Zachary Robertson

• Zachary Robertson 24 Apr 2021 15:48 UTC

  1 point

  in reply to: interstice’s comment on: NTK/​GP Models of Neu­ral Nets Can’t Learn Features

  I think that ‘universal function approximation’ and ‘feature learning’ are basically unrelated dimensions along which a learning algorithm can vary.

  We may have reached the crux here. Say you take a time series and extract the Fourier features. By universal approximation, these features will be sufficient for any downstream learning task. So the two are related. I agree that there is no learning taking place and that such a method may be inefficient. However, that goes beyond my original objection.

  This issue of ‘embedding efficiency’ seems only loosely related to the universal approximation property.

  This is not a trivial question. In the paper I referenced the authors show that approximation efficiency of NTK for deep and shallow are equivalent. However, infinitely differentiable activations can only approximate smooth functions. On the other hand, ReLU seems capable of approximating a larger class of potentially non-smooth functions.

• Zachary Robertson 24 Apr 2021 1:28 UTC

  1 point

  in reply to: interstice’s comment on: NTK/​GP Models of Neu­ral Nets Can’t Learn Features

  There’s a big difference between ‘universal learner’ and ‘fits any smooth function on a fixed input space’.

  Note I never said ‘universal learner’. What I actually said was,

  It’s clear enough that every finite embedding is a subspace of this embedding which sort of hints at the fact an infinite-width network is a universal function approximator.

  In the context of ML universal approximation, or more specifically, universal function approximation is the argument showing that NTK functions are dense in a certain sense. This was meant to address your request,

  You’ll also need an argument showing that their density in the NTK embedding is bounded above zero.

  This shows that the NTK functions in the associated reproducing space are dense in the smooth class of functions. I suspect I’m still not addressing your objection. If you could be more precise about your objection maybe we could get closer.

• Zachary Robertson 23 Apr 2021 20:39 UTC

  1 point

  in reply to: interstice’s comment on: NTK/​GP Models of Neu­ral Nets Can’t Learn Features

  The argument about sub-embeddings is meant to be a hand-wave. More precisely, the NTK kernel can be used to fit any smooth function. You can see precisely which class in this paper. I don’t have a proof on hand that the map is injective into the NTK space.

• Zachary Robertson 23 Apr 2021 16:36 UTC

  4 points

  on: NTK/​GP Models of Neu­ral Nets Can’t Learn Features

  This post argues that NTK/​GP models of neural nets can’t learn features. Feature learning here is explained as,

  By ‘feature learning’ I mean the typical practice whereby a neural net is trained on one task, then ‘fine-tuned’ with a lower learning to rate to fit another task, usually with less data than the first.

  It’s worth pointing out that the NTK can be seen as an embedding. This embedding does not vary during training instead features are weighted differently. It’s clear enough that every finite embedding is a subspace of this embedding which sort of hints at the fact an infinite-width network is a universal function approximator. Given, this I agree that it shouldn’t be surprising that NTK doesn’t learn features, but disagree on the reason. Namely, NTK doesn’t learn features because the feature class at initialization is a universal class and already has a good representation at initialization.

The Vari­a­tional Char­ac­ter­i­za­tion of Expectation

• Zachary Robertson 28 Mar 2021 22:00 UTC

  1 point

  in reply to: johnswentworth’s comment on: What is the Differ­ence Between Cheer­ful Price and Shadow Price?

  Your example is interesting and clarifies exchange rates. However,

  The shadow price quantifies the opportunity cost, so if I’m paid my shadow price, then that’s just barely enough to cover my opportunity cost.

  This is an interpretive point I’d like to focus on. When you move a constraint, in this case with price, the underlying equilibrium of the optimization shifts. From this perspective your usage of the word ‘barely’ stops making sense to me. If you were to ‘overshoot’ you wouldn’t be optimal in the new optimization problem.

