An intelligence explosion is what hypothetically happens when a clever creature finds that the best way to achieve its goals is to make itself even cleverer first, and then to do so again and again as its heightened intelligence makes the the further investment cheaper and cheaper. Eventually the creature becomes uberclever and can magically (from humans’ perspective) do most things, such as end humanity in pursuit of stuff it likes more. This is predicted by some to be the likely outcome for artificial intelligence, probably as an accidental result of a smart enough AI going too far with any goal other than forwarding everything that humans care about.
In trying to get to most goals, people don’t invest and invest until they explode with investment. Why is this? Because it quickly becomes cheaper to actually fulfil a goal at than it is to invest more and then fulfil it. This happens earlier the cheaper the initial goal. Years of engineering education prior to building a rocket will speed up the project, but it would slow down the building of a sandwich.
A creature should only invest in many levels of intelligence improvement when it is pursuing goals significantly more resource intensive than creating many levels of intelligence improvement. It doesn’t matter that inventing new improvements to artificial intelligence gets easier as you are smarter, because everything else does too. If intelligence makes other goals easier a the same rate as it makes building more intelligence easier, no goal which is cheaper than building a given amount of intelligence improvement with your current intelligence could cause an intelligence explosion of that size.
Plenty of questions anyone is currently looking for answers to, such as ‘how do we make super duper nanotechnology?’, ‘how do we cure AIDS?’, ‘how do I get really really rich?’ and even a whole bunch of math questions are likely easier than inventing multiple big advances in AI. The main dangerous goals are infinitely expensive questions such as ‘how many digits of pi can we work out?’ and ‘please manifest our values maximally throughout as much of the universe as possible’. If someone were to build a smart AI and set it to solve any of those relatively cheap goals, it would not accidentally lead to an intelligence explosion. The risk is only with the very expensive goals.
The relative safety of smaller goals here could be confused with the relative safety of goals that comprise a small part of human values. A big fear with an intelligence explosion is that the AI will only know about a few of human goals, so will destroy everything else humans care about in pursuit of them. Notice that these are two different parameters: the proportion of the set of important goals the intelligence knows about and the expense of carrying out the task. Safest are cheap tasks where the AI knows about many of our values it may influence. Worst are potentially infinitely expensive goals with a tiny set of relevant values, such as any variation on ‘do as much of x as you can’.