Implicit Regularization Properties of Variance Reduced Stochastic Mirror Descent
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Abstract (arXiv)
In machine learning and statistical data analysis, we often run into objective function that is a summation: the number of terms in the summation possibly is equal to the sample size, which can be enormous. In such a setting, the stochastic mirror descent (SMD) algorithm is a numerically efficient method -- each iteration involving a very small subset of the data. The variance reduction version of SMD (VRSMD) can further improve SMD by inducing faster convergence. On the other hand, algorithms such as gradient descent and stochastic gradient descent have the implicit regularization property that leads to better performance in terms of the generalization errors. Little is known on whether such a property holds for VRSMD. We prove here that the discrete VRSMD estimator sequence converges to the minimum mirror interpolant in the linear regression. This establishes the implicit regularization property for VRSMD. As an application of the above result, we derive a model estimation accuracy result in the setting when the true model is sparse. We use numerical examples to illustrate the empirical power of VRSMD.
Key facts. Cited by 1 paper in the map (data through 14 Oct 2026). Builds on 16 of its 22 references that are in the map. First posted on arXiv 29 Apr 2022.
Builds on
16 of 22 references.
References · 16
- 2016Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Conditioncited by 539
- 2019Implicit Regularization in Deep Matrix Factorizationcited by 405
- 2018SPIDER: Near-Optimal Non-Convex Optimization via Stochastic Path Integrated Differential Estimatorcited by 287
- 2021Deep learning: a statistical viewpointcited by 240
- 2017SARAH: A Novel Method for Machine Learning Problems Using Stochastic Recursive Gradientcited by 220
- 2021On the Origin of Implicit Regularization in Stochastic Gradient Descentcited by 184
- 2016Stochastic Variance Reduction for Nonconvex Optimizationcited by 145
- 2016Katyusha: The First Direct Acceleration of Stochastic Gradient Methodscited by 135
- 2019Implicit regularization for deep neural networks driven by an Ornstein-Uhlenbeck like processcited by 123
- 2019Implicit Regularization for Optimal Sparse Recoverycited by 80
- 2021On the Implicit Bias of Initialization Shape: Beyond Infinitesimal Mirror Descentcited by 70
- 2020The Implicit Regularization of Stochastic Gradient Flow for Least Squarescited by 59
4 more
- 2019Exact expressions for double descent and implicit regularization via surrogate random designcited by 35
- 2018Stochastic Gradient/Mirror Descent: Minimax Optimality and Implicit Regularizationcited by 29
- 2020The Statistical Complexity of Early-Stopped Mirror Descentcited by 19
- 2020Direction Matters: On the Implicit Bias of Stochastic Gradient Descent with Moderate Learning Ratecited by 12
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Citing papers per year: 2023: 1.
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