Original: Bayesian Risk Markov Decision Processes
Revised: Bayesian Risk Markov Decision Processes

Original: Anonymous Author(s)
Revised: Anonymous Author(s)

Original: AffiliationAddress email
Revised: AffiliationAddress email

Original: Abstract
Revised: Abstract

Original: We consider finite-horizon Markov Decision Processes where  parameters, such astransition probabilities, are unknown and estimated from data.
Revised: We consider finite-horizon Markov Decision Processes where distributional parameters, such as transition probabilities, are unknown and estimated from data.

Original: The popular distributionally robust approach to addressing the parameter uncertainty can sometimesbe overly conservative.
Revised: Thepopular distributionally robust approach to addressing the parameter uncertaintycan sometimes be overly conservative.

Original: In this paper, we propose a new formulation, Bayesianrisk Markov decision process (BR-MDP), to address parameter uncertainty in MDPs, where a risk functional is applied in nested form to the expected total costwith respect to the Bayesian posterior distributions of the unknown parameters.
Revised: In this paper, we propose a new formulation, Bayesian risk Markov decision process (BR-MDP), to address parameter uncertainty in MDPs, where a risk functional is applied in nested form to the expectedtotal cost with respect to the Bayesian posterior distributions of the unknown parameters.

Original: The proposed formulation provides more flexible risk attitudes towards parameteruncertainty and takes into account the availability of data in future time stages.
Revised: The proposed formulation provides more flexible risk attitudes towardsparameter uncertainty and takes into account the availability of data in future timestages.

Original: To solve the proposed formulation with the conditional value-at-risk (CVaR) riskfunctional, we propose an efficient approximation algorithm by deriving an analytical approximation of the value function and utilizing the convexity of CVaR.
Revised: To solve the proposed formulation with the conditional value-at-risk (CVaR)risk functional, we propose an efficient approximation algorithm by deriving ananalytical approximation of the value function and utilizing the convexity of CVaR.

Original: Wedemonstrate the empirical performance of the BR-MDP formulation and proposedalgorithms on a gambler’s betting problem and an inventory control problem.
Revised: We demonstrate the empirical performance of the BR-MDP formulation and theproposed algorithms on a gambler’s betting problem and an inventory controlproblem.

Original:  Introduction
Revised:  Introduction

Original: Markov decision process (MDP) is a paradigm for modeling sequential decision making underuncertainty.
Revised: Markov decision process (MDP) is a paradigm for modeling sequential decision making underuncertainty.

Original: From a modeling perspective, some parameters of MDPs are unknown and need tobe estimated from data.
Revised: From a modeling perspective, some parameters of MDPs are unknown and need tobe estimated from data.

Original: In this paper, we consider MDPs where transition probability and costparameters are not known.
Revised: In this paper, we consider MDPs where transition probability and costparameters are not known.

Original: A natural question would be: given a finite and probably small set of data,how does a decision maker find an “optimal” policy that minimizes the expected total cost under theuncertain transition probability and cost parameters?
Revised: A natural question would be: given a finite and probably small set of data,how does a decision maker find an “optimal” policy that minimizes the expected total cost under theuncertain transition probability and cost parameters?

Original: A possible approach that mitigates the parameter uncertainty (also known as epistemic uncertainty)lies in the framework ofdistributionally robust MDPs (DR-MDPs, [1]).
Revised: A possible approach that mitigates the parameter uncertainty lies in the framework of robust MDPs(e.g.

Original: DR-MDP regards theunknown parameters as random variables and assumes the associated distributions belong to anambiguity set that is constructed from the data.
Revised: In robust MDPs, parameters are usually assumed to belong to a known setreferred to as the ambiguity set, and the optimal decisions are chosen according to their performanceunder the worst possible parameter realizations within the ambiguity set. Later [6] extends thedistributionally robust approach to MDPs (DR-MDPs) with parameter uncertainty and utilizes theprobabilistic information of the unknown parameters. The distributionally robust approach originatesfrom stochastic optimization (e.g. [7], [8]), which regards the unknown parameters as randomvariables and assumes the associated distributions belong to an ambiguity set that is constructed fromthe data.

Original: DR-MDP then finds the optimal policy that minimizesthe expected total cost with the parameters following the most adversarial  distribution within theambiguity set.
Revised: DR-MDP then finds the optimal policy that minimizes the expected total cost with theparameters following the most adversarial (worst-case) distribution within the ambiguity set.

Original: However, distributionally robust approaches might yield overly conservative solutionsthat perform poorly for scenarios that are more likely to happen than the worst case.
Revised: However, distributionally robust approaches might yield overly conservative solutions that performpoorly for scenarios that are more likely to happen than the worst case.

Original: Moreover, aspointed out in [2], DR-MDP does not explicitly specify the dynamics of the considered problem inthe sense that the distribution of the unknown parameters does not depend on realizations of the dataprocess, and therefore is generally not time consistent (we refer the reader to [2] for details on timeconsistency).

Original: In view of the aforementioned drawbacks of DR-MDP, we propose anew formulation named as Bayesian risk MDP (BR-MDP), to address the parameter uncertainty in MDPs. BR-MDP takesa similar perspective as Bayesian risk optimization (BRO), which is a new framework proposedby [3, 4] for static (single stage) optimization.
Revised: In view of such drawback, [9,10] propose a Bayesian risk optimization (BRO) framework that reformulates a stochastic optimization problem with parameter uncertainty.

Original: They quantify the parameter uncertainty using a Bayesian posterior distribution  to replace the ambiguity set in DRO, and impose a risk functional onthe objective function with respect to the posterior distribution.
Revised: They quantify the parameter uncertainty using a Bayesianposterior distribution as opposed to an ambiguity set, and impose a risk functional on the objectivefunction with respect to the posterior distribution.

Original: To extend to BRO to MDPs and alsoensure time consistency, we propose to use a nested risk functional taken with respect to the processof Bayesian posterior distributions, where the posterior is updated with all realizations of randomnessup to each time stage. We show the proposed BR-MDP formulation is time consistent, and derivethe corresponding dynamic programming equation with an augmented state that incorporates theposterior information. The proposed framework works for an offline planning problem, where thedecision maker, after loaded with the learned optimal policy and put to the real environment, acts likeit adapts to the environment. To solve the proposed BR-MDP formulation, we develop an efficient algorithm by drawing aconnection between BR-MDP and partially observable MDP (POMDP) and utilizing the convexityof CVaR.
Revised: The risk functionals provide more flexible attitudestowards risk than the conservative worst-case measure.

Original: The connection between BR-MDP and POMDP is motivated by the observation that theposterior distribution inBR-MDP is exactly like the belief state (which is  the posterior distributionof the unobserved state given the history of observations) in a POMDP.
Revised: BRO is essentially seeking a trade-off betweenthe posterior expected performance and the robustness in the actual performance, which is measuredby the width of confidence intervals for the solution’s actual performance.

Original: The optimal value functionof a POMDP (in a minimization problem) can be expressed as an lower envelope of a set of linearfunctions (also called α -functions, see [5]). We show a similar α -function representation of thevalue function for BR-MDP, where the number of α -functions grow exponentially over time. Tohave computationally feasible algorithm, we further derive an analytical approximation of the valuefunction that keeps a constant number of α -functions over time. To summarize, the contributions of this paper are two folds.
Revised: Intuitively speaking,

Original: First, we propose a new time-consistentformulation BR-MDP to handle the parameter uncertainty in MDPs. Second, we propose an efficientalgorithm to solve the proposed formulation with a CVaR risk functional, and the algorithm can beeasily extended to other coherent risk measures (see [6] for an overview on coherent risk measures).
Revised: BRO framework tries to avoid a scenario where a solution performs well under the estimated modelbut performs badly under the true model, by possibly giving up some good expected performanceand trading for more confidence about the actual performance of a solution (in terms of width ofconfidence interval).

Original: Related Literature
Revised: Apart from the overly conservative concern, as pointed out in [11], DR-MDPdoes not explicitly specify the dynamics of the considered problem, in the sense that the distributionof the unknown parameters does not depend on realizations of the data process.

Original: One possible approach that mitigates the parameter uncertainty in MDPs lies in the framework ofrobust MDPs (e.g. [7, 8, 9, 10, 11]).
Revised: In this paper, we take a similar perspective as BRO in [9, 10] and propose a new Bayesian riskformulation for MDPs (BR-MDPs) to address the parameter uncertainty in MDPs.

