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parse/train/B8fp0LVMHa/B8fp0LVMHa.md

@@ -103,7 +103,7 @@ The bounds in equation 8 capture the main intuitions about the interplay between
 # Algorithm 1: Test-Time Policy $\pi _ { \mathrm { t e s t } }$
 
 Function TestEnsemble(values): return $\lambda \cdot \operatorname* { m i n } ( v a l u e s ) + ( 1 - \lambda ) \cdot \operatorname* { m a x } ( v a l u e s )$   
-Function $\pi _ { \mathrm { t e s t } } \left( s \right)$ : $\{ a _ { i } \} ^ { N } \sim \mu ( a | s )$ return arg $\operatorname* { m a x } _ { \{ a _ { i } \} ^ { N } }$ TestEnsemble[Qi(s, a) for all i]

+Function $\pi _ { \mathrm { t e s t } } \left( s \right)$ : $\{ a _ { i } \} ^ { N } \sim \mu ( a | s )$ return arg $\operatorname* { m a x } _ { \{ a _ { i } \} ^ { N } }$ TestEnsemble[Qi(s, a) for all i]

 
 close to $\pi _ { \underline { { \mu } } } ^ { * }$ . The two variants of $\Delta ( s , N )$ , based on $Q _ { \mu } ^ { * }$ or $Q _ { \mu } ^ { N }$ , suggest an intriguing notion of difficulty for an offline RL problem. If we could estimate either of these $\mathrm { Q }$ -value functions, then for a desired set of states (such as initial states) we could plot $\Delta ( s , N )$ as decreasing function of $N$ . The rate at which this function decreases could serve as an intuitive notion of difficulty for a given offline offline RL problem which consists of an MDP and a given behavior policy. While we leave theoretical investigations of this measure for future work, our empirical results in Section 5 demonstrate that the effective value of $N$ may be surprisingly small.
 

parse/train/cVwc7IHWEWi/cVwc7IHWEWi.md

@@ -42,7 +42,7 @@ Notation. For a positive integer $n$ , let $[ n ] : = \{ 1 , 2 , . . . , n \}$ .
 
 No-regret learning in games. We consider a game $G$ with $m \in \mathbb { N }$ players, where player $i \in [ m ]$ has action space $A _ { i }$ with $n _ { i } : = | A _ { i } |$ actions. We may assume that $\mathcal { A } _ { i } = [ n _ { i } ]$ for each player $i$ . The joint action space is $\mathcal { A } : = \mathcal { A } _ { 1 } \times \dots \times \mathcal { A } _ { m }$ . The specification of the game $G$ is completed by a collection of loss functions $\mathcal { L } _ { 1 } , \ldots , \mathcal { L } _ { m } : \mathcal { A }  [ 0 , 1 ]$ . For an action profile $a = ( a _ { 1 } , \ldots , a _ { m } ) \in \mathcal { A }$ and $i \in [ m ]$ , $\mathcal { L } _ { i } ( a )$ is the loss player $i$ experiences when each player $i ^ { \prime } \in [ m ]$ plays $a _ { i ^ { \prime } }$ . A mixed strategy $x _ { i } \in \Delta ( \mathcal { A } _ { i } )$ for player $i$ is a distribution over $A _ { i }$ , with the probability of playing action $j \in \mathcal A _ { i }$ given by $x _ { i } ( j )$ . Given a mixed strategy profile $\boldsymbol { x } = ( x _ { 1 } , \dots , x _ { m } )$ (or an action profile $\boldsymbol { a } = ( a _ { 1 } , \dots , a _ { m } ) )$ and a player $i \in [ m ]$ we let $x _ { - i }$ (or $a _ { - i }$ , respectively) denote the profile after removing the ith mixed strategy $x _ { i }$ (or the $i$ th action $a _ { i }$ , respectively).
 
