# Learnable Loss Geometries with Mirror Descent for Scalable and Convergent Meta-Learning

arXiv: https://arxiv.org/abs/2509.02418 · HTML: https://mlsanity.com/paper/2509.02418/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 34 of its 61 references that are in the map.
- First posted on arXiv 2 Sep 2025.

## Builds on (most cited first, 30 of 34)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [ImageNet Large Scale Visual Recognition Challenge](https://mlsanity.com/paper/1409.0575/) (2014)
- [Model-Agnostic Meta-Learning for Fast Adaptation of Deep Networks](https://mlsanity.com/paper/1703.03400/) (2017)
- [Matching Networks for One Shot Learning](https://mlsanity.com/paper/1606.04080/) (2016)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [A Closer Look at Few-shot Classification](https://mlsanity.com/paper/1904.04232/) (2019)
- [Meta-Learning with Implicit Gradients](https://mlsanity.com/paper/1909.04630/) (2019)
- [Meta-Learning with Latent Embedding Optimization](https://mlsanity.com/paper/1807.05960/) (2018)
- [Meta-Learning with Differentiable Convex Optimization](https://mlsanity.com/paper/1904.03758/) (2019)
- [Meta-Learning for Semi-Supervised Few-Shot Classification](https://mlsanity.com/paper/1803.00676/) (2018)
- [Input Convex Neural Networks](https://mlsanity.com/paper/1609.07152/) (2016)
- [Meta-learning with differentiable closed-form solvers](https://mlsanity.com/paper/1805.08136/) (2018)
- [Meta-SGD: Learning to Learn Quickly for Few-Shot Learning](https://mlsanity.com/paper/1707.09835/) (2017)
- [Rapid Learning or Feature Reuse? Towards Understanding the Effectiveness of MAML](https://mlsanity.com/paper/1909.09157/) (2019)
- [Bilevel Optimization: Convergence Analysis and Enhanced Design](https://mlsanity.com/paper/2010.07962/) (2020)
- [Bayesian Model-Agnostic Meta-Learning](https://mlsanity.com/paper/1806.03836/) (2018)
- [Cross-Domain Few-Shot Classification via Learned Feature-Wise Transformation](https://mlsanity.com/paper/2001.08735/) (2020)
- [Recasting Gradient-Based Meta-Learning as Hierarchical Bayes](https://mlsanity.com/paper/1801.08930/) (2018)
- [A Two-Timescale Framework for Bilevel Optimization: Complexity Analysis and Application to Actor-Critic](https://mlsanity.com/paper/2007.05170/) (2020)
- [On the Convergence Theory of Gradient-Based Model-Agnostic Meta-Learning Algorithms](https://mlsanity.com/paper/1908.10400/) (2019)
- [Meta-Learning for Low-Resource Neural Machine Translation](https://mlsanity.com/paper/1808.08437/) (2018)
- [Meta-Learning with Warped Gradient Descent](https://mlsanity.com/paper/1909.00025/) (2019)
- [Low Data Drug Discovery with One-shot Learning](https://mlsanity.com/paper/1611.03199/) (2016)
- [Meta-Curvature](https://mlsanity.com/paper/1902.03356/) (2019)
- [Convergence of Meta-Learning with Task-Specific Adaptation over Partial Parameters](https://mlsanity.com/paper/2006.09486/) (2020)
- [Learning where to learn: Gradient sparsity in meta and continual learning](https://mlsanity.com/paper/2110.14402/) (2021)
- [Meta-Learning with Network Pruning](https://mlsanity.com/paper/2007.03219/) (2020)
- [Meta-Learning with a Geometry-Adaptive Preconditioner](https://mlsanity.com/paper/2304.01552/) (2023)
- [Meta-learning the Learning Trends Shared Across Tasks](https://mlsanity.com/paper/2010.09291/) (2020)

## Abstract (arXiv)

Utilizing task-invariant knowledge acquired from related tasks as prior information, meta-learning offers a principled approach to learning a new task with limited data records. Sample-efficient adaptation of this prior information is a major challenge facing meta-learning, and plays an important role because it facilitates training the sought task-specific model with just a few optimization steps. Past works deal with this challenge through preconditioning that speeds up convergence of the per-task training. Though effective in representing locally quadratic loss curvatures, simple linear preconditioning can be hardly potent with complex loss geometries. Instead of relying on a quadratic distance metric, the present contribution copes with complex loss metrics by learning a versatile distance-generating function, which induces a nonlinear mirror map to effectively capture and optimize a wide range of loss geometries. With suitable parameterization, this generating function is effected by an expressive neural network that is provably a valid distance. Analytical results establish convergence of not only the proposed method, but also all meta-learning approaches based on preconditioning. To attain gradient norm less than ε, the convergence rate of O(ε-2) is on par with standard gradient-based meta-learning methods. Numerical tests on few-shot learning datasets demonstrate the superior empirical performance of the novel algorithm, as well as its rapid per-task convergence, which markedly reduces the number of adaptation steps, hence also accommodating large-scale meta-learning models.