Constrained Reinforcement Learning Has Zero Duality Gap (1910.13393v1)

Abstract: Autonomous agents must often deal with conflicting requirements, such as completing tasks using the least amount of time/energy, learning multiple tasks, or dealing with multiple opponents. In the context of reinforcement learning~(RL), these problems are addressed by (i)~designing a reward function that simultaneously describes all requirements or (ii)~combining modular value functions that encode them individually. Though effective, these methods have critical downsides. Designing good reward functions that balance different objectives is challenging, especially as the number of objectives grows. Moreover, implicit interference between goals may lead to performance plateaus as they compete for resources, particularly when training on-policy. Similarly, selecting parameters to combine value functions is at least as hard as designing an all-encompassing reward, given that the effect of their values on the overall policy is not straightforward. The later is generally addressed by formulating the conflicting requirements as a constrained RL problem and solved using Primal-Dual methods. These algorithms are in general not guaranteed to converge to the optimal solution since the problem is not convex. This work provides theoretical support to these approaches by establishing that despite its non-convexity, this problem has zero duality gap, i.e., it can be solved exactly in the dual domain, where it becomes convex. Finally, we show this result basically holds if the policy is described by a good parametrization~(e.g., neural networks) and we connect this result with primal-dual algorithms present in the literature and we establish the convergence to the optimal solution.

• The paper proves that constrained reinforcement learning problems have a zero duality gap under Slater's conditions, allowing them to be solved exactly via convex dual optimization.
• The theoretical result extends to parameterized policies, suggesting that near-optimal solutions are achievable with rich function approximators like neural networks.
• This finding enables efficient handling of multi-objective RL problems by automatically regulating conflicting goals through dual domain optimization, reducing the need for complex reward engineering.

Constrained Reinforcement Learning Has Zero Duality Gap: An Expert Overview

Reinforcement Learning (RL) presents numerous challenges, especially when autonomous agents face conflicting requirements. For instance, agents may need to complete tasks within minimal time or energy, handle multiple opponents, or learn various tasks. In RL, such multi-objective problems can often be addressed either by crafting a reward function that encapsulates all objectives or by aggregating modular value functions tied to individual tasks. However, designing these reward functions becomes increasingly difficult as the number of objectives increases, leading to potential training plateaus due to resource competition between conflicting goals.

The paper "Constrained Reinforcement Learning Has Zero Duality Gap" tackles these issues within the framework of constrained RL, proposing that such problems can be efficiently solved using dual optimization methods despite being inherently non-convex. The authors assert that constrained RL problems have zero duality gap, implying exact solvability in the dual domain, where the problem can be treated as convex.

Theoretical Foundations and Contributions

The paper first establishes a formal theory supporting the notion of zero duality gap for constrained RL. It proves that, under Slater's conditions, constrained RL presents no duality gap, thereby ensuring that the dual problem has the same solution as the primal problem. This theoretical result facilitates solving constrained RL in the dual domain, presenting practical advantages such as reduced dimensionality and guaranteed convexity.

The key contributions include:

1. Zero Duality Gap Assertion: Despite the non-convexity of constrained RL, there is a zero duality gap, ensuring equivalency between primal and dual solutions under stated conditions.
2. Parametrization Consideration: Extending the zero duality gap result to parametrized policies demonstrates that the suboptimality introduced by parametrizations, such as neural networks, is minimal if sufficiently rich approximators are used.
3. Primal-Dual Convergence: The authors leverage these theoretical insights to connect primal-dual algorithms in literature with robust convergence results, thus addressing the practicality of deploying such algorithms in RL environments.

Implications and Future Directions

This work has significant implications both theoretically and for practical applications. By proving the zero duality gap in constrained RL, the paper provides a solid foundation for designing RL systems that can efficiently handle multiple objectives without manually tuning reward functions. Practically, this alleviates one of RL's significant bottlenecks—balance between conflicting goals—by relying on the automatic regulation provided in the dual domain.

Further, the extension of results to parametrized policies reassures that popular neural network-based approaches can be used with little loss in performance. The paper catalyzes future research in developing more robust RL systems, integrating constrained optimization with RL tasks.

Speculation on AI Developments

This work underscores potential advancements in hierarchical and multi-objective RL systems, paving the way for agents capable of deploying strategic decision-making amidst complex requirements. Future developments may enhance RL systems by embedding improved parametric models or exploring other constraint optimization platforms to refine task achievement delicately.

Overall, this paper provides a compelling narrative on taming the complexity in RL through principled mathematical frameworks, promising efficient optimization methodologies that hold substantial promise for intelligent agent design.