A quantile is a popular performance measure for a stochastic system to evaluate its variability and risk. To reduce the risk, selecting the actions that minimize the tail quantiles of some loss distributions is typically of interest for decision makers. When the loss distribution is observed via simulations, evaluating and optimizing its quantile can be challenging, especially when the simulations are expensive as it may cost a large number of simulation runs to obtain accurate quantile estimators. In this work, we propose a multilevel metamodel (cokriging)-based algorithm to optimize quantiles more efficiently. Utilizing nondecreasing properties of quantiles, we first search on cheaper and informative lower quantiles, which are more accurate and easier to optimize. The quantile level iteratively increases to the objective level, and the search has a focus on the possible promising regions identified by the previous levels. This enables us to leverage the accurate information from the lower quantiles to find the optimums faster and improve algorithm efficiency.

It is common to encounter large-scale monotone inclusion problems where the objective has a finite sum structure. We develop a general framework for variance-reduced forward-backward splitting algorithms for this problem. This framework includes a number of existing deterministic and variance-reduced algorithms for function minimization as special cases, and it is also applicable to more general problems such as saddle-point problems and variational inequalities. With a carefully constructed Lyapunov function, we show that the algorithms covered by our framework enjoy a linear convergence rate in expectation under mild assumptions. We further consider Catalyst acceleration and asynchronous implementation to reduce the algorithmic complexity and computation time. We apply our proposed framework to a policy evaluation problem and a strongly monotone two-player game, both of which fall outside the realm of function minimization.

We motivate and propose a new model for non-cooperative Markov game which considers the interactions of risk-aware players. This model characterizes the time-consistent dynamic “risk” from both stochastic state transitions (inherent to the game) and randomized mixed strategies (due to all other players). An appropriate risk-aware equilibrium concept is proposed and the existence of such equilibria is demonstrated in stationary strategies by an application of Kakutani's fixed point theorem. We further propose a simulation-based Q-learning type algorithm for risk-aware equilibrium computation. This algorithm works with a special form of minimax risk measures which can naturally be written as saddle-point stochastic optimization problems, and covers many widely investigated risk measures. Finally, the almost sure convergence of this simulation-based algorithm to an equilibrium is demonstrated under some mild conditions. Our numerical experiments on a two player queuing game validate the properties of our model and algorithm, and demonstrate their worth and applicability in real life competitive decision-making.

We develop an optimal primal-dual first-order algorithm for a class of stochastic three-composite convex minimization problems. The convergence rate of our method not only improves upon the existing methods, but also matches a lower bound derived for all first-order methods that solve this problem. We extend our proposed algorithm to solve a composite stochastic program with any finite number of nonsmooth functions. In addition, we generalize an optimal stochastic alternating direction method of multipliers (SADMM) algorithm proposed for the two-composite case to solve this problem, and establish its connection to our optimal primal-dual algorithm. We perform extensive numerical experiments on a variety of machine learning applications to demonstrate the superiority of our method via-a-vis the state-of-the-art.

Fish stocks around the world are in danger from illegal fishing. In collaboration with the U.S. Coast Guard (USCG), we work to defend fisheries from illegal fisherman (henceforth called Lanchas) in the U.S. Gulf of Mexico. We have developed the COmPASS (Conservative Online Patrol ASSistant) system to design USCG patrols against the Lanchas. In this application, we face a population of Lanchas with heterogeneous behavior who fish frequently. We have some data about these Lanchas, but not enough to fit a statistical model. Previous security patrol assistants have focused on counterterrorism in one-shot games where adversaries are assumed to be perfectly rational, and much less data about their behavior is available. COmPASS is novel because: (i) it emphasizes environmental crime; (ii) it is based on a repeated Stackelberg game; (iii) it allows for bounded rationality of the Lanchas and it offers a robust approach against the heterogeneity of the Lancha population; and (iv) it can learn from sparse Lancha data. We report the effectiveness of COmPASS in the Gulf in our numerical experiments based on real fish data. The COmPASS system is to be tested by USCG.