I expect "slow takeoff," which we could operationalize as the economy doubling over some 4 year interval before it doubles over any 1 year interval. Lots of people in the AI safety community have strongly opposing views, and it seems like a really important and intriguing disagreement. I feel like I don't really understand the fast takeoff view.
(Below is a short post copied from Facebook. The link contains a more substantive discussion. See also: AI impacts on the same topic.)
I believe that the disagreement is mostly about what happens before we build powerful AGI. I think that weaker AI systems will already have radically transformed the world, while I believe fast takeoff proponents think there are factors that makes weak AI systems radically less useful. This is strategically relevant because I'm imagining AGI strategies playing out in a world where everything is already going crazy, while other people are imagining AGI strategies playing out in a world that looks kind of like 2018 except that someone is about to get a decisive strategic advantage.
Here is my current take on the state of the argument:
The basic case for slow takeoff is: "it's easier to build a crappier version of something" + "a crappier AGI would have almost as big an impact." This basic argument seems to have a great historical track record, with nuclear weapons the biggest exception.
On the other side there are a bunch of arguments for fast takeoff, explaining why the case for slow takeoff doesn't work. If those arguments were anywhere near as strong as the arguments for "nukes will be discontinuous" I'd be pretty persuaded, but I don't yet find any of them convincing.
I think the best argument is the historical analogy to humans vs. chimps. If the "crappier AGI" was like a chimp, then it wouldn't be very useful and we'd probably see a fast takeoff. I think this is a weak analogy, because the discontinuous progress during evolution occurred on a metric that evolution wasn't really optimizing: groups of humans can radically outcompete groups of chimps, but (a) that's almost a flukey side-effect of the individual benefits that evolution is actually selecting on, (b) because evolution optimizes myopically, it doesn't bother to optimize chimps for things like "ability to make scientific progress" even if in fact that would ultimately improve chimp fitness. When we build AGI we will be optimizing the chimp-equivalent-AI for usefulness, and it will look nothing like an actual chimp (in fact it would almost certainly be enough to get a decisive strategic advantage if introduced to the world of 2018).
In the linked post I discuss a bunch of other arguments: people won't be trying to build AGI (I don't believe it), AGI depends on some secret sauce (why?), AGI will improve radically after crossing some universality threshold (I think we'll cross it way before AGI is transformative), understanding is inherently discontinuous (why?), AGI will be much faster to deploy than AI (but a crappier AGI will have an intermediate deployment time), AGI will recursively improve itself (but the crappier AGI will recursively improve itself more slowly), and scaling up a trained model will introduce a discontinuity (but before that someone will train a crappier model).
I think that I don't yet understand the core arguments/intuitions for fast takeoff, and in particular I suspect that they aren't on my list or aren't articulated correctly. I am very interested in getting a clearer understanding of the arguments or intuitions in favor of fast takeoff, and of where the relevant intuitions come from / why we should trust them.

Not sure. I encountered this once in my research, but the preprint is not out yet (alas, I'm pretty sure that this will still be not enough to reach commercial viability, so pretty niche and academic and not a very strong example).
Regarding "this is not common": Of course not for problems many people care about. Once you are in the almost-optimal class, there are no more giant-sized fruit to pick, so most problems will experience that large jumps never, once or twice over all of expected human history (sorting is NlogN even if you are a super-intelligence) (pulling the numbers 0,1,2 out of my ass; feel free to do better). On the other hand, there is a long tail of problems very few people care about (e.g. because we have no fast solution and hence cannot incorporate a solver into a bigger construction). There, complexity-class jumps do not appear to be so uncommon.
Cheap prediction: Many visual machine-learning algos will gain in complexity class once they can handle compressed data (under the usual assumption that, after ordering by magnitude, coefficients will decay like a power-law for suitable wavelet-transforms of real-world input data; the "compression" introduces a cut-off / discretization, and a cool ML algo would only look at the coefficients it cares about and hence be able to e.g. output a classification in finite time for infinite input data). Also cheap prediction: This will turn out to be not as hot as it sounds, since GPUs (and to a lesser extent CPUs) are just too damn good at dealing with dense matrices and FFT is just too damn fast.
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Since I'm bad at computer history, some maybe-examples that spring to mind:
Probable example: Asymmetric crypto.
Very-maybe examples:
Fast fourier transform (except if you want to attribute it to Gauss). Linear programming (simplex algorithm). Multi-grid methods for PDE / the general shift from matrix-based direct solvers to iterative operator-based solvers. Probabilistic polynomial identity testing. Bitcoin (not the currency, rather the solution to trust-less sybil-safe global consensus that made the currency viable). Maybe Zk-snark (to my eternal shame I must admit that I don't understand the details of how they work).
For large economic impact, possibly discontinuous: MP3 / the moment where good audio compression suddenly became viable and made many applications viable. I think this was more of an engineering advance (fast software decoding, with large hardware overhang on general-purpose CPUs), but others here probably know more about the history.
Everyone, feel free to list other historic examples of kinda-discontinuous algorithmic advances and use your superior historical knowledge to tear down my probably very bad examples.
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Separate point: Many processes require "critical mass" (which I called "viability"), and such processes amplify discontinuities / should be modeled as discontinuities.
Physics / maths intuition pump would be phase transitions or bifurcations in dynamical systems; e.g. falling off a saddle-node looks quite discontinuous if you have a separation of time-scales, even if it is not discontinuous once you zoom in far enough. Recursive self-improvement / general learning does have some superficial resemblance to such processes, and does have a separation of time-scales. Tell me if you want me to talk about the analogies more, but I was under the impression that they have been discussed ad-nauseam.
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I'll continue to think about historic examples of complexity-class-like algorithmic advancements with economic impact and post if anything substantial comes to mind.