COMBINING GRADIENT-BASED OPTIMIZATION WITH STOCHASTIC SEARCH

ABSTRACT We propose a stochastic search algorithm for solving non-differentiable optimization problems. At each iteration, the algorithm searches the solution space by generating a population of candidate solutions from a parameterized sampling distribution. The basic idea is to convert the original optimization problem into a differentiable problem in terms of the parameters of the sampling distribution, and then use a quasiNewton-like method on the reformulated problem to find improved sampling distributions. The algorithm combines the strength of stochastic search from considering a population of candidate solutions to explore the solution space with the rapid convergence behavior of gradient methods by exploiting local differentiable structures. We provide numerical examples to illustrate its performance

Aerodynamic Shape Design of Nozzles Using a Hybrid Optimization Method

A hybrid design optimization method combining the stochastic method based on simultaneous perturbation stochastic approximation (SPSA) and the deterministic method of Broydon-Fletcher-Goldfarb-Shanno (BFGS) is developed in order to take advantage of the high efficiency of the gradient based methods and the global search capabilities of SPSA for applications in the optimal aerodynamic shape design of a three dimensional elliptic nozzle. The performance of this hybrid method is compared with that of SPSA, simulated annealing (SA) and gradient based BFGS method. The objective functions which are minimized are estimated by numerically solving the 3D Euler and Navier-Stokes equations using a TVD approach and a LU implicit scheme. Computed results show that the hybrid optimization method proposed in this study shows a promise of high computational efficiency and global search capabilities.Singapore-MIT Alliance (SMA

Asymptotic Bias of Stochastic Gradient Search

The asymptotic behavior of the stochastic gradient algorithm with a biased gradient estimator is analyzed. Relying on arguments based on the dynamic system theory (chain-recurrence) and the differential geometry (Yomdin theorem and Lojasiewicz inequality), tight bounds on the asymptotic bias of the iterates generated by such an algorithm are derived. The obtained results hold under mild conditions and cover a broad class of high-dimensional nonlinear algorithms. Using these results, the asymptotic properties of the policy-gradient (reinforcement) learning and adaptive population Monte Carlo sampling are studied. Relying on the same results, the asymptotic behavior of the recursive maximum split-likelihood estimation in hidden Markov models is analyzed, too.Comment: arXiv admin note: text overlap with arXiv:0907.102

Convergence and Convergence Rate of Stochastic Gradient Search in the Case of Multiple and Non-Isolated Extrema

The asymptotic behavior of stochastic gradient algorithms is studied. Relying on results from differential geometry (Lojasiewicz gradient inequality), the single limit-point convergence of the algorithm iterates is demonstrated and relatively tight bounds on the convergence rate are derived. In sharp contrast to the existing asymptotic results, the new results presented here allow the objective function to have multiple and non-isolated minima. The new results also offer new insights into the asymptotic properties of several classes of recursive algorithms which are routinely used in engineering, statistics, machine learning and operations research

Systems approaches and algorithms for discovery of combinatorial therapies

Effective therapy of complex diseases requires control of highly non-linear complex networks that remain incompletely characterized. In particular, drug intervention can be seen as control of signaling in cellular networks. Identification of control parameters presents an extreme challenge due to the combinatorial explosion of control possibilities in combination therapy and to the incomplete knowledge of the systems biology of cells. In this review paper we describe the main current and proposed approaches to the design of combinatorial therapies, including the empirical methods used now by clinicians and alternative approaches suggested recently by several authors. New approaches for designing combinations arising from systems biology are described. We discuss in special detail the design of algorithms that identify optimal control parameters in cellular networks based on a quantitative characterization of control landscapes, maximizing utilization of incomplete knowledge of the state and structure of intracellular networks. The use of new technology for high-throughput measurements is key to these new approaches to combination therapy and essential for the characterization of control landscapes and implementation of the algorithms. Combinatorial optimization in medical therapy is also compared with the combinatorial optimization of engineering and materials science and similarities and differences are delineated.Comment: 25 page