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Singularity or Bust

It’s a question of some interest whether the Singularity will consist of just more exponential growth, or whether some superexponential growth mode is likely to happen (or is even possible), such as would be required for a real mathematical singularity.

On the side of exponential growth, as I pointed out here, is the fact that just about every actual mechanism for growth is actually exponential. The exceptions are exogenous: instead of something growing, some outside outside phenomenon is adding to what is counted as the something. In some corners of economics, trading and speculation analysis in particular, this is known as “herding.”

Now I’ve argued for years that that would in fact happen in AI over approximately the coming decade. The cause would be that as the capabilities of AI moved up across some threshold of effectiveness, mainstream investors would become interested and begin to pour money into it.

A super-exponential is not easy to tell from an exponential looking at a regular graph.  They both take off through the roof on a big enough scale. But if you graph them on a log scale, the exponential is always a straight line, and the super-exponential curves upwards.  Thus, the shift between one ordinary exponential growth mode and another with a higher exponent will be a super-exponential fillet curve between the two straight ones representing the successive growth modes on a log scale.

It turns out, unfortunately, that super-exponentials are far from rare in the market, and they generally do not mark the shift from one growth mode to another. Far to the contrary, they usually reflect that self-organized form of a Ponzi scheme called a market bubble.

It turns out that the herding effects leading to bubbles can often be recognized by fitting the market to an oscillating super-exponential called a “log-periodic power law.”  Consider this paper, for example:

We present a self-consistent model for explosive financial bubbles, which combines a mean-reverting volatility process and a stochastic conditional return which reflects nonlinear positive feedbacks and continuous updates of the investors’ beliefs and sentiments. The conditional expected returns exhibit faster-than-exponential acceleration decorated by accelerating oscillations, called “log-periodic power law.” Tests on residuals show a remarkable low rate (0.2%) of false positives when applied to a GARCH benchmark. When tested on the S&P500 US index from Jan. 3, 1950 to Nov. 21, 2008, the model correctly identifies the bubbles ending in Oct. 1987, in Oct. 1997, in Aug. 1998 and the ITC bubble ending on the first quarter of 2000. Different unit-root tests confirm the high relevance of the model specification. Our model also provides a diagnostic for the duration of bubbles: applied to the period before Oct. 1987 crash, there is clear evidence that the bubble started at least 4 years earlier. We confirm the validity and universality of the volatility-confined LPPL model on seven other major bubbles that have occurred in the World in the last two decades. …

They’re not just blowing smoke. LPPL analysis predicted the US housing bubble in this 2005 paper, for example. More recently, it predicted the crash in Chinese equity markets with a remarkable timing accuracy. (h/t TR arXiv blog)

Here was the prediction:

LPPL curve to Ch equity

LPPL curve to Ch equity

and here’s what actually happened:

Ch equity Mar-Aug

Ch equity Mar-Aug

So the big question is, when we see super-exponentials developing in AI or nanotech, which will be happening?

2 Responses to “Singularity or Bust”

  1. John Novak Says:

    Seems like another useful question would be, “How can we tell, if we can tell at all, the difference between a super-exponential bubble, and the transition from one exponent to another as part of a (for lack of a better word) ‘phase change?’”

    I’m not a big believer in Robin Hanson’s works, but one of his more interesting and probably more solid notions is that different technological regimes lead to different economic exponential trends. If that’s taken as given, then AI and nanotechnology are both strong candidates for exponent-altering technologies. In combination with the work you reference, the question of bubble vs phase change discrimination looms rather large– whoever works it out correctly stands to gain staggering amounts of wealth earlier than everyone else, and thus gain staggering amounts of influence.

  2. dz Says:

    You will see both. Financial super exponentials turn into bubbles because they are based on beliefs, not actual performance. AI/nanotech super exponential will result from recursive improved performance in hardware (perhaps software as well, but probably not since hardware gains will make software improvement too expensive to pursue).

    The earliest AI will create a financial bubble like we saw with internet stocks and solar power. This bubble will collapse, but the bubble of AI performance will not. Nanotech will follow the same route.

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