Online Mirror Descent for Tchebycheff Scalarization in Multi-Objective Optimization

The goal of multi-objective optimization (MOO) is to learn under multiple, potentially conflicting, objectives. One widely used technique to tackle MOO is through linear scalarization, where one fixed preference vector is used to combine the objectives into a single scalar value for optimization. However, recent work (Hu et al., 2024) has shown linear scalarization often fails to capture the non-convex regions of the Pareto Front, failing to recover the complete set of Pareto optimal solutions. In light of the above limitations, this paper focuses on Tchebycheff scalarization that optimizes for the worst-case objective. In particular, we propose an online mirror descent algorithm for Tchebycheff scalarization, which we call OMD-TCH. We show that OMD-TCH enjoys a convergence rate of $O(\sqrt{\log m/T})$ where $m$ is the number of objectives and $T$ is the number of iteration rounds. We also propose a novel adaptive online-to-batch conversion scheme that significantly improves the practical performance of OMD-TCH while maintaining the same convergence guarantees. We demonstrate the effectiveness of OMD-TCH and the adaptive conversion scheme on both synthetic problems and federated learning tasks under fairness constraints, showing state-of-the-art performance.

LibMOON: A Gradient-based MultiObjective OptimizatioN Library in PyTorch

Multiobjective optimization problems (MOPs) are prevalent in machine learning, with applications in multi-task learning, learning under fairness or robustness constraints, etc. Instead of reducing multiple objective functions into a scalar objective, MOPs aim to optimize for the so-called Pareto optimality or Pareto set learning, which involves optimizing more than one objective function simultaneously, over models with millions of parameters. Existing benchmark libraries for MOPs mainly focus on evolutionary algorithms, most of which are zeroth-order methods that do not effectively utilize higher-order information from objectives and cannot scale to large-scale models with millions of parameters. In light of the above gap, this paper introduces LibMOON, the first multiobjective optimization library that supports state-of-the-art gradient-based methods, provides a fair benchmark, and is open-sourced for the community.

HeadGAP: Few-shot 3D Head Avatar via Generalizable Gaussian Priors

In this paper, we present a novel 3D head avatar creation approach capable of generalizing from few-shot in-the-wild data with high-fidelity and animatable robustness. Given the underconstrained nature of this problem, incorporating prior knowledge is essential. Therefore, we propose a framework comprising prior learning and avatar creation phases. The prior learning phase leverages 3D head priors derived from a large-scale multi-view dynamic dataset, and the avatar creation phase applies these priors for few-shot personalization. Our approach effectively captures these priors by utilizing a Gaussian Splatting-based auto-decoder network with part-based dynamic modeling. Our method employs identity-shared encoding with personalized latent codes for individual identities to learn the attributes of Gaussian primitives. During the avatar creation phase, we achieve fast head avatar personalization by leveraging inversion and fine-tuning strategies. Extensive experiments demonstrate that our model effectively exploits head priors and successfully generalizes them to few-shot personalization, achieving photo-realistic rendering quality, multi-view consistency, and stable animation.

Few for Many: Tchebycheff Set Scalarization for Many-Objective Optimization

Multi-objective optimization can be found in many real-world applications where some conflicting objectives can not be optimized by a single solution. Existing optimization methods often focus on finding a set of Pareto solutions with different optimal trade-offs among the objectives. However, the required number of solutions to well approximate the whole Pareto optimal set could be exponentially large with respect to the number of objectives, which makes these methods unsuitable for handling many optimization objectives. In this work, instead of finding a dense set of Pareto solutions, we propose a novel Tchebycheff set scalarization method to find a few representative solutions (e.g., 5) to cover a large number of objectives (e.g., $>100$) in a collaborative and complementary manner. In this way, each objective can be well addressed by at least one solution in the small solution set. In addition, we further develop a smooth Tchebycheff set scalarization approach for efficient optimization with good theoretical guarantees. Experimental studies on different problems with many optimization objectives demonstrate the effectiveness of our proposed method.

Smooth Tchebycheff Scalarization for Multi-Objective Optimization

Multi-objective optimization problems can be found in many real-world applications, where the objectives often conflict each other and cannot be optimized by a single solution. In the past few decades, numerous methods have been proposed to find Pareto solutions that represent different optimal trade-offs among the objectives for a given problem. However, these existing methods could have high computational complexity or may not have good theoretical properties for solving a general differentiable multi-objective optimization problem. In this work, by leveraging the smooth optimization technique, we propose a novel and lightweight smooth Tchebycheff scalarization approach for gradient-based multi-objective optimization. It has good theoretical properties for finding all Pareto solutions with valid trade-off preferences, while enjoying significantly lower computational complexity compared to other methods. Experimental results on various real-world application problems fully demonstrate the effectiveness of our proposed method.

