# Analyzing Monotonic Linear Interpolation in Neural Network Loss Landscapes

arXiv: https://arxiv.org/abs/2104.11044 · HTML: https://mlsanity.com/paper/2104.11044/

## Key facts

- Cited by 28 papers in the map (data through 14 Oct 2026).
- Builds on 47 of its 62 references that are in the map.
- First posted on arXiv 22 Apr 2021.

## Builds on (most cited first, 30 of 47)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [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)
- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [RoBERTa: A Robustly Optimized BERT Pretraining Approach](https://mlsanity.com/paper/1907.11692/) (2019)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [Densely Connected Convolutional Networks](https://mlsanity.com/paper/1608.06993/) (2016)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Unsupervised Cross-lingual Representation Learning at Scale](https://mlsanity.com/paper/1911.02116/) (2019)
- [Neural Tangent Kernel: Convergence and Generalization in Neural Networks](https://mlsanity.com/paper/1806.07572/) (2018)
- [The Lottery Ticket Hypothesis: Finding Sparse, Trainable Neural Networks](https://mlsanity.com/paper/1803.03635/) (2018)
- [Accurate, Large Minibatch SGD: Training ImageNet in 1 Hour](https://mlsanity.com/paper/1706.02677/) (2017)
- [Optimizing Neural Networks with Kronecker-factored Approximate Curvature](https://mlsanity.com/paper/1503.05671/) (2015)
- [Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent](https://mlsanity.com/paper/1902.06720/) (2019)
- [On Lazy Training in Differentiable Programming](https://mlsanity.com/paper/1812.07956/) (2018)
- [Linear Mode Connectivity and the Lottery Ticket Hypothesis](https://mlsanity.com/paper/1912.05671/) (2019)
- [Loss Surfaces, Mode Connectivity, and Fast Ensembling of DNNs](https://mlsanity.com/paper/1802.10026/) (2018)
- [Sharp Minima Can Generalize For Deep Nets](https://mlsanity.com/paper/1703.04933/) (2017)
- [Measuring the Intrinsic Dimension of Objective Landscapes](https://mlsanity.com/paper/1804.08838/) (2018)
- [Essentially No Barriers in Neural Network Energy Landscape](https://mlsanity.com/paper/1803.00885/) (2018)
- [Gaussian Process Behaviour in Wide Deep Neural Networks](https://mlsanity.com/paper/1804.11271/) (2018)
- [Exponential expressivity in deep neural networks through transient chaos](https://mlsanity.com/paper/1606.05340/) (2016)
- [Gradient Descent Happens in a Tiny Subspace](https://mlsanity.com/paper/1812.04754/) (2018)
- [The large learning rate phase of deep learning: the catapult mechanism](https://mlsanity.com/paper/2003.02218/) (2020)
- [Fixup Initialization: Residual Learning Without Normalization](https://mlsanity.com/paper/1901.09321/) (2019)
- [Qualitatively characterizing neural network optimization problems](https://mlsanity.com/paper/1412.6544/) (2014)
- [Deep learning versus kernel learning: an empirical study of loss landscape geometry and the time evolution of the Neural Tangent Kernel](https://mlsanity.com/paper/2010.15110/) (2020)
- [A mathematical theory of semantic development in deep neural networks](https://mlsanity.com/paper/1810.10531/) (2018)
- [The Break-Even Point on Optimization Trajectories of Deep Neural Networks](https://mlsanity.com/paper/2002.09572/) (2020)
- [On the linearity of large non-linear models: when and why the tangent kernel is constant](https://mlsanity.com/paper/2010.01092/) (2020)

