Private Continual Release of Real-Valued Data Streams - Date: February 26, 2019 Victor Perrier, Hassan Jameel Asghar, Dali Kaafar Macquarie ...
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Private Continual Release of
Real-Valued Data Streams
Date: February 26, 2019
Victor Perrier,
Hassan Jameel Asghar,
Dali Kaafar
Macquarie University, Data61 (CSIRO), ISAE-SUPAERO
Streaming Data and Statistics
• Real-time monitoring of customer data can improve services
• Real-time updates
• Analysts/planners can optimize services
Service Event Real-time statistics
Energy Smart-meter reading Electricity usage in a neighborhood
Transport Tap-on/off time Peak hour commute times
Retail Supermarket bill Average expenditure in a supermarket
Location Check-in/out time Average time spent in a restaurant
2
Issue: Privacy Smart meter readings
• Raw stats may reveal
sensitive events
• Unusual presence at home
(smart meter)
• Trip to beach instead of work
(transport)
• Events (observations) can be
linked to real-life activities
[MSF+10]
Unusual activity [MSF+10]
3
✏
on a given day. Notice that this is significantly less than
Since GS = B , we get a linear term in B . We will the total time possible in a 24 hour period, which is
use this algorithm as the building block, but we shall obviously 1,440 minutes.
Privacy-Preserving Statistics
find ways to improve the dependence on B . Our gain
will be through the assumption that actual distribution
of the string is concentrated far below B . We will CDF of observations
later justify this assumption by analyzing two real-world
datasets. Then, instead of using the global sensitivity,
• Differential privacy a natural candidate
we shall use smooth sensitivity tailored to a threshold
⌧ B , and therefore we expect considerable gain in
• Most⌧utility.
work on static databases
% readings
• Some work on binary data streams [DNPR10,
CSS11]
A. Motivation
We consider scenarios where the upper bound B on a
generic element of the incoming stream is overly conser-
• Our problem
vative resulting in utility that is begging to be improved
in practice. Consider the following likely scenarios
• Data from
• Thean event
bound is real-valued
B might withinFor
not be known in advance. a !
public upper a bound!
bound
instance, on the expenditure during a trip to public
Fig. 2: Histogram of journey times on Sydneybound
trains on
the supermarket. Any guess on the bound B would
• Release updated
be taking intosum/average
account instances atofeach event
unusually high
a particular day What is the scale of the x-axis.
• Event-level privacy
spendings. This will result in a very conservative
Likewise, we see a similar trend in the total expen-
upper bound.
• Peculiar events protected
• In some cases, a natural bound B exists. For in-
diture during a trip to a supermarket chain in Australia
time
on a given day. Converted into an empirical cumulative
stance, the commute time per day has a natural
distribution function (CDF), the two distributions
events 4 are
bound of B = 24 hours. However, most commute
shown in Figures 3 and 4 respectively. An interesting
for strings of length n, where trains on a particular day (including multiple trips). We
✓ r ◆ can see that the peak is around 20 to 30 mins, and very
1 1
↵ = O GS · 8 ln log2 (n)1.5 . few customers take more than 150 minutes of train ride
✏
on a given day. Notice that this is significantly less than
How to Release the Average?
Since GS = B , we get a linear term in B . We will
use this algorithm as the building block, but we shall
the total time possible in a 24 hour period, which i
obviously 1,440 minutes.
find ways to improve the dependence on B . Our gain
will be through the assumption that actual distribution
CDF of observations
• Basic: addofLaplace
the string noise of scalefar!below
is concentrated to each
B . We will
observation
later justify this assumption by analyzing two real-world
datasets.
• Error !" afterThen, instead of using the global sensitivity,
" events
we shall use smooth sensitivity tailored to a threshold
⌧ ⌧ B , and therefore we expect considerable gain in
• Generalized
utility.binary stream algorithm fairs better
% readings
• Error !log & " [DNPR10, CSS11]
A. Motivation
We consider scenarios where the upper bound B on a
• Problem: generic
error still proportional to !
element of the incoming stream is overly conser-
• In manyvative resulting !
situations in is toothat
utility loose or unknown
is begging to be improved
• E.g.,inUnlikely
practice.someone commuting
Consider the following for fullscenarios
likely 24 hours!
