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No Free Lunch theorem refesher: "if an algorithm performs well on a certain class of problems then it necessarily pays for that with degraded performance on th

No Free Lunch theorem refesher: "if an algorithm performs well on a certain class of problems then it necessarily pays for that with degraded performance on the set of all remaining problems" https://en.m.wikipedia.org/wiki/No_free_lunch_theorem https://en.m.wikipedia.org/wiki/No_free_lunch_theorem

CuriouslyC 10y ago
That doesn't apply to an ensemble of algorithms where the weights of a given member of the ensemble are adapted based on observations from the given domain. If it did, humans wouldn't be able to choose a good algorithm for specific cases, and obviously we can. Deep neural networks can be thought of as ensembles of smaller neural networks, though of course each member of the ensemble is going to share some degree of algorithmic bias. This suggests that perhaps deep neural networks with heterogeneous activation functions and branching structures will perform better than homogeneous networks.
- paulsutter 10y ago
  The human brain is exactly such an ensemble of algorithms, and look at all the cognitive biases we have[1]. That our brain was developed for foraging, hunting, and reproducing could explain why the code we write is so buggy. Look I'm just kidding around. Every algorithm has its advocates, and the arguments get tedious after a while. [1] https://en.m.wikipedia.org/wiki/List_of_cognitive_biases https://en.m.wikipedia.org/wiki/List_of_cognitive_biases
- IanCal 10y ago
  The No Free Lunch theorem doesn't really care how your algorithm does it, and an ensemble of algorithms is just another algorithm. It certainly applies. The reason it's not actually all that useful for real world applications is that problems we value are a subset of "all possible problems" for want of a better explanation. Problems are typically smooth, that is a small change in input variables results in a small change in 'fitness', which already limits how applicable the NFL theorem is. [disclaimer, based on lectures 10 years ago now]
- deong 10y ago
  > If it did, humans wouldn't be able to choose a good algorithm for specific cases, and obviously we can. This is a surprisingly commonly held fallacy in some AI circles. It's the idea that humans are mathematically perfect. When you phrase it that way, it's fairly obviously false, but you still see a lot of people argue things like "NFL doesn't apply to ensembles because humans..." or "machines can never be as intelligent as humans because...". The reality is that humans are subject to the same mathematical laws as machines. It's far more likely that my brain can't solve an NP-hard problem in polynomial time either. My brain can't beat random search on the set of all possible problems.
  - CuriouslyC 10y ago
    It doesn't imply that humans are mathematically perfect, but our brains are basically algorithm generating algorithms - we're not just weighting a set of preexisting solutions. To generalize quite a bit, saying NFL applies to general intelligence ends up implying there are problems for which efficient algorithms exist, but intelligence is literally incapable of discovering. That seems pretty absurd to me.
    - deong 10y ago
      The problem is that if our brains are not "magic" (essentially making this a religious argument), then they're operating via some algorithmic principles. If our brains are "algorithm generating algorithms", then I can in principle write an "algorithm generating algorithm" in silicon that does the same thing. And I know with mathematical certainty that my digital version is subject to NFL. So either my brain is too or we go back to religion to explain what non-physical process is responsible for our super-turing capabilities. The second point is that NFL is often interpreted in a weird way, where we only think about "interesting problems". It is defined on the set of all search or optimization problems. It says if your algorithm is better than exhaustive search over some subset of problems, it must be worse on the complement of that set. What does it mean for an algorithm to be efficient? Well, it's roughly speaking the number of steps it needed to take (assuming each step is the same amount of work, blah blah). OK, so an "efficient" algorithm must, by definition, prioritize some steps over others -- it's picking the "best" steps to take each time it has the choice. OK, so I'll just make up some instances of the problem that are custom-tailored so that what the algorithm thinks are the "best" steps always lead me in the wrong direction. You algorithm will then be worse than exhaustive search on my set of problems, precisely because it's choosing to avoid the steps I know to be good -- I defined the problem to make that happen. That is true of any algorithm you can conceive of. There will be problems for which the bias that makes the algorithm good on the problems you intended it to work on will be exactly the wrong bias. It doesn't matter if you say, "Aha, I'll let my algorithm generate new algorithms! Gotcha!" I'll just design a set of problems for which your algorithm generating algorithm will generate the wrong algorithms. Search is always about bias. Without bias, you have random search -- that's literally the textbook machine learning definition of bias. All NFL says is that if you have to worry about every possible problem, any bias you choose will be worse than random sometimes. There's no escape clause here. Your algorithm generating algorithm is still a search algorithm with its own biases, and it will still be worse than random on some subset of all possible problems.
