By Vivek S. Borkar
This straightforward, compact toolkit for designing and studying stochastic approximation algorithms calls for just a simple realizing of likelihood and differential equations. even supposing strong, those algorithms have functions up to speed and communications engineering, man made intelligence and fiscal modeling. certain subject matters comprise finite-time habit, a number of timescales and asynchronous implementation. there's a beneficial plethora of purposes, every one with concrete examples from engineering and economics. particularly it covers editions of stochastic gradient-based optimization schemes, fixed-point solvers, that are typical in studying algorithms for approximate dynamic programming, and a few versions of collective habit.
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Example text
For j ≥ nm , xj+1 ∗ (1 + a(j)K ) + a(j)K + a(j) Mj+1 ∗ √ ≤ xj ∗ (1 + a(j)K ) + a(j)K + a(j) K(1 + ( xj ≤ xj ∗ ∗ 2 1/2 ) ) (by the previous lemma) √ ≤ xj ∗ (1 + a(j)K ) + a(j)K + a(j) K(1 + xj ∗ ) = xj ∗ (1 + a(j)K1 ) + a(j)K1 , where K1 = K + √ K. 36 Lock-in Probability Applying this inequality repeatedly and using the fact that 1+aK1 ≤ eaK1 and the fact that if j < nm+1 , a(nm ) + a(nm + 1) + · · · + a(j) ≤ t(nm+1 ) − t(nm ) ≤ T + 1, we get xj+1 ∗ ≤ xnm ∗ K1 (T +1) e + K1 (T + 1)eK1 (T +1) for nm ≤ j < nm+1 .
Nδ (A) = {x : inf y∈A ||x − y|| < δ}. Fix some 0 < 1 < and δ > 0 such that Nδ (H 1 ) ⊂ H ⊂ Nδ (H ) ⊂ B. s. e. 2). Therefore, if we can show that with high probability ¯ then it follows that {xn } converges to {xn } remains inside the compact set B, H with high probability. We shall in fact show that x ¯(·), the piecewise linear and continuous curve obtained by linearly interpolating the points {xn } as in Chapter 2, lies inside Nδ (H ) ⊂ B with high probability from some time on. Let us define T = [maxx∈B¯ V (x)] − 1 .
Then h∞ (x) = −x, indicating that the scaling limit c → ∞ above basically picks the dominant term −x of h that essentially controls the behaviour far away from the origin. 3 Another stability criterion The second stability test we discuss is adapted from Abounady, Bertsekas and Borkar (2002). This applies to the case when stability for one initial condition implies stability for all initial conditions. e. 1) is assumed to converge to a bounded invariant set for all initial conditions. , a recursion which is reset to a bounded set whenever it exits from a larger prescribed bounded set containing the previous one.