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Polygonal Unadjusted Langevin Algorithms: Creating stable and efficient adaptive algorithms for neural networks

Dong-Young Lim, Sotirios Sabanis

TL;DR

This work tackles the instability of common adaptive optimizers when fine-tuning deep neural networks with nonconvex losses, where gradients can exhibit explosive or vanishing behavior. It introduces TH\\$\\varepsilon\\$O POULA (TheoPouLa), a polygonal unadjusted Langevin algorithm that combines element-wise taming and a boosting mechanism within Euler-Krylov polygonal drift approximations, along with a Langevin noise term to ensure exploration. The authors prove non-asymptotic convergence in Wasserstein distances and derive an excess risk bound, while demonstrating superior empirical performance on CIFAR-10/100 and Penn Treebank compared to Adam-type and SGD-based baselines. The results indicate TheoPouLa offers stable, efficient optimization with strong generalization in real-world neural network training, bridging the gap between theoretical guarantees and practical deep learning efficacy.

Abstract

We present a new class of Langevin based algorithms, which overcomes many of the known shortcomings of popular adaptive optimizers that are currently used for the fine tuning of deep learning models. Its underpinning theory relies on recent advances of Euler's polygonal approximations for stochastic differential equations (SDEs) with monotone coefficients. As a result, it inherits the stability properties of tamed algorithms, while it addresses other known issues, e.g. vanishing gradients in neural networks. In particular, we provide a nonasymptotic analysis and full theoretical guarantees for the convergence properties of an algorithm of this novel class, which we named TH$\varepsilon$O POULA (or, simply, TheoPouLa). Finally, several experiments are presented with different types of deep learning models, which show the superior performance of TheoPouLa over many popular adaptive optimization algorithms.

Polygonal Unadjusted Langevin Algorithms: Creating stable and efficient adaptive algorithms for neural networks

TL;DR

This work tackles the instability of common adaptive optimizers when fine-tuning deep neural networks with nonconvex losses, where gradients can exhibit explosive or vanishing behavior. It introduces TH\\O POULA (TheoPouLa), a polygonal unadjusted Langevin algorithm that combines element-wise taming and a boosting mechanism within Euler-Krylov polygonal drift approximations, along with a Langevin noise term to ensure exploration. The authors prove non-asymptotic convergence in Wasserstein distances and derive an excess risk bound, while demonstrating superior empirical performance on CIFAR-10/100 and Penn Treebank compared to Adam-type and SGD-based baselines. The results indicate TheoPouLa offers stable, efficient optimization with strong generalization in real-world neural network training, bridging the gap between theoretical guarantees and practical deep learning efficacy.

Abstract

We present a new class of Langevin based algorithms, which overcomes many of the known shortcomings of popular adaptive optimizers that are currently used for the fine tuning of deep learning models. Its underpinning theory relies on recent advances of Euler's polygonal approximations for stochastic differential equations (SDEs) with monotone coefficients. As a result, it inherits the stability properties of tamed algorithms, while it addresses other known issues, e.g. vanishing gradients in neural networks. In particular, we provide a nonasymptotic analysis and full theoretical guarantees for the convergence properties of an algorithm of this novel class, which we named THO POULA (or, simply, TheoPouLa). Finally, several experiments are presented with different types of deep learning models, which show the superior performance of TheoPouLa over many popular adaptive optimization algorithms.

Paper Structure

This paper contains 25 sections, 14 theorems, 157 equations, 4 figures, 10 tables.

Table of Contents

  1. Introduction
  2. Related work: Langevin-based algorithms and adaptive learning rate methods
  3. Our contributions
  4. New Algorithm: TH$\varepsilon$O POULA
  5. Preliminaries and Assumptions
  6. Mechanism of TH$\varepsilon$O POULA
  7. Convergence Analysis
  8. Comparison with Other Langevin-based algorithms
  9. Numerical Experiments
  10. Toy example
  11. Empirical performance on real data sets
  12. Image classification.
  13. Language modeling.
  14. Effect of $\lambda$, $\varepsilon$, and $\beta$ on the performance of TheoPouLa
  15. Additional Experiments
  16. ...and 10 more sections

Key Result

Proposition 2.7

Let Assumption ass:init_ass and ass:stoc_grad hold. Then, one obtains, for all $\theta$, $\theta' \in \mathbb{R}^d$, where $L_R= L_G\mathbb{E}[(1+|X_0|)^\rho](1+2|R|)^{q-1}>0$ and $R$ is given by

Figures (4)

  • Figure 1: A landscape of a loss function.
  • Figure 2: Performance of SGD, Adam, AMSGrad and TheoPouLa on an artificial example with the initial value $\theta_0 = 5.0$
  • Figure 3: Test accuracy for VGG11, ResNet34 and DenseNet121 on CIFAR-10 and CIFAR-100. TheoPouLa$^\dagger$ and TH$\varepsilon$OPOULA$^*$ represent the performances of TheoPouLa under the best and second best hyperparameters, respectively.
  • Figure 4: Test perplexity for 1, 2 and 3-layer AWD-LSTMs on PTB

Theorems & Definitions (24)

  • Remark 2.1
  • Remark 2.4
  • Remark 2.5
  • Remark 2.6
  • Proposition 2.7
  • Proposition 2.8
  • Remark 2.9
  • Remark 2.10
  • Theorem 2.11
  • Corollary 2.12
  • ...and 14 more