![Background Lasso and Enumeration
n Lasso Typical approach for feature selection
min
$
1
2
'( − * + + - ( . =: 1 ( , ', * ∈ ℝ5 ×7×ℝ5
n Enumeration for feature selection [Hara & Maehara, AAAI’17]
• Helpful for gaining more insights of data.
2
Ordinary Lasso
• One global optimum, i.e.,
one feature set, is obtained.
Enumeration of Lasso
• Several possible solutions, i.e.,
multiple feature sets, are obtained.
I found one feature set that is
helpful for predicting energy
consumption.
Found:
{Wall Area, Glazing Area}
I found several feature sets
that are helpful for predicting
energy consumption.
Found:
{Wall Area, Glazing Area},
{Wall Area, Overall Height},
{Roof Area, Glazing Area}, …](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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Convex Hull Approximation of Nearly Optimal Lasso Solutions
- 1. Convex Hull Approximation of Nearly Optimal Lasso Solutions 1 Satoshi Hara Takanori Maehara PRICAI’19
- 2. Background Lasso and Enumeration n Lasso Typical approach for feature selection min $ 1 2 '( − * + + - ( . =: 1 ( , ', * ∈ ℝ5 ×7×ℝ5 n Enumeration for feature selection [Hara & Maehara, AAAI’17] • Helpful for gaining more insights of data. 2 Ordinary Lasso • One global optimum, i.e., one feature set, is obtained. Enumeration of Lasso • Several possible solutions, i.e., multiple feature sets, are obtained. I found one feature set that is helpful for predicting energy consumption. Found: {Wall Area, Glazing Area} I found several feature sets that are helpful for predicting energy consumption. Found: {Wall Area, Glazing Area}, {Wall Area, Overall Height}, {Roof Area, Glazing Area}, …
- 3. Background Lasso and Enumeration n Example Lasso Enumeration for 20Newsdata • Identifying relevant words for article classification. 3 Selected words in Lasso solution adb apple bios bus cable com controller dos drivers duo fpu gateway ibm ide mac motherboard simm vlb vram windows
- 4. Background Lasso and Enumeration n Example Lasso Enumeration for 20Newsdata • Identifying relevant words for article classification. 4 Selected words in Lasso solution adb apple bios bus cable com controller dos drivers duo fpu gateway ibm ide mac motherboard simm vlb vram windows Model7 Remove motherboard cable adb drivers Model8 Remove motherboard cable adb drivers Model9 Remove motherboard cable adb drivers Model4 Remove motherboard cable adb drivers Model5 Remove motherboard cable adb drivers Model6 Remove motherboard cable adb drivers Model1 Remove motherboard cable adb drivers Model2 Remove motherboard cable adb drivers Model3 Remove motherboard cable adb drivers Enumerated Models
- 5. Background Lasso and Enumeration n Example Lasso Enumeration for 20Newsdata • Identifying relevant words for article classification. 5 Selected words in Lasso solution adb apple bios bus cable com controller dos drivers duo fpu gateway ibm ide mac motherboard simm vlb vram windows Model7 Remove motherboard cable adb drivers Model8 Remove motherboard cable adb drivers Model9 Remove motherboard cable adb drivers Model4 Remove motherboard cable adb drivers Model5 Remove motherboard cable adb drivers Model6 Remove motherboard cable adb drivers Model1 Remove motherboard cable adb drivers Model2 Remove motherboard cable adb drivers Model3 Remove motherboard cable adb drivers Enumerated Models Drawback of Enumeration Enumerated models can be just a combination of a few representative patterns. Exponentially many combinations of similar models can be found. These similar models are not helpful for gaining insights.
