# interstice comments on Understanding “Deep Double Descent”

• Yes, they do say they can make the minima flatter in section 5.1 -- but, as you say, only by re-parameterizing. This makes the result totally vacuous! Like yeah, obviously if you’re allowed to arbitrarily re-define the mapping from parameters to functions you can make the minima as “sharp” or as “flat” as you want. For the more substantive results in section 4, I do believe the direction is always flat --> sharp.

I’ve been thinking about the flatness results from a compression perspective. Being at a flat minimum reduces your description length since you only have to specify the coordinates of the basin you’re in, and flatter minima correspond to wider basins which can be specified with lower-precision coordinates. Re-parameterizing breaks this by introducing extra complexity into the parameter-function map. The ability to always obtain sharper but not flatter minima is kind of analogous to how, given a program, there are infinitely many longer programs replicating its behavior, but finitely many or none shorter such programs.

• Vacuous sure, but still true, and seems relevant to me. You initially wrote:

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.

But, allowing reparametrisation, this seems false? I don’t understand the step in your argument where you ‘rule out reparametrisation’, nor do I really understand what this would mean.

Your comment relating description length to flatness seems nice. To talk about flatness at all (in the definitions I have seen) requires a choice of parametrisation. And I guess your coordinate description is also using a fixed parametrisation, so this seems reasonable. A change of parametrisation will then change both flatness and description length (i.e. ‘required coordinate precision’).

• When I said they didn’t produce flat minima with poor generalization, I meant “they didn’t produce flat minima with poor generalization in the normal parameter space of a neural network”. This is what is relevant to the “flatness as an explanation for generalization” hypothesis, since that hypothesis is about how flatness correlates with generalization in neural networks in practice. That there exist other parameter-function mappings where this is not the case does not refute this. Of course, if you wished to produce a mathematical proof that flat minima generalize this would be an important example to keep in mind—but this post was about high-level scientific hypotheses about neural network generalization, not mathematical proofs. In that context I think it’s correct to say that the paper does not provide a meaningful counterexample.

• p.s.

For the more substantive results in section 4, I do believe the direction is always flat --> sharp.

I agree with this (with ‘sharp’ replaced by ‘generalise’, as I think you intend). It seems to me potentially interesting to ask whether this is necessarily the case.

• When I said “the direction is always flat-->sharp”, I meant that their theorems showed you could produce a sharp minimum given a flat one, but not the other way around, sorry if I was unclear.

Definitely agreed that “under which conditions does flatness imply generalization” is a very interesting question. I think this paper has a reasonably satisfying analysis, although I also have some reservations about “SGD as Bayesian sampler” picture.