Problem Statement i Give n possible shares to invest in provide a distribution of wealth on stocks ii The game is r epetitive each time we see how the stocks performed and change our distribution b roperties i Given no other information a reasona ID: 25570 Download Pdf

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Problem Statement i Give n possible shares to invest in provide a distribution of wealth on stocks ii The game is r epetitive each time we see how the stocks performed and change our distribution b roperties i Given no other information a reasona

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Decision Analysis (097328), Lecture 1 Notes by: Kira Radinsky Examples of decision problems a. Problem Statement: i. Give n possible shares to invest in , provide a distribution of wealth on stocks ii. The game is r epetitive each time we see how the stocks performed , and change our distribution b. roperties: i. Given no other information , a reasonable goal is to attain wealth comparable to the most profitable share in the period of the game. c. Problem Statement: i. Given a graph G=(V,E) , iteratively find a path between nodes a and b ii. nformation given : e only get to

see the length of the path that we chose d. Properties: i. t ii. Reasonable goal: converge to the path with minimum average length. iii. Complexity difficulty: possibly exponentially (in V) paths in the graph e. Problem Statement: i. Given n web search queries, produce a ranking of the documents by relevance ii. Information given: We only see partial information (whether the user clicked or not) , and derive the relevance from those clicks, f. Properties i. Complexity difficulty: n! options for sorting the documents. ii. Information difficulty (user clicked or not, and which document. Cannot

obtained information on unranked documents)

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Decision Analysis (097328), Lecture 1 Notes by: Kira Radinsky Expert Problem: The expert problem is one of the most basic problem s we shall consider, it does not include information or computational difficulties. Given a discrete choice p roblem over the set {A,B} and an ensemble of N experts . In each round, w e ask each expert about the choice he believes we should make . Then the correct answer (the right choice A or B) is revealed by the external source. The game is repetitive, and we would like to function at least as good as

the best expert in the ensemble. Let be the numb er of mistakes we make and let the m istakes of the best exper be denoted as , the goal is to design a strategy which bound s the number of mistakes by At time t, e ach expert is given a confidence weight, denoted . The online learning is done in rounds. 1. Initialization : 2. Decision 3. Update weights a. If decision was correct : do not change the weights , b. else: i. If expert was right : ii. If expert was wrong : ,

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Decision Analysis (097328), Lecture 1 Notes by: Kira Radinsky Theorem The algorithm satisfies: Proof: Let ,

and let If we are right: If we were wrong: Now we know: 1. iteratively applying Eq 2. (by definition of the updating of the weights) Therefore: sing Tay lor approximation : and

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Decision Analysis (097328), Lecture 1 Notes by: Kira Radinsky Generalized Experts Problem: The problem consists of N experts, each of which can choose from an infinite amount of choices. The algorithm is now allowed to choose according to a distribution on the experts (rather than choose A or B as before). This is given by a probability distribution which associates a value, in the range [0,1], with

each expert at time t. We al so define a gain/utility function denoted as decision was at time t. The total utility of the player at time is Intuition: The N experts can be considered as N stocks . T he for share is how much money was invested in the s tock , and is the change in the value of the s tock . The total utility of the player is the revenue he earns in the end of the game. Let be the share with the maximal gain, we wish to show that: For discrete time points: , the revenue at day t is Therefore, the total revenue is We wish to show that: , where is the best share. 1. Initialization

: 2. Distribution Setting: x 3. Update weights x

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Decision Analysis (097328), Lecture 1 Notes by: Kira Radinsky Theorem The algorithm satisfies: Proof: (1) (2) (3) Let , and let , we continue substituting: Using Taylor approximation: and

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