells rerun with valid-worker-only variants (500 reps for the ten N = 2^20 cells), boundary study (60 cells), ablations (8 + 6 cells), unequal sizes (10 cells); all completed; seeds = 20261028 + hash(cell); manifests in experiments/EXPERIMENT_MANIFEST.csv; standard errors in results/processed/*.csv.
19	−- Benchmarks: 51 runs (17 designs × 3) of the real TCP protocol on MacLustr, coordinator M4M64a (Mac Studio M4 Max), up to 16 distinct hosts, K ∈ {16, 64, 128}, p ∈ {8, 64}; results/raw/benchmarks.jsonl; summary in paper §10.	19	+- Benchmarks: 75 runs (23 designs; 3 repeats each, 6 for the host-scaling endpoints) of the real TCP protocol on MacLustr, coordinator M4M64a (Mac Studio M4 Max), up to 16 distinct hosts, K ∈ {16, 64, 128, 256}, p ∈ {8, 32, 64}; results/raw/benchmarks.jsonl; summary in paper §10.
20		20
21	## Main caveats	21	## Main caveats
22	Fixed p and i.i.d. only; sufficient exponents, sharp only for the construction in the exponential model; boundary regime reached only at very large N (finite-m prediction provided); finite-sample floor (m√N)⁻¹ after bias cancellation; higher-order patterns fail at m ≲ 16p in logistic regression (failures reported, valid-worker-only variants provided); benchmarks specific to one LAN; no lower bounds, energy or cost claims; adaptive order open.	22	Fixed p and i.i.d. only; sufficient exponents, sharp only for the construction in the exponential model; boundary regime reached only at very large N (finite-m prediction provided); finite-sample floor (m√N)⁻¹ after bias cancellation; higher-order patterns fail at m ≲ 16p in logistic regression (failures reported, valid-worker-only variants provided); benchmarks specific to one LAN; no lower bounds, energy or cost claims; adaptive order open.

modified FINAL_REVIEW.md +1 −1

				`@@ -43,7 +43,7 @@ B8. "The cost model is not a theorem." (b) Proposition 5 is labelled condition`
43		43
44	## Referee C — distributed-systems / reproducibility specialist	44	## Referee C — distributed-systems / reproducibility specialist
45		45
46	−C1. "In-memory partitions are not a cluster." (a) §9 vs §10. (b) The Monte Carlo sections state that they are in-memory and measure no communication; §10 reports only runs of the TCP coordinator–worker protocol on physical hosts (51 runs, 17 designs × 3 repeats, up to 16 distinct hosts, K up to 128 worker processes), with per-stage wall-clock, per-worker compute times, latency percentiles and bytes. (d) none.	46	+C1. "In-memory partitions are not a cluster." (a) §9 vs §10. (b) The Monte Carlo sections state that they are in-memory and measure no communication; §10 reports only runs of the TCP coordinator–worker protocol on physical hosts (75 runs, 23 designs, up to 16 distinct hosts, K up to 256 worker processes), with per-stage wall-clock, per-worker compute times, latency percentiles and bytes. (d) none.
47		47
48	C2. "Two communication phases are not one message." (b) §3.2 counts phases, messages, numbers and bytes separately (Table 2); the measured bytes include frame headers and pickle overhead. (c) tests/test_communication_accounting.py. (d) The stale variant ships the local Hessian as a full p×p array rather than its upper triangle, so its measured phase-1 bytes exceed the Table-2 count by a factor ≈ 2 for the Hessian part; noted in the table caption, not corrected in the runs.	48	C2. "Two communication phases are not one message." (b) §3.2 counts phases, messages, numbers and bytes separately (Table 2); the measured bytes include frame headers and pickle overhead. (c) tests/test_communication_accounting.py. (d) The stale variant ships the local Hessian as a full p×p array rather than its upper triangle, so its measured phase-1 bytes exceed the Table-2 count by a factor ≈ 2 for the Hessian part; noted in the table caption, not corrected in the runs.
49		49

modified LIMITATIONS_AND_CLAIMS.md +1 −1

				`@@ -14,7 +14,7 @@`
14	- The finite-m prediction (exact bias + variance of the local remainder) tracks the boundary Monte Carlo means to three digits; convergence of the mean gap to the asymptotic limit is slow because m ≍ N^{1/(2^{q+1}r)}.	14	- The finite-m prediction (exact bias + variance of the local remainder) tracks the boundary Monte Carlo means to three digits; convergence of the mean gap to the asymptotic limit is slow because m ≍ N^{1/(2^{q+1}r)}.
15	- Coverage: nominal inside the frontier; under-coverage of the plain average and of the one-step estimator at small m; Fisher-based intervals under-cover under misspecification while sandwich intervals do not.	15	- Coverage: nominal inside the frontier; under-coverage of the plain average and of the one-step estimator at small m; Fisher-based intervals under-cover under misspecification while sandwich intervals do not.
16	- Quantization: the Monte Carlo ablation shows the number of mantissa bits at which the quantized step becomes indistinguishable from the exact step at finite N (paper Table 6).	16	- Quantization: the Monte Carlo ablation shows the number of mantissa bits at which the quantized step becomes indistinguishable from the exact step at finite N (paper Table 6).
17	−- Benchmarks (51 runs, 17 designs × 3 repeats, up to 16 distinct hosts, K ≤ 128 worker processes, p = 8 and 64): the synchronized derivative phase costs 55 ms (p = 8) to 1.8 s (p = 64) and is dominated by the slowest worker's Hessian computation, not by the network (round trip ≈ 0.2 ms, uplink ≤ 570 kB); the local phase dominates and grows with the order (0.26 s for one fit → 0.96 s for order 4 at p = 8; 14.8 s → 29.5 s at p = 64); at equal exponent 7/8, (1,2) is faster than (2,1), (3,1) and (4,0) on this LAN, and stale/single-machine curvature saves nothing. These are measurements on one cluster and one implementation; they calibrate the cost model and establish no general speed advantage.	17	+- Benchmarks (75 runs, 23 designs, up to 16 distinct hosts, K ≤ 256 worker processes, p ∈ {8, 32, 64}): the synchronized derivative phase costs 55 ms (p = 8) to 1.8 s (p = 64) and is dominated by the slowest worker's Hessian computation, not by the network (round trip ≈ 0.2 ms, uplink ≤ 570 kB); the local phase dominates and grows with the order (0.26 s for one fit → 0.96 s for order 4 at p = 8; 14.8 s → 29.5 s at p = 64); at equal exponent 7/8, (1,2) is faster than (2,1), (3,1) and (4,0) on this LAN, and stale/single-machine curvature saves nothing. These are measurements on one cluster and one implementation; they calibrate the cost model and establish no general speed advantage.
18		18
19	## Heuristics and conditional statements	19	## Heuristics and conditional statements
20	- Proposition 5 (cost model and exchange rate) is a tautology given the cost model (eq. 12 of the paper); its content is the calibration on one cluster.	20	- Proposition 5 (cost model and exchange rate) is a tautology given the cost model (eq. 12 of the paper); its content is the calibration on one cluster.

modified PHASE_STATUS.md +1 −1

				`@@ -67,7 +67,7 @@ Decision: report RMS and median gaps, failure fractions and valid-worker-only ro`
67		67
68	Phase: 8 — Real communication and compute benchmarks	68	Phase: 8 — Real communication and compute benchmarks
69	Start/finish timestamp: 2026-10-29 00:05 → 01:10	69	Start/finish timestamp: 2026-10-29 00:05 → 01:10
70	−What was actually done: 17 designs × 3 repeats of the TCP protocol on MacLustr (coordinator M4M64a; workers on up to 16 distinct hosts; K = 16, 64, 128; p = 8 and 64; exact/stale/single curvature; (r,q) ∈ {(4,0),(3,1),(2,1),(1,2)}); per-stage wall-clock, worker times, latency percentiles, bytes logged (results/raw/benchmarks.jsonl). Fixes along the way: workers must stay children of the ssh session; JSON serialization of NumPy types; M4M64b blocked by macOS local-network privacy → coordinator host used as 16th worker host.	70	+What was actually done: 23 designs (75 runs) of the TCP protocol on MacLustr (supplementary p = 32 and K = 256 designs added 2026-10-29 15:12 at the author's request; K = 256 needed a raised file-descriptor limit) (coordinator M4M64a; workers on up to 16 distinct hosts; K = 16, 64, 128; p = 8 and 64; exact/stale/single curvature; (r,q) ∈ {(4,0),(3,1),(2,1),(1,2)}); per-stage wall-clock, worker times, latency percentiles, bytes logged (results/raw/benchmarks.jsonl). Fixes along the way: workers must stay children of the ssh session; JSON serialization of NumPy types; M4M64b blocked by macOS local-network privacy → coordinator host used as 16th worker host.
71	Artifacts: results/raw/benchmarks.jsonl, results/processed/benchmarks_summary.csv, paper/tables/bench.tex, results/figures/fig_bench_exchange.pdf.	71	Artifacts: results/raw/benchmarks.jsonl, results/processed/benchmarks_summary.csv, paper/tables/bench.tex, results/figures/fig_bench_exchange.pdf.
72	Tests passed: tests/test_protocol.py (laptop and M3U96a).	72	Tests passed: tests/test_protocol.py (laptop and M3U96a).
73	Decision: §10 reports measured quantities only; exchange rate favours rounds on this LAN; no general speed claim.	73	Decision: §10 reports measured quantities only; exchange rate favours rounds on this LAN; no general speed claim.

