Q-Learning

Reinforcement Learning for Monetary Policy Under Macroeconomic Uncertainty: Analyzing Tabular and Function Approximation Methods ArXiv ID: 2512.17929 “View on arXiv” Authors: Tony Wang, Kyle Feinstein, Sheryl Chen Abstract We study how a central bank should dynamically set short-term nominal interest rates to stabilize inflation and unemployment when macroeconomic relationships are uncertain and time-varying. We model monetary policy as a sequential decision-making problem where the central bank observes macroeconomic conditions quarterly and chooses interest rate adjustments. Using publicly accessible historical Federal Reserve Economic Data (FRED), we construct a linear-Gaussian transition model and implement a discrete-action Markov Decision Process with a quadratic loss reward function. We chose to compare nine different reinforcement learning style approaches against Taylor Rule and naive baselines, including tabular Q-learning variants, SARSA, Actor-Critic, Deep Q-Networks, Bayesian Q-learning with uncertainty quantification, and POMDP formulations with partial observability. Notably, despite its simplicity, standard tabular Q-learning achieved the best performance (-615.13 +- 309.58 mean return), outperforming both enhanced RL methods and traditional policy rules. Our results suggest that while sophisticated RL techniques show promise for monetary policy applications, simpler approaches may be more robust in this domain, highlighting important challenges in applying modern RL to macroeconomic policy. ...

Distributionally Robust Deep Q-Learning ArXiv ID: 2505.19058 “View on arXiv” Authors: Chung I Lu, Julian Sester, Aijia Zhang Abstract We propose a novel distributionally robust $Q$-learning algorithm for the non-tabular case accounting for continuous state spaces where the state transition of the underlying Markov decision process is subject to model uncertainty. The uncertainty is taken into account by considering the worst-case transition from a ball around a reference probability measure. To determine the optimal policy under the worst-case state transition, we solve the associated non-linear Bellman equation by dualising and regularising the Bellman operator with the Sinkhorn distance, which is then parameterized with deep neural networks. This approach allows us to modify the Deep Q-Network algorithm to optimise for the worst case state transition. We illustrate the tractability and effectiveness of our approach through several applications, including a portfolio optimisation task based on S&{“P”}~500 data. ...

Reinforcement Learning Methods for the Stochastic Optimal Control of an Industrial Power-to-Heat System ArXiv ID: 2411.02211 “View on arXiv” Authors: Unknown Abstract The optimal control of sustainable energy supply systems, including renewable energies and energy storage, takes a central role in the decarbonization of industrial systems. However, the use of fluctuating renewable energies leads to fluctuations in energy generation and requires a suitable control strategy for the complex systems in order to ensure energy supply. In this paper, we consider an electrified power-to-heat system which is designed to supply heat in form of superheated steam for industrial processes. The system consists of a high-temperature heat pump for heat supply, a wind turbine for power generation, a sensible thermal energy storage for storing excess heat and a steam generator for providing steam. If the system’s energy demand cannot be covered by electricity from the wind turbine, additional electricity must be purchased from the power grid. For this system, we investigate the cost-optimal operation aiming to minimize the electricity cost from the grid by a suitable system control depending on the available wind power and the amount of stored thermal energy. This is a decision making problem under uncertainties about the future prices for electricity from the grid and the future generation of wind power. The resulting stochastic optimal control problem is treated as finite-horizon Markov decision process for a multi-dimensional controlled state process. We first consider the classical backward recursion technique for solving the associated dynamic programming equation for the value function and compute the optimal decision rule. Since that approach suffers from the curse of dimensionality we also apply reinforcement learning techniques, namely Q-learning, that are able to provide a good approximate solution to the optimization problem within reasonable time. ...

Unified continuous-time q-learning for mean-field game and mean-field control problems ArXiv ID: 2407.04521 “View on arXiv” Authors: Unknown Abstract This paper studies the continuous-time q-learning in mean-field jump-diffusion models when the population distribution is not directly observable. We propose the integrated q-function in decoupled form (decoupled Iq-function) from the representative agent’s perspective and establish its martingale characterization, which provides a unified policy evaluation rule for both mean-field game (MFG) and mean-field control (MFC) problems. Moreover, we consider the learning procedure where the representative agent updates the population distribution based on his own state values. Depending on the task to solve the MFG or MFC problem, we can employ the decoupled Iq-function differently to characterize the mean-field equilibrium policy or the mean-field optimal policy respectively. Based on these theoretical findings, we devise a unified q-learning algorithm for both MFG and MFC problems by utilizing test policies and the averaged martingale orthogonality condition. For several financial applications in the jump-diffusion setting, we obtain the exact parameterization of the decoupled Iq-functions and the value functions, and illustrate our q-learning algorithm with satisfactory performance. ...

