Minima and Maxima
A minimum is a point where a function is lower than nearby points.
A maximum is a point where a function is higher than nearby points.
Local and Global
A local minimum is lower than nearby points.
A global minimum is lower than all points in the search space.
Machine learning often works with complicated loss surfaces where finding a good local minimum may be enough.
The word "nearby" matters. A local minimum may be the best point in its neighborhood, but there may be a lower point somewhere else.
The word "all" matters for global minima. To prove a point is global, we need to know that no other point in the entire search space is lower.
In deep learning, the goal is usually not to prove that training found the global minimum. The useful question is more practical: did optimization find parameters that work well on new data?
Why This Matters
Optimization needs a target shape.
If we minimize loss, we search for lower points. If we maximize reward or likelihood, we search for higher points.
Most training code is written as minimization. When we want to maximize something, we often minimize its negative instead. Maximizing reward is the same directional idea as minimizing negative reward.
This sign change is common because it lets one optimizer interface handle many goals. We change the objective, not the basic training loop.
If the goal is to minimize loss, should training prefer a lower loss value?
Enter 1 for yes, 0 for no.
Compute it first, then check your number.
Hint
Solution
Yes. Minimization prefers lower loss. Enter 1. The optimizer is being asked
to move toward smaller values of the objective.
A point is lower than nearby points, but another faraway point is lower still. Is the first point a local minimum or a global minimum?
Answer it first, then check.
Hint
Nearby points decide local behavior. The whole search space decides global behavior.
Solution
It is a local minimum because it is lower than nearby points, but it is not global because a faraway point is lower.
If we want to maximize a reward R, can we instead minimize -R?
Answer it first, then check.
Hint
Larger R means smaller -R.
Solution
Yes. Maximizing R is equivalent to minimizing -R. When R gets larger,
-R gets smaller, so the direction of preference is flipped.
To claim a point is a global minimum, do we need to know it beats every point in the search space?
Answer it first, then check.
Hint
Global means the whole search space, not only the neighborhood.
Solution
Yes. A global minimum must be no higher than every other point in the search space.
Enter 1 if a model can be useful even when we have not proved that training
found a global minimum.
Compute it first, then check your number.
Hint
Think about validation behavior rather than a mathematical proof over all parameters.
Solution
Enter 1. For many models, we care whether the learned parameters work well,
not whether we can prove they are globally optimal.
Before Moving On
Optimization needs us to know whether we are minimizing or maximizing.