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

Almost Optimal Distribution-free Junta Testing Nader H. Bshouty Technion

𝑙 -Junta A Boolean function 𝑔: 0,1 𝑜 → {0,1} is called 𝑙 − junta if it depends on at most 𝑙 variables/coordinates. Examples: 𝑔 𝑦 1 , 𝑦 2 , … , 𝑦 10000 = 𝑦 1111 ∧ 𝑦 9992 ⊕ 𝑦 3001 is 3 - junta Relevant variables/coordinates is 4-junta 1-junta is {𝑦 𝑗 , ഥ 𝑦 𝑗 , 0,1} 0-junta is {0,1} 𝑙 − Junta is the class of all 𝑙 − juntas.

Model of Testing Given a black box that contains a Boolean function 𝑔: 0,1 𝑜 → {0,1} Given two oracles: 1) when 𝑦 ∈ 0,1 𝑜 is queried, it returns 𝑔(𝑦) and 2) returns 𝑣 ∈ 0,1 𝑜 chosen acc. to unknown distribution 𝒠 A distribution-free testing algorithm 𝐵 for 𝑙 − Junta is an algorithm that, given an access to the two oracles and a distance parameter 𝜗 as an input , 1) if 𝑔 is 𝑙 -junta then 𝐵 outputs “ accept ” with probability at least 2/3. 2) if 𝑔 is 𝜗 -far from every 𝑙 -junta with respect to the distribution 𝒠 then 𝐵 outputs “ reject ” with probability at least 2/3. ∀ ℎ ∈ 𝑙 − 𝐾𝑣𝑜𝑢𝑏 𝑦~𝒠 𝑔 𝑦 ≠ ℎ 𝑦 Pr ≥ 𝜗

Model of Testing 𝑔 𝑙 -Junta 𝑙 -Junta 𝐵 outputs “ accept ” with probability at least 2/3.

Model of Testing 𝑔 𝑙 -Junta 𝑙 -Junta 𝐵 outputs “ reject ” with probability at least 2/3.

Model of Testing 𝑙 -Junta 𝑙 -Junta 𝑔 𝐵 halts and outputs either “ accept ” or “ reject ” .

Results in the uniform distribution Model Result ෩ 𝑷/෩ 𝛁 Reference Ada./NonAda Lo./Up. Lower bounds Adaptive Upper bounds NonAdaptive For the number of queries ෨ 𝑃 𝑈 = 𝑈 𝑞𝑝𝑚𝑧(log 𝑈) ෩ Ω 𝑈 = 𝑈 /𝑞𝑝𝑚𝑧(log 𝑈) 1 Time 𝑞𝑝𝑚𝑧 𝑜, 𝜗

Results in the uniform distribution Model Result ෩ 𝑷/෩ 𝛁 Reference Ada./NonAda Lo./Up. 𝑙/𝜗 Blais STOC 2009 Adaptive Upper

Results in the uniform distribution Model Result ෩ 𝑷/෩ 𝛁 Reference Ada./NonAda Lo./Up. 𝑙/𝜗 Blais STOC 2009 Adaptive Upper 𝑙 Saglam FOCS 2018 Adaptive Lower

Results in the uniform distribution Model Result ෩ 𝑷/෩ 𝛁 Reference Ada./NonAda Lo./Up. 𝑙/𝜗 Blais STOC 2009 Adaptive Upper 𝑙 Saglam FOCS 2018 Adaptive Lower 3 Blais APPROX 2008 NonAdaptive Upper 𝑙 2 /𝜗

Results in the uniform distribution Model Result ෩ 𝑷/෩ 𝛁 Reference Ada./NonAda Lo./Up. 𝑙/𝜗 Blais STOC 2009 Adaptive Upper 𝑙 Saglam FOCS 2018 Adaptive Lower 3 Blais APPROX 2008 NonAdaptive Upper 𝑙 2 /𝜗 3 Chen et al. CCC 2017 NonAdaptive Lower 𝑙 2 /𝜗

Results in the distribution-free model Result ෩ 𝑷/෩ 𝛁 Reference Ada./NonAda Lo./Up.

