Can a DiLoCo-style run hide its synchronisation by applying the outer update rounds late?

research/outer-delay

Overlapping the DiLoCo sync by applying the outer step τ rounds late does not pay: stable iff αs < 2 sin(π/(4τ+2)), best rate 1−2 sin(π/(4τ+2))/κ_H (exactly κ_H/(κ_H+1) at τ=1), momentum is useless once τ≥1 (69 → 1388 rounds at κ_H=100), hidden-sync gain ≤ (τ+1) sin(π/(4τ+2)) ≤ 1 (search: 1.5–4.9× slower than blocking averaging), fixed if the fresh pseudo-gradient has weight >½.

13 tests

What it found

Applying the outer step τ rounds late gives a mode the recursion x_t+1 = x_t − αs x_t−τ, stable iff αs < 2 sin(π/(4τ+2)) ≈ π/(2τ+1). The best step sits at that edge, so the rate is 1 − 2 sin(π/(4τ+2))/κ_H (exactly κ_H/(κ_H+1) at τ=1), and outer momentum stops helping the moment τ ≥ 1 (69 rounds at τ=0 become 1388 at κ_H=100). Even with the sync fully hidden the gain over blocking averaging is at most (τ+1) sin(π/(4τ+2)) ≤ 1, and an exact search over (H, τ) finds overlapped training 1.5–4.9× slower than blocking averaging and 2.3–31× slower than blocking with tuned momentum. Giving the fresh pseudo-gradient weight above ½ removes the penalty entirely (stable for every τ; at weight 0.75 and τ=1 the stale term acts as negative momentum, 2× faster than no delay). Stylised: noiseless quadratic, shared Hessian, one common delay.

A figure from its own output

Figure produced by research/outer-delay: stability_threshold — outer-delay/experiments/stability_threshold.svg, written by the project's own experiment code.

Paper, code and results

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Produced with AI assistance (Claude). Results are from the code in this repository and are simulated or stylised unless the abstract says otherwise; nothing here implies affiliation with or endorsement by any lab, researcher or company cited. MIT licensed.