Estimating the Hessian Matrix of Ranking Objectives for Stochastic Learning to Rank with Gradient Boosted Trees

TL;DR

This paper tackles the gap that stochastic learning-to-rank (LTR) methods have largely neglected gradient-boosted decision trees (GBDTs) due to the absence of Hessian information. It introduces a novel Hessian estimator for ranking objectives under Plackett-Luce (PL) models and integrates it with the PL-Rank framework to enable efficient second-order optimization. Empirical results across Yahoo!, MSLR, and Istella show that Hessian-enabled GBDTs achieve substantial NDCG$@K$ gains over Hessian-free counterparts and are competitive with neural networks on several datasets, with notably stable convergence. The work closes a critical gap between stochastic LTR and deterministic LTR by enabling robust, Hessian-informed GBDT optimization for ranking metrics.

Abstract

Stochastic learning to rank (LTR) is a recent branch in the LTR field that concerns the optimization of probabilistic ranking models. Their probabilistic behavior enables certain ranking qualities that are impossible with deterministic models. For example, they can increase the diversity of displayed documents, increase fairness of exposure over documents, and better balance exploitation and exploration through randomization. A core difficulty in LTR is gradient estimation, for this reason, existing stochastic LTR methods have been limited to differentiable ranking models (e.g., neural networks). This is in stark contrast with the general field of LTR where Gradient Boosted Decision Trees (GBDTs) have long been considered the state-of-the-art. In this work, we address this gap by introducing the first stochastic LTR method for GBDTs. Our main contribution is a novel estimator for the second-order derivatives, i.e., the Hessian matrix, which is a requirement for effective GBDTs. To efficiently compute both the first and second-order derivatives simultaneously, we incorporate our estimator into the existing PL-Rank framework, which was originally designed for first-order derivatives only. Our experimental results indicate that stochastic LTR without the Hessian has extremely poor performance, whilst the performance is competitive with the current state-of-the-art with our estimated Hessian. Thus, through the contribution of our novel Hessian estimation method, we have successfully introduced GBDTs to stochastic LTR.

Figures (3)

• Figure 1: The complete second-order derivative of the reward function $R(\pi)$ w.r.t. the scoring function $m(d)$.
• Figure 2: Learning curves of stochastic LTR methods in terms of NDCG$@K$ over a thousand epochs.
• Figure 3: Convergence analysis on the Yahoo! dataset.