6 ms·

Note that adversarial is kind of a red herring, not sure why they mentioned that. The number of flips is unbounded regardless. Which is why it's not really incr

Note that adversarial is kind of a red herring, not sure why they mentioned that. The number of flips is unbounded regardless. Which is why it's not really incredible that it can be done: it can't, not as the problem was originally stated. What can be done is solving a different (but useful) problem than the one originally posed. I realize this sounds like a minor detail to someone who finds this cool (and so do I), but I don't think it is. It's kind of frustrating to be told that your intuition is wrong by someone smarter than you, when your intuition is actually correct and the experts are moving the goalposts. IMO, it makes people lose respect for experts/authority.

QuesnayJr 1y ago
I have the exact opposite reaction, that if someone told me the answer is "no" because it requires an unbounded number of coin flips that they were the ones trying to bullshit me. In antic's formulation, nothing is said about requiring a bounded number of flips.
- dataflow 1y ago
  "Simulate a truly random coin" implies it IMO. You're not simulating a truly random coin if you need unbounded time for a single flip. The truly random coin definitely doesn't need that. It just feels like a scam if someone sold me such a machine with that description - I'd want my money back. I don't expect everyone would feel the same, but I think a lot of people would.
  - thaumasiotes 1y ago
    You don't need unbounded time for a single flip, that's all in your imagination. The worst-case time is unbounded, but you can't achieve the worst case.
    - falcor84 1y ago
      It's very easy to achieve if someone hands you a "coin" that is made such that it never lands on tails. Sorry for still being in the adversarial mindset, but this means that you essentially have to hardcode a maximum number of same-side flips after which you stop trusting the coin.
      - Dylan16807 1y ago
        That's not a situation of "can't be done", that's just a consequence of casually describing the problem instead of exhaustively specifying it. Yes the bias has to be finite / the entropy can't be zero, and the slowdown is related to how big the bias is / how low the entropy is.
    - dataflow 1y ago
      > You don't need unbounded time for a single flip, that's all in your imagination. The worst-case time is unbounded, but you can't achieve the worst case. There literally isn't a bound, it can be arbitrarily large. This isn't just in my head, it's a fact. If it's bounded in your mind then what is the bound?
      - Dylan16807 1y ago
        It's not a fact of the real world, at least. "You can't achieve it" is true. A pretty small number of failures and you're looking at a trillion years to make it happen. And if you buffer some flips that number gets even smaller.
        dataflow 1y ago
        > A pretty small number of failures and you're looking at a trillion years to make it happen. This depends on the bias of the original coin. P(H) can be arbitrarily large, making P(HH) the likeliest possibility even for a trillion years. "This wouldn't happen in the real world" would be a sorry excuse for the deliberate refusal to clearly state the problem assumptions upfront. IMO, if you really want to pleasantly surprise people, you need to be forthcoming and honest with them at the beginning about all your assumptions. There's really no good excuse to obfuscate the question and then move the goalposts when they (very predictably) fall into your trap.
        Dylan16807 1y ago
        > This depends on the bias of the original coin. P(H) can be arbitrarily large > There's really no good excuse to obfuscate the question and then move the goalposts when they (very predictably) fall into your trap. Interesting. Because I see the guy pulling out the one-in-a-million coin and expecting it to run at a similar speed to be doing a gotcha on purpose, not falling into a trap and having the goalposts moved. And I think "well if it's a million times less likely to give me a heads, then it takes a million times as many flips, but it's just as reliable" is an answer that preserves the impressiveness and the goalposts. It's fast relative to the bias. Which seems like plenty to me when the original claim never even said it was fast. (And if the coin never gives you a heads then I'd say it no longer qualifies as randomly flipping a coin.)
  - antics 1y ago
