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parse/train/6YEQUn0QICG/6YEQUn0QICG.md

@@ -111,7 +111,7 @@ Proof sketch The main idea follows Du et al. (2018); Dukler et al. (2020), that
 
 Based on our formulation, the convergence rate of FedAvg (Theorem 4.4) can be derived from Dukler et al. (2020) by considering non-identical covariance matrices. We derive the convergence rate of FedBN in Corollary 4.5. Our key result of comparing the convergence rates between FedAvg and FedBN is culminated in Corollary 4.6.
 
-Theorem 4.4 (G-dominated convergence for FedAvg Dukler et al. (2020)). Suppose network (4) is initialized as in (2) with $\alpha > 1$ , trained using gradient descent and Assumptions 4.1 holds. Given the loss function of training the neural network is the square loss with targets y satisfying $\| \mathbf { y } \| _ { \infty } = O ( 1 )$ . If $m = \Omega$ ma $\mathrm { x } \left\{ \tilde { N ^ { 4 } } M ^ { 4 } \log ( N M / \delta ) / \alpha ^ { 4 } \mu _ { 0 } ^ { 4 } , \hat { N ^ { 2 } } M ^ { 2 } \log ( N M / \delta ) \tilde { / } \mu _ { 0 } ^ { 2 } \right\} \nonumber$ , then with probability $1 - \delta$ ,
+Theorem 4.4 (G-dominated convergence for FedAvg Dukler et al. (2020)). Suppose network (4) is initialized as in (2) with $\alpha > 1$ , trained using gradient descent and Assumptions 4.1 holds. Given the loss function of training the neural network is the square loss with targets y satisfying $\| \mathbf { y } \| _ { \infty } = O ( 1 )$ . If $m = \Omega$ ma $\mathrm { x } \left\{ \tilde { N ^ { 4 } } M ^ { 4 } \log ( N M / \delta ) / \alpha ^ { 4 } \mu _ { 0 } ^ { 4 } , \hat { N ^ { 2 } } M ^ { 2 } \log ( N M / \delta ) \tilde { / } \mu _ { 0 } ^ { 2 } \right\} \nonumber$ , then with probability $1 - \delta$ ,
 
 1. For iterations $t = 0 , 1 , \cdots$ , the evolution matrix $\Lambda ( t )$ satisfies $\begin{array} { r } { \lambda _ { \operatorname* { m i n } } ( \mathbf { A } ( t ) ) \geq \frac { \mu _ { 0 } } { 2 } } \end{array}$
 

parse/train/EnmG3G5SYR/EnmG3G5SYR.md

@@ -232,7 +232,7 @@ After $T$ rounds of updates, the mirror descent algorithm that we use here readi
 
 # 4 Main results
 
-We now turn to the statement of a bound on the performance of the policy $\pi _ { \mathrm { A L G } }$ returned by PACLE. This upper bound involves three terms: an optimization error, an uncertainty term, and a model mis-specification term. The optimization error is given by $\textstyle { \mathcal { C } } ( T ) { \overset { d e f } { = } } 4 H { \sqrt { \frac { \log | A | } { T } } }$ log |A|T ; it captures the rate at which the error decreases as a function of the iterations of the actor. The mis-specification error $\begin{array} { r } { \mathcal { E } _ { \mathrm { m s p } } ( \nu ) \overset { d e f } { = } \sum _ { h = 1 } ^ { H } \nu _ { h } } \end{array}$ is simply the sum of all the stage-wise mis-specification errors; notice h=1  that the mis-specification error does depend on the choice of the radii for the critic $\rho _ { 1 } ^ { w } , \ldots , \rho _ { H } ^ { w }$ in a problem dependent way (cf. Assumption $\bigstar \bigstar \bigstar$ Finally, for each h, define the vector ¯⇡h def= $\mathbb { E } _ { ( S _ { h } , A _ { h } ) \sim \pi } [ \phi _ { h } ( S _ { h } , A _ { h } ) ]$ , where the expectation is over the state-action $\left( S _ { h } , A _ { h } \right)$ encountered at timestep $h$ upon following policy $\pi$ . In terms of these vectors, the uncertainty error is given by
+We now turn to the statement of a bound on the performance of the policy $\pi _ { \mathrm { A L G } }$ returned by PACLE. This upper bound involves three terms: an optimization error, an uncertainty term, and a model mis-specification term. The optimization error is given by $\textstyle { \mathcal { C } } ( T ) { \overset { d e f } { = } } 4 H { \sqrt { \frac { \log | A | } { T } } }$ log |A|T ; it captures the rate at which the error decreases as a function of the iterations of the actor. The mis-specification error $\begin{array} { r } { \mathcal { E } _ { \mathrm { m s p } } ( \nu ) \overset { d e f } { = } \sum _ { h = 1 } ^ { H } \nu _ { h } } \end{array}$ is simply the sum of all the stage-wise mis-specification errors; notice h=1  that the mis-specification error does depend on the choice of the radii for the critic $\rho _ { 1 } ^ { w } , \ldots , \rho _ { H } ^ { w }$ in a problem dependent way (cf. Assumption $\bigstar \bigstar \bigstar$ Finally, for each h, define the vector ¯⇡h def= $\mathbb { E } _ { ( S _ { h } , A _ { h } ) \sim \pi } [ \phi _ { h } ( S _ { h } , A _ { h } ) ]$ , where the expectation is over the state-action $\left( S _ { h } , A _ { h } \right)$ encountered at timestep $h$ upon following policy $\pi$ . In terms of these vectors, the uncertainty error is given by
 
 $$
 \mathcal { U } ( \pi ; \alpha ) \overset { d e f } { = } 2 \sum _ { h = 1 } ^ { H } \alpha _ { h } \| \bar { \phi } _ { h } ^ { \pi } \| _ { \Sigma _ { h } ^ { - 1 } } = 2 \sum _ { h = 1 } ^ { H } \alpha _ { h } \sqrt { ( \bar { \phi } _ { h } ^ { \pi } ) ^ { \top } \Sigma _ { h } ^ { - 1 } \bar { \phi } _ { h } ^ { \pi } } ,