Multivariable functions

01Understand the idea#

A model can use several inputs to produce one number: features in, prediction out, or many weights in, loss out. A multivariable function describes that relationship. An ordered input is one complete setting of all its variables, not a list of separate trials.

In $f(x,y)=2x-y$, the input $(3,1)$ has two coordinates. Assign $x=3$ and $y=1$; both belong to one evaluation.

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For $f(x,y)=2x-y$, substitute the coordinates before combining the terms. The output is a scalar.

If $f(x,y)=2x-y$, the input $(3,1)$ means $x=3$ and $y=1$. Substitute first: $2(3)-1=5$. Swapping to $(1,3)$ gives $2(1)-3=-1$. The positions carry meaning, just as swapping two feature columns changes which coefficient multiplies them.

02Coordinate order matters#

Coordinate order matters. For $f(x,y)=2x-y$, swapping the input usually changes the result.

03More input coordinates still can produce one number#

More input coordinates still can produce one number. For $S(a,b,c)=ab-c$, evaluate at $(3,2,1)$ by substituting all three entries.

04A closer look#

When testing a model at a particular parameter vector, give every parameter its value before evaluating the scalar loss. Later, partial derivatives vary one parameter while holding the others fixed; the function you evaluate remains the same.

Further questions