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>> The issue of unseen data arises for one simple reason: the model does not capture causal mechanisms in the DGP. The model instead learns a reduced form that

>> The issue of unseen data arises for one simple reason: the model does not capture causal mechanisms in the DGP. The model instead learns a reduced form that represents the training sample. Thanks for articulating it so clearly, that's very much how I think of it! >> This issue stands independent of the amount of data. Indeed, if you had a model that identified all relevant causal mechanisms, then of course more data is still better. Well, my point -and, as far as I understand it, Vapnik's point, in my link above- is that if a model can generalise well, then it doesn't need a lot of data, even if a lot of data is available. In terms of a model that can identify causal relations, then it should be able to do so without lots of data. My intuition for that is that if the model needs lots of data it will inevitably learn to represent the sampling error in the data generation process (DGP?). A model that can learn good generalisations from few data on the other hand is in a sense immunised against sampling error because it can ignore most of the data. Vapnik makes an analogy about a good teacher. He asks, what is it that a good teacher does that helps his students learn better? He answers it by saying that the good teacher gives his students "good predicates", what I would call background knowledge, from which the students can build good statistical invariants. But the students must also be able to build good statistical invariants _on their own_, otherwise it doesn't matter what good predicates the teacher can give them, the student's can't learn good invariants. I think that's the point of the challenge - Vapnik is asking for learning from few examples (not unreasonably few, I think) as a way to demonstrate that a system is learning good underlying principles- his good predicates and statistical invariants. In the past the machine learning community has sidestepped the issue by finding ways to augment data where it was scarce and then claim progress in certain problems for which there were initially few examples (for example, Bongard problems and Winograd schemas) but Vapnik is not only calling for few data, but also for good predicates, which is I think a better defense than just asking for few-shot learning. In a sense I think if we find that more data is better, then that means the system is still not learning the good bits that make more data unnecessary.

zwaps 5y ago
I really do like the practical challenge that Vapnik poses and based on what you wrote I support the agenda. In the face of the trial&error mentality prevalent in ML, it does strike me as a wise approach towards advancement. However, the minor technicality remains in my view: there exists a case where a sufficiently complex causal mechanism can be only asympotically identified from a given DGP. In that case, more data is better, and more data is perhaps even required - but also the model will be fully general and will not start to fit on the randomness in sample/DGP for this "mechanism". So indeed, we can theoretically construct situations where Vapnik's intuition does not hold. Then, "more data is better => model is fitting sampling error" is more of a heuristic - even if it is a good one! If the model manages to disentangle error from mechanism, then more data could still improve the model while retaining generality. Of course, one might propose that any causal mechanism will be identified "sufficiently well" in a finite sample, so there's a point where we have enough data. Perhaps this can even be proven, and my theoretical counterexample simply never exists. In that case, you'd be right: A model that keeps learning ad-infinum can not be causally identified. I am not aware of such a proof, but I'd not be surprised if it were the case. Or more practically speaking, one may be able to show that improvements to the causal estimates are, at some point, always inferior to the overfitting errors no matter the DGP/model. I would perhaps propose a reverse of the heuristic: We know a model is good, if it can demonstrate that more data does NOT influence its invariant (causally identified) parts. This will be the challenge to solve for DNNs, as the combination of model+ latent DGP necessarily makes this a matter of assumption. In classical models, we can formulate these assumptions mathematically e.g. as exclusion restrictions and the like. For DNNs, I think we do not have such results yet.

zwaps 5y ago

I really do like the practical challenge that Vapnik poses and based on what you wrote I support the agenda. In the face of the trial&error mentality prevalent in ML, it does strike me as a wise approach towards advancement. However, the minor technicality remains in my view: there exists a case where a sufficiently complex causal mechanism can be only asympotically identified from a given DGP. In that case, more data is better, and more data is perhaps even required - but also the model will be fully general and will not start to fit on the randomness in sample/DGP for this "mechanism". So indeed, we can theoretically construct situations where Vapnik's intuition does not hold. Then, "more data is better => model is fitting sampling error" is more of a heuristic - even if it is a good one! If the model manages to disentangle error from mechanism, then more data could still improve the model while retaining generality. Of course, one might propose that any causal mechanism will be identified "sufficiently well" in a finite sample, so there's a point where we have enough data. Perhaps this can even be proven, and my theoretical counterexample simply never exists. In that case, you'd be right: A model that keeps learning ad-infinum can not be causally identified. I am not aware of such a proof, but I'd not be surprised if it were the case. Or more practically speaking, one may be able to show that improvements to the causal estimates are, at some point, always inferior to the overfitting errors no matter the DGP/model. I would perhaps propose a reverse of the heuristic: We know a model is good, if it can demonstrate that more data does NOT influence its invariant (causally identified) parts. This will be the challenge to solve for DNNs, as the combination of model+ latent DGP necessarily makes this a matter of assumption. In classical models, we can formulate these assumptions mathematically e.g. as exclusion restrictions and the like. For DNNs, I think we do not have such results yet.