Differentially Private Algorithm and Auction Configuration
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1 Differentially Private Algorithm and Auction Configuration Ellen Vitercik CMU, Theory Lunch October 11, 2017 Joint work with Nina Balcan and Travis Dick
2 $1
3 Prices learned from purchase histories can reveal information about individual purchases. $100
4 We Prices need learned a way from to privately purchase set histories prices and can design reveal auctions information based about individual on purchase purchases. histories. $100
5
6 Dear Lab Technician, Please update your default CPLEX parameters to
7 Suppose a parameter correlates with a certain disease. An attacker can infer information about medical records used to tune those parameters.
8 Suppose We need a parameter way to correlates privately configure with a certain disease. algorithms. An attacker can infer information about medical records used to tune those parameters.
9 Suppose Many works a parameter have correlates shown that with is a certain disease. possible to invert a machine learning model to infer An sensitive attacker information can infer information about its training about set. medical records used to tune those parameters.
10 By observing a series of recommendations from websites such as Amazon, an adversary can infer individual users purchases. [Calandrino et al. 2011]
11 It is possible to extract images of training subjects from facial recognition models. The attacker has only the person s name and access to a facial recognition system that returns a class confidence score. [Fredrikson et al. 2015]
12 It is possible to invert a machine learning model to learn sensitive genomic information about individuals. [Fredrikson et al. 2014]
13 In response, computer scientists have developed private machine learning algorithms.
14 Existing private ML algorithms apply to optimization problems defined by well-studied, well-understood functions.
15 What if the objective is nonconvex and not differentiable?
16 We provide a private algorithm for maximizing datadependent piecewise Lipschitz functions. Applications in: Algorithm configuration and mechanism design Pricing reduce mechanism to maximizing data-dependent piecewise Lipschitz and auction functions. design
17 Introduction Setup Overview: Algorithm configuration and auction design Differential privacy Private pricing design The algorithm Privacy guarantees Utility guarantees Example: private multi-item pricing Summary
18 Learning-based algorithm configuration: Tune algorithm parameters to achieve high performance over a specific application domain. Led to breakthroughs in: Combinatorial auctions [Leyton-Brown et al., 2009] Scientific computing [Demmel et al., 2005] Vehicle routing [Caseau et al., 1999] SAT [Xu et al., 2008]
19 Learning-based algorithm configuration Application- Specific Distribution,, Algorithm Algorithm Designer How can I use the set of samples to find an algorithm that s best for my application domain?
20 We show that our private algorithm has strong utility guarantees for many algorithm configuration problems. Greedy algorithm configuration Hard combinatorial problems show up in diverse domains where privacy preservation is crucial.
21 We show that our private algorithm has strong utility guarantees for many algorithm configuration problems. Greedy algorithm configuration Hard These combinatorial are often solved problems by greedy show up in algorithms diverse domains where elements where privacy are preservation iteratively added is crucial. to a solution set according to a heuristic. E.g., in knapsack: size of item i (value ofitemi)
22 We show that our private algorithm has strong utility guarantees for many algorithm configuration problems. Greedy algorithm configuration Hard Gupta These combinatorial and are often Roughgarden solved problems by [2017] greedy show up in proposed algorithms diverse an domains where infinite elements where family privacy of are greedy preservation heuristics iteratively added for is the crucial. to knapsack a solution and set max weight according independent to a heuristic. set problems. E.g., in knapsack: size of item i (value of item i) ρ
23 We show that our private algorithm has strong utility guarantees for many algorithm configuration problems. Greedy algorithm configuration Hard Gupta These Our private combinatorial and are often Roughgarden algorithm solved problems uses by [2017] greedy sample show up in proposed algorithms problem diverse instances domains where infinite elements where to family find privacy a of are nearly greedy preservation heuristics iteratively optimal greedy added for is the crucial. heuristic to knapsack a solution for and the set max weight according specific independent application to a heuristic. domain. set problems. E.g., No sensitive in knapsack: information about the training set is size revealed. of item i (value of item i) ρ
24 We show that our private algorithm has strong utility guarantees for many algorithm configuration problems. Integer quadratic programming algorithm configuration x i IQPs are used in many applications where privacy preservation is essential, such as financial portfolio optimization. x j
25 We show that our private algorithm has strong utility guarantees for many algorithm configuration problems. Integer quadratic programming algorithm configuration x i IQPs are used often in approximated many applications by where solving privacy a semi-definite preservation program is and essential, rounding the such vectors as financial to integer portfolio optimization. values. x j There are many different rounding schemes with varying quality.
26 We show that our private algorithm has strong utility guarantees for many algorithm configuration problems. Integer quadratic programming algorithm configuration x i IQPs Our private are used often algorithm in approximated many uses applications sample by where solving IQP instances privacy a semi-definite to preservation find a nearly program is optimal and essential, rounding scheme the such vectors as financial for to the integer specific portfolio optimization. values. application domain. x j There No sensitive are many information different rounding about the schemes training set with is revealed. varying quality.