-The $m$ players play the game $G$ for a total of $T$ rounds. At the beginning of each round $t \in [ T ]$ , each player $i$ chooses a mixed strategy $x _ { i } ^ { ( t ) } \in \Delta ( \mathcal { A } _ { i } )$ . The loss vector of player $i$ , denoted $\ell _ { i } ^ { ( t ) } \in [ 0 , 1 ] ^ { n _ { i } }$ , is defined as $\ell _ { i } ^ { ( t ) } ( j ) = \mathbb { E } _ { a _ { - i } \sim x _ { - i } ^ { ( t ) } } [ \mathcal { L } _ { i } ( \dot { j } , a _ { - i } ) ]$ . As a matter of convention, set $\ell _ { i } ^ { ( 0 ) } = \mathbf { 0 }$ to be the all-zeros vector. We consider the full-information setting in this paper, meaning that player $i$ observes its full loss vector $\ell _ { i } ^ { ( t ) }$ for each round $t$ . Finally, player $i$ experiences a loss of $\langle \ell _ { i } ^ { ( t ) } , x _ { i } ^ { ( t ) } \rangle$ . The goal of each player $i$ is to minimize its regret, defined as: $\begin{array} { r } { \mathrm { R e g } _ { i , T } : = \sum _ { t \in [ T ] } \langle x _ { i } ^ { ( t ) } , \ell _ { i } ^ { ( t ) } \rangle - \operatorname* { m i n } _ { j \in [ n _ { i } ] } \sum _ { t \in [ T ] } \ell _ { i } ^ { ( t ) } ( j ) } \end{array}$ .
+The $m$ players play the game $G$ for a total of $T$ rounds. At the beginning of each round $t \in [ T ]$ , each player $i$ chooses a mixed strategy $x _ { i } ^ { ( t ) } \in \Delta ( \mathcal { A } _ { i } )$ . The loss vector of player $i$ , denoted $\ell _ { i } ^ { ( t ) } \in [ 0 , 1 ] ^ { n _ { i } }$ , is defined as $\ell _ { i } ^ { ( t ) } ( j ) = \mathbb { E } _ { a _ { - i } \sim x _ { - i } ^ { ( t ) } } [ \mathcal { L } _ { i } ( \dot { j } , a _ { - i } ) ]$ . As a matter of convention, set $\ell _ { i } ^ { ( 0 ) } = \mathbf { 0 }$ to be the all-zeros vector. We consider the full-information setting in this paper, meaning that player $i$ observes its full loss vector $\ell _ { i } ^ { ( t ) }$ for each round $t$ . Finally, player $i$ experiences a loss of $\langle \ell _ { i } ^ { ( t ) } , x _ { i } ^ { ( t ) } \rangle$ . The goal of each player $i$ is to minimize its regret, defined as: $\begin{array} { r } { \mathrm { R e g } _ { i , T } : = \sum _ { t \in [ T ] } \langle x _ { i } ^ { ( t ) } , \ell _ { i } ^ { ( t ) } \rangle - \operatorname* { m i n } _ { j \in [ n _ { i } ] } \sum _ { t \in [ T ] } \ell _ { i } ^ { ( t ) } ( j ) } \end{array}$ .
 
 Optimistic hedge. The Optimistic Hedge algorithm chooses mixed strategies for player $i \in [ m ]$ as follows: at time $t = 1$ , it sets $x _ { i } ^ { ( 1 ) } = ( 1 / n _ { i } , \ldots , 1 / n _ { i } )$ to be the uniform distribution on $\mathbf { \mathcal { A } } _ { i }$ . Then for all $t < T$ , player $i$ ’s strategy at iteration $t + 1$ is defined as follows, for $j \in [ n _ { i } ]$ :
 
@@ -52,7 +52,7 @@ $$
 
 Optimistic Hedge is a modification of Hedge, which performs the updates $x _ { i } ^ { ( t + 1 ) } ( j ) : =$ $\begin{array} { r l } & { \frac { x _ { i } ^ { ( t ) } ( j ) \cdot \exp ( - \eta \cdot \ell _ { i } ^ { ( t ) } ( j ) ) } { \sum _ { k \in [ n _ { i } ] } x _ { i } ^ { ( t ) } ( k ) \cdot \exp ( - \eta \cdot \ell _ { i } ^ { ( t ) } ( k ) ) } } \end{array}$ . The update (1) modifies the Hedge update by replacing the loss vector $\ell _ { i } ^ { ( t ) }$ 2 i   with a predictor of the following iteration’s loss vector, $\ell _ { i } ^ { ( t ) } + ( \ell _ { i } ^ { ( t ) } - \ell _ { i } ^ { ( t - 1 ) } )$ . Hedge corresponds to FTRL with a negative entropy regularizer (see, e.g., [Bub15]), whereas Optimistic Hedge corresponds to Optimistic FTRL with a negative entropy regularizer [RS13b, RS13a].
 
