Worse Than Random

And that's why I always say that the power of natural selection comes from the selection part, not the mutation part.

And the power of the internal combustion engine comes from the fuel part... Right, or at least, not even wrong. It seems that my congratulations a few months ago for your apparent immanent escape from simple reductionism were premature.

But it is an inherently odd proposition that you can get a better picture of the environment by adding noise to your sensory information - by deliberately throwing away your sensory acuity. This can only degrade the mutual information between yourself and the environment. It can only diminish what in principle can be extracted from the data.

It is certainly counterintuitive to think that, by adding noise, you can get more out of data. But it is nevertheless true.

Every detection system has a perceptual threshold, a level of stimulation needed for it to register a signal. If the system is mostly noise-free, this threshold is a ’sharp’ transition. If the system has a lot of noise, the theshold is ‘fuzzy’. The noise present at one moment might destructively interact with the signal, reducing its strength, or constructively interact, making it stronger. The result is that the threshold becomes an average; it is no longer possible to know whether the system will respond merely by considering the strength of the signal.

When dealing with a signal that is just below the threshold, a noiseless system won’t be able to perceive it at all. But a noisy system will pick out some of it - some of the time, the noise and the weak signal will add together in such a way that the result is strong enough for the system to react to it positively.

You can see this effect demonstrated at science museums. If an image is printed very, very faintly on white paper, just at the human threshold for visual detection, you can stare right at the paper and not see what’s there. But if the same image is printed onto paper on which a random pattern of grey dots has also been printed, we can suddenly perceive some of it - and extrapolate the whole from the random parts we can see. We are very good at extracting data from noisy systems, but only if we can perceive the data in the first place. The noise makes it possible to detect the data carried by weak signals.

When trying to make out faint signals, static can be beneficial. Which is why biological organisms introduce noise into their detection physiologies - a fact which surprised biologists when they first learned of it.

A combination dial often has a tolerance of 2 in either direction. 20-45-35 will open a lock set to 22-33-44.

I certainly hope not! I think you intended 20-35-45 for the first or 22-44-33 for the second.

You might want to footnote, before anyone starts making noise about the ant example, that colony selection is not a case of group selection but a case of individual selection on the queen (and drones), since the rest of the ants don't reproduce.

Daniel C. Dennett argues that random data is useful in optimization as it's absolutely key to experimentation -- you must "wiggle" different variables, as he says, and observe changes. This is a foundational part of his argumentation in Darwin's Dangerous Idea and Freedom Evolves.

Caledonian: couldn't you always do better in such a case, in principle (ignoring resource limits), by increasing resolution?

That is a fine observation from Caledonian. The noise is not being added to existing sense data, it's being added before the signal hits the receptors - but to stay in tune with the proposed scenario, we can easily imagine that the organism has already internalised the signals at that point.

Re: Daniel Dennet: that's not really right - random wiggling is better than no wiggling at all, but it is still an extremely bad way to generate variation.

Caledonian: couldn't you always do better in such a case, in principle (ignoring resource limits), by increasing resolution?

I double-checked the concept of 'optical resolution' on Wikipedia.Resolution is (roughly speaking) the ability to distinguish two dots that are close-together as different - the closer the dots can be and still distinguished, the higher the resolution, and the greater detail that can be perceived.

I think perhaps you mean 'sensitivity'. It's the ability to detect weak signals close to perceptual threshold that noise improves, not the detail.

I'm surprised at the pushback on this rather simple straightforward point. In fact, it seems kind of beaten to death so I hope it has some cool unexpected consequence soon-to-be revealed.

Besides being easy, randomness has the amusing property of helping overcome bias.

In fact, since random search is unbiased search, it now seems that overcoming bias should be frowned on! Perhaps "Finding Effective Biases" is a better title for this blog.

Caledonian: Yes, I did. So: can't you always do better in principle by increasing sensitivity?

drone: In fact, since random search is unbiased search, it now seems that overcoming bias should be frowned on! Perhaps "Finding Effective Biases" is a better title for this blog.

"Inductive Bias"

Re: Tim Tyler

Granted -- it's certainly an incredibly inefficient way of generating variation, but it's infinitely better than no wiggling at all in that no wiggling would seem to deliver you unto the "stupid" algorithm which gets stuck at the local maxima.

In simulating biological evolution I might argue that random noise is a prerequisite to true congruence, but if the goal isn't accurate emulation and instead is efficient optimization, more engineered noise seems to be the clear means to a trained algorithm.

This leads me to believe that, although 'you cannot do exceptionally well by finding a noise source that is exceptionally random,' there are specific use cases that make exceptional randomness desirable.

"This may not sound like a profound insight, since it is true by definition. But consider - how many comic books talk about "mutation" as if it were a source of power? Mutation is random. It's the selection part, not the mutation part, that explains the trends of evolution."