27 Learning-based mechanism design: Use information about past consumers to design mechanisms that extract high revenue from future consumers. Employed throughout industry. Garnered significant attention in TCS. [Elkind, 2007, Cole and Roughgarden, 2014, Huang et al., 2015, Medina and Mohri, 2014, Morgenstern and Roughgarden, 2015, Devanur et al., 2016, etc.]
28 Introduction Setup Overview: Algorithm configuration and auction design Differential privacy Private pricing design The algorithm Privacy guarantees Utility guarantees Example: private multi-item pricing Summary
29 An algorithm, given an input dataset D, is differentially private if the following holds: The output reveals (almost) nothing more about a record in D than the output would have if the record wasn t contained in D.
30 Alice Bob Claire David Algorithm
31 Alice Bob Claire David Algorithm
32 An algorithm A is (ε, δ)-differentially private if for all pairs of neighboring datasets D, D and all sets O of outputs, P A D O e ε P A D O + δ e ε 1 + ε
33 Introduction Setup Overview: Algorithm configuration and auction design Differential privacy Private pricing design The algorithm Privacy guarantees Utility guarantees Example: private multi-item pricing Summary
34 Single-item pricing problem: One good for sale
35 Single-item pricing problem: Distribution over buyers: ~D Buyers values are denoted as: value, Revenue, ρ = ρ 1 value, ρ Pricing algorithm receives a set S =,,, ~ D N Goal: maximize 1 Revenue, ρ + + Revenue, ρ N Learning theory tells us that this value of ρ approximately maximizes E Revenue, ρ asdfd
36 average revenue S = ~D value, = 3 price
37 average revenue S =, ~ D 2 value, = 3 value, = 4 price
38 average revenue S =, ~ D 2 value, = 3 value, = 4 price
39 average revenue We want to write an algorithm that gets average revenue as close to $3 as possible while preserving differential privacy. price
40 average revenue We want to write an algorithm that gets average revenue as close to $3 as possible while preserving differential privacy. Average utility U S (ρ) price ρ
41 General problem: Given a piecewise utility function U S ρ, privately find a parameter ρ that approximately maximizes U S ρ. S =,, U S Algorithm ρ
42 General problem: Given a piecewise utility function U S ρ, privately find a parameter ρ that approximately maximizes U S ρ. S =,, U S S =, U S Algorithm ρ
43 Introduction Setup Overview: Algorithm configuration and auction design Differential privacy Private pricing design The algorithm Privacy guarantees Utility guarantees Example: private multi-item pricing Summary
44 Ideally, we want to use the exponential mechanism [McSherry and Talwar 2007] Pick a parameter vector ρ with probability proportional to exp c ε U S ρ
45 average revenue Sampling from exp c ε U S ρ is easy when we can calculate its antiderivative and the parameter space is one-dimensional. price
46 What about when the parameter space is multi-dimensional?
47 Generally, sampling from a multi-dimensional distribution is hard.
48 Existing efficient sampling techniques work for logconcave distributions. [Lovász and Vempala 2006, 2007, Bassily et al. 2014] When U S ρ is piecewise concave, exp c ε U S ρ is logconcave on each piece.
49 Theorem (part 1) Suppose U S ρ is piecewise L-Lipschitz and concave over convex regions in R d. jhkj There is an ε, δ -DP algorithm that approximately maximizes U S ρ. Its running time polynomial in d, L, ε 1, log δ 1, and S.
50 Step 1: Estimate the probability mass that the exponential mechanism places on each piecewise portion of the domain. This requires an integral estimation algorithm. [Lovász and Vempala 07]
51 Step 2: Sample a portion of the domain according to that probability.
52 Step 3: Sample a point in that portion of the domain with probability approximately according to the exponential mechanism: exp c ε U S ρ This requires approximate sampling [Bassily et al. 2014]
53 Proof sketch: Show that the output distribution is almost exp c ε U S ρ. Lemma [Bassily et al. 2014] If an algorithm A s output distribution is ε-close ǁ to that of a (ε, 0)-differentially private algorithm for every input dataset S, then A is (2ε ǁ + ε, 0)-differentially private. (Close under the metric D μ f, μ g = sup log f(x)/g(x).) Since the exponential mechanism is (ε, 0)-differentially private, our algorithm is (O(ε), δ)-differentially private.
54 Theorem (part 1) Suppose U S ρ is piecewise L-Lipschitz and concave over convex regions in R d. jhkj There is an ε, δ -DP algorithm that approximately maximizes U S ρ. Its running time polynomial in d, L, ε 1, log δ 1, and S.
55 Theorem (part 1) Suppose U S ρ is piecewise L-Lipschitz and concave over convex regions in R d. jhkj There is an ε, δ -DP algorithm that approximately maximizes U S ρ. Its running time polynomial in d, L, ε 1, log δ 1, and S.
