Multi-objective optimization (MOO) is receiving more attention in various
fields such as multi-task learning. Recent works provide some effective
algorithms with theoretical analysis but they are limited by the standard
$L$-smooth or bounded-gradient assumptions, which are typically unsatisfactory
for neural networks, such as recurrent neural networks (RNNs) and transformers.
In this paper, we study a more general and realistic class of $\ell$-smooth
loss functions, where $\ell$ is a general non-decreasing function of gradient
norm. We develop two novel single-loop algorithms for $\ell$-smooth MOO
problems, Generalized Smooth Multi-objective Gradient descent (GSMGrad) and its
stochastic variant, Stochastic Generalized Smooth Multi-objective Gradient
descent (SGSMGrad), which approximate the conflict-avoidant (CA) direction that
maximizes the minimum improvement among objectives. We provide a comprehensive
convergence analysis of both algorithms and show that they converge to an
$\epsilon$-accurate Pareto stationary point with a guaranteed $\epsilon$-level
average CA distance (i.e., the gap between the updating direction and the CA
direction) over all iterations, where totally $\mathcal{O}(\epsilon^{-2})$ and
$\mathcal{O}(\epsilon^{-4})$ samples are needed for deterministic and
stochastic settings, respectively. Our algorithms can also guarantee a tighter
$\epsilon$-level CA distance in each iteration using more samples. Moreover, we
propose a practical variant of GSMGrad named GSMGrad-FA using only
constant-level time and space, while achieving the same performance guarantee
as GSMGrad. Our experiments validate our theory and demonstrate the
effectiveness of the proposed methods.

本文研究了多目标优化问题中更一般且更现实的平滑损失函数类别，在神经网络中，提出了两种新颖的单循环算法 GSMGrad 和 SGSMGrad，以逼近在所有目标之间最大化最小改进的冲突避免方向。

关于广义光滑下多目标优化的收敛性研究

On the Convergence of Multi-objective Optimization under Generalized  Smoothness

Multi-objective optimization (MOO) has become an influential framework in
many machine learning problems with multiple objectives such as learning with
multiple criteria and multi-task learning (MTL). In this paper, we propose a
new direction-oriented multi-objective problem by regularizing the common
descent direction within a neighborhood of a direction that optimizes a linear
combination of objectives such as the average loss in MTL. This formulation
includes GD and MGDA as special cases, enjoys the direction-oriented benefit as
in CAGrad, and facilitates the design of stochastic algorithms. To solve this
problem, we propose Stochastic Direction-oriented Multi-objective Gradient
descent (SDMGrad) with simple SGD type of updates, and its variant SDMGrad-OS
with an efficient objective sampling in the setting where the number of
objectives is large. For a constant-level regularization parameter $\lambda$,
we show that SDMGrad and SDMGrad-OS provably converge to a Pareto stationary
point with improved complexities and milder assumptions. For an increasing
$\lambda$, this convergent point reduces to a stationary point of the linear
combination of objectives. We demonstrate the superior performance of the
proposed methods in a series of tasks on multi-task supervised learning and
reinforcement learning. Code is provided at
this https URL

本文提出了基于方向的多目标问题、Stochastic Direction-oriented Multi-objective Gradient Descent 和 SDMGrad-OS 算法，实现了多任务学习和强化学习中的多目标优化。