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By Yacov Shlomo Crammer

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3. Now, setting γ = 1 we get that Lγ (w; (x, y)) = [1 − y w, x ]+ – the hinge loss for classification. 2 to obtain two loss bounds for the Hinge loss. First, note that by also setting ˆ ∗ /ˆ w∗ = w γ ∗ and thus γ ∗ = 1 we get that the second term on the left hand side of Eq. 16) vanishes as γ ∗ = γ = 1 and thus, m i=1 1 − y i w i , xi 2 + ≤ R2 w ∗ 2 = R2 . 23) We thus have obtained a bound on the squared hinge loss. The same bound was also derived by Herbster [52]. We can immediately use this bound to derive a mistake bound for the MIRA algorithm.

Xm , y m ) be an input sequence for the MIRA algorithm described in Fig. 2, where xi ∈ Rn and y i ∈ {±1}. Let w ∗ ∈ Rn be a vector, and fix some γ ∗ > 0. Assume that the MIRA algorithm is run with the margin parameter γ > 0 and 0 ≤ C < ∞. Denote the Hinge loss suffered on a single example to be, Lγ ∗ w∗ , (xi , y i ) = γ ∗ − y i w ∗ , xi + , and the cumulative loss over the sequence by, m L γ∗ ∗ Lγ ∗ w∗ , (xi , y i ) . (w ) = i=1 Then, the total sum of weights is upper bounded by, m i=1 αi ≤ 2C γ w∗ Lγ ∗ (w∗ ) + 2 γ∗ γ ∗2 2 .

19) Recall that y i xi is minus the gradient of the loss function L γ w; (xi , y i ) at wi . 1 we get that y i xi is the gradient of the loss function L γ ∗ w; (xi , y i ) at wi as well. 1) we get the inequality, Lγ ∗ w∗ ; (xi , y i ) − Lγ ∗ wi ; (xi , y i ) ≥ −y i xi , w∗ − wi . 1 and the assumption Lγ w∗ ; (xi , y i ) = 0 in Eq. 20) we have, ≥ Lγ ∗ wi ; (xi , y i ) − Lγ ∗ (w∗ ; (x, y)) = |γ ∗ − γ| + Lγ ∗ wi ; (xi , y i ) . 21) Combining Eq. 19) with Eq. 21) we get, y i xi , w ∗ − w i ∆i ≥ −αi 2 2 xi + 2αi Lγ wi ; (xi , y i ) + |γ ∗ − γ| = αi −αi xi 2 Plugging αi = Lγ wi ; (xi , y i ) / xi ∆i ≥ xi 2 2 + 2Lγ wi ; (xi , y i ) + 2|γ ∗ − γ| .

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