- 6. Goal of This Study n Goal Find small numbers of diverse models. large numbers similar n Overview of the Proposed Approach • Define a set of good models. ! " ≔ $: & $ ≤ " • Find vertices of ! " . Vertices = sparse models Vertices are distinct -> diversity 6 ! !Enumeration " "
- 7. Outline n Background and Overview n Problem Formulation n Proposed Method n Experiments n Summary 7
- 8. Properties of ! " n ! " ≔ $: & $ ≔ ' ( )$ − + ( + - $ ' ≤ " • A set of models with sufficiently small Lasso objectives. 1. ! " consists of smooth boundaries and non-smooth vertices. • Smooth boundaries = dense models • Non-smooth vertices = sparse models 2. A convex hull of the set of vertices / can approximate ! " well. • conv / ≈ ! " 8
- 9. Problem Approximation of ! " n Our Approach Approximate !(") by a set of % points & = () )*+ , . n To attain good approximation, the vertices - of !(") should be selected as &. 9
- 10. Problem Approximation of ! " n Our Approach Approximate !(") by a set of % points & = () )*+ , . n Question How to measure the approximation quality? 10 !(") How similar they are? We use Hausdorff distance. & = () )*+ ,
- 11. Problem Approximation of ! " n Def. Hausdorff distance between the two sets. • Maximum margin in the non-overlapping region. #$ %, %′ ≔ max sup /∈1 inf /5∈15 6 − 68 , sup /8∈18 inf /∈1 6 − 68 n We measure the approximation quality by using #$. Problem Minimization of Hausdorff distance min 9 #$ conv = , !(") , s. t. = ≤ C 11 % %′ !(") conv =Measure #$ = = EF FGH I
- 12. Outline n Background and Overview n Problem Formulation n Proposed Method n Experiments n Summary 12
- 13. Method Sampling + Greedy Selection n Step1 Sampling points from the boundary of ! " n Step2 Greedily select # points to minimize $%. 13 Step1 Sampling Step2 Greedy Selection
- 14. Step1 Sampling n Note Want to sample vertices as much as possible. n Proposed Sampling Method • Take a random direction. • Find an “edge” of ! " at that direction. 14 This method can sample vertices with high probabilities.
- 15. Step1 Sampling n Finding an “edge” max$ %&', s. t. ' ∈ -(/) (%: random direction) n Finding an “edge” by binary search • Dual Problem min 345 max $ %&' − 7(8 ' − /) • Find ' that satisfies 8 ' = / by finding optimal 7 by using binary search. 15 solvable with Lasso solvers large 7 small 7 optimal 7
- 16. Method Sampling + Greedy Selection n Step1 Sampling points from the boundary of ! " n Step2 Greedily select # points to minimize $%. 16 Step1 Sampling Step2 Greedy Selection
- 17. Step2 Greedy Selection n Original Problem min $ %& conv * , ,(.) , s. t. * ≤ 4 n Approximate , . with the sampled points 5. • , . ≈ conv 5 min $⊆8 %& conv * , conv 5 , s. t. * ≤ 4 • Remark %& conv * , conv 5 = max <∈8 min <>∈?@AB $ C − C′ 17 ,(.) conv(5) conv *Measure %& ≈
- 18. Step2 Greedy Selection n The problem is NP-hard in general. • min $⊆& '( conv , , conv . , s. t. , ≤ 3 n Our Approach Greedy Selection • Initialization step Select one point 4 ∈ . , 6 ← 4 , . ← . ∖ {4}, and ; ← 1 • While ; < 3 >4 ∈ max A∈& min AB∈CDEF $ G 4 − 4′ , JK6 ← , J ∪ >4 , . ← . ∖ { >4}, and ; ← ; + 1 18 conv(.) conv(,) Greedily add one point to , that minimizes the objective.
- 19. Step2 Greedy Selection n The problem is NP-hard in general. • min $⊆& '( conv , , conv . , s. t. , ≤ 3 n Our Approach Greedy Selection • Initialization step Select one point 4 ∈ . , 6 ← 4 , . ← . ∖ {4}, and ; ← 1 • While ; < 3 >4 ∈ max A∈& min AB∈CDEF $ G 4 − 4′ , JK6 ← , J ∪ >4 , . ← . ∖ { >4}, and ; ← ; + 1 19 conv(.) conv(,) Greedily add one point to , that minimizes the objective.
