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Calculus and Gradients
Curvature
Curvature describes how strongly a function bends, which affects how local directions change.
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Curvature
Definition
Curvature describes how strongly a function bends.
Low curvature:
nearly straight
High curvature:
bends sharply
Curvature is about bending, not just increasing or decreasing.
Slope vs Curvature
Slope tells how steep the function is at a point.
Curvature tells how quickly the slope changes.
Example:
straight ramp -> slope exists, curvature is low
bowl shape -> slope changes as you move, curvature exists
Slope is local direction; curvature is change in local direction.
Simple Examples
Linear function:
f(x) = 2x + 1
The slope is constant.
Curvature:
none
Quadratic function:
f(x) = x^2
The slope changes as x changes.
Curvature:
present
Curved functions have changing slope.
Why Curvature Affects Movement
If a function is nearly flat, a step may change the output slowly.
If a function bends sharply, the same size step may change behavior quickly.
Example:
gentle bowl -> gradual changes
narrow bowl -> rapid direction changes
Curvature affects how reliable a local direction remains after moving.
Curvature And Minima
Around a local minimum, curvature is often upward.
Shape:
bowl
But the bowl can be:
wide and shallow
narrow and steep
Both can have a minimum, but movement behaves differently around them.
The shape around a minimum matters, not only the minimum value.
Curvature In Multiple Dimensions
With multiple inputs, curvature can differ by direction.
Example:
steep in one direction
flat in another direction
This makes high-dimensional surfaces more complex than simple one-dimensional curves.
A surface can bend differently in different directions.
Why This Matters Later
Optimization uses local information to move through a function.
Curvature affects:
how quickly gradients change
how sensitive steps are
whether a region is flat or sharp
Curvature describes the terrain that optimization moves through.
Final Summary
- Curvature describes how strongly a function bends.
- Slope tells direction; curvature tells change in direction.
- Linear functions have no curvature.
- Curved functions have changing slope.
- Curvature can differ across directions.
- It affects how local movement behaves.
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Continue the series.
Convexity