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parse/train/gEzN9bBbLt8/gEzN9bBbLt8.md

@@ -169,7 +169,7 @@ Remark 3 (The Optimal Batch Sizes and Local Updates Trade-off). The parameter $\
 10: end for   
 11: Return: $\scriptstyle { \bar { x } } _ { a }$ where $a \sim \mathcal { U } \{ 1 , . . . , T \}$ .
 
-Remark 4 (The Sub-Optimal Batch Sizes and Local Updates Trade-off). From our proof (Theorem ${ \bf C . 1 0 }$ included in the supplemental material), we can see that STEM requires $\bar { \tilde { O } } ( \operatorname* { m a x } \big \{ ( b \cdot$ $I ) \epsilon ^ { - 1 } , K ^ { - 1 } \epsilon ^ { - 3 / 2 } \rbrace )$ samples and $\tilde { \mathcal { O } } \big ( \operatorname* { m a x } \big \{ \epsilon ^ { - 1 } , ( b \cdot I ) ^ { - 1 } K ^ { - 1 } \epsilon ^ { - 3 / 2 } \big \} \big )$  and communication rounds. According to the above expressions, if $b \cdot I$ increases beyond $\mathcal { O } ( K ^ { - 1 } \epsilon ^ { - 1 / 2 } )$ , then the sample complexity will increase from the optimal $\tilde { \mathcal { O } } ( \epsilon ^ { - 3 / 2 } )$ ; otherwise, the optimal sample complexity $\tilde { \mathcal O } ( \epsilon ^ { - 3 / 2 } )$ is maintained. On the other hand, if $b \cdot I$ decreases beyond $\mathcal { O } ( K ^ { - 1 } \epsilon ^ { - 1 / 2 } )$ , the communication complexity increases from $\tilde { \mathcal { O } } ( \epsilon ^ { - 1 } )$ . For instance, if we choose $b = \mathcal { O } ( 1 )$ and $I = { \mathcal { O } } ( 1 )$ the communication complexity becomes $\tilde { \mathcal { O } } ( \epsilon ^ { - 3 / 2 } )$ while the optimal sample complexity $\tilde { \mathcal { O } } ( \epsilon ^ { - 3 / 2 } )$ is maintained. This trade-off is illustrated in Figure $\boxed { 1 \mathrm { a } }$ where we maintain the optimal sample complexity, while changing $b$ and $I$ to generate the trade-off surface. □
+Remark 4 (The Sub-Optimal Batch Sizes and Local Updates Trade-off). From our proof (Theorem ${ \bf C . 1 0 }$ included in the supplemental material), we can see that STEM requires $\bar { \tilde { O } } ( \operatorname* { m a x } \big \{ ( b \cdot$ $I ) \epsilon ^ { - 1 } , K ^ { - 1 } \epsilon ^ { - 3 / 2 } \rbrace )$ samples and $\tilde { \mathcal { O } } \big ( \operatorname* { m a x } \big \{ \epsilon ^ { - 1 } , ( b \cdot I ) ^ { - 1 } K ^ { - 1 } \epsilon ^ { - 3 / 2 } \big \} \big )$  and communication rounds. According to the above expressions, if $b \cdot I$ increases beyond $\mathcal { O } ( K ^ { - 1 } \epsilon ^ { - 1 / 2 } )$ , then the sample complexity will increase from the optimal $\tilde { \mathcal { O } } ( \epsilon ^ { - 3 / 2 } )$ ; otherwise, the optimal sample complexity $\tilde { \mathcal O } ( \epsilon ^ { - 3 / 2 } )$ is maintained. On the other hand, if $b \cdot I$ decreases beyond $\mathcal { O } ( K ^ { - 1 } \epsilon ^ { - 1 / 2 } )$ , the communication complexity increases from $\tilde { \mathcal { O } } ( \epsilon ^ { - 1 } )$ . For instance, if we choose $b = \mathcal { O } ( 1 )$ and $I = { \mathcal { O } } ( 1 )$ the communication complexity becomes $\tilde { \mathcal { O } } ( \epsilon ^ { - 3 / 2 } )$ while the optimal sample complexity $\tilde { \mathcal { O } } ( \epsilon ^ { - 3 / 2 } )$ is maintained. This trade-off is illustrated in Figure $\boxed { 1 \mathrm { a } }$ where we maintain the optimal sample complexity, while changing $b$ and $I$ to generate the trade-off surface. □
 
 Remark 5 (Data Heterogeneity). The term $\begin{array} { r } { \tilde { \mathcal { O } } \biggl ( \frac { \zeta ^ { 2 } } { K ^ { 2 \nu / 3 } T ^ { 1 - \nu / 3 } } \biggr ) } \end{array}$ in the gradient bound $\textcircled{4}$ captures the effect of the heterogeneity of data across WNs, where $\zeta$ is the parameter characterizing the intra-node variance and has been defined in Assumption $\bigstar$ (ii). Highly heterogeneous data with large $\zeta ^ { 2 }$ can adversely impact the performance of STEM. Note that such a dependency on $\zeta$ also appears in other existing FL algorithms, such as $\boxed { 9 } \boxed { 1 4 } \boxed { 1 8 }$ . However, there is one special case of STEM that does not depend on the parameter $\zeta$ . This is the case where $I = 1$ , i.e., the minibatch SGD counterpart of STEM where only a single local iteration is performed between two communication rounds. We have the following corollary. □
 

parse/train/rkgKBhA5Y7/rkgKBhA5Y7.md

@@ -370,7 +370,7 @@ $\mathfrak { L } _ { d } [ R _ { \mathrm { M S E } } ( w + s d ) ] - R _ { \math
 
