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parse/train/6vaActvpcp3/6vaActvpcp3.md

@@ -165,7 +165,7 @@ Figure 1: Local coverage frequencies for adaptive conformal (blue), a non-adapti
 
 Daily open prices were obtained from publicly available datasets published by The Wall Street Journal. The realized local coverage frequencies for the non-adaptive and adaptive conformal methods on four different stocks are shown in Figure $1 .$ These stocks were selected out of a total of 12 stocks that we examined because they showed a clear failure of the non-adaptive method. Adaptive conformal inference was found to perform well in all cases (see Figure 9 in the appendix).
 
-As a visual comparator, the grey curves show the moving average 1500 Pt+250r=t250+1 Ir for sequences $\left\{ I _ { t } \right\} _ { 1 \leq t \leq T }$ that are i.i.d. Bernoulli(0.1). We see that the local coverage frequencies obtained by adaptive conformal inference (blue lines) always stay within the variation that would be expected from an i.i.d. Bernoulli sequence. On the other hand, the non-adaptive method undergoes large excursions away from the target level of $1 - \alpha = 0 . 9$ (red lines). For example, in the bottom right panel we can see that the non-adaptive method fails to cover the realized volatility of Fannie Mae during the 2008 financial crisis, while the adaptive method is robust to this event (see Figure 4 in the Appendix for a plot of the price of Fannie Mae over this time period).
+As a visual comparator, the grey curves show the moving average 1500 Pt+250r=t250+1 Ir for sequences $\left\{ I _ { t } \right\} _ { 1 \leq t \leq T }$ that are i.i.d. Bernoulli(0.1). We see that the local coverage frequencies obtained by adaptive conformal inference (blue lines) always stay within the variation that would be expected from an i.i.d. Bernoulli sequence. On the other hand, the non-adaptive method undergoes large excursions away from the target level of $1 - \alpha = 0 . 9$ (red lines). For example, in the bottom right panel we can see that the non-adaptive method fails to cover the realized volatility of Fannie Mae during the 2008 financial crisis, while the adaptive method is robust to this event (see Figure 4 in the Appendix for a plot of the price of Fannie Mae over this time period).
 
 # 3 Related Work
 
@@ -247,9 +247,9 @@ In this setting, $\{ ( \alpha _ { t } , A _ { t } ) \} _ { t \in \mathbb { N } }
 
 # 4.2.2 Large deviation bound for the errors
 
-Our first have that $\operatorname { e r r } _ { t }$ orrect aveand since value. More precisely, by is stationary it follows that $\boxed { 4 . 1 }$ weus, ${ \mathrm { l i m } } _ { T \to \infty } T ^ { - 1 } \sum _ { t = 1 } ^ { T } { \mathrm { e r r } } _ { t } \ { \overset { a . s . } { = } } \alpha$ $\operatorname { e r r } _ { t }$ $\mathbb { E } [ \mathsf { e r r } _ { t } ] = \alpha$ to understand the deviation of T 1 PTt= from $\alpha$ we simply need to characterize the dependence structure of $\{ \mathrm { e r r } _ { t } \} _ { t \in \mathbb { N } }$ .
+Our first have that $\operatorname { e r r } _ { t }$ orrect aveand since value. More precisely, by is stationary it follows that $\boxed { 4 . 1 }$ weus, ${ \mathrm { l i m } } _ { T \to \infty } T ^ { - 1 } \sum _ { t = 1 } ^ { T } { \mathrm { e r r } } _ { t } \ { \overset { a . s . } { = } } \alpha$ $\operatorname { e r r } _ { t }$ $\mathbb { E } [ \mathsf { e r r } _ { t } ] = \alpha$ to understand the deviation of T 1 PTt= from $\alpha$ we simply need to characterize the dependence structure of $\{ \mathrm { e r r } _ { t } \} _ { t \in \mathbb { N } }$ .
 
