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Yes. Specifically, the common argument has become that flat minima generalize well. As for the last part, when people talk about overfitting we're typically ig

Yes. Specifically, the common argument has become that flat minima generalize well. As for the last part, when people talk about overfitting we're typically ignoring distributional shift. Otherwise, models can achieve very good generalization.

breatheoften 9y ago
Is there a separate word from overfitting to refer to generalization performance in a way that explicitly includes distributional shift? Just curious as I never realized that overfitting was commonly constrained in the way you describe and would like to go back and relabel my knowledge to better distinguish between your definition of overfitting and a more general version.
- chillee 9y ago
  Generalization performance is typically defined in roughly this manner. Assume your training data consists of a sample S of m elements from your full distribution D. Let L(S) be the average loss for your model (hypothesis is also a common term) on S, and L(D) be the average loss for your model on D. Now, we say that training generalizes if your model that minimizes L(S) also achieves similarly low L(D). In practice, we detect generalization by taking a separate sample (validation set) S_2 from D and using L(S_2) to approximate L(D). I'm not familiar with any literature of generalization to arbitrary distributional shift between training and test. If you allow arbitrary distributional shift (say we decide that all dogs are now called cats and all cats are now called dogs), it's pretty clear that you can get arbitrarily poor "generalization".
  - breatheoften 9y ago
    Ok — but in many cases there’s more that can be said about the error in the approximation to D of S and S_2 which would allow discussion of various forms of bounded or constrained distributional shift. I don’t think the only choices are “no distributional shift” and “arbitrary distributional shift” are they? What’s the vocabulary to refer to poor generalization performance to datasets with some form of constrained or procedurally introduced distributional shift?
alok-g 9y ago
I would like to read more about this. Do you have some references? Thanks.
- chillee 9y ago
  Probably the single best article I saw recently was this: http://www.offconvex.org/2017/12/08/generalization1/ http://www.offconvex.org/2017/12/08/generalization1/ I think that blog post references most of the relevant papers as well, but if not, check the sibling comments I've made.
aoeusnth1 9y ago
Noob here - what do you mean by distributional shift?
- rocqua 9y ago
  In short, it's when your training data come from different conditions than your real world data. This means that the probability distribution your model was trained on is different from the one you are using it on. This means that an model that did great in cross-validation could still be useless in practice.
- chillee 9y ago
  Basically, it's when your distribution that your classes come from change between training and test. For example, if after training your classifier to distinguish between cats and dogs, all of a sudden society decides that all white haired dogs are now called cats, your classifier is clearly going to get "poor" performance.
YeGoblynQueenne 9y ago
I'm skeptical about both statements. I haven't heard the "comnon argument" you mention and I don't see how ignoring differences in distribution between laboratory and real-world data will improve generalisation. I find the last bit particularly hard to accept, but I'd like to hear why you say so, on both counts.
- chillee 9y ago
  Sorry, my last sentence > Otherwise, models can achieve very good generalization. was probably unclear. When we talk about generalization, we typically have to assume a lack of distributional shift. Otherwise, if the "truly unseen data" really is drawn from a different distribution (say one where all cats are now called "dogs" and all dogs are now called "cats"), we can get arbitrarily poor performance. As for the flat minima part, lemme pull up some papers from my notes. https://arxiv.org/abs/1609.04836 https://arxiv.org/abs/1609.04836 showed larger batch sizes lead to sharper minima. > ... and as is well known, sharp minima lead to poorer generalization. https://arxiv.org/abs/1711.01530 https://arxiv.org/abs/1711.01530 provides a parametrization-invariant notion of "flatness" that counters Dinh et al's point in https://arxiv.org/abs/1703.04933 https://arxiv.org/abs/1703.04933. I thought this paper looks pretty promising, although I haven't gone through the math. >Although flatness of the loss function has been linked to generalization [Hochreiter and Schmidhuber, 1997], existing definitions of flatness are neither invariant to nodewise re-scalings of ReLU nets nor general coordinate transformations [Dinh et al., 2017] of the parameter space, which calls into question their utility for describing generalization https://arxiv.org/abs/1703.11008 https://arxiv.org/abs/1703.11008 is also a really promising paper on generalization that basically shows another way to view flatness from a Bayesian perspective (they also show the most impressive generalization bound I've seen yet). And of course, the classic Schmidhuber paper on Flat Minima: http://www.bioinf.jku.at/publications/older/3304.pdf http://www.bioinf.jku.at/publications/older/3304.pdf I think it's clear from these papers that the "flat minima generalize well" argument is a common avenue of research with respect to generalization. Many of the generalization papers I've read recently are mostly related to different ways of calculating flatness, and pretty much all of them make reference and link their work to the 1st and 4th paper I linked above.

