Direction Matters: On the Implicit Bias of Stochastic Gradient Descent with Moderate Learning Rate

• URL: http://arxiv.org/abs/2011.02538v2
• Date: Mon, 29 Mar 2021 17:24:24 GMT
• Title: Direction Matters: On the Implicit Bias of Stochastic Gradient Descent with Moderate Learning Rate
• Authors: Jingfeng Wu, Difan Zou, Vladimir Braverman, Quanquan Gu
• Abstract summary: This paper attempts to characterize the particular regularization effect of SGD in the moderate learning rate regime. We show that SGD converges along the large eigenvalue directions of the data matrix, while GD goes after the small eigenvalue directions.
• Score: 105.62979485062756
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Understanding the algorithmic bias of \emph{stochastic gradient descent} (SGD) is one of the key challenges in modern machine learning and deep learning theory. Most of the existing works, however, focus on \emph{very small or even infinitesimal} learning rate regime, and fail to cover practical scenarios where the learning rate is \emph{moderate and annealing}. In this paper, we make an initial attempt to characterize the particular regularization effect of SGD in the moderate learning rate regime by studying its behavior for optimizing an overparameterized linear regression problem. In this case, SGD and GD are known to converge to the unique minimum-norm solution; however, with the moderate and annealing learning rate, we show that they exhibit different \emph{directional bias}: SGD converges along the large eigenvalue directions of the data matrix, while GD goes after the small eigenvalue directions. Furthermore, we show that such directional bias does matter when early stopping is adopted, where the SGD output is nearly optimal but the GD output is suboptimal. Finally, our theory explains several folk arts in practice used for SGD hyperparameter tuning, such as (1) linearly scaling the initial learning rate with batch size; and (2) overrunning SGD with high learning rate even when the loss stops decreasing.

Related papers

• Risk Bounds of Accelerated SGD for Overparameterized Linear Regression [75.27846230182885]
  Accelerated gradient descent (ASGD) is a workhorse in deep learning. Existing optimization theory can only explain the faster convergence of ASGD, but cannot explain its better generalization.
  arXiv  Detail & Related papers  (2023-11-23T23:02:10Z)
• Implicit Regularization or Implicit Conditioning? Exact Risk Trajectories of SGD in High Dimensions [26.782342518986503]
  gradient descent (SGD) is a pillar of modern machine learning, serving as the go-to optimization algorithm for a diverse array of problems. We show how to adapt the HSGD formalism to include streaming SGD, which allows us to produce an exact prediction for the excess risk of multi-pass SGD relative to that of streaming SGD.
  arXiv  Detail & Related papers  (2022-06-15T02:32:26Z)
• On the Double Descent of Random Features Models Trained with SGD [78.0918823643911]
  We study properties of random features (RF) regression in high dimensions optimized by gradient descent (SGD) We derive precise non-asymptotic error bounds of RF regression under both constant and adaptive step-size SGD setting. We observe the double descent phenomenon both theoretically and empirically.
  arXiv  Detail & Related papers  (2021-10-13T17:47:39Z)
• SGD: The Role of Implicit Regularization, Batch-size and Multiple-epochs [30.41773138781369]
  We present a multi-epoch variant of Gradient Descent (SGD) commonly used in practice. We prove that this is at least as good as single pass SGD in the worst case. For certain SCO problems, taking multiple passes over the dataset can significantly outperform single pass SGD.
  arXiv  Detail & Related papers  (2021-07-11T15:50:01Z)
• Understanding Long Range Memory Effects in Deep Neural Networks [10.616643031188248]
  textitstochastic gradient descent (SGD) is of fundamental importance in deep learning. In this study, we argue that SGN is neither Gaussian nor stable. Instead, we propose that SGD can be viewed as a discretization of an SDE driven by textitfractional Brownian motion (FBM)
  arXiv  Detail & Related papers  (2021-05-05T13:54:26Z)
• Benign Overfitting of Constant-Stepsize SGD for Linear Regression [122.70478935214128]
  inductive biases are central in preventing overfitting empirically. This work considers this issue in arguably the most basic setting: constant-stepsize SGD for linear regression. We reflect on a number of notable differences between the algorithmic regularization afforded by (unregularized) SGD in comparison to ordinary least squares.
  arXiv  Detail & Related papers  (2021-03-23T17:15:53Z)
• Least Squares Regression with Markovian Data: Fundamental Limits and Algorithms [69.45237691598774]
  We study the problem of least squares linear regression where the data-points are dependent and are sampled from a Markov chain. We establish sharp information theoretic minimax lower bounds for this problem in terms of $tau_mathsfmix$. We propose an algorithm based on experience replay--a popular reinforcement learning technique--that achieves a significantly better error rate.
  arXiv  Detail & Related papers  (2020-06-16T04:26:50Z)
• On Learning Rates and Schr\"odinger Operators [105.32118775014015]
  We present a general theoretical analysis of the effect of the learning rate. We find that the learning rate tends to zero for a broad non- neural class functions.
  arXiv  Detail & Related papers  (2020-04-15T09:52:37Z)
• Online stochastic gradient descent on non-convex losses from high-dimensional inference [2.2344764434954256]
  gradient descent (SGD) is a popular algorithm for optimization problems in high-dimensional tasks. In this paper we produce an estimator of non-trivial correlation from data. We illustrate our approach by applying it to a set of tasks such as phase retrieval, and estimation for generalized models.
  arXiv  Detail & Related papers  (2020-03-23T17:34:06Z)

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.