Limitations of Local Filters of Lipschitz and Monotone Functions

Limitations of Local Filtersof Lipschitz and Monotone Functions PranjalAwasthi Marco Molinaro MadhavJha SofyaRaskhodnikova

Local Filter • Locallytransforms function into to satisfy some property [Ailon-Chazelle-Comandur-Liu 08, Saks-Seshadhri 10, BGJJRW 12] filter .

Local Filter • Locallytransforms function into to satisfy some property [Ailon-Chazelle-Comandur-Liu 08, Saks-Seshadhri 10, BGJJRW 12] filter sublinearfor all reconstructed function • satisfies property • If satisfies property , then

Local Filter • Which properties? • Functions over • Lipschitz • Monotonicity • Local procedure to guarantee we have some property [ACCL 08, SS 10, BGJJRW 12] Sublinear? a,b,… filter client client client expects Lipschitz function reconstructed Lipschitz function 2 6 does not decrease • satisfies property • If satisfies property , then 5 1

Laplace mechanism Lipschitz filter any + noise Application: Privacy private database • Goal: Given private database and user func. , report without revealing identity of • Laplace mechanism and differential privacy [DMNS 06] • For general use local filter for Lipschitz [JhaRaskhodnikova11] Laplace mechanism Lipschitz + noise private database

What is Known • Monotonicity

What is Known • Monotonicity • Lipschitz • Related: property testing, local computation, graph reconstruction [ACCL 08, ARVX 12, CGR, KPS 08, Austin-Tao…]

Our Results • Open question: Can we get efficient adaptive local filters? Main result Every adaptive local filter for Lipschitz or monotonicity property needs to make queries There is input where makes queries

Hard Inputs Idea 1: Hard inputs that force structure on queries

Hard Inputs Idea 1: Hard inputs that force structure on queries Lipschitz not Lipschitz Lipschitz

Hard Inputs Lemma:The queries used for inputs andintersect at some point in Proof: By contradiction

Hard Inputs Lemma:The queries used for inputs andintersect at some point in Proof: By contradiction |y|-|x|+1

Hard Inputs Lemma:The queries used for inputs andintersect at some point in Proof: By contradiction Reconstruction is not Lipschitz Contradiction! |y|-|x|+1

Hard Inputs Lemma:The queries used for inputs andintersect at some point in Proof: By contradiction

Query Graph Query graph: Arcsfor queries in hard inputs queries for queries for query graph

Query Graph Query graph: Arcsfor queries in hard inputs queries for queries for query graph Great thing: suffices to show that there is node with high outdegree! Idea 2: Lower bound max outdegree by • Many pairs interacting. Uses localization of

Query Graph Query graph: Arcsfor queries in hard inputs Possible because of one hard input per point queries for queries for query graph Great thing: suffices to show that there is node with high outdegree! Idea 2: Lower bound max outdegree by • Many pairs interacting. Uses localization of

Query Graph Query graph: Arcsfor queries in hard inputs Possible because of one hard input per point Easier to make multiple hard inputs per point Good for non-adaptive queries for queries for query graph more structure, simpler Great thing: suffices to show that there is node with high outdegree! Idea 2: Lower bound max outdegree by • Many pairs interacting. Uses localization of

Conclusions • Exponential unconditional lower bound for local filters for Lipschitz and monotonicity • One hard input per point • Bounding outdegree with weaker structure • Helps narrowing down which relaxations of local filters allow efficient implementation • Relaxations with efficient filters? • Output satisfies property with constant probability • Enough to get -differential privacy • Output function can be -Lipschitz

Thank you!!

Limitations of Local Filters of Lipschitz and Monotone Functions