Online Learning and Simulation Based Algorithms for Stochastic Optimization

In many optimization problems, the relationship between the objective and parameters is not known. The objective function itself may be stochastic such as a long-run average over some random cost samples. In such cases finding the gradient of the objective is not possible. It is in this setting that stochastic approximation algorithms are used. These algorithms use some estimates of the gradient and are stochastic in nature. Amongst gradient estimation techniques, Simultaneous Perturbation Stochastic Approximation (SPSA) and Smoothed Functional(SF) scheme are widely used. In this thesis we have proposed a novel multi-time scale quasi-Newton based smoothed functional (QN-SF) algorithm for unconstrained as well as constrained optimization. The algorithm uses the smoothed functional scheme for estimating the gradient and the quasi-Newton method to solve the optimization problem. The algorithm is shown to converge with probability one. We have also provided here experimental results on the problem of optimal routing in a multi-stage network of queues. Policies like Join the Shortest Queue or Least Work Left assume knowledge of the queue length values that can change rapidly or hard to estimate. If the only information available is the expected end-to-end delay as with our case, such policies cannot be used. The QN-SF based probabilistic routing algorithm uses only the total end-to-end delay for tuning the probabilities. We observe from the experiments that the QN-SF algorithm has better performance than the gradient and Jacobi versions of Newton based smoothed functional algorithms. Next we consider constrained routing in a similar queueing network. We extend the QN-SF algorithm to this case. We study the convergence behavior of the algorithm and observe that the constraints are satisfied at the point of convergence. We provide experimental results for the constrained routing setup as well. Next we study reinforcement learning algorithms which are useful for solving Markov Decision Process(MDP) when the precise information on transition probabilities is not known. When the state, and action sets are very large, it is not possible to store all the state-action tuples. In such cases, function approximators like neural networks have been used. The popular Q-learning algorithm is known to diverge when used with linear function approximation due to the ’off-policy’ problem. Hence developing stable learning algorithms when used with function approximation is an important problem. We present in this thesis a variant of Q-learning with linear function approximation that is based on two-timescale stochastic approximation. The Q-value parameters for a given policy in our algorithm are updated on the slower timescale while the policy parameters themselves are updated on the faster scale. We perform a gradient search in the space of policy parameters. Since the objective function and hence the gradient are not analytically known, we employ the efficient one-simulation simultaneous perturbation stochastic approximation(SPSA)…