Layer-Wise Learning Rate Adaptation

The clinical management of gynecological diseases often relies on medical imaging for diagnosis, treatment planning, and follow-up. Segmentation in this setting is challenging because successive tasks may differ in imaging modality, target anatomy, pathology, and annotation structure. Continual learning allows models to adapt to new tasks without simultaneous access to previous datasets. However, when successive tasks differ substantially, learning a new task can degrade performance on earlier ones, a problem known as catastrophic forgetting. Understanding where adaptation disrupts previous knowledge can help guide the design of more targeted continual-learning strategies. We investigate how forgetting changes as different parts of an encoder--decoder network are allowed to adapt. We progressively expand the trainable region of a 3D nnU-Net backbone from the bottleneck toward input- and output-proximal blocks. Under a shared learning rate, adaptation near the bottleneck largely preserves previous-task performance but provides limited current-task learning, whereas broader adaptation improves current-task performance but sharply increases forgetting. This trade-off persists even when the average change in trainable backbone parameters is approximately comparable. Assigning different learning rates to different blocks substantially reduces forgetting when part of the backbone is trainable, although this changes both the size and location of the updates. Forgetting still increases as more blocks are trained and remains severe when the full backbone is updated. These results show that forgetting depends not only on how much the model changes, but also on which parts of the model are allowed to change.

Decentralized intelligence systems with heterogeneous devices and limited coordination increasingly rely on decentralized federated learning (DFL). However, DFL suffers from convergence inefficiency under data heterogeneity due to the use of a uniform learning rate (LR) that ignores layer-specific optimization needs. Foundational layers are responsible for maintaining network consensus, while specialized layers adapt to local data characteristics, leading to conflicting gradients and degraded performance under non-IID conditions. To address this fundamental tension, this work introduces FedA2L, a method that dynamically adjusts layer-wise LRs based on model divergence signals. By leveraging local update intensity and network consensus constraints, FedA2L seamlessly integrates into existing DFL protocols without additional communication or coordination. Extensive evaluations across DFL algorithms, various model architectures, and datasets demonstrate that FedA2L achieves up to 4.94 times faster convergence than vanilla DFL and reduces communication rounds by up to 59% compared to scheduler-based baselines. Furthermore, FedA2L exhibits resilience to severe data heterogeneity, larger network sizes, and sparse topologies, reducing communication overhead and establishing it as a versatile optimization tool for resource-constrained or large-scale distributed learning in edge and IoT deployments. The code is released at https://github.com/nclabteam/FedA2L.

Per-layer diagnostics reveal that, at the prescribed learning rate, Lion's effective scale is 2.6-2.8x too high for attention and MLP parameters and ~2x too high for normalization layers on ViT-Tiny/CIFAR-100; this 32% cross-layer-type disparity cannot be reproduced by a single global rate. The measurement comes from LionVote, a per-layer learning rate mechanism in which each parameter tensor maintains a compound level, a persistent integer updated every c epochs by two diagnostics (gradient direction stability and momentum health) resolved by a validation loss tiebreaker. Voting thresholds derive from geometric identities, the EMA time constant, and a noise-floor estimate; cadence is bounded structurally and selected by ablation. On ViT-Tiny/CIFAR-100, LionVote achieves 69.7% top-1 accuracy vs. Lion's 69.0% (p < 0.02, Welch's t-test) and AdamW's 68.8%. Per-layer adaptation value depends on both architectural heterogeneity and task; on uniform CNN architectures tuned SGD with cosine annealing remains dominant, and on ViT architectures gains are task-dependent.

Local sharpness, defined by the largest Hessian eigenvalue $λ_1$, sets the maximum stable gradient update size, but its computation would usually require running Lanczos or Hessian-vector products. However, we notice that even a single Armijo backtracking line search already contains this information with just a few forward passes, as the accepted step $α$ determines the directional curvature along the search direction up to the multiplicative band set by the backtracking factor. The correlation between $\logα$ and $\logλ_1$ on CIFAR-10, Fashion-MNIST and Imagenette reaches $-0.91$ to $-0.95$ in Pearson correlation, and even after removing the trend per run the correlation remains at $-0.60$ to $-0.70$. This allows for a cheap online Edge-of-Stability estimate of the slow sharpness component. The employed probing mechanism searches along Adam's first-step update direction, at initialisation and nine times over the course of the first 50 optimiser steps. The learning-rate cap is set as twice the smallest observed step size, and in the studied learning-rate ranges ($10^{-3}$ to $3.0$) and GPT-2 pretraining experiments all capped runs avoid divergence; in the general architecture analysis, one MLP architecture is still sensitive to the initial batch order. This probing protocol incurs an approximately one percent overhead, and using a non-binding cap means that the optimiser's state and first update are bit-identical. There is no fine-tuning of any of the protocol parameters to any specific architecture; this is the sense in which this is a calibration-free safeguard. It is meant as a way of avoiding divergence, not achieving accuracy. At GPT-2 scale, multiple measurements also illustrate why an initialisation-only cap is insufficient: directional curvature rises substantially in the first five optimiser steps, which motivates the short probationary window.

