Streaming -submodular Maximization under Noise subject to Size - PowerPoint PPT Presentation

Streaming 𝒍 -submodular Maximization under Noise subject to Size Constraint Lan N. Nguyen, My T. Thai University of Florida

𝒍 -submodular maximization s.t. size constraint ➢ 𝑙 -submodular function is a generalization of submodular function ❑ Submodular set function: input is a single subset 𝑊 𝑔 𝑌 + 𝑔 𝑍 ≥ 𝑔 𝑌 ∪ 𝑍 + 𝑔(𝑌 ∩ 𝑍) ❑ 𝑙 -submodular function: input is 𝑙 disjoint subsets of 𝑊 𝑔 𝐲 + 𝑔 𝐳 ≥ 𝑔 𝐲 ⊔ 𝐳 + 𝑔(𝐲 ⊓ 𝐳) ▪ 𝐲 = (𝑌 1 , … , 𝑌 𝑙 ) and 𝐳 = (𝑍 1 , … , 𝑍 𝑙 ) ▪ 𝐲 ⊔ 𝐳 = (𝑎 1 , … , 𝑎 𝑙 ) where 𝑎 𝑗 = 𝑌 𝑗 ∪ 𝑍 𝑗 ∖ (ڂ 𝑘≠𝑗 𝑌 𝑘 ∪ 𝑍 𝑘 ) ▪ 𝐲 ⊓ 𝐳 = (𝑌 1 ∩ 𝑍 1 , … , 𝑌 𝑙 ∩ 𝑍 𝑙 ) ➢ 𝑙 -submodular maximization s.t. size constraint ( M 𝒍 SC ) ❑ 𝑊 – a finite set of elements, 𝐶 – a positive integer. Find 𝐭 = (𝑇 1 , … , 𝑇 𝑙 ) s.t. 𝑙 + 1 𝑊 - a family of 𝑙 disjoint subsets of 𝑊 ❑ 𝐭 =ڂ 𝑗≤𝑙 𝑇 𝑗 ≤ 𝐶 that ❑ 𝑔: 𝑙 + 1 𝑊 → ℝ + - a 𝑙 -submodular function. maximizes 𝑔(𝐭)

𝒍 -submodular maximization s.t. size constraint ➢ Applications: ❑ Influence maximization with 𝑙 topics/products ❑ Sensor placement with 𝑙 kinds of sensors ❑ Coupled Feature Selection. ➢ Existing solutions (*) ❑ Greedy: 2 approximation ratio, 𝑃(𝑙𝑜𝐶) query complexity 𝐶 ❑ “Lazy” Greedy: 2 approximation ratio, 𝑃(𝑙 𝑜 − 𝐶 log 𝐶 log 𝜀 ) query complexity with probability at least 1 − 𝜀 (*) Ohsaka, Naoto, and Yuichi Yoshida. "Monotone k-submodular function maximization with size constraints." Advances in Neural Information Processing Systems . 2015.

Practical Challenges ➢ Noisy evaluation. ❑ In many applications (e.g. Influence Maximization), obtaining exact value for 𝑔(𝐭) is impractical. ❑ 𝑔 can only be queried through a noisy version 𝐺 1 − 𝜗 𝑔 𝐭 ≤ 𝐺 𝐭 ≤ 1 + 𝜗 𝑔(𝐭) for all 𝐭 ∈ 𝑙 + 1 𝑊 ➢ Streaming. ❑ Algorithms are required to take only one single pass over 𝑊 ▪ Produce solutions in a timely manner. ▪ Avoid excessive storage in memory.

Our contribution ➢ Two streaming algorithms for MkSC – DStream & RStream ❑ Take only 1 single scan over 𝑊 ❑ Access 𝐺 instead of 𝑔 ❑ Performance guarantee: ▪ Approximation ratio 𝑔 𝐭 /𝑔(𝐩) : 𝐩 - optimal solution. ▪ Query and memory complexity ➢ Experimental Evaluation ❑ Influence maximization with 𝑙 topics. ❑ Sensor placement with 𝑙 kinds of sensor.

