Efficient computation of exact solutions for quantitative model checking

Quantitative model checkers for Markov Decision Processes typically use finite-precision arithmetic. If all the coefficients in the process are rational numbers, then the model checking results are rational, and so they can be computed exactly. However, exact techniques are generally too expensive or limited in scalability. In this paper we propose a method for obtaining exact results starting from an approximated solution in finite-precision arithmetic. The input of the method is a description of a scheduler, which can be obtained by a model checker using finite precision. Given a scheduler, we show how to obtain a corresponding basis in a linear-programming problem, in such a way that the basis is optimal whenever the scheduler attains the worst-case probability. This correspondence is already known for discounted MDPs, we show how to apply it in the undiscounted case provided that some preprocessing is done. Using the correspondence, the linear-programming problem can be solved in exact arithmetic starting from the basis obtained. As a consequence, the method finds the worst-case probability even if the scheduler provided by the model checker was not optimal. In our experiments, the calculation of exact solutions from a candidate scheduler is significantly faster than the calculation using the simplex method under exact arithmetic starting from a default basis.Comment: In Proceedings QAPL 2012, arXiv:1207.055

Communication-Computation Efficient Gradient Coding

This paper develops coding techniques to reduce the running time of distributed learning tasks. It characterizes the fundamental tradeoff to compute gradients (and more generally vector summations) in terms of three parameters: computation load, straggler tolerance and communication cost. It further gives an explicit coding scheme that achieves the optimal tradeoff based on recursive polynomial constructions, coding both across data subsets and vector components. As a result, the proposed scheme allows to minimize the running time for gradient computations. Implementations are made on Amazon EC2 clusters using Python with mpi4py package. Results show that the proposed scheme maintains the same generalization error while reducing the running time by $32\%$ compared to uncoded schemes and $23\%$ compared to prior coded schemes focusing only on stragglers (Tandon et al., ICML 2017)

Hamiltonian cycles and subsets of discounted occupational measures

We study a certain polytope arising from embedding the Hamiltonian cycle problem in a discounted Markov decision process. The Hamiltonian cycle problem can be reduced to finding particular extreme points of a certain polytope associated with the input graph. This polytope is a subset of the space of discounted occupational measures. We characterize the feasible bases of the polytope for a general input graph $G$, and determine the expected numbers of different types of feasible bases when the underlying graph is random. We utilize these results to demonstrate that augmenting certain additional constraints to reduce the polyhedral domain can eliminate a large number of feasible bases that do not correspond to Hamiltonian cycles. Finally, we develop a random walk algorithm on the feasible bases of the reduced polytope and present some numerical results. We conclude with a conjecture on the feasible bases of the reduced polytope.Comment: revised based on referees comment