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Interpretation of Backtracking Line Search [RECAP] Interpretation of backtracking line search [RECAP] ∆x= direction of descent= f(x k) for gradient descent −∇ A different way of understanding the varying step size withβ: Multiplyingt byβ causes the interpolation to tilt as indicated in the figure Homeworks: Letf(x) =x 2 forx . Letx 0 = 2,∆x k = 1 for allk (since it is a valid ∈ℜk k − k descent direction ofx>0) andx = 1 + 2− . What is the step sizet implicitly being used. k Show that whilet satisifies the Armijo condition (determine ac 1), it does not satisfy the second Strong Wolfe condition in the following slides. Why is the choice of step size bad? March 19, 2018 107 / 188 Strong Wolfe conditions: Sufficient decrease BUT NOT at the cost of oscillations Ray Search for First Order Descent: Strong Wolfe Conditions [RECAP] Often, another condition is used for inexact line search in conjunction with the Armijo condition. ∆xT f(x+t∆x) c ∆xT f(x) (28) ∇ ≤ 2 ∇ where1>c 2 >c 1 >0. This condition ensures that the slope of the functionf(x+t∆x) att is less thanc 2 times that att=0. 1 The conditions in (27) and (28) are together called the strong Wolfe conditions. These conditions are particularly very important for non-convex problems. 2 While (??refeq:armijoCondition) ensures guaranteed decrease inf(x+δx), (28) provides guaranteed decrease in magnitude of slope and avoid too small steps. 3 Claim: If1>c 2 >c 1 >0 and the functionf(x) is convex and differentiable, there existst such that (27) and (28) are both satisfied for anyf. Hint: Use the Mean Value Theorem March 19, 2018 108 / 188 Convexity Strong Wolfe Conditions [RECAP] ⇒ Letϕ(t) =f(x k +t∆x k) f(x k) +t Tf(xk)∆xk (where the second inequality is by ≥ ∇ virtue of convexity). Remember that Tf(xk)∆xk <0 ∇ Since0<c <1, the linear approximationl(t) =f(x k) + tc Tf(xk)∆xk is unbounded 1 1∇ below and it can be shown to March 19, 2018 109 / 188 Convexity Strong Wolfe Conditions [RECAP] ⇒ Letϕ(t) =f(x k +t∆x k) f(x k) +t Tf(xk)∆xk (where the second inequality is by ≥ ∇ virtue of convexity). Remember that Tf(xk)∆xk <0 ∇ Since0<c <1, the linear approximationl(t) =f(x k) + tc Tf(xk)∆xk is unbounded 1 1∇ below and it can be shown to intersect the graph ofϕ atleast once. Lett ′ >0 be the smallest intersecting value oft, that is: k k T k k f(x+t ′∆x ) =f(x ) +t ′c f(x )∆x (29) 1∇ For allt [0,t ], ∈ ′ March 19, 2018 109 / 188 Convexity Strong Wolfe Conditions [RECAP] ⇒ Letϕ(t) =f(x k +t∆x k) f(x k) +t Tf(xk)∆xk (where the second inequality is by ≥ ∇ virtue of convexity). Remember that Tf(xk)∆xk <0 ∇ Since0<c <1, the linear approximationl(t) =f(x k) + tc Tf(xk)∆xk is unbounded 1 1∇ below and it can be shown to intersect the graph ofϕ atleast once. Lett ′ >0 be the smallest intersecting value oft, that is: k k T k k f(x+t ′∆x ) =f(x ) +t ′c f(x )∆x (29) 1∇ For allt [0,t ], ∈ ′ f(xk +t∆x k) f(x k) + tc Tf(xk)∆xk (30) ≤ 1∇ That is, there exists a non-empty set oft such that the first Wolfe condition is met. By the mean value theorem, t (0,t ) such that ∃ ′′ ∈ ′ k k k T k k k f(x +t ′∆x ) f(x ) =t ′ f(x +t ′′∆x )∆x (31) − ∇ March 19, 2018 109 / 188 Convexity Strong Wolfe Conditions (contd.) ⇒ Combining (29) and (31), and usingc <c ,and Tf(xk)∆xk <0 1 2 ∇ T k k k T k k T k k f(x +t ′′∆x )∆x =c f(x )∆x >c f(x )∆x (32) ∇ 1∇ 2∇ Again, since Tf(xk)∆xk <0, we get thet k =t satisfying (28) ∇ ′′ T k k k T k k f(x +t ′′∆x )∆x <c f(x )∆x (33) |∇ | 2|∇ | In fact, by continuity off(.), there exists an interval aroundt ′′ for which Strong Wolfe conditions hold. March 19, 2018 110 / 188 This is desirable: Neither too slow, nor too wayward.. Starts fast and then adaptively slows down Empirical Observations on Ray Search [RECAP] A finding that is borne out of plenty of empirical evidence is that exact ray searchdoes better than empirical ray search in a few cases only. Further, the exact choice of the value ofc 1 andc 2 seems to have little effect on the convergence of the overall descent method. The trend of specific descent methods has been like a parabola - starting with simple steepest descent techniques, then accomodating the curvature hessian matrix through a more sophisticated Newton’s method and finally, trying to simplify the Newton’s method through approximations to the hessian inverse, culminating in conjugate gradient techniques, that do away with any curvature matrix whatsoever, and form the internal combustion engine of many sophisticated optimization techniques today. We start the thread by describing the steepest descent methods. March 