# XSQ-Learning: Adaptive Similarity Thresholds for Accelerated and Stable Q-Learning
## 5 Conclusions  
In this paper, we proposed XSQ-Learning, a similarity-based RL algorithm designed to improve convergence speed while retaining the stability features of traditional Q-learning methods. Experiments conducted in three increasingly complex environments show that _XSQ-Learning_ outperforms classical and existing similarity-based methods under the tested conditions, with three key contributions emerging from this work:  
First, _XSQ-Learning_ addresses fundamental limitations of existing similarity-based methods through its adaptive threshold mechanism (\(\delta\)) and controlled update propagation (_sim_clip_value_). The results in Section 4 show that these innovations successfully preventthe over-generalization and policy degradation observed in methods like _QSLearning_, while maintaining computational efficiency. In Scenario 3 (the most complex environment), _XSQ-Learning_ reduced the required learning steps by 36.83% compared to standard Q-learning and by 24.43% versus the best similarity-based alternative (_QL-SmoothActions_). Second, the results indicate that the algorithm's performance scales positively with the complexity of the state space. The performance advantage grows linearly with state space complexity--from a 22.56% improvement in Scenario 1 to the substantial 58.30% gain in Scenario 3. This scaling property, combined with the algorithm's \(\mathcal{O}(|S||A|)\) complexity (Section 3), makes _XSQ-Learning_ particularly suitable for real-world applications where state spaces are typically large and high-dimensional. Third, post-convergence, _XSQ-Learning_ showed consistent performance across all test scenarios and random seeds (Figure 2). This stability is supported by the use of confidence-weighted updates (Equation (4)) and bounded propagation (Equation 6), which together help reduce interference effects commonly observed in other similarity-based algorithms. Despite these advantages, we acknowledge that the performance of _XSQ-Learning_ depends on the configuration of three key hyperparameters: the similarity threshold \(\delta\), the update clipping value \(sim\_clip\_value\), and the residual term \(\iota\). While our experiments suggest that effective performance can be achieved within moderate ranges (e.g., \(\delta\in[0.7,0.9]\), \(sim\_clip\_value\in[0.2,0.5]\), \(\iota\approx 10^{-5}\)), it is likely that tuning will be required when deploying the algorithm in new or highly heterogeneous environments. To mitigate this limitation, future research will explore automated hyperparameter optimization techniques such as Bayesian optimization and meta-learning, as well as adaptive scheduling strategies that gradually adjust \(\delta\) and \(sim\_clip\_value\) during training. These approaches aim to reduce manual tuning effort and enhance the algorithm's generalizability across tasks. The findings suggest multiple avenues for future research. First, while our grid-world experiments demonstrate promising results, testing _XSQ-Learning_ in physical systems such as robotic control or autonomous vehicles would validate its practical utility in real-world applications. Second, developing similarity functions capable of effectively handling mixed numerical and categorical state variables would significantly broaden the algorithm's applicability to more complex, heterogeneous state representations. Third, investigating methods to incorporate catastrophic state avoidance mechanisms during similarity-based updates could enhance the algorithm's robustness, particularly in safety-critical applications where harmful states must be avoided.