# Uniform convergence for a class of finite dimension

The following theorem is cited in Balcan, M.F., Sandholm, T. and Vitercik, E., 2019. Estimating approximate incentive compatibility which I am currently reading and it is referenced to David Pollard. Convergence of Stochastic Processes. Springer, 1984. However, I have trouble finding the proof. Anyone knows a reference?

Let $$\Phi$$ be a distribution over $$\mathcal{X}$$. With probability $$1-\delta$$ over $$x_1,...,x_N\sim\Phi$$, for all $$a\in\mathcal{A}$$ it holds that $$\left|\frac{1}{N}\sum\limits_{i=1}^na(x_i)-\mathbb{E}_{x\sim\Phi} a(x)\right|\le\sqrt{\frac{2d}{N}\ln\frac{eN}{d}}+\sqrt{\frac{1}{2N}\ln\frac{2}{\delta}}$$

where $$\mathcal{A}: \mathcal{X} \rightarrow \mathbb{R}$$ is a set of real-valued functions and $$d$$ is the psuedo-dimensionof $$\mathcal{A}$$.

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– D.W.
May 5, 2021 at 18:07
• Please make your question self-contained, and define all notation. What is Pdim? What is $\mathcal{A}$?
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May 5, 2021 at 18:08
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May 5, 2021 at 18:09
• – D.W.
May 5, 2021 at 18:10

The keyword to look for is Dudley's chaining integral, see e.g. Vershynin's book "High Dimensional Probability" which contains a chapter on the chaining technique. Chaining allows us to bound the empirical Rademacher complexity in terms of the empirical $$L_2$$ covering numbers of $$\mathcal{A}$$. Your result is directly obtained from the chaining integral and a result (which is think is credited to Dudley-Pollard) which relates the $$L_1$$ covering numbers (irrespective of the distribution) to the pseudo dimension, see e.g. theorem 4.2 in these notes. You still need to go from $$L_2$$ covering to $$L_1$$ somehow, this is easy when the functions are binary valued.