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Privately Detecting Changes in Unknown Distributions Wanrong Zhang, Georgia Tech joint work with Rachel Cummings, Sara Krehbiel, Yuliia Lut 1

Motivation I: Smart-home IoT devices 2

Motivation II: Disease outbreaks 3

Change-point problem: Identify distributional changes in stream of highly sensitive data Model: Data points 𝑦 " , … , 𝑦 % ∗ ∼ 𝑄 ) (pre-change) Need formal privacy 𝑦 % ∗ , … , 𝑦 * ∼ 𝑄 " (post-change) guarantees for change-point detection algorithms Question: Estimate the unknown change time 𝑙 ∗ Previous work: parametric model [CKM+18] ( 𝑄 ) and 𝑄 " known) Our work: nonparametric model (𝑄 ) and 𝑄 " unknown) 4

Differential privacy [DMNS ‘06] Bound the maximum amount that one person’s data can change the distribution of an algorithm’s output An algorithm 𝑁: 𝑈 * → 𝑆 is 𝝑 -differentially private if ∀ neighboring 𝑦, 𝑦′ ∈ 𝑈 * and ∀ 𝑇 ⊆ 𝑆 , 𝑄 𝑁 𝑦 ∈ 𝑇 ≤ 𝑓 ; 𝑄 𝑁 𝑦 < ∈ 𝑇 • 𝑇 as set of “bad outcomes” • Worst-case guarantee 5

Privately Detecting Changes in Unknown Distributions 1. Offline setting: dataset known in advance 2. Online setting: data points arrive one at a time 3. Drift change detection (in paper) 4. Empirical results (in paper) 6

Privately Detecting Changes in Unknown Distributions 1. Offline setting: dataset known in advance 2. Online setting: data points arrive one at a time 3. Drift change detection (in paper) 4. Empirical results (in paper) 7

Mann-Whitney test [MW ‘47] Datasets: 𝑦 " , 𝑦 = , … 𝑦 % ~𝑄 ) and 𝑦 %?" , 𝑦 %?= , … 𝑦 * ~𝑄 " 𝐼 ) : 𝑄 ) = 𝑄 " , 𝐼 " : 𝑄 ) ≠ 𝑄 " " * % %(*D%) ∑ ∑ Test statistic: 𝑊 𝑙 = 𝐽(𝑦 H > 𝑦 J ) JK%?" HK" Number of such pairs (𝑦 H , 𝑦 J ) such " Under 𝐼 " , require 𝑏: = 𝑄𝑠 N~O P ,Q~O R 𝑦 > 𝑧 ≠ that 𝑦 H > 𝑦 J = 8

Non-private nonparametric change-point detection [Darkhovsky ‘79] 1. F or every 𝑙 ∈ 𝜹𝒐 , … 𝟐 − 𝜹 𝒐 2. Compute 𝑊 𝑙 Can we compute V(𝑙) or ] = 𝑏𝑠𝑕𝑛𝑏𝑦 % 𝑊(𝑙) 3. Output 𝑙 arg max 𝑊(𝑙) privately? 𝑾(𝒍) 𝜹𝒐 𝒍 ∗ 𝟐 (𝟐 − 𝜹)𝒐 𝒐 9

Adding differential privacy Differentially private algorithms add noise that scale with the sensitivity of a query. Query sensitivity: The sensitivity of real-valued query 𝑔 is: i,i j *kHlmnopq 𝑔 𝑌 − 𝑔 𝑌 < Δ𝑔 = max . Laplace Mechanism: The mechanism 𝑁 𝑔, 𝑌, 𝜗 = 𝑔 𝑌 + Lap( xy z ) is 𝜗 -differentially private. 10

Offline PNCPD = Mann-Whitney + ReportNoisyMax Private Nonparametric Change-Point Detector: 𝑄𝑂𝐷𝑄𝐸(𝑌, 𝜗, 𝛿) 1. Input: database, privacy parameter 𝜗 , constraint parameter 𝛿 2. for k ∈ 𝛿𝑜 , … 1 − 𝛿 𝑜 3. Compute statistic 𝑊(𝑙) = 4. Sample 𝑎 % ~𝑀𝑏𝑞 ;…* † = 𝑏𝑠𝑕𝑛𝑏𝑦 % 𝑊 𝑙 + 𝑎 % 5. Output 𝑙 11

