MGDA Converges under Generalized Smoothness, Provably

• URL: http://arxiv.org/abs/2405.19440v4
• Date: Wed, 02 Oct 2024 18:51:01 GMT
• Title: MGDA Converges under Generalized Smoothness, Provably
• Authors: Qi Zhang, Peiyao Xiao, Shaofeng Zou, Kaiyi Ji,
• Abstract summary: 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. We study a more general and realistic class of generalized $ell$-smooth loss functions, where $ell$ is a general non-decreasing function of gradient norm.
• Score: 27.87166415148172
• License:
• Abstract: 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 typically do not hold for neural networks, such as Long short-term memory (LSTM) models and Transformers. In this paper, we study a more general and realistic class of generalized $\ell$-smooth loss functions, where $\ell$ is a general non-decreasing function of gradient norm. We revisit and analyze the fundamental multiple gradient descent algorithm (MGDA) and its stochastic version with double sampling for solving the generalized $\ell$-smooth MOO problems, which approximate the conflict-avoidant (CA) direction that maximizes the minimum improvement among objectives. We provide a comprehensive convergence analysis of these 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. We prove that they can also guarantee a tighter $\epsilon$-level CA distance in each iteration using more samples. Moreover, we analyze an efficient variant of MGDA named MGDA-FA using only $\mathcal{O}(1)$ time and space, while achieving the same performance guarantee as MGDA.

Related papers

• An Oblivious Stochastic Composite Optimization Algorithm for Eigenvalue Optimization Problems [76.2042837251496]
  We introduce two oblivious mirror descent algorithms based on a complementary composite setting. Remarkably, both algorithms work without prior knowledge of the Lipschitz constant or smoothness of the objective function. We show how to extend our framework to scale and demonstrate the efficiency and robustness of our methods on large scale semidefinite programs.
  arXiv  Detail & Related papers  (2023-06-30T08:34:29Z)
• On Convergence of Incremental Gradient for Non-Convex Smooth Functions [63.51187646914962]
  In machine learning and network optimization, algorithms like shuffle SGD are popular due to minimizing the number of misses and good cache. This paper delves into the convergence properties SGD algorithms with arbitrary data ordering.
  arXiv  Detail & Related papers  (2023-05-30T17:47:27Z)
• Multi-block-Single-probe Variance Reduced Estimator for Coupled Compositional Optimization [49.58290066287418]
  We propose a novel method named Multi-block-probe Variance Reduced (MSVR) to alleviate the complexity of compositional problems. Our results improve upon prior ones in several aspects, including the order of sample complexities and dependence on strongity.
  arXiv  Detail & Related papers  (2022-07-18T12:03:26Z)
• Sharper Convergence Guarantees for Asynchronous SGD for Distributed and Federated Learning [77.22019100456595]
  We show a training algorithm for distributed computation workers with varying communication frequency. In this work, we obtain a tighter convergence rate of $mathcalO!!!(sigma2-2_avg!! . We also show that the heterogeneity term in rate is affected by the average delay within each worker.
  arXiv  Detail & Related papers  (2022-06-16T17:10:57Z)
• An Improved Analysis of Gradient Tracking for Decentralized Machine Learning [34.144764431505486]
  We consider decentralized machine learning over a network where the training data is distributed across $n$ agents. The agent's common goal is to find a model that minimizes the average of all local loss functions. We improve the dependency on $p$ from $mathcalO(p-1)$ to $mathcalO(p-1)$ in the noiseless case.
  arXiv  Detail & Related papers  (2022-02-08T12:58:14Z)
• DoCoM: Compressed Decentralized Optimization with Near-Optimal Sample Complexity [25.775517797956237]
  This paper proposes the Doubly Compressed Momentum-assisted tracking algorithm $ttDoCoM$ for communication. We show that our algorithm outperforms several state-of-the-art algorithms in practice.
  arXiv  Detail & Related papers  (2022-02-01T07:27:34Z)
• On the Last Iterate Convergence of Momentum Methods [32.60494006038902]
  We prove for the first time that for any constant momentum factor, there exists a Lipschitz and convex function for which the last iterate suffers from an error. We show that in the setting with convex and smooth functions, our new SGDM algorithm automatically converges at a rate of $O(fraclog TT)$.
  arXiv  Detail & Related papers  (2021-02-13T21:16:16Z)
• Gradient-Based Empirical Risk Minimization using Local Polynomial Regression [39.29885444997579]
  A major goal of this literature has been to compare different algorithms, such as gradient descent (GD) or gradient descent (SGD) We demonstrate that when the loss function is smooth in the data, we can learn the oracle at every iteration and beat the oracle complexities of both GD and SGD.
  arXiv  Detail & Related papers  (2020-11-04T20:10:31Z)
• Optimal Robust Linear Regression in Nearly Linear Time [97.11565882347772]
  We study the problem of high-dimensional robust linear regression where a learner is given access to $n$ samples from the generative model $Y = langle X,w* rangle + epsilon$ We propose estimators for this problem under two settings: (i) $X$ is L4-L2 hypercontractive, $mathbbE [XXtop]$ has bounded condition number and $epsilon$ has bounded variance and (ii) $X$ is sub-Gaussian with identity second moment and $epsilon$ is
  arXiv  Detail & Related papers  (2020-07-16T06:44:44Z)
• Stochastic Proximal Gradient Algorithm with Minibatches. Application to Large Scale Learning Models [2.384873896423002]
  We develop and analyze minibatch variants of gradient algorithm for general composite objective functions with nonsmooth components. We provide complexity for constant and variable stepsize iteration policies obtaining that, for minibatch size $N$, after $mathcalO(frac1Nepsilon)$ $epsilon-$subity is attained in expected quadratic distance to optimal solution.
  arXiv  Detail & Related papers  (2020-03-30T10:43:56Z)
• Non-asymptotic Convergence of Adam-type Reinforcement Learning Algorithms under Markovian Sampling [56.394284787780364]
  This paper provides the first theoretical convergence analysis for two fundamental RL algorithms of policy gradient (PG) and temporal difference (TD) learning. Under general nonlinear function approximation, PG-AMSGrad with a constant stepsize converges to a neighborhood of a stationary point at the rate of $mathcalO(log T/sqrtT)$. Under linear function approximation, TD-AMSGrad with a constant stepsize converges to a neighborhood of the global optimum at the rate of $mathcalO(log T/sqrtT
  arXiv  Detail & Related papers  (2020-02-15T00:26:49Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.