Budget Feasible Mechanisms for Experimental Design Thibaut Horel - PowerPoint PPT Presentation

Budget Feasible Mechanisms for Experimental Design Thibaut Horel Joint work with Stratis Ioannidis and S. Muthukrishnan February 26, 2013 Thibaut Horel () February 26, 2013 1 / 1

Motivation Data mining engine Thibaut Horel () February 26, 2013 2 / 1

Motivation c1$ c2$ $ 3 c Data mining engine Thibaut Horel () February 26, 2013 2 / 1

Motivation c1$ B$ p1$ c2$ $ 3 c $ 2 p Data mining engine Thibaut Horel () February 26, 2013 2 / 1

Motivation B$ p1$ $ 2 p Data mining engine Thibaut Horel () February 26, 2013 2 / 1

Challenges Value of data? How to optimize it? Strategic users? Thibaut Horel () February 26, 2013 3 / 1

Challenges Value of data? How to optimize it? Strategic users? Thibaut Horel () February 26, 2013 3 / 1

Challenges Value of data? How to optimize it? Strategic users? Thibaut Horel () February 26, 2013 3 / 1

Challenges Value of data? How to optimize it? Strategic users? Thibaut Horel () February 26, 2013 3 / 1

Contributions case of the linear regression deterministic mechanism generalization (randomized mechanism) Thibaut Horel () February 26, 2013 4 / 1

Contributions case of the linear regression deterministic mechanism generalization (randomized mechanism) Thibaut Horel () February 26, 2013 4 / 1

Contributions case of the linear regression deterministic mechanism generalization (randomized mechanism) Thibaut Horel () February 26, 2013 4 / 1

Contributions case of the linear regression deterministic mechanism generalization (randomized mechanism) Thibaut Horel () February 26, 2013 4 / 1

Outline Thibaut Horel () February 26, 2013 5 / 1

Outline Thibaut Horel () February 26, 2013 6 / 1

Reverse auction set of N sellers: A = { 1 , . . . , N } ; a buyer V value function of the buyer, V : 2 A → R + c i ∈ R + price of seller’s i good B budget constraint of the buyer Goal Find S ⊂ A maximizing V ( S ) Find payment p i to seller i ∈ S Thibaut Horel () February 26, 2013 7 / 1

Reverse auction set of N sellers: A = { 1 , . . . , N } ; a buyer V value function of the buyer, V : 2 A → R + c i ∈ R + price of seller’s i good B budget constraint of the buyer Goal Find S ⊂ A maximizing V ( S ) Find payment p i to seller i ∈ S Thibaut Horel () February 26, 2013 7 / 1

Reverse auction set of N sellers: A = { 1 , . . . , N } ; a buyer V value function of the buyer, V : 2 A → R + c i ∈ R + price of seller’s i good B budget constraint of the buyer Goal Find S ⊂ A maximizing V ( S ) Find payment p i to seller i ∈ S Thibaut Horel () February 26, 2013 7 / 1

Reverse auction set of N sellers: A = { 1 , . . . , N } ; a buyer V value function of the buyer, V : 2 A → R + c i ∈ R + price of seller’s i good B budget constraint of the buyer Goal Find S ⊂ A maximizing V ( S ) Find payment p i to seller i ∈ S Thibaut Horel () February 26, 2013 7 / 1

Reverse auction set of N sellers: A = { 1 , . . . , N } ; a buyer V value function of the buyer, V : 2 A → R + c i ∈ R + price of seller’s i good B budget constraint of the buyer Goal Find S ⊂ A maximizing V ( S ) Find payment p i to seller i ∈ S Thibaut Horel () February 26, 2013 7 / 1

Objectives Payments ( p i ) i ∈ S must be: individually rational: p i ≥ c i , i ∈ S truthful: reporting one’s true cost is a dominant strategy budget feasible: � i ∈ S p i ≤ B Mechanism must be: computationally efficient: polynomial time good approximation: V ( OPT ) ≤ α V ( S ) with: � � � OPT = arg max V ( S ) | c i ≤ B S ⊂A i ∈ S Thibaut Horel () February 26, 2013 8 / 1

Objectives Payments ( p i ) i ∈ S must be: individually rational: p i ≥ c i , i ∈ S truthful: reporting one’s true cost is a dominant strategy budget feasible: � i ∈ S p i ≤ B Mechanism must be: computationally efficient: polynomial time good approximation: V ( OPT ) ≤ α V ( S ) with: � � � OPT = arg max V ( S ) | c i ≤ B S ⊂A i ∈ S Thibaut Horel () February 26, 2013 8 / 1

Objectives Payments ( p i ) i ∈ S must be: individually rational: p i ≥ c i , i ∈ S truthful: reporting one’s true cost is a dominant strategy budget feasible: � i ∈ S p i ≤ B Mechanism must be: computationally efficient: polynomial time good approximation: V ( OPT ) ≤ α V ( S ) with: � � � OPT = arg max V ( S ) | c i ≤ B S ⊂A i ∈ S Thibaut Horel () February 26, 2013 8 / 1

Objectives Payments ( p i ) i ∈ S must be: individually rational: p i ≥ c i , i ∈ S truthful: reporting one’s true cost is a dominant strategy budget feasible: � i ∈ S p i ≤ B Mechanism must be: computationally efficient: polynomial time good approximation: V ( OPT ) ≤ α V ( S ) with: � � � OPT = arg max V ( S ) | c i ≤ B S ⊂A i ∈ S Thibaut Horel () February 26, 2013 8 / 1

