Risk Aversion in Finite Markov Decision Processes Using Total Cost Criteria and Average Value at Risk
In this paper we present an algorithm to compute risk averse policies in Markov Decision Processes (MDP) when the total cost criterion is used together with the average value at risk (AVaR) metric. Risk averse policies are needed when large deviations from the expected behavior may have detrimental effects, and conventional MDP algorithms usually ignore this aspect. We provide conditions for the structure of the underlying MDP ensuring that approximations for the exact problem can be derived and solved efficiently. Our findings are novel inasmuch as average value at risk has not previously been considered in association with the total cost criterion. Our method is demonstrated in a rapid deployment scenario, whereby a robot is tasked with the objective of reaching a target location within a temporal deadline where increased speed is associated with increased probability of failure. We demonstrate that the proposed algorithm not only produces a risk averse policy reducing the probability of exceeding the expected temporal deadline, but also provides the statistical distribution of costs, thus offering a valuable analysis tool.
Markov Decision Processes (MDPs) are extensively used to solve sequential stochastic decision making problems in robotics  and other disciplines . A solution to an MDP problem instance provides a policy mapping states into actions with the property of optimizing (e.g., minimizing) in expectation a given objective function. In many practical situations a formulation based on expectation only is, however, not sufficient. This is the case, in particular, when variability in the system’s behavior can cause a highly undesirable outcome. For example, in an autonomous navigation system, a robot attempting to minimize the expected length of the traveled path will likely travel close to obstacles, and a large deviation from the planned path may result in a collision causing a huge loss (e.g., damage to an expensive robot or failure of the whole mission altogether). Metrics that study deviations from the expected value are often referred to as risk metrics.
Typical solution algorithms for MDP problems, like value iteration or policy iteration, aim exclusively at optimizing the expected cost. The computed policies are therefore labeled as risk neutral. The problem of quantifying risk associated with random variables has a rich history and is often related to the problem of managing financial assets . In fact, many risk-related studies motivated by financial problems have recently found applications in domains such as robotics . The term risk aversion refers to a preference for stochastic realizations with limited deviation from the expected value. In risk averse optimal control one may prefer a policy with higher cost in expectation but lower deviations to one with lower cost but possibly larger deviations. Particularly in the context of robotic planning, introducing risk aversion in MDPs is crucial to guarantee mission safety. However introducing risk aversion in MDPs creates a number of additional theoretical and computational hurdles. For example, in risk averse MDPs, optimal policies are not guaranteed to be Markov stationary but are instead history dependent.
Average Value at Risk (AVaR – also known as Conditional Value at Risk or CVaR) is a risk metric that has gained notable popularity in the area of risk averse control , . For a given random value and a predetermined confidence level, the AVaR is the tail average of the distribution exceeding a given confidence level (see Section 3 for a formal definition). Risk averse policies considering the AVaR metric have been studied for the case of MDPs with finite horizon and discounted infinite horizon cost criteria. In this paper we instead consider how the AVaR metric can be applied when an undiscounted, total cost criterion is considered. In fact, such cost criterion appears particularly useful and natural for robotic applications, whereby one is usually interested in optimizing the undiscounted, total cost accrued during a mission until a random, mission-dependent stopping time. (As an aside, the total cost criterion is the typical cost model for stochastic shortest path problems, see, e.g., [20, 21].) The contribution of this paper is three-fold:
We identify conditions for the underlying MDP ensuring that the AVaR MDP problem is well defined when the total cost criterion is used.
We define a surrogate MDP problem that can be efficiently solved and whose solution approximates the optimal policy for the original problem with arbitrary precision.
The rest of the paper is organized as follows. We discuss related work in Section 2 and provide some background about risk metrics and MDPs in Section 3. In Section 4 we formulate the risk-averse, total cost MDP problem we wish to solve. In Section 5 we propose and analyze an approximation strategy for the problem, and in Section 6 we provide an algorithmic solution. Simulation results for a rapid deployment problem are given in Section 7, and conclusions and future work are discussed in Section 8.