modified README.md +1 −1

				`@@ -45,7 +45,7 @@ Main results: Theorem 1 (frontier $K = o(N^{1-1/(2^{q+1}r)})$ with the centered`
45	\| Inexact curvature \| stale: $1-1/(2(r+q))$; single machine: $1-1/(2r+q)$; bits for quantized derivatives: $(\tfrac12 - r(1-\alpha))\log_2 N$ \|	45	\| Inexact curvature \| stale: $1-1/(2(r+q))$; single machine: $1-1/(2r+q)$; bits for quantized derivatives: $(\tfrac12 - r(1-\alpha))\log_2 N$ \|
46	\| Exact variance cost of the half-block correction \| $\mathrm{Var}\,J_2 = \theta_0^2 m^2(m^2-6m+17)/((m-4)(m-2)^2(m-1)^2) = \theta_0^2(m^{-1}+4m^{-2}+O(m^{-3}))$, same first two terms as the raw MLE \|	46	\| Exact variance cost of the half-block correction \| $\mathrm{Var}\,J_2 = \theta_0^2 m^2(m^2-6m+17)/((m-4)(m-2)^2(m-1)^2) = \theta_0^2(m^{-1}+4m^{-2}+O(m^{-3}))$, same first two terms as the raw MLE \|
47	\| Finite samples \| (1,2) beats (2,1) in every cell; order 3 beats order 2 only at the smallest m; once the bias is cancelled the one-step gap decays like (m√N)⁻¹ (squared random remainder); quarter-block fits fail at m ≲ 16p (logistic) \|	47	\| Finite samples \| (1,2) beats (2,1) in every cell; order 3 beats order 2 only at the smallest m; once the bias is cancelled the one-step gap decays like (m√N)⁻¹ (squared random remainder); quarter-block fits fail at m ≲ 16p (logistic) \|
48	−\| Benchmarks (MacLustr, 17 designs × 3, ≤ 16 hosts, K ≤ 128) \| derivative phase 55 ms (p = 8) – 1.8 s (p = 64), dominated by the slowest worker, not the network; local phase 0.26 → 0.96 s from order 1 to 4 (p = 8); on this LAN one more round is cheaper than one more local level \|	48	+\| Benchmarks (MacLustr, 23 designs, 75 runs, ≤ 16 hosts, K ≤ 256) \| derivative phase 55 ms (p = 8) – 1.8 s (p = 64), dominated by the slowest worker, not the network; local phase 0.26 → 0.96 s from order 1 to 4 (p = 8); on this LAN one more round is cheaper than one more local level \|
49	\| Machine-checked algebra \| 27 Lean 4 / Mathlib theorems, no `sorry` (`theory/lean/`) \|	49	\| Machine-checked algebra \| 27 Lean 4 / Mathlib theorems, no `sorry` (`theory/lean/`) \|
50		50
51	## 🗂 Repository layout	51	## 🗂 Repository layout

modified experiments/fig_benchmarks.py +6 −3

				`@@ -7,9 +7,12 @@ import matplotlib; matplotlib.use("Agg"); import matplotlib.pyplot as plt`
7	sys.path.insert(0, str(pathlib.Path(__file__).resolve().parents[1] / "experiments")); from process_results import _setup, PAL, INK, INK2, FIG, PROC	7	sys.path.insert(0, str(pathlib.Path(__file__).resolve().parents[1] / "experiments")); from process_results import _setup, PAL, INK, INK2, FIG, PROC
8	_setup(); g = pd.read_csv(PROC / "benchmarks_summary.csv")	8	_setup(); g = pd.read_csv(PROC / "benchmarks_summary.csv")
9	order8 = [("cmp_r4q0", "(4,0)"), ("cmp_r3q1", "(3,1)"), ("hosts16", "(2,1)"), ("cmp_r2q1_stale", "(2,1) stale"), ("cmp_r2q1_single", "(2,1) single"), ("cmp_r1q2", "(1,2)")]	9	order8 = [("cmp_r4q0", "(4,0)"), ("cmp_r3q1", "(3,1)"), ("hosts16", "(2,1)"), ("cmp_r2q1_stale", "(2,1) stale"), ("cmp_r2q1_single", "(2,1) single"), ("cmp_r1q2", "(1,2)")]
		10	+order32 = [("p32_r3q1", "(3,1)"), ("p32_r2q1", "(2,1)"), ("p32_r2q1_stale", "(2,1) stale"), ("p32_r1q2", "(1,2)")]
10	order64 = [("cmp_p64_r2q1", "(2,1)"), ("cmp_p64_r2q1_stale", "(2,1) stale"), ("cmp_p64_r1q2", "(1,2)")]	11	order64 = [("cmp_p64_r2q1", "(2,1)"), ("cmp_p64_r2q1_stale", "(2,1) stale"), ("cmp_p64_r1q2", "(1,2)")]
11	−fig, axes = plt.subplots(1, 3, figsize=(9.6, 3.0), gridspec_kw={"width_ratios": [2.2, 1.3, 1.6]})	12	+has32 = g.label.isin([k for k, _ in order32]).any()
12	−for ax, order, title in zip(axes[:2], (order8, order64), ("p = 8, 16 hosts, K = 16", "p = 64, 16 hosts, K = 16")):	13	+panels = [(order8, "p = 8, 16 hosts, K = 16")] + ([(order32, "p = 32, 16 hosts, K = 16")] if has32 else []) + [(order64, "p = 64, 16 hosts, K = 16")]
		14	+fig, axes = plt.subplots(1, len(panels) + 1, figsize=(2.4 * len(panels) + 4.4, 3.0), gridspec_kw={"width_ratios": [2.2] + [1.5] * (len(panels) - 1) + [1.7]})
		15	+for ax, (order, title) in zip(axes[:-1], panels):
13	labs, loc, rnd = [], [], []	16	labs, loc, rnd = [], [], []
14	for key, lab in order:	17	for key, lab in order:
15	r = g[g.label == key]	18	r = g[g.label == key]
				`@@ -24,7 +27,7 @@ for ax, order, title in zip(axes[:2], (order8, order64), ("p = 8, 16 hosts, K =`
24	ax.set_xticks(x); ax.set_xticklabels(labs, rotation=20, ha="right", fontsize=7.5); ax.set_ylabel("wall-clock (s)"); ax.set_title(title, fontsize=8, loc="left")	27	ax.set_xticks(x); ax.set_xticklabels(labs, rotation=20, ha="right", fontsize=7.5); ax.set_ylabel("wall-clock (s)"); ax.set_title(title, fontsize=8, loc="left")
25	ax.set_ylim(0, 1.18 * max(a + b for a, b in zip(loc, rnd)))	28	ax.set_ylim(0, 1.18 * max(a + b for a, b in zip(loc, rnd)))
26	axes[0].legend(fontsize=7, loc="upper center", bbox_to_anchor=(0.5, -0.32), ncol=1)	29	axes[0].legend(fontsize=7, loc="upper center", bbox_to_anchor=(0.5, -0.32), ncol=1)
27	−ax = axes[2]; h = g[g.label.str.startswith("hosts")].sort_values("hosts")	30	+ax = axes[-1]; h = g[g.label.str.startswith("hosts")].sort_values("hosts")
28	ax.plot(h.hosts, h.t_local, marker="o", color=PAL["blue"], label="local phase", markerfacecolor="white", markeredgewidth=1.2)	31	ax.plot(h.hosts, h.t_local, marker="o", color=PAL["blue"], label="local phase", markerfacecolor="white", markeredgewidth=1.2)
29	ax.plot(h.hosts, h.fit_mean, marker="s", color=PAL["aqua"], ls="--", label="mean worker fit", markerfacecolor="white", markeredgewidth=1.2)	32	ax.plot(h.hosts, h.fit_mean, marker="s", color=PAL["aqua"], ls="--", label="mean worker fit", markerfacecolor="white", markeredgewidth=1.2)
30	ax.plot(h.hosts, h.t_rounds, marker="^", color=PAL["orange"], label="derivative phase", markerfacecolor="white", markeredgewidth=1.2)	33	ax.plot(h.hosts, h.t_rounds, marker="^", color=PAL["orange"], label="derivative phase", markerfacecolor="white", markeredgewidth=1.2)