  At this point I understand that the cheerful price will be equivalent to or more than the shadow price. You want to be able to shift the equilibrium point and have slack left over. It just seems obvious, to me, that shadow price isn’t an exactly measurable thing in this context and so you’d naturally be led to make a confidence interval (belief) for it. Cheerful price is just the upper estimate on that. Hence, I’m surprised why this is being treated as a new /​ distinct concept.

• Zachary Robertson 28 Mar 2021 21:50 UTC

  4 points

  in reply to: Anon User’s comment on: What is the Differ­ence Between Cheer­ful Price and Shadow Price?

  I suppose this is the most correct answer. I’m not really updating very much though. From my perspective I’ll continue to see cheerful price as a psychological/​subjective reinvention of shadow price.

  Edit: It seems clear in this context, shadow price isn’t exactly measurable. Cheerful price is just the upper estimate on the shadow price.

[Question] What is the Differ­ence Between Cheer­ful Price and Shadow Price?

• Zachary Robertson 4 Mar 2021 5:48 UTC

  1 point

  in reply to: Joar Skalse’s comment on: Why Neu­ral Net­works Gen­er­al­ise, and Why They Are (Kind of) Bayesian

  You seem to have updated your opinion: overtraining does make difference, but it’s not ‘huge’. Have you run a significance test for your lines of best fit? The plots as presented suggest the effect is significant.

  Figure C.1.a indicates the tilting phenomena. Probabilities only go up to one so tilting down means that the most likely candidates from overstrained SGD are less likely with random sampling. Thus, unlikely random sampling candidates are more likely under SGD. At the tail, the opposite happens. Functions more likely with random sampling become less likely under SGD.

  While the optimizer has a larger effect, I think the subtle question is whether the overtraining tilts in the same way each time. Figure 16 indicates yes again. This phenomena you consider to be minor is what I found most interesting about the paper.

• Zachary Robertson 28 Feb 2021 18:22 UTC

  LW: 1 AF: 1

  AF

  in reply to: Joar Skalse’s comment on: Why Neu­ral Net­works Gen­er­al­ise, and Why They Are (Kind of) Bayesian

  The main point, as I see it, is essentially that functions with good generalisation correspond to large volumes in parameter-space, and that SGD finds functions with a probability roughly proportional to their volume.

  What I’m suggesting is that volume in high-dimensions can concentrate on the boundary. To be clear, when I say SGD only typically reaches the boundary, I’m talking about early stopping and the main experimental setup in your paper where training is stopped upon reaching zero train error.

  We have done overtraining, which should allow SGD to penetrate into the region. This doesn’t seem to make much difference for the probabilities we get.

  This does seem to invalidate the model. However, something tells me that the difference here is more about degree. Since you use the word ‘should’ I’ll use the wiggle room to propose an argument for what ‘should’ happen.

  If SGD is run with early stopping, as described above, then my argument is that this is roughly equivalent to random sampling via an appeal to concentration of measure in high-dimensions.

  If SGD is not run with early stopping, it’s enclosed by the boundary of zero train error functions. Because these are most likely in the interior these functions are unlikely to be produced by random sampling. Thus, on a log-log plot I’d expect overtraining to ‘tilt’ the correspondence between SGD and random sampling likelihoods downward.

  Falsifiable Hypothesis: Compare SGD with overtaining to the random sampling algorithm. You will see that functions that are unlikely to be generated by random sampling will be more likely under SGD with overtraining. Moreover, functions that are more likely with random sampling will be become less likely under SGD with overtraining.

Min­i­mal Map Constraints

Time to Count or Count to Time?

• Zachary Robertson 26 Jan 2021 15:56 UTC

  4 points

  in reply to: Eigil Rischel’s comment on: Rec­og­niz­ing Numbers

  A problem I’m finding with this formulation is that it moves the problem to something that is arguably harder. We’ve replaced the problem of recognizing numbers with the problem of recognizing sets. The main post does this as well. There’s nothing technically wrong with this, but then the immediate question is this: how do we know when sets are useful? If a similar logic applies: how do we create an abstraction of a set from observation(s)? George Cantor, one of the founders of set theory writes,

  A set is a gathering together into a whole of definite, distinct objects of our perception [Anschauung] or of our thought—which are called elements of the set.

