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(S)GD over Diagonal Linear Networks: Implicit Regularisation, Large Stepsizes and Edge of Stability

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arxiv 2302.08982 v2 pith:ABYR6WZO submitted 2023-02-17 cs.LG math.OCstat.ML

classification cs.LGmath.OCstat.ML
keywords stepsizesimplicitlargeregularisationdescentdiagonaledgegradient
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In this paper, we investigate the impact of stochasticity and large stepsizes on the implicit regularisation of gradient descent (GD) and stochastic gradient descent (SGD) over diagonal linear networks. We prove the convergence of GD and SGD with macroscopic stepsizes in an overparametrised regression setting and characterise their solutions through an implicit regularisation problem. Our crisp characterisation leads to qualitative insights about the impact of stochasticity and stepsizes on the recovered solution. Specifically, we show that large stepsizes consistently benefit SGD for sparse regression problems, while they can hinder the recovery of sparse solutions for GD. These effects are magnified for stepsizes in a tight window just below the divergence threshold, in the "edge of stability" regime. Our findings are supported by experimental results.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Criteria and Bias of Parameterized Linear Regression under Edge of Stability Regime

    math.OC 2024-12 conditional novelty 6.0 of 10

    Under specific conditions, gradient descent converges in the unstable edge-of-stability regime for a quadratic loss on a depth-2 diagonal linear network, with a bias bound depending on step size and initialization.

  2. Exponential Moving Average of Weights in Deep Learning: Dynamics and Benefits

    cs.LG 2024-11 conditional novelty 6.0 of 10

    An exponential moving average of weights consistently improves generalization, label-noise robustness, prediction consistency, calibration, and transfer learning for image classifiers.

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