-Distributions $\pmb { \& }$ divergences. For distributions $P , Q$ on a finite domain $[ n ]$ , the $K L$ divergence between $P , Q$ is $\begin{array} { r } { \mathrm { K L } ( P ; Q ) = \sum _ { j = 1 } ^ { n } P ( j ) \cdot \log \left( \frac { P ( j ) } { Q ( j ) } \right) } \end{array}$ . The chi-squared divergence between $P , Q$ is $\begin{array} { r } { \chi ^ { 2 } ( P ; Q ) = \sum _ { j = 1 } ^ { n } Q ( j ) \cdot \left( \frac { P ( j ) } { Q ( j ) } \right) ^ { 2 } - 1 = \sum _ { j = 1 } ^ { n } \frac { ( P ( j ) - Q ( j ) ) ^ { 2 } } { Q ( j ) } } \end{array}$ (P (j)Q(j))2 . For a distribution P on [n] and a vector $v \in \mathbb { R } ^ { n }$ , we write $\begin{array} { r } { \mathrm { V a r } _ { P } \left( v \right) : = \sum _ { j = 1 } ^ { n } P ( j ) \cdot \left( v ( j ) - \sum _ { k = 1 } ^ { n } P ( k ) v ( k ) \right) ^ { 2 } } \end{array}$ . Also define $\begin{array} { r } { \| v \| _ { P } : = \sqrt { \sum _ { j = 1 } ^ { n } P ( j ) \cdot v ( j ) ^ { 2 } } } \end{array}$ . If further $P$ has full support, then define $\begin{array} { r } { \left\| v \right\| _ { P } ^ { \star } = \sqrt { \sum _ { j = 1 } ^ { n } \frac { v ( j ) ^ { 2 } } { P ( j ) } } } \end{array}$ The above notations will often be used when $P$ is the mixed strategy profile $x _ { i }$ for some player $i$ and $v$ is a loss vector $\ell _ { i }$ ; in such a case the norms $\| v \| _ { P }$ and $\| v \| _ { P } ^ { \star }$ are often called local norms.
+Distributions $\pmb { \& }$ divergences. For distributions $P , Q$ on a finite domain $[ n ]$ , the $K L$ divergence between $P , Q$ is $\begin{array} { r } { \mathrm { K L } ( P ; Q ) = \sum _ { j = 1 } ^ { n } P ( j ) \cdot \log \left( \frac { P ( j ) } { Q ( j ) } \right) } \end{array}$ . The chi-squared divergence between $P , Q$ is $\begin{array} { r } { \chi ^ { 2 } ( P ; Q ) = \sum _ { j = 1 } ^ { n } Q ( j ) \cdot \left( \frac { P ( j ) } { Q ( j ) } \right) ^ { 2 } - 1 = \sum _ { j = 1 } ^ { n } \frac { ( P ( j ) - Q ( j ) ) ^ { 2 } } { Q ( j ) } } \end{array}$ (P (j)Q(j))2 . For a distribution P on [n] and a vector $v \in \mathbb { R } ^ { n }$ , we write $\begin{array} { r } { \mathrm { V a r } _ { P } \left( v \right) : = \sum _ { j = 1 } ^ { n } P ( j ) \cdot \left( v ( j ) - \sum _ { k = 1 } ^ { n } P ( k ) v ( k ) \right) ^ { 2 } } \end{array}$ . Also define $\begin{array} { r } { \| v \| _ { P } : = \sqrt { \sum _ { j = 1 } ^ { n } P ( j ) \cdot v ( j ) ^ { 2 } } } \end{array}$ . If further $P$ has full support, then define $\begin{array} { r } { \left\| v \right\| _ { P } ^ { \star } = \sqrt { \sum _ { j = 1 } ^ { n } \frac { v ( j ) ^ { 2 } } { P ( j ) } } } \end{array}$ The above notations will often be used when $P$ is the mixed strategy profile $x _ { i }$ for some player $i$ and $v$ is a loss vector $\ell _ { i }$ ; in such a case the norms $\| v \| _ { P }$ and $\| v \| _ { P } ^ { \star }$ are often called local norms.
 
