In recent years, multi-objective optimization (MOO) emerges as a foundational problem underpinning many multi-agent multi-task learning applications.

Sampling from an unnormalized target distribution is an essential problem with manyapplications inprobabilistic inference.

Missing data is a common problem across various scientific disciplines, including medical research (Bell et al., 2014), social sciences (Molenberghs et al., 2014), and astronomy (Ivezi c et al., 2020). To handle missing entries in the dataset, imputation (Grzesiak et al., 2025; Kim and Shao, 2021; Little and Rubin, 2019) is a popular approach that is widely accepted in practice. An imputation model generates plausible values for each missing entry, transforming an incomplete dataset into a complete one. The critical importance of this task has led to the development of a wide array of imputation models, grounded in various modeling assumptions. These range from traditional approaches like hot-deck imputation (Little and Rubin, 2019) to more sophisticated methods such as Multiple Imputation via Chained Equations (MICE; V an Buuren and Groothuis-Oudshoorn 2011), random forest imputation (Stekhoven and Bühlmann, 2012), techniques based on Markov assumptions on graphs (Y ang and Chen, 2025), and even generative adversarial networks (Y oon et al., 2018). Despite the proliferation of imputation models, the selection of an optimal imputation model for a given dataset remains a significant challenge, largely due to the unsupervised nature of the problem. Among the many proposed strategies for evaluating and selecting imputation models, masking has emerged as a particularly popular procedure (Gelman et al., 1998; Honaker et al., 2011; Leek et al., 2012; Qian et al., 2024; Troyanskaya et al., 2001; Wang et al., 2024). Masking involves intentionally creating missing values in observed entries to create a setting where imputation accuracy can be measured against a known ground truth. This approach has demonstrated remarkable success and power in other domains, notably in language modeling (Devlin et al., 2019; Y ang et al., 2019) and image recognition (Hondru et al., 2025; Vincent et al., 2010; Xie et al., 2022) and prediction-powered inference (Angelopoulos et al., 2023; Wang et al., 2020).

Recent advances in Large Language Models (LLMs) have raised interest in their formal reasoning capabilities, particularly in mathematics. While closed LLMs like GPT-4 perform well on mathematical benchmarks, e.g., GSM8K, it remains unclear whether small to medium-sized open LLMs can achieve similar performance, questioning their reliability. To close this gap, we propose a post-training approach leveraging a mixture of opinions (MoO) from weaker ancillary LLMs to enhance a (relatively) stronger LLM's reasoning. For that, each post-training sample is augmented with Chain-of-Thought (CoT) reasoning steps and answers from ancillary LLMs, enabling the main LLM to learn from diverse perspectives. We compare MoO with standard supervised fine-tuning (SFT), few-shot prompting, and the Mixture of Agents (MoA) method on mathematical reasoning benchmarks. Our results show that incorporating weaker LLMs' opinions improves mathematical reasoning by an average of 5%, highlighting the value of diverse perspectives in reasoning tasks.

Summary and Contributions: Multi-objective optimization (MOO) problems involve more than one objective function that are to be minimized or maximized. For non-trivial instances of MOO problems, no unique solution exists that simultaneously optimizes all objectives. In that case, the aim to identify the set of Pareto optimal solutions of the problem. Bayesian Optimization (BO) approaches rely on acquisition functions (AF), to evaluate promising query points for function evaluations. BO approaches for MOO require to define AFs that are applicable to the notion of Pareto optimality.

Recently, there has been an increasing interest in exploring the application of multiobjective optimization (MOO) in machine learning (ML). The interest is driven by the numerous situations in real-life applications where multiple objectives need to be optimized simultaneously. A key aspect of MOO is the existence of a Pareto set, rather than a single optimal solution, which illustrates the inherent trade-offs between objectives. Despite its potential, there is a noticeable lack of satisfactory literature that could serve as an entry-level guide for ML practitioners who want to use MOO. Hence, our goal in this paper is to produce such a resource. We critically review previous studies, particularly those involving MOO in deep learning (using Physics-Informed Neural Networks (PINNs) as a guiding example), and identify misconceptions that highlight the need for a better grasp of MOO principles in ML. Using MOO of PINNs as a case study, we demonstrate the interplay between the data loss and the physics loss terms. We highlight the most common pitfalls one should avoid while using MOO techniques in ML. We begin by establishing the groundwork for MOO, focusing on well-known approaches such as the weighted sum (WS) method, alongside more complex techniques like the multiobjective gradient descent algorithm (MGDA). Additionally, we compare the results obtained from the WS and MGDA with one of the most common evolutionary algorithms, NSGA-II. We emphasize the importance of understanding the specific problem, the objective space, and the selected MOO method, while also noting that neglecting factors such as convergence can result in inaccurate outcomes and, consequently, a non-optimal solution. Our goal is to offer a clear and practical guide for ML practitioners to effectively apply MOO, particularly in the context of DL.

In recent years, multi-objective optimization (MOO) emerges as a foundational problem underpinning many multi-agent multi-task learning applications. However, existing algorithms in MOO literature remain limited to centralized learning settings, which do not satisfy the distributed nature and data privacy needs of such multi-agent multi-task learning applications. This motivates us to propose a new federated multi-objective learning (FMOL) framework with multiple clients distributively and collaboratively solving an MOO problem while keeping their training data private. Notably, our FMOL framework allows a different set of objective functions across different clients to support a wide range of applications, which advances and generalizes the MOO formulation to the federated learning paradigm for the first time. For this FMOL framework, we propose two new federated multi-objective optimization (FMOO) algorithms called federated multi-gradient descent averaging (FMGDA) and federated stochastic multi-gradient descent averaging (FSMGDA). Both algorithms allow local updates to significantly reduce communication costs, while achieving the {\em same} convergence rates as those of their algorithmic counterparts in the single-objective federated learning. Our extensive experiments also corroborate the efficacy of our proposed FMOO algorithms.

We present a review that unifies decision-support methods for exploring the solutions produced by multi-objective optimization (MOO) algorithms. As MOO is applied to solve diverse problems, approaches for analyzing the trade-offs offered by MOO algorithms are scattered across fields. We provide an overview of the advances on this topic, including methods for visualization, mining the solution set, and uncertainty exploration as well as emerging research directions, including interactivity, explainability, and ethics. We synthesize these methods drawing from different fields of research to build a unified approach, independent of the application. Our goals are to reduce the entry barrier for researchers and practitioners on using MOO algorithms and to provide novel research directions.

Deep learning models form one of the most powerful machine learning models for the extraction of important features. Most of the designs of deep neural models, i.e., the initialization of parameters, are still manually tuned. Hence, obtaining a model with high performance is exceedingly time-consuming and occasionally impossible. Optimizing the parameters of the deep networks, therefore, requires improved optimization algorithms with high convergence rates. The single objective-based optimization methods generally used are mostly time-consuming and do not guarantee optimum performance in all cases. Mathematical optimization problems containing multiple objective functions that must be optimized simultaneously fall under the category of multi-objective optimization sometimes referred to as Pareto optimization. Multi-objective optimization problems form one of the alternatives yet useful options for parameter optimization. However, this domain is a bit less explored. In this survey, we focus on exploring the effectiveness of multi-objective optimization strategies for parameter optimization in conjunction with deep neural networks. The case studies used in this study focus on how the two methods are combined to provide valuable insights into the generation of predictions and analysis in multiple applications.