DStream ➢ Obtain 𝑝 such that 𝑔 𝑝 ≥ 𝑝 × 𝐶 ≥ 𝑔 𝑝 /(1 + 𝛿) ❑ Using lazy estimation (*) ➢ For a new element 𝑓 , if 𝒕 < 𝐶 𝑇 1 Find max 𝑗≤𝑙 𝐺(𝐭 ⊔ (𝑓, 𝑗)) 𝑇 2 𝑓 Disjoint subsets obtained by putting 𝑓 into 𝑇 𝑗 𝑇 3 (*) Badanidiyuru, Ashwinkumar, et al. "Streaming submodular maximization: Massive data summarization on the fly." Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining . 2014.

DStream ➢ Obtain 𝑝 such that 𝑔 𝑝 ≥ 𝑝 × 𝐶 ≥ 𝑔 𝑝 /(1 + 𝛿) ❑ Using lazy estimation (*) ➢ For a new element 𝑓 , if 𝒕 < 𝐶 𝑇 1 𝐺 𝐭⊔ 𝑓,𝑗 𝑝 𝑇 2 Putting 𝑓 to 𝑇 𝑗 if ≥ 𝐭 + 1 1−𝜗 𝑁 𝑓 𝑇 3 (*) Badanidiyuru, Ashwinkumar, et al. "Streaming submodular maximization: Massive data summarization on the fly." Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining . 2014.

DStream ➢ Obtain 𝑝 such that 𝑔 𝑝 ≥ 𝑝 × 𝐶 ≥ 𝑔 𝑝 /(1 + 𝛿) ❑ Using lazy estimation (*) ➢ For a new element 𝑓 , if 𝒕 < 𝐶 𝑇 1 𝐺 𝐭⊔ 𝑓,𝑗 𝑝 𝑇 2 Putting 𝑓 to 𝑇 𝑗 if ≥ 𝐭 + 1 1−𝜗 𝑁 𝑓 Largest possible value of 𝑔 𝐭 ⊔ 𝑓, 𝑗 𝑇 3 (*) Badanidiyuru, Ashwinkumar, et al. "Streaming submodular maximization: Massive data summarization on the fly." Proceedings of the 20th ACM SIGKDD international conference on Knowledge discovery and data mining . 2014.

DStream’s performance guarantee ➢ 𝐲 = (𝑌 1 , … , 𝑌 𝑙 ) can also be understood as a vector 𝐲: 𝑊 → [𝑙] 𝑓 1 𝑓 2 𝑓 3 … … 𝑓 … … … … 𝑘 x 1 0 4 … … 𝑗 … … … … 𝐲 𝑓 = ቊ 𝑗 if 𝑓 ∈ 𝑌 𝑗 0 if 𝑓 ∉ڂ 𝑗 𝑌 𝑗

DStream’s performance guarantee ➢ 𝐭 0 , 𝐭 1 , … , 𝐭 𝑢 - sequence of obtained solutions ❑ 𝐭 𝑗 - obtained solution after adding 𝑗 elements ( 𝒕 𝑗 = 𝑗 ) ➢ Construct a sequence 𝐩 0 , 𝐩 1 , … , 𝐩 𝑢 𝐩 𝑗 = (𝐩 ⊔ 𝐭 𝑗 ) ⊔ 𝐭 𝑗 𝐭 𝑗 1 0 2 3 0 0 0 0 𝐩 2 1 2 0 0 3 0 1 𝐩 𝑗 1 1 2 3 0 3 0 1

DStream’s performance guarantee 𝐩 0 𝐭 0 ➢ If in the end 𝐭 = 𝐶 𝐩 2 𝐭 1 𝑔 𝐭 ≥ 1 − 𝜗 𝑔(𝐩) 𝐩 2 𝐭 2 1 + 𝜗 1 + 𝛿 𝑁 𝐩 3 𝐭 3 𝐩 4 𝐭 4

