Our results If is computable by a read-k DT , then . If is computable DT with expected depth d , and satisfies , then . also proven by [CKLS13] The FEI + conjecture of [OT13] composes. also proven by [OT13]
Our technique • Want to show for certain Boolean functions .
Our technique • Want to show for certain Boolean functions . • Previous papers have studied the expression
Our technique • Want to show for certain Boolean functions . • Previous papers have studied the expression • We instead take an information theoretic approach via the Shannon Source Coding Theorem .
Shannon Source Coding Theorem Given a random variable , avg # of bits needed to communicate .
Shannon Source Coding Theorem Given a random variable , avg # of bits needed to communicate . Thus, to show , we need to construct an efficient protocol for communicating the value of .
Shannon Source Coding Theorem Given a random variable , avg # of bits needed to communicate . Thus, to show , we need to construct an efficient protocol for communicating the value of . ( efficient = bits on average)
a protocol for read-k DTs
Protocol for read-k DTs is computed by 0 0 1 1 0 1 0 1 0 1
Protocol for read-k DTs is computed by (a read-2 DT) 0 0 1 1 0 1 0 1 0 1
Protocol for read-k DTs is computed by 0 0 1 1 0 1 0 1 0 1
Protocol for read-k DTs is computed by Key fact : if is in the support 0 of , then the coordinates of appear in a root-to-leaf path in 0 1 1 . 0 1 0 1 0 1
Protocol for read-k DTs is computed by Key fact : if is in the support 0 of , then the coordinates of appear in a root-to-leaf path in 0 1 1 . 0 1 0 1 0 1 Some sets in the support of :
Protocol for read-k DTs is computed by Key fact : if is in the support 0 of , then the coordinates of appear in a root-to-leaf path in 0 1 1 . 0 1 0 1 0 1 Some sets in the support of :
Protocol for read-k DTs is computed by Key fact : if is in the support 0 of , then the coordinates of appear in a root-to-leaf path in 0 1 1 . 0 1 0 1 0 1 Some sets in the support of :
Protocol for read-k DTs is computed by Key fact : if is in the support 0 of , then the coordinates of appear in a root-to-leaf path in 0 1 1 . 0 1 0 1 0 1 Some sets in the support of :
Protocol for read-k DTs is computed by Key fact : if is in the support 0 of , then the coordinates of appear in a root-to-leaf path in 0 1 1 . 0 1 0 1 0 1 Some sets in the support of :
Protocol for read-k DTs is computed by Key fact : if is in the support 0 of , then the coordinates of appear in a root-to-leaf path in 0 1 1 . 0 1 0 1 0 1 Some sets in the support of :
Protocol for read-k DTs is computed by Key fact : if is in the support 0 of , then the coordinates of appear in a root-to-leaf path in 0 1 1 . 0 1 0 1 0 1 Some sets in the support of :
Protocol for read-k DTs is computed by Key fact : if is in the support 0 of , then the coordinates of appear in a root-to-leaf path in 0 1 1 . 0 1 0 1 0 1 Some sets in the support of :
Protocol for read-k DTs is computed by Key fact : if is in the support 0 of , then the coordinates of appear in a root-to-leaf path in 0 1 1 . 0 1 0 1 0 1 Some sets in the support of :
Protocol for read-k DTs is computed by Key fact : if is in the support 0 of , then the coordinates of appear in a root-to-leaf path in 0 1 1 . 0 1 0 1 0 1 Some sets in the support of :
Protocol for read-k DTs is computed by Key fact : if is in the support 0 of , then the coordinates of appear in a root-to-leaf path in 0 1 1 . 0 1 0 1 0 1 Some sets in the support of :
Protocol for read-k DTs is computed by Key fact : if is in the support 0 of , then the coordinates of appear in a root-to-leaf path in 0 1 1 . 0 1 0 1 0 1 Some sets in the support of :
Protocol for read-k DTs Given , what should our protocol output? 0 0 1 1 0 1 0 1 0 1
Protocol for read-k DTs Given , what should our protocol output? e.g., 0 0 1 1 0 1 0 1 0 1
