Convex Optimization. Dani Yogatama. School of Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA. February 12, 2014

Transcription

1 Convex Optimization Dani Yogatama School of Computer Science, Carnegie Mellon University, Pittsburgh, PA, USA February 12, 2014 Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

2 Key Concepts in Convex Analysis: Convex Sets Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

3 Key Concepts in Convex Analysis: Convex Functions Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

4 Key Concepts in Convex Analysis: Minimizers Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

5 Key Concepts in Convex Analysis: Strong Convexity Recall the definition of convex function: λ [0, 1], f (λx + (1 λ)x ) λf (x) + (1 λ)f (x ) convexity Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

6 Key Concepts in Convex Analysis: Strong Convexity Recall the definition of convex function: λ [0, 1], f (λx + (1 λ)x ) λf (x) + (1 λ)f (x ) A β strongly convex function satisfies a stronger condition: λ [0, 1] f (λx + (1 λ)x ) λf (x) + (1 λ)f (x ) β 2 λ(1 λ) x x 2 2 convexity strong convexity Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

7 Key Concepts in Convex Analysis: Strong Convexity Recall the definition of convex function: λ [0, 1], f (λx + (1 λ)x ) λf (x) + (1 λ)f (x ) A β strongly convex function satisfies a stronger condition: λ [0, 1] f (λx + (1 λ)x ) λf (x) + (1 λ)f (x ) β 2 λ(1 λ) x x 2 2 convexity Strong convexity strict convexity. strong convexity Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

8 Key Concepts in Convex Analysis: Subgradients Convexity continuity; convexity differentiability (e.g., f (w) = w 1 ). Subgradients generalize gradients for (maybe non-diff.) convex functions: Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

9 Key Concepts in Convex Analysis: Subgradients Convexity continuity; convexity differentiability (e.g., f (w) = w 1 ). Subgradients generalize gradients for (maybe non-diff.) convex functions: v is a subgradient of f at x if f (x ) f (x) + v (x x) Subdifferential: f (x) = {v : v is a subgradient of f at x} linear lower bound Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

10 Key Concepts in Convex Analysis: Subgradients Convexity continuity; convexity differentiability (e.g., f (w) = w 1 ). Subgradients generalize gradients for (maybe non-diff.) convex functions: v is a subgradient of f at x if f (x ) f (x) + v (x x) Subdifferential: f (x) = {v : v is a subgradient of f at x} If f is differentiable, f (x) = { f (x)} linear lower bound Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

11 Key Concepts in Convex Analysis: Subgradients Convexity continuity; convexity differentiability (e.g., f (w) = w 1 ). Subgradients generalize gradients for (maybe non-diff.) convex functions: v is a subgradient of f at x if f (x ) f (x) + v (x x) Subdifferential: f (x) = {v : v is a subgradient of f at x} If f is differentiable, f (x) = { f (x)} linear lower bound non-differentiable case Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

12 Key Concepts in Convex Analysis: Subgradients Convexity continuity; convexity differentiability (e.g., f (w) = w 1 ). Subgradients generalize gradients for (maybe non-diff.) convex functions: v is a subgradient of f at x if f (x ) f (x) + v (x x) Subdifferential: f (x) = {v : v is a subgradient of f at x} If f is differentiable, f (x) = { f (x)} linear lower bound non-differentiable case Notation: f (x) is a subgradient of f at x Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

13 Establishing convexity How to check if f (x) is a convex function? Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

14 Establishing convexity How to check if f (x) is a convex function? Verify definition of a convex function. Check if 2 f (x) greater than or equal to 0 (for twice differentiable 2 x function). Show that it is constructed from simple convex functions with operations that preserver convexity. nonnegative weighted sum composition with affine function pointwise maximum and supremum composition minimization perspective Reference: Boyd and Vandenberghe (2004) Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

15 Unconstrained Optimization Algorithms: First order methods (gradient descent, FISTA, etc.) Higher order methods (Newton s method, ellipsoid, etc.)... Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

16 Gradient descent Problem: min f (x) x Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

17 Gradient descent Problem: min f (x) x Algorithm: g t = f (xt) x. x t = x t 1 ηg t. Repeat until convergence. Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