Original: In robust MDPs, parameters are usually assumed to belong to aknown set referred to as the ambiguity set, and the  optimal decisions are chosen according to theirperformance under the worst possible parameter realizations within the ambiguity set. [1] furtherextends the distributionally robust approach to MDPs (DR-MDPs) with parameter uncertainty and utilizes the probabilistic information of the unknown parameters.
Revised: To clearly specifythe dynamics and the randomness in the system, we take the perspective of stochastic optimal controlto model the state transition by a difference equation that involves current state, next state, action, andthe randomness in the system (see Chapter 3.5 in [12]). We assume the distribution of the randomnessbelongs to some parametric family, and model the uncertainty over the unknown parameters via a

Original: Different from all the aforementionedworks, we take a Bayesian perspective in the BR-MDP formulation and seek a trade-off between theposterior expected performance and the robustness in the actual performance.
Revised: Bayesian posterior distribution that is updated based on the realization of the randomness at everytime stage.

Original: It is worth noting that applying the Bayesian approach to MDPs has been considered in [12], whichproposes a Bayes-adaptive MDP (BAMDP) formulation with an augmented state composed of theunderlying MDP state and the posterior distribution of the unknown parameters.
Revised: It is worth noting that applying the Bayesian approach to MDPs has been consideredin [13], which proposes a Bayes-adaptive MDP (BAMDP) formulation with an augmented statecomposed of the underlying MDP state and the posterior distribution of the unknown parameters.

Revised: In

Original: In BAMDP, eachtransition probability is treated as an unknown parameter associated with a Dirichlet prior distribution,and an expectation is taken with respect to the Dirichlet posterior on the expected total cost.
Revised: BAMDP, each transition probability is treated as an unknown parameter associated with a Dirichletprior distribution, and an expectation is taken with respect to the Dirichlet posterior on the expectedtotal cost.

Original: Incontrast, our BR-MDP formulation imposes a risk functional, taken with respect to the posteriordistribution (which could be chosen as Dirichlet distribution but is more general), on the expectedtotal cost in a nested form.
Revised: In contrast, our BR-MDP formulation imposes a risk functional, taken with respect to theposterior distribution (which could be chosen as Dirichlet distribution but is more general), on theexpected total cost in a nested form.

Original: On a related note, risk-averse decision making has been widely studied in MDPs.
Revised: On a related note, risk-averse decision making has been widely studied in MDPs.

Original: Apart from the robust MDPs that address the parameter uncertainty, risk-sensitive MDPs (e.g.[13, 14, 15, 16]) address the intrinsic uncertainty (also known as aleatoric uncertainty) that is dueto the inherent stochasticity of the underlying MDP, by replacing the risk-neutral expectation (withrespect to the state transition) by general risk measures, such as conditional value-at-risk (CVaR, see[17]).
Revised: Apart from therobust MDPs that address the parameter uncertainty (also termed as epistemic uncertainty), risksensitive MDPs (e.g. [14, 15, 16, 17]) address the intrinsic uncertainty (also termed as aleatoricuncertainty) that is due to the inherent stochasticity of the underlying MDP, by replacing the riskneutral expectation (with respect to the state transition) by general risk measures, such as conditionalvalue-at-risk (CVaR, see [18]).

Original: Most of the existing literature on risk-sensitive MDPs consider a static risk functional appliedto the total return (e.g. [18, 19, 20, 21]).
Revised: Most of the existing literature on risk-sensitive MDPs consider a staticrisk functional applied to the total return (e.g. [19, 20, 21, 22]).

Original: There are two recent works closely related to ours, both ofwhich apply a risk functional to BAMDP.
Revised: There are two recent works closelyrelated to ours, both of which apply a risk functional to BAMDP.

Original: Specifically, [22] formulates the risk-sensitive planningproblem as a two-player zero-sum game and then applies a static risk functional to the expected totalcost, which adds robustness to the incorrect priors over model parameters; [23] optimizes a CVaR riskfunctional over the total cost and simultaneously addresses both epistemic and aleatoric uncertainty.
Revised: Specifically, [23] formulates the risksensitive planning problem as a two-player zero-sum game and then applies a (static) risk functionalto the expected total cost, which adds robustness to the incorrect priors over model parameters; [24]optimizes a CVaR risk functional over the total cost and simultaneously addresses both epistemic andaleatoric uncertainty.

Original: In contrast, we consider a nested risk functional due to the  time consistency consideration discussedin previous section.
Revised: In contrast, we consider a nested risk functional and seek robustness with respectto the epistemic uncertainty. A primary motivation for considering such nested risk functional is theissue of time consistency (see [25, 26, 27, 11]) in the sense that the optimal solution to the formulatedproblem keeps optimal at every stage of the decision process with respect to the conditional riskfunctional. Optimizing a static risk measure can lead to “time-inconsistent” behavior, where theoptimal policy at the current time stage can become suboptimal in the next time stage simply becausea new piece of information is revealed (see [25, 26]). In this work, we show the proposed BR-MDPformulation is time consistent, and derive the corresponding dynamic programming equation withan augmented state that incorporates the posterior information.

Revised: The proposed framework works foran offline planning problem, where the decision maker, after loaded with the learned optimal policyand put to the real environment, acts like it adapts to the environment.

Original: Also note that our problem setting relies on partial knowledge of the model (state transition equation, cost function etc.) and works in an offline planning setting with no interactionwith the environment, and hence is different from Bayesian reinforcement learning (e.g. [24, 25]).
Revised: Also note that our problemsetting relies on partial knowledge of the model (state transition equation, cost function etc.) andworks in an offline planning setting with no interaction with the environment, and hence is differentfrom Bayesian reinforcement learning (e.g. [28, 29]).

Original: Preliminaries and Problem Formulation3.1 Preliminaries: BRO and CVaR
Revised: To solve the proposed BR-MDP formulation, we develop an efficient algorithm by drawing asimilarity between BR-MDP and partially observable MDP (POMDP) and utilizing the convexityof CVaR. In a POMDP, the optimal value function (in a minimization problem) can be expressedas an lower envelope of a set of linear functions (also called α -functions, see [30]). We show asimilar α -function representation of the value function for BR-MDP, where the number of α -functionsgrow exponentially over time. To have computationally feasible algorithm, we derive an analyticalapproximation of the value function that keeps a constant number of α -functions over time.

Revised: Notethat [31] also reformulates the BAMDP as a POMDP and proposes a point-based value iterationalgorithm.

Revised: However, their approach is not applicable for the risk-averse case, as it relies on thepiecewise linearity of the optimal value function, which does not hold for risk functionals other thanexpectation.

Revised: Additionally, [24] proposes an approximate algorithm based on Monte Carlo tree searchand Bayesian optimization to solve risk-averse BAMDP, but it works only for a static risk functionaland does not easily generalize to our formulation with a nested risk functional.

Revised: To summarize, the contributions of this paper are two folds.

Revised: First, we propose a Bayesian risk MDPformulation to handle the parameter uncertainty in MDPs.

Revised: Second, we propose an efficient algorithmto solve the proposed formulation with a CVaR risk functional, and the algorithm can be easilyextended to other coherent risk measures (see [32] for an overview on coherent risk measures).

Revised:  Preliminaries and problem formulation

Revised: 2.1 Preliminaries: Bayesian risk optimization and CVaR

Original: Bayesian risk optimization (BRO, see [3, 4]) considers a general stochastic optimization problem:min x E P c
Revised: Bayesian risk optimization (BRO, see [9, 10]) considers a general stochastic optimization problem:min x E P c

Original: [ h ( x, ξ )] , where x is the decision vector, ξ is a random vector with distribution P c , h isa deterministic cost function.
Revised: [ h ( x, ξ )] , where x is the decision vector, ξ is a random vector with distribution P c , h isa deterministic cost function.

Original: The true distribution P c is rarely known in practice and often needsto be estimated from data.
Revised: The true distribution P c is rarely known in practice and often needsto be estimated from data.

Original: It is very likely that the solution obtained from solving an estimatedmodel performs badly under the true model.
Revised: It is very likely that the solution obtained from solving an estimatedmodel performs badly under the true model.

Original: To avoid such a scenario, BRO seeks robustness inthe actual performance by imposing a risk functional to the expected cost function and solving thefollowing problem: min x ρ P n { E P θ [ h ( x, ξ )] } , where ρ is a risk functional, and P n is the posteriordistribution of θ after observing n data points.
Revised: To avoid such a scenario, BRO seeks robustness inthe actual performance by imposing a risk functional to the expected cost function and solving thefollowing problem: min x ρ P n { E P θ [ h ( x, ξ )] } , where ρ is a risk functional, and P n is the posteriordistribution of θ after observing n data points.

Original: It is assumed that the unknown distribution P c belongsto a parametric family { P θ | θ ∈ Θ } , where Θ is the parameter space and θ c ∈ Θ is the unknown trueparameter value.
Revised: It is assumed that the unknown distribution P c belongsto a parametric family { P θ | θ ∈ Θ } , where Θ is the parameter space and θ c ∈ Θ is the true parameter.

