Skip to main content
Article thumbnail
Location of Repository

Learning Policies for Contextual Submodular Prediction- Supplementary Material

By Stephane Ross, Jiaji Zhou, Yisong Yue, Debadeepta Dey and J. Andrew Bagnell

Abstract

This appendix contains the proofs of the various theoretical results presented in this paper. A.1. Preliminaries We begin by proving a number of lemmas about monotone submodular functions, which will be useful to prove our main results. Lemma 1. Let S be a set and f be a monotone submodular function defined on list of items from S. For any lists A, B, we have that: f(A ⊕ B) − f(A) ≤ |B|(E s∼U(B)[f(A ⊕ s)] − f(A)) for U(B) the uniform distribution on items in B. Proof. For any list A and B, let Bi denote the list of the first i items in B, and bi the i th item in B. We have that: f(A ⊕ B) − f(A) = ∑ |B| i=1 f(A ⊕ Bi) − f(A ⊕ Bi−1) ∑ |B| i=1 f(A ⊕ bi) − f(A) = |B|(Eb∼U(B)[f(A ⊕ b)] − f(A)) where the inequality follows from the submodularity property of f. Lemma 2. Let S be a set, and f a monotone submodular function defined on lists of items in S. Let A, B be any lists of items from S. Denote Aj the list of the first j items in A, U(B) the uniform distribution on items in B and define ɛj = Es∼U(B)[f(Aj−1 ⊕ s)] − f(Aj), the additive error term in competing with the average marginal benefits of the items in B when picking the jth item in A (which could be positive or negative)

Year: 2013
OAI identifier: oai:CiteSeerX.psu:10.1.1.352.9674
Provided by: CiteSeerX
Download PDF:
Sorry, we are unable to provide the full text but you may find it at the following location(s):
  • http://citeseerx.ist.psu.edu/v... (external link)
  • http://www.cs.cmu.edu/~sross1/... (external link)
  • Suggested articles


    To submit an update or takedown request for this paper, please submit an Update/Correction/Removal Request.