By Yacov Shlomo Crammer

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In this case, 1 − y i wi , xi + ≥ 1 and therefore the number of prediction mistakes is bounded by R 2 /(ˆ γ ∗ )2 . This bound is common to online algorithms for classifications such as ROMMA [63]. 2 to obtain a direct bound on the hinge loss. Using again γ = 1 and omitting the first term in the left hand side of Eq. 16) we get, m ∗ 2(γ − 1) i=1 1 − y i w i , xi + ≤ R2 w ∗ 2 . ˆ ∗ /ˆ By setting w ∗ = 2w γ ∗ , which implies that γ ∗ = 2, we can further simplify the above to get a bound on the cumulative hinge loss, T i=1 1 − y i w i , xi + ≤ 2 R2 .

29), together with the fact that α i ≤ C we get the following lower bound on ∆i , ∆i ≥ 0 − 2αi γ − 2CLγ ∗ w∗ , (xi , y i ) + 2αi γ ∗ . 31) Summing over i we get the following bound, m i=1 m ∆i ≥ 2(γ ∗ − γ) = 2(γ ∗ − γ) i=1 m i=1 αi − 2C Lγ ∗ w∗ , (xi , y i ) i αi − 2CLγ ∗ (w∗ ) . 32) Combining Eq. 27) with Eq. 32) we get, m ∗ 2(γ − γ) m i i=1 α − 2CL γ∗ ∗ (w ) ≤ w ∗ 2 ∗ ⇒ 2(γ − γ) i=1 αi ≤ 2CLγ ∗ (w∗ ) + w∗ 2 . 33) Let us fix c > 0, its exact value would be determined in the sequel. We replace w ∗ with cw ∗ and γ ∗ with cγ ∗ , and obtain, m 2(cγ ∗ − γ) i=1 αi ≤ 2CcLγ ∗ (w∗ ) + c2 w∗ 2 Substituting, c = 2γ/γ ∗ we obtain, m 2γ i=1 αi ≤ 4γC γ2 Lγ ∗ (w∗ ) + 4 w∗ γ∗ γ ∗2 2 , .

Since y ˆ is consistent with y we have that, y ˆai > y ˆbi and y ˆbi−1 > y ˆai for i = 1, . . , t. Applying the transitivity of the ”>” operator obtain that, yˆa1 > y ˆbt . This is a contradiction since a1 = bt . The lemma informs us that if we assume (as in the analysis of many online algorithms) that the target is consistent with some total ordering, then the target must be a semi-order. In this sense the class of semi-orders is the most general class which can be applied. 1 Examples In the examples below the input space X is fixed to be an n-dimensional vector space.