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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 = (𝑌1, … , 𝑌𝑙) and 𝐳 = (𝑍

1, … , 𝑍 𝑙)

▪ 𝐲 ⊔ 𝐳 = (𝑎1, … , 𝑎𝑙) where 𝑎𝑗 = 𝑌𝑗 ∪ 𝑍

𝑗 ∖ (ڂ𝑘≠𝑗 𝑌 𝑘 ∪ 𝑍 𝑘)

▪ 𝐲 ⊓ 𝐳 = (𝑌1 ∩ 𝑍

1, … , 𝑌𝑙 ∩ 𝑍 𝑙)

➢ 𝑙-submodular maximization s.t. size constraint (M𝒍SC) ❑ 𝑊 – a finite set of elements, 𝐶 – a positive integer. ❑ 𝑙 + 1 𝑊 - a family of 𝑙 disjoint subsets of 𝑊 ❑ 𝑔: 𝑙 + 1 𝑊 → ℝ+ - a 𝑙-submodular function.

Find 𝐭 = (𝑇1, … , 𝑇𝑙) s.t. 𝐭 =ڂ𝑗≤𝑙 𝑇𝑗 ≤ 𝐶 that 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 ❑ “Lazy” Greedy: 2 approximation ratio, 𝑃(𝑙 𝑜 − 𝐶 log 𝐶 log

𝐶 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

(*) 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.

𝑓 𝑇1 𝑇2 𝑇3

Find max

𝑗≤𝑙 𝐺(𝐭 ⊔ (𝑓, 𝑗))

Disjoint subsets obtained by putting 𝑓 into 𝑇𝑗

DStream

➢ Obtain 𝑝 such that 𝑔 𝑝 ≥ 𝑝 × 𝐶 ≥ 𝑔 𝑝 /(1 + 𝛿)

❑ Using lazy estimation (*)

➢ For a new element 𝑓, if 𝒕 < 𝐶

(*) 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.

𝑓 𝑇1 𝑇2 𝑇3

Putting 𝑓 to 𝑇𝑗 if

𝐺 𝐭⊔ 𝑓,𝑗 1−𝜗

≥ 𝐭 + 1

𝑝 𝑁

DStream

➢ Obtain 𝑝 such that 𝑔 𝑝 ≥ 𝑝 × 𝐶 ≥ 𝑔 𝑝 /(1 + 𝛿)

❑ Using lazy estimation (*)

➢ For a new element 𝑓, if 𝒕 < 𝐶

(*) 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.

𝑓 𝑇1 𝑇2 𝑇3

Putting 𝑓 to 𝑇𝑗 if

𝐺 𝐭⊔ 𝑓,𝑗 1−𝜗

≥ 𝐭 + 1

𝑝 𝑁

Largest possible value of 𝑔 𝐭 ⊔ 𝑓, 𝑗

DStream’s performance guarantee

➢ 𝐲 = (𝑌1, … , 𝑌𝑙) can also be understood as a vector 𝐲: 𝑊 → [𝑙]

𝑓1 𝑓2 𝑓3 … … 𝑓

𝑘

… … … … 1 4 … … 𝑗 … … … …

x 𝐲 𝑓 = ቊ 𝑗 if 𝑓 ∈ 𝑌𝑗 if 𝑓 ∉ڂ𝑗 𝑌𝑗

DStream’s performance guarantee

➢ 𝐭0, 𝐭1, … , 𝐭𝑢 - sequence of obtained solutions

❑ 𝐭𝑗 - obtained solution after adding 𝑗 elements ( 𝒕𝑗 = 𝑗)

➢ Construct a sequence 𝐩0, 𝐩1, … , 𝐩𝑢

𝐩𝑗 = (𝐩 ⊔ 𝐭𝑗) ⊔ 𝐭𝑗

1 2 3 2 1 2 3 1

𝐭𝑗 𝐩

1 1 2 3 3 1

𝐩𝑗

DStream’s performance guarantee

➢ If in the end 𝐭 = 𝐶 𝑔 𝐭 ≥ 1 − 𝜗 1 + 𝜗 𝑔(𝐩) 1 + 𝛿 𝑁

𝐭0 𝐩0 𝐭1 𝐩2 𝐭2 𝐩2 𝐭3 𝐩3 𝐭4 𝐩4

DStream’s performance guarantee

➢ If in the end 𝐭 = 𝑢 < 𝐶, with 𝑔 is monotone. ❑ Establish recursive relationship between 𝑝𝑘, 𝑡𝑘 𝑔 𝐩𝑘−1 + 𝑔 𝐭𝑘−1 ≤ 𝑔 𝐩𝑘 + 1 + 𝜗 1 − 𝜗 𝑔(𝐭𝑘) ❑ Bound 𝑔 𝐩 − 𝑔(𝐩𝑢) (∗) 𝑔 𝐩 − 𝑔 𝐩𝑢 ≤ 1 + 𝜗 + 2𝐶𝜗 1 − 𝜗 𝑔(𝐭) ❑ Bound 𝑔 𝐩𝑢 − 𝑔(𝐭) (∗∗) 𝑔 𝐩𝑢 − 𝑔 𝐭 ≤ 1 𝑁 𝑔 𝐩 + 2𝐶𝜗 1 − 𝜗 𝑔(𝐭) ❑ Discard 𝑔(𝐩𝑢) by combining ∗ and (∗∗) 𝑔 𝐩 ≤ 𝑁 𝑁 − 1 2 + 4𝐶𝜗 1 − 𝜗 𝑔(𝐭) 𝐭0 𝐩0 𝐭1 𝐩2 𝐭2 𝐩2 𝐭3 𝐩3

