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@@ -128,7 +128,7 @@ $$
 \bar { P } _ { e } \overset { < } { \le } 1 - \exp \bigg ( - \Big ( H ( T ) + I ( X ; S | T ) + I ( Z ; X | S , T ) - \hat { I } _ { \theta ^ { * } } ^ { ( n ) } ( Z _ { X } ; S ) + O \big ( \sqrt { \frac { d + \log ( 1 / \delta ) } { n } } \big ) \Big ) \bigg ) .
 $$
 
-Given arbitrary learned representations $( Z _ { X } )$ , Theorem 3 suggests the corresponding Bayes error rate $( P _ { e } )$ is small when: 1) the estimated mutual information $\big ( \widehat { I } _ { \theta ^ { * } } ^ { ( n ) } ( Z _ { X } ; S ) \big )$ is large; 2) a larger number of samples $n$ are used for estimating the mutual information; and 3) the task-irrelevant information the compression gap $I ( X ; S | T )$ and the superfluous information $I ( Z ; X | S , T )$ , defined in Theorem 2
 is small. The first and the second results supports the claim that maximizing $I ( Z _ { X } ; S )$ may learn the representations that are beneficial to downstream tasks. The third result implies the learned representations may perform better on the downstream task when the compression gap is small. Additionally, $Z ^ { \mathrm { s s l } _ { \mathrm { m i n } } }$ is preferable than $Z ^ { \mathrm { s s l } }$ since $I ( Z ^ { \mathrm { s s l } _ { \mathrm { m i n } } } ; X | S , T ) = 0$ and $I ( Z ^ { \mathrm { s s l } } ; \bar { X } | \bar { S } , T ) \ge 0$ .
+Given arbitrary learned representations $( Z _ { X } )$ , Theorem 3 suggests the corresponding Bayes error rate $( P _ { e } )$ is small when: 1) the estimated mutual information $\big ( \widehat { I } _ { \theta ^ { * } } ^ { ( n ) } ( Z _ { X } ; S ) \big )$ is large; 2) a larger number of samples $n$ are used for estimating the mutual information; and 3) the task-irrelevant information the compression gap $I ( X ; S | T )$ and the superfluous information $I ( Z ; X | S , T )$ , defined in Theorem 2
 is small. The first and the second results supports the claim that maximizing $I ( Z _ { X } ; S )$ may learn the representations that are beneficial to downstream tasks. The third result implies the learned representations may perform better on the downstream task when the compression gap is small. Additionally, $Z ^ { \mathrm { s s l } _ { \mathrm { m i n } } }$ is preferable than $Z ^ { \mathrm { s s l } }$ since $I ( Z ^ { \mathrm { s s l } _ { \mathrm { m i n } } } ; X | S , T ) = 0$ and $I ( Z ^ { \mathrm { s s l } } ; \bar { X } | \bar { S } , T ) \ge 0$ .
 
 Theorem 4 (Bayes Error Rates for Self-supervised Learned Representations). Let $P _ { e } ^ { \mathrm { s u p } } / P _ { e } ^ { \mathrm { s s l } } / P _ { e } ^ { \mathrm { s s l } _ { \mathrm { m i n } } }$ be the Bayes error rate of the supervised or the self-supervised learned representations $Z _ { X } ^ { \mathrm { s u p } } / Z _ { X } ^ { \mathrm { s s l } } / Z _ { X } ^ { \mathrm { s s l } _ { \mathrm { m i n } } }$ . Then, $P _ { e } ^ { \mathrm { s s l } } = \mathrm { T h } ( \bar { P } _ { e } ^ { \mathrm { s s l } } )$ and $P _ { e } ^ { \mathrm { s s l } _ { \mathrm { m i n } } } = \mathrm { T h } ( \bar { P } _ { e } ^ { \mathrm { s s l } _ { \mathrm { m i n } } } ) ~ w$ ith