We study optimal learning-rate selection in two-layer and three-layer linear neural networks trained to learn linear target functions. In particular, we derive the exact closed-form expressions for the gradients and test loss after one and two steps of gradient descent, enabling a precise characterization of early training dynamics. We characterize how learning rates should scale under the gradient approximation in the first two steps, and prove that performing updates with this approximation yields a tractable surrogate loss with a tight, small approximation error. This formulation enables the theoretical analysis of layer-wise learning rates and reveals a distinct early-training regime: test loss can be minimized by unequal learning rates at the initial step, while equal learning rates become optimal in subsequent steps. Our numerical experiments validate the theory and demonstrate the importance of balancing layer-wise learning rates early during training. The code is available at: https://github.com/TDCSZ327/Layer-Balancing.

Learning rate configuration is a fundamental aspect of modern deep learning. The prevailing practice of applying a uniform learning rate across all layers overlooks the structural heterogeneity of Transformers, potentially limiting their effectiveness as the backbone of Large Language Models (LLMs). In this paper, we introduce Layerwise Learning Rate (LLR), an adaptive scheme that assigns distinct learning rates to individual Transformer layers. Our method is grounded in Heavy-Tailed Self-Regularization (HT-SR) theory, which characterizes the empirical spectral density (ESD) of weight correlation matrices to quantify heavy-tailedness. Layers with weaker heavy-tailedness are assigned larger learning rates to accelerate training, while layers with stronger heavy-tailedness receive smaller learning rates. By tailoring learning rates in this manner, LLR promotes more balanced training across layers, leading to faster convergence and improved generalization. Extensive experiments across architectures ranging from LLaMA to GPT-nano, optimizers including AdamW and Muon, and model scales from 60M to 3B parameters with up to 100B training tokens demonstrate the effectiveness of LLR. LLR achieves up to 1.5x training speedup and consistently outperforms uniform-learning-rate baselines. In particular, it improves the average zero-shot accuracy of 1B models from 47.09% to 49.02%, and that of 3B models from 48.58% to 50.61%. A key advantage of LLR is its low tuning overhead: it can transfer nearly optimal learning-rate settings directly from the uniform baseline. Code is available at https://github.com/hed-ucas/Layer-wise-Learning-Rate.

Backpropagation with gradient descent is a common optimization strategy employed by most neural network architectures in machine learning. However, finding optimal hyperparameters to guide training has proven challenging. While it is widely acknowledged that selecting appropriate parameters is crucial for avoiding overfitting and achieving unbiased outcomes, this choice remains largely based on empirical experiments and experience. This paper presents a new probabilistic framework for the learning rate, a key parameter in stochastic gradient descent. The framework develops classic Bayesian statistics into a double-Bayesian decision mechanism involving two antagonistic Bayesian processes. A theoretically optimal learning rate can be derived from these two processes and used for stochastic gradient descent. Experiments across various classification, segmentation, and detection tasks corroborate the practical significance of the theoretically derived learning rate. The paper also discusses the ramifications of the proposed double-Bayesian framework for network training and model performance.

Neural networks are typically trained with a single learning rate across all layers. While recent empirical evidence suggests that assigning layer-specific learning rates can accelerate training, a principled understanding of the conditions and mechanisms under which non-uniform learning rates are beneficial remains limited. In this work, we investigate non-uniform learning rates through the lens of Stackelberg optimization. Specifically, we demonstrate that training neural networks with a smaller learning rate for the body layers and a larger learning rate for the final layer can be interpreted as a two-time-scale alternating gradient descent algorithm applied to a Stackelberg reformulation of the original objective. We establish finite-time convergence guarantees for the algorithm under broad conditions that accommodate constraint sets and non-smooth activation functions. Beyond convergence, we identify two mechanisms by which non-uniform learning rates can outperform uniform learning rates: (i) we show that certain problem instances induce a Stackelberg objective with stronger optimization structure than the original objective, yielding faster convergence to globally optimal solutions, (ii) our numerical analysis reveals that the Stackelberg objective can exhibit substantially sharper local curvature, especially in early training, which leads to more informative gradients and learning acceleration. Experiments in supervised learning and reinforcement learning support our findings.