Protocol for read-k DTs Given , what should our protocol output? e.g., 0 1.) find a path containing 0 1 1 0 1 0 1 0 1
Protocol for read-k DTs Given , what should our protocol output? e.g., 0 1.) find a path containing 0 1 1 0 1 0 1 0 1
Protocol for read-k DTs Given , what should our protocol output? e.g., 0 1.) find a path containing 0 1 1 2.) output the path’s description : 0 1 0 1 0 1
Protocol for read-k DTs Given , what should our protocol output? e.g., 0 1.) find a path containing 0 1 1 2.) output the path’s description : 0 1 0 1 0 1
Protocol for read-k DTs Given , what should our protocol output? e.g., 0 1.) find a path containing 0 1 1 2.) output the path’s description : 3.) indicate which nodes fall in : 0 1 0 1 0 1
Protocol for read-k DTs Given , what should our protocol output? e.g., 0 1.) find a path containing 0 1 1 2.) output the path’s description : 3.) indicate which nodes fall in : 0 1 0 1 0 1
Protocol for read-k DTs Given , what should our protocol output? e.g., 0 1.) find a path containing 0 1 1 2.) output the path’s description : 3.) indicate which nodes fall in : 0 1 0 1 0 1
Protocol for read-k DTs Given , what should our protocol output? e.g., 0 1.) find a path containing 0 1 1 2.) output the path’s description : 3.) indicate which nodes fall in : 0 1 0 1 0 1 4.) final output:
Protocol for read-k DTs Given , what should our protocol output? 0 0 1 1 0 1 0 1 0 1
Protocol for read-k DTs Given , what should our protocol output? e.g., 0 0 1 1 0 1 0 1 0 1
Protocol for read-k DTs Given , what should our protocol output? e.g., 0 1.) find a path containing 0 1 1 0 1 0 1 0 1
Protocol for read-k DTs Given , what should our protocol output? e.g., 0 1.) find a path containing 0 1 1 lots of choices! 0 1 0 1 0 1
Protocol for read-k DTs Given , what should our protocol output? e.g., 0 1.) find a path containing 0 1 1 lots of choices! 0 1 0 1 0 1
Protocol for read-k DTs Given , what should our protocol output? e.g., 0 1.) find a path containing 0 1 1 lots of choices! 0 1 0 1 0 1
Protocol for read-k DTs Given , what should our protocol output? e.g., 0 1.) find a path containing 0 1 1 lots of choices! 0 1 0 1 0 1
Protocol for read-k DTs Given , what should our protocol output? e.g., 0 1.) find the shortest path 0 1 1 containing 0 1 0 1 0 1
Protocol for read-k DTs Given , what should our protocol output? e.g., 0 1.) find the shortest path 0 1 1 containing 0 1 0 1 0 1
Protocol for read-k DTs Given , what should our protocol output? e.g., 0 1.) find the shortest path 0 1 1 containing 2.) output the path’s description : 0 1 0 1 0 1
Protocol for read-k DTs Given , what should our protocol output? e.g., 0 1.) find the shortest path 0 1 1 containing 2.) output the path’s description : 0 1 0 1 0 1 3.) indicate which nodes fall in :
Protocol for read-k DTs Given , what should our protocol output? e.g., 0 1.) find the shortest path 0 1 1 containing 2.) output the path’s description : 0 1 0 1 0 1 3.) indicate which nodes fall in : 4.) final output:
Analysis of protocol • A decision tree should be arranged with the most influential variables near the top.
Analysis of protocol • A decision tree should be arranged with the most influential variables near the top. • Since every path output is root-to-leaf, the variables near the top will contribute a lot of bits to the expectation.
Analysis of protocol • A decision tree should be arranged with the most influential variables near the top. • Since every path output is root-to-leaf, the variables near the top will contribute a lot of bits to the expectation. • In summary, contributes a lot to the expectation ⇒ is near the top of the tree ⇒ is highly influential
A bad tree Let be a decision tree.
A bad tree Let be a decision tree.
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