Almost Optimal Distribution-free Junta Testing Nader H. Bshouty - - PowerPoint PPT Presentation

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Almost Optimal Distribution-free Junta Testing Nader H. Bshouty - - PowerPoint PPT Presentation

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Almost Optimal Distribution-free Junta Testing Nader H. Bshouty Technion -Junta A Boolean function : 0,1 {0,1} is called junta if it depends on at most variables/coordinates. Examples: 1 , 2 , ,

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Almost Optimal Distribution-free Junta Testing

Nader H. Bshouty

Technion

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𝑙-Junta

A Boolean function 𝑔: 0,1 π‘œ β†’ {0,1} is called 𝑙 βˆ’junta if it depends on at most 𝑙 variables/coordinates. Examples: 𝑔 𝑦1, 𝑦2, … , 𝑦10000 = 𝑦1111 ∧ 𝑦9992 βŠ• 𝑦3001 is 3-junta is 4-junta 𝑙 βˆ’Junta is the class of all 𝑙 βˆ’juntas. Relevant variables/coordinates 1-junta is {𝑦𝑗, ΰ΄₯ 𝑦𝑗, 0,1} 0-junta is {0,1}

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Model of Testing

A distribution-free testing algorithm 𝐡 for 𝑙 βˆ’Junta is an algorithm that, given an access to the two oracles and a distance parameter πœ— as an input , 1) if 𝑔 is 𝑙-junta then 𝐡 outputs β€œaccept” with probability at least 2/3. 2) if 𝑔 is πœ—-far from every 𝑙-junta with respect to the distribution 𝒠 then 𝐡 outputs β€œreject” with probability at least 2/3. Given a black box that contains a Boolean function 𝑔: 0,1 π‘œ β†’ {0,1} Given two oracles: 1) when 𝑦 ∈ 0,1 π‘œ is queried, it returns 𝑔(𝑦) and 2) returns 𝑣 ∈ 0,1 π‘œ chosen acc. to unknown distribution 𝒠 βˆ€ β„Ž ∈ 𝑙 βˆ’ πΎπ‘£π‘œπ‘’π‘ Pr

𝑦~𝒠 𝑔 𝑦 β‰  β„Ž 𝑦

β‰₯ πœ—

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𝑙-Junta 𝑙-Junta 𝑔

Model of Testing

𝐡 outputs β€œaccept” with probability at least 2/3.

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𝑙-Junta 𝑙-Junta 𝑔

Model of Testing

𝐡 outputs β€œreject” with probability at least 2/3.

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𝑙-Junta 𝑙-Junta 𝑔

Model of Testing

𝐡 halts and outputs either β€œaccept” or β€œreject”.

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Results in the uniform distribution Model

Result ΰ·© 𝑷/ΰ·© 𝛁 Lo./Up. Ada./NonAda Reference

Adaptive NonAdaptive Lower bounds Upper bounds For the number of queries

ΰ·© Ξ© π‘ˆ = π‘ˆ /π‘žπ‘π‘šπ‘§(log π‘ˆ) ΰ·¨ 𝑃 π‘ˆ = π‘ˆ π‘žπ‘π‘šπ‘§(log π‘ˆ)

Time π‘žπ‘π‘šπ‘§ π‘œ,

1 πœ—

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Result ΰ·© 𝑷/ΰ·© 𝛁 Lo./Up. Ada./NonAda Reference 𝑙/πœ— Upper Adaptive Blais STOC 2009

Results in the uniform distribution Model

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Result ΰ·© 𝑷/ΰ·© 𝛁 Lo./Up. Ada./NonAda Reference 𝑙/πœ— Upper Adaptive Blais STOC 2009 𝑙 Lower Adaptive Saglam FOCS 2018

Results in the uniform distribution Model

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Result ΰ·© 𝑷/ΰ·© 𝛁 Lo./Up. Ada./NonAda Reference 𝑙/πœ— Upper Adaptive Blais STOC 2009 𝑙 Lower Adaptive Saglam FOCS 2018 𝑙