Continuous-time Risk-sensitive Reinforcement Learning via Quadratic Variation Penalty ArXiv ID: 2404.12598 “View on arXiv” Authors: Unknown Abstract This paper studies continuous-time risk-sensitive reinforcement learning (RL) under the entropy-regularized, exploratory diffusion process formulation with the exponential-form objective. The risk-sensitive objective arises either as the agent’s risk attitude or as a distributionally robust approach against the model uncertainty. Owing to the martingale perspective in Jia and Zhou (2023) the risk-sensitive RL problem is shown to be equivalent to ensuring the martingale property of a process involving both the value function and the q-function, augmented by an additional penalty term: the quadratic variation of the value process, capturing the variability of the value-to-go along the trajectory. This characterization allows for the straightforward adaptation of existing RL algorithms developed for non-risk-sensitive scenarios to incorporate risk sensitivity by adding the realized variance of the value process. Additionally, I highlight that the conventional policy gradient representation is inadequate for risk-sensitive problems due to the nonlinear nature of quadratic variation; however, q-learning offers a solution and extends to infinite horizon settings. Finally, I prove the convergence of the proposed algorithm for Merton’s investment problem and quantify the impact of temperature parameter on the behavior of the learning procedure. I also conduct simulation experiments to demonstrate how risk-sensitive RL improves the finite-sample performance in the linear-quadratic control problem. ...

On optimal tracking portfolio in incomplete markets: The reinforcement learning approach ArXiv ID: 2311.14318 “View on arXiv” Authors: Unknown Abstract This paper studies an infinite horizon optimal tracking portfolio problem using capital injection in incomplete market models. The benchmark process is modelled by a geometric Brownian motion with zero drift driven by some unhedgeable risk. The relaxed tracking formulation is adopted where the fund account compensated by the injected capital needs to outperform the benchmark process at any time, and the goal is to minimize the cost of the discounted total capital injection. When model parameters are known, we formulate the equivalent auxiliary control problem with reflected state dynamics, for which the classical solution of the HJB equation with Neumann boundary condition is obtained explicitly. When model parameters are unknown, we introduce the exploratory formulation for the auxiliary control problem with entropy regularization and develop the continuous-time q-learning algorithm in models of reflected diffusion processes. In some illustrative numerical example, we show the satisfactory performance of the q-learning algorithm. ...

Variations on the Reinforcement Learning performance of Blackjack ArXiv ID: 2308.07329 “View on arXiv” Authors: Unknown Abstract Blackjack or “21” is a popular card-based game of chance and skill. The objective of the game is to win by obtaining a hand total higher than the dealer’s without exceeding 21. The ideal blackjack strategy will maximize financial return in the long run while avoiding gambler’s ruin. The stochastic environment and inherent reward structure of blackjack presents an appealing problem to better understand reinforcement learning agents in the presence of environment variations. Here we consider a q-learning solution for optimal play and investigate the rate of learning convergence of the algorithm as a function of deck size. A blackjack simulator allowing for universal blackjack rules is also implemented to demonstrate the extent to which a card counter perfectly using the basic strategy and hi-lo system can bring the house to bankruptcy and how environment variations impact this outcome. The novelty of our work is to place this conceptual understanding of the impact of deck size in the context of learning agent convergence. ...

Continuous-time q-learning for mean-field control problems ArXiv ID: 2306.16208 “View on arXiv” Authors: Unknown Abstract This paper studies the q-learning, recently coined as the continuous time counterpart of Q-learning by Jia and Zhou (2023), for continuous time Mckean-Vlasov control problems in the setting of entropy-regularized reinforcement learning. In contrast to the single agent’s control problem in Jia and Zhou (2023), the mean-field interaction of agents renders the definition of the q-function more subtle, for which we reveal that two distinct q-functions naturally arise: (i) the integrated q-function (denoted by $q$) as the first-order approximation of the integrated Q-function introduced in Gu, Guo, Wei and Xu (2023), which can be learnt by a weak martingale condition involving test policies; and (ii) the essential q-function (denoted by $q_e$) that is employed in the policy improvement iterations. We show that two q-functions are related via an integral representation under all test policies. Based on the weak martingale condition and our proposed searching method of test policies, some model-free learning algorithms are devised. In two examples, one in LQ control framework and one beyond LQ control framework, we can obtain the exact parameterization of the optimal value function and q-functions and illustrate our algorithms with simulation experiments. ...

Evaluation of Reinforcement Learning Techniques for Trading on a Diverse Portfolio ArXiv ID: 2309.03202 “View on arXiv” Authors: Unknown Abstract This work seeks to answer key research questions regarding the viability of reinforcement learning over the S&P 500 index. The on-policy techniques of Value Iteration (VI) and State-action-reward-state-action (SARSA) are implemented along with the off-policy technique of Q-Learning. The models are trained and tested on a dataset comprising multiple years of stock market data from 2000-2023. The analysis presents the results and findings from training and testing the models using two different time periods: one including the COVID-19 pandemic years and one excluding them. The results indicate that including market data from the COVID-19 period in the training dataset leads to superior performance compared to the baseline strategies. During testing, the on-policy approaches (VI and SARSA) outperform Q-learning, highlighting the influence of bias-variance tradeoff and the generalization capabilities of simpler policies. However, it is noted that the performance of Q-learning may vary depending on the stability of future market conditions. Future work is suggested, including experiments with updated Q-learning policies during testing and trading diverse individual stocks. Additionally, the exploration of alternative economic indicators for training the models is proposed. ...