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@@ -43,7 +43,7 @@ Recall from Zhang (2002) that for the class of linear functions, ${ \mathcal { F
 
 Generalization bounds. This work focuses on providing log-covering number bounds, which determine the generalization error via standard Rademacher complexity and chaining arguments. The following lemma relates these quantities; we refer the reader to Appendix A.1 for more details.
 
-Lemma 2.2 (Generalization bound via covering number). Let $\mathcal { D }$ be a distribution over $\mathcal { X } \ \times$ $\mathbb { R }$ and let $\begin{array} { r l r } { \ell } & { { } : } & { \mathbb { R } \ \times \ \mathbb { R } } \end{array}$ be a $b$ -bounded loss function that is $L$ -Lipschitz in its first argument. For a given function class $\mathcal { F }$ and $f \in \mathcal F$ , let $\mathsf { r i s k } ( f ; \mathcal { D } ) \ : = \ \mathsf { \bar { E } } _ { ( x , y ) \sim \mathcal { D } } [ \ell ( f ( x ) , y ) ]$ and risk d f ; (z(i), y(i))mi=1
 := 1m Pmi=1 \`(f (z(i)), y(i)). Suppose F satisfies |f | ≤ A for all f ∈ F and $\log \mathcal { N } _ { \infty } ( \mathcal { F } ; \varepsilon ; x ^ { ( 1 ) } , \ldots , x ^ { ( m ) } ) \le C \varepsilon / \varepsilon ^ { 2 }$ for all for all $x ^ { ( 1 ) } , \ldots , x ^ { ( m ) } \in \mathcal { X } ^ { m }$ . Then for any $\delta > 0$ , with probability at least $1 - \delta$ , simultaneously for all $f \in { \mathcal { F } }$ and some constant $c > 0$ ,
+Lemma 2.2 (Generalization bound via covering number). Let $\mathcal { D }$ be a distribution over $\mathcal { X } \ \times$ $\mathbb { R }$ and let $\begin{array} { r l r } { \ell } & { { } : } & { \mathbb { R } \ \times \ \mathbb { R } } \end{array}$ be a $b$ -bounded loss function that is $L$ -Lipschitz in its first argument. For a given function class $\mathcal { F }$ and $f \in \mathcal F$ , let $\mathsf { r i s k } ( f ; \mathcal { D } ) \ : = \ \mathsf { \bar { E } } _ { ( x , y ) \sim \mathcal { D } } [ \ell ( f ( x ) , y ) ]$ and risk d f ; (z(i), y(i))mi=1
 := 1m Pmi=1 \`(f (z(i)), y(i)). Suppose F satisfies |f | ≤ A for all f ∈ F and $\log \mathcal { N } _ { \infty } ( \mathcal { F } ; \varepsilon ; x ^ { ( 1 ) } , \ldots , x ^ { ( m ) } ) \le C \varepsilon / \varepsilon ^ { 2 }$ for all for all $x ^ { ( 1 ) } , \ldots , x ^ { ( m ) } \in \mathcal { X } ^ { m }$ . Then for any $\delta > 0$ , with probability at least $1 - \delta$ , simultaneously for all $f \in { \mathcal { F } }$ and some constant $c > 0$ ,
 
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 \left| \mathrm { r i s k } ( f ; \mathcal { D } ) - \widehat { \mathrm { r i s k } } \left( f ; ( x ^ { ( i ) } , y ^ { ( i ) } ) _ { i = 1 } ^ { m } \right) \right| \leq 4 c L \sqrt { \frac { C _ { \mathcal { F } } } { m } } \left( 1 + \log \left( A \sqrt { m / C _ { \mathcal { F } } } \right) \right) + 2 b \sqrt { \frac { \log ( 1 / \delta ) } { 2 m } } .