56 Introduction Setup Overview: Algorithm configuration and auction design Differential privacy Private pricing design The algorithm Privacy guarantees Utility guarantees Example: private multi-item pricing Summary
57 Theorem (part 2) Suppose U S ρ is (w, k)-dispersed, piecewise L-Lipschitz, and the parameter space is contained in a ball of radius R. Let ρ R d be the parameter vector output by our ε, δ -DP algorithm. With probability at least 1 γ, U S ρ max U S ρ O H S ε d log R w + log 1 γ + Lw + Hk S (U S ρ is an average of functions with range in 0, H.)
58 Every private algorithm needs to add some noise to the parameter it outputs.
59 Therefore, nearby parameters should have similar utilities.
60 We often guarantee that nearby parameters have similar utilities by assuming the utility function is nice.
61 What does it mean for a piecewise function to be nice? U S ρ ρ
62 What does it mean for a piecewise function to be nice? U S ρ ρ
63 Let P = P 1,, P t be a partition of the parameter space such that U S ρ is piecewise L-Lipschitz within each P i. U S is w, k -dispersed if for any ball B of radius w, the number of sets in P that intersect B is at most k. U S ρ ρ
64 Let P = P 1,, P t be a partition of the parameter space such that U S ρ is piecewise L-Lipschitz within each P i. U S is w, k -dispersed if for any ball B of radius w, the number of sets in P that intersect B is at most k. U S ρ P 1 P 2 P 3 P 4 P 5 P 6 P 7 ρ
65 Let P = P 1,, P t be a partition of the parameter space such that U S ρ is piecewise L-Lipschitz within each P i. U S is w, k -dispersed if for any ball B of radius w, the number of sets in P that intersect B is at most k. U S ρ (1, 3)- dispersed 1 ρ
66 Theorem (part 2) Suppose U S ρ is (w, k)-dispersed, piecewise L-Lipschitz, and the parameter space is contained in a ball of radius R. Let ρ R d be the parameter vector output by our ε, δ -DP algorithm. With probability at least 1 γ, U S ρ max U S ρ O H S ε d log R w + log 1 γ + Lw + Hk S (U S ρ is an average of functions with range in 0, H.)
67 Let P = P 1,, P t be a partition of the parameter space such that U S ρ is piecewise L-Lipschitz within each P i. U S is w, k -dispersed if for any ball B of radius w, the number of sets in P that intersect B is at most k. U S ρ (1, 3)- dispersed 1 ρ
68 Let P = P 1,, P t be a partition of the parameter space such that U S ρ is piecewise L-Lipschitz within each P i. U S is w, k -dispersed if for any ball B of radius w, the number of sets in P that intersect B is at most k. U S ρ (2, 4)- dispersed ρ 2
69 Theorem (part 2) Suppose U S ρ is (w, k)-dispersed, piecewise L-Lipschitz, and the parameter space is contained in a ball of radius R. Let ρ R d be the parameter vector output by our ε, δ -DP algorithm. With probability at least 1 γ, U S ρ max U S ρ O H S ε d log R w + log 1 γ + Lw + Hk S (U S ρ is an average of functions with range in 0, H.)
70 Introduction Setup Overview: Algorithm configuration and auction design Differential privacy Private pricing design The algorithm Privacy guarantees Utility guarantees Example: private multi-item pricing Summary
71 Multi-item pricing problem: Multiple goods for sale
72 Multi-item pricing problem: Distribution over buyers: ~D Buyers values are denoted as: value, asdf asdf value, Revenue, ρ = ρ 1 1 value, ρ 1 + ρ 2 1 value, ρ 2 Pricing algorithm receives a set S =,,, ~ D N Goal: maximize 1 N Revenue, ρ + + Revenue, ρ
73 The average revenue over S =,,, of a pricing aesfe mechanism is piecewise linear and 1-Lipschitz. U S ρ = average revenue ρ 1 = price ρ 2 = price
74 How dispersed is U S ρ? Let f i (v) be the density of the distribution over the typical buyer s value for item i. U S ρ = average revenue Suppose there are m items and max f i(v) κ. i,v ρ 1 With probability at least 1 γ, U S ρ is w, k -dispersed with w = γ and k = m. S κ m ρ 2
75 Let f i (v) be the density of the distribution over the typical buyer s value for item i. U S ρ = average revenue Suppose there are m items and max f i(v) κ. i,v Let ρ R m be the price vector output by our DP algorithm. ρ 1 With probability 1 γ, U S ρ max U S ρ O Hm S ε + γ S κ m ρ 2
76 We also show that our DP algorithm has strong utility guarantees for many mechanism design problems. Multiple bidders Buyers with unit-demand valuations Buyers with general valuations 2 nd price auctions with reserve prices
77 Introduction Setup Overview: Algorithm configuration and auction design Differential privacy Private pricing design The algorithm Privacy guarantees Utility guarantees Example: private multi-item pricing Summary
78 We provide a private algorithm for maximizing datadependent piecewise Lipschitz functions. Utility function Piecewise concave and Lipschitz Piecewise Lipschitz Piecewise Lipschitz Parameter Dimension Multiple Privacy guarantee (ε, δ) Single (ε, 0) Multiple (ε, 0) Efficiency
79 Questions?
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