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See how a starting point changes the destination
See how a starting point changes the destination
A local update can follow the basin it starts in rather than locate the best solution overall.
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THE IDEA
Read the example.
Read the example.
This one-dimensional objective has two wells with different depths. The demo applies a small fixed-step gradient update from a chosen starting point. Different initial points can move towards different local minima.
f(x) = (x²−1)² + 0.15x
Use it well
Use it well
A one-dimensional picture simplifies high-dimensional optimisation. Plateaus, saddle points, noise and conditioning can also matter; a local minimum is not the only challenge.
TRY IT YOURSELF
A quick check.
A quick check.
Does a gradient near zero prove that the point is a global minimum?
Check your reasoning
Check your reasoning
No. It can indicate a local minimum, local maximum, saddle-type stationary point in higher dimensions, or numerical stagnation.
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