From Recency Bias to Stable Convergence Block Kaczmarz Methods for Online Preference Learning in Matchmaking Applications

Abstract: We present a family of Kaczmarz-based preference learning algorithms for real-time personalized matchmaking in reciprocal recommender systems. Post-step L2 normalization, common in Kaczmarz-inspired online learners, induces exponential recency bias: the influence of the t-th interaction decays as eta^n - t, reaching approximately 1e-6 after just 20 swipes at eta = 0.5. We resolve this by replacing the normalization step with a Tikhonov-regularized projection denominator that bounds step size analytically without erasing interaction history. When candidate tag vectors are not pre-normalized, as in realistic deployments where candidates vary in tag density, the Tikhonov denominator ||a||^2 + alpha produces genuinely per-candidate adaptive step sizes, making it structurally distinct from online gradient descent with any fixed learning rate. We further derive a block variant that processes full swipe sessions as a single Gram matrix solve. Population-scale simulation over 6,400 swipes reveals that Block Normalized Kaczmarz (BlockNK), which combines the batch Gram solve with post-session L2 normalization, achieves the highest preference alignment (Align@20 = 0.698), the strongest inter-session direction stability (delta = 0.994), and the flattest degradation profile under label noise across flip ratios p_flip in [0.10, 0.35]. Experiments under cosine similarity subsampling further show that adaptively filtering the candidate pool toward the current preference direction substantially improves asymptotic alignment, at the cost of introducing a feedback loop that may slow recovery from miscalibration. The sequential Tikhonov-Kaczmarz method performs comparably to K-NoNorm under our simulation conditions, suggesting the dominant practical gain over normalized Kaczmarz is the removal of per-step normalization rather than the Tikhonov constant alpha itself.

• The paper eliminates exponential recency bias by replacing ℓ2-normalization with Tikhonov regularization that preserves early feedback.
• It introduces block variants—especially Block-NK—that process swipe sessions collectively to enhance stability and robustness in preference learning.
• Empirical results show improved alignment (Align@20 = 0.698) and superior noise robustness compared to standard Kaczmarz methods.

Kaczmarz-Based Online Preference Learning for Reciprocal Matchmaking: Elimination of Recency Bias and Stable Convergence

Introduction

This work addresses the problem of real-time personalized preference learning in reciprocal recommender systems encountered in matchmaking applications, where recommendations are valid only if accepted reciprocally. The paper provides a thorough algorithmic and empirical analysis of Kaczmarz-style update rules for learning user preference vectors from one-sided (single-swipe) binary feedback, proposes Tikhonov-regularized and block variants that analytically resolve key pathologies of previously deployed online learning approaches, and introduces a principled framework for candidate sampling and label-noise robustness analysis.

Exponential Recency Bias in Normalized Kaczmarz

The standard Kaczmarz algorithm solves overdetermined linear systems via iterative projections. In preference learning, this is operationalized by updating a user's preference vector by projecting it onto the constraint imposed by the latest swipe. However, prior Kaczmarz-inspired online learners employ post-step $\ell_2$ normalization to control norm growth, which the authors show induces an exponential recency bias: the influence of any interaction decays as $\eta^{n-t}$, where $\eta$ is the step size.

Figure 1: Mean cosine alignment $\cos(\hat{v}_t, \bm{g}_u)$ over swipes, demonstrating that Block-NK (green) achieves the highest alignment while NK (blue) stagnates due to recency bias.

For $\eta = 0.5$, the impact of an interaction falls to $10^{-6}$ after just 20 swipes, causing the model to forget early interactions almost completely. This directly violates the desideratum that informative early feedback should persist and accumulate, especially critical in cold-start and sparse-interaction regimes.

