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COLOR TEST COLOR TEST COLOR TEST COLOR TEST. Dueling Algorithms. Nicole Immorlica , Northwestern University with A. Tauman Kalai , B. Lucier , A. Moitra , A. Postlewaite , and M. Tennenholtz. Social Contexts. Normal-form games :

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COLOR TESTCOLOR TESTCOLOR TESTCOLOR TEST


Dueling Algorithms

Nicole Immorlica, Northwestern University

with A. TaumanKalai, B. Lucier, A. Moitra, A. Postlewaite, and M. Tennenholtz


Social Contexts

Normal-form games:

Players choose strategies to maximize expected von Neumann-Morgenstern utility.

Social context games [AKT’08]:

Players choose strategies to achieve particular social status among peers.


Social Contexts

Ranking games [BFHS’08]:

Players choose strategies to achieve particular payoff rank among peers.


Two-Player Ranking Games

Bob

G

Alice and Bob play game:

Alice

1

Alice beats Bob in G

½

Alice ties Bob in G

RG payoff of Alice:

0

Alice loses to Bob in G


Implicit Representations

Succinct games [FIKU’08]:

Payoff matrix represented by boolean circuit. NE hard to solve or approximate.

Blotto games [B’21, GW’50, R’06, H’08]:

Distribute armies to battlefields.


Implicit Representations

Optimization duels [this work]:

Underlying game is optimization problem. Goal is to optimize better than opponent.


Ranking Duel

A search engine is an algorithm that inputs

  • set Ω = {1, 2, …, n} of items

  • probabilities p1 + … + pn = 1 of each

    and outputs a permutation π of Ω.

    Monopolist objective: minimize Ei~p[π(i)].


Ranking Duel

Competitive objective: Let the expected score of a ranking π versus a ranking π’ be

Pri~p[ π(i) < π’(i) ] + (½) Pri~p[ π(i) = π’(i) ].

Then objective is to output a π that maximizes expected score given algorithm of opponent.


Optimizing a Search Engine

?

User searches for object drawn according to known probability dist.


Greedy is optimal.

0.19

0.16

0.27

0.07

0.22

0.09

Search:

pretty shape

1.

(27%)

2.

(22%)

3.

(19%)

4.

(16%)

5.

(09%)

6.

(07%)


Choosing a Search Engine

Search for “pretty shape”.

See which search engine ranks my favorite shape higher.

Thereafter, use that one.


0.19

0.16

0.27

0.07

0.22

0.09

Search:

Search:

pretty shape

pretty shape

6.

1.

(27%)

(27%)

2.

1.

(22%)

(22%)

2.

3.

(19%)

(19%)

4.

3.

(16%)

(16%)

4.

5.

(09%)

(09%)

6.

5.

(07%)

(07%)


Questions

Can we efficiently compute an equilibrium of a ranking duel?

How poorly does greedy perform in a competitive setting?

What consequences does the duel have for the searcher?


Optimization Problems as Duels

Ranking

Binary Search

Routing

Finish

?

?

?

?

?

?

?

Start

Hiring

Compression

Parking


Duel Framework

Finite feasible set X of strategies.

Prob. distribution p over states of nature Ω.

Objective cost c: Ω × X R.

Monopolist: choose x to minimize Eω~p[cω(x)].


Duel Framework

1 if cω(x) < cω(x’)

v(x,x’) = Eω~p

0 if cω(x) > cω(x’)

½ if cω(x) = cω(x’)

Players select strategies x, x’ from X.

Nature selects state ωfrom Ωaccording to p.

Payoffs v(x,x’), (1-v(x,x’)) are realized.


Results: Computation

An LP-based technique to compute exact equilibria,

A low-regret learning technique to compute approximate equilibria,

… and a demonstration of these techniques in our sample settings


Computing Exact Equilibria

Formulate game as bilinear duel:

  • Efficiently map strategies to points X in Rn.

  • Define constraints describing K=convex-hull(X).

  • Define payoff matrix M that computes values.

  • Maps points in K back to strategies in original setting.