 # 3 Results
 
@@ -94,7 +94,7 @@ $$
 \mathrm { R e g } _ { i , T } \leq \frac { \log n _ { i } } { \eta } + \sum _ { t = 1 } ^ { T } \left. x _ { i } ^ { ( t ) } - \tilde { x } _ { i } ^ { ( t ) } \right. _ { x _ { i } ^ { ( t ) } } ^ { \times } \sqrt { \mathrm { V a r } _ { x _ { i } ^ { ( t ) } } \left( \ell _ { i } ^ { ( t ) } - \ell _ { i } ^ { ( t - 1 ) } \right) } - \frac { 1 } { \eta } \sum _ { t = 1 } ^ { T } \mathrm { K L } ( \tilde { x } _ { i } ^ { ( t ) } ; x _ { i } ^ { ( t ) } ) - \frac { 1 } { \eta } \sum _ { t = 1 } ^ { T } \mathrm { K L } ( \tilde { x } _ { i } ^ { ( t ) } ; x _ { i } ^ { ( t ) } ) ,
 $$
 
-2) thand $\mathrm { K L } ( \tilde { x } _ { i } ^ { ( t ) } ; x _ { i } ^ { ( t ) } )$ $\mathrm { K L } ( x _ { i } ^ { ( t ) } ; \tilde { x } _ { i } ^ { ( t - 1 ) } )$ may be lower bounded bytively. Note it is a standard $( 1 / 2 - O ( \eta ) ) \cdot \chi ^ { 2 } ( \tilde { x } _ { i } ^ { ( t ) } ; x _ { i } ^ { ( t ) } )$ $( 1 / 2 - O ( \eta ) ) \cdot \chi ^ { 2 } ( x _ { i } ^ { ( t ) } ; \tilde { x } _ { i } ^ { ( t - 1 ) } )$ fact that the KL divergence between two distributions is upper bounded by the chi-squared distribution between them; by contrast, Lemma A.2 can exploit that $x _ { i } ^ { ( t ) }$ , $\tilde { x } _ { i } ^ { ( t ) }$ t) and x˜ (t1)i a re close to each other between lower bo $x _ { i } ^ { ( t ) }$ i and x˜ $\tilde { x } _ { i } ^ { ( t - 1 ) }$ hat the , leadi $\chi ^ { 2 }$ -divergenco the term $\chi ^ { 2 } ( x _ { i } ^ { ( t ) } ; \tilde { x } _ { i } ^ { ( t - 1 ) } )$ $\left( 1 - O ( \eta ) \right) \cdot \eta ^ { 2 } \cdot \operatorname { V a r } _ { x _ { i } ^ { ( t ) } } \left( \ell _ { i } ^ { ( t - 1 ) } \right)$ $\begin{array} { r } { \frac { ( 1 - C \eta ) \eta } { 2 } \operatorname { V a r } _ { x _ { i } ^ { ( t ) } } \left( \ell _ { i } ^ { ( t - 1 ) } \right) } \end{array}$ being subtracted in (4). The $\chi ^ { 2 }$ -divergence $\chi ^ { 2 } ( \tilde { x } _ { i } ^ { ( t ) } ; x _ { i } ^ { ( t ) } )$ , as well as the term $\left\| \boldsymbol { x } _ { i } ^ { ( t ) } - \tilde { \boldsymbol { x } } _ { i } ^ { ( t ) } \right\| _ { \boldsymbol { x } _ { i } ^ { ( t ) } } ^ { \star }$ in (5) are bounded in a similar manner to obtain (4).
+2) thand $\mathrm { K L } ( \tilde { x } _ { i } ^ { ( t ) } ; x _ { i } ^ { ( t ) } )$ $\mathrm { K L } ( x _ { i } ^ { ( t ) } ; \tilde { x } _ { i } ^ { ( t - 1 ) } )$ may be lower bounded bytively. Note it is a standard $( 1 / 2 - O ( \eta ) ) \cdot \chi ^ { 2 } ( \tilde { x } _ { i } ^ { ( t ) } ; x _ { i } ^ { ( t ) } )$ $( 1 / 2 - O ( \eta ) ) \cdot \chi ^ { 2 } ( x _ { i } ^ { ( t ) } ; \tilde { x } _ { i } ^ { ( t - 1 ) } )$ fact that the KL divergence between two distributions is upper bounded by the chi-squared distribution between them; by contrast, Lemma A.2 can exploit that $x _ { i } ^ { ( t ) }$ , $\tilde { x } _ { i } ^ { ( t ) }$ t) and x˜ (t1)i a re close to each other between lower bo $x _ { i } ^ { ( t ) }$ i and x˜ $\tilde { x } _ { i } ^ { ( t - 1 ) }$ hat the , leadi $\chi ^ { 2 }$ -divergenco the term $\chi ^ { 2 } ( x _ { i } ^ { ( t ) } ; \tilde { x } _ { i } ^ { ( t - 1 ) } )$ $\left( 1 - O ( \eta ) \right) \cdot \eta ^ { 2 } \cdot \operatorname { V a r } _ { x _ { i } ^ { ( t ) } } \left( \ell _ { i } ^ { ( t - 1 ) } \right)$ $\begin{array} { r } { \frac { ( 1 - C \eta ) \eta } { 2 } \operatorname { V a r } _ { x _ { i } ^ { ( t ) } } \left( \ell _ { i } ^ { ( t - 1 ) } \right) } \end{array}$ being subtracted in (4). The $\chi ^ { 2 }$ -divergence $\chi ^ { 2 } ( \tilde { x } _ { i } ^ { ( t ) } ; x _ { i } ^ { ( t ) } )$ , as well as the term $\left\| \boldsymbol { x } _ { i } ^ { ( t ) } - \tilde { \boldsymbol { x } } _ { i } ^ { ( t ) } \right\| _ { \boldsymbol { x } _ { i } ^ { ( t ) } } ^ { \star }$ in (5) are bounded in a similar manner to obtain (4).
 
