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parse/train/SkxpDT4YvS/SkxpDT4YvS.md

@@ -651,7 +651,7 @@ $$
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 Lemma 4. Let $\zeta \quad > \quad \frac { 5 } { 3 2 }$ , the batch size of the trajectories of outer loop $\begin{array} { r l } { N _ { 1 } } & { { } = } \end{array}$   
-$\Bigl ( \frac { 1 } { 8 L \zeta ^ { 2 } } + \frac { 1 } { 2 ( \zeta - \frac { 5 } { 3 2 } ) } \bigl ( 1 + \frac { 1 } { 3 2 \zeta ^ { 2 } } \bigr ) \Bigr ) \sigma ^ { 2 }$ the iteration times of inner loop $\begin{array} { r l r } { m \mathrm { ~ - ~ } 1 } & { { } = } & { N _ { 2 } \quad = } \end{array}$ 1 1 )  1 + 132 ζ 2 
  8Lζ2 + 2(ζ− 532 σ , and step size $\begin{array} { r } { \alpha _ { k } = \frac { 1 } { 4 L } } \end{array}$ . For each $k$ and $t$ , $G _ { k , 0 }$ and $\theta _ { k , 0 }$ are    
+$\Bigl ( \frac { 1 } { 8 L \zeta ^ { 2 } } + \frac { 1 } { 2 ( \zeta - \frac { 5 } { 3 2 } ) } \bigl ( 1 + \frac { 1 } { 3 2 \zeta ^ { 2 } } \bigr ) \Bigr ) \sigma ^ { 2 }$ the iteration times of inner loop $\begin{array} { r l r } { m \mathrm { ~ - ~ } 1 } & { { } = } & { N _ { 2 } \quad = } \end{array}$ 1 1 )  1 + 132 ζ 2 
  8Lζ2 + 2(ζ− 532 σ , and step size $\begin{array} { r } { \alpha _ { k } = \frac { 1 } { 4 L } } \end{array}$ . For each $k$ and $t$ , $G _ { k , 0 }$ and $\theta _ { k , 0 }$ are    
 generated by Algorithm 2, then the following holds,
 
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@@ -782,7 +782,7 @@ $$
 \mathfrak { c } \Vert \tilde { g } _ { k , t } \Vert ^ { 2 } \leq \Big ( \frac { 1 } { ( m - 1 ) K \eta } + \frac { L ^ { 2 } \alpha ^ { 2 } } { K N _ { 2 } \eta \zeta ^ { 2 } } \Big ) \left( \mathbb { E } [ \mathcal { I } ( \tilde { \theta } _ { 0 } ) ] - \mathcal { I } ( \theta ^ { * } ) \right) + \Big ( \frac { L ^ { 2 } \alpha ^ { 3 } ( m - 1 ) } { 2 N _ { 2 } \eta \zeta ^ { 2 } } + \frac { \alpha } { 2 \zeta ^ { 2 } } + \frac { \alpha } { 2 \eta } \Big ) \epsilon _ { 1 } ^ { 2 } .
 $$
 
-1 8Lζ2 + − 532 )  1 + 32 ζ 22  2 2(ζ Recall $\begin{array} { r l r } { \alpha } & { { } = } & { \frac { 1 } { 4 L } } \end{array}$ , N1 $N _ { 2 }$ = m − 1 = $\frac { \sqrt { \left( \frac { 1 } { 8 L \zeta ^ { 2 } } + \frac { 1 } { 2 ( \eta - \frac { 5 } { 3 2 } ) } \left( 1 + \frac { 1 } { 3 2 \zeta ^ { 2 } } \right) \right) } \sigma } { \epsilon }$ , then we have
+1 8Lζ2 + − 532 )  1 + 32 ζ 22  2 2(ζ Recall $\begin{array} { r l r } { \alpha } & { { } = } & { \frac { 1 } { 4 L } } \end{array}$ , N1 $N _ { 2 }$ = m − 1 = $\frac { \sqrt { \left( \frac { 1 } { 8 L \zeta ^ { 2 } } + \frac { 1 } { 2 ( \eta - \frac { 5 } { 3 2 } ) } \left( 1 + \frac { 1 } { 3 2 \zeta ^ { 2 } } \right) \right) } \sigma } { \epsilon }$ , then we have
 
 $$
 \mathbb { E } \| \mathcal { G } _ { \alpha , \langle - \nabla J ( \bar { \theta } _ { K } ) , \theta \rangle } \| ^ { 2 } = \mathbb { E } \| \tilde { g } _ { k , t } \| ^ { 2 } \leq \frac { 4 L } { K ( m - 1 ) ( \zeta - \frac { 5 } { 3 2 } ) } \big ( 1 + \frac { 1 } { 1 6 \zeta ^ { 2 } } \big ) ( \mathbb { E } [ \mathcal { I } ( \tilde { \theta } _ { 0 } ) ] - \mathcal { I } ( \theta ^ { * } ) ) + \frac { 1 } { 2 } \epsilon ^ { 2 } .