    I'm not sure we're on the same page about what this result practically means, so let me re-state it a few different ways, so that people can draw their own conclusions: * The von Neumann approach will "appear to be O(1) (constant-time)" for any particular biased coin, but that constant might be big if the coin is very, VERY biased in one direction. * How can this be true? Every flip reduces the probability you do not have an answer exponentially. The "concentration" around the mean is very sharp—e.g., at 275 coin tosses for P(H)=0.5 (the fair case), the probability of not having an answer is smaller than 1 divided by the number of atoms in the known universe. It is technically possible, but I think most people would say that it's "effectively constant time" in the sense that we'd expect the coin to phase-shift through the desk before flipping it 275 times in a row and not getting answer. So it takes 275 flips, it's "constant" time! Interpret it how you like though. * As you make the coin more and more biased, that "horizon" increases linearly, in that 0.99^1,000 is approximately the same thing as 0.999^10,000. So, an order of magnitude increase in probability requires roughly an order of magnitude increase in the number of flips. This is why it's not useful for the adversarial case, and why adversarial extractors are held apart from normal extractors. Whether this is a "give me my money back" type thing is for you to decide. I think for most people the claim that you can simulate a fair coin from a biased coin in, effectively, O(1), and that the constant increases in O(n) in the bias, is plainly incredible. :)
  - dooglius 1y ago
    A real coin could repeatedly land on its side over and over indefinitely, every time you flipped it.
antics 1y ago
So, the problem in its original framing is: can we simulate a fair coin flip with an unfair coin? As stated, I do actually think the von Neumann response answer ("this is actually technically possible") is fair, in that if I wanted a solution in O(1), I think I should have to say so ahead of time. I suppose we'll have to disagree about whether this is incredible. The response shows that (1) this can be done at all, and (2) that the answer is exponentially likely as time goes on, not asymptotically, but for finite n. Incredible! You don't see finite-decay bounds very often! If you don't think that's incredible I invite you to ask a room full of people, even with the qualifications you deem appropriate, e.g., "solution does not need to be constant-time", or whatever.
- dataflow 1y ago
  What do you mean by "exponentially, not asymptotically, but for finite n"? Exponential is by definition asymptotic and continues infinitely, no? And to be clear, I'm not disagreeing (or agreeing) with the result being inherently incredible. I'm just saying it's not an incredible example of simulating a fair coin, because it just... isn't doing that. As an analogy: communicating with the Voyager spacecraft might be incredible, but it's not an incredible example of infinitely fast communication... because it just isn't. Telling me to go ask a room full of people whether they find either of these incredible is missing the point.
  - antics 1y ago
    > What do you mean by "exponentially, not asymptotically, but for finite n"? Exponential is by definition asymptotic and continues infinitely, no? In statistics, we generally separate asymptotic bounds—which usually make guarantees only asymptotically—from finite-decay bounds like Chernoff, which decay exponentially not only in the limit, but also at each particular finite n. The second is much rarer and much more powerful. This is an important distinction here: we are not talking about an asymptotic limit of coin flips. Each and every round of coin flips reduces the probability of not having an answer exponentially. > I'm just saying it's not an incredible example of simulating a fair coin, because it just... isn't doing that. As an analogy: communicating with the Voyager spacecraft might be incredible, but it's not an incredible example of infinitely fast communication... because it just isn't. Ok, no problem, it's up to you to decide if you want to use your own definition here. But, FYI, in computability theory, it is definitely fair game and very common to "simulate" some computation with something that is either much more memory-intensive or compute-intensive, in either the deterministic or probabilistic case. For example, it's pretty common to add or remove memory and see whether the new "simulated" routine takes more or less compute power than the "standard" algorithm, and that is kind of what people are up to with the PSPACE stuff that is going on right now. Using that lens—and this is entirely off the cuff, but in the 10 seconds of thinking, I believe probably mostly correct—this algorithm "simulates" a fair coin toss by using 1 extra bit of memory and O(p) compute, where p is the reciprocal of |0.5-<coin's bias>|. You can choose p to be infinite but you can do that for a sorting algorithm too (and that is why both sorting and this algorithm have DoS implications). Is this the best we can do? Well, that's the problem we're studying with randomness exractors: given this imperfect source of randomness, can we extract perfect randomness out of it at all?
dwattttt 1y ago
I don't think anyone would be surprised to hear that if a biased coin can only give one result, you can't extract randomness from it. And if it can only give the second result one in a million times, you could be flipping millions.