Original: Taking a Bayesian perspective, θ c is viewed as a realization of a random vector θ .
Revised: Through a Bayesian perspective, θ c can be viewed as a realization of a random variable θ .

Original: In particular, conditional value-at-risk (CVaR), a common coherent risk measure (see [6]), is considered for the risk functional.
Revised: In particular, conditional value at risk (CVaR), a common coherent risk measure (see [32]), isconsidered for the risk functional.

Original: For a random variable X defined on a probability space (Ω , F , P ) ,value-at-risk VaR α ( X ) is defined as the α -quantile of X , i.e., VaR α ( X ) :
Revised: For a random variable X defined on a probability space (Ω , F , P ) , VaR α ( X ) is defined as the α -quantile of X , i.e., VaR α ( X ) :

Original: = inf { t : P ( X ≤ t ) ≥ α } ,where the confidence level α ∈ (0 , 1) .
Revised: = inf { t : P ( X ≤ t ) ≥ α } , wherethe confidence level α takes value in (0 , 1) .

Original: Assuming there is no probability atom at VaR α ( X ) , CVaRat confidence level α is defined as the mean of the α -tail distribution of X , i.e., CVaR α ( X )=
Revised: Assuming there is no probability atom at VaR α ( X ) , CVaR at confidence level α is defined as the mean of the α -tail distribution of Z , i.e., CVaR α ( X )

Original: 107 E [ X | X ≥ VaR α ( X )] .
Revised: = 121 E [ X | X ≥ VaR α ( X )] .

Original: It is shown in [17] that CVaR can be written as a convex optimization:
Revised: It is shown in [18] that CVaR can be written as a convex optimization:

Original: [Equation]
Revised: [Equation]

Original: where ( · ) + stands for max(0 , · ) .

Original: 3.2 New formulation: Bayesian risk MDPs(BR-MDPs)
Revised: 2.2 New formulation: Bayesian risk MDPs

Original: Consider a finite-horizon MDP defined as ( S , A , P , C ) , where S is the state space, A is the actionspace, P is the transition probability with P ( s t +1 | s t , a t ) denoting the probability of transitioning tostate s t +1 from state s t when action a t is taken, C is the cost function with C t ( s t , a t , s t +1 ) denotingthe cost at time stage t when action a t is taken and state transitions from s t to s t +1 .
Revised: Consider a finite-horizon MDP defined as ( S , A , P , C ) , where S is the state space, A is the actionspace, P is the transition probability with P ( s t +1 | s t , a t ) denoting the probability of transitioning tostate s t +1 from state s t when action a t is taken, C is the cost function with C t ( s t , a t , s t +1 ) denotingthe cost at time stage t when action a t is taken and state transitions from s t to s t +1 .

Original: A Markoviandeterministic policy π is a function mapping from S to A .
Revised: A Markoviandeterministic policy π is a function mapping from S to A .

Original: Given an initial state s 0 , the goal is tofind an optimal policy that minimizes the expected total cost: min 
Revised: Given an initial state s 0 , the goal is tofind an optimal policy that minimizes the expected total cost: min π E π, P , C (cid:104)(cid:80) [Equation] s

Original: [Equation]
Revised: [Equation]

Original: where E π, P , C is the expectation with policy π when the transition probability is P and the cost is C .
Revised: ) (cid:105) ,where E π, P , C is the expectation with policy π when the transition probability is P and the cost is C .

Original: In practice, P and C are often unknown and estimated from data.
Revised: In practice, P and C are often unknown and estimated from data.

Original: To deal with the parameter uncertainty in MDPs, we propose anew formulation, Bayesian risk MDP (BR-MDP), by extending BRO  in static optimization to MDPs.
Revised: To deal with the parameter uncertainty in MDPs, we propose a Bayesian risk MDP formulation thatextends the BRO formulation in stochastic optimization to MDPs.

Original: We assume the state transitionis specified by the state equation s t +1 =
Revised: We assume the state transitionis specified by the state equation s t +1 =

Original: g t ( s t , a t , ξ t ) with a known transition function g t , whichinvolves state s t ∈ S⊆ R s , action a t ∈ A⊆ R a , and  randomness ξ t ∈ Ξ ⊆ R k , wheres, a, k are the dimensions of the state, action, and randomness, respectively.
Revised: g t ( s t , a t , ξ t ) with a known transition function g t , whichinvolves state s t ∈ S , action a t ∈ A , and the randomness ξ t ∈ Ξ in the system.

Original: We assume{ ξ  t , t =0 , · · · , T − 1 } are independent and identically distributed (i.i.d.).
Revised: We assume ξ tat time stages t = 0 , · · · , T − 1 to be independent and identically distributed.

Original: Note that the state equationtogether with the distribution of ξt uniquely determines the transition probability of the MDP, i.e., P ( s t +1∈ S ′ | s t , a t ) =
Revised: Note that the stateequation together with the distribution of ξ uniquely determines the transition probability of the MDP,i.e., P ( s t +1 | s t , a t ) =

Original: P ( { ξ t ∈ Ξ :g t ( s t , a t , ξ t ){++
+}
Revised: P (Ξ t ∈ { ξ t ∈ Ξ : s t {++
+}

Original: ∈ S ′ }| s t , a t) , where S ′ is a measurable set in S .
Revised: = g t ( s t , a t , ξ t ) } ) .

Original: We refer the readers to Chapter 3.5 in [26] for the equivalence between stochastic optimal control and MDP formulation.
Revised: We refer the readers to Chapter3.5 in [12] for the equivalence between stochastic optimal control and MDP formulation.

Original: We use the representation of state equations instead of transition probabilities in MDPs, for the purpose of decoupling the randomness and the policy, leading to a cleaner formulationin the nested form.
Revised: We usethe representation of state equation instead of transition probability in MDPs, for the purpose ofdecoupling the randomness and the policy, leading to a cleaner formulation in the nested form.

Original: We assume the distribution of ξt , denoted by f ( · ; θ c ) , belongs to a parametricfamily { f ( · ; θ ) | θ ∈ Θ } , where Θ⊆ R d is the parameter space,d is the dimension of the parameterθ , and θ c ∈ Θ is the true but unknown parameter value.
Revised: Weassume the distribution of ξ , denoted by f ( · ; θ c ) , belongs to a parametric family { f ( · ; θ ) | θ ∈ Θ } ,where Θ is the parameter space, and θ c ∈ Θ is the true but unknown parameter value.

Original: The parametric assumption is satisfied inmany problems; for example, in inventory control the customer demand is  often assumed to followa Poisson process (see [27]) with unknown arrival rate.
Revised: The parametricassumption implies that we have partial knowledge of the problem structure. For example, in inventorycontrol problems, the customer demand is unknown but often assumed to follow a Poisson process(see [33]).

Original: The cost at time stage t is assumed to be afunction of state s t , action a t , and randomness ξ t , i.e., C t ( s t , a t , ξ t ) .
Revised: The cost at time stage t is assumed to be a function of state s t , action a t , and randomnessξ t , i.e., C t ( s t , a t , ξ t ) .

Original: We start with a  prior distribution µ 0 over the parameter space Θ , which captures the initial uncertainty136 in the parameter estimate from an initial data set and can also incorporate expert opinion.
Revised: Now given a data set, we can compute the prior distribution µ 0 over the support Θ , which captures theinitial uncertainty in the parameter estimate.

Original: Then givenan observed realization of the data process, we update the posterior distribution µ t according to the138 Bayes’ rule.
Revised: Then given an observed realization of the data process,we update the posterior distribution µ t according to the Bayes’ rule.

Original: Let the policy be a sequence of mappings from  state s t and posterior µ t to the actionspace, i.e., π = { π t | 
Revised: Let the policy be a sequence ofmappings from physical state s t and posterior µ t to the action space, i.e., π = { π t | π t : S × M t →

Original: π t : S × M t → A , t = 0 , · · · , T − 1 } , where M t is the space of posterior140 distributions at time stage t .
Revised: A , t = 0 , · · · , T − 1 } , where M t is the space of posterior distributions at time stage t .

Original: Now we present the BR-MDP formulation  below.
Revised: The details ofthe BR-MDP formulation is presented below.

Original: [Equation]
Revised: [Equation]

Original: where a t = π t ( s t , µ t ) ,θ t is a random vector following distribution µ t , E f ( · ; θ t ) denotes the expectation with respect to ξ t ∼ f ( · ; θ t ) conditional on θ t , and ρ µ t denotes a risk functional  with respectto θ t ∼ µ t .We assume the last-stage cost only depends on the state, hence denoted by C T ( s T ) .
Revised: The cost at the last time stage only depends on the state, which is denoted by C T ( s T ) . Also notea t = π t ( s t , µ t ) , E f ( · ; θ t ) denotes the expectation with respect to ξ t ∼ f ( · ; θ t ) conditional on θ t , andρ µ t denotes a risk functional taken with respect to θ t ∼ µ t .

