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108 5 ⋅ an inner-outer iteration for pagerank Algorithm 2 – The basic inner-outer iteration. Input: P, v, α, τ, (β = 0.5, η = 10−2) Output: x 1: x ← v 2: y ← Px 3:while ∥αy+(1−α)v−x∥≥τ f ←(α−β)y+(1−α)v repeat 4: 5: 6: 7: 8: 9: 10: x ← αy + (1 − α)v end while x ← f + βy y ← Px until ∥f + βy − x∥ < η 5.3 algorithm discussion Algorithm 2 has a number of properties worth noting. Consider the al- gorithm with β = 0. The inner loop will always exit after a single iteration because x is set to f. In this case, f = αy + (1 − α)v, but y = Px from the previous iteration. Thus, the inner-outer iteration with β = 0 is just the power method! Now, consider the case β = α. We intuitively expect that this method reproduces the power method because the inner iteration with β = α is solved with a Richardson iteration, which is the same as the power method.5 Here f = (1 − α)v for every inner iteration and thus every inner iteration corresponds to a step of the power method. When ∥f − βy − x∥ < η, we will do only one inner iteration for each outer iteration, but again, each step is just a step of the power method. Thus, we expect no acceleration with β = 0 or β = α. We make one final observation about the basic inner-outer method. For each outer iteration, the first inner iteration is always a step of the power method. We use this fact repeatedly in the remainder of the chapter. 5 Check section 2.4.2 for the formal equivalence.PDF Image | Instagram Cheat Sheet
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