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"
"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"
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.
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.
Is there a resource I can refer to, that is clear and explicit about the reasoning?
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.
The subtle elements of the theorems (both for inference and search) have had huge impact on the field of optimization theory. They touch upon (Kolmogorov) Complexity, Incompleteness, Halting Problems, and later related work on the physical limits of inference: One can not know everything about the universe, if you are a part of it, and philosophy: Is the universe itself a computer?
That different algorithms and search strategies will succeed for different tasks, does not imply that we have to try them all: We can use prior knowledge on what worked on related tasks. Also the tasks we care about, is only a very small (explorable) subset of all possible tasks.
You can read more in: Antipredictable Sequences: Harder to Predict Than Random Sequences by Huaiyu Zhu and Wolfgang Kinzel
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.
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.
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.
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.
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
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]
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.
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.