Original: Equation (3) is the transition of the state s t , and without loss of generality we assume  the initial states 0 takes a deterministic value.
Revised: Equation (3) is the transition of thephysical state s t , and without loss of generality we assume that the initial state s 0 takes a deterministicvalue.

Original: Equation (4) is the updating of the posterior µ t  given the  prior µ 0 .
Revised: Equation (4) is the update of the posterior µ t , given the initial prior µ 0 that is constructed fromthe historical data set.

Original: 3.3 Time consistency and dynamic programming
Revised: 2.3 Time consistency and dynamic programming

Original: It is important to note that the BR-MDP formulation (2) takes a nested form of the risk functional.
Revised: It is important to note that the BR-MDP formulation (2) takes a nested form of the risk functional toensure time consistency of its optimal policy.

Original: Aprimary motivation for considering such nested risk functional is the issue of time consistency (see[28, 29, 30, 2]), which means that the optimal policy solved at time stage 0 is still optimal for anyremaining time stage t ≥ 1with respect to the conditional risk functional, even when the realizationof the randomness ξ t is revealed up to that time stage. 
Revised: Time consistency, simply put, means that the optimalpolicy solved at time stage 0 is still optimal for any remaining time stage t ≥ 1 even when the realization of the randomness ξ t is revealed up to that time stage. To illustrate, consider the simple caseof a two-stage problem where the risk functional ρ is CVaR with confidence level α .

Original: In contrast, optimizing a static risk functional can lead to “time-inconsistent” behavior, wheretheoptimal policy at the current time stage can become suboptimal in the next time stage simply becausea new piece of information is revealed (see [28, 29]).
Revised: Our BR-MDPsolves a nested formulation min a 0 ,a 1 ρ µ 0 E f ( · ; θ 0 ) [ C 0 ( s 0 , a 0 , ξ 0 ) + ρ µ 1 E f ( · ; θ 1 ) [ C 1 ( s 1 , a 1 , ξ 1 )]] , whilethe non-nested counterpart solves min a 0 ,a 1 ρ µ 0 E [ C 0 ( s 0 , a 0 , ξ 0 ) + C 1 ( s 1 , a 1 , ξ 1 )] , where the expectation is taken with respect to (w.r.t.)

Original: Another drawback of  the staticformulationfor a general risk functional is the lack of dynamic programming equation.
Revised: the joint distribution of ξ 0 and ξ 1 , and the static risk functional isapplied to the total cost. Then we have the following relation between the two formulations:

Original: In particular, [2] pointsout that for the static formulation, the derivation of the dynamic programming equation is based onthe interchangeability principle and the decomposability property of the risk functional, where suchdecomposability property holds only for expectation and max-type risk functionals.
Revised: [Equation] where the inequality is justified by CVaR being the right tail average of the distribution, and theequality follows from the tower property of conditional expectation. Note that for the non-nested formulation, the derivation of the dynamic programming equation isbased on the interchangeability principle and the decomposability property of the risk functional, where such decomposability property holds only for expectation and max-type risk functionals (see[34]).

Original: On the otherhand, for our nested formulation (2), the corresponding dynamic programming equation is easilyobtained as follows:
Revised: On the other hand, for our nested formulation (2), the corresponding dynamic programmingequation is easily obtained as follows.

Original: [Equation] [Equation]
Revised: [Equation] [Equation]

Original:  where s t and µ t follow equation (3) and (4), respectively.
Revised: ) where s t +1 = g t ( s t , a t , ξ t ) , µ t +1 ( θ ) = µ t ( θ ) f ( ξ t ; θ ) (cid:82) Θ µ t ( θ ) f ( ξ t ; θ ) dθ , for t = 0 , · · · , T − 1 .

Original: Therefore, our BR-MDP formulationprovides a time-consistent risk-averse framework to deal with epistemic uncertainty in MDPs.
Revised: Therefore, our BRMDP formulation provides a time-consistent risk-averse framework to deal with epistemic uncertaintyin MDPs.

Original: Theexact dynamic programming is summarized in Algorithm 1, for benchmark purpose.
Revised: The exact dynamic programming is summarized in Algorithm 1.

Original: input : finite horizon T , initial state s 0 , prior distribution µ 0 ; output : optimal value function V ∗ 0 ( s 0 , µ 0 ) and corresponding optimal policy π ∗ ; set V ∗ T ( s T , µ T ) =
Revised: input : finite horizon T , initial state s 0 , prior distribution µ 0 ; output : optimal value function V ∗ 0 ( s 0 , µ 0 ) and corresponding optimal policy π ∗ ; set V ∗ T ( s T , µ T ) =

Original: C T ( s T ) , ∀ ( s T , µ T ) ∈ S
Revised: C T ( s T ) , ∀ ( s T , µ T ) ∈ S

Original: × M t ; for t ← T − 1 to 0 do for each ( s t , µ t ) ∈ S
Revised: × M t ; for t ← T − 1 to 0 do for each ( s t , µ t ) ∈ S

Original: × M t do solve dynamic programming equation (5); set π ∗ t ( s t , µ t ) :
Revised: × M t do solve dynamic programming equation (5); set π ∗ t ( s t , µ t ) :

Original: = a ∗ t , where a ∗ t attains V ∗ t ( s t , µ t ) ; end end
Revised: = a ∗ t , where a ∗ t attains V ∗ t ( s t , µ t ) ; end end

Original:  An Analytical Approximate Solution to BR-MDPs
Revised:  An analytical approximate solution to BR-MDPs

Original: Although the exact dynamic programming works for a general risk functional, there are two challengesto carry it out.
Revised: Although the exact dynamic programming works for a general risk functional, there are two challengesto carry it out.

Original: First, the expectation and the risk functional are generally impossible to computeanalytically and estimation by (nested) Monte Carlo simulation can be computationally expensive.
Revised: First, the expectation and the risk functional are generally impossible to computeanalytically and estimation by (nested) Monte Carlo simulation can be computationally expensive.

Original: Second, the update of the posterior distribution µ t does not have a closed form in general and often results in an infinite-dimensional posterior.
Revised: Second, the update of the posterior distribution µ t does not have a closed form in general and oftenresults in an infinite-dimensional posterior.

Original: We circumvent the latter difficulty by using conjugatefamilies of distributions (see Chapter 5 in [31]), where the posterior distribution falls into the sameparametric family as the prior distribution, and thus maintain the dimensionality of the posterior to bethe finite (and often small) dimension of the parameter space of the conjugate distribution.
Revised: We circumvent the latter difficulty by using conjugatefamilies of distributions (see Chapter 5 in [35]), where the posterior distribution falls into the sameparametric family as the prior distribution, and thus maintain the dimensionality of the posterior to bethe finite (and often small) dimension of the parameter space of the conjugate distribution.

Original: However,the posterior parameters usually take continuous values and hence BR-MDP with the augmented state( s, µ ) is  a continuous-state MDP.
Revised: However,the posterior parameters usually take continuous values. Therefore, BR-MDP with the augmentedstate ( s, µ ) is in fact a continuous-state MDP.

Original: We note that this continuous-state MDP resembles a belief-MDP,which is the equivalent transformation of a POMDP by regarding the posterior distribution of theunobserved state as a (belief) state (see [32]), and our posterior distribution µ t is just like the beliefstate in POMDP in  the sense that both are posterior distributions updated via Baye’s rule.
Revised: The computational cost for solving a continuous-state MDP increases exponentially in dimension if we simply discretize the continuous state.

Original: Thisobservation motivates our algorithm development in this section. Specifically, we derive an efficient approximation algorithm for the BR-MDP with the risk functional CVaR.
Revised: In view of the computational challenges in carrying out the exact dynamic programming, we derive anefficient approximation algorithm for the BR-MDP with the risk functional CVaR.

Original: The main idea is that for CVaR, once we know the variable u in (1), it is reduced to anexpectation of a convex function.
Revised: The main idea isthat for CVaR, once we know the variable u in (1), it is reduced to an expectation of a convex function.

Original: If there is a way to approximate the value function for a given u , wecan utilize the convexity of CVaR and apply gradient descent to solve for u .
Revised: If there is a way to approximate the value function for a given u , we can utilize the convexity of CVaR and apply gradient descent to solve for u .

Original: Hence, we proceed byfirst deriving the approximate value function for a fixed u and time stage.
Revised: Hence, we proceed by first deriving the approximatevalue function for a fixed u and time stage.