Results in the distribution-free model Result ෩ 𝑷/෩ 𝛁 Reference Ada./NonAda Lo./Up. 2 𝑙 /𝜗 Halevy et al. APPROX 03 NonAdaptive Upper

Results in the distribution-free model Result ෩ 𝑷/෩ 𝛁 Reference Ada./NonAda Lo./Up. 2 𝑙 /𝜗 Halevy et al. APPROX 03 NonAdaptive Upper 2 𝑙/3 Chen et al. STOC 2018 NonAdaptive Lower

Results in the distribution-free model Result ෩ 𝑷/෩ 𝛁 Reference Ada./NonAda Lo./Up. 2 𝑙 /𝜗 Halevy et al. APPROX 03 NonAdaptive Upper 2 𝑙/3 Chen et al. STOC 2018 NonAdaptive Lower 𝑙 2 /𝜗 Chen et al. STOC 2018 Adaptive Upper

Results in the distribution-free model Result ෩ 𝑷/෩ 𝛁 Reference Ada./NonAda Lo./Up. 2 𝑙 /𝜗 Halevy et al. APPROX 03 NonAdaptive Upper 2 𝑙/3 Chen et al. STOC 2018 NonAdaptive Lower 𝑙 2 /𝜗 Chen et al. STOC 2018 Adaptive Upper 𝑙 Saglam FOCS 2018 Adaptive Lower

Results in the distribution-free model Result ෩ 𝑷/෩ 𝛁 Reference Ada./NonAda Lo./Up. 2 𝑙 /𝜗 Halevy et al. APPROX 03 NonAdaptive Upper 2 𝑙/3 Chen et al. STOC 2018 NonAdaptive Lower 𝑙 2 /𝜗 Chen et al. STOC 2018 Adaptive Upper 𝑙 Saglam FOCS 2018 Adaptive Lower 𝑙/𝜗 Ours Adaptive Upper

The algorithm 𝑔(𝑦 1 , … , 𝑦 𝑜 ) Choose a random uniform partition 𝑌 1 , 𝑌 2 , … , 𝑌 𝑠 of 𝑜 where 𝑠 = 2𝑙 2 𝑔(𝑦 𝑌 1 ∘ 𝑦 𝑌 2 ∘ ⋯ ∘ 𝑦 𝑌 𝑠 ) Why 2𝑙 2 ? If 𝑔 is k − junta, whp each 𝑌 𝑗 contains at most one relevant coordinate Find relevant sets Find relevant sets 𝑌 𝑗 1 , 𝑌 𝑗 2 , … , 𝑌 𝑗 𝑙′ 𝑔 𝑤 ≠ 𝑔 𝑤 𝑌 𝑗 ∘ 0 𝑌 𝑗 𝑌 𝑗 1 , 𝑌 𝑗 2 , … , 𝑌 𝑗 𝑘 𝑌 = 𝑌 𝑗 1 ∪ 𝑌 𝑗 2 ∪ ⋯ ∪ 𝑌 𝑗 𝑘 𝑔 𝑣 𝑌 ∘ 0 ത 𝑌 ? = 𝑔(𝑣) 𝑣~𝒠

Find relevant sets Find relevant sets 𝑌 𝑗 1 , 𝑌 𝑗 2 , … , 𝑌 𝑗 𝑙′ 𝑌 𝑗 1 , 𝑌 𝑗 2 , … , 𝑌 𝑗 𝑘 𝑌 = 𝑌 𝑗 1 ∪ 𝑌 𝑗 2 ∪ ⋯ ∪ 𝑌 𝑗 𝑘 𝑔 𝑣 𝑌 ∘ 0 ത 𝑌 ? = 𝑔(𝑣) 𝑣~𝒠 𝑔 𝑣 𝑌 ∘ 0 ത 𝑌 ≠ 𝑔(𝑣) Find a new relevant set 𝑌 𝑗 𝑘+1 = 𝑌 ℓ 𝑔 𝑣 𝑌 ∘ 𝑣 𝑍 1 ∘ 0 𝑍 2 ത 𝑌 = 𝑍 1 ∪ 𝑍 2