3 2/πœ—

Upper NonAdaptive Blais APPROX 2008

Results in the uniform distribution Model

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Result ΰ·© 𝑷/ΰ·© 𝛁 Lo./Up. Ada./NonAda Reference 𝑙/πœ— Upper Adaptive Blais STOC 2009 𝑙 Lower Adaptive Saglam FOCS 2018 𝑙

3 2/πœ—

Upper NonAdaptive Blais APPROX 2008 𝑙

3 2/πœ—

Lower NonAdaptive Chen et al. CCC 2017

Results in the uniform distribution Model

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Results in the distribution-free model

Result ΰ·© 𝑷/ΰ·© 𝛁 Lo./Up. Ada./NonAda Reference

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Results in the distribution-free model

Result ΰ·© 𝑷/ΰ·© 𝛁 Lo./Up. Ada./NonAda Reference 2𝑙/πœ— Upper NonAdaptive Halevy et al. APPROX 03

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Results in the distribution-free model

Result ΰ·© 𝑷/ΰ·© 𝛁 Lo./Up. Ada./NonAda Reference 2𝑙/πœ— Upper NonAdaptive Halevy et al. APPROX 03 2𝑙/3 Lower NonAdaptive Chen et al. STOC 2018

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Results in the distribution-free model

Result ΰ·© 𝑷/ΰ·© 𝛁 Lo./Up. Ada./NonAda Reference 2𝑙/πœ— Upper NonAdaptive Halevy et al. APPROX 03 2𝑙/3 Lower NonAdaptive Chen et al. STOC 2018 𝑙2/πœ— Upper Adaptive Chen et al. STOC 2018

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Results in the distribution-free model

Result ΰ·© 𝑷/ΰ·© 𝛁 Lo./Up. Ada./NonAda Reference 2𝑙/πœ— Upper NonAdaptive Halevy et al. APPROX 03 2𝑙/3 Lower NonAdaptive Chen et al. STOC 2018 𝑙2/πœ— Upper Adaptive Chen et al. STOC 2018 𝑙 Lower Adaptive Saglam FOCS 2018

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Results in the distribution-free model

Result ΰ·© 𝑷/ΰ·© 𝛁 Lo./Up. Ada./NonAda Reference 2𝑙/πœ— Upper NonAdaptive Halevy et al. APPROX 03 2𝑙/3 Lower NonAdaptive Chen et al. STOC 2018 𝑙2/πœ— Upper Adaptive Chen et al. STOC 2018 𝑙 Lower Adaptive Saglam FOCS 2018 𝑙/πœ— Upper Adaptive Ours

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The algorithm

Choose a random uniform partition π‘Œ1, π‘Œ2, … , π‘Œπ‘  of π‘œ where 𝑠 = 2𝑙2 𝑔(𝑦1, … , π‘¦π‘œ) 𝑔(π‘¦π‘Œ1 ∘ π‘¦π‘Œ2 ∘ β‹― ∘ π‘¦π‘Œπ‘ ) Find relevant sets π‘Œπ‘—1, π‘Œπ‘—2, … , π‘Œπ‘—π‘™β€² 𝑔 π‘£π‘Œ ∘ 0 ΰ΄€

π‘Œ ? = 𝑔(𝑣)

π‘Œπ‘—1, π‘Œπ‘—2, … , π‘Œπ‘—π‘˜ π‘Œ = π‘Œπ‘—1 βˆͺ π‘Œπ‘—2 βˆͺ β‹― βˆͺ π‘Œπ‘—π‘˜ 𝑣~𝒠 Why 2𝑙2? If 𝑔 is kβˆ’junta, whp each π‘Œπ‘— contains at most one relevant coordinate

Find relevant sets

𝑔 𝑀 β‰  𝑔 π‘€π‘Œπ‘— ∘ 0π‘Œπ‘—

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Find relevant sets

Find relevant sets π‘Œπ‘—1, π‘Œπ‘—2, … , π‘Œπ‘—π‘™β€² 𝑔 π‘£π‘Œ ∘ 0 ΰ΄€

π‘Œ ? = 𝑔(𝑣)

