Not sure if I agree regarding the real-world usefulness. For the non-IID case, PAC-Bayes bounds fail, and to re-instate them you'd need assumptions about how quickly the distribution changes, but then it's plausible that you could get high probability bounds based on the most recent performance. For small datasets, the PAC-Bayes bounds suffer because they scale as √KLN . (I may edit the post to be clearer about this)
Agreed that analyzing how the bounds change under different conditions could be insightful though. Ultimately I suspect that effective bounds will require powerful ways to extract 'the signal from the noise', and examining the signal will likely be useful for understanding if a model has truly learned what it is supposed to.
The neural tangent kernel guys have a paper where they give a heuristic argument explaining the double descent curve(in number of parameters) using the NTK.
Nice survey. The result about double descent even occurring in dataset size is especially surprising.
Regarding the 'sharp minima can generalize' paper, they show that there exist sharp minima with good generalization, not flat minima with poor generalization, so they don't rule out flatness as an explanation for the success of SGD. The sharp minima they construct with this property are also rather unnatural: essentially they multiply the weights of layer 1 by a constant α and divide the weights of layer 2 by the same constant. The piecewise linearity of ReLU means the output function is unchanged. For large α, the network is now highly sensitive to perturbations in layer 2. These solutions don't seem like they would be found by SGD, so it could still be that, for solutions found by SGD, flatness and generalization are correlated.
No worries, I also missed the earlier posts when I wrote mine. There's lots of stuff on this website.
I endorse your rephrasing of example 1. I think my position is that it's just not that hard to create a "self-consistent probability distribution". For example, say you trained an RNN to predict sequences, like in this post. Despite being very simple, it already implicitly represents a probability distribution over sequences. If you train it with back-propagation on a confusing article involving pyrite, then its weights will be updated to try to model the article better. However, if "pyrite" itself was easy to predict, then the weights that lead to it outputting "pyrite" will *not* be updated. The same thing holds for modern Transformer networks, which predict the next token based only on what it has seen so far. (Here is a paper with a recent example using GPT-2. Note the degeneracy of maximum likelihood sampling, but how this becomes less of a problem when just sampling from the implied distribution)
I agree that this sort of manipulative prediction could be a problem in principle, but it does not seem to occur in recent ML systems. (Although, there are some things which are somewhat like this; the earlier paper I linked and mode collapse do involve neglecting high-entropy components of the distribution. However, the most straightforward generation and training schemes do not incentivize this)
For example 2, the point about gradient descent is this: while it might be the case that outputting "Help I'm stuck in a GPU Factory000" would ultimately result in a higher accuracy, the way the gradient is propagated would not encourage the agent to behave manipulatively. This is because, *locally*, "Help I'm stuck in a GPU Factory" decreases accuracy, so that behavior(or policies leading to it) will be dis-incentivized by gradient descent. It may be the case that this will result in easier predictions later, but the structure of the reward function does not lead to any optimization pressure towards such manipulative strategies. Learning taking place over high-level abstractions doesn't change anything, because any high-level abstractions leading to locally bad behavior will likewise be dis-incentivized by gradient descent
Example 1 basically seems to be the problem of output diversity in generative models. This can be a problem in generative models, but there are ways around it. e.g. instead of outputting the highest-probability individual sequence, which will certainly look "manipulative" as you say, sample from the implied distribution over sequences. Then the sentence involving "pyrite" will be output with probability proportional to how likely the model thinks "pyrite" is on its own, disregarding subsequent tokens.
For example 2, I wrote a similar post a few months ago (and in fact, this idea seems to have been proposed and forgotten a few times on LW). But for gradient descent-based learning systems, I don't think the effect described will take place.
The reason is that gradient-descent-based systems are only updated towards what they actually observe. Let's say we're training a system to predict EU laws. If it predicts "The EU will pass potato laws..." but sees "The EU will pass corn laws..." the parameters will be updated to make "corn" more likely to have been output than "potato". There is no explicit global optimization for prediction accuracy.
As you train to convergence, the predictions of the model will attempt to approach a fixed point, a set of predictions that imply themselves. However, due to the local nature of the update, this fixed-point will not be selected to be globally minimal, it will just be the first minima the model falls into. (This is different from the problems with "local minima" you may have heard about in ordinary neural network training -- those go away in the infinite-capacity limit, whereas local minima among fixed-points do not) The fixed-point should look something like "what I would predict if I output [what I would predict if I output [what I would predict .. ]]]" where the initial prediction is some random gibberish. This might look pretty weird, but it's not optimizing for global prediction accuracy.
Yeah, if you train the algorithm by random sampling, the effect I described will take place. The same thing will happen if you use an RL algorithm to update the parameters instead of an unsupervised learning algorithm(though it seems willfully perverse to do so -- you're throwing away a lot of the structure of the problem by doing this, so training will be much slower)
I also just found an old comment which makes the exact same argument I made here. (Though it now seems to me that argument is not necessarily correct!)
Reflective Oracles are a bit of a weird case case because their 'loss' is more like a 0/1 loss than a log loss, so all of the minima are exactly the same(If we take a sample of 100000 universes to score them, the difference is merely incredibly small instead of 0). I was being a bit glib referencing them in the article; I had in mind something more like a model parameterizing a distribution over outputs, whose only influence on the world is via a random sample from this distribution. I think that such models should in general have fixed points for similar reasons, but am not sure. Regardless, these models will, I believe, favour fixed points whose distributions are easy to compute(But not fixed points with low entropy, that is they will punish logical uncertainty but not intrinsic uncertainy). I'm planning to run some experiments with VAEs and post the results later.
Is there a reason you think a reflective oracle (or equivalent) can't just be selected "arbitrarily", and will likely be selected to maximize some score?
The gradient descent is not being done over the reflective oracles, it's being done over some general computational model like a neural net. Any highly-performing solution will necessarily look like a fixed-point-finding computation of some kind, due to the self-referential nature of the predictions. Then, since this fixed-point-finder is *internal* to the model, it will be optimized for log loss just like everything else in the model.
That is, the global optimization of the model is distinct from whatever internal optimization the fixed-point-finder uses to choose the reflective oracle. The global optimization will favor internal optimizers that produce fixed-points with good score. So while fixed-point-finders in general won't optimize for anything in particular, the one this model uses will.
If we assume Sleeping Beauty has lots of information, we might expect that the shortest matching program will look like a simulation of physical law plus a "bridging law" that, given this simulation, tells you what symbols get written to the tape
I agree. I still think that the probabilities would be closer to 1/2, 1/4, 1/4. The bridging law could look like this: search over the universe for compact encodings of my memories so far, then see what is written next onto this encoding. In this case, it would take no more bits to specify waking up on Tuesday, because the memories are identical, in the same format, and just slightly later temporally.
In a naturalized setting, it seems like the tricky part would be getting the AIXI on Monday to care what happens after it goes to sleep. It 'knows' that it's going to lose consciousness(it can see that its current memory encoding is going to be overwritten) so its next prediction is undetermined by its world-model. There is one program that will give it the reward of its successor then terminates, as I described above, but it's not clear why the AIXI would favour that hypothesis. Maybe if it has been in situations involving memory-wiping before, or has observed other RO-AIXI's in such situations.
I still don't see how you're getting those probabilities. Say it takes 1 bit to describe the outcome of the coin toss, and assume it's easy to find all the copies of yourself(ie your memories) in different worlds. Then you need:
1 bit to specify if the coin landed heads or tails
If the coin landed tails, you need 1 more bit to specify if it's Monday or Tuesday.
So AIXI would give these scenarios P(HM)=0.50, P(TM)=0.25, P(TT)=0.25.