Main results: OfflinePNCPD Theorem: Offline𝑄𝑂𝐷𝑄𝐸 𝑌, 𝜗, 𝛿 is 𝜗 -differentially private and with † with probability 1 − 𝛾 , it outputs private change-point estimator 𝑙 error at most ".)" 1 ’ log 1 † − 𝑙 ∗ < 𝑃 𝑙 𝝑𝛿 • 𝑏 − 1/2 = 𝛾 Previous non-private analysis [Darkhovsky ‘76] § ] − 𝑙 ∗ < 𝑃(𝑜 =/“ ) 𝑙 Our improved non-private analysis: § 𝛿 • 𝑏 − 1/2 = log 1 1 ] − 𝑙 ∗ < 𝑃 𝑙 β = 𝑃 1 12

Privately Detecting Changes in Unknown Distributions 1. Offline setting: dataset known in advance 2. Online setting: data points arrive one at a time 3. Drift change detection (in paper) 4. Empirical results (in paper) 13

Online setting More challenging: must detect change quickly without much post- change data High Level Approach: 1. Privately detect online when V 𝑙 > 𝑈 in the center of a sliding window of last 𝑜 data points. 2. Run OfflinePNCPD on the identified window. Have DP algorithm (AboveNoisyThreshold) for this 14

Online setting More challenging: must detect change quickly without much post- change data Our Approach: 1. Run AboveNoisyThreshold on Mann-Whitney queries in the center of a sliding window of last 𝑜 data points. 2. Run OfflinePNCPD on the identified window. 𝑊 𝑙 + 𝑎 % < 𝑈 15

Online setting More challenging: must detect change quickly without much post- change data Our Approach: 1. Run AboveNoisyThreshold on Mann-Whitney queries in the center of a sliding window of last 𝑜 data points. 2. Run OfflinePNCPD on the identified window. 𝑊 𝑙 + 𝑎 % < 𝑈 16

Online setting More challenging: must detect change quickly without much post- change data Our Approach: 1. Run AboveNoisyThreshold on Mann-Whitney queries in the center of a sliding window of last 𝑜 data points. 2. Run OfflinePNCPD on the identified window. 𝑊 𝑙 + 𝑎 % ≥ 𝑈 17

OnlinePNCPD 1. Input: database 𝑌 = {𝑦 " , … } , privacy parameter 𝜗 , threshold 𝑈 ] = 𝑈 + Lap ˜ 2. Let 𝑈 z™ 3. For each new data point 𝑦 % : 4. Compute Mann-Whitney statistic 𝑊(𝑙) in center of last 𝑜 data points Sample 𝑎 % ∼ Lap Rš 5. z™ ] , then 6. If 𝑊 𝑙 + 𝑎 J > 𝑈 7. Run OfflinePNCPD on last 𝑜 data points with 𝜗/2 8. Else, output ⊥ 18

Main result: OnlinePNCPD Theorem: Online𝑄𝑂𝐷𝑄𝐸 𝑌, T, 𝜗, 𝛿 is 𝜗 -differentially private. For appropriate threshold T, with probability 1 − 𝛾 , it outputs private † with error at most change-point estimator 𝑙 † − 𝑙 ∗ < 𝑃 1 𝜗 log 𝑜 𝑙 𝛾 where 𝑜 is the window size. Choice of T • Can’t raise alarm too early (False positive: 𝑈 > 𝑈 • ) • Can’t fail to raise alarm at true change (False negative: 𝑈 < 𝑈 ž ) 19

Privately Detecting Changes in Unknown Distributions 1. Offline setting: dataset known in advance 2. Online setting: data points arrive one at a time 3. Drift change detection (in paper) 4. Empirical results (in paper) 20

References • Cummings, R., Krehbiel, S., Mei, Y., Tuo, R., & Zhang, W. Differentially private change-point detection. In Advances in Neural Information Processing Systems, NeurIPS’18 pp. 10848-10857,2018 • Dwork, C., McSherry, F., Nissim, K., & Smith, A. Calibrating noise to sensitivity in private data analysis. In Theory of cryptography conference, pp. 265-284, 2006. • Dwork, C., Roth, A. The algorithmic foundations of differential privacy. Foundations and Trends in Theoretical Computer Science , 9 (3– 4), 211-407, 2014. • Darkhovsky, B. A nonparametric method for the a posteriori detection of the ``disorder’’ time of a sequence of independent random variables. Theory of Probability & Its Applications , 21(1):178-183, 1976. • Mann, H.B. and Whitney, D.R. On a test of whether one of two random variables is stochastically larger than the other. The annals of mathematical statistics , pp 50-60, 1947. 21

Privately Detecting Changes in Unknown Distributions Wanrong Zhang, Georgia Tech joint work with Rachel Cummings, Sara Krehbiel, Yuliia Lut 22