GEV-Canonical Regression for Accurate Binary Class Probability Estimation when One Class is Rare
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1 GEV-Canonical Regression for Accurate Binary Class Probability Estimation when One Class is Rare ArpitAgarwal 1 HarikrishnaNarasimhan 1 ShivaramKalyanakrishnan 2 ShivaniAgarwal 1 1 Indian Institute of Science, Bangalore 2 Yahoo Labs, Bangalore
2 Binary Problems where One Class is Rare Fraud detection
3 Binary Problems where One Class is Rare Fraud detection Medical diagnosis
4 Binary Problems where One Class is Rare Fraud detection Medical diagnosis Webadvertising
5 Binary Problems where One Class is Rare + + +
6 Problem Setup Instance space, Label space Probability distribution on,
7 Problem Setup Instance space, Label space Probability distribution on, We are interested in settings where zzzzzzzzz
8 Problem Setup Instance space, Label space Probability distribution on, Goal:Given a training sample learn a good class probability estimation (CPE) model
9 Previous Approaches Weightingerrors on positive and negative examples differently (Provost, 2000; Japkowicz, 2000; Chawlaet al., 2004; Van Hulseet al., 2007; He & Garcia, 2009) Undersamplingmajority class to balance positive and negative examples (King & Zeng, 2001) Asymmetric `link function based on generalized extreme value (GEV) distribution (Wang & Dey, 2010; Calabrese & Osmetti, 2011)
10 Our Work We use tools from the theory of proper composite lossesto design a loss based on the GEV link termed GEV-canonical GEV-canonical loss is both flexible and convex We also propose the GEV-canonical regression algorithm for its minimization
11 Outline Proper Composite Loss Functions GEV-Canonical Loss Function & GEV-Canonical Regression Algorithm Experiments
12 Loss Functions for CPE A CPE loss function assigns a penalty for predicting when the true label is y
13 Loss Functions for CPE A CPE loss function assigns a penalty for predicting when the true label is y Can be defined by its partial losses and, given by
14 Proper Loss Functions A CPE loss function is proper if and strictly proper if the minimizer is unique
15 Example: Logarithmic Loss
16 Example: Logarithmic Loss Log loss is strictly proper
17 Link Functions Let A link function \psi:[0,1] V is any strictly increasing (and therefore invertible) function that maps probabilities in [0,1] to real-valued scores in
18 Example: LogitLink
19 Example: ProbitLink
20 Example: Complementary Log-Log Link
21 Proper Composite Loss Functions [Buja et al, 2005; Reid & Williamson, 2009, 2010] A loss function is said to be proper composite if a proper CPE loss and a link \psi:[0,1] s.t.
22 Canonical Proper Loss & Link Pairs [Buja et al, 2005; Reid & Williamson, 2009, 2010] For every link function there is a unique canonical proper loss function given by:
23 Canonical Proper Loss & Link Pairs [Buja et al, 2005; Reid & Williamson, 2009, 2010] For every link function there is a unique canonical proper loss function given by: The resulting proper composite losshas some nice properties, including convexity.
24 Example: Logistic Loss Log Loss + LogitLink = Logistic Loss
25 Example: Logistic Loss Log Loss + LogitLink = Logistic Loss Canonical pair
26 Outline Proper Composite Loss Functions GEV-Canonical Loss Function & GEV-Canonical Regression Algorithm Experiments
27 Generalized Extreme Value (GEV) Probability Distribution CDF of GEV distributionwith location parameter scale parameter and shape parameter : Used for modeling rare events in statistics
28 GEV Link Family (Parameterized by -----)
29 GEV-Log Loss Effectively Used in (Wang & Dey, 2010; Calabrese & Osmetti, 2011) Log Loss + GEV Link = GEV-Log Loss
30 GEV-Log Loss Effectively Used in (Wang & Dey, 2010; Calabrese & Osmetti, 2011) Log Loss + GEV Link = GEV-Log Loss
31 GEV-Log Loss Effectively Used in (Wang & Dey, 2010; Calabrese & Osmetti, 2011) Log Loss + GEV Link = GEV-Log Loss NOT a canonical pair; results in non-convex loss
32 Canonical Proper Loss for GEV Link
33 GEV-Canonical Loss (Canonical Loss) +GEV Link = GEV-Canonical Loss
34 GEV-Canonical Loss (Canonical Loss) +GEV Link = GEV-Canonical Loss Canonical pair by construction; results in convex loss!
35 GEV-Canonical Loss (Canonical Loss) +GEV Link = GEV-Canonical Loss Canonical pair by construction; results in convex loss!
36 GEV-Canonical Loss Can be tailoredfor the problem of CPE for varying degrees of rarity Not available in closed form. But, the gradient and Hessian are available in closed form Can be efficiently minimized using IRLS type algorithm. We term this GEV-canonical regression
37 GEV-Canonical Regression
38 Outline Proper Composite Loss Functions GEV-Canonical Loss Function & GEV-Canonical Regression Algorithm Experiments
39 Experiments We have conducted experiments with both synthetic and real data Parameter selected using a validation set. Results averaged over 10 experiments.
40 Experiments with Synthetic Data Evaluation Metric: Root Mean Square Error (RMSE) Dataset 1 : p= Dataset 2 : p= Dataset 3 : p= 0.095
41 Experiments with Real Data Experimented with 12 UCI data sets Evaluation Metric: Brier Score (Brier, 1950)
42 Summary
43 Conclusion and Future Work Proposed GEV-canonical regression algorithm using convex GEV-canonical lossfor the problem of CPE when one class is rare Future directions: extensions to large scale data statistical guarantees
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