18 Newton s method Problem: Assume f is twice differentiable. min f (x) x Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

19 Newton s method Problem: Assume f is twice differentiable. Algorithm: g t = f (xt) x. min f (x) x s t = H 1 g t, where H is the Hessian. x t = x t 1 ηs t. Repeat until convergence. Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

20 Newton s method Problem: Assume f is twice differentiable. Algorithm: g t = f (xt) x. min f (x) x s t = H 1 g t, where H is the Hessian. x t = x t 1 ηs t. Repeat until convergence. Newton s method is a special case of steepest descent using Hessian norm. Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

21 Primal problem: Duality min x f (x) subject to g i (x) 0 i = 1,..., m h i (x) = 0 i = 1,..., p for x X. Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

22 Primal problem: Duality min x f (x) subject to g i (x) 0 i = 1,..., m h i (x) = 0 i = 1,..., p for x X. Lagrangian: m L(x, λ, ν) = f (x) + λ i g i (x) + i=1 λ i and ν i are Lagrange multipliers. p ν i h i (x) i=1 Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

23 Primal problem: Duality min x f (x) subject to g i (x) 0 i = 1,..., m h i (x) = 0 i = 1,..., p for x X. Lagrangian: m L(x, λ, ν) = f (x) + λ i g i (x) + i=1 λ i and ν i are Lagrange multipliers. p ν i h i (x) Suppose x is feasible and λ 0, then we get the lower bound: f (x) L(x, λ, ν) x X, λ R m + i=1 Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

24 Duality Primal optimal: p = min x max L(x, λ, ν) λ 0,ν Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

25 Duality Primal optimal: p = min x max L(x, λ, ν) λ 0,ν Lagrange dual function: min x L(x, λ, ν) This is a concave function, regardless of whether f (x) convex or not. Can be for some λ and ν. Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

26 Duality Primal optimal: p = min x max L(x, λ, ν) λ 0,ν Lagrange dual function: min x L(x, λ, ν) This is a concave function, regardless of whether f (x) convex or not. Can be for some λ and ν. Lagrange dual problem: max L(x, λ, ν) subject to λ 0 λ,ν Dual feasible: if λ 0 and λ, ν dom L(x, λ, ν). Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

27 Duality Primal optimal: p = min x max L(x, λ, ν) λ 0,ν Lagrange dual function: min x L(x, λ, ν) This is a concave function, regardless of whether f (x) convex or not. Can be for some λ and ν. Lagrange dual problem: max L(x, λ, ν) subject to λ 0 λ,ν Dual feasible: if λ 0 and λ, ν dom L(x, λ, ν). Dual optimal: d = max min λ 0,ν x L(x, λ, ν) Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

28 Duality Weak duality p d always holds for convex and nonconvex problems Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

29 Duality Weak duality p d always holds for convex and nonconvex problems Strong duality p = d does not hold in general, but usually holds for convex problems. Strong duality is guaranteed by Slater s constraint qualification. Strong duality holds if the problem is strictly feasible, i.e. x int D s.t. g i (x) < 0, i = 1,..., m, h i (x) = 0, i = 1,..., p Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

30 Duality Weak duality p d always holds for convex and nonconvex problems Strong duality p = d does not hold in general, but usually holds for convex problems. Strong duality is guaranteed by Slater s constraint qualification. Strong duality holds if the problem is strictly feasible, i.e. x int D s.t. g i (x) < 0, i = 1,..., m, h i (x) = 0, i = 1,..., p Assume strong duality holds and p and d are attained. p = f (x ) = d = min L(x, λ, ν ) L(x, λ, ν ) f (x ) = p x Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

31 Duality Weak duality p d always holds for convex and nonconvex problems Strong duality p = d does not hold in general, but usually holds for convex problems. Strong duality is guaranteed by Slater s constraint qualification. Strong duality holds if the problem is strictly feasible, i.e. x int D s.t. g i (x) < 0, i = 1,..., m, h i (x) = 0, i = 1,..., p Assume strong duality holds and p and d are attained. p = f (x ) = d = min L(x, λ, ν ) L(x, λ, ν ) f (x ) = p x We have: x arg min x L(x, λ, ν ). λ i g i(x ) = 0 for i = 1,..., m (complementary slackness). Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

