Online Learning of Complex Categorical Problems by Yacov Shlomo Crammer

By Yacov Shlomo Crammer

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27) This provides an upper bound on i ∆i . In the following we prove a lower bound on ∆ i . Similar to Eq. 5) and Eq. 6) we obtain, 2 ∆i = −αi xi 2 2 − 2αi wi , y i xi + 2αi w∗ , y i xi ≥ −αi xi 2 − 2αi wi , y i xi − 2αi max{γ ∗ − w ∗ , y i xi , 0} + 2αi γ ∗ 1 2 = −2 αi xi 2 + αi wi , y i xi − αi γ 2 −2αi γ − 2αi max{γ ∗ − w∗ , y i xi , 0} + 2αi γ ∗ . 29) CHAPTER 2. ONLINE LEARNING 29 The definition of the algorithm implies, αi = min C, max 0, γ − w i , y i xi , 2 xi Specifically, 0 ≤ αi ≤ and thus, 1 i2 i α x 2 2 γ − w i , y i xi xi .

In his doctoral dissertation, under the supervision of Manfred Warmuth, Nick Littlestone presented and analyzed a whole range of online algorithms, including the famous WINNOW algorithm. Since then online algorithms has been a fertile field of research in machine learning. We would like to stress the difference between the online model and the batch model. In the online model the main focus is the learning process itself, as the goal is to minimize the cumulative loss during the training process.

25) ξ≥0. As in the separable case, we compute the dual of Eq. 25) and derive an appropriate update rule. Similar to the separable case the resulting update rule is conservative; that is, if the loss on example (xi , y i ) is zero then w i+1 = wi . When the loss is not zero the update takes the same form, wi+1 = wi + αi y i xi . However, the step size αi is clipped with the predefined parameter, αi = min C , Lγ wi ; (xi , y i ) xi 2 . 26) The goal of the clipping procedure is to control the diversity of the parameter vector w.

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