Revised: Similar to POMDP, by viewing the unknown parameterθ as unobservable state and physical state s as observable state, we maintain a belief (i.e. posteriordistribution) µ on θ .

Original: Similar to POMDP whosevalue function  can be represented by a set of so-called α -functions, we first show an α -functionrepresentation of the value function in BR-MDP, and then derive the approximate value functionbased on this representation.
Revised: It is well known that the value function in a POMDP can be represented bya set of so-called α -functions. We first show an α -function representation of the value function in BR-MDP, and then derive the approximate value function based on this representation.

Original: All proofs of the propositions/theorems below can be found in Appendix.

Original: 4.1 α -function representationof the value function
Revised: 3.1 α -function representation

Original: By the definition of CVaR (see (1)) with a confidence level α , we can rewrite the dynamic program- 191 ming equation (5) for the BR-MDP as:
Revised: By the definition of CVaR (see (1)) with a confidence level α , we can rewrite the dynamic program- 204 ming equation (5) for the BR-MDP as:

Original: [Equation]
Revised: [Equation]

Original: where we assume that ξ and θ take continuous values, and the integration can be numericallyapproximated by Monte Carlo sampling.
Revised: where we assume that ξ takes continuous values and the parameter space Θ is a finite set.

Original: If ξ and θ are discrete, the integral can be replaced bysummation.
Revised: If ξ isdiscrete, we can replace the integral by summation.

Revised: The assumption of a finite parameter space is practical in many real-world problems.

Revised: It also can be viewed as a discrete approximation of acontinuous parameter set and the discretization can be of any precision.

Original: The next proposition shows that the optimal value function corresponds to the lowerenvelope of a set of α -functions.
Revised: The next proposition showsthat the optimal value function corresponds to the lower envelope of a set of α -functions.

Original: Proposition 4.1 ( α -function representation) .
Revised: Proposition 3.1 ( α -function representation) .

Original: The optimal value function in (6) can be represented bythe lower envelope of a set of α -functions denoted by Γ t = { α t } a t ∈A , i.e.,
Revised: The optimal value function in (6) can be represented bythe lower envelope of a set of α -functions denoted by Γ t = { α t } a t ∈A , i.e.,

Original: [Equation]
Revised: [Equation]

Original: where α t ( s t , θ ) =

Original: u t + 1 1 − α (cid:16)(cid:82) Ξ f ( ξ ; θ ) (cid:16) C t ( s t , a t , ξ ) + min α t +1 (cid:82) Θ α [Equation] − u
Revised: The detailed proof of Proposition 3.1 can be found in the appendix.

Original: [Equation] + . Note that there is a major distinction between the α -function representations of a POMDP and a BR-MDP: the optimal value function in the risk-neutral POMDP is piecewise linear and convex inthe belief state (see [5]), whereas the optimal value function in BR-MDP is no longer piecewise linearin the posterior due to the ( · ) + operator in CVaR.
Revised: Note that there is a majordistinction between the α -function representations of a POMDP and a BR-MDP: the optimal valuefunction in the risk-neutral POMDP is piecewise linear and convex in the belief state (see [30]),whereas the optimal value function in BR-MDP is no longer piecewise linear in the posterior due tothe ( · ) + operator in CVaR.

Original: In addition, it is computationally impossible toobtain the α -functions except for the last time stage.
Revised: In addition, it is computationally impossible to obtain the α -functionsexcept for the last time stage.

Original: Specifically, denote the cardinality of the α t +1set as | Γ t +1 | .
Revised: Specifically, denote the cardinality of the α t +1 set as | Γ t +1 | .

Original: Note that for each realization of ξ there are | Γ t +1 | candidates forthe minimizer α∗ ( ξ ) t +1 ,which attains the minimum of (cid:82) Θ α t +1 ( s t +1 , θ ) µ t ( θ ) f ( ξ ; θ ) (cid:82) Θ µ t ( θ ) f ( ξ ; θ ) dθ dθ . There are a total of |A|| Γ t +1 | | Ξ |candidates for Γ t , let alone the optimization over u t . 
Revised: Since foreach realization of ξ there are | Γ t +1 | candidates for α t +1 , there are a total of |A|| Γ t +1 | | Ξ | candidatesfor α t , let alone the optimization over u t . When Ξ is uncountable (i.e., when ξ is continuous randomvariable), the set Γ t is also uncountable.

Original: To deal with the difficulty in computing theα -functions in POMDPs, [33] proposes several approximation algorithms, and later [34] extends theanalysis to a continuous-state optimal stopping problem by applying Jensen’s inequality to the exactvalue iteration in different ways and obtains a more computationally efficient approximation to theoptimal value function.
Revised: To deal with the difficulty in computing the α -functionsin POMDPs, [36] proposes several approximation algorithms, and later [37] extends the analysisto a continuous-state optimal stopping problem by applying Jensen’s inequality to the exact valueiteration in different ways and obtains a more computationally efficient approximation to the optimalvalue function.

Original: Inspired by these works, we derive an approximation approach next.
Revised: Inspired by these works, we derive the approximation approach in the next section.

Original: 4.2 α -function approximation
Revised: 3.2 α -function approximation

Original: In this section, we derive the α -function approximation for a fixed vector ( u 0 , u 1 , · · · , u T − 1 ) .
Revised: In this section, we derive the α -function approximation for a fixed vector ( u 0 , u 1 , · · · , u T − 1 ) .

Original: Without loss of generality, we assume the cost function at each time stage is non-negative (otherwise add a large constant to the cost at each time stage).
Revised: Without loss of generality, we assume the cost function at each time stage is nonnegative (otherwise add a large constant to the cost at each time stage).

Original: For ease of exposition,we rewrite (6) as V ∗ t ( s t , µ t ) = min a t ∈A ,u t ∈ R Q ∗ t ( s t , µ t , a t , u t ) , where Q ∗ t ( st , µ t , a t , u t ) =
Revised: For theease of exposition, we rewrite (6) as V ∗ t ( s t , µ t ) = min a t ∈A ,u t ∈ R Q ∗ t ( s t , µ t , a t , u t ) , where Q [Equation] s

Original: u t +1 1 − α (cid:82) Θ µ t ( θ ) (cid:0)(cid:82)
Revised: [Equation] + [Equation] +

Original: Ξ f ( ξ ; θ ) (cid:0) C t ( s t , a t , ξ ) + V ∗ t +1 ( s t +1 , µ t +1 ) (cid:1) dξ − u t (cid:1) +dθ .
Revised: [Equation] ( s [Equation] +

Original: Let V t ( s t , µ t ) :
Revised: [Equation] , µ [Equation] (cid:1)

Original: =min a t Q ∗ t ( s t , µ t , a t , u t ) be the “optimal” value function for a given u t .
Revised: [Equation] − u t (cid:1) + . Let V t ( s t , µ t ) : = min a t Q ∗ t ( s t , µ t , a t , u t ) be the “optimal” value function.

Original: Let V t ( s t , µ t ) :
Revised: Let V t ( s t , µ t ) :

Original: =min α t ∈ Γ t (cid:82) Θ α t ( s t , θ ) µ t ( θ )dθ , where Γ t = { α t } a t ∈A andα t ( s t , θ ) =
Revised: =min α t ∈ Γ t (cid:80) θ ∈ Θ α t ( s t , θ ) µ t ( θ ) , where Γ t = { α t } a t ∈A and

Original: u t + 1 − α (cid:90) Ξ (cid:18) C t ( s t , a t , ξ ) f ( ξ ; θ ) − u t + min α t +1 α t +1 ( s t +1 , θ ) f ( ξ ; θ ) (cid:19) dξ.
Revised: [Equation]

Original: V t ( s t , µ t ) serves as a lower bound for V t ( s t , µ t ) (see Proposition 4.2), and is similar to the fastinformed bound in POMDPs (see [33]).
Revised: V t ( s t , µ t ) serves as a lower bound for V t ( s t , µ t ) (see Proposition 3.2), and is similar to the fastinformed bound in POMDP (see [36]).

Original: Note that the set Γ t has a constant cardinality of |A| .
Revised: Note that the set Γ t has a constant cardinality of |A| .

Original: However, it involves a minimum within an integral, which can be hard to compute numerically.
Revised: However,it involves a minimum within an integral, which can be hard to compute numerically.

Original: Also,the lower bound is loose in the sense that it could be negative, while the true CVaR value is alwaysnon-negative (due to the non-negative cost).
Revised: Also, thelower bound is loose in the sense that it could be negative, while the true CVaR value is alwaysnon-negative (due to the non-negative cost).

