exercise:082aab4abb

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Let [math]K[/math] be a compact subset of the unit sphere of [math]\R^p[/math] that admits an [math]\eps[/math]-net [math]\cN_\eps[/math] with respect to the Euclidean distance of [math]\R^p[/math] that satisfies [math]|\cN_\eps|\le (C/\eps)^d[/math] for all [math]\eps \in (0,1)[/math]. Here [math]C \ge 1[/math] and [math]d\le p[/math] are positive constants. Let [math]X \sim \sg_p(\sigma^2)[/math] be a centered random vector.

Show that there exists positive constants [math]c_1[/math] and [math]c_2[/math] to be made explicit such that for any [math]\delta \in (0,1)[/math], it holds

[[math]] \max_{\theta \in K} \theta^\top X \le c_1\sigma\sqrt{d \log (2p/d)} +c_2\sigma\sqrt{ \log(1/\delta)} [[/math]]

with probability at least [math]1-\delta[/math]. Comment on the result in light of Theorem.