Scores and Boundaries

Log-Loss, Structurally

Log-loss combines labels with named sigmoid and logarithm operations. The lesson renders the structure without evaluating a transcendental value.

highlighted = computed this step

Log-loss structure

Log-loss combines the label, the named sigmoid value, and logarithm. The form is exact, but logarithm is named.

ℓ=−[yln⁡(σ)+(1−y)ln⁡(1−σ)]\ell=-[y\ln(\sigma)+(1-y)\ln(1-\sigma)]
Loss structureThe same score table supplies the named sigmoid inputs for log-loss.Classifier score tablexyzdecisionprob00-20σ(-2)10-10σ(-1)21011/23111σ(1)sigmoid is named except at zero

Why the value is named

The logarithm is another boundary operation. At z=0, the structural loss is ln 2, and no decimal is rendered.

z=0⇒ℓ=ln⁡2z=0\Rightarrow \ell=\ln 2
Loss structureThe same score table supplies the named sigmoid inputs for log-loss.Classifier score tablexyzdecisionprob00-20σ(-2)10-10σ(-1)21011/23111σ(1)sigmoid is named except at zero

Summary

The loss measures fit structurally, but this surface names the transcendental operations instead of evaluating them. Sigmoid and logarithm are named boundary operations here; the rendered structure is exact, but no decimal value is pinned for them.

render form, not float value\text{render form, not float value}
Loss structureThe same score table supplies the named sigmoid inputs for log-loss.Classifier score tablexyzdecisionprob00-20σ(-2)10-10σ(-1)21011/23111σ(1)sigmoid is named except at zero