towards practical differentially private convex optimization · contributions • new algorithm for...
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Towards Practical Differentially Private Convex Optimization
ROGER IYENGAR
CARNEGIE MELLON UNIVERSITY
JOSEPH P. NEAR UNIVERSITY OF
VERMONT
DAWN SONG UNIVERSITY OF
CALIFORNIA, BERKELEY
ABHRADEEP THAKURTA UNIVERSITY OF
CALIFORNIA, SANTA CRUZ
LUN WANG UNIVERSITY OF
CALIFORNIA, BERKELEY
OM THAKKAR BOSTON
UNIVERSITY
Contributions • NewAlgorithmforDifferentiallyPrivateConvexOptimization:ApproximateMinimaPerturbation(AMP)• Canleverageanyoff-the-shelfoptimizer• Worksforallconvexlossfunctions• Hasacompetitivehyperparameter-freevariant
• BroadEmpiricalStudy• 6state-of-the-arttechniques• 2models:LogisticRegression,andHuberSVM• 13datasets:9public(4high-dimensional),4real-worldusecases• Open-sourcerepo:https://github.com/sunblaze-ucb/dpml-benchmark
This Talk • WhyPrivacyforLearning?• Background• DifferentialPrivacy(DP)• ConvexOptimization
• ApproximateMinimaPerturbation(AMP)• BroadEmpiricalStudy
Why Privacy for Learning? SensitiveData𝐷
TrainingAlgorithm𝐴TrainedModel 𝜃 Input Output
• Modelscanleakinformationabouttrainingdata• Membershipinferenceattacks[ShokriStronatiSongShmatikov’17,CarliniLiuKosErlingssonSong’18,
MelisSongCristofaroShmatikov’18]• Modelinversionattacks[FredriksonJhaRistenpart’15,WuFredriksonJhaNaughton’16]
• Solution?
𝐏𝐫(𝑨(𝑫)= 𝜽 )
𝐷::
Differential Privacy [Dwork Mcsherry Nissim Smith ‘06]
Alice Bob Cathy Doug Emily Om
Randomized
Outcomes 𝜽 ∈𝚯
𝐴
Θ
𝐏𝐫(𝑨(𝑫)= 𝜽 )
𝐷↑′ :
Differential Privacy [Dwork Mcsherry Nissim Smith ‘06]
Alice Bob Cathy Doug Emily Om
Randomized
Outcomes 𝜽 ∈𝚯
𝐴
Θ
Felix
𝐏𝐫(𝑨(𝑫′)= 𝜽 )𝐷↑′ :
Differential Privacy [Dwork Mcsherry Nissim Smith ‘06]
Alice Bob Cathy Doug Emily Om
Randomized
Outcomes 𝜽 ∈𝚯
𝐴
ΘSmall
Felix
Differential Privacy [Dwork Mcsherry Nissim Smith ‘06] • Privacyparameters:(𝜀,𝛿)• Arandomizedalgorithm𝐴:𝒟↑𝑛 →𝑇is(𝜀,𝛿)-DPif• forallneighboringdatasets𝐷, 𝐷↑′ ∈ 𝒟↑𝑛 ,i.e.,𝑑𝑖𝑠𝑡(𝐷, 𝐷↑′ )=1• forallsetsofoutcomes𝑆⊆Θ,wehave