19, 2018 111 / 188 Algorithms: Steepest Descent [RECAP] The idea of steepest descent is to determine a descent direction such that for a unit step in that direction, the prediction of[ decrease in the] objective is maximized However, consider∆x= argmin 5 10 15 v v − ∞ = ∆x= ⇒ −∞ −∞ which is unacceptable Thus, there is a necessity to restrict the norm ofv The choice of the descent direction can be stated as: ∆x= argmin ⊤f(x)v v ∇ s.t. v =1 ∥ ∥ March 19, 2018 112 / 188 Algorithms: Steepest Descent [RECAP] Letv n be a unit vector under some norm. By first order convexity condition for ∈ℜ convex and differentiablef, f(x(k)) f(x (k) +v) Tf(x(k))v − ≤ −∇ For smallv, the inequality turns into approximate equality. The term Tf(x(k))v can −∇ be thought of as (an upper-bound on) the first order prediction of decrease. The idea in the steepest descent method is to choose a norm and then determine a descent direction such that for a unit step in that norm, the first order prediction of decrease is maximized. This choice of the descent direction can be stated as { } ∆x= argmin Tf(x)v v =1 ∇ | || || Empirical observation: If the norm chosen is aligned with the gross geometry of the sub-level sets3, the steepest descent method converges faster to the optimal solution. Else, it often amplifies the effect of oscillations. 3The alignment can be determined by fitting, for instance, a quadratic to a sample of the points. March 19, 2018 113 / 188 Various choices of the norm result in different solutions for∆x [RECAP] f(x(k)) For 2-norm,∆x= ∇ − f(x(k)) ∥∇ ∥2 (gradient descent) ( ) ∂f(x(k)) For 1-norm,∆x= sign (k) ei, wheree i is the ith standard basis vector − ∂xi (coordinate descent) You dont expect the steepest descent For -norm,∆x= sign( f(x (k))) ∞ − ∇ direction to have an acute angle with the gradient March 19, 2018 114 / 188 General Algorithm: Steepest Descent (contd) [RECAP] Find a starting pointx (0) . ∈D repeat { } 1. Set∆x (k) = argmin Tf(x(k))v v =1 . ∇ | || || 2. Choose a step sizet (k) >0 using exact or backtracking ray search. 3. Obtainx (k+1) =x (k) +t (k)∆x(k). 4. Setk=k+1. until stopping criterion (such as f(x(k+1)) ϵ) is satisfied ||∇ ||≤ Figure 8: The steepest descent algorithm. Two examples of the steepest descent method are the gradient descent method (for the eucledian orL 2 norm) and the coordinate-descent method (for theL 1 norm). One fact however is that no two norms should give exactly opposite steepest descent directions, though they may point in different directions. March 19, 2018 115 / 188 Convergence of Descent Algorithm To be revisited Consider the general descent algorithm ( Tf(xk)∆xk <0 for eachk) with each step: ∇ xk+1 =x k +t k∆xk. n ▶ Supposef is bounded below in and ℜ 0 ▶ is continuously differentiable in an open set containing the level set x f(x) f(x ) N { | ≤ } ▶ f is Lipschiz continuous. ∇∑ ∞ ( Tf(xk)∆xk)2 Then, ∇ < (that is, it is finite) ∆xk 2 ∞ An infinite sum will be finite k=1 ∥ ∥ Tf(xk)∆xk ∇ = 0 Thus, limk 0 ∆xk . → ∥ ∥ If we additionally assume that the descent direction is not orthogonal to the gradient, i.e., Tf(xk)∆xk ∇ Γ Γ>0 ( k) =0 ∆xk f(xk) for some , then, we can show that lim k 0 f x − ∥ ∥|∇ ∥ ≥ → ∥∇ ∥ Before we try and prove this result, let us discuss Lipschitz continuity (recall from midsem). March 19, 2018 116 / 188 Lipschitz Continuity March 19, 2018 117 / 188 Recall: Lipschitz Continuity off Formally,f(x): n is Lipschitz continuous if f(x) f(y) L x y for all D⊆ℜ → ℜ | − |≤ ∥ − ∥ x,y . ∈D A Lipschitz continuous function is limited in how fast it changes: there exists a definite positive real numberL>0 such that, for every pair of points on the graph of the function, the absolute value of the slope of the line connecting them is not greater than this real number. This bound is called the function’s Lipschitz constant,L>0. We showed that if a functionf: is convex in(α,β) it is Lipschitz continuous in ℜ→ℜ [γ,δ] whereα<γ<δ<β. We did not assume thatf is differentiable. Recap from midsem... March 19, 2018 118 / 188 Lipschitz continuity Intuitively, a Lipschitz continuous function is limited in how fast it changes: there exists a definite real numberL such that, for every pair of points on the graph of the gradient, the absolute value of the slope of the line connecting them is not greater than this real number ▶ This bound is called the function’s Lipschitz constant,L>0 ▶ The sum of two Lipschitz continuous functions is also Lipschitz continuous with the Lipschitz constant specified as the sum of the respective Lipschitz constants.
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