This is provably not true, and you can use the marginal likelihood / PAC-Bayes to prove it (or any other framework for measuring model quality). Increase the number of parameters in a linear model way beyond the point of interpolation, and concentrate the likelihood around the zero loss set. Then reduce the variance on a Gaussian prior. You can balance the two temperatures at exactly the right rate so that any measure of model quality will monotonically increase with model size and achieve a maximum at infinite model size.
Even easier, just take a limit of polynomial regression to a Gaussian process while optimizing the marginal likelihood over the prior temperature.
In all of these cases, the model with the least parameters is not the simplest in principle and does not have the best chance of not overfitting. The reality is significantly more nuanced.
> Then reduce the variance on a Gaussian prior.
Are you sure that doing this after seeing the data is valid and does not suffer from the equivalent of peeking-into-the-test-set problem ? There are ways to address the peeking problem but that requires additional machinery.
I don't dispute your broad claim but the first counterexample you quote seems problematic.
You can choose the prior according to any selection rule that does not see the data (actually, you can do more, but justifying this is the realm of empirical Bayes and requires some more precise arguments). In this case, you can choose it according to the model size and provided that your Jacobian is full rank, you will get increasing marginal likelihood.
What threw me off was the (possibly misunderstood) suggestion for minimizing the generalization bound over the prior after the data has been incorporated.
Ah, sorry for the misunderstanding, I can see how my comment reads that way. That is done in the Gaussian process context, not in my first example, and yes, it's a dirty idea, but you can justify it using differential privacy arguments (basically you are optimizing few parameters and these do not have full interaction with the data).
Yeah, I had read one of your parallel comments and understood what you had meant. Differential privacy is a good formulation (well, the only one I know) to deal with the peeking problem in general.
You are saying something interesting, but talking like Grok and skipping a lot of the details, without any references to common check-in points like terminology or specific studies.
> and concentrate the likelihood around the zero loss set. Then reduce the variance on a Gaussian prior.
Those phrases could mean a lot of different things. What are you proposing?
> so that any measure of model quality will monotonically increase with model size and achieve a maximum at infinite model size.
any measure of model quality? You must have some bounds of any measure, since trivially that's false because "fewer parameters is better" is a measure of model quality, even if dumb.
It's hard to even engage when you're being so imprecise, and not even giving one specific example.
Apologies, I'm skipping details, because that's how I speak with my colleagues, but I realize this is an external environment without context. No references since this is folklore (you can look at Hastie et al's Surprises in High-Dimensional Ridgeless Regression paper for the non-Bayesian version, Bruno Loureiro or Andrew Gordon Wilson probably have a paper with something similar).
Concentrating a density around a zero set means that I raise it to the power of 1/gamma (appropriately normalizing) and then take gamma to zero. If the likelihood was Gaussian, this would be equivalent to taking the variance to zero (yielding a point mass). But in overparameterized settings, this concentrates on a submanifold describing the set of interpolating solutions. In least-squares linear regression, that is the solution space. Reducing the variance on a Gaussian prior is treated as an asymptotic expansion by Laplace's method. If you choose the variance to decrease (inversely proportional to the parameter size, for example), then the marginal likelihood will increase monotonically with model size.
By any measure of model size, I mean that you can pick your favourite among the common ones, such as information metrics (e.g. mutual information / KL), statistical metrics (e.g. marginal likelihood), test error. You should be able to show the same phenomenon happening for all of them, so it isn't a quirk of marginal likelihood. It is concentration of measure working in your favor to reduce the variance in the estimator.
Okay, and that's all in-sample, which is the entire point, it won't necessarily hold out of sample.
E.g. over-fitting.
No, I am talking about out of sample error and estimates thereof. It is "overfitting" to data, but it also has lower out of sample error than the case where you do not "overfit".
This is why the notion of overfitting is not nearly as cut and dry as a basic ML course would have you believe. Just because you fit data exactly does not mean that your estimator has high error on out of sample data. A trivial counterexample is a spiking model that spikes to fit to the data but otherwise follows the correct trend outside of the dataset. The bias variance tradeoff gets thrown out at enormous scale and overfitting is not a meaningful concept. What matters is regularization and robustness, not how well you fit the data.
The reason why bias variance tradeoff and considerations of model size are a good approximation for smaller models is due to concentration of measure in the data which effectively kills any regularization in your modelling procedure. Once you enter settings where concentration of measure begins to bite in parameter space, everything changes. This isn't really that mysterious; any textbook on Gaussian processes (e.g. Rasmussen and Williams) will tell you this.
Nothing in your reply gets at the connection to out of sample data?