Finding specific preference-guided Pareto solutions that represent different trade-offs among multiple objectives is critical yet challenging in multi-objective problems. Existing methods are restrictive in preference definitions and/or their theoretical guarantees.In this work, we introduce a Flexible framEwork for pREfeRence-guided multi-Objective learning (FERERO) by casting it as a constrained vector optimization problem.Specifically, two types of preferences are incorporated into this formulation -- the relative preference defined by the partial ordering induced by a polyhedral cone, and the absolute preference defined by constraints that are linear functions of the objectives. To solve this problem, convergent algorithms are developed with both single-loop and stochastic variants. Notably, this is the first single-loop primal algorithm for constrained optimization to our knowledge. The proposed algorithms adaptively adjust to both constraint and objective values, eliminating the need to solve different subproblems at different stages of constraint satisfaction.

This paper examines the issue of fairness in the estimation of graphical models (GMs), particularly Gaussian, Covariance, and Ising models. These models play a vital role in understanding complex relationships in high-dimensional data. However, standard GMs can result in biased outcomes, especially when the underlying data involves sensitive characteristics or protected groups. To address this, we introduce a comprehensive framework designed to reduce bias in the estimation of GMs related to protected attributes. Our approach involves the integration of the pairwise graph disparity error and a tailored loss function into a nonsmooth multi-objective optimization problem, striving to achieve fairness across different sensitive groups while maintaining the effectiveness of the GMs.

The Sharpe ratio is an important and widely-used risk-adjusted return in financial engineering. In modern portfolio management, one may require an m-sparse (no more than m active assets) portfolio to save managerial and financial costs. We propose to convert the m-sparse fractional optimization problem into an equivalent m-sparse quadratic programming problem. The semi-algebraic property of the resulting objective function allows us to exploit the Kurdyka-Lojasiewicz property to develop an efficient Proximal Gradient Algorithm (PGA) that leads to a portfolio which achieves the globally optimal m-sparse Sharpe ratio under certain conditions. The convergence rates of PGA are also provided.

Collaborative learning is an important tool to train multiple clients more effectively by enabling communication among clients. Identifying helpful clients, however, presents challenging and often introduces significant overhead. In this paper, we model client-selection and model-training as two interconnected optimization problems, proposing a novel bilevel optimization problem for collaborative learning.We introduce CoBo, a scalable and elastic, SGD-type alternating optimization algorithm that efficiently addresses these problem with theoretical convergence guarantees. Empirically, CoBo achieves superior performance, surpassing popular personalization algorithms by 9.3% in accuracy on a task with high heterogeneity, involving datasets distributed among 80 clients.

When applying reinforcement learning from human feedback (RLHF), the reward is learned from data and, therefore, always has some error. It is common to mitigate this by regularizing the policy with KL divergence from a base model, with the hope that balancing reward with regularization will achieve desirable outcomes despite this reward misspecification. We show that when the reward function has light-tailed error, optimal policies under less restrictive KL penalties achieve arbitrarily high utility. However, if error is heavy-tailed, some policies obtain arbitrarily high reward despite achieving no more utility than the base model--a phenomenon we call catastrophic Goodhart. We adapt a discrete optimization method to measure the tails of reward models, finding that they are consistent with light-tailed error. However, the pervasiveness of heavy-tailed distributions in many real-world applications indicates that future sources of RL reward could have heavy-tailed error, increasing the likelihood of reward hacking even with KL regularization.

The nadir objective vector plays a key role in solving multi-objective optimization problems (MOPs), where it is often used to normalize the objective space and guide the search. The current methods for estimating the nadir objective vector perform effectively only on specific MOPs. This paper reveals the limitations of these methods: exact methods can only work on discrete MOPs, while heuristic methods cannot deal with the MOP with a complicated feasible objective region. To fill this gap, we propose a general and rigorous method, namely boundary decomposition for nadir objective vector estimation (BDNE). By utilizing bilevel optimization, boundary subproblems are optimized and adjusted alternately, thereby refining their optimal solutions to align with the nadir objective vector.

Zero-shot optimization involves optimizing a target task that was not seen during training, aiming to provide the optimal solution without or with minimal adjustments to the optimizer. It is crucial to ensure reliable and robust performance in various applications. Current optimizers often struggle with zero-shot optimization and require intricate hyperparameter tuning to adapt to new tasks. To address this, we propose a Pretrained Optimization Model (POM) that leverages knowledge gained from optimizing diverse tasks, offering efficient solutions to zero-shot optimization through direct application or fine-tuning with few-shot samples. Evaluation on the BBOB benchmark and two robot control tasks demonstrates that POM outperforms state-of-the-art black-box optimization methods, especially for high-dimensional tasks.

Hierarchical clustering has usually been addressed by discrete optimization using heuristics or continuous optimization of relaxed scores for hierarchies. In this work, we propose to optimize expected scores under a probabilistic model over hierarchies. EPH uses differentiable hierarchy sampling enabling end-to-end gradient descent based optimization, and an unbiased subgraph sampling approach to scale to large datasets. EPH outperforms all other approaches quantitatively and provides meaningful hierarchies in qualitative evaluations.

Reconstructing a continuous surface from a raw 3D point cloud is a challenging task. Latest methods employ supervised learning or pretrained priors to learn a signed distance function (SDF). However, neural networks tend to smooth local details due to the lack of ground truth signed distnaces or normals, which limits the performance of learning-based methods in reconstruction tasks. To resolve this issue, we propose a novel method, named MultiPull, to learn multi-scale implicit fields from raw point clouds to optimize accurate SDFs from coarse to fine. We achieve this by mapping 3D query points into a set of frequency features, which makes it possible to leverage multi-level features during optimization. Meanwhile, we introduce optimization constraints from the perspective of spatial distance and normal consistency, which play a key role in point cloud reconstruction based on multi-scale optimization strategies.

How to solve high-dimensional linear programs (LPs) efficiently is a fundamental question.Recently, there has been a surge of interest in reducing LP sizes using *random projections*, which can accelerate solving LPs independently of improving LP solvers. This paper explores a new direction of *data-driven projections*, which use projection matrices learned from data instead of random projection matrices.Given training data of n -dimensional LPs, we learn an n\times k projection matrix with n k . When addressing a future LP instance, we reduce its dimensionality from n to k via the learned projection matrix, solve the resulting LP to obtain a k -dimensional solution, and apply the learned matrix to it to recover an n -dimensional solution.On the theoretical side, a natural question is: how much data is sufficient to ensure the quality of recovered solutions? We address this question based on the framework of *data-driven algorithm design*, which connects the amount of data sufficient for establishing generalization bounds to the *pseudo-dimension* of performance metrics. We also provide an \Omega(nk) lower bound, implying our result is tight up to an \tilde{\mathrm{O}}(k) factor.