Original: Next, let ¯ V t ( s t , µ t ) 
Revised: Next, let ¯ V t ( s t , µ t ) :

Original: [Equation] =min [Equation] ¯Γ
Revised: =

Original: [Equation]

Original: s

Original: [Equation] , θ ) µ [Equation] ( θ )dθ ,where ¯Γ t = { ¯ α t } a t ∈A and¯ α t ( s t , θ )
Revised: min ¯ α t ∈ ¯Γ t (cid:80) θ ∈ Θ ¯ α t ( s t , θ ) µ t ( θ ) ,where ¯Γ t = { ¯ α t } a t ∈A and

Original: [Equation]
Revised: [Equation]

Original: t + [Equation] ¯ V t ( s t , µ t ) serves as an upper bound for V t ( s t , µ t ) (see Proposition 4.2), and is similar to the unobservable MDP bound in POMDPs (see [33]), obtained by discarding all observations available tothe decision maker.
Revised: ¯ V t ( s t , µ t ) serves as an upper bound for V t ( s t , µ t ) (see Proposition 3.2), and is similar to the unobservable MDP bound in POMDP (see [36]), obtained by discarding all observations available tothe decision maker.

Original: Suppose the cardinality at time stage t + 1 is | ¯Γ t +1 | , then ¯Γ t has a total numberof |A|| ¯Γ t +1 | candidates.
Revised: Suppose the cardinality at time stage t + 1 is | ¯Γ t +1 | , then ¯Γ t has a cardinalityof |A|| ¯Γ t +1 | .

Original: In the following, we derive another approximate value function (cid:101) V t that isbounded by V t and ¯ V t , and is at least better than one of the above bounds.
Revised: In the following, we derive another approximate value function (cid:101) V t that is boundedby V t and ¯ V t , and is at least better than one of the above bounds.

Original: It keeps a constant number|A| of α -functions at each time stage, thus drastically reducing the computational complexity.
Revised: It keeps a constant number|A| of α -functions at each time stage, thus drastically reducing the computational complexity.

Original: Let
Revised: Let

Revised: [Equation]

Revised: V t

Revised: [Equation]

Revised: s t ,

Original: [Equation]
Revised: [Equation]

Original: t
Revised: t

Original: [Equation]
Revised: [Equation]

Original: θ
Revised: =

Original: [Equation]
Revised: [Equation]

Original: dθ , where (cid:101) Γ t = { (cid:101) α t } a t ∈A and
Revised: t ( s t , θ ) µ t ( θ ) , where (cid:101) Γ t = { (cid:101) α t } a t ∈A and(cid:101) [Equation] t ( s t , θ [Equation]

Revised: =

Original: [Equation]
Revised: [Equation]

Original: Proposition 4.2.
Revised: Proposition 3.2.

Original: For all t < T and any given u t ∈ R , the following inequalities hold:
Revised: For all t < T and any given u t ∈ R , the following inequalities hold:

Original: [Equation]
Revised: [Equation]

Revised: The detailed proof of Proposition 3.2 can be found in the appendix.

Original: To have an implementable algorithm, we need to use the approximate updating of α -functionsiteratively and replace the true α -functions at the t + 1 time stage in (7) by the approximate (cid:101) αfunctions from the previous iteration.
Revised: To have an implementablealgorithm, we need to use the approximate updating of α -functions iteratively and replace thetrue α -functions at the t + 1 time stage in (7) by the approximate (cid:101) α -functions from the previousiteration.

Original: It is clear that the iterative approximations preserve thedirections of the inequalities.
Revised: It is clear that the iterative approximations preserve the directions of the inequalities.

Original: Define u t : := ( u t , · · · , u T − 1 ) .
Revised: Defineu t : := ( u t , · · · , u T − 1 ) .

Original: The approximate value functionat time stage t for a given u t : is given by (cid:101) V t
Revised: The approximate value function at time stage t for a given u t : is given by(cid:101)

Original:  ( s t , µ t , u t : ) 
Revised: V t ( s t , µ t , u t : ) =

Original: = min ˜
Revised: min ˜

Original: α t ∈ ˜Γ t (cid:82) Θ (cid:101) α t ( s t , θ ) µ t ( θ )dθ , where˜Γ t = { ˜ α t } a t ∈A and 
Revised: α t ∈ ˜Γ t (cid:80) θ ∈ Θ (cid:101) α t ( s t , θ ) µ t ( θ ) , where ˜Γ t = { ˜ α t } a t ∈A and [Equation] t

Original: [Equation]
Revised: [Equation]

Revised: s

Revised: [Equation]

Revised: [Equation]

Revised: ) = u

Revised: [Equation]

Original: 4.3 Approximate dynamic programming with gradient descent
Revised: 3.3 Approximate dynamic programming with gradient descent

Original: In this section, we incorporate α -function approximation with gradient descent on ( u 0 , u 1 , · · · , u T − 1 )based on the convexity of the approximate value function w.r.t.
Revised: In this section, we incorporate α -function approximation with gradient descent on ( u 0 , u 1 , · · · , u T − 1 )based on the convexity of the approximate value function w.r.t.

Original: ( u 0 , u 1 , · · · , u T − 1 ) , as formallyshown in the theorem below.
Revised: ( u 0 , u 1 , · · · , u T − 1 ) , as formallyshown in the theorem below.

Original: Theorem 4.3.
Revised: Theorem 3.3.

Original: Suppose the cost function C t ( s, a, ξ ) is jointly convex in ( s, a ) for any fixed ξ , and thestate transition function g t ( s, a, ξ ) is jointly convex in ( s, a ) for any fixed ξ .
Revised: Suppose the cost function C t ( s, a, ξ ) is jointly convex in ( s, a ) for any fixed ξ , and thestate transition function g t ( s, a, ξ ) is jointly convex in ( s, a ) for any fixed ξ .

Original: Then the approximatevalue function (cid:101) V t ( s t , µ t , u t : ) is convex in u t :, for all t < T .
Revised: Then the approximatevalue function (cid:101) V t ( s t , µ t , u t : ) is convex in u t : .

Original: The jointly convex assumption in Theorem 4.3 is common for gradient-based algorithms for solvingmulti-stage decision making problems (e.g. [35]). It is satisfied in many real-world problems such as while some stopping criterion is met inventory control (e.g. [36]) and portfolio optimization (e.g. [37]).
Revised: The detailed proof of Theorem 3.3 can be found in the appendix.

Original: We present the full algorithm in
Revised: We give the full algorithm in

Original: Algorithm 2.
Revised: Algorithm 2.

Revised: Algorithm 2: Approximate dynamic programming for finite-horizon CVaR BR-MDPs.

Revised: input : finite horizon T , initial state s 0 , prior distribution µ 0 , initial vector u 0 = ( u 00 , u 01 , · · · , u 0 T − 1 ) , gradient descent step size η k for k = 0 , 1 , · · · ; initialization : set (cid:101) α T ( s T , θ ) =

Revised: C T ( s T ) , ∀ s T ∈ S , ∀ θ ∈ Θ ; set k = 0 .

Original: Note that in Algorithm 2, when applying the gradient descent, we need to  compute  thegradient of theapproximate value function w.r.t.{+(cid:101) V 0 ( s 0 , µ 0 , u k ) : = min (cid:101) α 0 (cid:80) θ ∈ Θ (cid:101) α 0 ( s 0 , θ ) µ 0 ( θ ) ; compute the gradient ∂ (cid:101) V 
 ∂u k , update the vector u k +
+}
Revised: while some stopping criterion is met do for t ← T − 1 to 0 do for each action a t ∈ A , compute (cid:101) α t ( s t , θ ) according to (8); end approximate the value function {+(cid:101) V 0 ( s 0 , µ 0 , u k ) : = min (cid:101) α 0 (cid:80) θ ∈ Θ (cid:101) α 0 ( s 0 , θ ) µ 0 ( θ ) ; compute the gradient ∂ (cid:101) V 
 ∂u k , update the vector u k +
+}

Original: the vector ( u  0  , u 1 , · · · , u T − 1) . {+end output the approximate value function (cid:101) V 
+}
Revised: = u k − η k ∂ (cid:101) V 0 ∂u k , set k = k + 1 . {+end output the approximate value function (cid:101) V 
+}

Original: For ∂ (cid:101) V 0 ∂u 0 , we have∂ (cid:101)
Revised: ( s 0 , µ 0 , u k ) ; output the optimal policy (cid:101)

Original: [-V∂u 
 = (cid:90)-]
Revised: [-V∂u 
 = (cid:90)-]π t ( s t , µ t ) := arg min a t ∈A (cid:80) θ ∈ Θ (cid:101) α t ( s t , a t , θ ) µ t ( θ )

Original: [Equation]
Revised: [Equation]

Original: [Equation]
Revised: Note that in Algorithm 2, when applying the gradient descent, we need to compute the gradient of theapproximate value function w.r.t. the vector ( u 0 , u 1 , · · · , u T − 1 ) .