Find relevant sets Find relevant sets 𝑌 𝑗 1 , 𝑌 𝑗 2 , … , 𝑌 𝑗 𝑙′ 𝑌 𝑗 1 , 𝑌 𝑗 2 , … , 𝑌 𝑗 𝑘 𝑌 = 𝑌 𝑗 1 ∪ 𝑌 𝑗 2 ∪ ⋯ ∪ 𝑌 𝑗 𝑘 𝑔 𝑣 𝑌 ∘ 0 ത 𝑌 ? = 𝑔(𝑣) 𝑣~𝒠 𝑔 𝑣 𝑌 ∘ 0 ത 𝑌 ≠ 𝑔(𝑣) log 𝑠 = 𝑃 log 𝑙 Find a new relevant sets 𝑌 𝑗 𝑘+1 = 𝑌 ℓ If this is the (𝑙 + 1) -th relevant set then “ reject ’’ We also get a witness for 𝑌 ℓ ℓ ∘ 0 𝑌 ℓ 𝑔 𝑤 (ℓ) ≠ 𝑔 𝑤 𝑌 ℓ 1 For ෨ 𝑔 𝑣 𝑌 ∘ 0 ത 𝑌 = 𝑔(𝑣) 𝑃 𝜗 values of 𝑣~𝒠 𝑙 ෨ 𝑃 𝜗 queries 𝑦~𝒠 𝑔 𝑦 𝑌 ∘ 0 ത Pr 𝑌 ≠ 𝑔 𝑦 ≤ 𝜗/3

The algorithm 𝑌 𝑗 1 , 𝑌 𝑗 2 , … , 𝑌 𝑗 𝑙′ , 𝑙 ′ ≤ 𝑙 Each 𝑌 𝑗 𝑘 contains at least one relevant variable ≤ 𝜗 𝑌 = 𝑌 𝑗 1 ∪ 𝑌 𝑗 2 ∪ ⋯ ∪ 𝑌 𝑗 𝑙′ 𝑦~𝒠 𝑔 𝑦 𝑌 ∘ 0 ത Pr 𝑌 ≠ 𝑔 𝑦 3 𝜗 ℎ ≔ 𝑔(𝑦 𝑌 ∘ 0 ത 𝑌 ) is 3 − close to 𝑔 with respect to 𝒠 𝑔 𝑔 ℎ: = 𝑔(𝑦 𝑌 ∘ 0 ത 𝑌 ) 𝑙 -Junta 𝑙 -Junta ℎ: = 𝑔(𝑦 𝑌 ∘ 0 ത 𝑌 )

𝑌 𝑗 1 , 𝑌 𝑗 2 , … , 𝑌 𝑗 𝑙′ , 𝑙 ′ ≤ 𝑙 Each 𝑌 𝑗 𝑘 contains at least one relevant coordinte 𝑌 = 𝑌 𝑗 1 ∪ 𝑌 𝑗 2 ∪ ⋯ ∪ 𝑌 𝑗 𝑙′ 𝑔(𝑦) is 𝜗 − far from every 𝑙 − Junta with 𝑔 is 𝑙 − junta 𝑙 ෨ 𝑃 𝜗 queries respect to 𝒠 ℎ: = 𝑔(𝑦 𝑌 ∘ 0 ത 𝑌 ) is 𝑙 − junta 2𝜗 ℎ: = 𝑔(𝑦 𝑌 ∘ 0 ത 𝑌 ) is 3 − far Whp each 𝑌 𝑗 𝑘 contains from every 𝑙 − Junta with exactly one relevant coordinate respect to 𝒠 We also have witness for each relevant set 𝑌 𝑗 𝑘 – that is, 𝑘 ∘ 0 𝑌 𝑗𝑘 𝑔 𝑤 𝑘 ≠ 𝑔 𝑤 𝑌 𝑗𝑘