Bilinear Duels

If feasible strategies X are points in Rn, and payoff v(x, x’) is xtMx’ for some M in Rnxn, then

maxv,x v s.t.

xtMx’ ≥ v for all x’ in X

x is in K (=convex-hull(X))

Exponential, but equivalent poly-sized LP.


Ranking Duel

Formulate game as bilinear duel:

  • Efficiently map strategies to points X in Rn.

    X = set of permutation matrices

    (entry xij indicates item i placed in position j)

  • Define constraints describing K=convex-hull(X).

    K = set of doubly stochastic matrices

    (entry yij = prob. item i placed in position j)


Ranking Duel

Formulate game as bilinear duel:

  • Design “rounding alg.” that maps points in K back to strategies in original setting.

    Birkhoff–von Neumann Theorem: Can efficiently construct permutation basis for doubly stochastic matrix (e.g., via matching).


Ranking Duel

Formulate game as bilinear duel:

  • Define payoff matrix M that computes values.

    Ep,y,y’[v(x,x’)]

    = ∑i p(i) ( ½ Pry,y’ [xi= x’i] + Pry,y’ [xi> x’i])

    = ∑i p(i) (∑iyij ( ½ y’ij + ∑k>j y’ik))

    which is bilinear in y,y’ and so can be written ytMy’.


Ranking Duel

Result: Can reduce computation time to poly(n) versus poly(n!) with standard LP approach.

Technique also applies to hiring duel and binary search duel.


Compression Duel

data

(each with prob. p(.))

Goal: smaller compression (i.e., lower depth in tree).


Classical Algorithm

Huffman coding:

Repeatedly pair nodes with lowest probability.


Compression Duel

Formulate game as bilinear duel:

  • Efficiently map strategies to points X in Rn.

    X = subset of zero-one matrices*

    (entry xij indicates item i placed at depth j)

  • Define constraints describing K=convex-hull(X).

    K = subset of row-stochastic matrices*

    (entry yij = prob. item i placed at depth j)

    * Must correspond to depth profile of some binary tree!


Compression Duel

Formulate game as bilinear duel:

  • Define payoff matrix M that computes values.

    Ep,y,y’[v(x,x’)] = ∑i p(i) (∑iyij ( ½ y’ij + ∑k>j y’ik))

    which is bilinear in y,y’ and so can be written ytMy’.


Compression Duel

Bilinear Form:

maxv,x v s.t.

xtMx’ ≥ v for all x’ in X

x is in K (=convex-hull(X))

Problems:

1. How to round points in K back to a random binary tree with right depth profile?

2. How to succinctly express constraints describing K?


Approximate Minimax

Defn. For any ε > 0, an approximate minimaxstrategy guarantees payoff not worse than best possible value minus ε.

Defn. For any ε > 0, an approximate best response has payoff not worse than payoff of best response minus ε.


Best-Response Oracle

Idea. Use approximate best-response oracle to get approximate minimax strategies.

1. Low-regret learning: if x1,…,xT and x’1,…,x’T have low regret, then ave. is approx minimax.

2. Follow expected leader: on round t+1, play best-response to x1,…,xt to get low-regret.


Compression Best-Response

Multiple-choice Knapsack:

Given lists of items with values and weights, pick one from each list with max total value and total weight at most one.


Compression Best-Response

Depth:

1

2

3

4


Compression Best-Response

For j from 1..n, list of depth j:

v( ) = Pr[win at depth j | x’ ]

w( ) = 2-j

… Kraft inequality

(each with prob. p(.))

x’ in K


Other Duels

  • Hiring duel: constraints defining Euclidean subspace correspond to hiring probabilities.

  • Binary search duel: similar to hiring duel, but constraints defining Euclidean subspace more complex (must correspond to search trees).

  • Racing duel: seems computationally hard, even though single-player problem easy.


Conclusion

  • Every optimization problem has a duel.

  • Classic solutions (and all deterministic algorithms) can usually be badly beaten.

  • Duel can be easier or harder to solve, and can lead to inefficiencies.

    OPEN QUESTION: effect of duel on the solution to the optimization problem?


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