DStream’s performance guarantee

➢ If in the end 𝐭 = 𝑢 < 𝐶, with 𝑔 is non-monotone.

❑ 𝑔 is pairwise monotone Δ𝑓,𝑗𝑔 𝐲 + Δ𝑓,𝑘𝑔 𝐲 ≥ 0 ❑ Using the same framework as the monotone case but with different “math” 𝑔 𝐩 ≥ 𝑁 𝑁 − 1 (1 + 𝜗)(3 + 3𝜗 + 6𝐶𝜗) 1 − 𝜗 2 𝑔(𝐭) 𝐭0 𝐩0 𝐭1 𝐩2 𝐭2 𝐩2 𝐭3 𝐩3

DStream

Lazy estimation to obtain 𝑝

• 𝑔 𝐩 ∈ [Δ𝑚, 𝐶 × Δ𝑣]
• 𝑝 can be obtained by a value of 1 + 𝛿 𝑘 ∈

[

Δ𝑚 𝐶 , 𝑁(1 + 𝜗)Δ𝑣]

Query complexity 𝑃(𝑜𝑙 𝛿 log( 1 + 𝜗 (1 + 𝛿) 1 − 𝜗 𝐶𝑁)) Memory complexity 𝑃(𝐶 𝛿 log( 1 + 𝜗 (1 + 𝛿) 1 − 𝜗 𝐶𝑁))

DStream

Approximation ratio

1 + 𝜗 1 − 𝜗 min

𝑦∈(1,𝑁] max(𝑏 𝑦 , 𝑐(𝑦))

If 𝑔 is monotone

• 𝑏 𝑦 =

(1+𝛿)(1+𝜗) 1−𝜗

𝑦

• 𝑐 𝑦 =

2+4𝐶𝜗 1−𝜗 𝑦 𝑦−1

If 𝑔 is non-monotone

• 𝑏 𝑦 =

(1+𝛿)(1+𝜗) 1−𝜗

𝑦

• 𝑐 𝑦 =

(1+𝜗)(3+3𝜗+6𝐶𝜗) 1−𝜗 2 𝑦 𝑦−1

DStream’s weakness

𝑓 𝑇2 𝑇3

Putting 𝑓 to 𝑇𝑗 if

𝐺 𝐭⊔ 𝑓,𝑗 1−𝜗

≥ 𝐭 + 1

𝑝 𝑁

What if 𝑔 𝐭 ≥ 𝐭 + 1

𝑝 𝑁 ?

• 𝑓 may have no contribution to 𝐭
• Better consider marginal gain

RStream

➢ For a new element 𝑓, if 𝒕 < 𝐶

𝑓 𝑇1 𝑇2 𝑇3 𝑒𝑗 = 𝐺(𝐭 ⊔ (𝑓, 𝑗)) 1 − 𝜗 − 𝐺(𝐭) 1 + 𝜗

• 𝑒𝑗 is an upper bound on Δ𝑓,𝑗𝑔(𝐭)

RStream

➢ For a new element 𝑓, if 𝒕 < 𝐶

𝑓 𝑇1 𝑇2 𝑇3 𝑒𝑗 = 𝐺(𝐭 ⊔ (𝑓, 𝑗)) 1 − 𝜗 − 𝐺(𝐭) 1 + 𝜗

• 𝑒𝑗 is an upper bound on Δ𝑓,𝑗𝑔(𝐭)
• Filter out 𝑇𝑗 that 𝑒𝑗 ≤

𝑝 𝑁

• 𝑒𝑗 = 0 if 𝑒𝑗 ≤

𝑝 𝑁

• Otherwise 𝑒𝑗 keeps its value
• Randomly put 𝑓 into 𝑇𝑗 with probability

𝑒𝑗

𝑈−1/ ෍ 𝑘

𝑒𝑘

𝑈−1

• 𝑈 = |

𝑘 ∶ 𝑒𝑘 ≥

𝑝 𝑁

|

RStream

𝑓 𝑇1 𝑇2 𝑇3

What if 𝐺 𝐭 ≈ 𝑔 𝐭 = 𝑔(𝐭 ⊔ (𝑓, 𝑗)) ≈ 𝐺(𝐭 ⊔ (𝑓, 𝑗))