DStream’s performance guarantee ➢ If in the end 𝐭 = 𝑢 < 𝐶 , with 𝑔 is monotone . ❑ Establish recursive relationship between 𝑝 𝑘 , 𝑡 𝑘 𝑔 𝐩 𝑘−1 + 𝑔 𝐭 𝑘−1 ≤ 𝑔 𝐩 𝑘 + 1 + 𝜗 𝐩 0 𝐭 0 1 − 𝜗 𝑔(𝐭 𝑘 ) 𝐩 2 ❑ Bound 𝑔 𝐩 − 𝑔(𝐩 𝑢 ) (∗) 𝐭 1 𝑔 𝐩 − 𝑔 𝐩 𝑢 ≤ 1 + 𝜗 + 2𝐶𝜗 𝑔(𝐭) 𝐩 2 𝐭 2 1 − 𝜗 ❑ Bound 𝑔 𝐩 𝑢 − 𝑔(𝐭) (∗∗) 𝐩 3 𝐭 3 𝑔 𝐩 𝑢 − 𝑔 𝐭 ≤ 1 𝑁 𝑔 𝐩 + 2𝐶𝜗 1 − 𝜗 𝑔(𝐭) ❑ Discard 𝑔(𝐩 𝑢 ) by combining ∗ and (∗∗) 𝑁 2 + 4𝐶𝜗 𝑔 𝐩 ≤ 𝑔(𝐭) 𝑁 − 1 1 − 𝜗

DStream’s performance guarantee ➢ If in the end 𝐭 = 𝑢 < 𝐶 , with 𝑔 is non-monotone . 𝐩 0 𝐭 0 ❑ 𝑔 is pairwise monotone 𝐩 2 Δ 𝑓,𝑗 𝑔 𝐲 + Δ 𝑓,𝑘 𝑔 𝐲 ≥ 0 𝐭 1 ❑ Using the same framework as the 𝐩 2 𝐭 2 monotone case but with different “math” 𝐩 3 𝐭 3 𝑁 (1 + 𝜗)(3 + 3𝜗 + 6𝐶𝜗) 𝑔 𝐩 ≥ 𝑔(𝐭) 1 − 𝜗 2 𝑁 − 1

DStream Lazy estimation to obtain 𝑝 • 𝑔 𝐩 ∈ [Δ 𝑚 , 𝐶 × Δ 𝑣 ] • 𝑝 can be obtained by a value of 1 + 𝛿 𝑘 ∈ Δ 𝑚 [ 𝐶 , 𝑁(1 + 𝜗)Δ 𝑣 ] Query complexity 𝑃(𝑜𝑙 𝛿 log( 1 + 𝜗 (1 + 𝛿) 𝐶𝑁)) 1 − 𝜗 Memory complexity 𝑃(𝐶 𝛿 log( 1 + 𝜗 (1 + 𝛿) 𝐶𝑁)) 1 − 𝜗

DStream Approximation ratio 1 + 𝜗 1 − 𝜗 min 𝑦∈(1,𝑁] max(𝑏 𝑦 , 𝑐(𝑦)) If 𝑔 is monotone (1+𝛿)(1+𝜗) • 𝑏 𝑦 = 𝑦 1−𝜗 2+4𝐶𝜗 𝑦 • 𝑐 𝑦 = 1−𝜗 𝑦−1 If 𝑔 is non-monotone (1+𝛿)(1+𝜗) • 𝑏 𝑦 = 𝑦 1−𝜗 (1+𝜗)(3+3𝜗+6𝐶𝜗) 𝑦 • 𝑐 𝑦 = 1−𝜗 2 𝑦−1

DStream’s weakness 𝐺 𝐭⊔ 𝑓,𝑗 𝑝 Putting 𝑓 to 𝑇 𝑗 if ≥ 𝐭 + 1 𝑇 2 1−𝜗 𝑁 𝑓 𝑝 What if 𝑔 𝐭 ≥ 𝐭 + 1 𝑁 ? 𝑇 3 • 𝑓 may have no contribution to 𝐭 • Better consider marginal gain

