Level sets

01A level set collects all inputs that produce the same scalar output#

A level set collects all inputs that produce the same scalar output. On a map, a contour joins places with the same height. For a model’s loss, a contour joins parameter settings with the same error. This lets us discuss several-variable functions without drawing a surface in a higher-dimensional space.

For $f(x,y)=x^2+y^2$, many inputs give 4. The circle marks all those inputs in the input plane.

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Choose one output $c$. Its level set contains exactly the domain inputs that produce $c$.

For $f(x,y)=x^2+y^2$, the level 5 means solve $x^2+y^2=5$. The point $(1,2)$ belongs because $1+4=5$; $(2,2)$ does not because $4+4=8$. A level set is a collection of inputs, while 5 is their common output.

02For f(x,y)=x^2+y^2, both (2,0) and (0,2) give 4#

For $f(x,y)=x^2+y^2$, both $(2,0)$ and $(0,2)$ give 4. The origin gives 0, so it is outside level 4. A level set fixes one output; the full graph records every input and its output.

03To test a candidate, substitute it into the function and compare with the requested level#

To test a candidate, substitute it into the function and compare with the requested level. If a task lists candidates, check every one; that list need not describe the entire level set.

04A closer look#

Contour plots of training loss show where different parameter choices perform equally under that loss. Moving along a contour preserves the plotted value; moving across contours changes it. A contour plot shows only the coordinates or slice specified by its axes.

Further questions