I think this is a specific case of people treating optimization power as if it just drops out of the sky at random. This is certainly true for some individual humans (eg., winning the lottery), but as you point out, it can't be true for the system as a whole.

"These greedy algorithms work fine for some problems, but on other problems it has been found that greedy local algorithms get stuck in local minima."

Er, do you mean local maxima?

"When dealing with a signal that is just below the threshold, a noiseless system won’t be able to perceive it at all. But a noisy system will pick out some of it - some of the time, the noise and the weak signal will add together in such a way that the result is strong enough for the system to react to it positively."

In such a case, you can clearly affect the content of the signal, so why not just give it a blanket boost of ten points (or whatever), if the threshold is so high that you're missing desirable data?

So...noise can be useful in decision theory as long as you don't expect it to do any work. And the mistake gets easier to make the more complex your system. Sounds right enough to me.

[nitpick]

Your 'by definition' link needs a look, Eliezer.

Or imagine that the combination changes every second. In this case, 0-0-0-0, 0-0-0-0 is just as good as the randomized algorithm - no better and no worse.

If it changes every second, trying the same set of four over and over is marginally better than random.

If you've just entered 0-0-0-0 and got it wrong, then on the next try every sequence except 0-0-0-0 has a small chance of being the correct sequence from the previous, and hence is incorrect this try.

Anyone care to work out exactly how much better off 0-0-0-0 is than a random set in this case?

[/nitpick]

Another perspective on the issue comes from considering lazer printer drivers - another system with thresholding. Adding noise produces better dithering than not doing so. It's not as good as Floyd-Steinberg dithering, of course - but it's often "cheaper".

Caledonian: Yes, I did. So: can't you always do better in principle by increasing sensitivity?

There are lots of things we can do in principle if we ignore the fact that reality limits the principles that are valid.

As the saying goes: the difference between 'in principle' and 'in practice' is that in principle there is no difference between them, and in practice, there is.

If you remove the limitations on the amount and kind of knowledge you can acquire, randomness is inferior to the unrandom. But you can't remove those limitations.

A few posters might want to read up on Stochastic Resonance, which was surprisingly surprising a few decades ago. I'm getting a similar impression now from recent research in the field of Compressive Sensing, which ostensibly violates the Nyquist sampling limit, highlighting the immaturity of the general understanding of information-theory.

In my opinion, there's nothing especially remarkable here other than the propensity to conflate the addition of noise to data, with the addition of "noise" (a stochastic element) to (search for) data.

This confusion appears to map very well onto the cybernetic distinction between intelligently knowing the answer and intelligently controlling for the answer.

@Joshua_Simmons: I got to thinking about that idea as I read today's post, but I think Eliezer_Yudkowsky answered it therein: Yes, it's important to expirment, but why must your selection of what to try out, be random? You should be able to do better by exploiting all of your knowledge about the structure of the space, so as to pick better ways to experiment. To the extent that your non-random choices of what to test do worse than random, it is because your understanding of the problem is so poor as to be worse than random.

(And of course, the only time when searching the small space around known-useful points is a good idea, is when you already have knowledge of the structure of the space...)

@Caledonian: That's an interesting point. But are you sure the effect you describe (at science museums) isn't merely due to the brain now seeing a new color gradient in the image, rather than randomness as such? Don't you get the same effect from adding an orderly grid of dots? What about from aligning the dots along the lines of the image?

Remember, Eliezer_Yudkowsky's point was not that randomness can never be an improvement, but that it's always possible improve beyond what randomness would yield.

How would you categorize the practice of randomly selecting the pivot element in a quicksort?

Silas: @Caledonian: That's an interesting point. But are you sure the effect you describe (at science museums) isn't merely due to the brain now seeing a new color gradient in the image, rather than randomness as such? Don't you get the same effect from adding an orderly grid of dots? What about from aligning the dots along the lines of the image?

Yep. Adding a set of coherent modulations will do better than noise to improve your sensor, because you're guaranteed to get at least some modulations of a sufficiently high level, and you can subtract out the modulations afterward to arrive at a superior picture of the environment.

Remember, Eliezer_Yudkowsky's point was not that randomness can never be an improvement, but that it's always possible improve beyond what randomness would yield.

Lotta commenters seem to have entirely missed this.

Brian, the reason we do that is to avoid the quicksort algorithm being stupid and choosing the worst-case pivot every time. The naive deterministic choices of pivot (like "pick the first element") do poorly on many of the permutations of the input which are far more probable than 1/n! because of the types of inputs people give to sorting algorithm, namely, already or nearly-already sorted input. Picking the middle element does better because inputs sorted inside to outside are rarer, but they're still far more likely than 1/n! apiece. Picking a random element is a very easy way to say "hey, any simple algorithm I think up will do things that correlate with algorithms other people think up, so will hit worst-case running times more often than I'd like, so I'll avoid correlation with other people".