-We accomplish this in Theorem $^ { 4 . 1 , }$ which gives a large deviation bound on $\begin{array} { r } { | T ^ { - 1 } \sum _ { t = 1 } ^ { T } \mathrm { e r r } _ { t } - \alpha | } \end{array}$ The idea behind this result is to decompose the dependence in into two parts. First, there is dependence due to the fact that $\alpha _ { t }$ is a function of $\{ \mathrm { e r r } _ { r } \} _ { 1 \leq r \leq t - 1 }$ . In Section $\boxed { \mathbf { A } . 7 }$ in the Appendix we argue that this dependence induces a negative correlation and thus the errors concentrate around their expectation at a rate no slower than that of an i.i.d. Bernoulli sequence. This gives rise to the first term in $( 6 )$ , which is what would be obtained by applying Hoeffding’s inequality to an i.i.d. sequence. Second, there is dependence due to the fact that $A _ { t }$ depends on $A _ { t - 1 }$ . More specifically, consider a setting in which the distribution of $Y | X$ has more variability in some states than others. The goal of adaptive conformal inference is to adapt to the level of variability and thus return larger prediction sets in states where the distribution of $Y | X$ is more spread. However, this algorithm is not perfect and as a result there may be some states $a \in { \mathcal { A } }$ in which $\mathbb { E } [ \mathbf { e r r } _ { t } | A _ { t } = a ]$ is biased away from $\alpha$ . Furthermore, if the environment tends to spend long stretches of time in more variable (or less variable) states this will induce a positive dependence in the errors and cause T 1 PTt=1 to deviate from $\alpha$ . To control this dependence we use a Bernstein inequality for Markov chains to bound $\begin{array} { r } { | T ^ { - 1 } \sum _ { t = 1 } ^ { T } \mathbb { E } [ \mathrm { e r r } _ { t } | A _ { t } ] - \alpha | } \end{array}$ . This gives rise to the second term in $( 6 )$
+We accomplish this in Theorem $^ { 4 . 1 , }$ which gives a large deviation bound on $\begin{array} { r } { | T ^ { - 1 } \sum _ { t = 1 } ^ { T } \mathrm { e r r } _ { t } - \alpha | } \end{array}$ The idea behind this result is to decompose the dependence in into two parts. First, there is dependence due to the fact that $\alpha _ { t }$ is a function of $\{ \mathrm { e r r } _ { r } \} _ { 1 \leq r \leq t - 1 }$ . In Section $\boxed { \mathbf { A } . 7 }$ in the Appendix we argue that this dependence induces a negative correlation and thus the errors concentrate around their expectation at a rate no slower than that of an i.i.d. Bernoulli sequence. This gives rise to the first term in $( 6 )$ , which is what would be obtained by applying Hoeffding’s inequality to an i.i.d. sequence. Second, there is dependence due to the fact that $A _ { t }$ depends on $A _ { t - 1 }$ . More specifically, consider a setting in which the distribution of $Y | X$ has more variability in some states than others. The goal of adaptive conformal inference is to adapt to the level of variability and thus return larger prediction sets in states where the distribution of $Y | X$ is more spread. However, this algorithm is not perfect and as a result there may be some states $a \in { \mathcal { A } }$ in which $\mathbb { E } [ \mathbf { e r r } _ { t } | A _ { t } = a ]$ is biased away from $\alpha$ . Furthermore, if the environment tends to spend long stretches of time in more variable (or less variable) states this will induce a positive dependence in the errors and cause T 1 PTt=1 to deviate from $\alpha$ . To control this dependence we use a Bernstein inequality for Markov chains to bound $\begin{array} { r } { | T ^ { - 1 } \sum _ { t = 1 } ^ { T } \mathbb { E } [ \mathrm { e r r } _ { t } | A _ { t } ] - \alpha | } \end{array}$ . This gives rise to the second term in $( 6 )$
 
 Theorem 4.1 Assume that $\{ A _ { t } \} _ { t \in \mathbb { N } }$ has non-zero absolute spectral gap $1 - \eta > 0$ . Let
 

parse/train/rkxmPgrKwB/rkxmPgrKwB.md

@@ -286,7 +286,7 @@ Now we should consider other 2-nd order permutation points (at layer $k$ ) givin
 
 For this case, we have $\textstyle { \frac { n _ { k } ! } { 2 ! ^ { 2 } } }$ permutations given by permuting the neuron indices of layer $k$ instead of the usual $n _ { k }$ ! permutations since we should eliminate the equivalent permutations corresponding to the permutations among the two pairs of duplicated parameter vectors with a division by $2 ! ^ { 2 }$ . Therefore, this 2-nd order permutation point induces a permutation set with cardinality $\begin{array} { r } { | P ( \pmb { \theta } ) | = \frac { 1 } { 2 ! ^ { 2 } } \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! } \end{array}$
 