2 Related Work
For a general introduction to MDPs the reader is referred to textbooks such as  or more recent collections such as . As pointed out in the introduction, risk aversion in MDPs has been studied for over four decades, with earlier efforts focusing on exponential utility , mean-variance , and percentile risk criteria . With regard to mean-variance optimization in MDPs, it was recently shown that computing an optimal policy under a variance constraint is NP-hard . Recently, average value at risk was introduced in  in order to model the tail risk of a random outcome and to address some key limitations of the prevailing value-at-risk metric. Efficient methods to compute AVaR are discussed in . Leveraging the recent strides in AVaR risk modeling, there have been a number of efforts aimed at embedding the AVaR risk metric into risk-sensitive MDPs. In  the authors address the problem of minimizing the AVaR of the discounted cost over a finite and an infinite horizon, and propose a dynamic programming approach based on state augmentation. Similar techniques can be found in , where the authors propose a dynamic programming algorithm for finite-horizon, AVaR-constrained MDPs. The algorithm is proven to asymptotically converge to an optimal risk-constrained policy. However, the algorithm involves computing integrals over continuous variables (Algorithm 1 in ) and, in general, its implementation appears quite challenging. A different approach is taken by [16, 25, 7] where the authors consider a finite dimensional parameterization of the control policies, and show that an AVaR MDP can be optimized to a local optimum using stochastic gradient descent (policy gradient). However this approach imposes additional restrictions to the policy space and in general policy gradient algorithms only converge to a local optimum. Haskell and Jain recently considered the problem of risk aversion in MDPs using a framework based on occupancy measures  (closely connected to our recent works where constrained MDPs are used to solve the multirobot rapid deployment problem [8, 6]). While their findings are only valid for the case where an infinite horizon discounted cost criterion is considered, the solution we propose uses some of the ideas introduced in .
In this section we summarize some known concepts about risk metrics and MDPs. The reader is referred to the aforementioned references for more details.
Consider a probability space , and let be the space of all essentially bounded random variables on . A risk function (or risk metric) is a function that maps an uncertain outcome onto the real line . A risk function that is particularly popular in many financial applications is the value at risk. For the value at risk of at level is defined as
Here VaR represents the percentile value of outcome at confidence level . Despite its popularity, VaR has a number of limitations. In particular, VaR is not a coherent risk measure  and thus suffers from being unstable (high fluctuations under perturbations) when is not normally distributed. More importantly it does not quantify the losses that might be incurred beyond its value in the -tail of the distribution . An alternative measure that overcomes most shortcomings of VaR is the average value at risk, defined as
where is the confidence level as before. Intuitively, is the expectation of in the conditional distribution of its upper -tail. For this reason, it can be interpreted as a metric of “how bad is bad.” can be equivalently written as 
Furthermore, it has been recently shown in  that optimizing the CVaR of total reward is equivalent to optimizing the worst-case (robust) expected total reward of a system whose model uncertainty is subjected to a trajectory budget. This finding corroborates the fact that a CVaR risk metric models both the variability of random costs, as well as the robustness to system transition errors.
3.2 Total Cost, Transient Markov Decision Processes
For a finite set , let indicate the set of mass distributions with support on . A finite, discrete-time Markov Decision Process (MDP) is a tuple where
, the state space, is a finite set comprising elements.
, the control space, is a collection of finite sets . Set , , represents the actions that can be applied when in state . The set of allowable state/action pairs is defined as
is the transition probability from state to state when action is applied. According to our definitions, .
is a non-negative cost function. Specifically, is the cost incurred when executing action at state .
Let and note that the maximum is attained as is a finite set. Define the set of admissible histories up to time by , for , and . An element of has the form , and records all states traversed and actions taken up to time . In the most general case a policy is a function , i.e., it decides which action to take in state considering the entire state-action history. Note that according to this definition a policy is in general randomized. Let be the set of all policies, i.e., including history-dependent, randomized policies. It is well known that in the standard MDP setting where an expected cost is minimized there is no loss of optimality in restricting the optimization over deterministic, stationary Markovian policies, i.e., policies of the type . However, in the risk-averse setting one needs to consider the more general class of history-dependent policies . This is achieved through a state augmentation process described later.