There are variants of quicksort that completely avoid worst-case complexity by choosing the true median of the list each time. They incur an extra cost that makes average case worse, though, and they're usually not the best choice because we're almost always not trying to avoid the worst-case for a single run, we're actually trying to make the average case faster.

Consider this scenario:

There are a large number of agents independently working on the same problem (for example, trying to find a string that hash-collides with some given string), but they cannot communicate in any way, they don't have any identification information about each other, they don't know how many other agents there are working on the problem (they aren't even sure there are any). It seems to me that each agent should decide at random where to start searching, not to fool each other but to avoid pointlessly duplicating each others' work.

Are you sure there is always something better than randomness?

It's not always possible to improve beyond what randomness would yield. Consider, for example, the coin toss predicting game. Or the face-the-firing-squad game.

GreedyAlgorithm, yes that's mostly why it's done. I'd add that it applies even when the source of the ordering is not a person. Measurement data can also follow the type of patterns you'd get by following a simple, fixed rule.

But I'd like to see it analyzed Eliezer's way.

How does the randomness tie in to acquired knowledge, and what is the superior non-random method making better use of that knowledge?

Using the median isn't it, because that generally takes longer to produce the same result.

"""What about from aligning the dots along the lines of the image?"""

Wouldn't you need to find them first?

To be precise, in every case where the environment only cares about your actions and not what algorithm you use to produce them, any algorithm that can be improved by randomization can always be improved further by derandomization.

Or you may have heard people talking about "emergence" as if it could explain complex, functional orders. People will say that the function of an ant colony emerges - as if, starting from ants that had been selected only to function as solitary individuals, the ants got together in a group for the first time and the ant colony popped right out. But ant colonies have been selected on as colonies by evolution. Optimization didn't just magically happen when the ants came together.

I don't think the point of stressing emergence is to explain via the conjuring of magic. The point is to counter the idea that something as simple and stupid as ants couldn't possibly do something complex other than by magic. It's people’s lack of appreciation for emergent behavior that is the problem. They see the simple but can't understand how to get the complex out of it. They then believe that there must be some intelligent force behind the emergent behavior.

We are currently living through a crisis that is in large part due to this lack of appreciation for emergent behavior. Not only people in general but trained economists, even Nobel laureates like Paul Krugman, lack the imagination to understand the emergent behavior of free monetary systems. Lacking the belief that such systems could actually operate without some outside intelligence in control they set up central planning agencies like the Fed. Then like any central planning agency trying to control a market it will fail, precisely because the emergent behavior of the market is more powerful, more intelligent, in solving the problem of resource allocation than any committee.

Even with all the evidence staring them in the face they will still not grasp their mistake. It's obvious to those who comprehend the emergent behavior that interest rates have been set way below market rates, for too long, and that is the cause of the current crisis. The committee made the mistake of thinking it could use general price signals directly to decide on the price signal for interest rates. Price stability, keeping inflation within certain bounds was believed to be the control metric to follow. Unfortunately "the market" was trying to deflate prices due to productivity increases caused by the Reagan/Thatcher revolution. Holding prices steady (to slight inflation) was contrary to market forces and therefore the wrong move.

Free markets are emergent behavior. It is quite amazing that complex coordination can operate on such simple principles without some central agency. The fact that it works better than any central agency could is even more amazing, to most people. Once you understand it then it's not so amazing but it is very difficult to understand. Ben Bernanke doesn't understand and Alan Greenspan didn't understand before him. Emergent behavior is non-magic masquerading as magic.

So emergent behavior is a useful concept when you know what it's about. It's a bias checker.

@Mike Plotz: It's true that you can't do better than random in predicting (theoretical nonphysical) coin tosses, but you also can't do worse than random. As Eliezer pointed out, the claim isn't "it is always possible to to better than random", but "any algorithm which can be improved by adding randomness, can be improved even more without adding randomness."

Brian: How does the randomness tie in to acquired knowledge, and what is the superior non-random method making better use of that knowledge?

On a random list, one choice of pivot is as good as another. On a sorted list, though, the ends are always a bad choice. Picking the first item is thus stupider than average, and you can do better with a random pivot. But you can do even better by picking the middle item: it's always the median when the list is sorted, and it's no worse than random when the list is random.

What if you have an intelligent adversary trying to mount a denial-of-service attack against your quicksort? Then you should expect to get inputs that your algorithm is likely to do poorly on. If you pick pivots randomly then your expected time to sort a list is completely independent of the list's initial ordering. Even though the number of permutations that cause degenerate O(n^2) behavior is the same, you've made it impossible to pick one ahead of time. Randomness can defeat an intelligent adversary for the same reason that optimization processes don't work in a completely random universe.

Anyone care to work out exactly how much better off 0-0-0-0 is than a random set in this case? The probability of success at each cycle goes up to 1/9999 from 1/10000 (after the first cycle).

