Sublinear r Space Pri rivate Algori rithms Under r the Sliding - - PowerPoint PPT Presentation

Sublinear r Space Pri rivate Algori rithms Under r the Sliding Win Window M Mod

• del

Jalaj Upadhyay

Differential Privacy

A

!" !# !$

⋮

A

!" !#

&

!$

⋮

Differential Privacy

A

!" !# !$

⋮

queries/tasks

&(()

private random coin

A

!" !#

*

!$

⋮

queries/tasks

&((′)

private random coin

Differential Privacy

Output distribution is close

A

!" !# !$

⋮

queries/tasks

&(()

private random coin

A

!" !#

*

!$

⋮

queries/tasks

&((′)

private random coin

Differential Privacy

! and !’ are neighbor if they differ in one data point Output distribution is close

A

"# "$ "%

⋮

queries/tasks

'(!)

private random coin

A

"# "$

*

"%

⋮

queries/tasks

'(!′)

private random coin

Differential Privacy

! and !’ are neighbor if they differ in one data point Differential Privacy [DMNS06] Algorithm " is #-differentially private if

• for all neighboring data sets ! and !$
• for all possible outputs %,

Pr " ! ∈ S ≤ +, ⋅ Pr " !$ ∈ % Output distribution is close

A

./ .0 .1

⋮

queries/tasks

3(!)

private random coin

A

./ .0

$

.1

⋮

queries/tasks

3(!′)

private random coin

Differential Privacy

! and !’ are neighbor if they differ in one data point Differential Privacy [DMNS06] Algorithm " is #-differentially private if

• for all neighboring data sets ! and !$
• for all possible outputs %,

Pr " ! ∈ S ≤ +, ⋅ Pr " !$ ∈ % Output distribution is close

# = 0: perfect privacy no utility As # increases, less privacy more utility

A

01 02 03

⋮

queries/tasks

5(!)

private random coin

A

01 02

$

03

⋮

queries/tasks

5(!′)

private random coin

Differential Privacy

! and !’ are neighbor if they differ in one data point Differential Privacy [DMNS06] Algorithm " is #-differentially private if

• for all neighboring data sets ! and !$
• for all possible outputs %,

Pr " ! ∈ S ≤ +, ⋅ Pr " !$ ∈ % Output distribution is close Allows utility- privacy trade-off

# = 0: perfect privacy no utility As # increases, less privacy more utility

A

01 02 03

⋮

queries/tasks

5(!)

private random coin

A

01 02

$

03

⋮

queries/tasks

5(!′)

private random coin

Differential Privacy Under Sliding Window

• Differential privacy overview of Apple

“Apple retains the collected data for a maximum of three months”

Differential Privacy Under Sliding Window

• Differential privacy overview of Apple

“Apple retains the collected data for a maximum of three months”

Differential Privacy Under Sliding Window

• Differential privacy overview of Apple

“Apple retains the collected data for a maximum of three months”

Goal of this paper

• Formalize privacy under

sliding window model

• Design sublinear space

private algorithms in the sliding window model

Problem Studied: Private ℓ" heavy hitters

• # be an $-dimensional vector
• Output all indices % ∈ [$], #* ≥ , ∥ # ∥" and estimate of #*
• Allowed to accept % ∈ [$] if #* ≥ (, − 0) ∥ # ∥"

Problem Studied: Private ℓ" heavy hitters

• # be an $-dimensional vector
• Output all indices % ∈ [$], #* ≥ , ∥ # ∥" and estimate of #*
• Allowed to accept % ∈ [$] if #* ≥ (, − 0) ∥ # ∥"

Main Theorem

There is an efficient 2(3) space (4, 5)-DP algorithm that returns a set of indices, ℐ, and estimates 7 #* for % ∈ ℐ,

• If #* ≥ , ∥ # ∥", then #* − 7

#* ≤ 0 ∥ # ∥" + :

" ; log 3

• Does not include any % if #* < , − 3 0 ∥ # ∥" + :

A ; log 3

Problem Studied: Private ℓ" heavy hitters

• # be an $-dimensional vector
• Output all indices % ∈ [$], #* ≥ , ∥ # ∥" and estimate of #*
• Allowed to accept % ∈ [$] if #* ≥ (, − 0) ∥ # ∥"

Main Theorem

There is an efficient 2(3) space (4, 5)-DP algorithm that returns a set of indices, ℐ, and estimates 7 #* for % ∈ ℐ,

• If #* ≥ , ∥ # ∥", then #* − 7

#* ≤ 0 ∥ # ∥" + :

" ; log 3

• Does not include any % if #* < , − 3 0 ∥ # ∥" + :

A ; log 3

Price of privacy

Other Results and Open Problems

• Algorithm extends to continual observation under sliding window
• Current non-private framework do not extend to privacy
• Lower bound using standard packing argument
• Space lower bound on estimating ℓ"-heavy hitters
• Reduction to communication complexity problem

Other Results and Open Problems

• Algorithm extends to continual observation under sliding window
• Current non-private framework do not extend to privacy
• Lower bound using standard packing argument
• Space lower bound on estimating ℓ"-heavy hitters
• Reduction to communication complexity problem

Characterize what is possible to compute privately under the sliding window model