Following , we define the countable space , where , is the sample space and is the Borel field on . Specific trajectories in the MDP are written as , and we denote by and the state and actions at time along trajectory . In general the exact initial state is unknown. Rather it is described by an initial mass distribution over , i.e., . A policy and initial distribution induce a probability distribution over , that we will indicate as .
In this paper we focus on transient total cost MDPs, defined as follows. Consider a partition of into sets and , i.e., and . A transient MDP is an MDP where each policy satisfies the following two properties:
for each , i.e., the state will eventually enter set , and
for each , , , i.e., once the state enters it cannot leave it.
A transient, total cost MDP is a transient MDP where
for each , i.e., once the state enters no additional cost is incurred,
the cost associated with each trajectory is given by
Note that the cost is a random variable depending on both the policy and the initial distribution . The name total cost stems from the fact that an (undiscounted) cost is incurred throughout the “lifetime” of the system (i.e., until the state hits the absorbing set).
Transient, total cost MDPs (closely related to stochastic shortest path problems, e.g., [20, 21]) represent an alternative to the more commonly used discounted, infinite-horizon MDPs or finite horizon MDPs. As outlined in the introduction, for many robotic applications the total cost, i.e., , is the most appropriate cost function. We justify this statement by noting that most robotic tasks have finite duration but such duration is usually not known in advance. In these circumstances the finite horizon cost is inappropriate because one cannot define the length of the finite horizon up front. Similarly, the discounted infinite horizon cost is also ill suited because the task does not continue forever and the cost will not be exponentially discounted over time.
Without loss of generality, we assume that set consists of a single absorbing state equipped with a single action , i.e., and with . In the following, with a slight abuse of notation, we denote by the set , i.e., we exclude the absorbing state from the definition of . Moreover, we assume that for a transient, total cost MDP , i.e., the probability of starting at the absorbing state is zero. In fact, whenever the resulting state trajectory will deterministically remain in , and the corresponding cost is zero.
4 Problem formulation
Our problem formulation relies on the following two technical assumptions necessary to establish an a-priori upper bound on the total cost incurred by any trajectory obtained under any policy, and to define an approximate problem that can be efficiently solved.
The first assumption simply requires that all costs in the transient states are positive (recall that we excluded when re-defining .) As it will be shown later, this assumption ensures a non-zero discretization step when approximating the cumulative cost accrued by a system throughout the trajectory until it is absorbed in .
Assumption 1 (Positivity of costs)
All costs in except for state are positive and bounded, i.e., .
When considering cost criteria like finite horizon or discounted infinite
horizon with a finite state space, an a-priori upper bound on the accrued cost can be immediately
established assuming that all costs are finite (a fact crucially exploited in ). However, the situation is more complex
when considering the total cost case, because without introducing
further hypotheses on the structure of the MDP a malicious adversary could
establish an history-dependent policy capable of invalidating any
a-priori established bound on the cost
Assumption 2 (Reachability of MDP)
Let be the Markov chain induced by the inputs . Then the absorbing state , under Markov chain , is reachable from any state , for all .
We recall that a state in a Markov chain is said reachable from another state if there exists an integer such that the probability that the chain will be in state after transitions is positive . Note that when Assumption 2 holds, under every policy there is a path of non-zero probability connecting every state to . Therefore, it is impossible to devise a policy that prevents sure absorption for an arbitrary number of steps. This holds for all policies, including history dependent policies (see Figure 1).
Building upon the previous material, we can now define the problem we aim to solve in this paper:
Note that, under the assumption of transient total cost MDP, one can easily verify that . Since, by equation (1), , one obtains for all , as well. However, to derive an optimization algorithm for the computation of it is necessary to formulate an a-priori upper bound for the optimal cost in (2). Assumptions 1 and 2 are introduced to ensure that such bound exists and can be computed.