- 20. Step2 Greedy Selection n The problem is NP-hard in general. • min $⊆& '( conv , , conv . , s. t. , ≤ 3 n Our Approach Greedy Selection • Initialization step Select one point 4 ∈ . , 6 ← 4 , . ← . ∖ {4}, and ; ← 1 • While ; < 3 >4 ∈ max A∈& min AB∈CDEF $ G 4 − 4′ , JK6 ← , J ∪ >4 , . ← . ∖ { >4}, and ; ← ; + 1 20 conv(.) conv(,) Greedily add one point to , that minimizes the objective.
- 21. Step2 Greedy Selection n The problem is NP-hard in general. • min $⊆& '( conv , , conv . , s. t. , ≤ 3 n Our Approach Greedy Selection • Initialization step Select one point 4 ∈ . , 6 ← 4 , . ← . ∖ {4}, and ; ← 1 • While ; < 3 >4 ∈ max A∈& min AB∈CDEF $ G 4 − 4′ , JK6 ← , J ∪ >4 , . ← . ∖ { >4}, and ; ← ; + 1 21 conv(.) conv(,) Greedily add one point to , that minimizes the objective.
- 22. Step2 Greedy Selection n Details of computing !" ∈ max '∈( min '+∈,-./ 0 1 " − "′ n 1. Computing min Quadratic Programming (QP) • min '+∈,-./ 0 1 " − "′ ⇔ min 5 " − 6 7 8797 , s. t. 8 ≥ 0, 6 7 87 = 1 n 2. Computing max Lazy Update • A naïve implementation requires searching over all " ∈ B. • By using a monotonicity of the Hausdorff distance, we can skip redundant computations and accelerate the search. 22
- 23. Method Sampling + Greedy Selection n Step1 Sampling points from the boundary of ! " • Sampling random directions + Lasso + Binary Search n Step2 Greedily select # points to minimize $%. • Greedy selection 23 Step1 Sampling Step2 Greedy Selection
- 24. Outline n Background and Overview n Problem Formulation n Proposed Method n Experiments n Summary 24
- 25. Synthetic Experiment Visualization of ! " and # n Synthetic Problems • 2D ver. $ = 1 1 1 1 + 1/40 , , = 1 1 • 3D ver. $ = 1 1 1 1 1 + 1/40 1 1 1 1 + 2/40 , , = 1 1 1 n Results 25 2D ver. 3D ver. Hausdorff dist. 2D ver. 3D ver.
- 26. Synthetic Experiment High-dimensional Data n Synthetic data • ! = #$% + ' • % ∼ ) 0, , , ,-. = exp −0.1|6 − 7| • dimensionality of % = 100 n Result • Huadorff dist. decreases as 8 increases. • Huadorff dist. decreases as the sampling size 9 increases. The effect is marginal, though. In practice, 9 ≈ 1,000 would suffice. 26
- 27. Real-Data Experiment Diversity verification n Data: 20Newsgroups • Classification of news articles into two categories. (ibm or mac) • Feature selection = Identification of important words. ! ∈ ℝ$$%&': tf-idf weighted bag-of-words ( ∈ {0, 1}: categories of articles # of data: 1168 n Model • Linear logistic regression + ℓ$ n Baseline Methods [Hara & Maehara, AAAI’17] • Enumeration Exact enumeration of top-K models • Heuristic Skip similar models while enumeration. 27
- 28. Real-Data Experiment Diversity verification n Comparison of the found 500 models n Visualization with PCA • Projected found models with PCA. • The proposed method attained the largest diversity. 28 Found Words Enumeration 39 Heuristic 63 Proposed 889 apple macs macintosh Enumeration ✘ ✘ Heuristic ✘ Proposed Baseline methods found combinations of a few representative patterns only. Baseline methods missed some important words.
- 29. Summary n Our Goal • Find small numbers of diverse models for Lasso. n Our Method • Find “vertices” of a set of models ! " ≔ $: & $ ≤ " • Problem: Hausdorff distance minimization. • Method: Sampling + Greedy Selection n Verified the effectiveness of the proposed method. • The proposed method could find points that can well approximate ! " . obtain diverse models than the existing enumeration methods. 29 GitHub: /sato9hara/LassoHull