Isotonic Hawkes Processes

0 g⇤(w⇤ ·xt)dt = P j2Si aijg ⇤ (w⇤ ·xj). Set y⇤ i = g ⇤ (w⇤ ·xi) to be the expected value of each yi. Let ̄ Ni be the expected value of Ni. Then we have ̄ Ni = P j2Si aijy ⇤ j . Clearly we do not have access to ̄ Ni. However, consider a hypothetical call to the algorithm with input {(xi, ̄ Ni)}i=1 and suppose it returns ḡk. In this case, we define ȳk i = ḡk(w̄k · xi). Next we begin the proof and introduce Lemma 3-5. Analysis roadmap. To prove Theorem 6, we establish several lemmas. The heart of the proof is Lemma 3, in which we show a property of the learned parameters ŵk at iteration k. That is, the squared distance kŵk w⇤k2 between ŵk and the true direction w⇤ decreases at each iteration at a rate which depends on "(ĝk, ŵk) and some other additive error terms ⌘