 # A.7 INCLUDING HIGH LEARNING RATE ITERATES INTO SWA
 
-As discussed in Mandt et al. (2017), under certain assumptions SGD samples from a Gaussian distribution centered at the optimum of the loss $w _ { 0 }$ with covariance proportional to the learning rate. Suppose then that we have $n$ weights sampled at learning rate $\eta _ { 1 }$ , $w _ { i } ^ { ( 1 ) } \stackrel { i i d } { \sim } \mathcal { N } ( w _ { 0 } , \eta _ { 1 } \Sigma )$ and $m$ weights sampled with the higher learning rate $\eta _ { 2 }$ , $w _ { j } ^ { ( 2 ) } \stackrel { i i d } { \sim } \mathcal { N } ( w _ { 0 } , \eta _ { 2 } \Sigma )$ . For the SWA estimator $\begin{array} { r } { \hat { w } _ { \mathrm { S W A } } = \frac { 1 } { n } \sum _ { i } w _ { i } ^ { ( 1 ) } , \mathbb { E } [ \| \hat { w } _ { \mathrm { S W A } } - w _ { 0 } \| ^ { 2 } ] = \mathrm { t r } ( \mathrm { C o v } ( \hat { w } _ { \mathrm { S W A } } ) ) = \frac { \eta _ { 1 } } { n } \mathrm { t r } ( \Sigma ) } \end{array}$ . But if we include the high variance points in the average, as in fast-SWA, wˆfSWA = 1n+m  $\begin{array} { r } { \hat { w } _ { \mathrm { f S W A } } = \frac { 1 } { n + m } \big ( \sum _ { i } w _ { i } ^ { ( 1 ) } + \sum _ { j } w _ { j } ^ { ( 2 ) } \big ) } \end{array}$ , then $\begin{array} { r } { \mathbb { E } [ \| \hat { w } _ { \mathrm { f S W A } } - w _ { 0 } \| ^ { 2 } ] = \frac { n \eta _ { 1 } + m \eta _ { 2 } } { ( n + m ) ^ { 2 } } \mathrm { t r } ( \Sigma ) } \end{array}$ . If $\begin{array} { r } { \frac { n \eta _ { 1 } + m \eta _ { 2 } } { ( n + m ) ^ { 2 } } < \frac { \eta _ { 1 } } { n } } \end{array}$ then including the high learning rate points decreases the MSE of the estimator for $\begin{array} { r } { m > n \bigl ( \frac { \eta _ { 2 } } { \eta _ { 1 } } - 2 \bigr ) } \end{array}$ . If we include enough points, we will still improve the estimate.
+As discussed in Mandt et al. (2017), under certain assumptions SGD samples from a Gaussian distribution centered at the optimum of the loss $w _ { 0 }$ with covariance proportional to the learning rate. Suppose then that we have $n$ weights sampled at learning rate $\eta _ { 1 }$ , $w _ { i } ^ { ( 1 ) } \stackrel { i i d } { \sim } \mathcal { N } ( w _ { 0 } , \eta _ { 1 } \Sigma )$ and $m$ weights sampled with the higher learning rate $\eta _ { 2 }$ , $w _ { j } ^ { ( 2 ) } \stackrel { i i d } { \sim } \mathcal { N } ( w _ { 0 } , \eta _ { 2 } \Sigma )$ . For the SWA estimator $\begin{array} { r } { \hat { w } _ { \mathrm { S W A } } = \frac { 1 } { n } \sum _ { i } w _ { i } ^ { ( 1 ) } , \mathbb { E } [ \| \hat { w } _ { \mathrm { S W A } } - w _ { 0 } \| ^ { 2 } ] = \mathrm { t r } ( \mathrm { C o v } ( \hat { w } _ { \mathrm { S W A } } ) ) = \frac { \eta _ { 1 } } { n } \mathrm { t r } ( \Sigma ) } \end{array}$ . But if we include the high variance points in the average, as in fast-SWA, wˆfSWA = 1n+m  $\begin{array} { r } { \hat { w } _ { \mathrm { f S W A } } = \frac { 1 } { n + m } \big ( \sum _ { i } w _ { i } ^ { ( 1 ) } + \sum _ { j } w _ { j } ^ { ( 2 ) } \big ) } \end{array}$ , then $\begin{array} { r } { \mathbb { E } [ \| \hat { w } _ { \mathrm { f S W A } } - w _ { 0 } \| ^ { 2 } ] = \frac { n \eta _ { 1 } + m \eta _ { 2 } } { ( n + m ) ^ { 2 } } \mathrm { t r } ( \Sigma ) } \end{array}$ . If $\begin{array} { r } { \frac { n \eta _ { 1 } + m \eta _ { 2 } } { ( n + m ) ^ { 2 } } < \frac { \eta _ { 1 } } { n } } \end{array}$ then including the high learning rate points decreases the MSE of the estimator for $\begin{array} { r } { m > n \bigl ( \frac { \eta _ { 2 } } { \eta _ { 1 } } - 2 \bigr ) } \end{array}$ . If we include enough points, we will still improve the estimate.
 
 # A.8 NETWORK ARCHITECTURES