QuesnayJr 1y ago

I have the exact opposite reaction, that if someone told me the answer is "no" because it requires an unbounded number of coin flips that they were the ones trying to bullshit me. In antic's formulation, nothing is said about requiring a bounded number of flips.

dataflow 1y ago
"Simulate a truly random coin" implies it IMO. You're not simulating a truly random coin if you need unbounded time for a single flip. The truly random coin definitely doesn't need that. It just feels like a scam if someone sold me such a machine with that description - I'd want my money back. I don't expect everyone would feel the same, but I think a lot of people would.
- thaumasiotes 1y ago
  You don't need unbounded time for a single flip, that's all in your imagination. The worst-case time is unbounded, but you can't achieve the worst case.
  - falcor84 1y ago
    It's very easy to achieve if someone hands you a "coin" that is made such that it never lands on tails. Sorry for still being in the adversarial mindset, but this means that you essentially have to hardcode a maximum number of same-side flips after which you stop trusting the coin.
    - Dylan16807 1y ago
      That's not a situation of "can't be done", that's just a consequence of casually describing the problem instead of exhaustively specifying it. Yes the bias has to be finite / the entropy can't be zero, and the slowdown is related to how big the bias is / how low the entropy is.
  - dataflow 1y ago
    > You don't need unbounded time for a single flip, that's all in your imagination. The worst-case time is unbounded, but you can't achieve the worst case. There literally isn't a bound, it can be arbitrarily large. This isn't just in my head, it's a fact. If it's bounded in your mind then what is the bound?
    - Dylan16807 1y ago
      It's not a fact of the real world, at least. "You can't achieve it" is true. A pretty small number of failures and you're looking at a trillion years to make it happen. And if you buffer some flips that number gets even smaller.
      - dataflow 1y ago
        > A pretty small number of failures and you're looking at a trillion years to make it happen. This depends on the bias of the original coin. P(H) can be arbitrarily large, making P(HH) the likeliest possibility even for a trillion years. "This wouldn't happen in the real world" would be a sorry excuse for the deliberate refusal to clearly state the problem assumptions upfront. IMO, if you really want to pleasantly surprise people, you need to be forthcoming and honest with them at the beginning about all your assumptions. There's really no good excuse to obfuscate the question and then move the goalposts when they (very predictably) fall into your trap.
        Dylan16807 1y ago
        > This depends on the bias of the original coin. P(H) can be arbitrarily large > There's really no good excuse to obfuscate the question and then move the goalposts when they (very predictably) fall into your trap. Interesting. Because I see the guy pulling out the one-in-a-million coin and expecting it to run at a similar speed to be doing a gotcha on purpose, not falling into a trap and having the goalposts moved. And I think "well if it's a million times less likely to give me a heads, then it takes a million times as many flips, but it's just as reliable" is an answer that preserves the impressiveness and the goalposts. It's fast relative to the bias. Which seems like plenty to me when the original claim never even said it was fast. (And if the coin never gives you a heads then I'd say it no longer qualifies as randomly flipping a coin.)
- antics 1y ago
  I'm not sure we're on the same page about what this result practically means, so let me re-state it a few different ways, so that people can draw their own conclusions: * The von Neumann approach will "appear to be O(1) (constant-time)" for any particular biased coin, but that constant might be big if the coin is very, VERY biased in one direction. * How can this be true? Every flip reduces the probability you do not have an answer exponentially. The "concentration" around the mean is very sharp—e.g., at 275 coin tosses for P(H)=0.5 (the fair case), the probability of not having an answer is smaller than 1 divided by the number of atoms in the known universe. It is technically possible, but I think most people would say that it's "effectively constant time" in the sense that we'd expect the coin to phase-shift through the desk before flipping it 275 times in a row and not getting answer. So it takes 275 flips, it's "constant" time! Interpret it how you like though. * As you make the coin more and more biased, that "horizon" increases linearly, in that 0.99^1,000 is approximately the same thing as 0.999^10,000. So, an order of magnitude increase in probability requires roughly an order of magnitude increase in the number of flips. This is why it's not useful for the adversarial case, and why adversarial extractors are held apart from normal extractors. Whether this is a "give me my money back" type thing is for you to decide. I think for most people the claim that you can simulate a fair coin from a biased coin in, effectively, O(1), and that the constant increases in O(n) in the bias, is plainly incredible. :)
- dooglius 1y ago
  A real coin could repeatedly land on its side over and over indefinitely, every time you flipped it.

