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

James Nguyen

cs.LGApr 11, 2026

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

Authors: James Nguyen

Organizations: Department of Mathematics, University of California, Irvine.

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.

Figures & tables

$\alpha$	Step fraction $\frac{1}{1+\alpha}$	Equivalent OGD $\eta$
0.01	99.0%	0.990
0.10	90.9%	0.909
1.0	50.0%	0.500
9.0	10.0%	0.100

Table 1: Effect of

\alpha

on step fraction and OGD equivalence (pre-normalized candidates only; see remark 1 ).

Figure 1: Left: mean cosine alignment $\cos(\hat{\bm{v}}_{t},\bm{g}_{u})$ under row-norm sampling ( $p_{\text{flip}}=0.2$ ), averaged over 100 users ( $\pm 1$ std). Right: alignment under two-stage cosine similarity subsampling (pool $=32\to 16$ ), with session count doubled to equalize total swipes. Block-NK (green) achieves the highest alignment under both strategies; NK (blue) remains low throughout due to recency bias.

Method	Like rate	Align@20	$\Delta_{s}$
TK ( $\alpha=1$ )	0.800	0.710	0.948
Block-TK ( $\alpha=1$ )	0.800	0.698	0.968
Block-NK	0.800	0.698	0.994
NK ( $\eta=0.5$ )	0.800	0.626	0.738
K-NoNorm	0.800	0.710	0.948
OGD-0.1	0.800	0.716	0.933

Table 2: Summary statistics under row-norm sampling,

p_{\text{flip}}=0.2

T=6{,}400

swipes.

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Preference learning infers people’s values and trade-offs from choices, comparisons, rankings, and feedback to guide predictions and decisions.

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Abstract

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