32 Karush-Kuhn-Tucker condition For a differentiable g(x) and h(x), the KKT conditions are: g i (x ) 0, h i (x ) = 0, λ i 0, λ i g i (x ) = 0, primal feasibility dual feasibility complementary slackness L(x, λ, ν ) x=x = 0 Lagrangian stationarity x Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

33 Karush-Kuhn-Tucker condition For a differentiable g(x) and h(x), the KKT conditions are: g i (x ) 0, h i (x ) = 0, λ i 0, λ i g i (x ) = 0, primal feasibility dual feasibility complementary slackness L(x, λ, ν ) x=x = 0 Lagrangian stationarity x If ˆx, ˆλ, ˆν satify the KKT for a convex problem, they are optimal. Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

34 Support Vector Machines Primal problem (hard constraint): 1 min w 2 w 2 2 subject to y i x i, w 1, i = 1,..., n Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

35 Support Vector Machines Primal problem (hard constraint): 1 min w 2 w 2 2 subject to y i x i, w 1, i = 1,..., n Lagrangian: L(w, λ) = 1 n 2 w 2 2 λ i (y i x i, w 1) i=1 Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

36 Support Vector Machines Primal problem (hard constraint): 1 min w 2 w 2 2 subject to y i x i, w 1, i = 1,..., n Lagrangian: L(w, λ) = 1 n 2 w 2 2 λ i (y i x i, w 1) i=1 Minimizing with respect to w, we have: L(w,λ) w = 0 w n i=1 λ iy i x i = 0 n w = λ i y i x i i=1 Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

37 Plug this back into the Lagrangian: Support Vector Machines L(λ) = n λ i 1 2 1=1 n n y i y j λ i λ j xi i=1 j=1 x j Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

38 Plug this back into the Lagrangian: Support Vector Machines L(λ) = Lagrange dual problem is: n λ i 1 2 1=1 n n y i y j λ i λ j xi i=1 j=1 x j max λ subject to n λ i 1 2 1=1 n n y i y j λ i λ j xi i=1 j=1 λ i 0, i = 1,..., n n λ i y i = 0 i=1 x j Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

39 Plug this back into the Lagrangian: Support Vector Machines L(λ) = Lagrange dual problem is: n λ i 1 2 1=1 n n y i y j λ i λ j xi i=1 j=1 x j max λ subject to n λ i 1 2 1=1 n n y i y j λ i λ j xi i=1 j=1 λ i 0, i = 1,..., n n λ i y i = 0 i=1 x j Since this problem only depends on xi x j, we can use kernels and learn in high dimensional space without having to explicitly represent φ(x). Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

40 Support Vector Machines Primal problem (soft constraint): min w 1 2 w C subject to n i=1 y i x i, w 1 ξ i, i = 1,..., n ξ i ξ i 0, i = 1..., n Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

41 Support Vector Machines Lagrange dual problem for the soft constraint: max λ n λ i 1 n n y i y j λ i λ j xi x j (1) 2 1=1 i=1 j=1 subject to 0 λ i C, i = 1,..., n (2) n λ i y i = 0 (3) i=1 Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

42 Support Vector Machines Lagrange dual problem for the soft constraint: max λ n λ i 1 n n y i y j λ i λ j xi x j (1) 2 1=1 i=1 j=1 subject to 0 λ i C, i = 1,..., n (2) n λ i y i = 0 (3) i=1 KKT conditions, for all i: λ i = 0 y i x i, w 1 (4) 0 < λ i < C y i x i, w = 1 (5) λ i = C y i x i, w 1 (6) Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

43 Sequential Minimal Optimization (Platt, 1998) An efficient way to solve SVM dual problem. Break a large QP program into a series of smallest possible QP problems. Solve these small subproblems analytically. Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

44 Sequential Minimal Optimization (Platt, 1998) An efficient way to solve SVM dual problem. Break a large QP program into a series of smallest possible QP problems. Solve these small subproblems analytically. In a nutshell Choose two Lagrange multipliers λ i and λ j. Optimize the dual problem with respect to these two Lagrange multipliers, holding others fixed. Repeat until convergence. Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