 # 4.2 Finite differences
 
@@ -190,7 +190,7 @@ $$
 \sum _ { t = 1 } ^ { T - h - 1 } \operatorname { V a r } _ { x _ { i } ^ { ( t ) } } \left( ( \operatorname { D } _ { h + 1 } \ell _ { i } ) ^ { ( t ) } \right) \leq O ( 1 / H ) \cdot \sum _ { t = 1 } ^ { T - h } \operatorname { V a r } _ { x _ { i } ^ { ( t ) } } \left( ( \operatorname { D } _ { h } \ell _ { i } ) ^ { ( t ) } \right) + \tilde { O } \left( \mu ^ { 2 h } \right) ,
 $$
 
-where $\tilde { O }$ hides factors polynomial in $\log T$ . The validity of (12) for $h = 0$ implies Lemma 4.2. On implies that the other hand, as long we choose the value $\begin{array} { r } { \sum _ { t = 1 } ^ { T - H } \operatorname { V a r } _ { x _ { i } ^ { ( t ) } } \left( \left( \operatorname { D } _ { H } \ell _ { i } \right) ^ { ( t ) } \right) \leq O ( \mu ^ { 2 H } ) } \end{array}$ $\mu$ in (12) to satisfy . This gives that (12) holds for $\mu \geq m \eta H ^ { \Omega ( 1 ) }$ , then Lemma 4.4 $h = H - 1$ . To show that (12) holds for all $H - 1 > h \geq 0$ , we use downwards induction; fix any $h$ , and assume that (12) has been shown for all $h ^ { \prime }$ satisfying $h < h ^ { \prime } \leq H - 1$ . Our main tool in the inductive step is to apply Lemma 4.7 below. To state it, for $\zeta > 0 , n \in \mathbb { N }$ , we say that a sequence of distributions $P ^ { ( 1 ) } , \ldots , P ^ { ( T ) } \in \Delta ^ { n }$ is $\zeta$ -consecutively close if for each $1 \leq t < T$ , it holds that max $\left\{ \left\| \frac { P ^ { ( t ) } } { P ^ { ( t + 1 ) } } \right\| _ { \infty } , \left\| \frac { P ^ { ( t + 1 ) } } { P ^ { ( t ) } } \right\| _ { \infty } \right\} \leq 1 + \zeta$ . 4 Lemma 4.7 shows that given a sequence of vectors for which the variances of its second-order finite differences are bounded by the variances of its first-order finite differences, a similar relationship holds between its first- and zeroth-order finite differences.
+where $\tilde { O }$ hides factors polynomial in $\log T$ . The validity of (12) for $h = 0$ implies Lemma 4.2. On implies that the other hand, as long we choose the value $\begin{array} { r } { \sum _ { t = 1 } ^ { T - H } \operatorname { V a r } _ { x _ { i } ^ { ( t ) } } \left( \left( \operatorname { D } _ { H } \ell _ { i } \right) ^ { ( t ) } \right) \leq O ( \mu ^ { 2 H } ) } \end{array}$ $\mu$ in (12) to satisfy . This gives that (12) holds for $\mu \geq m \eta H ^ { \Omega ( 1 ) }$ , then Lemma 4.4 $h = H - 1$ . To show that (12) holds for all $H - 1 > h \geq 0$ , we use downwards induction; fix any $h$ , and assume that (12) has been shown for all $h ^ { \prime }$ satisfying $h < h ^ { \prime } \leq H - 1$ . Our main tool in the inductive step is to apply Lemma 4.7 below. To state it, for $\zeta > 0 , n \in \mathbb { N }$ , we say that a sequence of distributions $P ^ { ( 1 ) } , \ldots , P ^ { ( T ) } \in \Delta ^ { n }$ is $\zeta$ -consecutively close if for each $1 \leq t < T$ , it holds that max $\left\{ \left\| \frac { P ^ { ( t ) } } { P ^ { ( t + 1 ) } } \right\| _ { \infty } , \left\| \frac { P ^ { ( t + 1 ) } } { P ^ { ( t ) } } \right\| _ { \infty } \right\} \leq 1 + \zeta$ . 4 Lemma 4.7 shows that given a sequence of vectors for which the variances of its second-order finite differences are bounded by the variances of its first-order finite differences, a similar relationship holds between its first- and zeroth-order finite differences.
 
