Online Learning: Theory, Algorithms, and Applications - Ph.D by Shai Shalev-Shwartz

By Shai Shalev-Shwartz

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11) t=1 t=1 To simplify our derivation, we focus on the problem of binary classification with the hinge-loss. Formally, let (xt , yt ) ∈ Rn × {±1} be the tth classification example. We set gt (w) = [γ − yt w, xt ]+ , where γ > 0 is a margin parameter, and [a]+ = max{0, a} is the hinge function. 3, the Fenchel conjugate of gt (w) is the function  −γ α gt (λ) = ∞ if λ ∈ {−αyt xt : α ∈ [0, 1]} otherwise Since our goal is to maximize the dual objective, we can restrict ourselves to the case λt = −αt yt xt , where αt ∈ [0, 1], and rewrite the dual objective as a function of the vector α = (α1 , .

Let f be a differentiable convex function over a set S. Then, f induces the following Bregman divergence over S: Bf (u v) = f (u) − f (v) − u − v, ∇f (v) . 1) For example, the function f (v) = 12 v 22 yields the divergence Bf (u v) = 12 u − v 22 . Since f is convex, the Bregman divergence is non-negative. We recall that a function f is strongly convex over S with respect to a norm · if ∀u, v ∈ S, ∀λ ∈ ∂f (v), f (u) − f (v) − u − v, λ ≥ 34 u−v 2 2 . CHAPTER 4. LOGARITHMIC REGRET FOR STRONGLY CONVEX FUNCTIONS 35 PARAMETERS : A function f : S → R and a scalar σ > 0 I NITIALIZE : w1 ∈ S F OR t = 1, 2, .

15) CHAPTER 5. DERIVED ALGORITHMS 53 that we utilize is that ∆t ≥ 0 for all t. Furthermore, if t ∈ M then √ √ U Xt UX xt 2 xt 2 ∆t ≥ gt (wt ) − ≥ γ− ≥ γ− ≥ γ− . 2 ct 2 ct 2 |Mt | 2 |Mt | Therefore, √ T D(α T +1 ) ≥ ∆t ≥ t=1 ∆t ≥ γ|M| − t∈M UX 2 |M| i=1 √ 1 √ ≥ γ|M| − U X i |M| . Next, we upper bound D(αT +1 ). Without loss of generality, we assume that T ∈ M (otherwise, we can simply ignore the last rounds). Therefore, T D(αT +1 ) ≤ cT f (u) + T gt (u) = X |M|/U f (u) + t=1 T gt (u) ≤ X t=1 U |M| + gt (u) .

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