5 Approximation Strategy
In this section we study an approximation strategy for the risk averse total cost MDP in equation (2). Similar to the method presented in , we aim at solving the problem by using the concept of occupation measures. However, unlike for the cases studied in , in total cost MDPs an explicit upper bound for the accrued cost is not available, which makes the solution strategy in  not applicable. Our strategy is to find a surrogate to problem (2). By imposing an effective horizon, we construct a total cost MDP with time-out and recast this problem into a bilinear programming problem. Furthermore we characterize the sub-optimality gap for such surrogate approximation. We start with a technical result characterizing the convergence rate to the absorbing state.
5.1 Convergence rate to the absorbing state
Consider a selection of inputs and the corresponding Markov chain . For each state , let denote the simple (i.e., without cycles) path from to of lowest, strictly positive probability. Note that exists due to Assumption 2. Let be the probability of the path, i.e., the product of the probabilities of all the transitions along the path. Since there are nodes and by definition the path is simple, includes at most transitions between nodes. Let
Note that the minimum is achieved as the minimization is over a finite set, and that is strictly positive due to Assumption 2. The constant lower bounds the probability that, under any policy , the absorbing state is reached in no more than steps, from any state . We are now in a position to characterize the convergence rate to the absorbing state.
Lemma 1 (Number of stages to reach the absorbing set)
For any policy and initial distribution ,
The claim is proven by induction on . Base case: we prove that
Indeed, . Because of Assumption 1, for any policy , , and the base case follows.
For the inductive step, assume that , for some . Then, . By definition of ,
and the claim follows.
5.2 Surrogate problem and approximation bounds
Our solution strategy is to solve a surrogate problem, whereby after a
deterministic number of steps, the state moves to the absorbing state surely. In other words, acts as a “timeout” for the MDP
The surrogate problem is simpler to solve, and we will show in the following that
its solution can approximate the solution of the original problem with
We want to characterize the relation between (i.e., the risk for the surrogate problem) and (i.e., the risk for the original problem). To this end, let be the total cost for the original problem up to time , i.e.,
The following lemma shows the equivalence between and .
Lemma 2 (Correspondence of costs)
For any policy and any trajectory , .
Given a policy , for any trajectory , the cost cumulated up to time is the same for both the original and the surrogate problem. After time , both and do not cumulate any additional cost, then the claim follows.
The following theorem represents the main result of this section
Theorem 3 (Suboptimality bound)
The surrogate problem approximates the original problem according to
The left inequality is proven by noticing that , where the equality follows from Lemma 2.
We now prove the right inequality. For any and policy , one has
Let be the tail cumulated cost, and, as before, . Since the function is sub-additive, i.e., and the expectation operator preserves monotonicity, one obtains the inequality
Furthermore, for each trajectory in the event set , one has
Collecting the results so far, one has the following inequalities:
Equation (4) implies
where the second to last equality follows from Lemma 2. We are left with the task of upper bounding . To this purpose, one can write
Note that the result in Lemma 1 implies
The claim then follows immediately, as the above upper bound is policy-independent.
Note that according to Theorem 3, as , the optimal cost of the surrogate problem recovers the optimal cost of the original problem, i.e., the surrogate problem provides a consistent approximation to the original problem, with a sub optimality factor that is computable from problem data.