45 Sequential Minimal Optimization (Platt, 1998) An efficient way to solve SVM dual problem. Break a large QP program into a series of smallest possible QP problems. Solve these small subproblems analytically. In a nutshell Choose two Lagrange multipliers λ i and λ j. Optimize the dual problem with respect to these two Lagrange multipliers, holding others fixed. Repeat until convergence. There are heuristics to choose Lagrange multipliers that maximizes the step size towards the global maximum. The first one is chosen from examples that violate the KKT condition. The second one is chosen using approximation based on absolute difference in error values (see Platt (1998)). Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

46 Sequential Minimal Optimization (Platt, 1998) Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

47 Sequential Minimal Optimization (Platt, 1998) For any two Lagrange multipliers, the constraints are:: 0 < λ i, λ j < C (7) y i λ i + y j λ j = k i,j y kλ k = γ (8) Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

48 Sequential Minimal Optimization (Platt, 1998) For any two Lagrange multipliers, the constraints are:: 0 < λ i, λ j < C (7) y i λ i + y j λ j = k i,j y kλ k = γ (8) Express λ i in terms of λ j λ i = γ λ jy j y i Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

49 Sequential Minimal Optimization (Platt, 1998) For any two Lagrange multipliers, the constraints are:: 0 < λ i, λ j < C (7) y i λ i + y j λ j = k i,j y kλ k = γ (8) Express λ i in terms of λ j λ i = γ λ jy j y i Plug this back into our original function. We are then left with a very simple quadratic problem with one variable λ j Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

50 Sequential Minimal Optimization (Platt, 1998) Solve for the second Lagrange multiplier λ j. Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

51 Sequential Minimal Optimization (Platt, 1998) Solve for the second Lagrange multiplier λ j. If y i y j, the following bounds apply to λ j : L H = max(0, λ t 1 j = min(c, C + λ t 1 j λ y 1 i ) (9) λ y 1 i ) (10) If y i = y j, the following bounds apply to λ j : L H = max(0, λ t 1 j = min(c, λ t 1 j + λ y 1 i C) (11) + λ y 1 i ) (12) Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

52 Sequential Minimal Optimization (Platt, 1998) Solve for the second Lagrange multiplier λ j. If y i y j, the following bounds apply to λ j : L H = max(0, λ t 1 j = min(c, C + λ t 1 j λ y 1 i ) (9) λ y 1 i ) (10) If y i = y j, the following bounds apply to λ j : L H = max(0, λ t 1 j = min(c, λ t 1 j + λ y 1 i C) (11) + λ y 1 i ) (12) The solution is: H λ j = λ j L ifλ j > H ifl λ j H ifλ j < L Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

53 Fenchel duality If a convex conjugate of f (x) is known, the dual function can be easily derived. The convex conjugate of a function f is: f (y) = max y, x f (x) (13) x Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

54 Fenchel duality If a convex conjugate of f (x) is known, the dual function can be easily derived. The convex conjugate of a function f is: For a generic problem f (y) = max y, x f (x) (13) x min x subject to f (x) Ax b Cx = d The dual function is: f ( A λ C ν) b λ d ν Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

55 Fenchel duality If a convex conjugate of f (x) is known, the dual function can be easily derived. The convex conjugate of a function f is: For a generic problem f (y) = max y, x f (x) (13) x min x subject to f (x) Ax b Cx = d The dual function is: f ( A λ C ν) b λ d ν There are many functions whose conjugate are easy to compute: Exponential Logistic Quadratic form Log determinant... Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

56 Parting notes Dual formulation is useful. Give new insights into our problem, Allow us to develop better optimization methods and use kernel tricks. Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

57 Thank you! Questions? Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26

58 References I Some slides are from an upcoming EACL 2014 tutorial with Andre F. T. Martins, Noah A. Smith, and Mario F. T. Figueiredo Boyd, S. and Vandenberghe, L. (2004). Convex Optimization. Cambridge University Press. Platt, J. (1998). Fast training of support vector machines using sequential minimal optimization. In Scholkopf, B., Burges, C., and Smola, A., editors, Advances in Kernel Methods - Support Vector Learning. MIT Press. Dani Yogatama (Carnegie Mellon University) Convex Optimization February 12, / 26