End-to-end deep learning has achieved impressive results but often relies on large labeled datasets, exhibits limited generalization to unseen scenarios, and incurs substantial computational cost. Classical optimization methods, in contrast, are more data-efficient and lightweight but frequently suffer from slow convergence. Learned optimizers aim to bridge this gap, yet existing approaches have focused primarily on first-order methods, while learned second-order optimization has received much less attention. We introduce L-SR1, a learned second-order optimizer inspired by the classical Symmetric Rank-One (SR1) method. At its core, L-SR1 employs a Projection-Guided Secant Mechanism (PGSM) that generates positive semi-definite preconditioners and biases meta-training toward the quasi-Newton secant relation. Through controlled analytic benchmarks, we study stability, generalization across problem dimensions, and search direction quality, and further evaluate L-SR1 on Monocular Human Mesh Recovery (HMR), where it outperforms both classical and learned optimization-based baselines. With a compact model and no reliance on task-specific fine-tuning or annotated data, L-SR1 demonstrates strong generalization and can be integrated into a broad range of iterative optimization problems to accelerate convergence and reduce the required number of iterations.
Geometry-aware optimizers such as Newton and natural gradient can improve conditioning in deep learning, but scalable variants such as K-FAC, Shampoo, and related preconditioners usually impose structural approximations early, often discarding cross-layer interactions induced by the network computation. We introduce Layerwise LQR (LLQR), a framework for learning structured inverse preconditioners under a global layerwise optimal-control objective. The starting point is an exact equivalence: the steepest-descent step under a broad class of divergence-induced quadratic models--including Newton, Gauss-Newton, Fisher/natural-gradient, and intermediate-layer metrics--can be written as a finite-horizon Linear Quadratic Regulator (LQR) problem. This formulation serves as a reference that exposes the layerwise dynamics and cost matrices encoding the original dense geometry. We then derive a scalable relaxation that learns diagonal, (E-)Kronecker-factored, or other structured inverse preconditioners by minimizing the LQR objective and reusing them across iterations. The resulting optimizer wraps standard methods while retaining a principled connection to second-order geometry, without forming or inverting the global curvature matrix. Experiments on ResNets and Transformers show that LLQR improves optimization dynamics and often translates these gains into improved final test performance, while adding only modest wall-clock overhead. It establishes LLQR as a practical framework for geometry-aware second-order methods and a reference for evaluating scalable approximations.
Simon Dufort-Labbé, Pierre-Luc Bacon, Razvan Pascanu +2
Low-Rank Adaptation (LoRA) significantly reduces compute and memory costs for finetuning Deep Learning models but is often harder to tune than dense training: when using factor-wise optimizers such as AdamW, it is sensitive to initialization choices, its optimal learning rates transfer poorly across ranks, and it often fails to beat dense baselines. We derive LoRA-Muon by applying the Muon optimizer's spectral steepest-descent rule to the low-rank setting. Along with our split weight-decay rule, our main claim is that LoRA-Muon is a good low-rank proxy for full-rank Muon and Shampoo-family optimizers. Its optimal learning rates transfer across rank, width, depth, and factor-rescaling. In our compute-matched TinyShakespeare study, a rank-2 proxy recovers the dense best tested learning rate, and a rank-32 LoRA-Muon run attains lower mean validation loss than the dense baseline in the seed-averaged sweep. We further show that the Spectron optimizer depends on arbitrary factor scaling, so it would likely be a poor fit when finetuning starts from badly imbalanced factors, and that LoRA-RITE's simplified QR-coordinate core implements the same spectral update. LoRA-Muon computes that update without QR-decomposition and avoids storing second moments, making it more accelerator-friendly and memory-efficient.
Franz Louis Cesista, Katherine Crowson, Cédric Simal +1
Deep learning models are usually trained with stochastic gradient descent-based algorithms, but these optimizers face inherent limitations, such as slow convergence and stringent assumptions for convergence. In particular, data heterogeneity arising from distributed settings poses significant challenges to their theoretical and numerical performance. This paper develops an algorithm, PISA (Preconditioned Inexact Stochastic Alternating Direction Method of Multipliers). Grounded in rigorous theoretical guarantees, the algorithm converges under the sole assumption of Lipschitz continuity of the gradient on a bounded region, thereby removing the need for other conditions commonly imposed by stochastic methods. This capability enables the proposed algorithm to tackle the challenge of data heterogeneity effectively. Moreover, the algorithmic architecture enables scalable parallel computing and supports various preconditions, such as second-order information, second moment, and orthogonalized momentum by Newton-Schulz iterations. Incorporating the latter two preconditions in PISA yields two computationally efficient variants: SISA and NSISA. Comprehensive experimental evaluations for training or fine-tuning diverse deep models, including vision models, large language models, reinforcement learning models, generative adversarial networks, and recurrent neural networks, demonstrate superior numerical performance of SISA and NSISA compared to various state-of-the-art optimizers.