How can AIC converge in the limit when even 2 parameter models can have infinite VC dimension? AIC-based model-selection converges to zero error in the limit, and also has finite-sample convergence that is rate-optimal with respect to worst case minimax error [1].  (Note that AIC refers to Akaike Information Criterion)

However, AIC relies solely on the number of parameters in the model to assess/penalize model complexity.  And yet models can have infinite representational capacity even with just 2 parameters, as measured by VC-dimension [2], e.g. {sign(sin(ωx + θ) : ω}.  So this infinite VC model can act as a lookup table and perfectly overfit the data, while only having 2 parameters in the penalty term, so it would have low AIC, but no guarantee of low error out-of-sample.

I'm assuming that the issue is that the AIC convergence theorems don't apply to all models, and in particular perhaps these infinite VC-dimension models violate one of the assumptions.  But at least based on the papers I've looked at so far it's not clear to me what assumption that might be (if any)?

By the way, according to this review article [1] the convergence theorems apply quite broadly e.g.:


  When the true model is not in the candidate model set the AIC is
  efficient, in that it will asymptotically choose whichever model
  minimizes the mean squared error of prediction/estimation...
  The AIC's minimax rate of convergence in risk (and the BIC's lack of
  it) is a highly general result in regression and density estimation,
  and holds under more general circumstances than required for
  consistency properties discussed above. For example, the AIC is
  minimax-rate optimal regardless of whether the true model is finitely
  or infinitely dimensional; it is optimal whether the true model is
  among the candidates or not; and it extends to highly non-linear
  models, such as functions defined in Sobolev spaces


[1] https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3366160/

[2] http://www.cs.huji.ac.il/~shashua/papers/class11-PAC2.pdf
I tried to find a simple and accessible analysis of AIC. A definitive work seems to be Barron, Birgé, Massart, "Risk bounds for model selection via penalization"
https://link.springer.com/article/10.1007/s004400050210
though I can't vouch for accessibility.

Instead, let's analyze your sinusoids example. Suppose I have a parametric class of densities over $[0,1]$, parametrized by a single parameter $\alpha$, which is the frequency of the sinusoid. I take the positive part of the sinusoid and normalize the function to integrate to $1$ -- now it's a density. I can choose the $\alpha$ via any of the model selection techniques, including maximum likelihood and AIC. For any fixed $\alpha$, I think all of these will converge (maybe even almost surely) to the true $\alpha$. So why doesn't this contradict the fact that this family of functions (when thresholded at $0$) has infinite VC-dimension?

It's all about the order of quantifiers. Finite VC-dimension means that you can specify a sample size $m_0(\epsilon,\delta)$ that will suffice to achieve $\epsilon$-accuracy at $\delta$-confidence -- in advance of seeing any data. In our model selection example above, you will not be able to specify such an $m_0(\epsilon,\delta)$.

There are certainly infinite-VC classes that are learnable in this sense, which is weaker than PAC. For any target concept and distribution on the integers, you will eventually see enough examples to know the true concept to any desired accuracy -- you will just not be able to specify this sample size independently of the distribution and the target concept. (See Ex. 0.2(c) here
https://www.cs.bgu.ac.il/~asml162/wiki.files/hwk0.pdf
).