eutectic 10y ago
The No Free lunch theorem is basically a consequence of the fact that almost all problems 'look random'; it doesn't really apply to the tiny subset that are of interest to humans.
- cgearhart 10y ago
  > it doesn't really apply to the tiny subset that are of interest to humans. That's a very misleading TL;DR. NFLT certainly _applies_ -- deep networks are not immune to NFLT -- it's just that NFLT isn't very useful because we can't use it as a basis for decisions. You can't detect that your algorithm's performance is being limited as a consequence of NFLT; and even if it were consequential, you would just see that the algorithm wasn't working very well, and there are _much_ more likely causes for that than NFLT.
  - eutectic 10y ago
    If you have two predictors p0 and p1 then it can be shown that the mixture p(data) = (p0(data) + p1(data)) / 2 incurs at most 1 bit of loss compared to the better of p0 and p1. In other words, for any pair of machine learning algorithms there is an algorithm which only performs marginally worse than either of the two on any given problem, and which may perform arbitrarily better. NFL equalises these two cases by smearing tiny losses over vast regions of possibility-space that are vanishingly unlikely.
    - ssivark 10y ago
      I often hear the claim that NFL just implies that different algorithms and learning strategies will succeed for different tasks. And that NFL has no practical consequences apart from that. Is there a resource I can refer to, that is clear and explicit about the reasoning?
      - eutectic 10y ago
        The fact that different algorithms are best for different tasks has more to do with computational complexity. If we didn't care about resource usage then we could just pick a sufficiently expressive model class (e.g. all algorithms) to mix over, and perform Bayesian inference. You can look up AIXI and Solomonoff induction for more thoughts along these lines.
mljoe 10y ago
"All remaining problems" in this case means the universe of all potential functions. Everything. Only a vanishingly small subset of every kind of function or dataset are of interest or even observable in nature.
Houshalter 10y ago
There is a theoretically optimal machine learning algorithm called Solomonoff Induction. Solomonoff Induction assumes a prior over all possible computer programs that could have produced the data. And it assumes that shorter, simpler programs are more likely than longer, more complex ones. Under such a prior, the no free lunch theorem doesn't apply. NFLT assumes that all machine learning problems have infinite information. Infinite Kolmogorov complexity. That all positive and negative examples are labelled completely randomly without any underlying pattern or reason. Which is obviously untrue.
- deong 10y ago
  Solomonoff Induction isn't really an "algorithm" in the way we normally think of algorithms, as it isn't computable. There are computable approximations, but at that point, you lose the claim of theoretically optimal.
  - Xcelerate 10y ago
    > There are computable approximations, but at that point, you lose the claim of theoretically optimal. True, but I'm surprised more work isn't being done on "searching the space of programs that produce the data". There's very little research on this topic other than a few papers on minimum description length (MDL). I feel that this is probably the eventual route to AGI. We know what the "optimal" predictor is in theory; now work out the best time/memory approximation to it for practical purposes.
    - Cybiote 10y ago
      There's lots of work that's being done related to program synthesis and inductive logic programming. You can even view the more sophisticated recurrent Neural nets, with more complex memory structures, as differentiable programs. Then you're searching the space of programs guided by gradients. SGD effectively acts as an additional prior (assumption) by the kind of solutions it tends towards. The reason (the general) you don't hear much about them if you don't go looking is that the state of the art hasn't budged much for the past couple decades. It's the same graph algorithms, searching and sorting toy problems. The search space over programs is difficult to traverse and it remains to be seen what the added compute power + gradients gets us. On a more practical level, the learning 2 search paradigm can be viewed as also searching for a particular program under certain strict constraints that make search tractable. Probabilistic programming where the priors and likelihoods are themselves complex programs instead of simple distributions from the exponential family are effectively also searching for programs.