Original: (cid:26)(cid:90) Ξ
Revised: For ∂ (cid:101) V 0 ∂u 0 , we have

Original: [Equation]
Revised: [Equation]

Original: where (cid:101) a ∗ 0 = arg min a 0 ∈A (cid:82) Θ (cid:101)
Revised: where (cid:101) a ∗ 0 = arg min a 0 ∈A (cid:80) θ ∈ Θ (cid:101)

Original: α 

Revised: α

Original: ( s 0 , θ ) µ 0 ( θ )dθ .
Revised: ( s 0 , θ ) µ 0 ( θ ) .

Original: For ∂ (cid:101) V 0 ∂u t , t = 1 , · · · , T − 1 , we have∂
Revised: For ∂ (cid:101) V 0 ∂u t , t = 1 , · · · , T − 1 , we have

Original: [Equation]
Revised: [Equation]

Original: V

Original: [Equation]
Revised: [Table]

Original: where ∂ (cid:101) α l ∂u t can be computed recursively from ∂ (cid:101)
Revised: where ∂ (cid:101) α l ∂u t can be computed recursively from ∂ (cid:101)

Original: α l +1 ∂u t for l = 1 , · · · , t − 1 .
Revised: α l +1 ∂u t for l = 1 , · · · , t − 1 .

Original: The approximate value function output by Algorithm 2 provides an upper bound on the optimal valuefunction, which is shown in the theorem below.
Revised: The approximate value function output by Algorithm 2 provides an upper bound on the optimal valuefunction, which is shown in the theorem below.

Original: Theorem 4.4.
Revised: Theorem 3.4.

Original: min u t : (cid:101) V t ( s t , µ t , u t : ) is an upper bound for the optimal value function V ∗ t ( s t , µ t ) .
Revised: min u t : (cid:101) V t ( s t , µ t , u t : ) is an upper bound for the optimal value function V ∗ t ( s t , µ t ) .

Revised: The detailed proof of Theorem 3.4 can be found in the appendix.

Original: Even though the approximate value function is an upper bound on the optimal value function,wewill later show in the numerical experiments that the gap between these two is small.
Revised: We will later show in the numericalexperiments that even though the approximate value function is an upper bound on the optimalvalue function, the gap between these two is small.

Original: As a final note,even though we develop the algorithm for the risk functional CVaR, it can be extended easily to othercoherent risk measures.
Revised: As a final note, even though we develop thealgorithm for the risk functional CVaR, it can be extended easily to other coherent risk measures.

Original: The detailed discussion can be found in Appendix.
Revised: More specifically, for a class of coherent risk measures which can be represented in the followingparametric form R ( Z ) : = inf λ ∈ Λ E [Ψ( Z, λ )] , where Ψ : R × Λ

Revised: → R is a real-valued function andΨ( z, λ ) is convex in ( z, λ ) , we can apply the same technique to approximate the α -functions for agiven vector λ 0 , · · · , λ T − 1 , and then apply gradient descent on the approximate value function.

Revised: Theconvergence is guaranteed due to the convexity.

Revised: We refer the readers to [38] for more discussion onthis class of coherent risk measures.

Original:  Numerical experiments
Revised:  Numerical experiments

Revised: Code in Python for the numerical experiments is included in the supplementary.

Revised: Computational timeis reported for a 1.4 GHz Intel Core i5 processor with 8 GB memory.

Original: We illustrate the performance of our proposed formulation and algorithms with two offline planningproblems.
Revised: We illustrate the performanceof our proposed formulation and algorithms with two offline planning problems.

Original: Gambler’s betting problem .
Revised: (1) Gambler’sbetting problem .

Original: Consider a gambler betting in the casino with initialmoney of s 0 .
Revised: Consider a gambler betting in the casino with initial money of s 0 .

Original: At each time stage the gambler chooses how much to bet from a set { 0 , 1 , 2 ,3 , 5  } .
Revised: At each timestage the gambler chooses how much to bet from a set { 0 , 1 , 2 , 5 , 10 } .

Original: The gambler bets for T = 6 rounds. The cost at each time stage is  − a · ξ , where a stands foraction of how much to bet, ξ = 2 stands for a win, ξ = − 1 stands for a loss, and the winningrate θ c = P ( ξ = 2) is unknown. We add a constant c = 10 to make the adjusted cost c − a · ξnon-negative to run our algorithm (since the algorithm requires non-negative stage-wise cost),and then subtract cT from the resultant total adjusted cost to recover the total cost.
Revised: To ensure the cost for eachtime stage is non-negative, we add a constant c = 10 to each time stage, and the cost is given byc − a · ξ , where a stands for action of how much to bet, ξ = 1 stands for a win and ξ = − 1 standsfor a loss, and the winning rate θ c = P ( ξ = 1) is unknown. The gambler bets for T = 6 rounds.

Original: The data setconsists of historical betting records with size N .
Revised: The data set consists of historical betting record with size 10.

Original: (2) Inventory control problem .
Revised: (2) Inventory control problem .

Original: Consider awarehouse manager with initial inventory level s 0 .
Revised: Consider a warehouse manager with initial inventory level s 0 .

Original: At each time stage the manager chooses howmuch to replenish from the set { 0 , 1 , · · · , M − s } , where M= 15 is the storage capacity.
Revised: At each time stage the managerchooses how much to replenish from the set { 0 , 1 , · · · , M − s } , where M is the storage capacity.

Original: Thecustomer demand is a random variable ξ following a Poisson distribution with parameter θ c truncatedbelow M C= 20 ,which is the maximal customer demand the warehouse can handle.
Revised: The customer demand is a random variable ξ , which is assumed to follow a Poisson distributionwith parameter θ c that is truncated below M C , the maximal customer demand the warehouse canhandle.

Original: The statetransition is given by s t +1 =max
Revised: The state transition is given by s t +1 =

Original: ( s t + a t − ξ t , 0) , the cost function is given by C t ( s t , a t , ξ t ) =
Revised: ( s t + a t − ξ t ) + , the cost function is given by C t ( s t , a t , ξ t ) =

Original: h t · max( s t + a t − ξ t , 0) + p t · max( ξ t − s t − a t , 0) , where h t is the holding cost and p t is the penalty cost.
Revised: h t · ( s t + a t − ξ t ) + + p t · ( ξ t − s t − a t ) + , where h t is the holding cost and p t is thepenalty cost.

Original: The final stage cost is set to 0 for simplicity.
Revised: The final stage cost is set to 0 for simplicity.

Original: The manager has to plan for T = 6 timestages.
Revised: The manager has to plan for T = 6 timestages.

Original: The data set consists of historical customer demands with size N .
Revised: The data set consists of historical customer demand with size 10.

Original: We compare the following approaches.
Revised: We compare the following approaches.

Original: (1)  BR-MDP (e.): exact dynamic programming (Algorithm 1),where the state s t and µ t are discretized to a fine grid to carry out the dynamic programming (see
Revised: (1) CVaR BR-MDP (exact): exact dynamic programming (Algorithm 1) presented in this paper.

Original: Appendix for details). (2)  BR-MDP (a.): approximate dynamic programming (Algorithm 2).
Revised: (2) CVaR BR-MDP (approx): approximate dynamic programming(Algorithm 2) presented in this paper.

Original:  Nominal: maximal likelihood estimation (MLE) estimator θ MLE is computed from the given data, andthen a policy is obtained by solving the MDP with parameter θ MLE .
Revised: (3) Nominal: maximal likelihood estimation (MLE) estimatorθ MLE is computed from the given data. The policy is obtained by solving an MDP with parameterθ MLE .

Original: (4) DR-MDP: distributionallyrobust MDP presented in [1].
Revised: (4) DR-MDP: distributionally robust MDP presented in [6].

Original: For each of the considered approaches, we obtain the correspondingoptimal policy for a same data set, and then evaluate the actual performance of the obtained policyon the true system, i.e., MDP with the true parameter θ c .
Revised: For each of the consideredapproaches, we obtain the corresponding optimal policy for a given data set, and then evaluate theactual performance of the obtained policy on the true system, i.e., MDP with the true distributionalparameter θ c .

Original: This is referred to as one replication, andwe repeat the experiments for 100 replications on different independent data sets.
Revised: This is referred to as one replication, and we repeat the experiments for 100 replications.

Original: Results for thegambler’s betting problem can be found in Table 1a,Table 2, Figure 1a and Figure 1b.
Revised: Results for the gambler’s betting problem can be found in Table 1a, Figure 1 and Figure 1b.