-Again, we should consider other 2-nd order permutation points (at layer $k$ ) giving rise to the same network function and corresponding to the same unordered partition. Note that we can choose two parameter vectors to duplicate out of $n _ { k } - 2$ in ${ \binom { n _ { k } - 2 } { 2 } }$ ways and there are 12!2 Qd−1j=1 nj ! many points in each one of the permutation sets. Therefore, we end up having nk−22 
 12!2 Qd−1j=1 nj ! many 2-nd order permutation points (at layer $k$ ) corresponding to this unordered partition. Overall, we have $\begin{array} { r } { \binom { n _ { k } - 2 } { 1 } \frac { 1 } { 3 ! } \prod _ { j = 1 } ^ { d - 1 } n _ { j } \bar { ! } + \binom { n _ { k } - 2 } { 2 } \frac { \mathrm { i } } { 2 ! ^ { 2 } } \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! } \end{array}$ many 2-nd order permutation points at layer $k$ .
+Again, we should consider other 2-nd order permutation points (at layer $k$ ) giving rise to the same network function and corresponding to the same unordered partition. Note that we can choose two parameter vectors to duplicate out of $n _ { k } - 2$ in ${ \binom { n _ { k } - 2 } { 2 } }$ ways and there are 12!2 Qd−1j=1 nj ! many points in each one of the permutation sets. Therefore, we end up having nk−22 
 12!2 Qd−1j=1 nj ! many 2-nd order permutation points (at layer $k$ ) corresponding to this unordered partition. Overall, we have $\begin{array} { r } { \binom { n _ { k } - 2 } { 1 } \frac { 1 } { 3 ! } \prod _ { j = 1 } ^ { d - 1 } n _ { j } \bar { ! } + \binom { n _ { k } - 2 } { 2 } \frac { \mathrm { i } } { 2 ! ^ { 2 } } \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! } \end{array}$ many 2-nd order permutation points at layer $k$ .
 
 # (3) The case $K = 3$ :
 
@@ -294,7 +294,7 @@ There are three ways to have $n _ { k } - 3$ distinct vectors out of $n _ { k }$
 
 (i) $n _ { k } = 4 + 1 + . . . + 1$ For this case, we have $\textstyle { \frac { n _ { k } ! } { 4 ! } }$ permutations given by permuting the neuron indices of layer $k$ . Therefore, this 3-rd order permutation point induces a permutation set with cardinality $\begin{array} { r } { | P ( \pmb { \theta } ) | = \frac { 1 } { 4 ! } \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! } \end{array}$ .
 
-As usual, we should consider other 3-rd order permutation points (at layer $k$ ) giving rise to the same network function and corresponding to the same unordered partition. If we had chosen another parameter vector to replicate four times, this 3-rd order permutation point would induce another permutation set. Note that we can choose the parameter vector to replicate out of nk − 3 in nk−31 
 ways and there ar e 14! Qd−1j=1 nj ! many points in each one of the permutation sets. Therefore, we end up having $\begin{array} { r } { \left( { \overset { n _ { k } - 3 } { \ 1 } } \right) { \frac { 1 } { 4 ! } } \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! } \end{array}$ many 3-rd order permutation points (at layer $k$ ) corresponding to this unordered partition.
+As usual, we should consider other 3-rd order permutation points (at layer $k$ ) giving rise to the same network function and corresponding to the same unordered partition. If we had chosen another parameter vector to replicate four times, this 3-rd order permutation point would induce another permutation set. Note that we can choose the parameter vector to replicate out of nk − 3 in nk−31 
 ways and there ar e 14! Qd−1j=1 nj ! many points in each one of the permutation sets. Therefore, we end up having $\begin{array} { r } { \left( { \overset { n _ { k } - 3 } { \ 1 } } \right) { \frac { 1 } { 4 ! } } \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! } \end{array}$ many 3-rd order permutation points (at layer $k$ ) corresponding to this unordered partition.
 
 For this case, we have $\textstyle { \frac { n _ { k } ! } { 3 ! 2 ! } }$ permutations given by permuting the neuron indices of layer $k$ . Therefore, this 3-rd order permutation point induces a permutation set with cardinality $\begin{array} { r } { | P ( \pmb { \theta } ) | = \frac { 1 } { 3 ! 2 ! } \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! } \end{array}$
 
@@ -304,7 +304,7 @@ As usual, we should consider other 3-rd order permutation points (at layer $k$ )
 
 For this case, we have $\frac { n _ { k } ! } { 2 ! ^ { 3 } }$ permutations given by permuting the neuron indices of layer $k$ . Therefore, this 3-rd order permutation point induces a permutation set with cardinality $\begin{array} { r } { | P ( \pmb { \theta } ) | = \frac { 1 } { 2 ! ^ { 3 } } \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! } \end{array}$
 
