1. Suppose that a list contains the values 20 44 48 55 62 66 74 88 93 99 at index positions 0 through 9. Trace the values of the variables….

## Prove Claim 14.10.

1 Prove Claim 14.10. *Hint: *Extend the proof of Lemma 13.5.

2 Prove Corollary 14.14.

3 Perceptron as a subgradient descent algorithm: Let *S *= ((x1*, **y*1)*, . . .,*(x*m**, **ym*)) ∈ (R*d *×{±1})*m*. Assume that there exists w ∈ R*d *such that for every *i *∈ [*m*] we have *yi *_w*,*x*i*_ ≥ 1, and let w*_ *be a vector that has the minimal norm among all vectors that satisfy the preceding requirement. Let *R *= max*i * x*i* . Define a function *f *(w) = max *i*∈[*m*] *(*1− *yi *_w*,*x*i *_*) *. _ Show that minw: w ≤ w*_* *f *(w) = 0 and show that any w for which *f *(w) *<>*1 separates the examples in *S*. _ Show how to calculate a subgradient of *f *. _ Describe and analyze the subgradient descent algorithm for this case. Compare the algorithm and the analysis to the Batch Perceptron algorithm given in

Section 9.1.2.