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Regret in the On-Line Decision Problem Dean Foster & Rakesh Vohara Presented by: Tom Whipple 2/7/2006

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Outline Intuition Notation Definition of Regret –Internal Regret –No Regret –External Regret Theorem 2 How do we use Regret? –Stock market example

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What is Regret? Informal: I wish I had decided x instead of y. Goal: Quantify this, accounting for: –“How bad?” (I could have made $1,000,000) –“How likely?” (But it wasn’t very likely)

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Notation the set of possible decisions. Loss for decision i at time t. stochastic vector s.t. is chosen at time t with probability S denotes an arbitrary “scheme” for making decisions T is the total number of time steps t.

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Definition: Regret Total Regret from using scheme S is: Weighted by probability of making decision j at time t. Difference in loss from deciding j instead of i. If < 0, i is better than j at time t. Regret cannot be negative for any decision.

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No (Internal) Regret If the scheme S is asymptotically close to the number of time intervals T then S is said to have no internal regret. “Internal” refers to the regret of decisions within S.

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No External Regret Uses Expected Loss Note that “External” refers to comparing scheme S to other schemes. Intuition: The expected loss from S is no more than the expected loss from the best P. Scheme S has no external regret if

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Theorem 2 “Given any finite set of decision schemes F, there exists a (randomized) decision scheme S with no external regret w.r.t. F ” Intuition: There is some S, not in F that is at least as good (asymptotically) as the best scheme in F.

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Application: Stock Selection Theorem 2 shows we can approach best constant portfolio. –Constant refers to relative allocation (BCRP). –This is the same result we saw in previous paper. In the context of this paper, is constant for

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