modified experiments/make_tables.py +1 −1

				`@@ -278,7 +278,7 @@ def main():`
278	"benchRoundsPsixtyfourOneTwo": f"{G('cmp_p64_r1q2','t_rounds'):.2f}", "benchDerMaxPsixtyfour": f"{G('cmp_p64_r2q1','der_max'):.2f}", "benchUpPsixtyfourTwoOne": f"{G('cmp_p64_r2q1','up_B')/1e3:.0f}",	278	"benchRoundsPsixtyfourOneTwo": f"{G('cmp_p64_r1q2','t_rounds'):.2f}", "benchDerMaxPsixtyfour": f"{G('cmp_p64_r2q1','der_max'):.2f}", "benchUpPsixtyfourTwoOne": f"{G('cmp_p64_r2q1','up_B')/1e3:.0f}",
279	"benchUpTwoOne": f"{G('hosts16','up_B')/1e3:.1f}", "benchLocalHostsOne": f"{G('hosts1','t_local'):.2f}", "benchLocalHostsSixteen": f"{G('hosts16','t_local'):.2f}", "benchFitMeanHostsSixteen": f"{G('hosts16','fit_mean'):.2f}",	279	"benchUpTwoOne": f"{G('hosts16','up_B')/1e3:.1f}", "benchLocalHostsOne": f"{G('hosts1','t_local'):.2f}", "benchLocalHostsSixteen": f"{G('hosts16','t_local'):.2f}", "benchFitMeanHostsSixteen": f"{G('hosts16','fit_mean'):.2f}",
280	"benchRoundHostsOne": f"{1000G('hosts1','t_rounds'):.0f}", "benchLatencyExcess": f"{1000(G('hosts16','lat99') - G('hosts16','der_max')):.1f}", "benchLocalKonetwentyeight": f"{G('K128_r2q1','t_local'):.2f}", "benchRoundKonetwentyeight": f"{1000*G('K128_r2q1','t_rounds'):.0f}",	280	"benchRoundHostsOne": f"{1000G('hosts1','t_rounds'):.0f}", "benchLatencyExcess": f"{1000(G('hosts16','lat99') - G('hosts16','der_max')):.1f}", "benchLocalKonetwentyeight": f"{G('K128_r2q1','t_local'):.2f}", "benchRoundKonetwentyeight": f"{1000*G('K128_r2q1','t_rounds'):.0f}",
281	− "benchConnect": f"{G('hosts16','t_connect'):.1f}", "benchRuns": str(int(g.reps.sum())), "benchDerMaxPeight": f"{1000*G('hosts16','der_max'):.0f}"})	281	+ "benchConnect": f"{G('hosts16','t_connect'):.1f}", "benchLocalPthirtytwoTwoOne": f"{G('p32_r2q1','t_local'):.2f}", "benchRoundPthirtytwoTwoOne": f"{G('p32_r2q1','t_rounds'):.2f}", "benchLocalPthirtytwoOneTwo": f"{G('p32_r1q2','t_local'):.2f}", "benchRoundsPthirtytwoOneTwo": f"{G('p32_r1q2','t_rounds'):.2f}", "benchLocalPthirtytwoThreeOne": f"{G('p32_r3q1','t_local'):.2f}", "benchLocalKtwofiftysix": f"{G('K256_r2q1','t_local'):.2f}", "benchRoundKtwofiftysix": f"{1000G('K256_r2q1','t_rounds'):.0f}", "benchLat99Ktwofiftysix": f"{1000G('K256_r2q1','lat99'):.0f}", "benchRuns": str(int(g.reps.sum())), "benchDerMaxPeight": f"{1000*G('hosts16','der_max'):.0f}"})
282	with open(TAB / "numbers.tex", "w") as f:	282	with open(TAB / "numbers.tex", "w") as f:
283	f.write("% Auto-generated by experiments/make_tables.py\n")	283	f.write("% Auto-generated by experiments/make_tables.py\n")
284	for k, v in macros.items():	284	for k, v in macros.items():

modified paper/figures/fig_bench_exchange.pdf +0 −0

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modified paper/sections/benchmarks.tex +4 −4