 Lemma 4.7. There is a sufficiently large constant $C _ { 0 } > 1$ so that the following holds. For any $M , \zeta , \alpha > 0$ and $n \in \mathbb N$ , suppose that $P ^ { ( 1 ) } , \ldots , P ^ { ( T ) } \in \Delta ^ { n }$ and $Z ^ { ( 1 ) } , \ldots , Z ^ { ( \breve { T } ) } \in [ - M , M ] ^ { \breve { n } }$ satisfy the following conditions:
 
@@ -216,7 +216,7 @@ $$
 \begin{array} { r } { ( \mathrm { D } _ { h } ^ { \circ } L ) ^ { ( t ) } = \left\{ \begin{array} { l l } { \left( \mathrm { D } _ { h - 1 } ^ { \circ } L \right) ^ { ( t + 1 ) } - \left( \mathrm { D } _ { h - 1 } ^ { \circ } L \right) ^ { ( t ) } } & { : 0 \le t \le S - 2 } \\ { \left( \mathrm { D } _ { h - 1 } ^ { \circ } L \right) ^ { ( 1 ) } - \left( \mathrm { D } _ { h - 1 } ^ { \circ } L \right) ^ { ( T ) } } & { : t = S - 1 . } \end{array} \right. } \end{array}
 $$
 
-Circular finite differences for a sequence $L ^ { ( 0 ) } , \ldots , L ^ { ( S - 1 ) }$ are defined similarly to finite differences (Definition 4.1) except that unlike for finite differences, where (Dh L) (Sh) , $\left( \mathrm { D } _ { h } \cal L \right) ^ { ( S - h ) } , . . . , \left( \mathrm { D } _ { h } \cal L \right) ^ { ( S - 1 ) }$ are not defined, (Dh L) (Sh) , $\big ( \mathrm { D } _ { h } ^ { \circ } L \big ) ^ { ( S - h ) } , \ldots , \big ( \mathrm { D } _ { h } ^ { \circ } L \big ) ^ { ( S - 1 ) }$ are defined by “wrapping around” back to the beginning of the sequence. The above-described reductions, which are worked out in detail in Section C.2, allow us to reduce proving Lemma 4.7 to proving the following simpler lemma:
+Circular finite differences for a sequence $L ^ { ( 0 ) } , \ldots , L ^ { ( S - 1 ) }$ are defined similarly to finite differences (Definition 4.1) except that unlike for finite differences, where (Dh L) (Sh) , $\left( \mathrm { D } _ { h } \cal L \right) ^ { ( S - h ) } , . . . , \left( \mathrm { D } _ { h } \cal L \right) ^ { ( S - 1 ) }$ are not defined, (Dh L) (Sh) , $\big ( \mathrm { D } _ { h } ^ { \circ } L \big ) ^ { ( S - h ) } , \ldots , \big ( \mathrm { D } _ { h } ^ { \circ } L \big ) ^ { ( S - 1 ) }$ are defined by “wrapping around” back to the beginning of the sequence. The above-described reductions, which are worked out in detail in Section C.2, allow us to reduce proving Lemma 4.7 to proving the following simpler lemma:
 
 Lemma 4.8. Suppose $\mu \in \mathbb { R } , \alpha > 0$ , and $W ^ { ( 0 ) } , \ldots , W ^ { ( S - 1 ) } \in \mathbb { R }$ is a sequence of reals satisfying