RStream ➢ For a new element 𝑓 , if 𝒕 < 𝐶 𝑒 𝑗 = 𝐺(𝐭 ⊔ (𝑓, 𝑗)) − 𝐺(𝐭) 𝑇 1 1 − 𝜗 1 + 𝜗 • 𝑒 𝑗 is an upper bound on Δ 𝑓,𝑗 𝑔(𝐭) 𝑇 2 𝑓 𝑇 3

RStream ➢ For a new element 𝑓 , if 𝒕 < 𝐶 𝑒 𝑗 = 𝐺(𝐭 ⊔ (𝑓, 𝑗)) − 𝐺(𝐭) 𝑇 1 1 − 𝜗 1 + 𝜗 • 𝑒 𝑗 is an upper bound on Δ 𝑓,𝑗 𝑔(𝐭) 𝑇 2 𝑓 𝑝 • Filter out 𝑇 𝑗 that 𝑒 𝑗 ≤ 𝑁 𝑝 • 𝑒 𝑗 = 0 if 𝑒 𝑗 ≤ 𝑁 • Otherwise 𝑒 𝑗 keeps its value 𝑇 3 • Randomly put 𝑓 into 𝑇 𝑗 with probability 𝑈−1 / ෍ 𝑈−1 𝑒 𝑗 𝑒 𝑘 𝑘 𝑝 • 𝑈 = | 𝑘 ∶ 𝑒 𝑘 ≥ | 𝑁

RStream 𝑒 𝑗 = 𝐺(𝐭 ⊔ (𝑓, 𝑗)) − 𝐺(𝐭) 𝑇 1 1 − 𝜗 1 + 𝜗 • 𝑒 𝑗 is an upper bound on Δ 𝑓,𝑗 𝑔(𝐭) 𝑇 2 𝑓 What if 𝐺 𝐭 ≈ 𝑔 𝐭 = 𝑔(𝐭 ⊔ (𝑓, 𝑗)) ≈ 𝐺(𝐭 ⊔ (𝑓, 𝑗)) • 𝑓 has no contribution 𝑇 3 2𝜗 𝑝 • But 𝑒 𝑗 ≈ 1−𝜗 2 𝑔 𝐭 ≥ 𝑁

RStream 𝑒 𝑗 = 𝐺(𝐭 ⊔ (𝑓, 𝑗)) − 𝐺(𝐭) 𝑇 1 1 − 𝜗 1 + 𝜗 • 𝑒 𝑗 is an upper bound on Δ 𝑓,𝑗 𝑔(𝐭) 𝑇 2 𝑓 (Denoise) Run multiple instances, each instance assumes 𝐺 is less noisy than it is. 𝒆 𝒋,𝝑 ′ = 𝐺(𝐭 ⊔ (𝑓, 𝑗)) − 𝐺(𝐭) 𝑇 3 1 − 𝝑 ′ 1 + 𝝑 ′ where 𝜗 ′ = 0, 𝜗 2𝜗 𝜃−1 , 𝜃−1 , … , 𝜗 𝜃 – adjustable parameter, controlling number of instances

(Denoise) Run multiple instances, each instance assumes 𝐺 is less noisy than it actually is.

Lazy estimation: Δ 𝑣 is much larger than the one in DStream in order to bound 𝑒 𝑗 s’ value. Query complexity 𝛿 log(( 1 + 𝜗 2 + 4𝐶𝜗)(1 + 𝛿) 𝑃(𝑜𝑙𝜃 𝐶𝑁)) 1 − 𝜗 2 Memory complexity 𝛿 log(( 1 + 𝜗 2 + 4𝐶𝜗)(1 + 𝛿) 𝑃(𝜃𝐶 𝐶𝑁)) 1 − 𝜗 2