Local Minimum

Definition

A local minimum is a point where the function value is lower than nearby values.

Basic shape:

down -> bottom -> up

Local minimum = lowest point in a nearby region.

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Simple Example

Function:

f(x) = x^2

At x = 0:

f(0) = 0

Nearby values are higher:

f(-1) = 1
f(1) = 1

So x = 0 is a local minimum.

The function is lower there than around it.

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Local vs Global Minimum

A local minimum is best nearby.

A global minimum is best everywhere in the whole domain.

local minimum -> lowest in a neighborhood
global minimum -> lowest overall

A function can have many local minima.

Only one or several may be global minima.

Local best does not always mean overall best.

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Derivative Around A Local Minimum

For a smooth one-dimensional function, the derivative often changes like this:

negative -> zero -> positive

Meaning:

function decreases before the point
function is flat at the point
function increases after the point

A local minimum often appears where decreasing turns into increasing.

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Local Minimum In A Surface

For a two-input function, imagine a bowl.

At the bottom:

moving in nearby directions raises the value

That point is a local minimum.

In multiple dimensions, a local minimum is lower than nearby points in all directions.

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Why Local Minima Matter

If a loss function measures error, a lower value is better.

Then a local minimum can represent a point where nearby parameter changes do not reduce loss.

This can be useful if the loss is low.

It can be limiting if a better region exists elsewhere.

A local minimum can be a solution, or just a locally stable point.

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Not Every Flat Point Is A Minimum

If the derivative is zero, the point is flat.

But flat does not always mean minimum.

It could be:

local maximum
saddle point
flat plateau

A local minimum depends on surrounding values, not just zero slope.

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Final Summary

• A local minimum is lower than nearby values.
• It is not necessarily the lowest value overall.
• Smooth local minima often have zero derivative.
• In multiple dimensions, nearby movement in every direction raises the value.
• Local minima matter when trying to reduce a loss or objective.