-As always, we should consider the other 3-rd order permutation points (at layer $k$ ) giving rise to the same network function and corresponding to the same unordered partition. Note that we can choose three parameter vectors to duplicate out of $n _ { k } - 3$ in $\binom { n _ { k } - 3 } { 3 }$ ways and there are $\begin{array} { r } { { \frac { 1 } { 2 ^ { 3 } } } \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! } \end{array}$ many points in each one of the permutation sets. Therefore, we end up having $\begin{array} { r } { \binom { n _ { k } - 3 } { 3 } \frac { 1 } { 2 ^ { 3 } } \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! } \end{array}$ many 3-rd order permutation points (at layer $k$ ) corresponding to this unordered partition. Overall, we hav e nk−31 
 14! Qd−1j=1 nj ! +  $\begin{array} { r } { \binom { n _ { k } - 3 } { 1 } \frac 1 { 4 ! } \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! + \binom { n _ { k } - 3 } { 2 } \frac 1 { 3 ! } \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! + \binom { n _ { k } - 3 } { 3 } \frac 1 { 2 ! ^ { 3 } } \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! } \end{array}$ nk−33 
 12!3 Qd−1j=1 nj ! many 3-rd order permutation points at layer $k$ .
+As always, we should consider the other 3-rd order permutation points (at layer $k$ ) giving rise to the same network function and corresponding to the same unordered partition. Note that we can choose three parameter vectors to duplicate out of $n _ { k } - 3$ in $\binom { n _ { k } - 3 } { 3 }$ ways and there are $\begin{array} { r } { { \frac { 1 } { 2 ^ { 3 } } } \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! } \end{array}$ many points in each one of the permutation sets. Therefore, we end up having $\begin{array} { r } { \binom { n _ { k } - 3 } { 3 } \frac { 1 } { 2 ^ { 3 } } \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! } \end{array}$ many 3-rd order permutation points (at layer $k$ ) corresponding to this unordered partition. Overall, we hav e nk−31 
 14! Qd−1j=1 nj ! +  $\begin{array} { r } { \binom { n _ { k } - 3 } { 1 } \frac 1 { 4 ! } \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! + \binom { n _ { k } - 3 } { 2 } \frac 1 { 3 ! } \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! + \binom { n _ { k } - 3 } { 3 } \frac 1 { 2 ! ^ { 3 } } \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! } \end{array}$ nk−33 
 12!3 Qd−1j=1 nj ! many 3-rd order permutation points at layer $k$ .
 
 # (4) A note on the general closed form formula for $T ( K , n _ { k } )$ :
 
@@ -312,7 +312,7 @@ For a general integer $K$ there is no closed-form formula for the number of part
 
 # (5) A lower bound for general $K$ :
 
-For general $K$ , we have $l = n _ { k } - K$ distinct parameter vectors in the small network. There are many ways to partition $n _ { k }$ into $l$ positive integers without respecting order. Since we are interested in a lower bound, we only consider the following unordered partition: $n _ { k } = 2 + . . . + 2 + 1 + . . . + 1$ , i.e. we have $K$ duplicated parameter vectors and $n _ { k } - 2 K$ parameter vectors that appear once. For this unordered partition, we have ${ \binom { n _ { k } - K } { K } }$ ways to choose the duplicated parameter vectors. For each one of these choices, we can permute the neuron indices in $\textstyle { \frac { n _ { k } ! } { 2 ^ { K } } }$ different ways. Including the permutations in other layers j 6= k, we end up with nk−KK 
 $\begin{array} { r } { \left( { \overset { n _ { k } - K } { K } } \right) { \frac { 1 } { 2 ^ { K } } } \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! } \end{array}$ points in the permutation set. The number is a lower bound of $T ( K , n _ { k } )$ , because other unordered partitions of $n _ { k }$ give rise to other $K ^ { \mathrm { t h } }$ -order permutation points at layer $k$ .
+For general $K$ , we have $l = n _ { k } - K$ distinct parameter vectors in the small network. There are many ways to partition $n _ { k }$ into $l$ positive integers without respecting order. Since we are interested in a lower bound, we only consider the following unordered partition: $n _ { k } = 2 + . . . + 2 + 1 + . . . + 1$ , i.e. we have $K$ duplicated parameter vectors and $n _ { k } - 2 K$ parameter vectors that appear once. For this unordered partition, we have ${ \binom { n _ { k } - K } { K } }$ ways to choose the duplicated parameter vectors. For each one of these choices, we can permute the neuron indices in $\textstyle { \frac { n _ { k } ! } { 2 ^ { K } } }$ different ways. Including the permutations in other layers j 6= k, we end up with nk−KK 
 $\begin{array} { r } { \left( { \overset { n _ { k } - K } { K } } \right) { \frac { 1 } { 2 ^ { K } } } \prod _ { j = 1 } ^ { d - 1 } n _ { j } ! } \end{array}$ points in the permutation set. The number is a lower bound of $T ( K , n _ { k } )$ , because other unordered partitions of $n _ { k }$ give rise to other $K ^ { \mathrm { t h } }$ -order permutation points at layer $k$ .
 
 # C.5 PROOF OF LEMMA 2