We explore the match prediction problem where one seeks to estimate the likelihood of a group of M items preferred over another, based on partial group comparison data. Challenges arise in practice. As existing state-of-the-art algorithms are tailored to certain statistical models, we have different best algorithms across distinct scenarios. Worse yet, we have no prior knowledge on the underlying model for a given scenario. These call for a unified approach that can be universally applied to a wide range of scenarios and achieve consistently high performances. To this end, we incorporate deep learning architectures so as to reflect the key structural features that most state-of-the-art algorithms, some of which are optimal in certain settings, share in common. This enables us to infer hidden models underlying a given dataset, which govern in-group interactions and statistical patterns of comparisons, and hence to devise the best algorithm tailored to the dataset at hand. Through extensive experiments on synthetic and real-world datasets, we evaluate our framework in comparison to state-of-the-art algorithms. It turns out that our framework consistently leads to the best performance across all datasets in terms of cross entropy loss and prediction accuracy, while the state-of-the-art algorithms suffer from inconsistent performances across different datasets. Furthermore, we show that it can be easily extended to attain satisfactory performances in rank aggregation tasks, suggesting that it can be adaptable for other tasks as well. The most elementary form of comparisons is pairwise: we often compare a pair of items and make judgments as to which one is of higher utility or simply preferable over the other. With a large amount of such comparison data, one can consider various interesting tasks. One may wish to predict future outcomes of unseen matches, and also to rank alternatives in order of utility or preference. Challenges arise in carrying out these tasks. Almost all existing state-of-the-art algorithms have been developed under the assumption that given a scenario, there exists a certain underlying model which governs statistical patterns of comparison data (see Section 2 for details). As such, we have different best-performing algorithms across distinct scenarios. This traditional approach, which begins by assuming certain models to develop algorithms, comes with limitations in practice. First, it gives rise to inconsistent performances. No single algorithm can perform consistently well in a wide range of scenarios, since it has been tailored to a specific model. Second, it is hard to know the underlying model without expert domain knowledge. In its absence, we have little choice but to find an appropriate algorithm via trial-and-error. Third, the model can be inherently complex for any existing algorithm to be effective. Sometimes groups of items are compared, thus the effects of interactions among in-group items come into play, further complicating the model. In this work, we propose a unified algorithmic framework aimed to overcome these barriers. We focus on the match prediction problem where one wishes to estimate the likelihood of a group of M items preferred over another, based on partially observed group comparison data among a collection of n items. One can imagine that such group comparison data may bear complex statistical patterns due to a combination of two underlying models: the interaction model which governs the effects of in-group interactions in determining the utility or preference of a group; and the comparison model which governs the statistical patterns of pairwise group comparison data. Hence, designing a novel framework hinges heavily upon accurate inference of these underlying models. Main contribution. We incorporate deep learning techniques into our framework design. This enables us to infer the underlying models from real-world data obtained from a given application, and thus to achieve consistently high performances on a variety of datasets from diverse real-world applications where match prediction tasks are of interest. To this end, we build on progress made through analysis in well-defined statistical models. We gain insights instrumental to the progress by looking into existing state-of-the-art algorithms in related and long-studied tasks such as rank aggregation (Negahban et al., 2016; Hunter, 2004; Huang et al., 2006; . We find that most of them share a key element. They all exhibit so-called reward-andpenalty mechanisms in estimating the utilities of individual items. To be more specific, they reward an item more greatly for winning (or being more preferred) in a disadvantageous comparison where its group is weaker than the counterpart. Likewise, they penalize it more greatly for losing (or being less preferred) in an advantageous one. In addition, the magnitudes of rewards and penalties are proportional to the contribution of the individual item to its group. This structural similarity across the state-of-the-art algorithms has attracted our attention. Through some manipulation, we find that they all employ the same basic rule for estimating individual utilities (see (6) in Section 4 for details). The terms corresponding to rewards and penalties turn out to vary as either one of the two underlying models changes. This observation has inspired us to incorporate neural networks into our framework design. The novelty of our design is salient in an ablation study where we compare it with a simple design. As an initial effort, a single-layer neural network has been employed to predict winning