Preference-Aligned BestFit Reranking

• The paper introduces a convex optimization framework that directly fits preference orders instead of raw scores, mitigating overfitting in noisy data.
• It employs nuclear norm regularization and isotonic regression to ensure low-rank structure and order consistency across incomplete rankings.
• Empirical evaluations on datasets like Movielens100K and Neurosynth validate enhanced ranking accuracy and generalization over standard methods.

Preference-Aligned BestFit Reranking is a paradigm in learning-to-rank and collaborative filtering that seeks to reorder a set of candidate items or responses so that the final ranking most effectively aligns with the underlying (often latent) user or entity preferences, rather than with potentially noisy, arbitrarily scaled, or partially observed numerical scores. This framework is especially pertinent in matrix/tensor completion, recommendation, and collaborative ranking settings, as well as in downstream applications where full or partial ordering information is available and fidelity to the order structure takes precedence over fidelity to observed numeric values.

1. Problem Definition and Motivation

Traditional approaches to collaborative filtering and matrix completion often attempt to impute or fit the numerical affinity or score matrix between entities (such as users) and items, commonly employing least-squares or regression losses. However, in many scenarios, the recorded affinity values are subject to idiosyncratic monotonic transformations and can be highly noisy, so matching these values exactly may result in overfitting and poor generalization. Crucially, the predictive quantity of operational interest is the ranking order induced by preferences, not the exact numerical magnitude. Consequently, "Preference-Aligned BestFit Reranking" reorients the estimation objective to directly fit the observed (partial) preference order and uses regularization to ensure learnability and generalization, even under limited or arbitrary observation and supervision.

2. Algorithmic Formulation

The core estimator is defined as a convex optimization problem that recovers a preference matrix $X$ such that its observed entries are as close as possible—in ranking order, not in value—to the ground-truth (and often monotonic-transformed) preferences. For a universe of entities (columns) and items (rows), suppose that for each entity $j$ a partial ordering over a subset of items is observed, represented as $\mathbf{y}^{(j)}$ (typically a noisy monotonic transformation plus sub-sampling) across indices $\Omega_j$. The estimator solves:

$$X = \underset{X}{\arg\min} \min_{z \in \mathbb{R}^{|\Omega|}} \left\{ \lambda \|X\|_* + \frac{1}{2} \|z - P_\Omega(X)\|_2^2 \right\} \quad\text{subject to}\quad \forall j,\; z_{\Omega_j} \in \varepsilon^{(n_j)}(\mathbf{y}^{(j)})$$

where:

• $P_\Omega(X)$ extracts the observed entries of $X$,
• $\|X\|_*$ is the nuclear norm (convex surrogate for low-rank constraint),
• $\lambda$ is the regularization parameter,
• $\varepsilon^{(n)}(y)$ is the $\varepsilon$-margin isotonic set defined by

$$\varepsilon^n(y) = \{x \in \mathbb{R}^n: \forall i,k,\, y_i < y_k \Rightarrow x_i \leq x_k - \varepsilon\}$$

• $\varepsilon > 0$ is a margin parameter that enforces strict separation in the predicted orderings.

The optimization is performed by alternating minimization: updating $X$ (using proximal gradient steps with singular value thresholding for the nuclear norm) and updating $z$ by projection onto the constraint set (which may involve isotonic regression per entity).

3. Preference Order Fitting and Robustness

The estimator's core property is that it fits only the internal order of recorded preferences—respecting inequalities and their $\varepsilon$-margin—while permitting the numerical predictions to deviate from the actual (potentially distorted) affinity values. This detaches the model from the pitfalls of overfitting to arbitrary or over-parameterized observations. The nuclear norm constraint on $X$ ensures a latent low-rank structure, reflecting the existence of unobserved global factors modulating entity–item preferences. By regularizing the model's complexity, this approach also prevents overfitting to the training data, yielding better generalization to unobserved pairs.

4. Support for Partial and Directed Acyclic Ranking Structures

A major strength of Preference-Aligned BestFit Reranking is its flexibility in accepting both total and partial orders, the latter of which may be encoded as a directed acyclic graph (DAG) $G([n], E)$. Here, an edge $(i, k) \in E$ indicates that item $i$ is ranked below $k$. The order constraint is then efficiently expressed as

$\{x : A_G^T x \leq -\varepsilon \mathbf{1} \}$

where $A_G$ is the graph's incidence matrix. Projection onto such DAG-constrained isotonic sets can be accomplished by computing longest chains or by extended isotonic regression algorithms, making the method broadly applicable across highly incomplete or inherently partial supervision regimes.

5. Computational Complexity

Despite introducing order constraints and the potential for rich supervision structures (DAGs, blocks, etc.), the algorithm retains efficiency:

• For strict or blockwise total orders, projection can be handled in $O(n)$ (total) or $O(n\log n)$ (blockwise) time using variants of the Pool-Adjacent-Violators algorithm.
• The alternating minimization converges in $O(1/\varepsilon)$ iterations, with each main step dominated by singular value decomposition (as in standard low‐rank matrix completion).
• Overall, the computational burden of extending from basic matrix completion to partial-ranking-aware preference alignment is within a logarithmic factor (i.e., an additional $O(\log|\Omega|)$) of established nuclear norm–regularized approaches.

6. Empirical Evidence and Application to High-Dimensional, Real-World Data

The methodology has been empirically validated on standard datasets (e.g., Movielens100K) as well as on challenging neuroimaging meta-analysis data from Neurosynth. On the Neurosynth task—ranking cognitive concept associations for each brain region (“reverse inference” from voxel to term) in a regime involving a dense, high‐ranked item matrix—the preference-aligned estimator (Retargeted Matrix Completion, RMC) outperformed standard matrix completion and collaborative rankers (CoFi-Rank) with respect to NDCG, precision@N, and rank correlation metrics (Spearman, Kendall). The improvement is attributed to direct enforcement of ordering constraints and the increased robustness granted by not fitting raw scores, especially when the true target is order preservation. Notably, the RMC estimator was consistently superior even in highly structured or nearly–total‐order regime, evidencing its capacity for both flexibility and statistical power.

7. Generalization, Overfitting, and Theoretical Guarantees

The framework includes non-asymptotic generalization error bounds, which assert that the test loss remains bounded provided the empirical surrogate ranking loss and nuclear norm of the estimator are controlled. In particular, the loss function

$$\Phi(x, y) = \min_{z \in \varepsilon^n(y)} \|x - z\|_2$$

directly measures the fit to any monotonic transformation preserving the order, obviating the need for strong parametric assumptions about the affinity values. Theoretical analysis proves that, for bounded nuclear norm and small empirical ranking error, the probability of large test error decreases rapidly as the number of observed entries grows. This demonstrates that preference-aligned best-fit reranking avoids overfitting while being robust to the unknown, possibly highly nonlinear transformations in the observed data.

---

Preference-Aligned BestFit Reranking, as instantiated by the described convex estimator with nuclear norm constraints and order-based surrogate loss, provides a computationally efficient and statistically robust method for collaborative ranking and completion. Its adoption is particularly warranted in settings involving noisy, partial, or arbitrary-valued recorded affinities, as it guarantees order-consistent recovery of preferences while preventing overfitting—a critical property in domains characterized by diverse, partial, and high-dimensional user–item interactions (Gunasekar et al., 2016).