## Built on by (most cited first, 28 of 28)

- [Editing Models with Task Arithmetic](https://mlsanity.com/paper/2212.04089/) (2022)
- [Robust fine-tuning of zero-shot models](https://mlsanity.com/paper/2109.01903/) (2021)
- [Fusing finetuned models for better pretraining](https://mlsanity.com/paper/2204.03044/) (2022)
- [Model Zoos: A Dataset of Diverse Populations of Neural Network Models](https://mlsanity.com/paper/2209.14764/) (2022)
- [Merging by Matching Models in Task Parameter Subspaces](https://mlsanity.com/paper/2312.04339/) (2023)
- [What can linear interpolation of neural network loss landscapes tell us?](https://mlsanity.com/paper/2106.16004/) (2021)
- [Proving Linear Mode Connectivity of Neural Networks via Optimal Transport](https://mlsanity.com/paper/2310.19103/) (2023)
- [The Empirical Impact of Neural Parameter Symmetries, or Lack Thereof](https://mlsanity.com/paper/2405.20231/) (2024)
- [Phase diagram of early training dynamics in deep neural networks: effect of the learning rate, depth, and width](https://mlsanity.com/paper/2302.12250/) (2023)
- [A Second-Order Perspective on Model Compositionality and Incremental Learning](https://mlsanity.com/paper/2405.16350/) (2024)
- [Towards Noise-adaptive, Problem-adaptive (Accelerated) Stochastic Gradient Descent](https://mlsanity.com/paper/2110.11442/) (2021)
- [Special Properties of Gradient Descent with Large Learning Rates](https://mlsanity.com/paper/2205.15142/) (2022)
- [Plateau in Monotonic Linear Interpolation -- A "Biased" View of Loss Landscape for Deep Networks](https://mlsanity.com/paper/2210.01019/) (2022)
- [Edit at your own risk: evaluating the robustness of edited models to distribution shifts](https://mlsanity.com/paper/2303.00046/) (2023)
- [On Linear Mode Connectivity of Mixture-of-Experts Architectures](https://mlsanity.com/paper/2509.11348/) (2025)
- [Linear Interpolation In Parameter Space is Good Enough for Fine-Tuned Language Models](https://mlsanity.com/paper/2211.12092/) (2022)
- [No Wrong Turns: The Simple Geometry Of Neural Networks Optimization Paths](https://mlsanity.com/paper/2306.11922/) (2023)
- [EMA Policy Gradient: Taming Reinforcement Learning for LLMs with EMA Anchor and Top-k KL](https://mlsanity.com/paper/2602.04417/) (2026)
- [Functional Equivalence in Attention: A Comprehensive Study with Applications to Linear Mode Connectivity](https://mlsanity.com/paper/2606.17830/) (2026)
- [FuNNscope: Visual microscope for interactively exploring the loss landscape of fully connected neural networks](https://mlsanity.com/paper/2204.04511/) (2022)
- [Extra-Merge: Tracing the Rank-1 Subspace of Model Merging in Language Model Pre-Training](https://mlsanity.com/paper/2605.26484/) (2026)
- [Class Interference of Deep Neural Networks](https://mlsanity.com/paper/2211.01370/) (2022)
- [Visualizing, Rethinking, and Mining the Loss Landscape of Deep Neural Networks](https://mlsanity.com/paper/2405.12493/) (2024)
- [In Search of the Successful Interpolation: On the Role of Sharpness in CLIP Generalization](https://mlsanity.com/paper/2410.16476/) (2024)
- [Visualizing the loss landscapes of physics-informed neural networks](https://mlsanity.com/paper/2602.05849/) (2026)
- [On Accelerating Distributed Convex Optimizations](https://mlsanity.com/paper/2108.08670/) (2021)
- [Hyper-Representations: Learning from Populations of Neural Networks](https://mlsanity.com/paper/2410.05107/) (2024)
- [Finetune Once: Decoupling General & Domain Learning with Dynamic Boosted Annealing](https://mlsanity.com/paper/2509.26242/) (2025)

## Abstract (arXiv)

Linear interpolation between initial neural network parameters and converged parameters after training with stochastic gradient descent (SGD) typically leads to a monotonic decrease in the training objective. This Monotonic Linear Interpolation (MLI) property, first observed by Goodfellow et al. (2014) persists in spite of the non-convex objectives and highly non-linear training dynamics of neural networks. Extending this work, we evaluate several hypotheses for this property that, to our knowledge, have not yet been explored. Using tools from differential geometry, we draw connections between the interpolated paths in function space and the monotonicity of the network - providing sufficient conditions for the MLI property under mean squared error. While the MLI property holds under various settings (e.g. network architectures and learning problems), we show in practice that networks violating the MLI property can be produced systematically, by encouraging the weights to move far from initialization. The MLI property raises important questions about the loss landscape geometry of neural networks and highlights the need to further study their global properties.