Pr�(𝐴(𝐷)∈𝑆) ≤ 𝑒↑𝜀 Pr�(𝐴(𝐷↑′ )∈𝑆) + 𝛿
𝜀:Multiplicativechange.Typically,𝜀=𝑂(1)
𝛿:Additivechange.Typically,𝛿=𝑂(1/ 𝑛↑2 )
Convex Optimization • Input:
• Dataset𝐷∈ 𝒟↑𝑛 • Lossfunction𝐿(𝜃,𝐷),where
• 𝜃∈ ℝ↑𝑝 isamodel• Loss𝐿isconvexinthefirstparameter𝜃
• Goal:Outputmodel𝜃 suchthat 𝜃 ∈ min┬𝜃∈ ℝ↑𝑝 �𝐿(𝜃,𝐷)
• Applications:• MachineLearning,DeepLearning,CollaborativeFiltering,etc. 𝜃
𝐿(𝜃,𝐷)
𝜃
DP Convex Optimization - Prior Work SensitiveData𝐷
TrainingAlgorithm𝐴 TrainedModel 𝜃 Input Output
ObjectivePerturbation[ChaudhuriMonteleoniSarwate’11,KiferSmith
Thakurta’12,JainThakurta’14]
DPGD/SGD[SongChaudhuri
Sarwate’13,BassilySmithThakurta’14,AbadiChuGoodfellowMcMahan
MironovTalwarZhang’16]
DPFrankWolfe
[TalwarThakurtaZhang’14]
OutputPerturbation[CMS’11,KST’12,JT’14]
DPPermutation-basedSGD[WuLiKumarChaudhuri
JhaNaughton’17]
-Requiresminimaofloss-Requirescustomoptimizer
• Input:• Dataset𝐷,Lossfunction:𝐿(𝜃,𝐷)• Privacyparameters:𝑏=(𝜖, 𝛿)• Gradientnormbound𝛾
• Algorithm(high-level):1. Splitprivacybudgetinto2parts𝑏↓1 and 𝑏↓2 2. Perturbloss: 𝐿↓𝑝𝑟𝑖𝑣 (𝜃,𝐷)=𝐿(𝜃,𝐷)+𝑅𝑒𝑔(𝜃, 𝑏↓1 )
𝐿↓𝑝𝑟𝑖𝑣 (𝜃
,𝐷)
𝐿(𝜃,𝐷)
Approximate Minima Perturbation (AMP)
𝜃𝜃↓𝑝𝑟𝑖𝑣 𝜃 SimilartostandardObjectivePerturbation[KST’12]
• Input:• Dataset𝐷,Lossfunction:𝐿(𝜃,𝐷)• Privacyparameters:𝑏=(𝜖, 𝛿)• Gradientnormbound𝛾
• Algorithm(high-level):1. Splitprivacybudgetinto2parts𝑏↓1 and 𝑏↓2 2. Perturbloss: 𝐿↓𝑝𝑟𝑖𝑣 (𝜃,𝐷)=𝐿(𝜃,𝐷)+𝑅𝑒𝑔(𝜃, 𝑏↓1 )3. Let 𝜃↓𝑎𝑝𝑝𝑟𝑜𝑥 =𝜃s.t. ‖∇𝐿↓𝑝𝑟𝑖𝑣 (𝜃,𝐷)‖↓2 ≤𝛾4. Output𝜃↓𝑎𝑝𝑝𝑟𝑜𝑥 +𝑁𝑜𝑖𝑠𝑒(𝑏↓2 ,𝛾)
𝐿↓𝑝𝑟𝑖𝑣 (𝜃
,𝐷)
Approximate Minima Perturbation (AMP)
𝜃𝜃↓𝑝𝑟𝑖𝑣
‖∇𝐿↓𝑝𝑟𝑖𝑣 (𝜃,𝐷)‖↓2 ≤𝛾
𝜃↓𝑎𝑝𝑝𝑟𝑜𝑥
SimilartostandardObjectivePerturbation[KST’12]
Utility guarantees • Let 𝜃 minimize𝐿(𝜃;𝐷),andtheregularizationparameterΛ= Θ (𝜉√�𝑝 /𝜖𝑛‖𝜃 ‖ ).
• ObjectivePerturbation[KST’12]:If𝜃↓𝑝𝑟𝑖𝑣 istheoutputofobj.pert.: 𝔼(𝐿(𝜃↓𝑝𝑟𝑖𝑣 ;𝐷)−𝐿(𝜃 ;𝐷))= 𝑂 (𝜉√�𝑝 ‖𝜃 ‖/𝜖𝑛 ).• AMP(adaptedfrom[KST’12]):Foroutput 𝜃↓𝐴𝑀𝑃 :
𝔼(𝐿(𝜃↓𝐴𝑀𝑃 ;𝐷)−𝐿(𝜃 ;𝐷))= 𝑂 (𝜉√�𝑝 ‖𝜃 ‖/𝜖𝑛 +‖𝜃 ‖𝛾𝑛).• For𝛾=𝑂(1/𝑛↑2 ),theutilityofAMPisasymptoticallythesameasthatofObj.Pert.
