Anthropomorphic AI: A reasonably efficient strategy for AI is to use a design loosely inspired by the human brain.

This is a rather anthropocentric view.

Yes, but intentionally so. ;)

We are getting into a realm where its important to understand background assumptions, which is why I listed some of mine. But notice I did quality with 'reasonably efficient' and 'loosely inspired'.

The human brain is a product of natural selection and is far from perfect.

'Perfect' is a pretty vague qualifier. If we want to talk in quantitative terms about efficiency and performance, we need to look at the brain in terms of circuit complexity theory and evolutionary optimization.

Evolution as a search algorithm is known (from what I remember from studying CS theory a while back) to be optimal in some senses: given enough time and some diversity considerations in can find global maxima in very complex search spaces.

For example, if you want to design a circuit for a particular task and you have a bunch of CPU time available, you can run a massive evolutionary search using a GA (genetic algorithm) or variant thereof. The circuits you will eventually get are the best known solutions, and in many cases incorporate bizarre elements that are even difficult for humans to understand.

Now, that same algorithm is what has produced everything from insect ganglions to human brains.

Look at the wiring diagram for a cockroach or a bumblebee compared to what it actually does, and if you compare that circuit to equivalent complexity computer circuits for robots we can build, it is very hard to say that the organic circuit design could be improved on. An insect ganglion's circuit organization, is in some sense perfect. (keep in mind organic circuits runs at less than 1khz). Evolution has had a long long time to optimize these circuits.

Can we improve on the brain - eventually we can obviously beat the brain by making bigger and faster circuits, but that would be cheating to some degree, right?

A more important question is: can we beat the cortex's generic learning algorithm.

The answer today is: no. Not yet. But the evidence trend looks like we are narrowing down on a space of algorithms that are similar to the cortex (deep belief networks, hierarchical temporal etc etc).

Many of the key problems in science and engineering can be thought of as search problems. Designing a new circuit is a search in the vast space of possible arrangement of molecules on a surface.

So we can look at how the brain compares to our best algorithms in smaller constrained search worlds. For smaller spaces (such as checkers), we have much simpler serial algorithms that can win by a landslide. For more complex search spaces, like chess the favor shifts somewhat but even desktop PC's can now beat grandmasters. Now go up one more complexity jump to a game like Go and we are still probably years away from an algorithm that can play at top human level.

Most interesting real world problems are many steps up the complexity ladder past Go.

Also remember this very important principle: the brain runs at only a few hundred hertz. So computers are cheating - they are over a million times faster.

So to compare the brains algorithms for a fair comparison, you would need to compare the brain to a large computer cluster that runs at only 500hz or so. Parallel algorithms do not scale nearly as well, so this is a huge handicap - and yet the brain still wins by a landslide in any highly complex search spaces.

Our neurons are believed to calculate using continuous values, but our computers are assemblies of discrete on/off switches. A properly structured AI could make much better use of this fact, not to mention be better at mental arithmetic than us.

Neurons mainly do calculate in analog space, but that is because this is vastly more efficient for probabilistic approximate calculation, which is what the brain is built on. A digital multiplier is many orders of magnitude less circuit space efficient than an analog multiplier - it pays a huge cost for its precision.

The brain is a highly optimized specialized circuit implementation of a very general universal intelligence algorithm. Also, the brain is Turing complete - keep that in mind.

The human mind is a small island in a massive mindspace, and the only special thing about it is that it's the first sentient mind we have encountered.

mind != brain

The brain is the hardware and the algorithms, the mind is the actual learned structure, the data, the beliefs, ideas, personality - everything important. Very different concepts.