It's obvious to those who comprehend the emergent behavior that interest rates have been set way below market rates, for too long, and that is the cause of the current crisis. By "comprehend the emergent behavior" do you mean that you have a vague intuitive feel for this, or that you have the equations relating interest rates to other factors, along with enough mathematical theory to make specific quatitative predictions? If you (or people like you who "comprehend the emergent behavior") did not make a lot of money out of the current crisis, then your statement is wrong. Explanations after the fact are simply stories.

Would you discourage, where resources exist to find alternative strategies, use of any random element in choosing samples for statistical hypothesis testing?

Don't you get the same effect from adding an orderly grid of dots?

If the static could change over time, you could get a better sense of where the image lies. It's cheaper and easier - and thus 'better' - to let natural randomness produce this static, especially since significant resources would have to be expended to eliminate the random noise.

What about from aligning the dots along the lines of the image?

To be precise, in every case where the environment only cares about your actions and not what algorithm you use to produce them, any algorithm that can be improved by randomization can always be improved further by derandomization.

It is not clear this can be shown to be true. 'Improvement' depends on what is valued, and what the context permits. In the real world, the value of an algorithm depends on not only its abstract mathematical properties but the costs of implementing it in an environment for which we have only imperfect knowledge.

"It is not clear this can be shown to be true. 'Improvement' depends on what is valued, and what the context permits. In the real world, the value of an algorithm depends on not only its abstract mathematical properties but the costs of implementing it in an environment for which we have only imperfect knowledge."

Eliezer specifically noted this in the post:

"Sometimes it is too expensive to take advantage of all the knowledge that we could, in theory, acquire from previous tests. Moreover, a complete enumeration or interval-skipping algorithm would still end up being stupid. In this case, computer scientists often use a cheap pseudo-random algorithm, because the computational cost of using our knowledge exceeds the benefit to be gained from using it. This does not show the power of randomness, but, rather, the predictable stupidity of certain specific deterministic algorithms on that particular problem."

@Caledonian and Tiiba: If we knew where the image was, we wouldn't need the dots.

Okay, let's take a step back: the scenario, as Caledonian originally stated, was that the museum people could make a patron better see the image if the museum people put random dots on the image. (Pronouns avoided for clarity.) So, the problem is framed as whether you can make someone else see an image that you already know is there, by somehow exploiting randomness. My response is that, if you already know the image is there, you can improve beyond randomness, but just putting the dots there in a way that highlights the hidden image's lines. In any case, from that position, Eliezer_Yudkowsky is correct in that you can only improve the patron's detection ability for that image, by exploiting your non-random knowledge about the image.

Now, if you want to reframe that scenario, you have to adjust the baselines appropriately. (Apples to apples and all.) Let's look at a different version:

I don't know if there are subtle, barely-visible images that will come up in my daily life, but if there are, I want to see them. Can I make myself better off by adding random gray dots to my vision? By scattering physical dots wherever I go?

I can's see how it would help, but feel free to prove me wrong.

To be precise, in every case where the environment only cares about your actions and not what algorithm you use to produce them, any algorithm that can be improved by randomization can always be improved further by derandomization.

Consider the heads-tails-edge game. Betting on edge is a deterministic strategy with a low payoff. It can be improved on by randomisation: randomly betting on heads with p=0.5 and tails with p=0.5 is a stochastic strategy which offers improved returns - and there is no deterministic strategy which produces superior results to it.

To be precise, in every case where the environment only cares about your actions and not what algorithm you use to produce them, any algorithm that can be improved by randomization can always be improved further by derandomization.

Isn't this trivially true? Isn't the most (time) efficient algorithm always a giant lookup table?

Isn't the most (time) efficient algorithm always a giant lookup table?

That approach doesn't help an embedded observer much, if they find that they cannot determine the entries to the table - because of the uncertainty principle.

"We are currently living through a crisis that is in large part due to this lack of appreciation for emergent behavior. Not only people in general but trained economists, even Nobel laureates like Paul Krugman, lack the imagination to understand the emergent behavior of free monetary systems."

"Emergence", in this instance, is an empty buzzword, see http://lesswrong.com/lw/iv/the_futility_of_emergence/. "Imagination" also seems likely to be an empty buzzword, in the sense of http://lesswrong.com/lw/jb/applause_lights/.

"precisely because the emergent behavior of the market is more powerful, more intelligent, in solving the problem of resource allocation than any committee."

Markets do not allocate resources anywhere near optimally, and sometimes they do even worse than committees of bureaucrats; the bureaucrats, for instance, may increase utility by allocating more resources to poor people on grounds of higher marginal utility per dollar per person.

"Once you understand it then it's not so amazing but it is very difficult to understand. Ben Bernanke doesn't understand and Alan Greenspan didn't understand before him."

If you think you know more than Bernanke, then why haven't you become rich by making better-than-expected bets?

