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parse/train/Re_VXFOyyO/Re_VXFOyyO.md

@@ -221,7 +221,7 @@ $$
 
 for some constants $C _ { 1 } , . . , C _ { 4 } > 0 .$ . In case $j = 0$ , we default $\textstyle \sum _ { j ^ { \prime } = 1 } ^ { 0 } \cdot = 0$
 
-The expression of constants $C _ { i }$ ’s are complicated, we provide their detailed formula in the appendix. If we set H = O log(1/) 
 and $\begin{array} { r } { \delta \le \frac { 1 } { 2 H \ell _ { \psi } } } \end{array}$ , then $C _ { i }$ only depends polynomially on the Lipschitz constants, $\log ( \epsilon ^ { - 1 } )$ , and $( 1 - \gamma ) ^ { - 1 }$ . Combining Lemma 5.5, 5.8, and 5.4 gives Theorem 5.9.
+The expression of constants $C _ { i }$ ’s are complicated, we provide their detailed formula in the appendix. If we set H = O log(1/) 
 and $\begin{array} { r } { \delta \le \frac { 1 } { 2 H \ell _ { \psi } } } \end{array}$ , then $C _ { i }$ only depends polynomially on the Lipschitz constants, $\log ( \epsilon ^ { - 1 } )$ , and $( 1 - \gamma ) ^ { - 1 }$ . Combining Lemma 5.5, 5.8, and 5.4 gives Theorem 5.9.
 
 Theorem 5.9. For Algorithm $^ { l }$ , we choose $\begin{array} { r } { H = \frac { 2 \log ( 1 / \epsilon ) } { 1 - \gamma } } \end{array}$ , $\begin{array} { r } { \delta = \frac { 1 } { 2 H \ell _ { \psi } } } \end{array}$ , $B = m = \epsilon ^ { - 1 }$ , $N = \epsilon ^ { - 2 }$ η = 11+(C3+C4)/L2θ · 12Lθ . After running the algorithm for T = −1 epochs and output θout from $\{ \theta _ { j } ^ { i } \} _ { j = 0 , \cdots , m - 1 } ^ { i = 1 , \cdots , T }$ uniformly at random, we have $\mathbb { E } [ \| \mathcal { G } _ { \eta } ( \theta _ { o u t } ) \| ] \le \mathcal { O } ( \epsilon )$ . The total number of samples is $\dot { T } \times ( ( m - 1 ) B + N ) \times H = \tilde { \mathcal { O } } ( \epsilon ^ { - 3 } )$ . By Lemma 5.4, we also have $\mathbb { E } [ \| \nabla _ { \theta } F ( \lambda ( \theta _ { o u t } ) ) \| ] \le { \mathcal { O } } ( \epsilon )$ .