π‘Œπ‘—1, π‘Œπ‘—2, … , π‘Œπ‘—π‘˜ π‘Œ = π‘Œπ‘—1 βˆͺ π‘Œπ‘—2 βˆͺ β‹― βˆͺ π‘Œπ‘—π‘˜ Find a new relevant set π‘Œπ‘—π‘˜+1 = π‘Œβ„“ 𝑔 π‘£π‘Œ ∘ 0 ΰ΄€

π‘Œ β‰  𝑔(𝑣)

𝑔 π‘£π‘Œ ∘ 𝑣𝑍

1 ∘ 0𝑍 2

𝑣~𝒠 ΰ΄€ π‘Œ = 𝑍

1 βˆͺ 𝑍 2

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Find relevant sets

Find relevant sets π‘Œπ‘—1, π‘Œπ‘—2, … , π‘Œπ‘—π‘™β€² 𝑔 π‘£π‘Œ ∘ 0 ΰ΄€

π‘Œ ? = 𝑔(𝑣)

π‘Œπ‘—1, π‘Œπ‘—2, … , π‘Œπ‘—π‘˜ π‘Œ = π‘Œπ‘—1 βˆͺ π‘Œπ‘—2 βˆͺ β‹― βˆͺ π‘Œπ‘—π‘˜ Find a new relevant sets π‘Œπ‘—π‘˜+1 = π‘Œβ„“ 𝑔 π‘£π‘Œ ∘ 0 ΰ΄€

π‘Œ β‰  𝑔(𝑣)

𝑔 π‘£π‘Œ ∘ 0 ΰ΄€

π‘Œ = 𝑔(𝑣)

𝑣~𝒠 For ΰ·¨ 𝑃

1 πœ— values of 𝑣~𝒠

If this is the (𝑙 + 1)-th relevant set then β€œreject’’ Pr

𝑦~𝒠 𝑔 π‘¦π‘Œ ∘ 0 ΰ΄€ π‘Œ β‰  𝑔 𝑦

≀ πœ—/3 log 𝑠 = 𝑃 log 𝑙 ΰ·¨ 𝑃

𝑙 πœ— queries

We also get a witness for π‘Œβ„“ 𝑔 𝑀(β„“) β‰  𝑔 π‘€π‘Œβ„“

β„“ ∘ 0π‘Œβ„“

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The algorithm

π‘Œπ‘—1, π‘Œπ‘—2, … , π‘Œπ‘—π‘™β€² , 𝑙′ ≀ 𝑙 Pr

𝑦~𝒠 𝑔 π‘¦π‘Œ ∘ 0 ΰ΄€ π‘Œ β‰  𝑔 𝑦

≀ πœ— 3 π‘Œ = π‘Œπ‘—1 βˆͺ π‘Œπ‘—2 βˆͺ β‹― βˆͺ π‘Œπ‘—π‘™β€² β„Ž ≔ 𝑔(π‘¦π‘Œ ∘ 0 ΰ΄€

π‘Œ) is πœ— 3 βˆ’close to 𝑔 with respect to 𝒠

Each π‘Œπ‘—π‘˜ contains at least one relevant variable 𝑙-Junta 𝑙-Junta 𝑔 𝑔 β„Ž: = 𝑔(π‘¦π‘Œ ∘ 0 ΰ΄€

π‘Œ)

β„Ž: = 𝑔(π‘¦π‘Œ ∘ 0 ΰ΄€

π‘Œ)

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π‘Œπ‘—1, π‘Œπ‘—2, … , π‘Œπ‘—π‘™β€² , 𝑙′ ≀ 𝑙 π‘Œ = π‘Œπ‘—1 βˆͺ π‘Œπ‘—2 βˆͺ β‹― βˆͺ π‘Œπ‘—π‘™β€² Each π‘Œπ‘—π‘˜ contains at least one relevant coordinte 𝑔 is 𝑙 βˆ’junta 𝑔(𝑦) is πœ— βˆ’far from every 𝑙 βˆ’Junta with respect to 𝒠 β„Ž: = 𝑔(π‘¦π‘Œ ∘ 0 ΰ΄€

