Optimal Stopping with Entropy Regularization: A Singular Control Formulation

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This paper proposes a reinforcement learning framework for continuous-time and state-space optimal stopping problems by introducing entropy regularization to encourage exploration and facilitate learning.

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The paper explores continuous-time and state-space optimal stopping (OS) problems from a reinforcement learning (RL) perspective. It begins by formulating the stopping problem using randomized stopping times, where the decision maker's control is represented by the probability of stopping within a given time. To encourage exploration and facilitate learning, the authors introduce a regularized version of the problem by penalizing the performance criterion with the cumulative residual entropy of the randomized stopping time.

The regularized problem takes the form of an (n+1)-dimensional degenerate singular stochastic control with finite-fuel. The authors address this through the dynamic programming principle, which enables them to identify the unique optimal exploratory strategy. For a specific real option problem, they derive a semi-explicit solution to the regularized problem, allowing them to assess the impact of entropy regularization and analyze the vanishing entropy limit.

Finally, the authors propose a reinforcement learning algorithm based on policy iteration. They show both policy improvement and policy convergence results for the proposed algorithm.

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Viktiga insikter från

Exploratory Optimal Stopping: A Singular Control Formulation

by Jodi Dianett... på arxiv.org 10-03-2024

https://arxiv.org/pdf/2408.09335.pdf

Djupare frågor

How can the proposed entropy regularization framework be extended to other types of optimal control problems beyond optimal stopping?

The entropy regularization framework introduced in the context of optimal stopping problems can be effectively extended to various other types of optimal control problems, such as impulse control, switching control, and even more complex stochastic control scenarios. The core idea of incorporating an entropy term to encourage exploration can be adapted to these settings by modifying the objective function to include a regularization term that penalizes deterministic or overly conservative strategies.
For instance, in impulse control problems, where decisions are made at discrete time points to adjust the state of the system, the entropy regularization can be applied to the decision-making process to promote exploration of different impulse actions. This can be achieved by defining a cumulative residual entropy term that quantifies the uncertainty in the choice of impulses, thus incentivizing the agent to explore various impulse strategies rather than sticking to a fixed policy.
Similarly, in switching control problems, where the control strategy involves switching between different modes of operation, the entropy regularization can be utilized to encourage the exploration of different switching sequences. By incorporating an entropy term that reflects the uncertainty in the switching decisions, the framework can help balance the trade-off between exploitation of known effective strategies and exploration of potentially better alternatives.
Moreover, the framework can be generalized to multi-agent systems, where the interactions between agents can be modeled using entropy regularization to promote diverse strategies among agents, thereby enhancing overall system performance. The flexibility of the entropy regularization approach allows it to be tailored to various control problems, making it a versatile tool in the realm of stochastic control and reinforcement learning.

What are the potential limitations or drawbacks of the entropy regularization approach compared to other exploration techniques in reinforcement learning?

While the entropy regularization approach offers several advantages, such as promoting exploration and addressing the sparsity of rewards in optimal stopping problems, it also has potential limitations and drawbacks compared to other exploration techniques in reinforcement learning (RL).
One significant limitation is the computational complexity associated with estimating the entropy term and optimizing the regularized objective function. The introduction of the cumulative residual entropy can lead to more complex optimization landscapes, which may require sophisticated numerical methods or approximations to solve effectively. This added complexity can result in longer training times and increased resource requirements, particularly in high-dimensional state spaces.
Additionally, the choice of the temperature parameter λ, which balances exploration and exploitation, can be challenging. If λ is set too high, the agent may overly prioritize exploration at the expense of exploiting known strategies, leading to suboptimal performance. Conversely, if λ is too low, the agent may converge prematurely to a deterministic policy, negating the benefits of exploration. This sensitivity to parameter tuning can complicate the implementation of the entropy regularization approach in practice.
Furthermore, while entropy regularization encourages exploration, it may not always lead to the most efficient exploration strategies. Other exploration techniques, such as epsilon-greedy methods, Upper Confidence Bound (UCB) strategies, or Thompson sampling, may provide more targeted exploration based on the agent's current knowledge of the environment. These methods can be more effective in certain scenarios, particularly when the underlying dynamics of the environment are well understood.
Lastly, the entropy regularization approach may not be suitable for all types of optimal control problems. In cases where the optimal control strategy is inherently deterministic or where exploration is less critical, the added complexity of entropy regularization may not yield significant benefits compared to simpler exploration techniques.

How can the insights from this work on continuous-time optimal stopping be leveraged to develop reinforcement learning algorithms for high-dimensional or partially observable optimal stopping problems?

The insights gained from the study of continuous-time optimal stopping problems can be instrumental in developing reinforcement learning (RL) algorithms tailored for high-dimensional or partially observable optimal stopping scenarios. Several key strategies can be derived from this work:

Regularization Techniques: The use of entropy regularization to encourage exploration can be adapted to high-dimensional settings where the state space is large and complex. By incorporating a regularization term that penalizes deterministic policies, RL algorithms can be designed to explore a broader range of actions, thereby improving the agent's ability to discover optimal stopping strategies in high-dimensional environments.

Dynamic Programming Approaches: The dynamic programming principle (DPP) utilized in the analysis of the entropy-regularized optimal stopping problem can be extended to high-dimensional or partially observable settings. By formulating the value function in terms of a recursive relationship, RL algorithms can leverage DPP to efficiently compute value estimates and optimal policies, even in complex state spaces.

Policy Iteration Algorithms: The policy iteration framework developed in the context of the entropy-regularized optimal stopping problem can be adapted for high-dimensional scenarios. By iteratively updating the policy based on the value function estimates, RL algorithms can converge to optimal stopping strategies while effectively managing the exploration-exploitation trade-off.

Handling Partial Observability: Insights from the work can inform the design of algorithms that address partial observability by incorporating belief states or using techniques such as Partially Observable Markov Decision Processes (POMDPs). The entropy regularization can be applied to the belief state updates, encouraging exploration of different states based on the uncertainty in observations.

Sample-Based Approaches: The sample-based policy iteration algorithm proposed in the context of the entropy-regularized problem can be particularly useful in high-dimensional settings. By utilizing trajectory samples to estimate value functions and update policies, RL algorithms can effectively learn optimal stopping strategies without requiring full knowledge of the underlying dynamics.

Convergence Analysis: The theoretical convergence results established in the study provide a foundation for ensuring that RL algorithms converge to optimal solutions in high-dimensional or partially observable settings. By leveraging the established properties of the entropy-regularized framework, researchers can develop robust algorithms with guaranteed performance.

In summary, the insights from continuous-time optimal stopping problems, particularly the use of entropy regularization and dynamic programming techniques, can significantly enhance the development of RL algorithms for high-dimensional and partially observable optimal stopping scenarios, ultimately leading to more effective decision-making in complex environments.