Rethinking Neural Network Learning Rates: A Stackelberg Perspective

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.

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.