breatheoften 9y ago

Is there a separate word from overfitting to refer to generalization performance in a way that explicitly includes distributional shift? Just curious as I never realized that overfitting was commonly constrained in the way you describe and would like to go back and relabel my knowledge to better distinguish between your definition of overfitting and a more general version.

chillee 9y ago
Generalization performance is typically defined in roughly this manner. Assume your training data consists of a sample S of m elements from your full distribution D. Let L(S) be the average loss for your model (hypothesis is also a common term) on S, and L(D) be the average loss for your model on D. Now, we say that training generalizes if your model that minimizes L(S) also achieves similarly low L(D). In practice, we detect generalization by taking a separate sample (validation set) S_2 from D and using L(S_2) to approximate L(D). I'm not familiar with any literature of generalization to arbitrary distributional shift between training and test. If you allow arbitrary distributional shift (say we decide that all dogs are now called cats and all cats are now called dogs), it's pretty clear that you can get arbitrarily poor "generalization".
- breatheoften 9y ago
  Ok — but in many cases there’s more that can be said about the error in the approximation to D of S and S_2 which would allow discussion of various forms of bounded or constrained distributional shift. I don’t think the only choices are “no distributional shift” and “arbitrary distributional shift” are they? What’s the vocabulary to refer to poor generalization performance to datasets with some form of constrained or procedurally introduced distributional shift?

alok-g 9y ago

I would like to read more about this. Do you have some references? Thanks.

chillee 9y ago
Probably the single best article I saw recently was this: http://www.offconvex.org/2017/12/08/generalization1/ http://www.offconvex.org/2017/12/08/generalization1/ I think that blog post references most of the relevant papers as well, but if not, check the sibling comments I've made.

aoeusnth1 9y ago

Noob here - what do you mean by distributional shift?

rocqua 9y ago
In short, it's when your training data come from different conditions than your real world data. This means that the probability distribution your model was trained on is different from the one you are using it on. This means that an model that did great in cross-validation could still be useless in practice.
chillee 9y ago
Basically, it's when your distribution that your classes come from change between training and test. For example, if after training your classifier to distinguish between cats and dogs, all of a sudden society decides that all white haired dogs are now called cats, your classifier is clearly going to get "poor" performance.

YeGoblynQueenne 9y ago

I'm skeptical about both statements. I haven't heard the "comnon argument" you mention and I don't see how ignoring differences in distribution between laboratory and real-world data will improve generalisation. I find the last bit particularly hard to accept, but I'd like to hear why you say so, on both counts.

chillee 9y ago
Sorry, my last sentence > Otherwise, models can achieve very good generalization. was probably unclear. When we talk about generalization, we typically have to assume a lack of distributional shift. Otherwise, if the "truly unseen data" really is drawn from a different distribution (say one where all cats are now called "dogs" and all dogs are now called "cats"), we can get arbitrarily poor performance. As for the flat minima part, lemme pull up some papers from my notes. https://arxiv.org/abs/1609.04836 https://arxiv.org/abs/1609.04836 showed larger batch sizes lead to sharper minima. > ... and as is well known, sharp minima lead to poorer generalization. https://arxiv.org/abs/1711.01530 https://arxiv.org/abs/1711.01530 provides a parametrization-invariant notion of "flatness" that counters Dinh et al's point in https://arxiv.org/abs/1703.04933 https://arxiv.org/abs/1703.04933. I thought this paper looks pretty promising, although I haven't gone through the math. >Although flatness of the loss function has been linked to generalization [Hochreiter and Schmidhuber, 1997], existing definitions of flatness are neither invariant to nodewise re-scalings of ReLU nets nor general coordinate transformations [Dinh et al., 2017] of the parameter space, which calls into question their utility for describing generalization https://arxiv.org/abs/1703.11008 https://arxiv.org/abs/1703.11008 is also a really promising paper on generalization that basically shows another way to view flatness from a Bayesian perspective (they also show the most impressive generalization bound I've seen yet). And of course, the classic Schmidhuber paper on Flat Minima: http://www.bioinf.jku.at/publications/older/3304.pdf http://www.bioinf.jku.at/publications/older/3304.pdf I think it's clear from these papers that the "flat minima generalize well" argument is a common avenue of research with respect to generalization. Many of the generalization papers I've read recently are mostly related to different ways of calculating flatness, and pretty much all of them make reference and link their work to the 1st and 4th paper I linked above.