CuriouslyC 10y ago

That doesn't apply to an ensemble of algorithms where the weights of a given member of the ensemble are adapted based on observations from the given domain. If it did, humans wouldn't be able to choose a good algorithm for specific cases, and obviously we can. Deep neural networks can be thought of as ensembles of smaller neural networks, though of course each member of the ensemble is going to share some degree of algorithmic bias. This suggests that perhaps deep neural networks with heterogeneous activation functions and branching structures will perform better than homogeneous networks.

paulsutter 10y ago
The human brain is exactly such an ensemble of algorithms, and look at all the cognitive biases we have[1]. That our brain was developed for foraging, hunting, and reproducing could explain why the code we write is so buggy. Look I'm just kidding around. Every algorithm has its advocates, and the arguments get tedious after a while. [1] https://en.m.wikipedia.org/wiki/List_of_cognitive_biases https://en.m.wikipedia.org/wiki/List_of_cognitive_biases
IanCal 10y ago
The No Free Lunch theorem doesn't really care how your algorithm does it, and an ensemble of algorithms is just another algorithm. It certainly applies. The reason it's not actually all that useful for real world applications is that problems we value are a subset of "all possible problems" for want of a better explanation. Problems are typically smooth, that is a small change in input variables results in a small change in 'fitness', which already limits how applicable the NFL theorem is. [disclaimer, based on lectures 10 years ago now]
deong 10y ago
> If it did, humans wouldn't be able to choose a good algorithm for specific cases, and obviously we can. This is a surprisingly commonly held fallacy in some AI circles. It's the idea that humans are mathematically perfect. When you phrase it that way, it's fairly obviously false, but you still see a lot of people argue things like "NFL doesn't apply to ensembles because humans..." or "machines can never be as intelligent as humans because...". The reality is that humans are subject to the same mathematical laws as machines. It's far more likely that my brain can't solve an NP-hard problem in polynomial time either. My brain can't beat random search on the set of all possible problems.
- CuriouslyC 10y ago
  It doesn't imply that humans are mathematically perfect, but our brains are basically algorithm generating algorithms - we're not just weighting a set of preexisting solutions. To generalize quite a bit, saying NFL applies to general intelligence ends up implying there are problems for which efficient algorithms exist, but intelligence is literally incapable of discovering. That seems pretty absurd to me.
  - deong 10y ago
    The problem is that if our brains are not "magic" (essentially making this a religious argument), then they're operating via some algorithmic principles. If our brains are "algorithm generating algorithms", then I can in principle write an "algorithm generating algorithm" in silicon that does the same thing. And I know with mathematical certainty that my digital version is subject to NFL. So either my brain is too or we go back to religion to explain what non-physical process is responsible for our super-turing capabilities. The second point is that NFL is often interpreted in a weird way, where we only think about "interesting problems". It is defined on the set of all search or optimization problems. It says if your algorithm is better than exhaustive search over some subset of problems, it must be worse on the complement of that set. What does it mean for an algorithm to be efficient? Well, it's roughly speaking the number of steps it needed to take (assuming each step is the same amount of work, blah blah). OK, so an "efficient" algorithm must, by definition, prioritize some steps over others -- it's picking the "best" steps to take each time it has the choice. OK, so I'll just make up some instances of the problem that are custom-tailored so that what the algorithm thinks are the "best" steps always lead me in the wrong direction. You algorithm will then be worse than exhaustive search on my set of problems, precisely because it's choosing to avoid the steps I know to be good -- I defined the problem to make that happen. That is true of any algorithm you can conceive of. There will be problems for which the bias that makes the algorithm good on the problems you intended it to work on will be exactly the wrong bias. It doesn't matter if you say, "Aha, I'll let my algorithm generate new algorithms! Gotcha!" I'll just design a set of problems for which your algorithm generating algorithm will generate the wrong algorithms. Search is always about bias. Without bias, you have random search -- that's literally the textbook machine learning definition of bias. All NFL says is that if you have to worry about every possible problem, any bias you choose will be worse than random sometimes. There's no escape clause here. Your algorithm generating algorithm is still a search algorithm with its own biases, and it will still be worse than random on some subset of all possible problems.