A causal-decoder block is hierarchical: lower layers build the residual basis that upper layers attend over. We identify a failure mode in GPT pretraining: upper layers commit to sharp attention patterns before lower-layer features stabilize. We call this premature upper-layer attention specialization. Temporarily slowing only upper-layer Q/K projections during early training improves final perplexity and downstream accuracy without altering other parameters; it prevents upper attention from collapsing onto an immature residual basis. In LLaMA-style blocks, the same intervention is nearly unnecessary. Through ablations, we isolate multiplicative gated FFNs (not RMSNorm or bias removal) as the component that suppresses the upstream residual writes driving the failure. A pathwise analysis unifies both findings: the learning-rate intervention reduces a step-size factor, while gated FFNs reduce a residual-energy factor on the same growth pathway. Our results identify upper-layer Q/K timing as a concrete interaction point between decoder architecture and optimization.

Muon fixes the \emph{direction} of every matrix-valued update at the polar factor of its momentum, while each layer's step \emph{magnitude} is addressed only by a static shape correction. We derive a dynamic per-layer scalar by adapting the LARS/LAMB trust-ratio principle to the orthogonalized setting, where the standard denominator candidates---the raw momentum norm or the polar-factor norm---either live in the wrong unit space or carry no update-scale information. The resulting method, \emph{OrScale}, uses the norm of the parameter-space direction actually applied and anchors each layer's ratio at one via a per-layer calibration, so that the Moonlight recipe (tuned for AdamW, shared with Muon via RMS matching) transfers with \emph{no additional sweep}; a component ablation confirms each design choice is individually load-bearing. Theoretically, OrScale retains a nuclear-norm $O(1/\sqrt{T})$ convergence rate for any clipped multiplier and achieves a strict layer-adaptive descent gain $κ_{\mathrm{eff}}>1$ under two conditions estimable from standard training diagnostics---a bound that predicts the gain should \emph{grow with architectural heterogeneity}. Experiments confirm the prediction: with every hyperparameter inherited verbatim from the Moonlight recipe, OrScale matches or beats Muon+Moonlight across dense 125M--1.1B FineWeb-Edu pre-training, and on a 16B-A3B mixture-of-experts model---where the logged trust ratios separate cleanly by layer class---the gap widens by an order of magnitude to $0.130$ nats ($3.8\%$ relative) at parity wall-clock cost.

Learning rate scheduling has evolved from the single global fixed rate of early SGD to sophisticated layer-wise adaptive strategies. We systematize this evolution into five generations: (Gen1) global fixed learning rates, (Gen2) global scheduling, (Gen3) parameter-level adaptation, (Gen4) layer-level differentiation, and (Gen5) joint layer-time scheduling. We trace the fundamental motivation behind each transition, showing how the shift from one-size-fits-all to tailoring by layer and time addresses the impossible trinity of transfer learning: lower layers require small updates to preserve general knowledge while higher layers need large updates to adapt to new tasks. Building on this taxonomy, we propose Discriminative Adaptive Layer Scaling (DALS), a unified framework that integrates phase-adaptive cosine scheduling, depth-aware Grokfast gradient filtering, and LARS-style trust ratios into a single coherent optimizer. We benchmark 18 strategies including three DALS variants across all five generations on five datasets: synthetic, CIFAR-10 (from scratch), RTE, TREC-6, and IMDb (fine-tuning). On synthetic, DALS achieves the best accuracy at 98.0%, while DALS-Fast reaches 90% in just 3 epochs. The cross-dataset analysis reveals striking regime-dependent patterns -- no single strategy wins across all regimes. Critically, STLR+Discriminative, the ULMFiT champion, catastrophically fails on from-scratch tasks (43.6% on TREC-6 from scratch vs. 96.8% with RAdam), confirming that directional decay biases are harmful without pretrained features. DALS avoids either extreme, achieving the best synthetic result while maintaining competitive fine-tuning performance.