Objectives Payments ( p i ) i ∈ S must be: individually rational: p i ≥ c i , i ∈ S truthful: reporting one’s true cost is a dominant strategy budget feasible: � i ∈ S p i ≤ B Mechanism must be: computationally efficient: polynomial time good approximation: V ( OPT ) ≤ α V ( S ) with: � � � OPT = arg max V ( S ) | c i ≤ B S ⊂A i ∈ S Thibaut Horel () February 26, 2013 8 / 1

Objectives Payments ( p i ) i ∈ S must be: individually rational: p i ≥ c i , i ∈ S truthful: reporting one’s true cost is a dominant strategy budget feasible: � i ∈ S p i ≤ B Mechanism must be: computationally efficient: polynomial time good approximation: V ( OPT ) ≤ α V ( S ) with: � � � OPT = arg max V ( S ) | c i ≤ B S ⊂A i ∈ S Thibaut Horel () February 26, 2013 8 / 1

Objectives Payments ( p i ) i ∈ S must be: individually rational: p i ≥ c i , i ∈ S truthful: reporting one’s true cost is a dominant strategy budget feasible: � i ∈ S p i ≤ B Mechanism must be: computationally efficient: polynomial time good approximation: V ( OPT ) ≤ α V ( S ) with: � � � OPT = arg max V ( S ) | c i ≤ B S ⊂A i ∈ S Thibaut Horel () February 26, 2013 8 / 1

Known results When V is submodular: randomized budget feasible mechanism, approximation ratio: 7 . 91 (Chen et al., 2011) deterministic mechanisms for: √ ◮ Knapsack: 2 + 2 (Chen et al., 2011) ◮ Matching: 7.37 (Singer, 2010) ◮ Coverage: 31 (Singer, 2012) Thibaut Horel () February 26, 2013 9 / 1

Known results When V is submodular: randomized budget feasible mechanism, approximation ratio: 7 . 91 (Chen et al., 2011) deterministic mechanisms for: √ ◮ Knapsack: 2 + 2 (Chen et al., 2011) ◮ Matching: 7.37 (Singer, 2010) ◮ Coverage: 31 (Singer, 2012) Thibaut Horel () February 26, 2013 9 / 1

Known results When V is submodular: randomized budget feasible mechanism, approximation ratio: 7 . 91 (Chen et al., 2011) deterministic mechanisms for: √ ◮ Knapsack: 2 + 2 (Chen et al., 2011) ◮ Matching: 7.37 (Singer, 2010) ◮ Coverage: 31 (Singer, 2012) Thibaut Horel () February 26, 2013 9 / 1

Outline Thibaut Horel () February 26, 2013 10 / 1

Linear Regression x1 y1 x2 y2 Linear x3 y3 regression N users Thibaut Horel () February 26, 2013 11 / 1

Linear Regression x1 y1 x2 y2 Linear x3 y3 regression x i : public features (e.g. age, gender, height, etc.) Thibaut Horel () February 26, 2013 11 / 1

Linear Regression x1 y1 x2 y2 Linear x3 y3 regression y i : private data (e.g. disease, etc.) Thibaut Horel () February 26, 2013 11 / 1

Linear Regression x1 y1 x2 y2 Linear x3 y3 regression Gaussian Linear model: y i = β T x i + ε i β ∗ = arg min � | y i − β T x i | 2 β i Thibaut Horel () February 26, 2013 11 / 1

Linear Regression c1$ x1 y1 c2$ $ 3 x2 y2 c Linear x3 y3 regression Thibaut Horel () February 26, 2013 11 / 1

Linear Regression c1$ B$ x1 y1 p1$ c2$ $ 3 x2 y2 c $ 2 p Linear x3 y3 regression Thibaut Horel () February 26, 2013 11 / 1

Linear Regression c1$ B$ x1 y1 p1$ c2$ $ 3 x2 y2 c $ 2 p Linear x3 y3 regression Thibaut Horel () February 26, 2013 11 / 1

Experimental design Public vector of features x i ∈ R d Private data y i ∈ R Gaussian linear model: y i = β T x i + ε i , β ∈ R d , ε i ∼ N ( 0 , σ 2 ) Which users to select? Experimental design ⇒ D-optimal criterion Experimental Design � � � x i x T maximize V ( S ) = log det I d + subject to | S | ≤ k i i ∈ S Thibaut Horel () February 26, 2013 12 / 1

Experimental design Public vector of features x i ∈ R d Private data y i ∈ R Gaussian linear model: y i = β T x i + ε i , β ∈ R d , ε i ∼ N ( 0 , σ 2 ) Which users to select? Experimental design ⇒ D-optimal criterion Experimental Design � � � x i x T maximize V ( S ) = log det I d + subject to | S | ≤ k i i ∈ S Thibaut Horel () February 26, 2013 12 / 1

Experimental design Public vector of features x i ∈ R d Private data y i ∈ R Gaussian linear model: y i = β T x i + ε i , β ∈ R d , ε i ∼ N ( 0 , σ 2 ) Which users to select? Experimental design ⇒ D-optimal criterion Experimental Design � � � x i x T maximize V ( S ) = log det I d + subject to | S | ≤ k i i ∈ S Thibaut Horel () February 26, 2013 12 / 1