# Mask in the Mirror: Implicit Sparsification

arXiv: https://arxiv.org/abs/2408.09966 · HTML: https://mlsanity.com/paper/2408.09966/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- Builds on 64 of its 81 references that are in the map.
- First posted on arXiv 19 Aug 2024.

## Builds on (most cited first, 30 of 64)

- [Hierarchical Text-Conditional Image Generation with CLIP Latents](https://mlsanity.com/paper/2204.06125/) (2022)
- [Convex optimization](https://mlsanity.com/paper/2106.01946/) (2021)
- [The Lottery Ticket Hypothesis: Finding Sparse, Trainable Neural Networks](https://mlsanity.com/paper/1803.03635/) (2018)
- [Learning both Weights and Connections for Efficient Neural Networks](https://mlsanity.com/paper/1506.02626/) (2015)
- [SNIP: Single-shot Network Pruning based on Connection Sensitivity](https://mlsanity.com/paper/1810.02340/) (2018)
- [Linear Mode Connectivity and the Lottery Ticket Hypothesis](https://mlsanity.com/paper/1912.05671/) (2019)
- [Learning Sparse Neural Networks through L0 Regularization](https://mlsanity.com/paper/1712.01312/) (2017)
- [Learning Structured Sparsity in Deep Neural Networks](https://mlsanity.com/paper/1608.03665/) (2016)
- [Linear Convergence of Gradient and Proximal-Gradient Methods Under the Polyak-Lojasiewicz Condition](https://mlsanity.com/paper/1608.04636/) (2016)
- [Rigging the Lottery: Making All Tickets Winners](https://mlsanity.com/paper/1911.11134/) (2019)
- [Pruning neural networks without any data by iteratively conserving synaptic flow](https://mlsanity.com/paper/2006.05467/) (2020)
- [Picking Winning Tickets Before Training by Preserving Gradient Flow](https://mlsanity.com/paper/2002.07376/) (2020)
- [Implicit Regularization in Matrix Factorization](https://mlsanity.com/paper/1705.09280/) (2017)
- [Kernel and Rich Regimes in Overparametrized Models](https://mlsanity.com/paper/2002.09277/) (2020)
- [Implicit Bias of Gradient Descent for Wide Two-layer Neural Networks Trained with the Logistic Loss](https://mlsanity.com/paper/2002.04486/) (2020)
- [Characterizing Implicit Bias in Terms of Optimization Geometry](https://mlsanity.com/paper/1802.08246/) (2018)
- [Sustainable AI: Environmental Implications, Challenges and Opportunities](https://mlsanity.com/paper/2111.00364/) (2021)
- [Comparing Rewinding and Fine-tuning in Neural Network Pruning](https://mlsanity.com/paper/2003.02389/) (2020)
- [Sparse Networks from Scratch: Faster Training without Losing Performance](https://mlsanity.com/paper/1907.04840/) (2019)
- [Proving the Lottery Ticket Hypothesis: Pruning is All You Need](https://mlsanity.com/paper/2002.00585/) (2020)
- [Power Hungry Processing: Watts Driving the Cost of AI Deployment?](https://mlsanity.com/paper/2311.16863/) (2023)
- [Pruning Neural Networks at Initialization: Why are We Missing the Mark?](https://mlsanity.com/paper/2009.08576/) (2020)
- [Drawing Early-Bird Tickets: Towards More Efficient Training of Deep Networks](https://mlsanity.com/paper/1909.11957/) (2019)
- [Soft Threshold Weight Reparameterization for Learnable Sparsity](https://mlsanity.com/paper/2002.03231/) (2020)
- [Implicit Bias of SGD for Diagonal Linear Networks: a Provable Benefit of Stochasticity](https://mlsanity.com/paper/2106.09524/) (2021)
- [Winning the Lottery with Continuous Sparsification](https://mlsanity.com/paper/1912.04427/) (2019)
- [The Unreasonable Effectiveness of Random Pruning: Return of the Most Naive Baseline for Sparse Training](https://mlsanity.com/paper/2202.02643/) (2022)
- [Optimal Lottery Tickets via SubsetSum: Logarithmic Over-Parameterization is Sufficient](https://mlsanity.com/paper/2006.07990/) (2020)
- [Implicit Regularization for Optimal Sparse Recovery](https://mlsanity.com/paper/1909.05122/) (2019)
- [Gradient Descent Can Take Exponential Time to Escape Saddle Points](https://mlsanity.com/paper/1705.10412/) (2017)

## Built on by (most cited first, 14 of 14)

- [Mirror, Mirror of the Flow: How Does Regularization Shape Implicit Bias?](https://mlsanity.com/paper/2504.12883/) (2025)
- [Sign-In to the Lottery: Reparameterizing Sparse Training From Scratch](https://mlsanity.com/paper/2504.12801/) (2025)
- [Never Saddle for Reparameterized Steepest Descent as Mirror Flow](https://mlsanity.com/paper/2603.02064/) (2026)
- [Neural Graduated Assignment for Maximum Common Edge Subgraphs](https://mlsanity.com/paper/2505.12325/) (2025)
- [Differentiable Sparsity via D-Gating: Simple and Versatile Structured Penalization](https://mlsanity.com/paper/2509.23898/) (2025)
- [Sparse Training of Neural Networks based on Multilevel Mirror Descent](https://mlsanity.com/paper/2602.03535/) (2026)
- [Implicit Bias of Mirror Flow in Homogeneous Neural Networks: Sparse and Dense Feature Learning](https://mlsanity.com/paper/2605.19458/) (2026)
- [More Experts Than Galaxies: Conditionally-overlapping Experts With Biologically-Inspired Fixed Routing](https://mlsanity.com/paper/2410.08003/) (2024)
- [The Graphon Limit Hypothesis: Understanding Neural Network Pruning via Infinite Width Analysis](https://mlsanity.com/paper/2510.17515/) (2025)
- [HORST: Composing Optimizer Geometries for Sparse Transformer Training](https://mlsanity.com/paper/2605.21104/) (2026)
- [Theoretical Analysis of Sparse Optimization with Reparameterization, Weight Decay, and Adaptive Learning Rate](https://mlsanity.com/paper/2605.25134/) (2026)
- [SparseOpt: Addressing Normalization-induced Gradient Skew in Sparse Training](https://mlsanity.com/paper/2605.27541/) (2026)
- [Mind the Drift: Diagonal Linear Networks Under Large Learning Rates](https://mlsanity.com/paper/2610.06120/) (2026)
- [From Parallel to Sequential: Small-Batch Noise Restores Simplicity Bias of Muon](https://mlsanity.com/paper/2610.16868/) (2026)

## Abstract (arXiv)

Continuous sparsification strategies are among the most effective methods for reducing the inference costs and memory demands of large-scale neural networks. A key factor in their success is the implicit L1 regularization induced by jointly learning both mask and weight variables, which has been shown experimentally to outperform explicit L1 regularization. We provide a theoretical explanation for this observation by analyzing the learning dynamics, revealing that early continuous sparsification is governed by an implicit L2 regularization that gradually transitions to an L1 penalty over time. Leveraging this insight, we propose a method to dynamically control the strength of this implicit bias. Through an extension of the mirror flow framework, we establish convergence and optimality guarantees in the context of underdetermined linear regression. Our theoretical findings may be of independent interest, as we demonstrate how to enter the rich regime and show that the implicit bias can be controlled via a time-dependent Bregman potential. To validate these insights, we introduce PILoT, a continuous sparsification approach with novel initialization and dynamic regularization, which consistently outperforms baselines in standard experiments.