Best-of-N (BoN) sampling, a common strategy for test-time scaling of Large Language Models (LLMs), relies on reward models to select the best candidate solution from multiple generations. However, traditional reward models often assign arbitrary and inconsistent scores, limiting their effectiveness. To address this, we propose a Pairwise Reward Model (Pairwise RM) combined with a knockout tournament for BoN sampling. Instead of assigning absolute scores, given one math problem, Pairwise RM evaluates two candidate solutions' correctness simultaneously. This approach eliminates the need for arbitrary scoring and enables cross-validation of solutions through parallel comparison. In the knockout tournament, Pairwise RM conducts pairwise comparisons between candidate solutions and eliminates the incorrect ones iteratively. We construct \ourdataset, a large-scale dataset of 443K pairwise comparisons derived from NumiaMath and annotated using \texttt{gemini-1.5-flash}, and train the Pairwise RM via supervised fine-tuning. Experiments on MATH-500 and the Olympiad Bench demonstrate significant improvements over traditional discriminative reward models. And a 40\% to 60\% relative improvement is achieved on the top 50\% challenging problems.

Semantic Embedding Model (SEM), a neural network-based Siamese architecture, is gaining momentum in information retrieval and natural language processing. In order to train SEM in a supervised fashion for Web search, the search engine query log is typically utilized to automatically formulate pairwise judgments as training data. Despite the growing application of semantic embeddings in the search engine industry, little work has been done on formulating effective pairwise judgments for training SEM. In this paper, we make the first in-depth investigation of a wide range of strategies for generating pairwise judgments for SEM. An interesting (perhaps surprising) discovery reveals that the conventional pairwise judgment formulation strategy wildly used in the field of pairwise Learning-to-Rank (LTR) is not necessarily effective for training SEM. Through a large-scale empirical study based on query logs and click-through activities from a major commercial search engine, we demonstrate the effective strategies for SEM and highlight the advantages of a hybrid heuristic (i.e., Clicked > Non-Clicked) in comparison to the atomic heuristics (e.g., Clicked > Skipped) in LTR. We conclude with best practices for training SEM and offer promising insights for future research.

We revisit the notion of individual fairness proposed by Dwork et al. A central challenge in operationalizing their approach is the difficulty in eliciting a human specification of a similarity metric. In this paper, we propose an operationalization of individual fairness that does not rely on a human specification of a distance metric. Instead, we propose novel approaches to elicit and leverage side-information on equally deserving individuals to counter subordination between social groups. We model this knowledge as a fairness graph, and learn a unified Pairwise Fair Representation(PFR) of the data that captures both data-driven similarity between individuals and the pairwise side-information in fairness graph. We elicit fairness judgments from a variety of sources, including humans judgments for two real-world datasets on recidivism prediction (COMPAS) and violent neighborhood prediction (Crime & Communities). Our experiments show that the PFR model for operationalizing individual fairness is practically viable.