antics 1y ago

So, the problem in its original framing is: can we simulate a fair coin flip with an unfair coin? As stated, I do actually think the von Neumann response answer ("this is actually technically possible") is fair, in that if I wanted a solution in O(1), I think I should have to say so ahead of time. I suppose we'll have to disagree about whether this is incredible. The response shows that (1) this can be done at all, and (2) that the answer is exponentially likely as time goes on, not asymptotically, but for finite n. Incredible! You don't see finite-decay bounds very often! If you don't think that's incredible I invite you to ask a room full of people, even with the qualifications you deem appropriate, e.g., "solution does not need to be constant-time", or whatever.

dataflow 1y ago
What do you mean by "exponentially, not asymptotically, but for finite n"? Exponential is by definition asymptotic and continues infinitely, no? And to be clear, I'm not disagreeing (or agreeing) with the result being inherently incredible. I'm just saying it's not an incredible example of simulating a fair coin, because it just... isn't doing that. As an analogy: communicating with the Voyager spacecraft might be incredible, but it's not an incredible example of infinitely fast communication... because it just isn't. Telling me to go ask a room full of people whether they find either of these incredible is missing the point.
- antics 1y ago
  > What do you mean by "exponentially, not asymptotically, but for finite n"? Exponential is by definition asymptotic and continues infinitely, no? In statistics, we generally separate asymptotic bounds—which usually make guarantees only asymptotically—from finite-decay bounds like Chernoff, which decay exponentially not only in the limit, but also at each particular finite n. The second is much rarer and much more powerful. This is an important distinction here: we are not talking about an asymptotic limit of coin flips. Each and every round of coin flips reduces the probability of not having an answer exponentially. > I'm just saying it's not an incredible example of simulating a fair coin, because it just... isn't doing that. As an analogy: communicating with the Voyager spacecraft might be incredible, but it's not an incredible example of infinitely fast communication... because it just isn't. Ok, no problem, it's up to you to decide if you want to use your own definition here. But, FYI, in computability theory, it is definitely fair game and very common to "simulate" some computation with something that is either much more memory-intensive or compute-intensive, in either the deterministic or probabilistic case. For example, it's pretty common to add or remove memory and see whether the new "simulated" routine takes more or less compute power than the "standard" algorithm, and that is kind of what people are up to with the PSPACE stuff that is going on right now. Using that lens—and this is entirely off the cuff, but in the 10 seconds of thinking, I believe probably mostly correct—this algorithm "simulates" a fair coin toss by using 1 extra bit of memory and O(p) compute, where p is the reciprocal of |0.5-<coin's bias>|. You can choose p to be infinite but you can do that for a sorting algorithm too (and that is why both sorting and this algorithm have DoS implications). Is this the best we can do? Well, that's the problem we're studying with randomness exractors: given this imperfect source of randomness, can we extract perfect randomness out of it at all?

dwattttt 1y ago

I don't think anyone would be surprised to hear that if a biased coin can only give one result, you can't extract randomness from it. And if it can only give the second result one in a million times, you could be flipping millions.