Spectral Analysis of Dueling Q-Learning

Abstract: Q-learning is a fundamental algorithm in reinforcement learning (RL) for solving discounted Markov decision processes (MDPs) when the transition kernel is unknown. The deep Q-network (DQN) extends Q-learning by using a deep neural network for Q-function approximation, which makes Q-learning applicable to more practical high-dimensional problems. Dueling Q-learning decomposes the Q-function into a value function and an advantage function and learns the two components jointly, which can improve learning efficiency. However, the theoretical understanding of dueling Q-learning is still limited. Recent work has initiated an analysis of tabular dueling Q-learning, but existing guarantees focus on a regularized formulation and leave the pure tabular update less completely understood. This paper strengthens that line of analysis by adding a direct interpretation of the centered tabular decomposition and by establishing convergence guarantees for the unregularized, unprojected constant step-size recursion. In particular, we derive an exact switching linear system representation for deterministic dueling Q-learning and a finite-time error bound in expectation for the sampled stochastic version. The analysis clarifies how the value and advantage updates act as different gains on the action-common (value function) and action-differential (advantage function) components of the Q-function.

• The paper presents a spectral analysis of dueling Q-learning using SLS and JSR theory to certify convergence and establish finite-time error bounds.
• It decomposes the Q-function into state-value and advantage components, enabling separate tuning of contraction rates for improved performance.
• Explicit finite-time error bounds in both deterministic and stochastic settings guide practical step-size selection and algorithm design.

Spectral Analysis of Dueling Q-Learning: An Expert Overview

Introduction and Context

This work delivers a rigorous spectral analysis of the tabular dueling Q-learning algorithm, focusing on its convergence and error dynamics under deterministic and stochastic settings. Dueling Q-learning, originally proposed as an architecture for deep Q-networks (DQNs), decomposes the action-value function into state-value and advantage components. While empirical evidence supports its improved efficiency over classical Q-learning, theoretical guarantees remain incomplete—particularly for unregularized, constant step-size, tabular versions. This paper closes key gaps by applying tools from switching linear system (SLS) theory, especially joint spectral radius (JSR) techniques, to analyze and certify convergence, contraction, and finite-time error bounds for dueling Q-learning recursions.

Theoretical Framework

The analysis proceeds by decomposing the Q-function into two orthogonal components—a state-wise action-common mean ($V$) and a centered action-differential (advantage, $A$) part—via orthogonal projections. This decomposition yields the update:

$Q = V + A,\quad V = \Pi Q,\quad A = (I-\Pi)Q,$

where $\Pi$ is the state-wise mean projection. This structure allows for assigning distinct learning rates for the $V$ and $A$ components ($|A|\alpha$ and $\beta$ respectively), introducing flexibility in tuning their contraction rates. Both standard Q-learning ($\eta$ step size applied globally) and dueling Q-learning (decoupled rates) are subsumed.

The evolution of the vectorized error $(Q_k - Q^*)$ is expressed through a deterministic or stochastic SLS, with mode switching governed by the greedy policy induced by the current Q-function. The central technical tool is the JSR of the matrix family induced by these dynamics. Explicit expressions for the mode matrices are derived for both the common/differential (block) structure and the Q-space recursion.

Key SLS and JSR Results

• Convergence Guarantee: The Q-learning and dueling Q-learning recursions are recast as switched linear or affine systems. Under the condition that the JSR of the mode family is $A$0, exponential convergence of iterates to $A$1 is established in the sense of a Lyapunov norm, with explicit rates.
• Component-Wise Convergence: The block structure analysis exposes separate contraction rates for the action-common ($A$2-space) and action-differential ($A$3-space) components. This demonstrates that appropriate tuning can accelerate the slowest component, which is typically the bottleneck.

Finite-Time and Stochastic Analysis

For the sampled, stochastic version (with i.i.d. data), the analysis yields an explicit finite-time error bound. Using martingale concentration and contractive drift properties, the result shows that for sufficiently small constant step-sizes, the average error is driven toward a $A$4 neighborhood of $A$5. The constants involved depend on MDP parameters and the fixed ratio between value and advantage gains.

$A$6

The explicit form of the error bound quantifies the tradeoff between contraction rate and stochasticity-induced error floor. This formalizes previously anecdotal observations on the practical step-size regime for dueling Q-learning.

Comparative Characterization

Dueling Q-learning, as clarified here, applies larger and independent gains to the action-common and action-differential directions compared to standard Q-learning. The spectral analysis in the one-state example proves that, in high-discount or ill-conditioned regimes, substantial practical acceleration is achievable for the limiting convergence rate. This distinguishes dueling from prior "value-augmented" methods such as QV and VA-learning, with sharper convergence diagnostics.

Moreover, the analysis directly connects the regularized and unregularized forms, showing equivalence in error dynamics under appropriate initialization, and generalizes to families of algorithms with mixed gain schedules.

Practical and Theoretical Implications

This work provides a rigorous, operator-theoretic, and spectral foundation for practical tuning of dueling Q-learning algorithms in tabular settings. From a practical standpoint, the results guide the selection of component-wise step sizes to optimize convergence subject to stability. The deterministic and stochastic theory extends to finite sample and finite memory analysis under mild conditions (i.i.d. or Markovian sampling), and can be extended to projected or regularized variants as used in function approximation.

Theoretically, the SLS/JSR methodology offers a unifying perspective for temporal-difference learning algorithms with mode-dependent nonlinearities, enabling future extensions to non-tabular settings and to other operator-splitting RL approaches. The coupling between projections, mode selection, and contraction properties is laid bare, supporting more refined algorithm design.

Conclusion

This paper establishes, for the first time, exact convergence and error bounds for unregularized, constant step-size dueling Q-learning in tabular MDPs, harnessing the analytical power of JSR and SLS theory. The results not only bridge a significant theoretical gap but also provide practical tools for algorithm tuning and design. The approach sets a foundation for more advanced operator splitting and spectral certification for value-based RL algorithms. Future work should address extensions to function approximation, non-i.i.d. sampling, and the development of adaptive gain schedules.

Reference: "Spectral Analysis of Dueling Q-Learning" (2607.08340)