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 = ((x1y1), . . .,(xmym)) ∈ (R×{±1})m. Assume that there exists w ∈ Rsuch that for every ∈ [m] we have yi _w,xi_ ≥ 1, and let wbe a vector that has the minimal norm among all vectors that satisfy the preceding requirement. Let = max      xi             . Define a function (w) = max i∈[m(1− yi _w,x_. _ Show that minw:          w            ≤             w_          (w) = 0 and show that any w for which (w) <>1 separates the examples in S. _ Show how to calculate a subgradient of . _ 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.

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