• PrivatePSGD[WLK↑+ 17]:Foroutput𝜃↓𝑃𝑆𝐺𝐷 ,andmodelspaceradius𝑅:𝔼(𝐿(𝜃↓𝑃𝑆𝐺𝐷 ;𝐷)−𝐿(𝜃 ;𝐷))= 𝑂 (𝜉√�𝑝 𝑅/𝜖√�𝑛 ).
• For𝛾=𝑂(1/𝑛↑2 ),theutilityofAMPhasabetterdependenceon𝑛thanPrivatePSGD.thanPrivatePSGD.
AMP - Takeaways • Canleverageanyoff-the-shelfoptimizer• Worksforallstandardconvexlossfunctions• For𝛾=𝑂(1/𝑛↑2 ),theutilityofAMP:
• isasymptoticallythesameasObjectivePerturbation[KST’12]• hasabetterdependenceon𝑛thanPrivatePSGD[WLK↑+ 17]
• 𝛾= 1/𝑛↑2 achievableusingstandardPythonlibraries
Empirical Evaluation • Algorithmsevaluated:
• ApproximateMinimaPerturbation(AMP)• PrivateSGD[BST↑′ 14, ACG↑+ 17]
• PrivateFrank-Wolfe(FW)[TTZ↑′ 14]
• PrivatePermutation-basedSGD(PSGD)[WLK↑+ 17]
• PrivateStrongly-convex(SC)PSGD[WLK↑+ 17]
• Hyperparameter-free(HF)AMP• Splittingtheprivacybudget:Weprovideascheduleforlow-andhigh-dim.databyevaluatingAMPonlyonsyntheticdata
• Non-private(NP)Baseline
Empirical Evaluation • Lossfunctionsconsidered:
• Logisticloss• HuberSVM
• Procedure:• 80/20train/testrandomsplit• Fix𝛿= 1/𝑛↑2 ,andvary𝜖from0.01to10• Measureaccuracyoffinaltuned*privatemodelovertestset• Reportthemeanaccuracyandstd.dev.over10independentruns
Thistalk
*DoesnotapplytoHyperparameter-freeAMP.
Synthetic Datasets Synthetic-L(10k ×20)
LegendNPBaseline
AMP
HFAMP
PrivateSGD
PrivatePSGD
PrivateSCPSGD
PrivateFW
- Synthetic-Hishigh-dimensional,butlow-rank- PrivateFrank-WolfeperformsthebestonSynthetic-H
Synthetic-H(2k ×2k)
High-dimensional Datasets Real-sim(72k×21k)
LegendNPBaseline
AMP
HFAMP
PrivateSGD
PrivatePSGD
PrivateSCPSGD
PrivateFW
- BothvariantsofAMPalmostalwaysprovidethebestperformance
RCV-1(50k×47k)
Real-world Use Cases (Uber) Dataset1(4m×23)
LegendNPBaseline
AMP
HFAMP
PrivateSGD
PrivatePSGD
PrivateSCPSGD
PrivateFW
- DPasaregularizer[BST’14,DworkFeldmanHardtPitassiReingoldRoth’15]- Evenfor𝜖= 10↑−2 ,accuracyofAMPisclosetonon-privatebaseline
Dataset2(18m×294)
Conclusions • Forlargedatasets,costofprivacyislow
• Privatemodeliswithin4%accuracyofthenon-privateonefor𝜖=0.01,andwithin2%for𝜖=0.1
• AMPalmostalwaysprovidesthebestaccuracy,andiseasilydeployableinpractice
• Hyperparameter-freeAMPiscompetitivew.r.t.tunedstate-of-the-artprivatealgorithms
• Open-sourcerepo:https://github.com/sunblaze-ucb/dpml-benchmark
ThankYou!