  To gather distinct perceptions together requires unity of apperception or a single ‘I think’ to be attached to each perception so that they may be brought under a category/​set/​etc.

• Zachary Robertson 18 Jan 2021 22:22 UTC

  5 points

  in reply to: alkexr’s comment on: What is go­ing on in the world?

  I like this one because you can generate more using GPT3 (that doesn’t imply they make sense)

A Toy-Model of In­stru­men­tal Abstraction

• Zachary Robertson 28 Dec 2020 23:55 UTC

  7 points

  on: The Good Try Rule

  I think this post does a good job of motivating a definition for “good try”. It also seems possible to think of habit changes as examples of goals. I personally find the SMART goal system to be useful and related to the discussion. SMART goals should be Specific, Measurable, Attainable, Reasonable, Timely. The approach is to specify why the habit change goal meets each of the SMART criteria.

  I’d think that giving something a “good try” is similar enough to trying habit change with a SMART goal that I mention this. This makes it clearer (at least for me) that what we’re talking about is creating some sort of prediction about how a successful habit change will proceed and then testing the prediction by attempting the habit change according to the plan. I think this also opens up the opportunity for giving something multiple “good tries” before evaluating success/​failure.

• Zachary Robertson 8 Dec 2020 16:06 UTC

  5 points

  in reply to: johnswentworth’s comment on: Min­i­mal Maps, Semi-De­ci­sions, and Neu­ral Representations

  I’m going to have to spend some time unpacking the very compact notation in the post, but here are my initial reactions.

  I should apologize a bit for that. To a degree I wasn’t really thinking about any of the concepts in the title and only saw the connection later.

  First, very clean proof of the lemma, well done there.

  Thanks!

  Second… if I’m understanding this correctly, each neuron activation (or set of neuron activations?) would contain all the information from some-part-of-data relevant to some-other-part-of-data and the output.

  To be honest, I haven’t thought about interpreting the monad beyond the equivalence with neural networks. One thing I noticed early on is that you can create sequences of activations ${\sigma }_n$ that delete information in the limit. For example, the ReLU activation is the limit of the SoftMax (change log base). I think something like this could be seen as abstracting away unnecessary data.

  Better yet, it looks like the OP gives a recipe for unpacking those natural abstractions?

  I’m not sure. I do think the method can justify the reuse of components (queries) and I wouldn’t be surprised if this is a pre-requisite for interpreting network outputs. Most of my interest comes from trying to formalize the (perhaps obvious) idea that anything that can be reduced to a sequence of classifications can be used to systematically translate high-level reasoning about these processes into a neural networks.

  I guess it’s best to give an example of how I currently think about abstraction. Say we take the position that every object $x\in X$ is completely determined by the information contained in a set of queries $y\in Y$ such that $|y|\ll |x|$. For a picture, consider designing a game-avatar (mii character) by fiddling around with some knobs. The formalism lets us package observations as queries using return. Thus, we’re hypothesizing that we can take a large collection of queries and make them equivalent to a small set of queries. Said another way, we can answer a large collection of queries by answering a much smaller set of ‘principle’ queries. In fact, if our activation was linear we’d be doing PCA. How we decide to measure success determines what abstraction is learned. If we only use the $y$ to answer a few queries then we’re basically doing classification. However, if the $y$ have to be able to answer every query about $x$ then we’re doing auto-encoding.

Min­i­mal Maps, Semi-De­ci­sions, and Neu­ral Representations

How to Cat­alyze Co­op­er­a­tion in Me­dia Sharing

A Toy Model for Me­dia Sharing