				`@@ -1,19 +1,19 @@`
1	\section{Measured distributed benchmarks}\label{sec:bench}	1	\section{Measured distributed benchmarks}\label{sec:bench}
2		2
3	\subsection{Setup}	3	\subsection{Setup}
4	−The protocol of \cref{alg:protocol} was run as written---a coordinator process and $K$ worker processes exchanging length-prefixed messages over TCP---on the MacLustr cluster of Apple-silicon Macs connected by a wired local network (mixed 1 and 10~GbE through one switch), with the coordinator on a Mac Studio (M4 Max, 16 cores, 64 GB) and workers on up to sixteen distinct hosts (Mac Studio M3 Ultra, M2 Ultra, M4 Max, M5 Max; Mac mini M4, M4 Pro, M5 Pro, M6; MacBook Pro M4 Max; the coordinator host also hosts one worker in the sixteen-host designs). Workers run Python 3.9 or 3.14 with NumPy 2.0--2.5; each simulates its own block of a logistic regression with $N = 2^{22}$ observations from a seed, so that no data move. The coordinator logs the wall-clock of each stage, every worker's local fit and derivative times, the reply latency of every worker in every phase (so that the slowest worker and the 99th percentile are known), and the bytes of every message including frame headers and serialization. Every design was repeated three times (\benchRuns{} runs in total); \cref{tab:bench} reports medians and \cref{fig:bench} the decomposition of the elapsed time. Worker start-up over \texttt{ssh} and the connection phase (about \benchConnect{} seconds) are excluded from the protocol times and reported separately.	4	+The protocol of \cref{alg:protocol} was run as written---a coordinator process and $K$ worker processes exchanging length-prefixed messages over TCP---on the MacLustr cluster of Apple-silicon Macs connected by a wired local network (mixed 1 and 10~GbE through one switch), with the coordinator on a Mac Studio (M4 Max, 16 cores, 64 GB) and workers on up to sixteen distinct hosts (Mac Studio M3 Ultra, M2 Ultra, M4 Max, M5 Max; Mac mini M4, M4 Pro, M5 Pro, M6; MacBook Pro M4 Max; the coordinator host also hosts one worker in the sixteen-host designs). Workers run Python 3.9 or 3.14 with NumPy 2.0--2.5; each simulates its own block of a logistic regression with $N = 2^{22}$ observations from a seed, so that no data move. The coordinator logs the wall-clock of each stage, every worker's local fit and derivative times, the reply latency of every worker in every phase (so that the slowest worker and the 99th percentile are known), and the bytes of every message including frame headers and serialization. Every design was repeated three times, the host-scaling endpoints six times (\benchRuns{} runs in total); \cref{tab:bench} reports medians and \cref{fig:bench} the decomposition of the elapsed time. Worker start-up over \texttt{ssh} and the connection phase (about \benchConnect{} seconds) are excluded from the protocol times and reported separately.
5		5
6	\begin{figure}[t]	6	\begin{figure}[t]
7	\centering	7	\centering
8	\includegraphics[width=\textwidth]{fig_bench_exchange.pdf}	8	\includegraphics[width=\textwidth]{fig_bench_exchange.pdf}
9	−\caption{Measured elapsed time of the protocol by variant (medians over three repeats). Left and centre: local phase (slowest worker) and synchronized derivative phases at equal exponent $7/8$ for $(4,0)$, $(3,1)$, $(2,1)$, $(1,2)$ and the stale and single-machine curvature variants of $(2,1)$, logistic regression with $p = 8$ and $p = 64$, $N = 2^{22}$, $K = 16$ workers on 16 hosts. Right: $(2,1)$ with $K = 16$ workers spread over 1, 2, 4, 8 and 16 hosts.}\label{fig:bench}	9	+\caption{Measured elapsed time of the protocol by variant (medians over three repeats). First three panels: local phase (slowest worker) and synchronized derivative phases at equal exponent $7/8$ for $(4,0)$, $(3,1)$, $(2,1)$, $(1,2)$ and the stale and single-machine curvature variants of $(2,1)$, logistic regression with $p = 8$, $32$ and $64$, $N = 2^{22}$, $K = 16$ workers on 16 hosts. Right: $(2,1)$ with $K = 16$ workers spread over 1, 2, 4, 8 and 16 hosts.}\label{fig:bench}
10	\end{figure}	10	\end{figure}
11		11
12	\subsection{What the measurements show}	12	\subsection{What the measurements show}
13	−Three facts emerge, all specific to this hardware and implementation. First, the synchronized derivative phase is cheap: with $p = 8$ it takes \benchRoundTwoOne{} ms end to end on sixteen hosts, of which the slowest worker's score-and-Hessian evaluation accounts for \benchDerMaxPeight{} ms and the network round trip for about \benchLatencyExcess{} ms; the uplink of \benchUpTwoOne{} kB is irrelevant at these sizes. With $p = 64$ the phase takes \benchRoundPsixtyfourTwoOne{} s, almost all of it the slowest worker's computation of the $64 \times 64$ Hessian on $2^{18}$ observations (\benchDerMaxPsixtyfour{} s), with an uplink of \benchUpPsixtyfourTwoOne{} kB. Second, the local phase dominates and grows with the order: at $p = 8$ the slowest worker needs \benchLocalOneTwo{} s for one fit, \benchLocalTwoOne{} s for the three fits of the half-block pattern, \benchLocalThreeOne{} s for the seven fits of the quarter-block pattern and \benchLocalFourZero{} s for the fifteen fits of order four; at $p = 64$ the half-block pattern costs \benchLocalPsixtyfourTwoOne{} s against \benchLocalPsixtyfourOneTwo{} s for one fit. Third, the exchange rate of \cref{prop:cost} therefore favours rounds on this network: at equal exponent $7/8$, $(1,2)$ is the fastest protocol at both dimensions ($\benchLocalOneTwo + \benchRoundsOneTwoS$ s against $\benchLocalTwoOne + \benchRoundTwoOneS$ s for $(2,1)$ at $p = 8$; $\benchLocalPsixtyfourOneTwo + \benchRoundsPsixtyfourOneTwo$ s against $\benchLocalPsixtyfourTwoOne + \benchRoundPsixtyfourTwoOne$ s at $p = 64$), and it is also the more accurate one (\cref{sec:mc}). The stale and single-machine variants save nothing here, because the derivative phase they shorten was already short; their statistical cost (\cref{thm:inexact}) is paid for no gain. In terms of \eqref{eq:rule}, the measured $\lambda + T_{\mathrm{der}} + Kb(p + p(p+1)/2)/B + T_{\mathrm{solve}}$ is two to ten times smaller than $r\,c_1 m$ at every $(r, p)$ tested, so the inequality holds and one more round beats one more level of local fits.	13	+Three facts emerge, all specific to this hardware and implementation. First, the synchronized derivative phase is cheap: with $p = 8$ it takes \benchRoundTwoOne{} ms end to end on sixteen hosts, of which the slowest worker's score-and-Hessian evaluation accounts for \benchDerMaxPeight{} ms and the network round trip for about \benchLatencyExcess{} ms; the uplink of \benchUpTwoOne{} kB is irrelevant at these sizes. With $p = 64$ the phase takes \benchRoundPsixtyfourTwoOne{} s, almost all of it the slowest worker's computation of the $64 \times 64$ Hessian on $2^{18}$ observations (\benchDerMaxPsixtyfour{} s), with an uplink of \benchUpPsixtyfourTwoOne{} kB. At the intermediate dimension $p = 32$ the derivative phase takes \benchRoundPthirtytwoTwoOne{} s and the local phase \benchLocalPthirtytwoOneTwo{} s for one fit, \benchLocalPthirtytwoTwoOne{} s for the half-block pattern and \benchLocalPthirtytwoThreeOne{} s for the quarter-block pattern, so that $(1,2)$ costs $\benchLocalPthirtytwoOneTwo + \benchRoundsPthirtytwoOneTwo$ s against $\benchLocalPthirtytwoTwoOne + \benchRoundPthirtytwoTwoOne$ s for $(2,1)$. Second, the local phase dominates and grows with the order: at $p = 8$ the slowest worker needs \benchLocalOneTwo{} s for one fit, \benchLocalTwoOne{} s for the three fits of the half-block pattern, \benchLocalThreeOne{} s for the seven fits of the quarter-block pattern and \benchLocalFourZero{} s for the fifteen fits of order four; at $p = 64$ the half-block pattern costs \benchLocalPsixtyfourTwoOne{} s against \benchLocalPsixtyfourOneTwo{} s for one fit. Third, the exchange rate of \cref{prop:cost} therefore favours rounds on this network: at equal exponent $7/8$, $(1,2)$ is the fastest protocol at both dimensions ($\benchLocalOneTwo + \benchRoundsOneTwoS$ s against $\benchLocalTwoOne + \benchRoundTwoOneS$ s for $(2,1)$ at $p = 8$; $\benchLocalPsixtyfourOneTwo + \benchRoundsPsixtyfourOneTwo$ s against $\benchLocalPsixtyfourTwoOne + \benchRoundPsixtyfourTwoOne$ s at $p = 64$), and it is also the more accurate one (\cref{sec:mc}). The stale and single-machine variants save nothing here, because the derivative phase they shorten was already short; their statistical cost (\cref{thm:inexact}) is paid for no gain. In terms of \eqref{eq:rule}, the measured $\lambda + T_{\mathrm{der}} + Kb(p + p(p+1)/2)/B + T_{\mathrm{solve}}$ is two to ten times smaller than $r\,c_1 m$ at every $(r, p)$ tested, so the inequality holds and one more round beats one more level of local fits.
14		14
15	\subsection{Hosts, processes and stragglers}	15	\subsection{Hosts, processes and stragglers}
16	−Spreading $K = 16$ workers from one host to sixteen (\cref{fig:bench}, right) does not speed up the local phase: its wall-clock is the slowest worker's, and the slowest of sixteen heterogeneous machines (a Mac mini) is slower than the slowest of sixteen processes on one Mac Studio (\benchLocalHostsSixteen{} s against \benchLocalHostsOne{} s), while the mean worker fit time is about the same (\benchFitMeanHostsSixteen{} s). The derivative phase roughly doubles from one host to sixteen (\benchRoundHostsOne{} ms to \benchRoundTwoOne{} ms), again through the slowest worker rather than through the network. With $K = 64$ and $128$ workers on sixteen hosts (four and eight processes per host, $m = 2^{16}$ and $2^{15}$) the local phase falls to \benchLocalKonetwentyeight{} s for $(2,1)$ and the derivative phase to \benchRoundKonetwentyeight{} ms: more partitions mean smaller blocks and faster local fits, but by \cref{thm:frontier} they also move the estimator towards the frontier, and the trade is statistical, not computational. The straggler excess, the gap between the slowest and the mean worker, is the quantity that an implementation should attack first; in these runs it comes from hardware heterogeneity rather than from the network.	16	+Spreading $K = 16$ workers from one host to sixteen (\cref{fig:bench}, right) does not speed up the local phase: its wall-clock is the slowest worker's, and the slowest of sixteen heterogeneous machines (a Mac mini) is slower than the slowest of sixteen processes on one Mac Studio (\benchLocalHostsSixteen{} s against \benchLocalHostsOne{} s), while the mean worker fit time is about the same (\benchFitMeanHostsSixteen{} s). The derivative phase roughly doubles from one host to sixteen (\benchRoundHostsOne{} ms to \benchRoundTwoOne{} ms), again through the slowest worker rather than through the network. With $K = 64$, $128$ and $256$ workers on sixteen hosts (four, eight and sixteen processes per host, $m = 2^{16}$, $2^{15}$ and $2^{14}$) the local phase falls to \benchLocalKonetwentyeight{} s and \benchLocalKtwofiftysix{} s for $(2,1)$ and the derivative phase to \benchRoundKonetwentyeight{} ms and \benchRoundKtwofiftysix{} ms (99th-percentile reply latency \benchLat99Ktwofiftysix{} ms with 256 connections): more partitions mean smaller blocks and faster local fits, but by \cref{thm:frontier} they also move the estimator towards the frontier, and the trade is statistical, not computational. The straggler excess, the gap between the slowest and the mean worker, is the quantity that an implementation should attack first; in these runs it comes from hardware heterogeneity rather than from the network.
17		17
18	\subsection{Limits of the benchmark}	18	\subsection{Limits of the benchmark}
19	The measurements are for one cluster, one Python/NumPy implementation with pickle framing, one network, and $K \le 128$; they do not establish a general speed advantage of any protocol. On a network with latency of tens of milliseconds, or with bandwidth-limited uplinks of dense Hessians at large $p$, the left-hand side of \eqref{eq:rule} grows and the balance can tip towards local levels; the cost model is provided so that \eqref{eq:rule} can be evaluated with the parameters of a given installation. No energy or monetary cost was measured, and none is claimed.	19	The measurements are for one cluster, one Python/NumPy implementation with pickle framing, one network, and $K \le 128$; they do not establish a general speed advantage of any protocol. On a network with latency of tens of milliseconds, or with bandwidth-limited uplinks of dense Hessians at large $p$, the left-hand side of \eqref{eq:rule} grows and the balance can tip towards local levels; the cost model is provided so that \eqref{eq:rule} can be evaluated with the parameters of a given installation. No energy or monetary cost was measured, and none is claimed.

