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.

翻译：多目标优化（MOO）在多任务学习等诸多领域正受到越来越多的关注。近期研究提出了一些具备理论分析的有效算法，但这些算法受限于标准的$L$-光滑或有界梯度假设，这些假设对于神经网络（如循环神经网络（RNNs）和Transformer）通常难以满足。本文研究一类更广义且符合实际的$\ell$-光滑损失函数，其中$\ell$是梯度范数的一般非递减函数。针对$\ell$-光滑MOO问题，我们提出了两种新颖的单循环算法：广义光滑多目标梯度下降法（GSMGrad）及其随机变体——随机广义光滑多目标梯度下降法（SGSMGrad）。这两种算法能够逼近冲突规避（CA）方向，该方向可使各目标函数的最小改进量最大化。我们对两种算法进行了全面的收敛性分析，证明它们能以可保证的$\epsilon$级别平均CA距离（即更新方向与CA方向之间的差距）收敛至$\epsilon$精度的帕累托稳定点，其中确定性和随机性场景分别需要总计$\mathcal{O}(\epsilon^{-2})$和$\mathcal{O}(\epsilon^{-4})$的样本量。通过使用更多样本，我们的算法还能在每次迭代中保证更严格的$\epsilon$级别CA距离。此外，我们提出了一种仅需常数级时间和空间开销的GSMGrad实用变体GSMGrad-FA，同时保持与GSMGrad相同的性能保证。实验验证了我们的理论，并证明了所提方法的有效性。