Risk-Sensitive and Robust Decision-Making: a CVaR Optimization Approach (1506.02188v1)

Abstract: In this paper we address the problem of decision making within a Markov decision process (MDP) framework where risk and modeling errors are taken into account. Our approach is to minimize a risk-sensitive conditional-value-at-risk (CVaR) objective, as opposed to a standard risk-neutral expectation. We refer to such problem as CVaR MDP. Our first contribution is to show that a CVaR objective, besides capturing risk sensitivity, has an alternative interpretation as expected cost under worst-case modeling errors, for a given error budget. This result, which is of independent interest, motivates CVaR MDPs as a unifying framework for risk-sensitive and robust decision making. Our second contribution is to present an approximate value-iteration algorithm for CVaR MDPs and analyze its convergence rate. To our knowledge, this is the first solution algorithm for CVaR MDPs that enjoys error guarantees. Finally, we present results from numerical experiments that corroborate our theoretical findings and show the practicality of our approach.

• The paper proposes using Conditional Value-at-Risk (CVaR) optimization within Markov Decision Processes (MDPs) to unify risk sensitivity and robustness by interpreting CVaR as both a risk measure and a measure of robustness to model parameter perturbations.
• The research introduces the first approximate value-iteration algorithm for CVaR MDPs with finite-time error guarantees, utilizing a state-augmentation technique to handle the continuous CVaR confidence interval.
• Numerical experiments in a grid-world environment demonstrate that decreasing the CVaR confidence level effectively trades off path efficiency for collision risk and validates the robustness of the resulting policies under perturbed conditions.

Analyzing CVaR Optimization in Risk-Sensitive and Robust Decision-Making for MDPs

The paper "Risk-Sensitive and Robust Decision-Making: a CVaR Optimization Approach" presents a methodological advancement in decision-making processes under uncertainty through the use of a Conditional Value-at-Risk (CVaR) objective within Markov Decision Processes (MDPs). This approach is positioned against the standard practice of using risk-neutral expectations for MDPs, which do not account for the variability in outcomes or model uncertainties.

Key Contributions

The research contributes significantly in two primary areas:

1. Unifying Risk and Robustness: The authors propose that the CVaR objective can be interpreted not only as a measure of risk sensitivity but also as a measure of robustness. They demonstrate that the CVaR of a discounted cost in an MDP corresponds with the expected cost under worst-case model parameter perturbations, providing the perturbations do not exceed a specified error budget. This insight positions CVaR MDPs as a comprehensive framework for planning under uncertainty, accommodating both risk variability and parametric robustness.
2. Algorithmic Advancement: The paper introduces an approximate value-iteration algorithm tailored for CVaR MDPs. This represents the first algorithm with finite-time error guarantees for CVaR MDPs, implementing a state-augmentation technique to address the continuous nature of the CVaR confidence interval. The convergence of this algorithm is proven, with explicit error bounds derived from contraction arguments. This algorithm simplifies solving for globally optimal policies over CVaR confidence intervals, outperforming existing methods that are often complex and computationally intensive.

Theoretical Implications and Numerical Experiments

The theoretical underpinning is substantiated by reinterpreting CVaR in the context of robustness to modeling errors, a perspective previously unexplored to this extent. By using a reformulation that relates CVaR to adversarial perturbations, the work supplements existing robustness literature within MDPs, often constrained by conservative assumptions such as rectangular uncertainty sets.

The paper supports its theoretical contributions with numerical experiments in a grid-world environment, illustrating how decreasing the CVaR confidence level trades off between path efficiency (fuel consumption) and risk (collision avoidance). The robustness aspect is experimentally validated by comparing trajectories under nominal and perturbed conditions, revealing the risk-averse policy's superiority in terms of robustness.

Future Directions

The authors indicate potential extensions, notably in addressing large state spaces through sampling-based approximate DP methods. Given the demonstrated contractive properties of the CVaR BeLLMan equation, approaches like approximate policy iteration could leverage the findings for scalable applications.

Conclusion

This paper adds substantial depth to the decision-making literature by bridging risk sensitivity and robustness in MDPs through the CVaR framework. It not only offers a novel theoretical perspective but also backs it with practical algorithmic solutions, setting the groundwork for further exploration in AI-driven decision processes under uncertainty. The implications, notably in fields like autonomous systems and finance, underscore the importance of considering both stochastic risks and model uncertainties in operational strategies.