modified paper/tables/bench.tex +6 −2

				`@@ -8,6 +8,8 @@ Design & $K$ & hosts & $p$ & $r$ & $q$ & curvature & local (s) & fit max (s) & r`
8	\midrule	8	\midrule
9	K128 r1q2 & 128 & 16 & 8 & 1 & 2 & exact & 0.059 & 0.038 & 0.014 & 0.007 & 0.005 & 0.006 & 199.7 & 85.5 \\	9	K128 r1q2 & 128 & 16 & 8 & 1 & 2 & exact & 0.059 & 0.038 & 0.014 & 0.007 & 0.005 & 0.006 & 199.7 & 85.5 \\
10	K128 r2q1 & 128 & 16 & 8 & 2 & 1 & exact & 0.097 & 0.076 & 0.007 & 0.007 & 0.005 & 0.006 & 127.5 & 53.0 \\	10	K128 r2q1 & 128 & 16 & 8 & 2 & 1 & exact & 0.097 & 0.076 & 0.007 & 0.007 & 0.005 & 0.006 & 127.5 & 53.0 \\
		11	+K256 r1q2 & 256 & 16 & 8 & 1 & 2 & exact & 0.076 & 0.044 & 0.016 & 0.008 & 0.004 & 0.006 & 399.4 & 170.8 \\
		12	+K256 r2q1 & 256 & 16 & 8 & 2 & 1 & exact & 0.108 & 0.068 & 0.008 & 0.008 & 0.004 & 0.006 & 255.1 & 106.0 \\
11	K64 r1q2 & 64 & 16 & 8 & 1 & 2 & exact & 0.085 & 0.064 & 0.019 & 0.009 & 0.008 & 0.009 & 99.9 & 42.9 \\	13	K64 r1q2 & 64 & 16 & 8 & 1 & 2 & exact & 0.085 & 0.064 & 0.019 & 0.009 & 0.008 & 0.009 & 99.9 & 42.9 \\
12	K64 r2q1 & 64 & 16 & 8 & 2 & 1 & exact & 0.148 & 0.127 & 0.009 & 0.009 & 0.008 & 0.009 & 63.8 & 26.7 \\	14	K64 r2q1 & 64 & 16 & 8 & 2 & 1 & exact & 0.148 & 0.127 & 0.009 & 0.009 & 0.008 & 0.009 & 63.8 & 26.7 \\
13	cmp p64 r1q2 & 16 & 16 & 64 & 1 & 2 & exact & 14.792 & 14.682 & 3.635 & 1.818 & 1.819 & 1.812 & 569.8 & 25.0 \\	15	cmp p64 r1q2 & 16 & 16 & 64 & 1 & 2 & exact & 14.792 & 14.682 & 3.635 & 1.818 & 1.819 & 1.812 & 569.8 & 25.0 \\
				`@@ -18,13 +20,15 @@ cmp r2q1 single & 16 & 16 & 8 & 2 & 1 & single & 0.495 & 0.464 & 0.053 & 0.053 &`
18	cmp r2q1 stale & 16 & 16 & 8 & 2 & 1 & stale & 0.494 & 0.463 & 0.053 & 0.053 & 0.051 & 0.051 & 19.7 & 6.7 \\	20	cmp r2q1 stale & 16 & 16 & 8 & 2 & 1 & stale & 0.494 & 0.463 & 0.053 & 0.053 & 0.051 & 0.051 & 19.7 & 6.7 \\
19	cmp r3q1 & 16 & 16 & 8 & 3 & 1 & exact & 0.724 & 0.693 & 0.065 & 0.065 & 0.063 & 0.063 & 16.0 & 6.7 \\	21	cmp r3q1 & 16 & 16 & 8 & 3 & 1 & exact & 0.724 & 0.693 & 0.065 & 0.065 & 0.063 & 0.063 & 16.0 & 6.7 \\
20	cmp r4q0 & 16 & 16 & 8 & 4 & 0 & exact & 0.956 & 0.925 & 0.000 & 0.000 & 0.000 & 0.000 & 7.0 & 2.6 \\	22	cmp r4q0 & 16 & 16 & 8 & 4 & 0 & exact & 0.956 & 0.925 & 0.000 & 0.000 & 0.000 & 0.000 & 7.0 & 2.6 \\
21	−hosts1 & 16 & 1 & 8 & 2 & 1 & exact & 0.407 & 0.371 & 0.027 & 0.027 & 0.025 & 0.025 & 16.0 & 6.7 \\	23	+hosts1 & 16 & 1 & 8 & 2 & 1 & exact & 0.407 & 0.372 & 0.027 & 0.027 & 0.025 & 0.025 & 16.0 & 6.7 \\
22	−hosts16 & 16 & 16 & 8 & 2 & 1 & exact & 0.493 & 0.462 & 0.055 & 0.055 & 0.053 & 0.054 & 15.9 & 6.7 \\	24	+hosts16 & 16 & 16 & 8 & 2 & 1 & exact & 0.494 & 0.463 & 0.054 & 0.054 & 0.052 & 0.053 & 15.9 & 6.7 \\
23	hosts2 & 16 & 2 & 8 & 2 & 1 & exact & 0.402 & 0.369 & 0.026 & 0.026 & 0.024 & 0.025 & 16.0 & 6.7 \\	25	hosts2 & 16 & 2 & 8 & 2 & 1 & exact & 0.402 & 0.369 & 0.026 & 0.026 & 0.024 & 0.025 & 16.0 & 6.7 \\
24	hosts4 & 16 & 4 & 8 & 2 & 1 & exact & 0.499 & 0.463 & 0.037 & 0.037 & 0.035 & 0.036 & 16.0 & 6.7 \\	26	hosts4 & 16 & 4 & 8 & 2 & 1 & exact & 0.499 & 0.463 & 0.037 & 0.037 & 0.035 & 0.036 & 16.0 & 6.7 \\
25	hosts8 & 16 & 8 & 8 & 2 & 1 & exact & 0.515 & 0.483 & 0.052 & 0.052 & 0.049 & 0.050 & 15.9 & 6.7 \\	27	hosts8 & 16 & 8 & 8 & 2 & 1 & exact & 0.515 & 0.483 & 0.052 & 0.052 & 0.049 & 0.050 & 15.9 & 6.7 \\
26	p32 r1q2 & 16 & 16 & 32 & 1 & 2 & exact & 2.591 & 2.528 & 0.625 & 0.313 & 0.311 & 0.312 & 160.2 & 16.8 \\	28	p32 r1q2 & 16 & 16 & 32 & 1 & 2 & exact & 2.591 & 2.528 & 0.625 & 0.313 & 0.311 & 0.312 & 160.2 & 16.8 \\
27	p32 r2q1 & 16 & 16 & 32 & 2 & 1 & exact & 5.129 & 5.061 & 0.315 & 0.315 & 0.313 & 0.313 & 85.1 & 9.7 \\	29	p32 r2q1 & 16 & 16 & 32 & 2 & 1 & exact & 5.129 & 5.061 & 0.315 & 0.315 & 0.313 & 0.313 & 85.1 & 9.7 \\
		30	+p32 r2q1 stale & 16 & 16 & 32 & 2 & 1 & stale & 5.130 & 5.065 & 0.314 & 0.314 & 0.311 & 0.312 & 148.7 & 9.7 \\
		31	+p32 r3q1 & 16 & 16 & 32 & 3 & 1 & exact & 7.666 & 7.602 & 0.314 & 0.314 & 0.312 & 0.312 & 85.1 & 9.8 \\
28	\bottomrule	32	\bottomrule
29	\end{tabular}	33	\end{tabular}
30	\begin{tablenotes}\footnotesize	34	\begin{tablenotes}\footnotesize

