November 1997 LIDS-P-2404 GRADIENT CONVERGENCE IN GRADIENT METHODS

For the classical gradient method Xt+l = xt -ytVf(xt) and several deterministic and stochastic variants, we discuss the issue of convergence of the gradient sequence Vf(xt) and the attendant issue of stationarity of limit points of xt. W;"e assume that Vf is Lipschitz continuous, and that the stepsize at diminishes to 0 and satisfies standard stochastic approximation conditions. We show that either f(xt) -oo or else f(xt) converges to a finite value and Vf(.t) -0 (with probability 1 in the stochastic case). Existing results assume various boundedness conditions such as boundedness from below of f, or boundedness of Vf(xt), or boundedness of Xt. ' Research supported by NSF under Grant DMII-9625489 2 Dept. of Electrical Engineering and Computer Science, M.I.T., Cambridge, A/lass., 02139. 1