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@@ -169,7 +169,7 @@ Remark 3 (The Optimal Batch Sizes and Local Updates Trade-off). The parameter $\
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 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 $\left| \overline { { \mathbf { C . 1 0 } } } \right|$ 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 $\left| \overline { { \mathbf { C . 1 0 } } } \right|$ 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 $[ \bigcirc , \bigcirc , \bigcirc , \bigcirc , \bigcirc ]$ . 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. □