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"Enhancing Feature Selection with Submodularity in Predictive Models"

This document explores the concept of feature selection in predictive modeling, emphasizing the challenges posed by uncertainty in various parameters. It discusses the use of Informative Gain (IG) to identify the most informative features from a set of random variables, while illustrating the principles of submodularity. Key examples such as the impact of fever, rash, and gender on predicting sickness are analyzed, highlighting the diminishing returns property in the selection process and its implications for model performance.

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"Enhancing Feature Selection with Submodularity in Predictive Models"

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  1. Y“Sick” X1“Fever” X2“Rash” X3“Male” Uncertaintybefore knowing XA Uncertaintyafter knowing XA Example: Feature selection • Given random variables Y, X1, … Xn • Want to predict Y from subset XA = (Xi1,…,Xik) Want k most informative features: A* = argmax IG(XA; Y) s.t. |A| · k where IG(XA; Y) = H(Y) - H(Y | XA) Problem inherently combinatorial! Naïve BayesModel

  2. Key property: Diminishing returns Selection A = {} Selection B = {X2,X3} Y“Sick” Y“Sick” X2“Rash” X3“Male” X1“Fever” Adding X1will help a lot! Adding X1doesn’t help much New feature X1 + s B Large improvement Submodularity: A + s Small improvement For Aµ B, z(A [ {s}) – z(A) ¸ z(B [ {s}) – z(B)

  3. A [ B AÅB Submodular set functions • Set function z on V is called submodular if For all A,B µ V: z(A)+z(B) ¸ z(A[B)+z(AÅB) • Equivalent diminishing returns characterization: + ¸ + B A + S B Large improvement Submodularity: A + S Small improvement For AµB, sB, z(A [ {s}) – z(A) ¸ z(B [ {s}) – z(B)

  4. Example: Set cover Want to cover floorplan with discs Place sensorsin building Possiblelocations V For A µ V: z(A) = “area covered by sensors placed at A” Node predicts values of positions with some radius Formally: W finite set, collection of n subsets Siµ W For A µ V={1,…,n} define z(A) = |i2 A Si|

  5. S’ S’ Set cover is submodular A={S1,S2} S1 S2 z(A[{S’})-z(A) ¸ z(B[{S’})-z(B) S1 S2 S3 S4 B = {S1,S2,S3,S4}

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