Introduction to Machine Learning Lecture 7. Mehryar Mohri Courant Institute and Google Research
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1 Introduction to Machine Learning Lecture 7 Mehryar Mohri Courant Institute and Google Research mohri@cims.nyu.edu
2 Convex Optimization
3 Differentiation Definition: let f : X R N R be a differentiable function, then the gradient of f at x X is defined by f(x) = f x 1 (x). f x N (x) Definition: let f : X R N R be a twice differentiable function, then the Hessian of f at is defined by 2 f(x) =. 2 f (x) x i x j 1 i,j N. x X 3
4 Unconstrained Optimization Theorem: let f : X R N R be a differentiable function. If f admits a local extremum at x X, then x is a stationary point. f(x )=0. a local minimum is a global minimum if the function is convex. (Fermat, 1629) 4
5 Convexity Definition: X R N is said to be convex if for any two points x, y X the segment [x, y] lies in X: {αx +(1 α)y, 0 α 1} X. Definition: let X be a convex set. A function is said to be convex if for all x, y X and α [0, 1], f(αx +(1 α)y) αf(x)+(1 α)f(y). f : X R With a strict inequality, f is said to be strictly convex. f is said to be concave when f is convex. 5
6 Properties of Convex Functions Theorem: let is convex iff f be a differentiable function. Then, f is convex and dom(f) x, y dom(f), f(y) f(x) f(x) (y x). f(y) (x, f(x)) f(x)+ f(x) (y x). Theorem: let f be a twice differentiable function. Then, f is convex iff its Hessian is positive semidefinite: x dom(f), 2 f(x) 0. 6
7 Constrained Optimization Problem Problem: Let X R N and f,g i : X R, i [1,m]. A constrained optimization problem has the form: min x X f(x) subject to: g i (x) 0,i [1,m]. no convexity assumption. primal problem. optimal value p. can be augmented with equality constraints. 7
8 Lagrangian/Lagrange Function Definition: the Lagrange function or Lagrangian associated to a constraint problem is the function defined by: x X, α 0,L(x, α) =f(x)+ m α i g i (x). i=1 α i s are called Lagrange or dual variables. 8
9 Lagrange Dual Function Definition: the (Lagrange) dual function associated to the constraint optimization problem is defined by F is always concave: Lagrangian is linear with respect to α and inf preserves concavity. α 0,F(α) p : for a feasible x, f(x)+ α 0,F(α) = inf L(x, α) x X = inf f(x)+ m α i g i (x). x X m i=1 i=1 α i g i (x) f(x). 9
10 Dual Optimization Problem Definition: the dual (optimization) problem associated to the constraint optimization is max F (α) α subject to: α 0. always a convex optimization problem. optimal value d. 10
11 Weak and Strong Duality Weak duality: d p. always holds (clear from previous observations). Strong duality: d = p. does not hold in general. holds for convex problems with constraint qualifications. 11
12 Constraint Qualification Definition: Assume that intx =. Then, the following is the strong constraint qualification or Slater s condition: x intx: g(x) < 0. Definition: Assume that intx =. Then, the following is the weak constraint qualification or Slater s condition: x intx: i [1,m], g i (x) < 0 g i (x) =0 g i affine. 12
13 Saddle Point - Sufficient Condition Theorem: Let P be a constrained optimization problem over X =R N. If (x, α ) is a saddle point, that is then it is a solution of P. Proof: By the first inequality, In view of that, the second inequality gives (Lagrange, 1797) x R N, α 0, L(x, α) L(x, α ) L(x, α ), α 0,L(x, α) L(x, α ) α 0, α g(x ) α g(x ) (use α + then α 0) g(x ) 0 α g(x )=0. x,l(x, α ) L(x, α ) x,f(x ) f(x)+α g(x). Thus, for all x such that, g(x) 0 f(x ) f(x). 13
14 Saddle Point - Necessary Conditions Theorem: Assume that f and g i, i [1,m], are convex functions and that Slater s condition holds. If x is a solution of the constrained optimization problem, then there exists α 0 such that (x, α) is a saddle point of the Lagrangian. Theorem: Assume that f and g i, i [1,m], are convex differentiable functions and that the weak Slater s condition holds. If x is a solution of the constrained optimization problem, then there exists α 0 such that (x, α) is a saddle point of the Lagrangian. 14
15 Kuhn-Tucker s Theorem (Karush 1939; Kuhn-Tucker, 1951) Theorem: Assume that f,g i : X R, i [1,m] are convex and differentiable and that the constraints are qualified. Then x is a solution of the constrained program iff there exists α 0 such that: x L(x, α) = x f(x)+α x g(x) =0 α L(x, α) =g(x) 0 m α g(x) = α i g(x i )=0. i=1 Note: Last two conditions equivalent to g(x) 0 i [1,m], ᾱi g i (x) =0. complementary conditions 15 KKT conditions
16 Since the constraints are qualified, if x is solution, then there exists α such that (x, α) is a saddle point. In that case, the three conditions are verified (for the 3rd condition see proof of sufficient condition slide). Conversely, assume that the conditions are verified. Then, for any xsuch that g(x)<0, f(x) f(x) x f(x) (x x) (convexity of f) m α i x g i (x) (x x) (first condition) i=1 m α i [g i (x) g i (x)] (convexity of g i s) i=1 m α i g i (x) 0. i=1 (third and second condition) 16
17 References Stephen Boyd and Lieven Vandenberghe. Convex Optimization, Cambridge University Press, Courant Institute, NYU
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