' !
• The bound B might not be known in advance. For threshold
• Most readings concentrated
instance, a bound on thebelow during a trip to'
a threshold
expenditure public
Fig. 2: Histogram of journey times on Sydney bound
trains on
the supermarket. Any guess on the bound B would
a particular day What is the scale of the x-axis.
• be taking into account
If ' known, error is only 'log & " instances of unusually high
spendings.
• Significant This will result in a very conservative
if !: ' large Likewise, we see a similar trend in the total expen
upper bound.
diture during a trip to a supermarket chain5 in Australia
• In some cases, a natural bound B exists. For in-
Validation of Data Concentration
Train trips
• Is data really concentrated well below a
conceivable !?
% trips
• Train trips dataset
• 50 million trips over four weeks (Sydney, Australia) minutes
• Conceivable bound ! = 24 hours
Supermarket
• Supermarket dataset
% transactions
• 140,000 transactions by 1,000 customers (Australia)
• Conceivable bound ! = ?
dollars
6
How to Estimate Threshold with Privacy?
• Need to observe a subset ! of observations
– time lag Stream Readings
90
80
• Time lag needs to be optimized for accuracy 70
60
• Too early: high outlier error 50
• Too late: marginal gain (may just use " as 40
30
estimate) 20
10
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
• Naively estimating # violates privacy !
Time lag
$
• E.g., maximum of ! observations is an exact
event!
7
Our Work
• A method to estimate threshold ! using a subset of observations
• With differential privacy
• and utility optimized for moving average
• Mechanism is generic – can also be used for
• Average over a sliding window
• Releasing histogram of streaming data
• Estimating scale of distribution
8
Background: Binary Tree Algorithm
• Binary tree (BT) algorithm [DNPR10, CSS11]
• Find at most log $% nodes in tree whose union
equals sum up to & events
'()*+ , ',
• Add Laplace noise of scale -
instead of -
• Goal: Use BT as sub-module but noise
scaled to . instead of /
Computing private sum of first 7 observations
9
Global Mechanism
1. Estimate threshold ! using first " observations using budget #$
%
2. Use Laplace noise with scale to release sum of first "
&'
observations
3. Update & release sum for each event after " with Laplace noise of
scale ! log + ,/# using BT algorithm
• Overall: (#, 0)-differential privacy
10What are the Choices for Threshold?
• False starts
• Differentially private max of ! values?
• max function is highly sensitive
• Adjacent streams can differ by any value in [0, %] Distribution of '
• Standard deviation of distribution of '?
• Need to know distribution in advance
( fraction of values
• Statistic of choice: (-quantile
• E.g., ( = 0.005 (0.5% of values)
,(
(-quantile
11Privately Estimating !-Quantile
• Need to estimate !-quantile through first " readings
• Satisfying # ≫ " ≫ 1/! required for stable estimate of !
• Roadmap
• Obtain the empirical estimate (' ! of (!
• Add differentially private noise to (' !
• Set the result as threshold )
• Complication: cannot use Global Sensitivity (GS) for DP noise
• Maximum change in function over all adjacent streams
• GS of !-quantile is close to *
12Using Smooth Sensitivity
• Local sensitivity (LS)
• Maximum change in !-quantile over streams adjacent to input stream only
• Unfortunately, LS itself can be sensitive
• E.g., big differences in LS over nearby streams
• Smooth sensitivity (SS) [NRS07]
• " #, # % : Hamming distance between streams # and # %
234 .,. /
• SS(#, () = max./ {1 5 LS./ }
• Smooths out change in LS as we move away from input stream
13Privately Obtaining the Threshold
• Obtain threshold as
! = $# %+ Laplace noise with SS
• We have swept some details under the
rug
• $# % and ! should be ≥ $% to bound error
• We assume $# % ≥ $%
14Utility Analysis
• Light-tailed distributions Train trips dataset
• Lighter than exponential distribution with the
same !-quantile
• True for train trips and supermarket datasets
for sufficiently small !
