eq-6fdcaa1434be58mathematical expression
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The average uses a symmetric window of w observations around time t. For odd w, there are (w−1)/2 samples on each side and one center sample. Because future samples enter the window, the smoothed series is a retrospective descriptive summary rather than an immediately available online detector.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| g_t | Recorded gradient norm |
| w | Odd window length |
| g_bar | Smoothed gradient norm |
How it works
Read the definition from its right-hand expression to the quantity on the left. Its surrounding source records the intended input, state, or parameter context. Where the source leaves an operation unspecified, this page preserves that gap.
Assumptions & limits
End-of-record handling must specify whether incomplete windows are omitted or treated differently.
Original context
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4.1 Gradient and loss summaries
Let gt denote the trainer’s recorded gradient norm. Its raw maximum is reported alongside complete-window centered means at widths w∈{5,11,21,31}:
gtw=1wj=−w−1/2w−1/2gt+j.
Centered means use future observations relative to their nominal center. They are descriptive summaries, not online onset detectors. The field is interpreted as the trainer’s reported gradient magnitude, not a direct measurement of electrical energy, accumulated information, or the magnitude of the eventual parameter update. Optimizer state, learning rate, and clipping may distinguish those quantities.
The adjacent windows 161–180 and 181–200 summarize the early episode. Their relative decline is 1001−g181:200/g161:180. This comparison is retrospective and motivated by the already published transition region; it is not a prospectively predicted effect or an estimated causal effect of declaration.
For successive ten-step losses, define qt=ℓt−10−ℓt. A positive qt indicates lower recorded loss. The largest decrement locates an interval, not a unique intervening event. The full series, including any late increases, is retained.Exact original notation
{\hat{g}}_{t}^{\left(w\right)}=\frac{1}{w}\sum _{j=-\left(w-1\right)/2}^{\left(w-1\right)/2}{{g}_{t+j}}.Related builds
Sources & provenance
01
Declaration_and_State_Change_Final.docx
paragraph 68, math 1 · Sep 17, 2026
{\hat{g}}_{t}^{\left(w\right)}=\frac{1}{w}\sum _{j=-\left(w-1\right)/2}^{\left(w-1\right)/2}{{g}_{t+j}}.
02
Declaration_and_State_Change_Final.docx
paragraph 68, math 1 · Sep 17, 2026
{\hat{g}}_{t}^{\left(w\right)}=\frac{1}{w}\sum _{j=-\left(w-1\right)/2}^{\left(w-1\right)/2}{{g}_{t+j}}.