PMGDA: A Preference-based Multiple Gradient Descent Algorithm

It is desirable in many multi-objective machine learning applications, such as multi-task learning with conflicting objectives and multi-objective reinforcement learning, to find a Pareto solution that can match a given preference of a decision maker. These problems are often large-scale with available gradient information but cannot be handled very well by the existing algorithms. To tackle this critical issue, this paper proposes a novel predict-and-correct framework for locating a Pareto solution that fits the preference of a decision maker. In the proposed framework, a constraint function is introduced in the search progress to align the solution with a user-specific preference, which can be optimized simultaneously with multiple objective functions. Experimental results show that our proposed method can efficiently find a particular Pareto solution under the demand of a decision maker for standard multiobjective benchmark, multi-task learning, and multi-objective reinforcement learning problems with more than thousands of decision variables. Code is available at: https://github.com/xzhang2523/pmgda. Our code is current provided in the pgmda.rar attached file and will be open-sourced after publication.}

UMOEA/D: A Multiobjective Evolutionary Algorithm for Uniform Pareto Objectives based on Decomposition

Multiobjective optimization (MOO) is prevalent in numerous applications, in which a Pareto front (PF) is constructed to display optima under various preferences. Previous methods commonly utilize the set of Pareto objectives (particles on the PF) to represent the entire PF. However, the empirical distribution of the Pareto objectives on the PF is rarely studied, which implicitly impedes the generation of diverse and representative Pareto objectives in previous methods. To bridge the gap, we suggest in this paper constructing \emph{uniformly distributed} Pareto objectives on the PF, so as to alleviate the limited diversity found in previous MOO approaches. We are the first to formally define the concept of ``uniformity" for an MOO problem. We optimize the maximal minimal distances on the Pareto front using a neural network, resulting in both asymptotically and non-asymptotically uniform Pareto objectives. Our proposed method is validated through experiments on real-world and synthetic problems, which demonstrates the efficacy in generating high-quality uniform Pareto objectives and the encouraging performance exceeding existing state-of-the-art methods. The detailed model implementation and the code are scheduled to be open-sourced upon publication.

Panacea: Pareto Alignment via Preference Adaptation for LLMs

Current methods for large language model alignment typically use scalar human preference labels. However, this convention tends to oversimplify the multi-dimensional and heterogeneous nature of human preferences, leading to reduced expressivity and even misalignment. This paper presents Panacea, an innovative approach that reframes alignment as a multi-dimensional preference optimization problem. Panacea trains a single model capable of adapting online and Pareto-optimally to diverse sets of preferences without the need for further tuning. A major challenge here is using a low-dimensional preference vector to guide the model's behavior, despite it being governed by an overwhelmingly large number of parameters. To address this, Panacea is designed to use singular value decomposition (SVD)-based low-rank adaptation, which allows the preference vector to be simply injected online as singular values. Theoretically, we prove that Panacea recovers the entire Pareto front with common loss aggregation methods under mild conditions. Moreover, our experiments demonstrate, for the first time, the feasibility of aligning a single LLM to represent a spectrum of human preferences through various optimization methods. Our work marks a step forward in effectively and efficiently aligning models to diverse and intricate human preferences in a controllable and Pareto-optimal manner.

CivRealm: A Learning and Reasoning Odyssey in Civilization for Decision-Making Agents

The generalization of decision-making agents encompasses two fundamental elements: learning from past experiences and reasoning in novel contexts. However, the predominant emphasis in most interactive environments is on learning, often at the expense of complexity in reasoning. In this paper, we introduce CivRealm, an environment inspired by the Civilization game. Civilization's profound alignment with human history and society necessitates sophisticated learning, while its ever-changing situations demand strong reasoning to generalize. Particularly, CivRealm sets up an imperfect-information general-sum game with a changing number of players; it presents a plethora of complex features, challenging the agent to deal with open-ended stochastic environments that require diplomacy and negotiation skills. Within CivRealm, we provide interfaces for two typical agent types: tensor-based agents that focus on learning, and language-based agents that emphasize reasoning. To catalyze further research, we present initial results for both paradigms. The canonical RL-based agents exhibit reasonable performance in mini-games, whereas both RL- and LLM-based agents struggle to make substantial progress in the full game. Overall, CivRealm stands as a unique learning and reasoning challenge for decision-making agents. The code is available at https://github.com/bigai-ai/civrealm.

Evolutionary Pareto Set Learning with Structure Constraints

The multiobjective evolutionary optimization algorithm (MOEA) is a powerful approach for tackling multiobjective optimization problems (MOPs), which can find a finite set of approximate Pareto solutions in a single run. However, under mild regularity conditions, the Pareto optimal set of a continuous MOP could be a low dimensional continuous manifold that contains infinite solutions. In addition, structure constraints on the whole optimal solution set, which characterize the patterns shared among all solutions, could be required in many real-life applications. It is very challenging for existing finite population based MOEAs to handle these structure constraints properly. In this work, we propose the first model-based algorithmic framework to learn the whole solution set with structure constraints for multiobjective optimization. In our approach, the Pareto optimality can be traded off with a preferred structure among the whole solution set, which could be crucial for many real-world problems. We also develop an efficient evolutionary learning method to train the set model with structure constraints. Experimental studies on benchmark test suites and real-world application problems demonstrate the promising performance of our proposed framework.