"It can be improved on by randomisation: randomly betting on heads with p=0.5 and tails with p=0.5 is a stochastic strategy which offers improved returns - and there is no deterministic strategy which produces superior results to it."

Eliezer has already noted that it is possible for a random strategy to be superior to a stupid deterministic strategy:

"But it is possible in theory, since you can have things that are anti-optimized. Say, the average state has utility -10, but the current state has an unusually low utility of -100. So in this case, a random jump has an expected benefit. If you happen to be standing in the middle of a lava pit, running around at random is better than staying in the same place. (Not best, but better.) A given AI algorithm can do better when randomness is injected, provided that some step of the unrandomized algorithm is doing worse than random."

The point of the post is that a random strategy is never better than the best possible deterministic strategy. And assuming that you're betting on real, physical coinflips, a random strategy is actually worse than the deterministic strategy of betting that the coin will come up heads if it started as heads and vice versa (see http://www.npr.org/templates/story/story.php?storyId=1697475).

Daniel I. Lewis, as I said, lists can have structure even when that structure is not chosen by a person.

"Let's say, for the sake of argument, that you get sorted lists (forwards or backwards) more often than chance, and the rest of the time you get a random permutation."

Let's not say that, because it creates an artificial situation. No one would select randomly if we could assume that, yet random selection is done. In reality, lists that are bad for selecting from the middle are more common than by random chance, so random beats middle.

If you put the right kind of constraints on the input, it's easy to find a nonrandom algorithm that beats random. But those same constraints can change the answer. In your case, part of the answer was the constraint that you added.

I was hoping for an answer to the real-world situation.

Brian, you want an answer to the real-world situation? Easy. First assume you have a source of inputs that is not antagonistic, as discussed. Then measure which deterministic pivot-choice algorithms would work best on large samples of the inputs, and use the best. Median-of-three is a great pivot choosing algorithm in practice, we've found. If your source of inputs is narrower than "whatever people anywhere using my ubergeneral sort utility will input" then you may be able to do better. For example, I regularly build DFAs from language data. Part of this process is a sort. I could implement this plan and possibly find that, I don't know, median of first, last, and about-twenty-percent-of-the-way-in is in general better. I anticipate the effort would not be worth the cost, so I don't, but there you are.

You don't have to put constraints on the input, you can just measure them (or guess well!). They're probably already there in real-world situations.

Tom, I made it clear which comments I was addressing by quoting them.

If you really want to hack my example, you should bear in mind that the "heads", "tails", and "edge" values are produced by taking bytes from a cryptographic source of randomness, and using heads: more 0s than 1s; tails: more 1s than 0s; edge: other cases.

GreedyAlgorithm,

"If your source of inputs is narrower than 'whatever people anywhere using my ubergeneral sort utility will input' then you may be able to do better."

That's actually the scenario I had in mind, and I think it's the most common. Usually, when someone does a sort, they do it with a general-purpose library function or utility.

I think most of those are actually implemented as a merge sort, which is usually faster than quicksort, but I'm not clear on how that ties in to the use of information gained during the running of the program.

What I'm getting at is that the motivation for selecting randomly and any speedup for switching to merge sort don't seem to directly match any of the examples already given.

In his explanation and examples, Eliezer pointed to information gained while the algorithm is running. Choosing the best type of selection in a quicksort is based on foreknowledge of the data, with random selection seeming best when you have the least foreknowledge.

Likewise, the difference between quicksort and other sorts that may be faster doesn't have an obvious connection to the type information that would help you choose between different selections in a quicksort.

I'm not looking for a defense of nonrandom methods. I'm looking for an analysis of random selection in quicksort in terms of the principles that Eliezer is using to support his conclusion.

Brian: FYI, mergesort isn't faster than a good quicksort. Skiena's The Algorithm Design Manual, for example, says that "experiments show that a where a properly implemented quicksort is implemented well, it is typically 2-3 times faster than mergesort or heapsort" (2nd ed., p. 129).

I'm still waiting to hear what Eliezer says about your example too, as quicksort does seem to be an example of the best known implementation making necessary use of (pseudo-) randomness, and quicksort is of course extremely well-understood.

Brian, a very simple analysis would say something like, "If you think the middle element is likely to be an unusually bad pivot, exclude it as a possible pivot during random selection. If you have no such belief, why not go ahead and use them? If you do have such a belief, why let random selection occasionally select bad pivots?"

Stuart Armstrong ,

"By 'comprehend the emergent behavior' do you mean that you have a vague intuitive feel for this, or that you have the equations relating interest rates to other factors, along with enough mathematical theory to make specific quantitative predictions?"

If I believe that a individual or committee cannot determine a market price other than by actually observing one then why on earth do you think I am claiming to be able to "make specific quantitative predictions?"

Those economists that make the mistake of thinking they can make a killing in the market are notoriously bad at it. Greenspan being an example.

Austrian theory holds that you cannot make the kind of quantitative predictions you expect to. So when you can predictibly and consistently make quantitative predictions on your economic theory then you can prove Austrian theory wrong.

