All fairly straightforward, however, there's a very important

point here that's perhaps so obvious that you might have missed it.

Imagine standing on the surface with its hills and valleys, and

we're trying to climb to the top of the highest peak.

That's no problem, we just look around, spot the tallest mountain and

walk straight towards it.

But what if we're walking at night?

This would be much like the scenario where we didn't have a nice

analytical expression for our function.

So we simply aren't able to plot the whole function and look around.

Perhaps each data point is the result of a week-long simulation on a super computer

or may be the outcome of an actual real world experiment.

These night-time scenarios very commonly arise in optimisation and

can be very challenging to solve.

However, if we're lucky, we might find that using a torch,

we can see the Jacobian vectors painted on road signs all around us and

each one would say, peak - this way.

We would have to remember that although the Jacobians all point uphill,

they don't necessarily point to the top of the tallest hill.

And you could find yourself walking up to one of the local maxima at C or E.

And even worse, when you get there,

you'll find that all of the road signs are pointing directly at you.

This nighttime hill-walking analogy is often used when discussing the problem of

optimisation.

However, it does have some misleading features.

Such as the fact that when you're really evaluating a function,

it's no problem to effectively transport all over the map by teleportation.

As you can try the function at many different places, but

there's no need to evaluate everywhere in between.

And the calculation costs the same, essentially,

no matter how far apart the points are, so we're not really walking.

Instead, we're going to switch to the analogy of a sandpit with an uneven base.

In the following exercises, you're going to try to find the deepest

point of a sandpit by measuring the depth of various points using a long stick.

This is a very deep sandpit, so once you push the stick down to the bottom,

there's no way to move it around sideways.

You just have pull it out and try somewhere else.

Also crucially, just like our nightwalking scenario,

you will have no idea what the peaks and troughs looks like at the bottom of

the pit because you can't see, the sand is in the way.

As you work through the exercise,

I'm hoping that you will start to pick up on a few of the subtleties of optimisation.

And hopefully, leave you with a few new questions as well,

see you next time and have fun in the sandpit.