probabilities of unseen group matches (Menke & Martinez, 2008) . It has shown a promising result, demonstrating prediction accuracy to be improved on a real-world online game dataset, but also exhibited a scalability issue. It requires one input node per item, making it prohibitive to be extended to realworld applications with a large number of items. Leveraging more advanced architectures (see Figures 1 and 2 ) motivated by observant analysis as emphasized, our design not only addresses such a scalability issue by design, but also outperforms the single-layer neural network. The merits of our design are evaluated against the single-layer neural network and other state-of-the-art algorithms through extensive experiments on a variety of synthetic and real-world datasets (see Section 5). Using synthetic datasets, we demonstrate that our approach can achieve the performances of the state-of-the-art algorithms in the models for which they have been specifically developed. We investigate four models. Three consider various extensions of the Bradley-Terry-Luce model (Bradley & Terry, 1952) to the group comparison scenario. The other is a generalized version of the Thurstone model (Herbrich et al., 2007) widely used in skill rating systems of online games. As a result, we show that our framework consistently yields the best performances across all of these datasets (nearbest in some cases), while the other state-of-the-art algorithms suffer from inconsistent performances across different models. Using real-world datasets, we also demonstrate that our framework performs consistently well across diverse real-world applications. We investigate five real-world datasets (sources in Footnote 6). One is a crowd-sourced image classification dataset, another is a collection of movie ratings, and the other three are match records from online games. We consider, in addition to the cross entropy loss, the prediction accuracy as another metric (defined in (9)). As a result, we show that our framework consistently yields almost the best performances across all of these datasets in terms of both metrics. We also show that our framework can be easily extended to achieve the best performance in rank aggregation tasks where one seeks to rank items in order of utility or preference. Using a realworld dataset of movie ratings, we demonstrate that our framework yields the best performances in terms of two well-known metrics (see Footnote 10): Kendall tau distance (Kendall, 1938) and normalized discounted cumulative gain (Järvelin & Kekäläinen, 2002) . This result suggests that it can potentially be adaptable for other tasks as well. We investigate the match prediction problem where the task is to predict the preference of one group over the other given an unseen pair of groups based on the observed group comparison data. Facing with real-world challenges that underlying models that govern in-group interactions and group comparisons are unknown and complex, we develop an algorithm that employs deep neural networks to infer such latent models from data specific to a given application. As a result, we show that our algorithm can show consistently best prediction performances compared to other state-ofthe-art algorithms on multiple datasets across various domains. We also demonstrate that it can be applied to the rank aggregation task, which implies its potentially broader application to other tasks. In view of it, we consider the following task as one possible direction for future work. The task is to predict whether multiple items that constitute a group would make an effective combination producing positive synergies, and thus lead to a desired outcome. Bundling strategies in e-commerce can be a real-world example: multiple items are bundled as a package and offered to the potential buyer with a discount. The goal is to figure out which set of items would appeal most to the buyer given past sales data. We expect that our current architecture can be extended to this task. Among a number of items, some will contribute positively to the group (rewards) and some negatively (penalties). Our modules R and P can be applied to measure them. Our module G can be applied to govern how these rewards and penalties manifest collectively as a group outcome. We also expect that our work can appear frequently in other tasks as well where in-group interactions are critically concerned, but their statistical patterns are unknown in practice. 10 Kendall tau distance is defined as |{(i, j) : i < j, (τ1(i ) < τ1(j ) ∧ τ2(i ) > τ2(j )) ∨ (τ1(i ) > τ1(j ) ∧ τ2(i ) < τ2(j))}| where τ1(i) and τ2(i) are the rankings of item i in τ1 and τ2. In words , it counts the number of item pairs that are ranked reversely in the two rankings. In NDCG , items are associated with relevance scores. In our case, items ranked higher in the ground-truth ranking have higher scores. Let reli be the score of the item ranked i-th in a ranking. NDCG discounts reli by log 2 (i + 1) to "penalyze" the quality of a ranking for placing a high-relevance item at a low rank. NDCG@K is normalized DCG@K defined as K i=1 reli/log 2 (i + 1).
We investigate the merits of employing neural networks in the match prediction problem where one seeks to estimate the likelihood of a group of M items preferred over another, based on partial group comparison data. This paper proposes a deep neural network solution to the set ranking problem and designs a architecture for this task inspired by previous manually designed algorithms. This paper provides a technique to solve the match prediction problem using a deep learning architecture.