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Connections between Learning Theory, Game Theory, and Optimization Lecture 1, August 24 th 2010 Maria Florina (Nina) Balcan Big Picture Over the past decades, many important and deep connections between: machine learning theory algorithmic game theory combinatorial optimization

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Connections between learning theory game theory and optimization l.jpg

Connections between Learning Theory, Game Theory, and Optimization

Lecture 1, August 24th2010

Maria Florina (Nina) Balcan


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Big Picture

Over the past decades, many important and deep connections between:

  • machine learning theory

  • algorithmic game theory

  • combinatorial optimization

We will explore such connections, discussing:

  • fundamental topics in each area.

  • how ideas from each area can shed light on the others.


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Outline

Online learning. Combining expert advice.

Regret minimization (no external regret and no internal regret). Bandit algorithms.

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Zero sum games. Nash equilibria.

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Experts learning & Minimax theorem.

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Nash equilibria and approximate nash equilibria in general sum bimatrix games.


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Outline

Learning in a distributional setting.

Sample complexity results.

Weak-learning vs. Strong-learning.

Boosting with connections to game theory.

Quality of equilibria (Price of anarchy/stability).

Games with many players. Potential games.

Dynamics in games and the price of learning.


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Outline

Mechanism design (MD).

Combinatorial auctions. [Social welfare; revenue maximization]

Auctions for digital goods.

  • Reductions from MD to algorithm design using machine learning.

Algorithmic pricing problems.

  • Online learning for designing online pricing schemes.


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Outline

Submodularity with connections to game theory and machine learning.

  • Combinatorial auctions with submodular valuations

  • Learning submodular functions

  • Other optimization pbs involving submodularity (ranking, clustering, etc.)


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Admin

http://www.cc.gatech.edu/~ninamf/LGO10/

  • Course web page:

  • 3 hwk assignments. Exercises/problems (pencil-and-paper problem-solving variety).

  • Project: explore a theoretical question, try some experiments, or read a couple of papers and explain the idea. Writeup and class presentation. Groups ok.

[50%]

[50%]

  • “Algorithmic Game Theory”, Nisan, Roughgarden, Tardos, Vazirani

  • Other papers, surveys, and tutorials


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Online learning, minimizing regret, and combining expert advice.

  • “The weighted majority algorithm”

N. Littlestone & M. Warmuth

  • “Online Algorithms in Machine Learning” (survey)

A. Blum

  • Algorithmic Game Theory, Nisan, Roughgarden, Tardos, Vazirani (eds) [Chapters 4]

  • Prediction, Learning, and Games, Cesa-Bianchi, Lugosi


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Online learning, minimizing regret, and combining expert advice.

Expert 3

Expert 2

Expert 1


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Using “expert” advice

Assume we want to predict the stock market.

  • Will the market go up or down?

  • We solicit n “experts” for their advice.

  • We then want to use their advice somehow to make our prediction. E.g.,

Can we do nearly as well as best in hindsight?

Note: “expert” ´ someone with an opinion.

[Not necessairly someone who knows anything.]


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Formal model

  • For each round t=1,2, …, T

  • There are n experts.

  • Each expert makes a prediction in {0,1}

  • The learner (using experts’ predictions) makes a prediction in {0,1}

  • The learner observes the actual outcome. There is a mistake if the predicted outcome is different form the actual outcome.

Can we do nearly as well as best in hindsight?


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Weighted Majority Algorithm

Deterministic Majority Algorithm

  • Start with all experts having weight 1.

  • Predict based on weighted majority vote.

  • If

  • then predict 1

  • else predict 0

  • Penalize mistakes by cutting weight in half.

Randomized versions of this algorithm can provide surprisingly strong guarantees


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Weighted Majority Algorithm

  • E[# mistakes] ·(1+e)OPT + e-1log(n).

  • If set =(log(n)/OPT)1/2 to balance the two terms out (or use guess-and-double), get bound of

  • E[mistakes]·OPT+2(OPT¢log n)1/2

Note: Of course we might not know OPT, so if running T time steps, since OPT · T, set ² to get additive loss (2T log n)1/2

regret

  • E[mistakes]·OPT+2(T¢log n)1/2

  • So, regret/T ! 0.

[no regret algorithm]


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Many other useful extensions

E.g., what if have n options, not n predictors?

  • We’re not combining n experts, we’re choosing one.

  • Nice feature of RWM: can be applied when experts are n different options

  • E.g., n different ways to drive to work each day, n different ways to invest our money.

Other generalizations as well.

Other notions of no regret (e.g., no internal regret).


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Online Learning, Game Theory, and Minimax Optimality

“Game Theory, On-line Prediction, and Boosting”, Freund & Schapire, GEB


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Zero Sum Games

Game defined by a matrix M.

Assume wlog entries in [0,1].

Scissors

Rock

Paper

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Rock

Paper

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Row player (Mindy) chooses row i.

Scissors

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Column player (Max) chooses column j (simultaneously).

Mindy’s goal: minimize her loss M(i,j).

Max’s goal: maximize this loss (zero sum).


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Randomized Play

Mindy chooses a distribution P over rows.

Max chooses a distribution Q over columns [simultaneously]

Mindy’s expected loss:

If i,j = pure strategies, and P,Q = mixed strategies

M(P,j) - Mindy’s expected loss when she plays P and Max plays j

M(i,Q) - Mindy’s expected loss when she plays i and Max plays Q


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Sequential Play

Say Mindy plays before Max. If Mindy chooses P, then Max will pick Q to maximize M(P,Q), so the loss will be

So, Mindy should pick P to minimize L(P). Loss will be:

Similarly, if Max plays first, loss will be:


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Minimax Theorem

Playing second cannot be worse than playing first

Mindy plays first

Mindy plays second

Von Neumann’s minimax theorem:

No advantage to playing second!


