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

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The Power of Selective Memory

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The power of selective memory

The Power of Selective Memory

Shai Shalev-Shwartz

Joint work with

Ofer Dekel, Yoram Singer

Hebrew University, Jerusalem


Outline

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


Online learning

Online Learning

  • For

    • Get an instance

    • Predict a target based on

    • Get true update and suffer loss

    • Update prediction mechanism


Analysis of online algorithm

Analysis of Online Algorithm

  • Relative loss bounds (external regret):

    For any fixed hypothesis h :


P rediction s uffix t ree pst

Prediction Suffix Tree (PST)

Each hypothesis is parameterized by a triplet:

context function


Pst example

PST Example

0

-3

-1

1

4

-2

7


Margin of prediction

Margin of Prediction

  • Margin of prediction

  • Hinge loss


Complexity of hypothesis

Complexity of hypothesis

  • Define the complexity of hypothesis as

  • We can also extend g s.t. and get


Algorithm i learning unbounded depth pst

Algorithm I :Learning Unbounded-Depth PST

  • Init:

  • For t=1,2,…

    • Get and predict

    • Get and suffer loss

    • Set

    • Update weight vector

    • Update tree


Example

Example

y = ?

y =

0


Example1

Example

y = ?

y = +

0


Example2

Example

y = ??

y = +

0


Example3

Example

y = ??

y = +-

0

+

-.23


Example4

Example

y = ???

y = +-

0

+

-.23


Example5

Example

y = ???

y = +-+

0

+

-

.23

-.23

+

.16


Example6

Example

y = ???-

y = +-+

0

+

-

.23

-.23

+

.16


Example7

Example

y = ???-

y = +-+-

0

+

-

.23

-.42

+

-

.16

-.14

+

-.09


Example8

Example

y = ???-+

y = +-+-

0

+

-

.23

-.42

+

-

.16

-.14

+

-.09


Example9

Example

y = ???-+

y = +-+-+

0

+

-

.41

-.42

+

-

.29

-.14

-

+

.09

-.09

+

.06


Analysis

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,


Proof sketch

Proof Sketch

  • Define

  • Upper bound

  • Lower bound

  • Upper + lower bounds give the bound in the theorem


Proof sketch cont

Proof Sketch (Cont.)

Where does the lower bound come from?

  • For simplicity, assume that and

  • Define a Hilbert space:

  • The context function gt+1is the projection of gtonto the half-space where f is the function


Example revisited

Example revisited

y = +-+-+-+-

  • The following hypothesis has cumulative loss of 2 and complexity of 2. Therefore, the number of mistakes is bounded above by 12.


Example revisited1

Example revisited

y = +-+-+-+-

  • 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

Problem: The tree we learned is much more deeper !


Geometric intuition

Geometric Intuition


Geometric intuition cont

Geometric Intuition (Cont.)

Lets force gt+1 to be sparse by “canceling” the new coordinate


Geometric intuition cont1

Geometric Intuition (Cont.)

Now we can show that:


Trading margin for sparsity

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


Automatic calibration

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


Algorithm ii learning self bounded depth pst

Algorithm II :Learning Self Bounded-Depth PST

  • Init:

  • For t=1,2,…

    • Get and predict

    • Get and suffer loss

    • If do nothing! Otherwise:

      • Set

      • Set

      • Set

      • Update w and the tree as in Algo. I, up to depth dt


Analysis loss bound

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,


Analysis bounded depth

Analysis – Bounded depth

  • Under the previous conditions, the depth of all the trees learned by the algorithm is bounded above by


Example revisited2

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

Example revisited

0

-

+

.55

-.55

+

-

-. 22

.39

-

+

.07

-.07

+

-

.05

-.05

-

.03


Conclusions

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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