"If you (or people like you who "comprehend the emergent behavior") did not make a lot of money out of the current crisis, then your statement is wrong."

Not at all. One can predict that the actions of say, Mugabe in Zimbabwe, would be an economic disaster without being able to capitalize on it.

"Explanations after the fact are simply stories."

But the explainations were given before the fact. Austrian theory exists as a model and it predicts certain outcomes given certain actions.

In the theory, below market interest rates result in low savings, overborrowing, trade deficits, asset inflation, and market bubbles. Everything that has been occuring makes sense in light of the theory.

BTW, that theory predicted the Great Depression before the fact, and this crash before the fact. It also predicted the existence of stagflation before the fact.

Likewise it predicts that the current actions of the Fed if continued are going to lead to inflation, all other things being equal.

The current crisis was caused by fractional reserve banking and monetary inflation and the current solutions being proposed and acted on by the likes of Krugman are to inject more money. These are precisely the wrong things to do.

Interest rates are prices like any other. It's well understood that when you put a price ceiling on a good that you get shortages as consumers try to consume more at the lower price and producers produce less. That is exactly what we are experiencing now, a shortage of capital due to a price ceiling on interest rates. This is a simple case of trying to violate economic law and being stung by it.

Austrian theory has more to say on the matter in that this kind of monetary inflation causes misallocation of resources but I'll refrain from writing a long article. It isn't at all about "vague intuitive" feelings either. It consists of clear, understandable mechanisms by which each result can be deduced from the model.

"'Emergence';, in this instance, is an empty buzzword.

Buzzword in this instance is a buzzword. This sentence is merely an assertion. I read that article before I wrote my argument. The phrase, "emergent behavior" and the word "emergence" have a specific meaning and it isn't about giving a "mysterious answer to a mysterious queston".

For example, Mises can and does give a complete and non-mysterious explaination of how the business cycle is a result of fractional reserve banking. Likewise, he can explain how market prices arise, and why markets clear at the market price. All in a very reductionist fashion.

"Imagination" also seems likely to be an empty buzzword, ..."

No, it's has the same exact meaning as in "Creationist lack the imagination to understand how evolution works." or "Behe, lacking the imagination to understand how eyes arose proposes the concept of irreducible complexity".

"Markets do not allocate resources anywhere near optimally, and sometimes they do even worse than committees of bureaucrats; the bureaucrats, for instance, may increase utility by allocating more resources to poor people on grounds of higher marginal utility per dollar per person."

I didn't use the word "optimally" anywhere in my comment. I said it "solved the problem of resource allocation."

The rest of you statement is just a bald assertion. In fact, "allocating resources optimally", is an ill defined concept. Allocated optimally in reference to what value system? The very concept of thinking you can make a utility function in the way you construct it is absurd, and ignores factors of that wealth redistribution that would harm the very poor it was suppose to help. Actual real world experimentation with redistributive systems like communism have shown it to be a bust.

Your statement is true in the same sense that it is possible by brownian motion for an elephant to fly. Markets are analogous to distributed supercomputers where each individual participates as a processor and prices are the signaling mechanisms between the processors. If you mess with those signals you get predictable results, depending on what you do and what kind of price you mess with.

"If you think you know more than Bernanke, then why haven't you become rich by making better-than-expected bets?"

Wow, you assume a lot. Firstly, my mother was a sharecropper, and my father a poorly paid college professor. I paid my own way through college. So it's not like I had a big nest egg to invest.

I did however predict the surge in sliver prices. I did converted my IRA to bullion. I did quadruple my investment in four years.

Besides, it's not the position of my theory that you get rich by understanding economics. That's your ridiculous claim. Did you apply that theory to Greenspan or Bernanke? Why in the world aren't all these economists retired rich? I mean that's your theory, right?

Dear Brian,

Sorry if I was overly brusque in my response. you have obviously given the subject more thought than I gave you credit for.

The line you wrote here is the critical one: BTW, that theory predicted the Great Depression before the fact, and this crash before the fact. It also predicted the existence of stagflation before the fact.

Where were these predictions? Did they represent a majority view of Austrian School economics? How is their false positive, false negative score? (basically, I'm asking for a statistical survey of Austrian school economics track record; that seems to be the minimum requirement to endorse the statement above). If their track record is so demonstrably superior, someone should have done such a survey.

Because for the moment a simple "classical economics + power law of random fluctuations (possibly giving way somewhat to a gaussian distribution for rare events)" seems a much more economical theory fitting the data (and yes, those were my economic opinions before the crash; I didn't act on it, because I have no money to spare :-).