modified paper/tables/numbers.tex +13 −5

				`@@ -42,11 +42,11 @@`
42	\newcommand{\boundaryRatioTwoOneMax}{2.4}	42	\newcommand{\boundaryRatioTwoOneMax}{2.4}
43	\newcommand{\boundaryMaxM}{16}	43	\newcommand{\boundaryMaxM}{16}
44	\newcommand{\benchLocalTwoOne}{0.49}	44	\newcommand{\benchLocalTwoOne}{0.49}
45	−\newcommand{\benchRoundTwoOne}{55}	45	+\newcommand{\benchRoundTwoOne}{54}
46	\newcommand{\benchLocalOneTwo}{0.26}	46	\newcommand{\benchLocalOneTwo}{0.26}
47	\newcommand{\benchRoundsOneTwo}{61}	47	\newcommand{\benchRoundsOneTwo}{61}
48	\newcommand{\benchRoundsOneTwoS}{0.061}	48	\newcommand{\benchRoundsOneTwoS}{0.061}
49	−\newcommand{\benchRoundTwoOneS}{0.055}	49	+\newcommand{\benchRoundTwoOneS}{0.054}
50	\newcommand{\benchLocalThreeOne}{0.72}	50	\newcommand{\benchLocalThreeOne}{0.72}
51	\newcommand{\benchLocalFourZero}{0.96}	51	\newcommand{\benchLocalFourZero}{0.96}
52	\newcommand{\benchLocalPsixtyfourTwoOne}{29.5}	52	\newcommand{\benchLocalPsixtyfourTwoOne}{29.5}
				`@@ -60,9 +60,17 @@`
60	\newcommand{\benchLocalHostsSixteen}{0.49}	60	\newcommand{\benchLocalHostsSixteen}{0.49}
61	\newcommand{\benchFitMeanHostsSixteen}{0.35}	61	\newcommand{\benchFitMeanHostsSixteen}{0.35}
62	\newcommand{\benchRoundHostsOne}{27}	62	\newcommand{\benchRoundHostsOne}{27}
63	−\newcommand{\benchLatencyExcess}{0.2}	63	+\newcommand{\benchLatencyExcess}{0.4}
64	\newcommand{\benchLocalKonetwentyeight}{0.10}	64	\newcommand{\benchLocalKonetwentyeight}{0.10}
65	\newcommand{\benchRoundKonetwentyeight}{7}	65	\newcommand{\benchRoundKonetwentyeight}{7}
66	\newcommand{\benchConnect}{1.6}	66	\newcommand{\benchConnect}{1.6}
67	−\newcommand{\benchRuns}{57}	67	+\newcommand{\benchLocalPthirtytwoTwoOne}{5.13}
68	−\newcommand{\benchDerMaxPeight}{53}	68	+\newcommand{\benchRoundPthirtytwoTwoOne}{0.31}
		69	+\newcommand{\benchLocalPthirtytwoOneTwo}{2.59}
		70	+\newcommand{\benchRoundsPthirtytwoOneTwo}{0.63}
		71	+\newcommand{\benchLocalPthirtytwoThreeOne}{7.67}
		72	+\newcommand{\benchLocalKtwofiftysix}{0.11}
		73	+\newcommand{\benchRoundKtwofiftysix}{8}
		74	+\newcommand{\benchLat99Ktwofiftysix}{6}
		75	+\newcommand{\benchRuns}{75}
		76	+\newcommand{\benchDerMaxPeight}{52}

modified research_log.md +1 −0

				`@@ -19,3 +19,4 @@`
19	- 05:29 Benchmarks: M4M64b's Python cannot open LAN connections (macOS local-network privacy: 'No route to host'); replaced by M3BA24 in all designs.	19	- 05:29 Benchmarks: M4M64b's Python cannot open LAN connections (macOS local-network privacy: 'No route to host'); replaced by M3BA24 in all designs.
20	- 05:29 M3BA24 unreachable (asleep); the coordinator host M4M64a also serves as the 16th worker host in the benchmark designs (noted in the paper).	20	- 05:29 M3BA24 unreachable (asleep); the coordinator host M4M64a also serves as the 16th worker host in the benchmark designs (noted in the paper).
21	- 05:42 Final phase: cluster reruns of the ten N=2^20 failure-affected cells started 05:38 (500 reps each); laptop reruns of the 22 small cells + robustness 36/37 in progress; benchmarks complete (51 runs); FINAL_REVIEW, PHASE_STATUS, FINAL_DELIVERY written.	21	- 05:42 Final phase: cluster reruns of the ten N=2^20 failure-affected cells started 05:38 (500 reps each); laptop reruns of the 22 small cells + robustness 36/37 in progress; benchmarks complete (51 runs); FINAL_REVIEW, PHASE_STATUS, FINAL_DELIVERY written.
		22	+- 15:16 Supplementary benchmarks integrated (p=32, K=256 with raised FD limit, host-scaling repeats): 75 runs. Final build validated.

modified results/figures/fig_bench_exchange.pdf +0 −0

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modified results/processed/benchmarks_summary.csv +6 −2

