BIZNIQUE / CALCULUS / 50
Explore a non-convex objective
Explore a non-convex objective
Separate the shape of the objective from the behaviour of one chosen update rule.
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THE IDEA
Read the example.
Read the example.
A non-convex function can violate the chord inequality and contain multiple stationary points. This example provides a small landscape in which the starting point and local update affect the final region reached.
f(x) = (x²−1)² + 0.15x
Use it well
Use it well
Non-convex does not mean impossible to optimise, and a visually smooth descent is not a certificate of global optimality. Appropriate guarantees depend on additional structure.
TRY IT YOURSELF
A quick check.
A quick check.
Why does this function fail the convexity test between −1 and 1?
Check your reasoning
Check your reasoning
The midpoint value f(0) = 1 lies above the average endpoint value, which is 0.
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