momentum_step

std.optim.momentum_step · Level L2

One step of gradient descent with heavy-ball momentum, on the state [params; velocity]: v ← β·v + grad, then params ← params − lr·v.

v ← β·v + g, θ ← θ − η·v

Signature

momentum_step(state: f64[2, n], grad: f64[n], lr: f64[], beta: f64[]) → f64[2, n]

Structure

The function as NOVA stores it: one box per input, operation and output, and arrows that carry values. A double border marks another library function this one runs — called once, or by Scan once per element; select it to open that function.

input
operation
constant
call
output

Verification

Signature proven by NOVA’s shape solver, for every size.
Equal to the reference v = beta*v + grad; p = p - lr*v in exact rational arithmetic, on all 40 test cases.
All 322 float64 results inside the running error bound; the closest uses 92% of it.
Interpreter and NumPy backend return bit-identical results.
At float32 too: the 40 test cases rounded to float32, and 40 of them inside the running error bound at u = 2⁻²⁴; the closest uses 82% of it.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 83%
bit-equal to the NumPy formula in float64: 100%
largest error, in units in the last place: 18

Large ulp counts appear where a result is tiny next to the numbers it is computed from (after cancellation, for example), so one unit in the last place is tiny too; the absolute error is still inside the bound. Results within their own error of zero are not counted.

Identity

Calls

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Called by

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sha256:d01d1e26dd77f770ab0d67e486cdb9a1800eda4b1531981d5d77308caf8152cf

The semantic hash of the graph. It changes when the program changes, and never when only its documentation does.

Control handle

Symbol: Ω:std.optim.momentum_step · Ω:momentum_step
Pins: sha256:3bde2f544d8c48583b864bb9240bf845430a387aa2516da6dff09cda6652befdthis graph alone
Evidence: sha256:1872461464502c92c8bdca68f63d833d213fade7ba3ee9a0c0efd7a8ebe97e0dthe hash of its verification record
Needs: no capability: a pure function

Through NOVA’s control layer, the symbol launches this function only while the program still matches what it pins: a change to this graph, or to any graph it reaches, needs a migration first.