Add files using upload-large-folder tool

parse/train/SkA-IE06W/SkA-IE06W.md

@@ -114,7 +114,7 @@ $$
 
 Note both $\gamma ( \phi )$ and $L ( \phi )$ increases as $\phi$ decreases so we can choose a constant step size $\begin{array} { r } { \eta _ { t } \ = \ \Theta \left( \frac { \gamma ( \phi _ { 0 } ) } { ( L ( 0 ) + \beta ) ^ { 2 } } \right) } \end{array}$ This theorem implies that we can find the $\epsilon$ -close solution of $\mathbf { w } _ { \ast }$ in $\begin{array} { r } { O \left( \frac { ( L ( 0 ) + \beta ) ^ { 2 } } { \gamma ^ { 2 } ( \phi _ { 0 } ) } \log \left( \frac { 1 } { \epsilon } \right) \right) } \end{array}$ iterations. It also suggests a direct relation between the smoothness of the distribution and the convergence rate. For smooth distribution where $\gamma ( \phi )$ and $L ( \phi )$ are close and $\beta$ is small then $\frac { ( L ( 0 ) + \beta ) ^ { 2 } } { \gamma ^ { 2 } ( \phi _ { 0 } ) }$ is relatively small and we need fewer iterations. On the other hand, if $L ( \phi )$ or $\beta$ is much larger than $\gamma ( \phi )$ , we will need more iterations. We verify this intuition in Section 4.
 
-If we are able to choose the step sizes adaptively $\begin{array} { r } { \eta _ { t } \ = \ \Theta \left( \frac { \gamma ( \phi _ { t } ) } { ( L ( \phi _ { t } ) + \beta ) ^ { 2 } } \right) } \end{array}$  γ(φt)(L(φt)+β)2 , like using methods proposed by Lin & Xiao (2014), we may improve the computational complexity to $\begin{array} { r } { O \left( \operatorname* { m a x } _ { \phi \leq \phi _ { 0 } } \frac { \left( L \left( \phi \right) + \beta \right) ^ { 2 } } { \gamma ^ { 2 } \left( \phi \right) } \log \left( \frac { 1 } { \epsilon } \right) \right) , } \end{array}$ 0 (L(φ)+β)2γ2(φ) log  1 
. This justifies the use of two-stage learning rate strategy proposed by He et al. (2016); Szegedy et al. (2017) where at the beginning we need to choose learning to be small because $\frac { \gamma ( \phi _ { 0 } ) } { 2 ( L ( \phi _ { 0 } ) + 2 \beta ) ^ { 2 } }$ is small and later we can choose a large learning rate because as the angle between $\mathbf { w } _ { t }$ and $\mathbf { w } _ { \ast }$ becomes smaller, $\frac { \gamma ( \phi _ { t } ) } { 2 ( L ( \phi _ { t } ) + 2 \beta ) ^ { 2 } }$ becomes bigger.
+If we are able to choose the step sizes adaptively $\begin{array} { r } { \eta _ { t } \ = \ \Theta \left( \frac { \gamma ( \phi _ { t } ) } { ( L ( \phi _ { t } ) + \beta ) ^ { 2 } } \right) } \end{array}$  γ(φt)(L(φt)+β)2 , like using methods proposed by Lin & Xiao (2014), we may improve the computational complexity to $\begin{array} { r } { O \left( \operatorname* { m a x } _ { \phi \leq \phi _ { 0 } } \frac { \left( L \left( \phi \right) + \beta \right) ^ { 2 } } { \gamma ^ { 2 } \left( \phi \right) } \log \left( \frac { 1 } { \epsilon } \right) \right) , } \end{array}$ 0 (L(φ)+β)2γ2(φ) log  1 
. This justifies the use of two-stage learning rate strategy proposed by He et al. (2016); Szegedy et al. (2017) where at the beginning we need to choose learning to be small because $\frac { \gamma ( \phi _ { 0 } ) } { 2 ( L ( \phi _ { 0 } ) + 2 \beta ) ^ { 2 } }$ is small and later we can choose a large learning rate because as the angle between $\mathbf { w } _ { t }$ and $\mathbf { w } _ { \ast }$ becomes smaller, $\frac { \gamma ( \phi _ { t } ) } { 2 ( L ( \phi _ { t } ) + 2 \beta ) ^ { 2 } }$ becomes bigger.
 
 The theorem requires the initialization satisfying $\left\| \mathbf { w } _ { 0 } - \mathbf { w } _ { * } \right\| _ { 2 } < \left\| \mathbf { w } _ { * } \right\| _ { 2 }$ , which can be achieved by random initialization with constant success probability. See Section 3.2 for a detailed discussion.
 
@@ -162,7 +162,7 @@ $$
 \| \mathbf { w } _ { t + 1 } - \mathbf { w } _ { * } \| _ { 2 } ^ { 2 } \leq \left( 1 - \frac { \eta ( \gamma ( \phi _ { t } ) - 6 L _ { \mathrm { c r o s s } } ) } { 2 } \right) \| \mathbf { w } _ { t } - \mathbf { w } _ { * } \| _ { 2 } ^ { 2 }
 $$
 
-Our theorem suggests if the initialization satisfies $\gamma ( \phi _ { 0 } ) ~ > ~ 6 L _ { \mathrm { c r o s s } }$ , we obtain linear convergence rate. In Section 3.1, we give a concrete example showing closeness of patches implies large $\gamma ( \phi )$ and small $L _ { \mathrm { c r o s s } }$ . Similar to Theorem 2.2, if the step size is chosen so that $\eta _ { t } =$ find the  γ(φ0)−6Lcross(LS(w,w∗)(0)+10Lcross+4β)2 , in O  γ(φ0)−6LcrossLS(w,w∗)(0)+10Lcross+4β 2 log  1 
 iterations, we can
+Our theorem suggests if the initialization satisfies $\gamma ( \phi _ { 0 } ) ~ > ~ 6 L _ { \mathrm { c r o s s } }$ , we obtain linear convergence rate. In Section 3.1, we give a concrete example showing closeness of patches implies large $\gamma ( \phi )$ and small $L _ { \mathrm { c r o s s } }$ . Similar to Theorem 2.2, if the step size is chosen so that $\eta _ { t } =$ find the  γ(φ0)−6Lcross(LS(w,w∗)(0)+10Lcross+4β)2 , in O  γ(φ0)−6LcrossLS(w,w∗)(0)+10Lcross+4β 2 log  1 
 iterations, we can
 
 In practice,we never get a true population gradient but only stochastic gradient $g ( \mathbf { w } )$ (c.f. Equation (2)). The following theorem shows SGD also recovers the underlying filter.