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Optimal Play

Von Neumann’s minimax theorem:

Value of the game

Optimal strategies:

Min-max strategy

Max-min strategy

We will show how to use WM to prove this!

And to also find approximate min-max strategies quickly.


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Optimal Play

Von Neumann’s minimax theorem:

Value of the game

Optimal strategies:

Min-max strategy

Max-min strategy

(P*, Q*) is Nash Equilibria (No player has an incentive to unilateraly deviate)

Central solution

concept we will study


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Games with many players with interesting structure

"Potential Games", D. Monderer and L, S. Shapley , Games and Economic Behavior


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Fair cost-sharing

Fair cost-sharing: n players in weighted directed graph G. Player i wants to get from si to ti, and they share cost of edges they use with others.

G


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Fair cost-sharing

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  • n players in directed graph G, each edge e costsce.

  • Player i wants to get fromsito ti.

  • All players share cost of edges they use with others.

  • Each player wants to minimize his own cost.

Good equilibrium: all use edge of cost 1.

(paying 1/n each)

Bad equilibrium: all use edge of cost n.

(paying 1 each)


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Inefficiency of equilibria, PoA and PoS

Price of Anarchy (PoA): ratio of worst Nash equilibrium to OPT.

[Koutsoupias-Papadimitriou’99]

Price of Stability (PoS): ratio of best Nash equilibrium to OPT.

[Anshelevich et. al, 2004]

E.g., for fair cost-sharing, PoS is log(n), whereas PoA is n.

Significant effort spent on understanding these in CS.

“Algorithmic Game Theory”, Nisan, Roughgarden, Tardos, Vazirani


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Congestion games

  • Nice general class of games with many players.

  • Always have a pure-strategy equilibrium.

  • Have a potential functions.t. whenever a player switches, potential drops by exactly that player’s improvement.

  • We will analyze dynamics in these games!!!

  • What happens if players follow natural learning dynamics!!!


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Learning in a distributional setting.

[With feature information]


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Used all over CS and Science

Image Classification

Document Categorization

Speech Recognition

Protein Classification

Spam Detection

Branch Prediction

Fraud Detection


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Example: Supervised Classification

Decide which emails are spam and which are important.

Supervised classification

Not spam

spam

Goal: use emails seen so far to produce good prediction rule for future data.


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Example: Supervised Classification

Represent each message by features. (e.g., keywords, spelling, etc.)

example

label

Reasonable RULES:

Predict SPAM if unknown AND (money OR pills)

Predict SPAM if 2money + 3pills –5 known > 0

Linearly separable


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Two Main Aspects of Supervised Learning

Algorithm Design. How to optimize?

Automatically generate rules that do well on observed data.

Optimization played a significant role in the recent years.

Confidence Bounds, Generalization Guarantees, Sample Complexity

Confidence for rule effectiveness on future data.

Well understood for passive supervised learning.


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Standard Passive Supervised Learning

  • S={(x, l)} - set of labeled examples

  • X – feature space

  • drawn i.i.d. from distr. D over X and labeled by target concept c*

  • Do optimization over S, find hypothesis h 2 C.

  • Goal: h has small error over D.

  • err(h)=Prx 2 D(h(x)  c*(x))

c*

h

  • c* in C, realizable case; else agnostic


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Standard Passive Supervised Learning

Classic models: PAC (Valiant), SLT (Vapnik)

  • Sample Complexity, Finite Hypothesis Spaces, Realizable Case

  • In in the non-realizable case, replace \epsilon with \epsilon ^2.


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Standard Passive Supervised Learning

Classic models: PAC (Valiant), SLT (Vapnik)

  • Sample Complexity, Finite Hypothesis Spaces, Realizable Case

  • Such ideas/techniques useful in Auction design, Learning submodular functions, etc.


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Boosting & game theory

  • Suppose I have an algorithm A that for any distribution (weighting fn) over a dataset S can produce a rule h2H that gets < 40% error.

  • Adaboost gives a way to use such an A to get error ! 0 at a good rate, using weighted votes of rules produced.

  • We can show that this is in principle possible by using the minimax theorem!


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Supermarket Pricing Problem

  • A supermarket trying to decide on how to price the goods.

Seller’s Goal: set prices to maximize revenue.

  • Simple case: customers make separate decisions on each item.

  • Harder case: customers buy everything or nothing based on

  • sum of prices in list.

  • Or could be even more complex.


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Supermarket Pricing Problem

Algorithmic

  • Seller knows the market well.

Incentive Compatible Auction

  • Must be in customers’ interest (dominant strategy) to report truthfully.

Online Pricing

  • Customers arrive one at a time, buy what they want at current prices. Seller modifies prices over time.

  • Techniques from learning will be useful here.


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Submodular functions

V={1,2, …, n}, f : 2V!R

Submodularity:

  • Concave Functions Let h : R!R be concave.For each S µ V, let f(S) = h(|S|)

f(S)+f(T) ¸ f(S Å T) + f(S [ T) 8 S,Tµ V

Equivalent

Decreasing marginal values:

f(S [ {x})-f(S) ¸ f(T [ {x})-f(T) 8SµTµV, xT

Examples:

  • Vector Spaces Let V={v1,,vn}, each vi2Rn.For each S µ V, let f(S) = rank(V[S])


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Submodular functions

  • Strong connection between optimization and submodularity

    • e.g.: minimization [C’85,GLS’87,IFF’01,S’00,…],maximization [NWF’78,V’07,…]

  • Algorithmic game theory

  • Submodular utility functions

  • Much interest in Machine Learning community recently

  • Tutorials at major conferences: ICML, NIPS, etc.

  • www.submodularity.org is a Machine Learning site

  • Interesting to understand their learnability


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