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Register a new userRisk-Constrained Reinforcement Learning with Percentile Risk Criteria
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Yinlam Chow
Publication date
2015
fields
Informatics Engineering
and research's language is
English
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In many sequential decision-making problems one is interested in minimizing an expected cumulative cost while taking into account emph{risk}, i.e., increased awareness of events of small probability and high consequences. Accordingly, the objective of this paper is to present efficient reinforcement learning algorithms for risk-constrained Markov decision processes (MDPs), where risk is represented via a chance constraint or a constraint on the conditional value-at-risk (CVaR) of the cumulative cost. We collectively refer to such problems as percentile risk-constrained MDPs. Specifically, we first derive a formula for computing the gradient of the Lagrangian function for percentile risk-constrained MDPs. Then, we devise policy gradient and actor-critic algorithms that (1) estimate such gradient, (2) update the policy in the descent direction, and (3) update the Lagrange multiplier in the ascent direction. For these algorithms we prove convergence to locally optimal policies. Finally, we demonstrate the effectiveness of our algorithms in an optimal stopping problem and an online marketing application.
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We motivate and propose a new model for non-cooperative Markov game which considers the interactions of risk-aware players. This model characterizes the time-consistent dynamic risk from both stochastic state transitions (inherent to the game) and randomized mixed strategies (due to all other players). An appropriate risk-aware equilibrium concept is proposed and the existence of such equilibria is demonstrated in stationary strategies by an application of Kakutanis fixed point theorem. We further propose a simulation-based Q-learning type algorithm for risk-aware equilibrium computation. This algorithm works with a special form of minimax risk measures which can naturally be written as saddle-point stochastic optimization problems, and covers many widely investigated risk measures. Finally, the almost sure convergence of this simulation-based algorithm to an equilibrium is demonstrated under some mild conditions. Our numerical experiments on a two player queuing game validate the properties of our model and algorithm, and demonstrate their worth and applicability in real life competitive decision-making.
We propose a new risk-constrained reformulation of the standard Linear Quadratic Regulator (LQR) problem. Our framework is motivated by the fact that the classical (risk-neutral) LQR controller, although optimal in expectation, might be ineffective under relatively infrequent, yet statistically significant (risky) events. To effectively trade between average and extreme event performance, we introduce a new risk constraint, which explicitly restricts the total expected predictive variance of the state penalty by a user-prescribed level. We show that, under rather minimal conditions on the process noise (i.e., finite fourth-order moments), the optimal risk-aware controller can be evaluated explicitly and in closed form. In fact, it is affine relative to the state, and is always internally stable regardless of parameter tuning. Our new risk-aware controller: i) pushes the state away from directions where the noise exhibits heavy tails, by exploiting the third-order moment (skewness) of the noise; ii) inflates the state penalty in riskier directions, where both the noise covariance and the state penalty are simultaneously large. The properties of the proposed risk-aware LQR framework are also illustrated via indicative numerical examples.
The standard approach to risk-averse control is to use the Exponential Utility (EU) functional, which has been studied for several decades. Like other risk-averse utility functionals, EU encodes risk aversion through an increasing convex mapping $varphi$ of objective costs to subjective costs. An objective cost is a realization $y$ of a random variable $Y$. In contrast, a subjective cost is a realization $varphi(y)$ of a random variable $varphi(Y)$ that has been transformed to measure preferences about the outcomes. For EU, the transformation is $varphi(y) = exp(frac{-theta}{2}y)$, and under certain conditions, the quantity $varphi^{-1}(E(varphi(Y)))$ can be approximated by a linear combination of the mean and variance of $Y$. More recently, there has been growing interest in risk-averse control using the Conditional Value-at-Risk (CVaR) functional. In contrast to the EU functional, the CVaR of a random variable $Y$ concerns a fraction of its possible realizations. If $Y$ is a continuous random variable with finite $E(|Y|)$, then the CVaR of $Y$ at level $alpha$ is the expectation of $Y$ in the $alpha cdot 100 %$ worst cases. Here, we study the applications of risk-averse functionals to controller synthesis and safety analysis through the development of numerical examples, with emphasis on EU and CVaR. Our contribution is to examine the decision-theoretic, mathematical, and computational trade-offs that arise when using EU and CVaR for optimal control and safety analysis. We are hopeful that this work will advance the interpretability and elucidate the potential benefits of risk-averse control technology.
Reinforcement Learning (RL) is emerging as tool for tackling complex control and decision-making problems. However, in high-risk environments such as healthcare, manufacturing, automotive or aerospace, it is often challenging to bridge the gap between an apparently optimal policy learnt by an agent and its real-world deployment, due to the uncertainties and risk associated with it. Broadly speaking RL agents face two kinds of uncertainty, 1. aleatoric uncertainty, which reflects randomness or noise in the dynamics of the world, and 2. epistemic uncertainty, which reflects the bounded knowledge of the agent due to model limitations and finite amount of information/data the agent has acquired about the world. These two types of uncertainty carry fundamentally different implications for the evaluation of performance and the level of risk or trust. Yet these aleatoric and epistemic uncertainties are generally confounded as standard and even distributional RL is agnostic to this difference. Here we propose how a distributional approach (UA-DQN) can be recast to render uncertainties by decomposing the net effects of each uncertainty. We demonstrate the operation of this method in grid world examples to build intuition and then show a proof of concept application for an RL agent operating as a clinical decision support system in critical care
Recently equal risk pricing, a framework for fair derivative pricing, was extended to consider dynamic risk measures. However, all current implementations either employ a static risk measure that violates time consistency, or are based on traditional dynamic programming solution schemes that are impracticable in problems with a large number of underlying assets (due to the curse of dimensionality) or with incomplete asset dynamics information. In this paper, we extend for the first time a famous off-policy deterministic actor-critic deep reinforcement learning (ACRL) algorithm to the problem of solving a risk averse Markov decision process that models risk using a time consistent recursive expectile risk measure. This new ACRL algorithm allows us to identify high quality time consistent hedging policies (and equal risk prices) for options, such as basket options, that cannot be handled using traditional methods, or in context where only historical trajectories of the underlying assets are available. Our numerical experiments, which involve both a simple vanilla option and a more exotic basket option, confirm that the new ACRL algorithm can produce 1) in simple environments, nearly optimal hedging policies, and highly accurate prices, simultaneously for a range of maturities 2) in complex environments, good quality policies and prices using reasonable amount of computing resources; and 3) overall, hedging strategies that actually outperform the strategies produced using static risk measures when the risk is evaluated at later points of time.
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