Markov Decision Problems Where Means Bound Variances

We identify a rich class of finite-horizon Markov decision problems (MDPs) for which the variance of the optimal total reward can be bounded by a simple linear function of its expected value. The class is characterized by three natural properties: reward nonnegativity and boundedness, existence of a do-nothing action, and optimal action monotonicity. These properties are commonly present and typically easy to check. Implications of the class properties and of the variance bound are illustrated by examples of MDPs from operations research, operations management, financial engineering, and combinatorial optimization

State-Augmentation Transformations for Risk-Sensitive Reinforcement Learning

In the framework of MDP, although the general reward function takes three arguments-current state, action, and successor state; it is often simplified to a function of two arguments-current state and action. The former is called a transition-based reward function, whereas the latter is called a state-based reward function. When the objective involves the expected cumulative reward only, this simplification works perfectly. However, when the objective is risk-sensitive, this simplification leads to an incorrect value. We present state-augmentation transformations (SATs), which preserve the reward sequences as well as the reward distributions and the optimal policy in risk-sensitive reinforcement learning. In risk-sensitive scenarios, firstly we prove that, for every MDP with a stochastic transition-based reward function, there exists an MDP with a deterministic state-based reward function, such that for any given (randomized) policy for the first MDP, there exists a corresponding policy for the second MDP, such that both Markov reward processes share the same reward sequence. Secondly we illustrate that two situations require the proposed SATs in an inventory control problem. One could be using Q-learning (or other learning methods) on MDPs with transition-based reward functions, and the other could be using methods, which are for the Markov processes with a deterministic state-based reward functions, on the Markov processes with general reward functions. We show the advantage of the SATs by considering Value-at-Risk as an example, which is a risk measure on the reward distribution instead of the measures (such as mean and variance) of the distribution. We illustrate the error in the reward distribution estimation from the direct use of Q-learning, and show how the SATs enable a variance formula to work on Markov processes with general reward functions

TA11 -8~30 VARIABILITY SENSITIVE MARKOV DECISION PROCESSES*

Abstract The time-average Markov Decisiou Procesncn with fiuitc state and action spaces are considered. Several definitionn of varialdity are introduced aid coinparetl. It in nliown that a ntatioiiary policy niaxiinizcn oue of these criteria, iiaincly, the cxpcctlcd long-run average variability. Furtlicriuorc, ail algoritllul is given wliicli prodiiccs such an optiuial statiouary policy

Trading Performance for Stability in Markov Decision Processes

We study the complexity of central controller synthesis problems for finite-state Markov decision processes, where the objective is to optimize both the expected mean-payoff performance of the system and its stability. We argue that the basic theoretical notion of expressing the stability in terms of the variance of the mean-payoff (called global variance in our paper) is not always sufficient, since it ignores possible instabilities on respective runs. For this reason we propose alernative definitions of stability, which we call local and hybrid variance, and which express how rewards on each run deviate from the run's own mean-payoff and from the expected mean-payoff, respectively. We show that a strategy ensuring both the expected mean-payoff and the variance below given bounds requires randomization and memory, under all the above semantics of variance. We then look at the problem of determining whether there is a such a strategy. For the global variance, we show that the problem is in PSPACE, and that the answer can be approximated in pseudo-polynomial time. For the hybrid variance, the analogous decision problem is in NP, and a polynomial-time approximating algorithm also exists. For local variance, we show that the decision problem is in NP. Since the overall performance can be traded for stability (and vice versa), we also present algorithms for approximating the associated Pareto curve in all the three cases. Finally, we study a special case of the decision problems, where we require a given expected mean-payoff together with zero variance. Here we show that the problems can be all solved in polynomial time.Comment: Extended version of a paper presented at LICS 201

Adaptive Experimental Design with Temporal Interference: A Maximum Likelihood Approach

Suppose an online platform wants to compare a treatment and control policy, e.g., two different matching algorithms in a ridesharing system, or two different inventory management algorithms in an online retail site. Standard randomized controlled trials are typically not feasible, since the goal is to estimate policy performance on the entire system. Instead, the typical current practice involves dynamically alternating between the two policies for fixed lengths of time, and comparing the average performance of each over the intervals in which they were run as an estimate of the treatment effect. However, this approach suffers from *temporal interference*: one algorithm alters the state of the system as seen by the second algorithm, biasing estimates of the treatment effect. Further, the simple non-adaptive nature of such designs implies they are not sample efficient. We develop a benchmark theoretical model in which to study optimal experimental design for this setting. We view testing the two policies as the problem of estimating the steady state difference in reward between two unknown Markov chains (i.e., policies). We assume estimation of the steady state reward for each chain proceeds via nonparametric maximum likelihood, and search for consistent (i.e., asymptotically unbiased) experimental designs that are efficient (i.e., asymptotically minimum variance). Characterizing such designs is equivalent to a Markov decision problem with a minimum variance objective; such problems generally do not admit tractable solutions. Remarkably, in our setting, using a novel application of classical martingale analysis of Markov chains via Poisson's equation, we characterize efficient designs via a succinct convex optimization problem. We use this characterization to propose a consistent, efficient online experimental design that adaptively samples the two Markov chains