• If distribution is light-tailed
• We show that error "log & '/) (as required)
• Note: Privacy definition not dependent on
distribution assumption ! = 0.005-quantile
15Utility Analysis for Light-tailed Distributions
• Exponential distribution has the property
!" # $ ≥ !&' for all $ ≥ 1
• For light-tailed distributions: !) " # $ ≥ !&'
• Idea:
• Estimate "-quantile using 1/" readings
!" !& '
• Set threshold + to !) " # $
• Benefits: ≈ + = !) " # $
• Estimate threshold with a much smaller time lag ,
• Minimise outlier error project
. estimate
• - log 3 4
/
16What Values to Use in Practice?
• Improvement Factor (IF) metric
• Ratio of error through BT versus Impact of epsilon Impact of time lag
our method
• Epsilon: IF increases with larger !
but then drops
• Due to truncation: any value greater
than threshold is fixed to threshold
• Time lag: Noticeable increase in
impact factor with " ≈ 50,000
! "
17Heuristics for Choosing Parameters
• Optimization suggests
Parameter Interpretation Value
! !-quantile 0.005
" Shifting !-quantile Between 1 and 2
#$ Budget to estimate threshold 0.8 of overall privacy budget
#% Budget to release sum of first & terms Derive from #$
& Time lag 50,000
18Experimental Evaluation
Train trips dataset
• Max error on the sum (at step !)
• 20k repetitions
• Train trips
• ! = 250, 000, 000
• " = 50,000
• # = 1440 mins (24 hrs)
• Improvement factor: 3.5
Supermarket dataset
• Supermarkets
• ! = 150, 000
• " = 50,000
• # = 3,000 dollars
• Improvement factor: 9
19Discussion
• Improved private release of moving average if distributions are light-tailed
• Question: which data have light-tailed distribution?
• Any data coming from short-lived, time constrained events
• Smart-meter data
• Phone-call durations
• Length of posts (on social media)
• Daily average inter-arrivals of check-in times
• Heavy-tailed distributions are not “directly” time-constrained
• Income distribution
• File sizes in computer systems
20Conclusion
• Shown a way to privately estimate the bulk of a distribution of streaming real-
valued data
• Can be estimated by sacrificing a time lag
• Heuristics for choosing parameters in practice
• In worst-case, threshold is close to public bound !
• We do not need to abort as in the propose-test-release approach [DL09]
• Moving average release is just one application – can be used in other applications
21Questions
22References
• [CSS11]Chan, T.H.H., Shi, E. and Song, D., 2011. Private and continual release of
statistics.ACM Transactions on Information and System Security (TISSEC), 14(3), p.26.
• [DL09] Dwork, C. and Lei, J., 2009, May. Differential privacy and robust statistics.
In STOC (Vol. 9, pp. 371-380).
• [DNPR10] Dwork, C., Naor, M., Pitassi, T. and Rothblum, G.N., 2010, June. Differential
privacy under continual observation. In Proceedings of the forty-second ACM symposium
on Theory of computing (pp. 715-724). ACM.
• [MSF+10] Molina-Markham, A., Shenoy, P., Fu, K., Cecchet, E. and Irwin, D., 2010,
November. Private memoirs of a smart meter. In Proceedings of the 2nd ACM workshop
on embedded sensing systems for energy-efficiency in building (pp. 61-66). ACM.
• [NRS07] Nissim, K., Raskhodnikova, S. and Smith, A., 2007, June. Smooth sensitivity and
sampling in private data analysis. In Proceedings of the thirty-ninth annual ACM symposium on
Theory of computing (pp. 75-84). ACM.
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