This talk of "general principles" is rubbish from a physical scientist's perspective; were what you were saying a general principle there would be no such thing as stochastic resonance, nor would Monte Carlo integration be so useful a technique. And if there is some "derandomized" alternative to simulated annealing that is superior in finding minima (or maxima) in ugly solution spaces, it hasn't been found. "So what?" you might ask, but consider that tens or hundreds of thousands of engineers, scientists, and mathematicians have thought about this problem before you 'blogged it.

Yours is an interesting post, certainly, but perhaps you should substitute modesty for flowery finishes and hasty elevation of a minor finding to "general principle" status.

To those discussing randomized Quicksort: relevant to your discussion about more intelligent behaivour might be the Introsort algorithm.

See https://secure.wikimedia.org/wikipedia/en/wiki/Introsort

"Introsort or introspective sort is a sorting algorithm designed by David Musser in 1997. It begins with quicksort and switches to heapsort when the recursion depth exceeds a level based on (the logarithm of) the number of elements being sorted. It is the best of both worlds, with a worst-case O(n log n) runtime and practical performance comparable to quicksort on typical data sets. Since both algorithms it uses are comparison sorts, it too is a comparison sort."

"Or you may have heard people talking about "emergence" as if it could explain complex, functional orders. People will say that the function of an ant colony emerges - as if, starting from ants that had been selected only to function as solitary individuals, the ants got together in a group for the first time and the ant colony popped right out. But ant colonies have been selected on as colonies by evolution. Optimization didn't just magically happen when the ants came together."

There is nothing magical about emergence and emergence is a well document phenomena. In fact what is stated above is purely false the emergence in the ant colony example comes from the simpler less effectual units working together as more complex highly effectual whole. The idea of emergence is what forms part of the basis for swarm logic.

""Lo! When we make our artificial neurons noisy, just like biological neurons, they work better! Behold the healing life-force of entropy!""

One has to consider when one models a system what features one is trying to model and what ones to generalize away.

[some ad hominems deleted from an over-long comment]

THERE ARE BOOKS ON RANDOMIZED ALGORITHMS. READ THEM BEFORE YOU POST THIS GARBAGE AND DELETE THIS BLOG COMMENT BECAUSE IT DISAGREES WITH YOUR SENSELESS ACT OF NOT LOOKING ANYTHING UP BEFORE YOU WRITE.

There is nothing magical about emergence and emergence is a well document phenomena.

Unfortunately, the term "emergence" has come to have two meanings. One is uncontroversial. The other is drivel.

"Sorry if I was overly brusque in my response."
No, I don't believe you are sorry. I think you have a particular view on economics that colors your questions. You're looking for some angle to justify those beliefs. It's quite clear that the Austrians are correct about what is occuring right now.

"Because for the moment a simple "classical economics + power law of random fluctuations (possibly giving way somewhat to a gaussian distribution for rare events)" seems a much more economical theory fitting the data ..."

Really? The "classical economics" was abandoned long ago by mainstream economists. Therefore I'm not sure what you mean by it. Since there is no school of economics that has any way of predicting prices that even somewhat fit I'm thinking that your just rationalizing here. Since prices, from the standpoint of ignorance, already looks gaussian it seems that almost any old theory could make this claim you are making. Doesn't really provide a means to test between one theory and the next.

Why should I be satisified with a theory that claims to predict prices, but can't, and only "fits" if one assumes that it's wrong by a gaussian corrective value? Meanwhile it has no explanatory value, and very often is not only counterintuitive but self contradictory?

That's a way to falsify a theory, self-contradiction, that many schools of economics completely ignore. The "liquidity trap" being an example of this kind of crazy thinking. Failing to have a proper theory some schools of economics are forced to accept what is obviously a ridiculous and self contradictory explanation of events. They drop fallacies like a chicken it a factory farm. Krugman for example actually believes in the brokent window fallacy in addition to the "liquidity trap" fallacy.

It's quite clear what is going to happen next with the economy. I personally predicted that the Fed was going to take the actions it is now taking back in 2003, choose inflation. Their backs are against the wall for prior stupid actions and now they have choosen to inflate. They increased the base money supply over the past two months at a rate of 341%. I laugh at the people who think they can just "mop this up later". Mop it up with what, their credibility? They are figuratively in the position of someone trying to pull themselves up by grabbing their bootstraps.

This is one of my favorite posts. I'm disappointed that I never grasped this during my computer science education.

"it's not impossible, but it could happen" - typo? Should be "It's not very likely, but it could happen" or something like that?

Here's an algorithm that I've heard is either really hard to derandomize, or has been proven impossible to derandomize. (I couldn't find a reference for the latter claim.) Find an arbitrary prime between two large numbers, like 10^500 - 10^501. The problem with searching sequentially is that there are arbitrarily long stretches of composites among the naturals, and if you start somewhere in one of those you'll end up spending a lot more time before you get to the end of the stretch.

I recently ran into a nice example of how randomness can be improved upon but you may not want to de-randomize.

One of my personal tools is a setup which archives & preserves URLs for later reference, since pretty much every web page will disappear or die eventually. The core is a loop which reads a URL out of a text file and then run the Internet requests to archive it.

