# A Stochastic Bundle Method for Interpolating Networks

arXiv: https://arxiv.org/abs/2201.12678 · HTML: https://mlsanity.com/paper/2201.12678/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 46 of its 63 references that are in the map.
- First posted on arXiv 29 Jan 2022.

## Builds on (most cited first, 30 of 46)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [SGDR: Stochastic Gradient Descent with Warm Restarts](https://mlsanity.com/paper/1608.03983/) (2016)
- [Going Deeper with Convolutions](https://mlsanity.com/paper/1409.4842/) (2014)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Wide Residual Networks](https://mlsanity.com/paper/1605.07146/) (2016)
- [A large annotated corpus for learning natural language inference](https://mlsanity.com/paper/1508.05326/) (2015)
- [Decoupled Weight Decay Regularization](https://mlsanity.com/paper/1711.05101/) (2017)
- [Adafactor: Adaptive Learning Rates with Sublinear Memory Cost](https://mlsanity.com/paper/1804.04235/) (2018)
- [On the Convergence of Adam and Beyond](https://mlsanity.com/paper/1904.09237/) (2019)
- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [On the Variance of the Adaptive Learning Rate and Beyond](https://mlsanity.com/paper/1908.03265/) (2019)
- [Convex Optimization: Algorithms and Complexity](https://mlsanity.com/paper/1405.4980/) (2014)
- [signSGD: Compressed Optimisation for Non-Convex Problems](https://mlsanity.com/paper/1802.04434/) (2018)
- [Supervised Learning of Universal Sentence Representations from Natural Language Inference Data](https://mlsanity.com/paper/1705.02364/) (2017)
- [Characterizing Adversarial Subspaces Using Local Intrinsic Dimensionality](https://mlsanity.com/paper/1801.02613/) (2018)
- [The Marginal Value of Adaptive Gradient Methods in Machine Learning](https://mlsanity.com/paper/1705.08292/) (2017)
- [Adaptive Gradient Methods with Dynamic Bound of Learning Rate](https://mlsanity.com/paper/1902.09843/) (2019)
- [On the Convergence of A Class of Adam-Type Algorithms for Non-Convex Optimization](https://mlsanity.com/paper/1808.02941/) (2018)
- [Gradient Descent with Early Stopping is Provably Robust to Label Noise for Overparameterized Neural Networks](https://mlsanity.com/paper/1903.11680/) (2019)
- [On the Convergence of Stochastic Gradient Descent with Adaptive Stepsizes](https://mlsanity.com/paper/1805.08114/) (2018)
- [The Power of Interpolation: Understanding the Effectiveness of SGD in Modern Over-parametrized Learning](https://mlsanity.com/paper/1712.06559/) (2017)
- [Fast and Faster Convergence of SGD for Over-Parameterized Models and an Accelerated Perceptron](https://mlsanity.com/paper/1810.07288/) (2018)
- [Stochastic Polyak Step-size for SGD: An Adaptive Learning Rate for Fast Convergence](https://mlsanity.com/paper/2002.10542/) (2020)
- [Painless Stochastic Gradient: Interpolation, Line-Search, and Convergence Rates](https://mlsanity.com/paper/1905.09997/) (2019)
- [Online Learning Rate Adaptation with Hypergradient Descent](https://mlsanity.com/paper/1703.04782/) (2017)
- [Stochastic (Approximate) Proximal Point Methods: Convergence, Optimality, and Adaptivity](https://mlsanity.com/paper/1810.05633/) (2018)
- [Variants of RMSProp and Adagrad with Logarithmic Regret Bounds](https://mlsanity.com/paper/1706.05507/) (2017)
- [Training Neural Networks for and by Interpolation](https://mlsanity.com/paper/1906.05661/) (2019)

## Built on by (most cited first, 3 of 3)

- [Function Value Learning: Adaptive Learning Rates Based on the Polyak Stepsize and Function Splitting in ERM](https://mlsanity.com/paper/2307.14528/) (2023)
- [Nonconvex Stochastic Bregman Proximal Gradient Method with Application to Deep Learning](https://mlsanity.com/paper/2306.14522/) (2023)
- [Bundle Network: a Machine Learning-Based Bundle Method](https://mlsanity.com/paper/2509.24736/) (2025)

## Abstract (arXiv)

We propose a novel method for training deep neural networks that are capable of interpolation, that is, driving the empirical loss to zero. At each iteration, our method constructs a stochastic approximation of the learning objective. The approximation, known as a bundle, is a pointwise maximum of linear functions. Our bundle contains a constant function that lower bounds the empirical loss. This enables us to compute an automatic adaptive learning rate, thereby providing an accurate solution. In addition, our bundle includes linear approximations computed at the current iterate and other linear estimates of the DNN parameters. The use of these additional approximations makes our method significantly more robust to its hyperparameters. Based on its desirable empirical properties, we term our method Bundle Optimisation for Robust and Accurate Training (BORAT). In order to operationalise BORAT, we design a novel algorithm for optimising the bundle approximation efficiently at each iteration. We establish the theoretical convergence of BORAT in both convex and non-convex settings. Using standard publicly available data sets, we provide a thorough comparison of BORAT to other single hyperparameter optimisation algorithms. Our experiments demonstrate BORAT matches the state-of-the-art generalisation performance for these methods and is the most robust.