π‘Œ) is 𝑙 βˆ’junta

Whp each π‘Œπ‘—π‘˜ contains exactly one relevant coordinate β„Ž: = 𝑔(π‘¦π‘Œ ∘ 0 ΰ΄€

π‘Œ) is 2πœ— 3 βˆ’far

from every 𝑙 βˆ’Junta with respect to 𝒠 We also have witness for each relevant set π‘Œπ‘—π‘˜β€“ that is, 𝑔 𝑀 π‘˜ β‰  𝑔 π‘€π‘Œπ‘—π‘˜

π‘˜ ∘ 0π‘Œπ‘—π‘˜

ΰ·¨ 𝑃

𝑙 πœ— queries

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𝑔 is 𝑙 βˆ’junta 𝑔(𝑦) is πœ— βˆ’far from every 𝑙 βˆ’Junta with respect to 𝒠 β„Ž: = 𝑔(π‘¦π‘Œ ∘ 0 ΰ΄€

π‘Œ) is 𝑙 βˆ’junta

Whp each π‘Œπ‘—π‘˜ contains exactly one relevant variable β„Ž: = 𝑔(π‘¦π‘Œ ∘ 0 ΰ΄€

π‘Œ) is 2πœ— 3 βˆ’far

from every 𝑙 βˆ’Junta with respect to 𝒠 𝑔 𝑀 π‘˜ β‰  𝑔 π‘€π‘Œπ‘—π‘˜

π‘˜ ∘ 0π‘Œπ‘—π‘˜

𝑔 π‘€π‘Œπ‘—π‘˜

π‘˜ ∘ π‘¦π‘Œπ‘—π‘˜

is equal to π‘¦πœ π‘˜ or π‘¦πœ π‘˜ 𝑔 π‘€π‘Œπ‘—π‘˜

π‘˜ ∘ π‘¦π‘Œπ‘—π‘˜

is (1/15)-close to a literal in {π‘¦πœ π‘˜ , π‘¦πœ π‘˜ } according to the uniform distribution ΰ·¨ 𝑃 𝑙 queries 1 βˆ’Junta\0 βˆ’Junta 𝑃(log π‘œ) ΰ·¨ 𝑃 1 queries

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Ξ“ = {𝜐(1), 𝜐(2), … , 𝜐(𝑙′)} 𝑕 = β„Ž(𝑦Γ ∘ 𝑧ΰ΄₯

Ξ“)

β„Ž(𝑦) is

2πœ— 3 βˆ’far from any 𝑙-junta with respect to 𝒠

Pr

𝑣~𝒠 β„Ž 𝑣 β‰  β„Ž 𝑣Γ ∘ 𝑧ΰ΄₯ Ξ“

β‰₯ 2πœ— 3 Pr

𝑣~𝒠,𝑧~𝑉 β„Ž 𝑣 β‰  β„Ž 𝑣Γ ∘ 𝑧ΰ΄₯ Ξ“

β‰₯ 2πœ— 3 β„Ž: = 𝑔(π‘¦π‘Œ ∘ 0 ΰ΄€

π‘Œ) is either 𝑙 βˆ’junta that depends on

β„Ž(𝑦) is 𝑙 βˆ’junta β„Ž 𝑦 = β„Ž(𝑦Γ ∘ 𝑧ΰ΄₯

Ξ“)

