Stochastic Approximation: A Dynamical Systems Viewpoint by Vivek S. Borkar

By Vivek S. Borkar

This straightforward, compact toolkit for designing and interpreting stochastic approximation algorithms calls for just a uncomplicated figuring out of chance and differential equations. even supposing robust, those algorithms have functions on top of things and communications engineering, man made intelligence and fiscal modeling. detailed themes contain finite-time habit, a number of timescales and asynchronous implementation. there's a valuable plethora of functions, every one with concrete examples from engineering and economics. particularly it covers variations of stochastic gradient-based optimization schemes, fixed-point solvers, that are normal in studying algorithms for approximate dynamic programming, and a few types of collective habit.

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E. x˙ = h∞ (x) with initial condition x. Lemma 1. There exists a T > 0 such that for all initial conditions x on the unit sphere, φ∞ (t, x) < 81 for all t > T . Proof. Since asymptotic stability implies Liapunov stability (see Appendix B), there is a δ > 0 such that any trajectory starting within distance δ of the origin stays within distance 81 thereof. For an initial condition x on the unit sphere, let Tx be a time at which the solution is within distance δ/2 of the origin. Let y be any other initial condition on the unit sphere.

Also, the time dependence of its dynamics is via the continuous dependence of its coefficients on x(·), which lies in a compact set. Hence the smallest eigenvalue of Φ(t, s)ΦT (t, s), being a continuous function of its entries, is bounded away from zero. e. 2) on [t(n), ∞) with xn (t(n)) = x ¯(t(n)). Recall that xn (t(j + 1)) = xn (t(j)) + a(j)h(xn (t(j))) + O(a(j)2 ). 1) and using Taylor expansion, one has yj+1 = yj + a(j)(∇h(xn (t(j)))yj + κj ) + a(j)Mj+1 + O(a(j)2 ), where κj = o(||yj ||). In particular, iterating the expression above leads to ym(n)+i m(n)+i−1 = Πj=m(n) (1 + a(j)∇h(xn (t(j))))ym(n) m(n)+i−1 m(n)+i−1 (1 + a(k)∇h(xn (t(k))))κj m(n)+i−1 (1 + a(k)∇h(xn (t(k))))Mj+1 a(j)Πk=j+1 + j=m(n) m(n)+i−1 a(j)Πk=j+1 + j=m(n) + O(a(m(n))).

Appendix C): Consider the filtration F1 ⊂ F2 ⊂ · · · ⊂ F. Let S1 , . . , Sn be a (scalar) martingale with respect to this filtration, with Y1 = S1 , Yk = Sk − Sk−1 (k ≥ 2) the corresponding martingale difference sequence. Let ck ≤ Yk ≤ bk . Then −2t2 P ( max |Sk | ≥ t) ≤ 2e k≤n (bk −ck ) 2 1≤k≤n . If B ∈ F1 , we can state a conditional version of this inequality as follows: −2t2 P ( max |Sk | ≥ t|B) ≤ 2e k≤n (bk −ck ) 1≤k≤n 2 . , ||x||∞ = maxi |xi |. Note that √ √ for d v ∈ R , ||v||∞ ≤ ||v|| ≤ d||v||∞ .

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