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110 5 ⋅ an inner-outer iteration for pagerank The following argument summarizes the convergence theory for inexact inner iterations. First, we provide an expression for the error in the linear system during the inner iterations. Then we derive a monotonic bound on the error during the outer iterations. Let g( j) = y( j) − x be the error after the jth inner iteration. Lemma 16. In the inner-outer iteration with 0 < α < 1 and β < α, when the inner iterations are solved using a Richardson iteration, after j inner iterations, ⎛ α−βj−1 ⎞ g(j) =y(j)−x= αβj−1Pj +( )∑βlPl g(0) j≥1, ⎝ βl=1⎠ where g(0) = y(0) − x = x(k) − x is the error after the kth outer iteration. Proof. We proceed by induction. In the base case, g(1) = αPg(0) follows because one inner iteration is identical to an iteration of the power method. Consider g( j+1). First, expand g( j+1) = βPy( j) + (α − β)Px(k) + (1 − α)v − x = βPg( j) + (α − β)Pg(0) . Now apply the induction hypothesis and simplify: ⎛ α−βj−1 ⎞ g(j+1)=βP (αβj−1Pj+ )∑βlPl g(0)+(α−β)Pg(0) = = ⎝ βl=1⎠ ⎛ j−1⎞ αβjPj+1 +(α−β)P+(α−β)∑βlPl+1 g(0) ⎝ l=1⎠ ⎛α−βj⎞ αβjPj+1 +( )∑βlPl ⎝ βl=1⎠ g(0).PDF Image | Instagram Cheat Sheet
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