the current function f ( · ; θ)
noisy data — fit it by hand
everything this family can reach
Drag to rotate · turn the knobs until the surface hugs the dots · R random θ · N new data
Knobs
How good is your fit?
TOTAL LOSS 𝓛(θ) — SUM OF SQUARED MISSES
–
turn the knobs, make this number small. You are doing — by hand — exactly what gradient descent will do for you.
The family
θ — ONE POINT IN ℝ³
(0.5, 1.2, −0.8)
Hypothesis class, as a set
the gray region is every value this family can reach — with θ unbounded, not just this page's sliders. Linear and waves: everything. Exponential: only y > 0 — that wall is intrinsic; no θ ever crosses zero. Negative data + exponential family = wrong hypothesis class.