[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86264-en":3,"doc-seo-86264-105":30,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},86264,687197207919,"Theodora","https://ap-avatar.wpscdn.com/avatar/a000253d6f5f7c60be?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779446848396160552",8,"Research & Report","Advancing Optimal Subset Oracle via Learning Relaxation of Neural Set Functions","Learning neural set functions is central to applications such as compound selection in AI-driven drug discovery and product recommendation. Prior approaches use optimal subset oracles with variational inference, optimizing parameters via mean-field variational ELBO. However, Monte Carlo sampling is required to estimate ELBO gradients, causing repeated sampling overhead and destabilizing stochastic optimization. This work replaces sampling-based gradient estimation with a learned continuous relaxation surrogate objective, yielding stable, efficient gradients and faster inference. It also proves approximation guarantees under submodular maximization and links to variational free energy.","arXiv :2607 . 1 1555v 1 [ cs .LG] 13 Jul 2026  \nADVANCING OPTIMAL SUBSET ORACLE VIA  \nLEARNING RELAXATION OF NEURAL SET FUNCTIONS  \nYongquan Shi 1 , Zijing Ou2 , Shiping Wang 1 , Yatao Bian3  \n1Fuzhou University, 2Imperial College London, 3National University of Singapore [losparksayoji@outlook.com](losparksayoji@outlook.com)  \nABSTRACT  \nLearning neural set functions is pivotal to a wide range of important applications, including compound selection in AI-driven drug discovery and product recommendation. Recent work has introduced optimal subset oracles to implicitly learn set functions under practical weakly supervised settings, where model parameters are optimized through mean-field variational inference. However, these frameworks rely on Monte Carlo sampling to estimate gradients of the evidence lower bound when updating the variational distribution. Repeated sampling across iterations incurs substantial computational overhead, while the resulting stochasticity can destabilize the optimization trajectory. In this work, we reinterpret the evidence lower bound as a continuous relaxation of the set function and learn a surrogate objective that replaces sampling-based ELBO gradient estimation during variational optimization. The learned surrogate provides stable and efficient gradients throughout the continuous domain, thereby reducing computational overhead and accelerating inference. Furthermore, we establish an approximation guarantee for the proposed framework under submodular maximization and characterize its connection to variational free energy. Experiments on a variety of real-world tasks demonstrate consistent improvements over existing baselines.  \n1 INTRODUCTION  \nSet-value prediction has a wide range of applications in real-world scenarios and plays a crucial role in many tasks. For example, recommendation systems select products that are likely to interest a user (Coppolillo et al., 2024), anomaly detection identifies outliers from the majority of observations (Zhang et al., 2020), and AI-driven drug discovery prioritizes promising compounds from large candidate databases (Gimeno et al., 2019) . These tasks require explicitly or implicitly learning a set function (Rezatofighi et al., 2017 ; Zaheer et al., 2017) that assigns a utility value to each candidate subset, with more desirable subsets receiving higher values.  \nMore formally, our objective is to select an optimal subset S ∗ from a given large ground set V , such that it attains the highest utility value among all candidate subsets according to a set function Fθ (S;V ) parameterized by θ . This process can be understood as optimizing the following criteria:  \nS ∗ = arg max Fθ (S;V ) . (1)  \nS∈2V  \nA direct approach is to learn Fθ (S;V ) in a supervised manner from tuples (Vi , Si , Ui), where Ui denotes the utility of the candidate subset Si , a setting commonly referred to as a function-value (FV) oracle (Balcan & Harvey, 2018) . However, this training paradigm is often prohibitively expensive, as it requires collecting utility annotations for a sufficiently large and diverse set of candidate subsets (Ou et al., 2022) . To overcome this limitation, an alternative approach to optimizing objective (1) is to implicitly learn the set function from a probabilistic perspective (Tschiatschek et al., 2018) . This approach estimates the parameter θ in a supervised manner using pairs { (Vi , S)}, where S denotes the optimal subset corresponding to Vi , serving as an optimal subset (OS) oracle. With limited data sampled from the underlying distribution P (S, V ), the OS oracle learns latent patterns by maximizing the empirical log-likelihood P log pθ (S|Vi) over the observed data. Compared with the FV oracle, the OS oracle is generally more practical, as it avoids explicit utility labeling for candidate subsets, thereby reducing annotation costs in practice.  \nFigure 1: (left) Optimization trajectories over the same number of steps on a black-box objective: Monte Carl","cbCaieGtSfOYDu2c","https://ap.wps.com/l/cbCaieGtSfOYDu2c","pdf",4097973,3,1,32,"English","en",105,"# Introduction\n## Set-value prediction and set functions\n## FV oracle vs OS oracle\n## Variational inference and ELBO optimization\n## Monte Carlo sampling limitations\n## Core idea and contribution outline","[{\"question\":\"What problem does the paper address in learning neural set functions?\",\"answer\":\"It targets the computational cost and training instability caused by Monte Carlo sampling when estimating ELBO gradients in variational optimal-subset-oracle frameworks.\"},{\"question\":\"How does the proposed method avoid Monte Carlo sampling for ELBO gradients?\",\"answer\":\"It reinterprets the ELBO as a continuous relaxation of the set function and learns a surrogate objective that directly provides stable gradients over the continuous domain.\"},{\"question\":\"What theoretical and practical results does the paper claim?\",\"answer\":\"The framework includes an approximation guarantee under submodular maximization, characterizes a connection to variational free energy, and achieves consistent improvements over baselines on multiple real-world 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problem does the paper address in learning neural set functions?","Question",{"text":75,"@type":76},"It targets the computational cost and training instability caused by Monte Carlo sampling when estimating ELBO gradients in variational optimal-subset-oracle frameworks.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How does the proposed method avoid Monte Carlo sampling for ELBO gradients?",{"text":80,"@type":76},"It reinterprets the ELBO as a continuous relaxation of the set function and learns a surrogate objective that directly provides stable gradients over the continuous domain.",{"name":82,"@type":73,"acceptedAnswer":83},"What theoretical and practical results does the paper claim?",{"text":84,"@type":76},"The framework includes an approximation guarantee under submodular maximization, characterizes a connection to variational free energy, and achieves consistent improvements over baselines on multiple real-world 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