The problem is that I want to archive hundreds & thousands of URLs a week, but I can only archive 1 every 20 seconds. The file is sorted to eliminate duplicates. The obvious approach is to archive the first URL in the file, which is also alphabetically first.

The problem with this is that as ever more URLs are dumped into the file, we wind up with a sort of resource starvation where URLs which begin with an 'x' or 'z' may never get archived because the archiving keeps on processing the 'a' and 'b' URLs first. We don't want to disproportionately favor any particular domain.

So what do we do? Select a random URL, of course. This avoids the bias problem - now the 'x' URLs have the same chance as a similar number of 'a' URLs.

But is random selection optimal? Nope. As with the lock example, we can improve the distribution by adding some sort of 'memory' about what domains have already gotten a URL archived; so one could randomly select a URL - as long as its domain hasn't gotten a URL archived in the last n archives.

Why haven't I done this? Because the archiver is meant to be able to survive frequent crashes and shutdowns (it runs on my laptop), and to get a memory over more than a few days, I would have to implement code to regularly serialize out the memory and so on. This wouldn't be too bad in Haskell (perhaps another 5 or 6 lines to the code), but that represents a pretty big increase in source code and the complexity of the design, and the benefit is fairly small. In this case, I'm happy to use the simpler dumber randomized solution.

As a general principle, on any problem for which you know that a particular unrandomized algorithm is unusually stupid - so that a randomized algorithm seems wiser - you should be able to use the same knowledge to produce a superior derandomized algorithm.

This claim is at least as strong as BPP = P, which while suspected is definitely not proven, and appears to be very difficult to prove. In particular, even if BPP = P, current knowledge has no way of de-randomizing all (or even most) polynomial-time randomized algorithms.

Am I misunderstanding something here?

Omega creates one more copy of you, then he asks each of you to say either “zero” or “one”; he will give each of you $10 unless you both pick the same number. You have no idea which copy you are, and cannot communicate with the other copy before you both make your choice.

The strategies “choose 0” and “choose 1” both lose, but the strategy “choose 0 with 50% probability and 1 with 50% probability” has a 50% probability of winning.

"Randomness hath no power"? This may be true for 1-player games (i.e. noncompetitive decision theory problems), but in 2-player or multiplayer game theory situations, a mixed strategy can be optimal, and in fact in some cases a pure strategy may be the worst thing one can do (rock paper scissors is a great example). Randomness hath power, it just varies by the field one enters. In fact, according to the 2nd law of thermodynamics, randomness may be the most difficult power to resist, for the universe is constantly losing to it.

It is theoretically possible for a poorly designed "pseudo-random algorithm" to be stupid relative to the search space; for example, it might always jump in the same direction. But the "pseudo-random algorithm" has to be really shoddy for that to happen. You're only likely to get stuck with that problem if you reinvent the wheel instead of using a standard, off-the-shelf solution.

You don't need that much shoddiness to get very weird things in certain situations (IIRC there was a PRNG getting some value wrong by 20 standard deviations in a Monte Carlo simulation of the Ising model because of a non-zero three-way correlation between the n-th, (n + 63)-th, and (n + 126)-th outputs or something like that), and some off-the-shelf solutions are shoddier than you may think.

OK, let me give you another example of the lock device. Each time a code is tried, the correct code changes to (previous code) + 2571 mod 10000. You don't know this. You won't find out before opening the door, because of limited feedback. Sequential check of every code will fail, but let you know that the correct code changes (if there is a correct code). Constantly guessing the same code because you think it'll randomly change to that one will fail. Random guessing will eventually succeed. Using randomness prevents you from getting stuck due to your own stupidity or an opponent. There is no method for always beating randomness, making it a reliable failsafe against always losing.

You won't always have knowledge about the problem, and on occasion what you think is knowledge is wrong. Random may be stupid, but it has a lower bound for stupidity.

What if the lock has multiple combinations that are close to each other. In this case the brute force does much worse than a random search. Because the correct combinations could be 98,99,100. This is true for real combination locks btw, they usually aren't too sensitive to being off by a single number, and so multiple combinations do work.

Or an even simpler case, you need to create a single algorithm to break a large number of locks. And all the locks have the same combination. And you can't save memory or modify the algorithm after it it's finished, so you have to pick the best one.

If you pick a deterministic algorithm, there is a significant chance that the combination could be something like 99999. Then you have the worst case time. Whereas the time for the random algorithm will be a Gaussian distribution around the average case.

The expected completion time for each is the same, but the deterministic algorithm has a significantly higher chance of hitting a worst case scenario.

Now a random search is bad because it can forget what it's already tried, but you can just add a memory or some other way of creating a random ordering.

And in general, there exists a pattern that will disrupt any deterministic algorithm and give it much worse performance. And they aren't rare pathological cases either.