Is the test error unbiased for these programs? 2017 Kevin Jamieson

Transcription

1 Is the test error unbiased for these programs? 2017 Kevin Jamieson 1

2 Is the test error unbiased for this program? 2017 Kevin Jamieson 2

3 Simple Variable Selection LASSO: Sparse Regression Machine Learning CSE546 Kevin Jamieson University of Washington October 9,

4 Sparsity bw LS = arg min w y i x T i w 2 Vector w is sparse, if many entries are zero Very useful for many tasks, e.g., Efficiency: If size(w) = 100 Billion, each prediction is expensive: If part of an online system, too slow If w is sparse, prediction computation only depends on number of non-zeros 4

5 Sparsity bw LS = arg min w y i x T i w 2 Vector w is sparse, if many entries are zero Very useful for many tasks, e.g., Efficiency: If size(w) = 100 Billion, each prediction is expensive: If part of an online system, too slow If w is sparse, prediction computation only depends on number of non-zeros Interpretability: What are the relevant dimension to make a prediction? E.g., what are the parts of the brain associated with particular words? 5Figure from Tom Mitchell

6 Sparsity bw LS = arg min w y i x T i w 2 Vector w is sparse, if many entries are zero Very useful for many tasks, e.g., Efficiency: If size(w) = 100 Billion, each prediction is expensive: If part of an online system, too slow If w is sparse, prediction computation only depends on number of non-zeros Interpretability: What are the relevant dimension to make a prediction? E.g., what are the parts of the brain associated with particular words? How do we find best subset among all possible? 6Figure from Tom Mitchell

7 Greedy model selection algorithm Pick a dictionary of features e.g., cosines of random inner products Greedy heuristic: Start from empty (or simple) set of features F 0 = Run learning algorithm for current set of features F t Obtain weights for these features Select next best feature h i (x) * e.g., h j (x) that results in lowest training error learner when using F t + {h j (x) * } F t+1! F t + {h i (x) * } Recurse 7

8 Greedy model selection Applicable in many other settings: Considered later in the course: Logistic regression: Selecting features (basis functions) Naïve Bayes: Selecting (independent) features P(X i Y) Decision trees: Selecting leaves to expand Only a heuristic! Finding the best set of k features is computationally intractable! Sometimes you can prove something strong about it 8

9 When do we stop??? Greedy heuristic: Select next best feature X i * E.g. h j (x) that results in lowest training error learner when using F t + {h j (x) * } Recurse When do you stop??? When training error is low enough? When test set error is low enough? Using cross validation? Is there a more principled approach? 9

10 Recall Ridge Regression Ridge Regression objective: bw ridge = arg min w = y i x T i w 2 + w

11 Ridge vs. Lasso Regression Ridge Regression objective: bw ridge = arg min w = y i x T i w 2 + w 2 2 Lasso objective: bw lasso = arg min w y i x T i w 2 + w

12 Penalized Least Squares Ridge : r(w) = w 2 2 Lasso : r(w) = w 1 bw r = arg min w y i x T i w 2 + r(w) 12

13 Penalized Least Squares Ridge : r(w) = w 2 2 Lasso : r(w) = w 1 bw r = arg min w y i x T i w 2 + r(w) For any 0 for which bw r achieves the minimum, there exists a 0 such that bw r = arg min w y i x T i w 2 subject to r(w) apple 13

14 Penalized Least Squares Ridge : r(w) = w 2 2 Lasso : r(w) = w 1 bw r = arg min w y i x T i w 2 + r(w) For any 0 for which bw r achieves the minimum, there exists a 0 such that bw r = arg min w y i x T i w 2 subject to r(w) apple 14

15 Optimizing the LASSO Objective LASSO solution: bw lasso, b b lasso = arg min w,b b blasso = arg min w,b 1 n y i (x T i w + b) 2 + w 1 y i x T i bw lasso ) 15

16 Optimizing the LASSO Objective LASSO solution: bw lasso, b b lasso = arg min w,b b blasso = arg min w,b 1 n y i (x T i w + b) 2 + w 1 y i x T i bw lasso ) So as usual, preprocess to make sure that 1 n so we don t have to worry about an o set. y i =0, 1 n x i = 0 16