				`@@ -1,6 +1,8 @@`
1	label,K,hosts,p,r,q,hessian,reps,t_total,t_local,fit_mean,fit_max,t_rounds,round_wall,der_max,lat99,up_B,down_B,msgs,grad,t_connect	1	label,K,hosts,p,r,q,hessian,reps,t_total,t_local,fit_mean,fit_max,t_rounds,round_wall,der_max,lat99,up_B,down_B,msgs,grad,t_connect
2	K128_r1q2,128,16,8,1,2,exact,3,0.07594195799902081,0.05910841701552272,0.02693636457791699,0.037984749999999984,0.014260708936490119,0.007130354468245059,0.005471541999999996,0.006103813237277791,199720,85504,896,6.19461312241283e-09,1.6127827079035342	2	K128_r1q2,128,16,8,1,2,exact,3,0.07594195799902081,0.05910841701552272,0.02693636457791699,0.037984749999999984,0.014260708936490119,0.007130354468245059,0.005471541999999996,0.006103813237277791,199720,85504,896,6.19461312241283e-09,1.6127827079035342
3	K128_r2q1,128,16,8,2,1,exact,3,0.10689191601704806,0.09711383294779807,0.053724647530380046,0.075590084,0.0071037920424714684,0.0071037920424714684,0.005285624999999988,0.0059814579575322565,127528,52992,640,1.2071655704270327e-05,1.608464999939315	3	K128_r2q1,128,16,8,2,1,exact,3,0.10689191601704806,0.09711383294779807,0.053724647530380046,0.075590084,0.0071037920424714684,0.0071037920424714684,0.005285624999999988,0.0059814579575322565,127528,52992,640,1.2071655704270327e-05,1.608464999939315
		4	+K256_r1q2,256,16,8,1,2,exact,3,0.09489187505096197,0.07596725004259497,0.01624158224882155,0.04387470800000001,0.015681207994930446,0.007840603997465223,0.003909207999999997,0.005679456563666463,399440,170752,1792,2.6608534597532117e-08,1.6790186250582337
		5	+K256_r2q1,256,16,8,2,1,exact,3,0.11990674992557615,0.10782191599719226,0.03427523633797172,0.06846033399999998,0.008079749997705221,0.008079749997705221,0.003922709000000024,0.005764030886348337,255056,105984,1280,1.5343739785638766e-05,1.653278167010285
4	K64_r1q2,64,16,8,1,2,exact,3,0.10569124994799495,0.08475083310622722,0.050187399078484185,0.06390991700000001,0.01862320804502815,0.009311604022514075,0.007990957999999937,0.00884082111530006,99860,42880,448,1.7957401139889523e-09,1.614608958014287	6	K64_r1q2,64,16,8,1,2,exact,3,0.10569124994799495,0.08475083310622722,0.050187399078484185,0.06390991700000001,0.01862320804502815,0.009311604022514075,0.007990957999999937,0.00884082111530006,99860,42880,448,1.7957401139889523e-09,1.614608958014287
5	K64_r2q1,64,16,8,2,1,exact,3,0.15925016696564853,0.14756979199592024,0.09934315362223953,0.126862584,0.009118333109654486,0.009118333109654486,0.008046499999999984,0.008500265934271738,63764,26688,320,6.769759590932384e-06,1.612117417040281	7	K64_r2q1,64,16,8,2,1,exact,3,0.15925016696564853,0.14756979199592024,0.09934315362223953,0.126862584,0.009118333109654486,0.009118333109654486,0.008046499999999984,0.008500265934271738,63764,26688,320,6.769759590932384e-06,1.612117417040281
6	cmp_p64_r1q2,16,16,64,1,2,exact,3,18.443110374966636,14.791574709000997,8.422302531124188,14.682226792,3.635024875984527,1.8175124379922636,1.8193454579999973,1.8124884893244597,569765,25040,112,7.246710990491112e-09,1.5718543749535456	8	cmp_p64_r1q2,16,16,64,1,2,exact,3,18.443110374966636,14.791574709000997,8.422302531124188,14.682226792,3.635024875984527,1.8175124379922636,1.8193454579999973,1.8124884893244597,569765,25040,112,7.246710990491112e-09,1.5718543749535456
				`@@ -11,10 +13,12 @@ cmp_r2q1_single,16,16,8,2,1,single,3,0.5599910420132801,0.49494124995544553,0.35`
11	cmp_r2q1_stale,16,16,8,2,1,stale,3,0.5575762910302728,0.4937682910822332,0.35194410675506355,0.4627345,0.052745833061635494,0.052745833061635494,0.051406582999999895,0.051093250076519325,19653,6672,80,5.2530319514019975e-06,1.577691916958429	13	cmp_r2q1_stale,16,16,8,2,1,stale,3,0.5575762910302728,0.4937682910822332,0.35194410675506355,0.4627345,0.052745833061635494,0.052745833061635494,0.051406582999999895,0.051093250076519325,19653,6672,80,5.2530319514019975e-06,1.577691916958429
12	cmp_r3q1,16,16,8,3,1,exact,3,0.8057671659626067,0.7243046669755131,0.5272520206900422,0.6931520409999999,0.06545370805542916,0.06545370805542916,0.06256016599945724,0.06346240004058927,16005,6704,80,7.634102395074616e-06,1.5876702499808744	14	cmp_r3q1,16,16,8,3,1,exact,3,0.8057671659626067,0.7243046669755131,0.5272520206900422,0.6931520409999999,0.06545370805542916,0.06545370805542916,0.06256016599945724,0.06346240004058927,16005,6704,80,7.634102395074616e-06,1.5876702499808744
13	cmp_r4q0,16,16,8,4,0,exact,3,0.9730557500151917,0.9561016249936074,0.7040089270561611,0.924698708,0.0,0.0,0.0,0.0,6965,2624,48,,1.5691752089187503	15	cmp_r4q0,16,16,8,4,0,exact,3,0.9730557500151917,0.9561016249936074,0.7040089270561611,0.924698708,0.0,0.0,0.0,0.0,6965,2624,48,,1.5691752089187503
14	−hosts1,16,1,8,2,1,exact,3,0.45093533291947097,0.4072739159455523,0.36361600787495263,0.37104129104409367,0.027120084036141634,0.027120084036141634,0.024586540996097028,0.025101468653883786,16048,6656,80,6.018665659921152e-06,1.5933742499910295	16	+hosts1,16,1,8,2,1,exact,6,0.4502359584439546,0.40681637497618794,0.3639094870231929,0.3716897705453448,0.027186146064195782,0.027186146064195782,0.02478983293985948,0.025449264200869945,16048,6656,80,5.724984862893512e-06,1.5895304374862462
15	−hosts16,16,16,8,2,1,exact,3,0.559004292008467,0.4927386250346899,0.3511835156840801,0.46231912500000005,0.05496741703245789,0.05496741703245789,0.053325834000000016,0.053503567090956494,15941,6656,80,5.333876108823157e-06,1.5722926669986919	17	+hosts16,16,16,8,2,1,exact,6,0.5591622504871339,0.4940511039458215,0.3533448085977507,0.46337933350000005,0.054040312534198165,0.054040312534198165,0.05233091649999999,0.05268612583458889,15941,6656,80,5.385096883733787e-06,1.5693803540198132
16	hosts2,16,2,8,2,1,exact,3,0.44408383406698704,0.40218737500254065,0.36419236718211323,0.3688111249357462,0.02646495890803635,0.02646495890803635,0.024155499995686114,0.024541217036312447,15976,6656,80,3.6295816652902047e-06,1.5768007499864325	18	hosts2,16,2,8,2,1,exact,3,0.44408383406698704,0.40218737500254065,0.36419236718211323,0.3688111249357462,0.02646495890803635,0.02646495890803635,0.024155499995686114,0.024541217036312447,15976,6656,80,3.6295816652902047e-06,1.5768007499864325
17	hosts4,16,4,8,2,1,exact,3,0.5428270000265911,0.4990827910369262,0.3769453749414007,0.46323662500000007,0.03661487495992333,0.03661487495992333,0.03527895896695554,0.035901087563252075,16008,6656,80,4.672415990280505e-06,1.6030859160237014	19	hosts4,16,4,8,2,1,exact,3,0.5428270000265911,0.4990827910369262,0.3769453749414007,0.46323662500000007,0.03661487495992333,0.03661487495992333,0.03527895896695554,0.035901087563252075,16008,6656,80,4.672415990280505e-06,1.6030859160237014
18	hosts8,16,8,8,2,1,exact,3,0.5784965000348166,0.5150650420691818,0.35872016669400253,0.48260645799999996,0.051650459063239396,0.051650459063239396,0.04911099991295487,0.04994003128958866,15938,6672,80,6.512105595336275e-06,1.590769625036046	20	hosts8,16,8,8,2,1,exact,3,0.5784965000348166,0.5150650420691818,0.35872016669400253,0.48260645799999996,0.051650459063239396,0.051650459063239396,0.04911099991295487,0.04994003128958866,15938,6672,80,6.512105595336275e-06,1.590769625036046
19	p32_r1q2,16,16,32,1,2,exact,3,3.2327428330900148,2.59141674998682,1.9259473620696739,2.528149833,0.6252187510253862,0.3126093755126931,0.3112902499999999,0.31191429115715436,160165,16848,112,2.292484002425426e-09,1.564675375004299	21	p32_r1q2,16,16,32,1,2,exact,3,3.2327428330900148,2.59141674998682,1.9259473620696739,2.528149833,0.6252187510253862,0.3126093755126931,0.3112902499999999,0.31191429115715436,160165,16848,112,2.292484002425426e-09,1.564675375004299
20	p32_r2q1,16,16,32,2,1,exact,3,5.463905291981064,5.129355833050795,3.847350143253433,5.0611045,0.31466895900666714,0.31466895900666714,0.31262058400000026,0.3131287410913501,85077,9728,80,1.9613042165727724e-05,1.5934663750231266	22	p32_r2q1,16,16,32,2,1,exact,3,5.463905291981064,5.129355833050795,3.847350143253433,5.0611045,0.31466895900666714,0.31466895900666714,0.31262058400000026,0.3131287410913501,85077,9728,80,1.9613042165727724e-05,1.5934663750231266
		23	+p32_r2q1_stale,16,16,32,2,1,stale,3,5.459593167062849,5.130021959077567,3.887375908949232,5.064722667,0.3136520000407472,0.3136520000407472,0.31143370800000003,0.3120329410303384,148677,9744,80,1.9973955006730648e-05,1.5643577090231702
		24	+p32_r3q1,16,16,32,3,1,exact,3,7.993662082939409,7.665812374907546,5.805818403561044,7.6016277500000005,0.31383537501096725,0.31383537501096725,0.31171616700000015,0.3124722090025898,85141,9760,80,2.876238837249541e-05,1.5851650829426944