6 Solution Algorithm
Leveraging the surrogate problem from the previous section, we can now adapt the results proposed in  to solve problem (2). An essential step to solve this optimization problem is to compute , which entails deriving the probability distribution for the possible costs generated by the random variable . This problem can be solved by suitably augmenting the state space as described in the following, and then using occupancy measures. In the space of occupancy measures, an optimal policy is determined through the solution of a bilinear program, as explained below. For a given policy and initial distribution , we define the occupancy measure for as
Note that is non negative but is in general not a probability itself. In the following we will use occupancy measures to determine the probability distribution of the total costs and then to compute the needed expectation. According to the definition, occupancy measures depend on the policy and the initial distribution . Given an absorbing MDP , we define a new state-augmented absorbing MDP with additional state components that track the cumulated total cost and current stage. Although the original MDP is finite and absorbing, the set of costs generated by all possible policies can be very large, and this can subsequently lead to a linear program with an unmanageable number of decision variables. To counter this problem, we introduce a discretized approximation for whose error can be arbitrarily bounded. To this end, we set , where is a parameter describing the desired number of discretized values for the cumulated cost. Due to Assumption 1, is strictly positive. The effective number of different values is This value may be higher than due to our definition of . We then define a new MDP as follows. Its state space is , where and . Elements in the augmented states will be indicated as . As clarified in the following, the two additional components store the cumulated running cost () and current stage (). Recall that in the surrogate problem, after steps, the state is guaranteed to have entered the absorbing set, i.e., it is guaranteed that . Thus the value of the component is in . On the other hand the input sets are defined as . and induce a new set . The new cost function is . The transition probability function is modified as follows:
As evident from the definition of the new transition function, the new variables included in the state stores the discretized
Note that the properties of carry over to . In particular, if Assumptions 1-2
hold for then they hold for too,
and, if is absorbing, then is also absorbing. Thus we indicate with its transient set of states.
For each and each , there exists a discretization step such that .
Let be the approximation error at time . Note
that by definition and .
From the definition of the transition probability function, , it follows that ,
which implies . Since for we have
, the claim follows.
A key step towards the solution of problem in (5) is therefore to derive the statistical description of the discretized total cost that is used to approximate . This objective can be achieved by exploiting the occupancy measures for the state-augmented MDP . For , the occupancy measure on induced by a policy and an initial distribution is given as:
The occupancy measure, , is a vector in , i.e., it is a vector with non negative components. The set of legitimate occupancy vectors is constrained by the initial distribution and defined by the policy . It is well known  that these constraints can be expressed as follows:
where if and only if . For we introduce random variables with the property that . This is easily achieved using occupancy measures, i.e., where is the indicator function equal to 1 when its argument is true, and 0 otherwise. Note that by definition is equal to , and by Theorem 4 approximates with arbitrary precision. Combining the above definitions we then get to the following problem whose solution approximates the solution to (5):
When comparing this last optimization problem with (5), the reader will note that the variable is constrained in the interval . Indeed, the objective function is continuous with respect to , and it is straightforward to verify that the partial derivative of the objective function with respect to is negative for and positive for . The objective function given in Eq. (7) is concave with respect to and is defined over a convex feasibility set . To the best of our knowledge, there exist no efficient methods to determine the global minimum for this class of problems. Hence, the problem is approximately solved fixing different values of within the range , and then solving the corresponding linear problem over the optimization variables and . Comparing the problem in Eq. (7) with the one in Eq. (2) one might initially think that the objective function in Eq. (2) does not depend on the policy . However, the dependency on is carried over by the occupancy measure , as evident from Eq. (6). Moreover, it is well known from the theory of constrained MDPs  that there is a one to one correspondence between policies and occupancy measures, i.e., every policy defines a unique occupancy measure and every occupancy measure induces a policy.
7 Numerical Experiments
To illustrate the performance of the proposed algorithm, we adopt the rapid deployment scenario considered in [6, 8]. A graph is used to abstract and model the connectivity of a given map of an environment (see, e.g., ). One robot is positioned at a start vertex and is tasked to reach the goal vertex within a given temporal deadline while providing some guarantee about its probability of successfully completing the task. When moving from vertex to vertex, the robot can choose from a set of actions, each trading off speed with probability of success. In particular, actions with rapid transitions between two vertices have higher probability of failure; and conversely when the robot moves slowly between two vertices it has a higher probability of success. In this scenario, failure means that the robot does not move (e.g., fails to pass through an opening), so elapsed time increases without making progress towards the goal. With a given temporal deadline and success probability , the robot is tasked to reach the target vertex “safely” (such that the true mission success probability is at least ), while satisfying the temporal constraint. From a design perspective it is of interest to know if there exists a policy achieving this objective, and to compute it. If the policy does not exist, it is of interest to know how to modify the parameters in order to make the task feasible.