𝑒𝑗 = 𝐺(𝐭 ⊔ (𝑓, 𝑗)) 1 − 𝜗 − 𝐺(𝐭) 1 + 𝜗

• 𝑒𝑗 is an upper bound on Δ𝑓,𝑗𝑔(𝐭)
• 𝑓 has no contribution
• But 𝑒𝑗 ≈

2𝜗 1−𝜗2 𝑔 𝐭 ≥ 𝑝 𝑁

RStream

𝑓 𝑇1 𝑇2 𝑇3 𝑒𝑗 = 𝐺(𝐭 ⊔ (𝑓, 𝑗)) 1 − 𝜗 − 𝐺(𝐭) 1 + 𝜗

• 𝑒𝑗 is an upper bound on Δ𝑓,𝑗𝑔(𝐭)

(Denoise) Run multiple instances, each instance assumes 𝐺 is less noisy than it is. 𝒆𝒋,𝝑′ = 𝐺(𝐭 ⊔ (𝑓, 𝑗)) 1 − 𝝑′ − 𝐺(𝐭) 1 + 𝝑′ where 𝜗′ = 0,

𝜗 𝜃−1 , 2𝜗 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: Δ𝑣 is much larger than the

• ne in DStream in order to bound 𝑒𝑗s’ value.

Query complexity 𝑃(𝑜𝑙𝜃 𝛿 log(( 1 + 𝜗 2 + 4𝐶𝜗)(1 + 𝛿) 1 − 𝜗 2 𝐶𝑁)) Memory complexity 𝑃(𝜃𝐶 𝛿 log(( 1 + 𝜗 2 + 4𝐶𝜗)(1 + 𝛿) 1 − 𝜗 2 𝐶𝑁))

Approximation ratio

1 + 𝜗 1 − 𝜗 min

𝑦∈(1,𝑁] max(𝑏 𝑦 , 𝑐(𝑦))

If 𝑔 is monotone

• 𝑏 𝑦 =

(1+𝛿)(1+𝜗+2𝐶𝜗) 1−𝜗

𝑦

• 𝑐 𝑦 = (

1+𝜗 2+4𝐶𝜗 1−𝜗2

1 −

1 𝑙 + 1) 𝑙𝑦 𝑙𝑦−𝑙−1

If 𝑔 is non-monotone

• 𝑏 𝑦 =

(1+𝛿)(1+𝜗+2𝐶𝜗) 1−𝜗

𝑦

• 𝑐 𝑦 =

3𝑙−2 1+𝜗 2+ 8𝑙−8 𝐶𝜗 1−𝜗 2 𝑦 𝑙𝑦−𝑙−2

Experimental Evaluation

➢ Influence Maximization with 𝑙 topics

❑ 𝑙 influence spread processes occur independently in a social network. ❑ Find 𝑇1, … , 𝑇𝑙 that maximize the number of active users

▪ An active user is a user who is activated by at least 1 topics. ▪ 𝑇𝑗 - a seed set of users who start spreading topic 𝑗 ▪ 𝑇1 ∪ ⋯ ∪ 𝑇𝑙 ≤ 𝐶

➢ Social network: Facebook dataset from SNAP

❑ Leskovec, Jure, and Rok Sosič. "Snap: A general-purpose network analysis and graph-mining library." ACM Transactions on Intelligent Systems and Technology (TIST) 8.1 (2016): 1-20.

➢ Influence model: Linear Threshold

❑ Kempe, David, Jon Kleinberg, and Éva Tardos. "Maximizing the spread of influence through a social network." Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining. 2003.

Influence Maximization with 𝒍 topics

➢ Compared algorithms

❑ Greedy (Ohsaka, Naoto, and Yuichi Yoshida et al. NIPS’15) ❑ IM: randomly select 1 topic and solve classical Influence Maximization problem ❑ SGr: simple streaming, pick 𝑓 with prob.

𝐶 𝑜 and put to 𝑇𝑗 that maximizes 𝐺(𝑡 ⊔ (𝑓, 𝑗))

Influence Maximization with 𝒍 topics

➢ DStream and RStream (𝜃 = 2)

❑ Returned solutions approximately to Greedy, outperformed IM in most cases. ❑ Outperformed Greedy in # queries by a huge margin.

Influence Maximization with 𝒍 topics

➢ Denoise step helped RStream improve performance.

❑ 𝜃 = 1 causes RStream terminate prematurely and perform worse than DStream ❑ 𝜃 = 2 helps RStream improve solution quality but take 4 times more queries than DStream.

Influence Maximization with 𝒍 topics

➢ The larger 𝜹 is, the lower solution quality and the fewer queries the

algorithms obtained.

➢ The smaller M is, the lower solution quality and the fewer queries

the algorithms obtained.

Conclusion

➢ We propose 2 streaming algorithms with theoretical performance guarantee to solve MkSC under noise. ➢ In comparison with Greedy, our algorithms

❑ Take much fewer queries ❑ Obtain comparable solutions in term of quality.

➢ Thanks! Questions?

❑ lan.nguyen@ufl.edu