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Power of Selective Memory. Slide 1 The Power of Selective Memory Shai Shalev-Shwartz Joint work with Ofer Dekel, Yoram Singer Hebrew University, Jerusalem

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Power of Selective Memory. Slide 2 Outline Online learning, loss bounds etc. Hypotheses space – PST Margin of prediction and hinge-loss An online learning algorithm Trading margin for depth of the PST Automatic calibration A self-bounded online algorithm for learning PSTs

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Power of Selective Memory. Slide 3 Online Learning For Get an instance Predict a target based on Get true update and suffer loss Update prediction mechanism

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Power of Selective Memory. Slide 4 Analysis of Online Algorithm Relative loss bounds (external regret): For any fixed hypothesis h :

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Power of Selective Memory. Slide 5 Prediction Suffix Tree (PST) Each hypothesis is parameterized by a triplet: context function

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Power of Selective Memory. Slide 6 PST Example 0 -3 1 4 -2 7

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Power of Selective Memory. Slide 7 Margin of Prediction Margin of prediction Hinge loss

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Power of Selective Memory. Slide 8 Complexity of hypothesis Define the complexity of hypothesis as We can also extend g s.t. and get

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Power of Selective Memory. Slide 9 Algorithm I : Learning Unbounded-Depth PST Init: For t=1,2,… Get and predict Get and suffer loss Set Update weight vector Update tree

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Power of Selective Memory. Slide 10 Example y = 0 y = ?

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Power of Selective Memory. Slide 11 Example y = + 0 y = ?

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Power of Selective Memory. Slide 12 Example y = + 0 y = ??

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Power of Selective Memory. Slide 13 Example y = +- 0 y = ?? -.23 +

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Power of Selective Memory. Slide 14 Example y = +- 0 y = ??? -.23 +

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Power of Selective Memory. Slide 15 Example y = +-+ 0 y = ??? -.23 +.23.16 + -

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Power of Selective Memory. Slide 16 Example y = +-+ 0 y = ???- -.23 +.23.16 + -

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Power of Selective Memory. Slide 17 Example y = +-+- 0 y = ???- -.42 +.23.16 + - -.14 -.09 + -

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Power of Selective Memory. Slide 18 Example y = +-+- 0 y = ???-+ -.42 +.23.16 + - -.14 -.09 + -

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Power of Selective Memory. Slide 19 Example y = +-+-+ 0 y = ???-+ -.42 +.41.29 + - -.14 -.09 + -.09.06 + -

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Power of Selective Memory. Slide 20 Analysis Let be a sequence of examples and assume that Let be an arbitrary hypothesis Let be the loss of on the sequence of examples. Then,

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Power of Selective Memory. Slide 21 Proof Sketch Define Upper bound Lower bound Upper + lower bounds give the bound in the theorem

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Power of Selective Memory. Slide 22 Proof Sketch (Cont.) Where does the lower bound come from? For simplicity, assume that and Define a Hilbert space: The context function g t+1 is the projection of g t onto the half-space where f is the function

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Power of Selective Memory. Slide 23 Example revisited The following hypothesis has cumulative loss of 2 and complexity of 2. Therefore, the number of mistakes is bounded above by 12. y = +-+-+-+-

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Power of Selective Memory. Slide 24 Example revisited The following hypothesis has cumulative loss of 1 and complexity of 4. Therefore, the number of mistakes is bounded above by 18. But, this tree is very shallow 0 1.41 -1.41 + - y = +-+-+-+- Problem: The tree we learned is much more deeper !

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Power of Selective Memory. Slide 25 Geometric Intuition

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Power of Selective Memory. Slide 26 Geometric Intuition (Cont.) Lets force g t+1 to be sparse by “canceling” the new coordinate

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Power of Selective Memory. Slide 27 Geometric Intuition (Cont.) Now we can show that:

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Power of Selective Memory. Slide 28 Trading margin for sparsity We got that If is much smaller than we can get a loss bound ! Problem: What happens if is very small and therefore ? Solution: Tolerate small margin errors ! Conclusion: If we tolerate small margin errors, we can get a sparser tree

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Power of Selective Memory. Slide 29 Automatic Calibration Problem: The value of is unknown Solution: Use the data itself to estimate it ! More specifically: Denote If we keep then we get a mistake bound

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Power of Selective Memory. Slide 30 Algorithm II : Learning Self Bounded-Depth PST Init: For t=1,2,… Get and predict Get and suffer loss If do nothing! Otherwise: Set Update w and the tree as in Algo. I, up to depth d t

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Power of Selective Memory. Slide 31 Analysis – Loss Bound Let be a sequence of examples and assume that Let be an arbitrary hypothesis Let be the loss of on the sequence of examples. Then,

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Power of Selective Memory. Slide 32 Analysis – Bounded depth Under the previous conditions, the depth of all the trees learned by the algorithm is bounded above by

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Power of Selective Memory. Slide 33 Example revisited Performance of Algo. II y = + - + - + - + - … Only 3 mistakes The last PST is of depth 5 The margin is 0.61 (after normalization) The margin of the max margin tree (of infinite depth) is 0.7071 0 -.55 +.55.39 + - -. 22 -.07 + -.07.05 -.03 -.05 - + -

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Power of Selective Memory. Slide 34 Conclusions Discriminative online learning of PSTs Loss bound Trade margin and sparsity Automatic calibration Future work Experiments Features selection and extraction Support vectors selection

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