Nonparametric Bellman Mappings for Reinforcement Learning: A Robust Adaptive Filtering Solution

Core Concepts

This paper designs novel nonparametric Bellman mappings in reproducing kernel Hilbert spaces (RKHSs) for reinforcement learning (RL) and applies them to offer a solution for countering outliers in adaptive filtering, without any prior knowledge on the statistics of the outliers and without training data.

Abstract

The paper proposes a novel family of Bellman mappings (B-Maps) defined in RKHSs to take advantage of the rich approximating properties of RKHSs and the flexibility an RKHS inner product brings into the design of loss functions and constraints. The proposed B-Maps possess ample degrees of freedom, and by appropriately designing their free parameters, several popular B-Map designs are shown to fall as special cases.

The key highlights and insights are:

1. The proposed B-Maps are nonparametric, with no need for statistical priors and assumptions on the data, to reduce the bias inflicted on data modeling. To address the "curse of dimensionality" issue, a dimensionality-reduction strategy based on random Fourier features is offered.

2. The B-Maps allow for sampling on-the-fly, do not require any knowledge on transition probabilities of Markov decision processes, and enable computationally lightweight operations to fit into the online or time-adaptive learning required by the adaptive filtering problem.

3. For the first time in the literature, the paper offers an RL-based solution to the problem of countering outliers in adaptive filtering. The proposed solution, built on a continuous state space and a discrete action space, adopts the well-known policy-iteration strategy and defines a quadratic loss on the Q-functions via the proposed B-Maps.

4. Theoretical properties of the proposed B-Maps, such as Lipschitz continuity and consistency of their fixed points, are established. A performance analysis of the proposed RL algorithm is provided, and numerical tests on synthetic data demonstrate its superior performance over several RL and non-RL schemes.

Stats

s'(1)
n-1 = log |en|^2
s'(2)
n-1 = (1/Mav) * sum_{m=1}^Mav log |yn-m - θ_n(an-1)^T x_n-m|^2 / ||x_n-m||_2^2
s'(3)
n-1 = log ||x_n||_2
s'(4)
n-1 = ϖ s_n-1^(4) + (1-ϖ) log(1 / (ρ ||θ_n(an-1) - θ_n-1||_2))

Quotes

None

Key Insights Distilled From

Nonparametric Bellman Mappings for Reinforcement Learning

by Yuki Akiyama... at arxiv.org 04-01-2024

https://arxiv.org/pdf/2403.20020.pdf

Deeper Inquiries

How can the proposed nonparametric Bellman mappings be extended to handle other types of loss functions, such as robust losses, to further improve the outlier-resilience of the adaptive filtering solution

The proposed nonparametric Bellman mappings can be extended to handle other types of loss functions, such as robust losses, by modifying the definition of the loss function in the variational framework presented in Proposition 1. By adjusting the loss function L(γ, Υ) and the regularizing function R(γ, Υ) in the variational problem (10), the Bellman mappings can be tailored to optimize for robust losses. Robust losses, such as ℓ1-norm loss, can be incorporated into the framework to enhance the outlier-resilience of the adaptive filtering solution. By appropriately tuning the loss functions, the Bellman mappings can be customized to address specific challenges related to outliers in the data, leading to improved performance in adaptive filtering applications.

What are the potential applications of the proposed nonparametric Bellman mappings beyond the adaptive filtering problem, and how can they be adapted to those applications

The proposed nonparametric Bellman mappings have potential applications beyond adaptive filtering in various fields where reinforcement learning is utilized. These mappings can be adapted to applications in autonomous navigation, robotics, resource planning, sensor networks, biomedical imaging, and gaming, among others. In autonomous navigation, the Bellman mappings can be used to optimize decision-making processes for self-driving vehicles. In robotics, they can enhance the learning capabilities of robotic systems for complex tasks. For resource planning, the mappings can aid in efficient allocation of resources based on dynamic environments. In sensor networks, they can optimize data collection and processing strategies. In biomedical imaging, the mappings can assist in image analysis and interpretation. In gaming, they can be employed to enhance the intelligence of game agents for more challenging gameplay experiences.

Given the connections between the proposed Bellman mappings and the classical temporal-difference learning, how can the insights from this work be leveraged to develop new reinforcement learning algorithms with improved sample efficiency and convergence properties

The insights from the proposed nonparametric Bellman mappings, which have connections to classical temporal-difference learning, can be leveraged to develop new reinforcement learning algorithms with improved sample efficiency and convergence properties. By incorporating the principles of Bellman mappings into the design of reinforcement learning algorithms, researchers can create more robust and adaptive systems that learn efficiently from limited data. The use of nonparametric mappings in reinforcement learning can lead to more flexible and adaptable algorithms that can handle complex and dynamic environments effectively. By integrating the concepts of Bellman mappings with state-of-the-art reinforcement learning techniques, novel algorithms can be developed that offer superior performance in terms of sample efficiency, convergence speed, and overall learning capabilities.