𝑔(𝑦) is 𝜗 − far from every 𝑙 − Junta with 𝑔 is 𝑙 − junta respect to 𝒠 ℎ: = 𝑔(𝑦 𝑌 ∘ 0 ത 𝑌 ) is 𝑙 − junta 2𝜗 ℎ: = 𝑔(𝑦 𝑌 ∘ 0 ത 𝑌 ) is 3 − far Whp each 𝑌 𝑗 𝑘 contains from every 𝑙 − Junta with exactly one relevant variable respect to 𝒠 𝑘 ∘ 0 𝑌 𝑗𝑘 𝑔 𝑤 𝑘 ≠ 𝑔 𝑤 𝑌 𝑗𝑘 𝑘 ∘ 𝑦 𝑌 𝑗𝑘 ෨ 𝑔 𝑤 𝑌 𝑗𝑘 is equal to 𝑦 𝜐 𝑘 or 𝑦 𝜐 𝑘 𝑃 𝑙 queries 𝑃(log 𝑜) 1 − Junta\ 0 − Junta 𝑘 ∘ 𝑦 𝑌 𝑗𝑘 𝑔 𝑤 𝑌 𝑗𝑘 is (1/15)-close to a literal in {𝑦 𝜐 𝑘 , 𝑦 𝜐 𝑘 } ෨ 𝑃 1 queries according to the uniform distribution

Γ = {𝜐(1), 𝜐(2), … , 𝜐(𝑙 ′ )} ℎ: = 𝑔(𝑦 𝑌 ∘ 0 ത 𝑌 ) is either 𝑙 − junta that depends on 2𝜗 𝑌 𝑗 1 , 𝑌 𝑗 2 , … , 𝑌 𝑗 𝑙 ′ 3 − far from every 𝑙 − junta w.r.t. 𝒠 Or Fix any 𝑧 ∈ 0,1 𝑜 𝑕 = ℎ(𝑦 Γ ∘ 𝑧 ഥ Γ ) is 𝑙 -junta 2𝜗 ℎ(𝑦) is 3 − far from any 𝑙 -junta with respect to 𝒠 ℎ(𝑦) is 𝑙 − junta ≥ 2𝜗 ℎ 𝑦 = ℎ(𝑦 Γ ∘ 𝑧 ഥ Γ ) 𝑣~𝒠 ℎ 𝑣 ≠ ℎ 𝑣 Γ ∘ 𝑧 ഥ Pr Γ 3 ≥ 2𝜗 𝑣~𝒠,𝑧~𝑉 ℎ 𝑣 ≠ ℎ 𝑧 |𝑧 Γ = 𝑣 Γ = 0 Pr 𝑣~𝒠,𝑧~𝑉 ℎ 𝑣 ≠ ℎ 𝑣 Γ ∘ 𝑧 ഥ Pr Γ 3 𝑙 ෨ 𝑣~𝒠,𝑧~𝑉 ℎ 𝑣 ≠ ℎ 𝑧 |𝑧 Γ = 𝑣 Γ ≥ 2𝜗 𝑃 𝜗 queries 1 Pr ෨ 𝑃 𝜗 queries 3 Given 𝑣 ? ෨ 𝑃 𝑙 queries How to draw a random uniform 𝑧 such that 𝑧 Γ = 𝑣 Γ ?

Γ = {𝜐 1 , 𝜐 2 , … , 𝜐(𝑙 ′ )} ℎ: = 𝑔(𝑦 𝑌 ∘ 0 ത 𝑌 ) Given 𝑣 ? How to draw a random uniform 𝑧 such that 𝑧 Γ = 𝑣 Γ ? 𝑘 ∘ 𝑦 𝑌 𝑗𝑘 𝑔 𝑤 𝑌 𝑗𝑘 is (1/15)-close to a literal in {𝑦 𝜐 𝑘 , 𝑦 𝜐 𝑘 } wrt uniform dist. → A procedure that given 𝑣 finds 𝑣 𝜐 𝑘 with high probability ෨ 𝑃 1 queries 𝑧 = 𝑧 𝑌 𝑗1 ∘ 𝑧 𝑌 𝑗2 ∘ ⋯ ∘ 𝑧 𝑌 𝑗𝑙′ ∘ 𝑧 ത 𝑌 Draw a random uniform 𝑧 𝑌 𝑗 𝑘 ෨ 𝑃 𝑙 queries If 𝑧 𝜐 𝑘 = 𝑣 𝜐 𝑘 then output( 𝑧 𝑌 𝑗𝑘 ) Chen, Liu, Servedio, Sheng, Xie 2018 If 𝑧 𝜐 𝑘 ≠ 𝑣 𝜐 𝑘 then output( 𝑧 𝑌 𝑗𝑘 )