4 ms·

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 bett

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.
- BickNowstrom 10y ago
  NFL is probably the worst mis-quoted theorem in ML. 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.

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.
BickNowstrom 10y ago
NFL is probably the worst mis-quoted theorem in ML. 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.