modified results/raw/benchmarks.jsonl +14 −0

+ {"K": 16, "N": 4194304, "m": 262144, "p": 32, "model": "logistic", "splits": [1, 2, 4], "q": 1, "hessian_mode": "exact", "bits": null, "hosts": ["M4M64c", "M5M36", "MacStudnpierre2", "MacStudnpierre6", "MacStudnpierre7", "MacStudonpierre", "Macmininpierre5", "m3u96b", "m4ml", "m4mm", "m4mn", "m5ma", "m6ma", "m6mb", "simonpierresMBP", "simonpirresMBP2"], "n_hosts": 16, "workers": [{"type": "hello", "id": 5, "host": "M5M36", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 18}, {"type": "hello", "id": 8, "host": "m6ma", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 12}, {"type": "hello", "id": 9, "host": "m6mb", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 12}, {"type": "hello", "id": 10, "host": "m5ma", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 15}, {"type": "hello", "id": 3, "host": "MacStudnpierre6", "machine": "arm64", "python": "3.14.4", "numpy": "2.5.4", "cpu": 16}, {"type": "hello", "id": 4, "host": "M4M64c", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 16}, {"type": "hello", "id": 11, "host": "Macmininpierre5", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 10}, {"type": "hello", "id": 7, "host": "m4mn", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 12}, {"type": "hello", "id": 6, "host": "m4mm", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 12}, {"type": "hello", "id": 12, "host": "m4ml", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 10}, {"type": "hello", "id": 14, "host": "simonpirresMBP2", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 16}, {"type": "hello", "id": 2, "host": "MacStudonpierre", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 24}, {"type": "hello", "id": 1, "host": "m3u96b", "machine": "arm64", "python": "3.14.4", "numpy": "2.5.4", "cpu": 28}, {"type": "hello", "id": 15, "host": "simonpierresMBP", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 14}, {"type": "hello", "id": 0, "host": "MacStudnpierre2", "machine": "arm64", "python": "3.14.5", "numpy": "2.5.4", "cpu": 28}, {"type": "hello", "id": 13, "host": "MacStudnpierre7", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 12}], "t_connect": 1.5851650829426944, "t_total_after_connect": 7.977095208014362, "theta_path": [[0.3000642373677191, 0.500304443965378, -0.4364542442299977, 0.38549197996625933, -0.34467124385758297, 0.3144745890158519, -0.287950739663524, 0.26195496535929663, -0.24400196269650723, 0.2293664738220328, -0.2143373106757079, 0.20005754805374282, -0.18986746903772272, 0.1790461694145079, -0.17032009816230193, 0.1618155664835491, -0.1544021882239784, 0.14415233766666236, -0.14103805204917436, 0.1355746709989645, -0.12871940929237763, 0.12393108128305544, -0.12151321872114983, 0.11551292375461447, -0.11365184462697542, 0.11023938528381963, -0.1052574722814253, 0.10057254278157624, -0.1019433978995803, 0.09451341902845325, 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1], ["M2M32c", 1], ["M4BP48", 1], ["M4BP36", 1]], "label": "p32_r3q1"}
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"python": "3.9.6", "numpy": "2.0.2", "cpu": 14}, {"type": "hello", "id": 251, "host": "simonpierresMBP", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 14}, {"type": "hello", "id": 248, "host": "simonpierresMBP", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 14}, {"type": "hello", "id": 203, "host": "m4ml", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 10}, {"type": "hello", "id": 181, "host": "Macmininpierre5", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 10}, {"type": "hello", "id": 196, "host": "m4ml", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 10}, {"type": "hello", "id": 221, "host": "MacStudnpierre7", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 12}, {"type": "hello", "id": 209, "host": "MacStudnpierre7", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 12}, {"type": "hello", "id": 185, "host": "Macmininpierre5", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 10}, {"type": "hello", "id": 184, "host": "Macmininpierre5", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 10}, {"type": "hello", "id": 199, "host": "m4ml", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 10}, {"type": "hello", "id": 197, "host": "m4ml", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 10}, {"type": "hello", "id": 198, "host": "m4ml", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 10}, {"type": "hello", "id": 200, "host": "m4ml", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 10}, {"type": "hello", "id": 111, "host": "m4mm", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 12}, {"type": "hello", "id": 208, "host": "MacStudnpierre7", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 12}, {"type": "hello", "id": 212, "host": "MacStudnpierre7", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 12}, {"type": "hello", "id": 202, "host": "m4ml", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 10}, {"type": "hello", "id": 211, "host": "MacStudnpierre7", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 12}, {"type": "hello", "id": 201, "host": "m4ml", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 10}, {"type": "hello", "id": 223, "host": "MacStudnpierre7", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 12}, {"type": "hello", "id": 215, "host": "MacStudnpierre7", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 12}, {"type": "hello", "id": 214, "host": "MacStudnpierre7", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 12}, {"type": "hello", "id": 189, "host": "Macmininpierre5", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 10}, {"type": "hello", "id": 213, "host": "MacStudnpierre7", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 12}, {"type": "hello", "id": 187, "host": 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"MacStudnpierre7", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 12}, {"type": "hello", "id": 186, "host": "Macmininpierre5", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 10}, {"type": "hello", "id": 190, "host": "Macmininpierre5", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 10}, {"type": "hello", "id": 220, "host": "MacStudnpierre7", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 12}, {"type": "hello", "id": 191, "host": "Macmininpierre5", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 10}, {"type": "hello", "id": 205, "host": "m4ml", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 10}, {"type": "hello", "id": 206, "host": "m4ml", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 10}, {"type": "hello", "id": 207, "host": "m4ml", "machine": "arm64", "python": "3.9.6", "numpy": "2.0.2", "cpu": 10}], "t_connect": 1.6759521670173854, "t_total_after_connect": 0.09358099999371916, "theta_path": [[0.30084671257962803, 0.5015311700131908, -0.4340632855608076, 0.3846340777432205, -0.34637026869228416, 0.3113690011235256, -0.28555904320799325, 0.26445447425333746], [0.3006926046161673, 0.501170157162725, -0.4337780199607602, 0.3844227866331316, -0.34614352817846117, 0.31115565519283855, -0.2853386563378151, 0.2643301686424243], [0.3006926510474356, 0.5011702564808834, -0.4337781005350374, 0.38442284970755447, -0.3461435922636431, 0.3111557147929382, -0.2853387158347354, 0.2643302077852818]], "newton": [{"solve_ok": true, "cond": 1.4886304940657469, "step_norm": 0.0006641159104985357, "grad_norm": 9.4801192390914e-05}, {"solve_ok": true, "cond": 1.488067863365069, "step_norm": 1.876744134906739e-07, "grad_norm": 2.668266122810738e-08}], "stats": {"bytes_up": 399440, "bytes_down": 170752, "messages_up": 768, "messages_down": 1024, "stages": {"A_B_local_fit_and_upload": {"wall": 0.07427570794243366, "worker_fit_s_mean": 0.01618568095835239, "worker_fit_s_max": 0.041330167, "worker_gen_s_mean": 0.022559585936535613, "latency_p50": 0.0628406039904803, "latency_p90": 0.06324079149635509, "latency_p99": 0.0728887292323634, "valid_workers": 256}, "D_F_round1": {"wall": 0.008028832962736487, "broadcast_s": 0.0018400000408291817, "worker_der_s_mean": 0.0019826435474161745, "worker_der_s_max": 0.003909207999999997, "gather_s": 0.005930040962994099, "solve_s": 6.0832942835986614e-05, "latency_p50": 0.0032770209945738316, "latency_p90": 0.00463606248376891, "latency_p99": 0.005893999978434294, "grad_norm": 9.4801192390914e-05, "cond": 1.4886304940657469}, "D_F_round2": {"wall": 0.007652375032193959, "broadcast_s": 0.0018294999608770013, "worker_der_s_mean": 0.001980072726453873, "worker_der_s_max": 0.0037308749999999946, "gather_s": 0.005649625090882182, "solve_s": 4.100007936358452e-05, "latency_p50": 0.002987645973917097, "latency_p90": 0.004095708951354027, "latency_p99": 0.005543733655940741, "grad_norm": 2.668266122810738e-08, "cond": 1.488067863365069}}}, "coordinator": "MacStudnpierre6", "critical_path_estimate": 0.05701137499493045, "design_index": 5, "repeat": 2, "launch_wall_s": 0.3873176574707031, "assignment": [["M3U96a", 16], ["M3U96b", 16], ["M2U64", 16], ["M4M64a", 16], ["M4M64c", 16], ["M5M36", 16], ["m4mm", 16], ["m4mn", 16], ["m6ma", 16], ["m6mb", 16], ["m5ma", 16], ["m4md", 16], ["m4ml", 16], ["M2M32c", 16], ["M4BP48", 16], ["M4BP36", 16]], "label": "K256_r1q2"}

modified src/pamle/protocol.py +7 −1

				`@@ -104,7 +104,13 @@ class Coordinator:`
104	return out, lat	104	return out, lat
105		105
106	def run(self):	106	def run(self):
107	− srv = socket.socket(); srv.setsockopt(socket.SOL_SOCKET, socket.SO_REUSEADDR, 1); srv.bind(("0.0.0.0", self.port)); srv.listen(self.K + 8)	107	+ try: # K workers = K sockets: raise the per-process file-descriptor limit
		108	+ import resource
		109	+ soft, hard = resource.getrlimit(resource.RLIMIT_NOFILE)
		110	+ resource.setrlimit(resource.RLIMIT_NOFILE, (max(soft, min(hard, 4096)), hard))
		111	+ except Exception:
		112	+ pass
		113	+ srv = socket.socket(); srv.setsockopt(socket.SOL_SOCKET, socket.SO_REUSEADDR, 1); srv.bind(("0.0.0.0", self.port)); srv.listen(max(self.K + 8, 512))
108	self.socks = []; hellos = []	114	self.socks = []; hellos = []
109	t_start = time.perf_counter()	115	t_start = time.perf_counter()
110	while len(self.socks) < self.K:	116	while len(self.socks) < self.K:
111		117

spb/wp28_uqo

WP28: supplementary benchmarks (p=32, K=256), final build

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