Take the next learning step

Choose an initial value and step size. Step through a quadratic objective or play the whole trajectory.

Teaching model · computed in your browser

Follow the gradient

Choose a starting point and learning rate, then inspect every update toward a quadratic minimum.

f(x)=\tfrac12(x-a)^2,\qquad x_{t+1}=x_t-\eta(x_t-a)

a is the location of the minimum; x is the current parameter; η is the learning rate. The derivative is x − a. For this unit-curvature quadratic, an initial error contracts precisely when 0 < η < 2. This condition does not automatically apply to a different loss function.

Try this: keep the same starting point and compare η = 0.5, 1, 2, and 2.5. Watch the distance from the minimum rather than the learning rate alone.

Experiment controls

Starting x -2-5 to 8 · step 0.5Minimum location a 3-3 to 3 · step 0.5Learning rate η 0.250 to 3 · step 0.05

Playback starts only when you choose Play. It pauses when this tab is hidden or the lab leaves view. Reduced-motion preferences slow the updates; Step always gives you direct control. Runs stop after 40 updates or at the display limit.

The default worked example is shown below.

Completed updates0 / 40

Current x-2

Loss ½(x − a)²12.5

Error multiplier0.75

The solid trace gives x after each update; the dashed line marks the minimum at a = 3. Convergent: each update moves toward the minimum without crossing it. The axis rescales to include all completed updates. Solid: x · Dashed: minimum location a

Convergent: each update moves toward the minimum without crossing it. Press Step to compute x₁ = x₀ − η(x₀ − a).

Computed update history
Update	x	Gradient x − a	Loss ½(x − a)²
0	-2	-5	12.5

Turn exploration into understanding

Learn the ideas behind this lab.

Follow a worked example, practice on your own, and return to the original project sources.

Progress is kept in this browser. How learning works