Pr

𝑣~𝒠,𝑧~𝑉 β„Ž 𝑣 β‰  β„Ž 𝑧 |𝑧Γ = 𝑣Γ β‰₯ 2πœ—

3 Given 𝑣? How to draw a random uniform 𝑧 such that 𝑧Γ = 𝑣Γ? Or

2πœ— 3 βˆ’far from every 𝑙 βˆ’junta w.r.t. 𝒠

ΰ·¨ 𝑃 𝑙 queries ΰ·¨ 𝑃

1 πœ— queries

ΰ·¨ 𝑃

𝑙 πœ— queries

Pr

𝑣~𝒠,𝑧~𝑉 β„Ž 𝑣 β‰  β„Ž 𝑧 |𝑧Γ = 𝑣Γ = 0

is 𝑙-junta Fix any 𝑧 ∈ 0,1 π‘œ π‘Œπ‘—1, π‘Œπ‘—2, … , π‘Œπ‘—π‘™β€²

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Ξ“ = {𝜐 1 , 𝜐 2 , … , 𝜐(𝑙′)} β„Ž: = 𝑔(π‘¦π‘Œ ∘ 0 ΰ΄€

π‘Œ)

Given 𝑣? How to draw a random uniform 𝑧 such that 𝑧Γ = 𝑣Γ? 𝑔 π‘€π‘Œπ‘—π‘˜

π‘˜ ∘ π‘¦π‘Œπ‘—π‘˜

is (1/15)-close to a literal in {π‘¦πœ π‘˜ , π‘¦πœ π‘˜ } wrt uniform dist. β†’A procedure that given 𝑣 finds π‘£πœ π‘˜ with high probability Draw a random uniform π‘§π‘Œπ‘—π‘˜ If π‘§πœ π‘˜ = π‘£πœ π‘˜ then output(π‘§π‘Œπ‘—π‘˜) If π‘§πœ π‘˜ β‰  π‘£πœ π‘˜ then output(π‘§π‘Œπ‘—π‘˜) 𝑧 = π‘§π‘Œπ‘—1 ∘ π‘§π‘Œπ‘—2 ∘ β‹― ∘ π‘§π‘Œπ‘—π‘™β€² ∘ 𝑧 ΰ΄€

π‘Œ

ΰ·¨ 𝑃 𝑙 queries ΰ·¨ 𝑃 1 queries Chen, Liu, Servedio, Sheng, Xie 2018

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Results in the distribution-free model

Result ΰ·© 𝑷/ΰ·© 𝛁 Lo./Up. Ada./NonAda Reference 2𝑙/πœ— Upper NonAdaptive Halevy et al. APPROX 03 2𝑙/3 Lower NonAdaptive Chen et al. STOC 2018 𝑙/πœ— Upper Adaptive Ours CCC 2019 𝑙 Lower Adaptive Saglam FOCS 2018

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Result ΰ·© 𝑷/ΰ·© 𝛁 Lo./Up. Ada./NonAda Reference 2𝑙/πœ— Upper NonAdaptive Halevy et al. APPROX 03 2𝑙/3 Lower NonAdaptive Chen et al. STOC 2018 𝑙/πœ— Upper Adaptive Ours CCC 2019 𝑙 Lower Adaptive Saglam FOCS 2018

Open Problems

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Open Problems

Result ΰ·© 𝑷/ΰ·© 𝛁 Lo./Up. Ada./NonAda Reference 2𝑙/πœ— Upper NonAdaptive Halevy et al. APPROX 03 2𝑙/3 Lower NonAdaptive Chen et al. STOC 2018 ? 𝑃(1)-Round 𝑙/πœ— Upper Adaptive Ours CCC 2019 𝑙 Lower Adaptive Saglam FOCS 2018

Almost non-Adaptive

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Open Problems

Result ΰ·© 𝑷/ΰ·© 𝛁 Lo./Up. Ada./NonAda Reference 2𝑙/πœ— Upper NonAdaptive Halevy et al. APPROX 03 2𝑙/3 Lower NonAdaptive Chen et al. STOC 2018 ? 𝑃(1)-Round poly(𝑙/πœ—) ?-Round 𝑙/πœ— Upper Adaptive Ours CCC 2019 𝑙 Lower Adaptive Saglam FOCS 2018

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