Potential-Based Shaping and Q-Value Initialization are Equivalent (1106.5267v1)

Abstract: Shaping has proven to be a powerful but precarious means of improving reinforcement learning performance. Ng, Harada, and Russell (1999) proposed the potential-based shaping algorithm for adding shaping rewards in a way that guarantees the learner will learn optimal behavior. In this note, we prove certain similarities between this shaping algorithm and the initialization step required for several reinforcement learning algorithms. More specifically, we prove that a reinforcement learner with initial Q-values based on the shaping algorithm's potential function make the same updates throughout learning as a learner receiving potential-based shaping rewards. We further prove that under a broad category of policies, the behavior of these two learners are indistinguishable. The comparison provides intuition on the theoretical properties of the shaping algorithm as well as a suggestion for a simpler method for capturing the algorithm's benefit. In addition, the equivalence raises previously unaddressed issues concerning the efficiency of learning with potential-based shaping.

• The paper demonstrates the equivalence between potential-based shaping rewards and initial Q-value settings in reinforcement learning algorithms.
• It shows that Q-value evolution remains identical when initialized with potential functions versus using shaping rewards under advantage-based policies.
• Practically, Q-value initialization can replace shaping in discrete-state RL for simpler algorithms, while shaping may suit continuous spaces.

Equivalence of Potential-Based Shaping and Q-Value Initialization in Reinforcement Learning

The contribution by Eric Wiewiora in this paper explores a noteworthy equivalence between potential-based shaping and Q-value initialization within the context of reinforcement learning (RL). Specifically, the research explores how manipulating these two distinct approaches effectively yields the same outcome in terms of learning dynamics. This paper not only provides insights into theoretical properties, but also suggests practical considerations in the implementation of RL algorithms.

Paper Overview

The primary assertion of the paper is the equivalence between potential-based shaping rewards and initial Q-value settings when applied to reinforcement learning algorithms. Initial Q-values or quality values are estimations of the expected future rewards for executing a particular action in a specified state, a critical aspect in the field of reinforcement learning for the derivation of optimal policies.

Results and Theoretical Implications

The rigorous analysis presented establishes that when Q-values are initialized in alignment with potential functions, the learning updates parallel those received with potential-based shaping rewards. Concretely, two learners are defined: one executing traditional Q-learning with shaping rewards and the other initialized with potential functions, yet devoid of shaping influences. Through logical deductions and mathematical induction, Wiewiora demonstrates that their Q-value evolution remains identical under equivalent experiential inputs. This underscores that both approaches are interchangeable for the learning process.

It's crucial to note the paper's focus on advantage-based policies—a broad category encompassing the majority of policies, including greedy and exploratory approaches like ε-greedy and Boltzmann softmax. Wiewiora's findings indicate that under these policies, action selection probabilities remain unaffected by additive constants, such as the state potentials.

Moreover, in goal-directed tasks where reaching a goal quickly is pivotal, the equivalence outlined suggests significant efficiency in using Q-value initialization. This is particularly meaningful given the potential for exponential learning times with suboptimal initial Q-values compared with the polynomial time guaranteed by strategically initialized Q-values in deterministic environments.

Practical Implications

From a practical standpoint, these findings advocate for initializing Q-values using potential functions in scenarios with discrete-state environments. This approach circumvents the necessity of integrating additional shaping rewards into the agent's learning algorithm. However, when dealing with continuous state-spaces, potential-based shaping could still hold advantages due to the capacity for function continuity over the state space. This adaptability makes it suitable for agents operating under restricted state representation paradigms.

Future Directions

The research invites further examination into the application of these findings in continuous state-spaces, especially considering the versatility of continuous potential functions. Investigating potential function selection in non-discrete environments could yield advanced insights and refinements to RL strategies.

In summary, Wiewiora's paper offers a substantiated perspective on the equivalence of potential-based shaping and Q-value initialization within RL, providing theoretical clarity and practical strategies for implementing these insights effectively. This work encourages leveraging Q-value initialization as a simplified yet equivalent alternative to potential-based shaping, particularly in environments where this substitution is operationally viable.