In our previous work we solved this problem by modeling it
using Constrained Markov Decision Processes (CMDP).
In the CMDP approach, one maximizes the probability of success
while imposing a constraint on the temporal deadline.
However, this method only returns risk-neutral policies, i.e., the resultant policies only guarantee that the temporal deadline is met in expectation, and there is no explicit control on the tail probability of the constraint.
As a radical departure from the original problem formulation, the AVaR minimization method proposed in Eq. (2) searches for a policy that is feasible with respect to the temporal deadline constraint
To illustrate the performance of risk-averse deployment, two different policies are compared. Here both policies are computed using unconstrained stochastic control methodologies for which the immediate cost is the travel time between two vertices, and the actions correspond to all possible node transitions on the graph. The first is the classic risk-neutral policy obtained with value iteration. The second is a risk averse policy obtained with the algorithm presented in this paper using . For each policy, 1000 executions are run, and the distribution of total travel time is reported. Figures 2 and 3 show the distributions for the two cases. The risk-neutral policy obtains a lower expected cost, but has a longer tail, as evidenced by the 61 instances with a cost larger than or equal to 15 (notice that is the desired time of completion in this example). Moreover, as evidenced by the shape of the histogram, costs are more spread out. The risk averse policy, on the other hand, results in less variability as desired. Less than 30 instances have a cost larger or equal than 15, a reduction of the weight of the tail by more than one half.
Importantly, when computing a risk-neutral policy using classic methods like policy iteration or value iteration, one is merely provided with a policy that minimizes the expected cost (in our case time to completion), and no additional information is readily available. With our approach instead, one not only obtains a policy minimizing the AVaR criterion, but a statistical description of the costs is also obtained as a byproduct. That is to say, for each discretized completion time , the probability is computed as well, thus unveiling the relationship between the time to complete the deployment task and its probability. This is shown in Figure 4 for different values of . Hence, if the computed policy does not meet the desired performance, the designer has information on how to tune and .
In this paper we have considered how risk aversion in MDPs can be introduced jointly with the AVaR risk metric under the total cost criterion. Our results advance the state of the art because AVaR has only been previously considered in MDPs with finite horizon or discounted infinite horizon cost criteria. Such extension is important as the total cost criterion appears as a natural model for robotic applications, and is non-straightforward as current algorithms, e.g., from  and , only work with bounded cumulated costs (which is not the case for total cost formulations). Under two mild assumptions, an approximation algorithm with provable sub-optimality gap was provided. Furthermore, a rapid deployment scenario was used to demonstrate that risk-aversion gives more informative policies when compared to traditional risk-neutral formulations. While our findings focus on risk averse MDPs with an AVaR risk metric, our approach can be easily extended along multiple dimensions. In particular, by exploiting the results presented in , it is possible to use our approximation for a broader range of risk metrics, i.e., metrics that are uniformly continuous and law invariant. Moreover, since the algorithm we considered is based on occupancy measures, it can be easily extended to the CMDP case. This will be the focus of future work.
- First note that we are seeking a uniform upper bound for all possible policies, including history dependent policies. Hence, given a tentative bound , in the general case one could devise a history dependent policy ensuring that every trajectory generated by the policy is not absorbed in in less than steps, thus invalidating the bound.
- An alternative strategy would be to investigate reductions of problem (2) to an equivalent risk-averse, discounted, infinite-horizon problem by using, e.g., the results recently presented in , and then apply the approach in  to the reformulated problem. This is an interesting direction left for future research.
- To be precise, the discretized running cost is scaled by .
- margin: [ZS]: Why is this needed? It doesn’t appear to be used anywhere. If you keep it, “its” is ambiguous. Do you mean ’s set of transient states?
- For any random variable with finite expectation, for . Therefore, if the solution to the AVaR minimization problem is bounded above by the temporal deadline, then the corresponding minimizer is also a feasible policy to the original problem.
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