17 Optimizing the LASSO Objective LASSO solution: bw lasso, b b lasso = arg min w,b b blasso = arg min w,b 1 n y i (x T i w + b) 2 + w 1 y i x T i bw lasso ) So as usual, preprocess to make sure that 1 n so we don t have to worry about an o set. y i =0, 1 n x i = 0 bw lasso = arg min w How do we solve this? y i x T i w 2 + w 1 17

18 Coordinate Descent Given a function, we want to find minimum Often, it is easy to find minimum along a single coordinate: How do we pick next coordinate? Super useful approach for *many* problems Converges to optimum in some cases, such as LASSO 18

19 Optimizing LASSO Objective One Coordinate at a Time Fix any j 2 {1,...,d} y i x T i w 2 + w 1 = y i 0 X y i x i,k w k k6=j! 2 dx dx x i,k w k + w k k=1 k=1 1 2 x i,j w j A + X w k + w j k6=j 19

20 Optimizing LASSO Objective One Coordinate at a Time Fix any j 2 {1,...,d} y i x T i w 2 + w 1 = y i 0 X y i x i,k w k k6=j! 2 dx dx x i,k w k + w k k=1 k=1 1 2 x i,j w j A + X w k + w j k6=j Initialize bw k = 0 for all k 2 {1,...,d} Loop over j 2 {1,...,n}: r (j) i = y i X k6=j bw j = arg min w j x i,j bw k r (j) i x i,j w j 2 + wj 20

21 Convex Functions Equivalent definitions of convexity: y x x f convex: f ( x +(1 )y) apple f(x)+(1 )f(y) 8x, y, 2 [0, 1] f(y) f(x)+rf(x) T (y x) 8x, y Gradients lower bound convex functions and are unique at x iff function differentiable at x Subgradients generalize gradients to non-differentiable points: Any supporting hyperplane at x that lower bounds entire function g is a subgradient at x if f(y) f(x)+g T (y x) 21

22 Taking the Subgradient bw j = arg min w j r (j) i x i,j w j 2 + wj g is a subgradient at x if f(y) f(x)+g T (y x) Convex function is minimized at w if 0 is a sub-gradient at wj w j wj n X r (j) i x i,j w j 2 = 22

23 Setting Subgradient to wj a j =( n X! 2 r (j) i x i,j w j + wj x 2 i,j) c j = 2( r (j) i x i,j ) = 8 >< a j w j c j if w j < 0 [ c j, c j + ] if w j =0 >: a j w j c j + if w j > 0 23

24 Setting Subgradient to wj a j =( n X! 2 r (j) i x i,j w j + wj x 2 i,j) c j = 2( bw j = arg min w j r (j) i x i,j ) = 8 >< a j w j c j if w j < 0 [ c j, c j + ] if w j =0 >: a j w j c j + if w j > 0 r (j) i x i,j w j 2 + wj w is a minimum if 0 is a sub-gradient at w bw j = 8 >< (c j + )/a j if c j < 0 if c j apple >: (c j )/a j if c j > 24

25 Soft Thresholding a j = bw j = x 2 i,j 8 >< (c j + )/a j if c j < 0 if c j apple >: (c j )/a j if c j > X c j =2 y i x i,k w k x i,j k6=j 25

26 Coordinate Descent for LASSO (aka Shooting Algorithm) Repeat until convergence (initialize w=0) Pick a coordinate l at (random or sequentially) Set: 8 >< (c j + )/a j if c j < bw j = 0 if c j apple >: (c j )/a j if c j > Where: a j = x 2 i,j c j =2 X y i x i,k bw k x i,j k6=j For convergence rates, see Shalev-Shwartz and Tewari 2009 Other common technique = LARS Least angle regression and shrinkage, Efron et al

27 Recall: Ridge Coefficient Path From Kevin Murphy textbook Typical approach: select λ using cross validation 27

28 Now: LASSO Coefficient Path From Kevin Murphy textbook 28

29 What you need to know Variable Selection: find a sparse solution to learning problem L 1 regularization is one way to do variable selection Applies beyond regression Hundreds of other approaches out there LASSO objective non-differentiable, but convex Use subgradient No closed-form solution for minimization Use coordinate descent Shooting algorithm is simple approach for solving LASSO 29