Estimating Risk Measures in Discounted Markov Cost Processes: Lower and Upper Bounds

Core Concepts

The authors derive minimax sample complexity lower bounds as well as upper bounds for estimating risk measures such as variance, Value-at-Risk (VaR), and Conditional Value-at-Risk (CVaR) in discounted Markov cost processes.

Abstract

The key points of the content are:

Lower bounds:

The authors derive a minimax sample complexity lower bound of Ω(1/ε^2) for estimating VaR, CVaR, and variance in two types of Markov cost process (MCP) problem instances: one with deterministic costs and the other with stochastic costs.
The lower bound proofs involve novel techniques, including solving a constrained optimization problem for the deterministic costs case.
The lower bound for mean estimation also improves upon the existing Ω(1/ε) bound.

Upper bounds:

Using a truncation scheme, the authors derive an upper bound of Õ(1/ε^2) for CVaR and variance estimation, matching the corresponding lower bounds up to logarithmic factors.
They also propose an extension of the estimation scheme to cover a broader class of Lipschitz-continuous risk measures, such as spectral risk measures and utility-based shortfall risk.

Significance:

To the best of the authors' knowledge, their work is the first to provide lower and upper bounds for estimating any risk measure beyond the mean within a Markovian setting.
The lower bounds establish the sample complexity requirements for accurately estimating various risk measures in discounted Markov cost processes.
The upper bounds provide practical estimation schemes that achieve the optimal sample complexity, up to logarithmic factors.

Stats

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Quotes

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Key Insights Distilled From

Risk Estimation in a Markov Cost Process

by Gugan Thoppe... at arxiv.org 04-12-2024

https://arxiv.org/pdf/2310.11389.pdf

Deeper Inquiries

How can the lower bound results be extended to other risk measures beyond VaR, CVaR, and variance

The lower bound results obtained in the context of estimating risk measures within a Markov Cost Process can be extended to cover other risk measures beyond VaR, CVaR, and variance by considering risk measures that satisfy a certain continuity criterion. In the context of Lipschitz risk measures, the lower bounds can be applied to estimate risk measures such as spectral risk measures and utility-based shortfall risk. These risk measures can be accommodated within the estimation framework by ensuring that they meet the Lipschitz continuity criterion. By extending the estimation scheme to cover a broader class of risk measures, the lower bounds derived in the study can provide insights into the sample complexity required for estimating a wider range of risk measures accurately within a Markovian setting.

What are the implications of the derived bounds on the design and analysis of risk-sensitive reinforcement learning algorithms

The implications of the derived bounds on the design and analysis of risk-sensitive reinforcement learning algorithms are significant. By establishing lower and upper bounds for estimating risk measures such as variance, VaR, and CVaR within a Markov Cost Process, the study provides valuable insights into the sample complexity required for accurate risk estimation. These bounds can guide the development of risk-sensitive reinforcement learning algorithms by informing the design of algorithms that optimize risk measures beyond the mean in practical applications. Understanding the sample complexity constraints can help in the efficient implementation of risk-aware decision-making processes in various domains such as finance, transportation, and other risk-sensitive applications.

Can the techniques used in this work be applied to obtain sample complexity bounds for risk estimation in partially observable Markov decision processes

The techniques used in this work can potentially be applied to obtain sample complexity bounds for risk estimation in partially observable Markov decision processes (POMDPs). By adapting the methodology and proof techniques employed in estimating risk measures within a Markov Cost Process, researchers can explore the sample complexity required for accurate risk estimation in POMDPs. Considering the inherent uncertainty and partial observability in POMDPs, understanding the sample complexity bounds for risk estimation can enhance the development of risk-aware decision-making algorithms in dynamic and uncertain environments. By leveraging the insights and methodologies from this study, researchers can extend the analysis to address the challenges of risk estimation in partially observable settings.