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03-error-sensitivity-feedback.md

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03 — Error, sensitivity, feedback

This is the conceptual bridge of the whole ecosystem.

A system becomes controllable when output becomes error, error meets sensitivity, and feedback informs the next run.

The loop

x → f(x) → ŷ → error → f′ → feedback

Made explicit, every stage is its own inspectable value:

Input x
  ↓
Transform f(x)
  ↓
Prediction ŷ
  ↓
Error E = y - ŷ
  ↓
Sensitivity f′(x)
  ↓
Update Signal = E · f′(x)
  ↓
Feedback Action
  • x — the input.
  • f(x) = ŷ — the prediction (forward pass).
  • error = y − ŷ — the gap between target and prediction.
  • f′(x) — the local sensitivity: how much the output moves when the input moves.
  • update signal = error · f′(x) — error scaled by sensitivity.

The sensitivity comes from the output

For sigmoid, the derivative has a clean identity in terms of the output itself:

ŷ = f(x)        ⇒        f′(x) = ŷ (1 − ŷ)

This is what errorSensitivity in @composable-model-graph/math uses (activation.derivativeFromOutput): right after a forward pass you already hold ŷ, so you can read the sensitivity straight off the trace without recomputing the pre-activation.

Why both terms matter

  • Error alone tells you that something is wrong, but not where a change would help.
  • Sensitivity alone tells you where change has leverage, but not whether anything is wrong.
  • error × sensitivity combines them into a directed signal: change the things that are both wrong and influential.

Worked values

Taking the prediction from Example 02 (ŷ ≈ 0.492144, a sigmoid output) with target = 1:

ŷ             = 0.492144
error E       = 1 − ŷ       ≈ 0.507856
f′            = ŷ (1 − ŷ)   ≈ 0.249938
update signal = E · f′       ≈ 0.126932

The update signal is largest when the prediction is both wrong and sits in a sensitive region of the activation (near ŷ = 0.5), and it collapses to zero when either the error is zero or the activation is saturated (f′ → 0).

Not training (yet)

This loop does not update any weights. It only exposes the feedback signal. Turning the signal into a weight update is backpropagation, which is out of scope for v1.

Example

See Example 03 — Error Sensitivity Feedback, which runs the neural graph, then prints the prediction, the target, the error E, the sensitivity f′, and the E · f′ update signal, before mapping the run to a feedback action.