2.5 Random descent probabilities Consider the quadratic function g(w) = w/w +2, which we aim to...

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2.5 Random descent probabilities Consider the quadratic function g(w) = w/w +2, which we aim to minimize using random search starting at w defined in Equation (2.31), with a = 1 and |||d|| = 1. (a) When N = 2, show that the probability of descent - i.e., the probability that g(w + ad) < g(w) for a randomly chosen unit direction d - is upper-bounded by. Hint: see Figure 2.13. (b) Extend your argument in part (a) to find an upper-bound on the probability of descent for general N. 8(w) Figure 2.13 Figure associated with Exercise 2.5. VS 2 2.5 Random descent probabilities Consider the quadratic function g(w) = w/w +2, which we aim to minimize using random search starting at w defined in Equation (2.31), with a = 1 and |||d|| = 1. (a) When N = 2, show that the probability of descent - i.e., the probability that g(w + ad) < g(w) for a randomly chosen unit direction d - is upper-bounded by. Hint: see Figure 2.13. (b) Extend your argument in part (a) to find an upper-bound on the probability of descent for general N. 8(w) Figure 2.13 Figure associated with Exercise 2.5. VS 2

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Lets start by calculating the expression gw0 alpha d0 gw0 for N 2 Given that gw wTw 2 we have gw0 al... View the full answer