Tikhonov-Regularized Kaczmarz: Analytical Step Control

To resolve this, the authors replace normalization with a Tikhonov-regularized denominator so that step size is bounded analytically based on each candidate's tag density:

$v^{(t+1)} = v^{(t)} + \frac{r_t}{\norm{\bm{a}_{j(t)}}^2 + \alpha} \bm{a}_{j(t)}$

This update admits two major contributions:

• Recency bias elimination: Interaction history persists as no normalization-induced decay occurs.
• Per-candidate adaptive step size: Step magnitude adapts to the norm of the presented candidate's tag vector, which is crucial under realistic, non-uniform tag densities.

Notably, with pre-normalized candidates, this update collapses to standard online gradient descent (OGD); under real-world, variable candidate norms, the method is structurally distinct and non-replicable via any fixed-rate OGD.

Block Kaczmarz Variants and Stability

To further enhance stability, the Block-TK and Block-NK variants are introduced. These process entire swipe sessions as a single regularized Gram matrix solve, performing aggregation and projection onto the joint constraint space of a session. Block-NK, in particular, combines batch Gram solution and post-session normalization (once per session, not per swipe), crucially allowing within-session learning to persist and mitigating flip-induced drift.

Empirical results robustly demonstrate that Block-NK achieves:

• Highest early-stage alignment (Align@20 = 0.698) and final stability ($\Delta_s = 0.994$).
• Superior noise robustness, with almost flat degradation even at high flip rates.

Empirical Evaluation and Numerical Results

Large-scale simulations (over 6,400 swipes per user) are conducted on synthetic populations with simulated ground-truth soulmate vectors, using both row-norm sampling and adaptive cosine-based subsampling.

Figure 2: Final cosine alignment $\cos(\hat{v}_T, \bm{g}_u)$ versus label flip ratio, indicating Block-NK's superior robustness across increasing noise.

Key quantitative findings include:

• NK (Normalized Kaczmarz) underperforms: It exhibits poor alignment (Align@20 = 0.626) and unstable session-to-session directions ($\Delta_s = 0.738$), directly resulting from recency bias.
• Block-NK dominates stability and robustness: Stable across high label noise and consistently best sessional direction stability.
• Adaptive candidate pool subsampling via cosine similarity accelerates convergence, with Block-NK achieving $\eta^{n-t}$0 by swipe 3,000, but introduces a positive feedback loop that can impede recovery from miscalibration.
• Sequential Tikhonov-Kaczmarz and K-NoNorm perform equivalently when candidate vectors are not normalized, indicating that the primary benefit of Tikhonovization here is step control per tag density, not the addition of the regularization constant $\eta^{n-t}$1 alone.

Implications and Prospective Developments

The research has both practical and theoretical consequences:

• Reciprocal recommenders in cold-start scenarios ought to prefer Tikhonovized and block Kaczmarz approaches—particularly Block-NK—when individual learning must be robust and sample-efficient, and when swipe sessions are naturally batched.
• Adaptive step-size control via per-candidate geometry yields both improved learning dynamics and analytical norm bounds, enhancing interpretability.
• Adaptive candidate subsampling demonstrates that careful design of the presentation policy can exploit current model beliefs for faster convergence, but system designers must monitor for self-reinforcing bias and be ready to revert to informativeness-driven sampling as needed.

Future extensions can pursue formal convergence analysis of Block-NK under adversarial noise, explore adaptive $\eta^{n-t}$2 and batch size $\eta^{n-t}$3 based on Gram matrix spectra, recalibrate label generation against raw dot product statistics, and conduct human subject trials in live systems with evolving preference drift.

Conclusion

This work systematically diagnoses and resolves the principal failure mode of existing Kaczmarz-based online preference learners—exponential recency bias introduced by post-update normalization—by advocating for Tikhonov-regularized, per-candidate step control and block solution frameworks. Block-NK, in particular, delivers both empirically verified and theoretically motivated improvements in alignment, stability, and robustness. These algorithmic insights are broadly applicable to real-world reciprocal recommendation systems, especially in data-sparse, privacy-preserving, or cold-start regimes, and suggest multiple promising avenues for further refinement and deployment in interactive, real-time applications.