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Submodular Observation Selection and Information Gathering for Quadratic Models Abolfazl Hashemi ∗ , Mahsa Ghasemi, Haris Vikalo, and Ufuk Topcu ICML, Wednesday June 12, 2019

Observation Selection and Information Gathering • Resource-constrained inference problems ◦ Target tracking, experimental design, sensor networks • Access to expensive / limited observations ◦ Communication cost, power consumption, computational burden Goal Cost-effectively identify the most useful subset of information 1/6 Hashemi et al.: Submodular Observation Selection for Quadratic Models

Observation Selection for Quadratic Models • Quadratic relation between observations and unknown parameters y i = 1 2 x ⊤ Z i x + h ⊤ i x + v i , i ∈ { 1 , 2 , . . . , n } � �� � g i ( x ) (a) Phase retrieval: y i = 1 (b) Localization: y i = 1 2 � h i − x � 2 2 x ∗ ( z i z ∗ i ) x + v i 2 + v i (Figures from [Candes’15] and [Gezici’05]) 2/6 Hashemi et al.: Submodular Observation Selection for Quadratic Models

Locally-Optimal Observation Selection • Challenge: Unknown optimal estimator and error covariance matrix • Locally-optimal observation selection [Flaherty’06, Krause’08]: Linearize around a guess x 0 y i := y i − g i ( x 0 ) ≈ ∇ g i ( x 0 ) ⊤ x + v i , ˆ and find an approximate covariance matrix: � � − 1 1 � ˆ Σ − 1 ∇ g i ( x 0 ) ∇ g i ( x 0 ) ⊤ P S = + x σ 2 i i ∈S • Observation selection task � � ˆ minimize Tr P S S s.t. S ⊂ [ n ] , |S| = K 3/6 Hashemi et al.: Submodular Observation Selection for Quadratic Models

Proposed Approach: VTB for Quadratic Models Main Idea Exploiting Van Trees’ bound (VTB) on error covariance of weakly biased estimators 4/6 Hashemi et al.: Submodular Observation Selection for Quadratic Models

Proposed Approach: VTB for Quadratic Models Main Idea Exploiting Van Trees’ bound (VTB) on error covariance of weakly biased estimators • A closed-form expression for VTB of quadratic models Theorem 1 For any weakly biased estimator ˆ x S with error covariance P S it holds that �� � − 1 1 � � Z i Σ x Z ⊤ i + h i h ⊤ P S � + I x = B S σ 2 i i i ∈S 4/6 Hashemi et al.: Submodular Observation Selection for Quadratic Models

Proposed Approach: VTB for Quadratic Models Main Idea Exploiting Van Trees’ bound (VTB) on error covariance of weakly biased estimators • A closed-form expression for VTB of quadratic models Theorem 1 For any weakly biased estimator ˆ x S with error covariance P S it holds that �� � − 1 1 � � Z i Σ x Z ⊤ i + h i h ⊤ P S � + I x = B S σ 2 i i i ∈S • Proposed method: Find S by greedily maximizing Tr(.) scalarization of B S : f A ( S ) := Tr ( I − 1 − B S ) x 4/6 Hashemi et al.: Submodular Observation Selection for Quadratic Models

Characterizing f A ( S ) • Submodularity: f j ( S ) ≥ f j ( T ) for all S ⊆ T ⊂ X and j ∈ X\T • α f -Weak Submodularity [Zhang’16, Chamon17]: α f × f j ( S ) ≥ f j ( T ) where α f > 1 for all S ⊆ T ⊂ X and j ∈ X\T 5/6 Hashemi et al.: Submodular Observation Selection for Quadratic Models

Characterizing f A ( S ) • Submodularity: f j ( S ) ≥ f j ( T ) for all S ⊆ T ⊂ X and j ∈ X\T • α f -Weak Submodularity [Zhang’16, Chamon17]: α f × f j ( S ) ≥ f j ( T ) where α f > 1 for all S ⊆ T ⊂ X and j ∈ X\T • Greedy maximization performance: f ( S ) ≥ ( 1 − e − 1 α f ) f ( O ) 5/6 Hashemi et al.: Submodular Observation Selection for Quadratic Models

Characterizing f A ( S ) • Submodularity: f j ( S ) ≥ f j ( T ) for all S ⊆ T ⊂ X and j ∈ X\T • α f -Weak Submodularity [Zhang’16, Chamon17]: α f × f j ( S ) ≥ f j ( T ) where α f > 1 for all S ⊆ T ⊂ X and j ∈ X\T • Greedy maximization performance: f ( S ) ≥ ( 1 − e − 1 α f ) f ( O ) Theorem 2 f A ( S ) is a normalized, monotone set function with bounded α f A . 5/6 Hashemi et al.: Submodular Observation Selection for Quadratic Models

Characterizing f A ( S ) • Submodularity: f j ( S ) ≥ f j ( T ) for all S ⊆ T ⊂ X and j ∈ X\T • α f -Weak Submodularity [Zhang’16, Chamon17]: α f × f j ( S ) ≥ f j ( T ) where α f > 1 for all S ⊆ T ⊂ X and j ∈ X\T • Greedy maximization performance: f ( S ) ≥ ( 1 − e − 1 α f ) f ( O ) Theorem 2 f A ( S ) is a normalized, monotone set function with bounded α f A . • Interpretation of bound on α f A as SNR condition 5/6 Hashemi et al.: Submodular Observation Selection for Quadratic Models

Evaluation of Theoretical Results • Phase retrieval problem with n = 12 observations 4.5 10 0 4 3.5 3 10 -1 2.5 2 10 -2 1.5 1 10 -3 0.5 10 -3 10 -2 10 -1 10 0 10 1 3 4 5 6 7 8 9 10 11 (c) Tightness of VTB (d) Bound on α f A • Asymptotic tightness of VTB • Tightness of weak submodularity bound in low SNR regime 6/6 Hashemi et al.: Submodular Observation Selection for Quadratic Models

Thank you! Submodular Observation Selection and Information Gathering for Quadratic Models Poster # 167 Wed Jun 12th 06:30 PM – 09:00 PM @ Pacific Ballroom Correspondance: Abolfazl Hashemi (email: abolfazl@utexas.edu) Mahsa Ghasemi (email: mahsa.ghasemi@utexas.edu)