Original: Results for theinventory control problem can be found in Table 1b.
Revised: Resultsfor the inventory control problem can be found in Table 1b.

Original: [Table] [Table] Table 1 reports the averagecomputation time to obtain the optimal policy and the mean and varianceof the actual performance of the obtained policy over the 100 replications.
Revised: Table 1 reports the average time to obtain the optimal policy and the average actual performance ofthe obtained policy over the 100 replications.

Original: Table 2 reports the actualperformance of different formulations over 100 replications with different data size N = 5 , 10 , 100 . We have the following observations. (1) Robustness of BR-MDP : BR-MDP is the most robust inthe sense of balancing the mean (smaller mean cost) and variability (smaller variance) of the actualperformance ofits solution. In contrast, the nominal approach has much larger variance, especiallywhen the data size is small, indicating it is not robust against parameter uncertainty.
Revised: We have the following observations: (1) The nominalapproach has the largest variability (see standard deviation of the actual performance).

Original: On the otherhand, DR-MDP is overly conservative since the variance of actual performance is 0 and the mean isthe largest (i.e., the worst) among all approaches.
Revised: (2) DR-MDPis the most conservative since the standard deviation is 0, but this conservativeness is not alwaysdesirable.

Original: This conservativeness is often not desirable: forexample, in the betting problem,DR-MDP always chooses the conservative action “not bet”, whichis obviously not optimal when θ c (probability of winning) is large and  the goal is to minimize theexpected cost.
Revised: In the betting problem, the conservative action is not to bet. DR-MDP will always choosenot to bet, which is not optimal when θ c > 0 . This can be seen from the worst actual performanceof DR-MDP in the betting problem when θ c = 0 . 55 and in the inventory control problem when

Original: (2) Efficiency of the approximation algorithm for BR-MDP : the computation timeof the approximation algorithm for BR-MDP is less than 1/10 of that of the exact algorithm, whilethe performance (mean, variance) is not much different.
Revised: [Table] [Figure] [Figure] [Figure]

Original: (3) Larger data size reduces parameteruncertainty: as expected, as we have more data, the uncertainty about model parameters reduces.
Revised: [Figure] [Figure] [Table] [Figure]

Original: Hence, the benefit of considering future data realization in BR-MDP decreases compared to thenominal approach, resulting in their similar performance when the data size is large ( N = 100 inour examples).
Revised: [Table] [Figure]

Original: The reason is that both the posterior distribution (used in BR-MDP) and the MLEestimator (used in the nominal approach ) converge to the true parameter as data size goes to infinity.
Revised: θ c = 12 . (3) For CVaR BR-MDP, the computational time for the approximate algorithm is greatlyreduced compared to the exact algorithm.

Original: Figure 1a shows the histogram of actual performance over 100 replications for the nominal approachand CVaR BR-MDP (exact) with different confidence levels α = 0 .
Revised: Figure 1a shows the histogram of actual performance over100 replications for the nominal approach and CVaR BR-MDP (exact) with different confidence levelsα = 0 .

Original:  under  θ c = 0 .
Revised: 1 , 0 . 5 , 0 . 99 under distributional parameter θ c = 0 .

Original: We have the following observations combining Table 1 and Figure 1a: (1) Robustness of BR-MDP: BR-MDP  (both exact and approximate) produce more consistent solution performance across awide range of input data compared to the nominal approach, which can be seen from the smallervariance in Table 1 and more concentrated distribution of the actual performance in Figure 1a. 
Revised: We have the following observationscombining Table 1 and Figure 1a: (1) Our CVaR BR-MDP formulation (both exact and approximate)produces more consistent solution performance across a wide range of input data compared to thenominal approach, which can be seen from the smaller standard deviation in Table 1 and moreconcentrated distribution of the actual performance in Figure 1a. (2) Compared with DR-MDP, our

Original: Benefit of learning from future data realization (time consistency): Figure 1a shows that in thebetting problem with θ c = 0 .
Revised: BR-MDP formulation provides less conservative solution by not fixating on the worst-case scenario,which can be seen from the better actual performance in the inventory control problem (see Table 1b)and the betting problem (see Table 1a when θ c = 0 . 55 ). As such, our CVaR BR-MDP formulationbalances the trade-off between the expected performance and robustness in the actual performance. Figure 1a shows that in the betting problem with θ c = 0 .

Original: 45 , the nominal approach has 40 replications where MLE estimatorθ MLE < 0 .
Revised: 45 , the nominal approach has 20 replicationswhere MLE estimator θ MLE > 0 .

Original: 33 and the gambler choosesnot to bet, which is not the optimal action.
Revised: 5 and the gambler chooses to bet, which is not the optimal action.

Original: In contrast,  BR-MDP formulation learns from the future data realization and updates its posterior distribution onθ .
Revised: In contrast, CVaR BR-MDP formulation will learn from the future data realization and update itsposterior distribution on θ .

Original: As a result, in those 40 replications, the gambler initially chooses not to bet, but after some timechooses to bet, which results in the left-shift of the actual performance distribution.
Revised: As a result, in those 20 replications, the gambler initially chooses tobet, but after some time chooses not to bet, which results in the left-shift of the actual performancedistribution.

Original: This illustratestime consistency (or in other words, adaptivity to the data process) of our BR-MDP formulation.
Revised: This illustrates time consistency (or in other words, adaptivity to the data process) of our BR-MDP formulation.

Original: This illustration is even more evident by the comparison between DR-MDP and BR-MDP withα = 1 (CVaR with α = 1 corresponds to the worst-case measure), where the only difference is that BR-MDP takes a nested form of risk functional while DR-MDP uses a static one.
Revised: This illustration is even more evident by the comparison between DR-MDPand CVaR BR-MDP with α = 1 (CVaR with α = 1 corresponds to the worst-case measure), wherethe only difference is that BR-MDP takes a nested form of risk functional while DR-MDP uses astatic one.

Original: (3) Effect of risklevel: risk level α affects the conservativeness ofBR-MDP; as α increases, the gambler is morelikely to take a conservative action (which is not to bet), so the actual performance distribution willshift more to the right. (4) Effectiveness of the approximation algorithm for BR-MDP: Figure 1bplots the value function V ∗ 0 ( s 0 , µ 0 ) of BR-MDP (exact) and ˜ V ∗ 0 ( s 0 , µ 0 ) of BR-MDP (approx) underdifferent prior distributions µ 0 with θ c = 0 . 45 , verifying Theorem 4.4 that ˜ V ∗ 0 ( s 0 , µ 0 ) is indeed anupper bound for V ∗ 0 ( s 0 , µ 0 ) but the difference between these two is small. (a) Histogram  Conclusion
Revised: The confidence level α affects the conservativeness of the CVaR BR-MDP formulation.

Original: In this paper, we propose a new formulation, coined as Bayesian Risk MDP (BR-MDP), to providerobustness against parameter uncertainty in MDPs. BR-MDP is  a time-consistent formulation with adynamic risk functional that seeks the trade-off between the posterior expected performance and therobustness in the actual performance.
Revised: As α increases, the gambler is more likely to take a conservative action (which is not to bet), so theactual performance distribution will shift more to the left.

Original: For finite-horizon BR-MDP with the CVaR risk functional,we develop an efficient approximation algorithm by drawing a connection between BR-MDPs and POMDPs and deriving an approximate alpha-function representation that remains a low computationalcost. Our experiment results demonstrate the efficiency of the proposed approximate algorithm, andshow the robustness and the adaptivity to future data realization of the BR-MDP formulation. One of the limitations of our work is the parametric assumption on the distribution of randomness.
Revised: Lastly, Figure 1b plots the value function

Original: In 342 the future work, we wish to extend the BR-MDP formulation to non-parametric Bayesian setting, andevaluate the performance of the proposed formulation and algorithm on real-world data sets in more 344 challenging problems. In addition, the proposed alpha-function approximation algorithm provides anupper bound of the exact value, while there is no theoretical guarantee on the gap between the two. In 346 future we will develop more efficient approximation algorithms with a convergence guarantee, such asmethods based on stochastic dual dynamic programming. There are also other interesting directions, 348 such as extending the BR-MDP formulation to an infinite horizon problem and utilizing functionapproximation to improve the scalability of the proposed approach to more complex domains.
Revised: V ∗ 0 ( s 0 , µ 0 ) of CVaR BR-MDP (exact) and ˜ V ∗ 0 ( s 0 , µ 0 ) of CVaR BR-MDP (approx) under differentprior distributions µ 0 with θ c = 0 . 45 , verifying Theorem 3.4 that ˜ V ∗ 0 ( s 0 , µ 0 ) is indeed an upperbound for V ∗ 0 ( s 0 , µ 0 ) but the difference between these two is small.

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