Do Less, Get More: Streaming Submodular Maximization with Subsampling - PowerPoint PPT Presentation

Do Less, Get More: Streaming Submodular Maximization with Subsampling Moran Feldman 1 Amin Karbasi 2 Ehsan Kazemi 2 1 Open University of Israel and 2 Yale University

Data Summarization Large set of images Videos Sensor data fMRI parcellation � 2

Submodularity Diminishing returns property for set functions. { } V = ({ }) ({ }) ≥ f — f ({ }) ({ }) f — f ∀ A ⊆ B ⊆ V and x ∉ B f ( A ∪ { x }) - f ( A ) ≥ f ( B ∪ { x } ) - f ( B) � 3

Submodularity Diminishing returns property for set functions. { } V = ({ }) ({ }) ≥ f — f ({ }) ({ }) f — f ∀ A ⊆ B ⊆ V and x ∉ B f ( A ∪ { x }) - f ( A ) ≥ f ( B ∪ { x } ) - f ( B) � 3

Streaming Algorithms • Many practical scenarios we need to use streaming algorithms: • the data arrives at a very fast pace • there is only time to read the data once • random access to the entire data is not possible and only a small fraction of the data can be loaded to the main memory Surveillance camera Summary � 4


 Streaming Algorithms • Many practical scenarios we need to use streaming algorithms: • the data arrives at a very fast pace • there is only time to read the data once Key challenge: 
 • random access to the entire data is not possible and only a small fraction of the data can be loaded to the Extract small, representative subset 
 main memory out of a massive stream of data Surveillance camera Summary � 4

<latexit sha1_base64="Pu/j5s7RZ6u4tiUqNEnw8D7iAT0=">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</latexit> Constrained Non-Monotone Submodular Maximization constraints Set system : a pair ( 𝓞 , 𝓙 ), where 𝓞 is the ground set and 𝓙 ⊆ 2 𝓞 is the set of independent sets p -matchoid : a set system ( 𝓞 , 𝓙 ) where there exist m matroids ( 𝓞 i , 𝓙 i ) such that every element of 𝓞 appears in the ground set of at most p matroids and I = { S ⊆ 2 N | ∀ 1 ≤ i ≤ m S ∩ N i ∈ I i } [Chekuri et al., 2015] � 5

The Sample-Streaming Algorithm Data Stream u i u i +1 u i +2 u i +3 u i +5 1 Keep with probability q = p + √ p ( p +1)+1 U i ← Exchange-Candidate ( S i − 1 , u i ) S i − 1 � 6

Constrained Submodular Maximization � 7

Constrained Submodular Maximization � 7

Constrained Submodular Maximization � 7

Conclusion • Our algorithm provides the best of three worlds: • the tightest approximation guarantees in various settings • minimum memory requirement • fewest queries per element Poster: Today (Thu Dec 6th) 10:45 AM-12:45 PM @ Room 210 & 230 AB #75 � 8