eutectic 10y ago

The No Free lunch theorem is basically a consequence of the fact that almost all problems 'look random'; it doesn't really apply to the tiny subset that are of interest to humans.

cgearhart 10y ago
> it doesn't really apply to the tiny subset that are of interest to humans. That's a very misleading TL;DR. NFLT certainly _applies_ -- deep networks are not immune to NFLT -- it's just that NFLT isn't very useful because we can't use it as a basis for decisions. You can't detect that your algorithm's performance is being limited as a consequence of NFLT; and even if it were consequential, you would just see that the algorithm wasn't working very well, and there are _much_ more likely causes for that than NFLT.
- eutectic 10y ago
  If you have two predictors p0 and p1 then it can be shown that the mixture p(data) = (p0(data) + p1(data)) / 2 incurs at most 1 bit of loss compared to the better of p0 and p1. In other words, for any pair of machine learning algorithms there is an algorithm which only performs marginally worse than either of the two on any given problem, and which may perform arbitrarily better. NFL equalises these two cases by smearing tiny losses over vast regions of possibility-space that are vanishingly unlikely.
  - ssivark 10y ago
    I often hear the claim that NFL just implies that different algorithms and learning strategies will succeed for different tasks. And that NFL has no practical consequences apart from that. Is there a resource I can refer to, that is clear and explicit about the reasoning?
    - eutectic 10y ago
      The fact that different algorithms are best for different tasks has more to do with computational complexity. If we didn't care about resource usage then we could just pick a sufficiently expressive model class (e.g. all algorithms) to mix over, and perform Bayesian inference. You can look up AIXI and Solomonoff induction for more thoughts along these lines.

mljoe 10y ago

"All remaining problems" in this case means the universe of all potential functions. Everything. Only a vanishingly small subset of every kind of function or dataset are of interest or even observable in nature.

Houshalter 10y ago

There is a theoretically optimal machine learning algorithm called Solomonoff Induction. Solomonoff Induction assumes a prior over all possible computer programs that could have produced the data. And it assumes that shorter, simpler programs are more likely than longer, more complex ones. Under such a prior, the no free lunch theorem doesn't apply. NFLT assumes that all machine learning problems have infinite information. Infinite Kolmogorov complexity. That all positive and negative examples are labelled completely randomly without any underlying pattern or reason. Which is obviously untrue.

deong 10y ago
Solomonoff Induction isn't really an "algorithm" in the way we normally think of algorithms, as it isn't computable. There are computable approximations, but at that point, you lose the claim of theoretically optimal.
- Xcelerate 10y ago
  > There are computable approximations, but at that point, you lose the claim of theoretically optimal. True, but I'm surprised more work isn't being done on "searching the space of programs that produce the data". There's very little research on this topic other than a few papers on minimum description length (MDL). I feel that this is probably the eventual route to AGI. We know what the "optimal" predictor is in theory; now work out the best time/memory approximation to it for practical purposes.
  - Cybiote 10y ago
    There's lots of work that's being done related to program synthesis and inductive logic programming. You can even view the more sophisticated recurrent Neural nets, with more complex memory structures, as differentiable programs. Then you're searching the space of programs guided by gradients. SGD effectively acts as an additional prior (assumption) by the kind of solutions it tends towards. The reason (the general) you don't hear much about them if you don't go looking is that the state of the art hasn't budged much for the past couple decades. It's the same graph algorithms, searching and sorting toy problems. The search space over programs is difficult to traverse and it remains to be seen what the added compute power + gradients gets us. On a more practical level, the learning 2 search paradigm can be viewed as also searching for a particular program under certain strict constraints that make search tractable. Probabilistic programming where the priors and likelihoods are themselves complex programs instead of simple distributions from the exponential family are effectively also searching for programs.