Yumeng Fu, Jiayin Zhu, Lingling Zhang, Wenjun Wu, Bo Zhao, Shaoxuan Ma, Yushun Zhang, Jun Liu. Proceedings of the 64th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers). 2026.

We propose Adam-mini, an optimizer that achieves on-par or better performance than AdamW with $50$% less memory footprint. Adam-mini reduces memory by cutting down the learning rate resources in Adam (i.e., $1/\sqrt{v}$). By delving into the Hessian structure of neural nets, we find Adam’s $v$ might not function at its full potential as effectively as we expected. We find that $\geq 99.9$% of these learning rates in $v$ could be harmlessly removed if we (1) carefully partition the parameters into blocks following our proposed principle on Hessian structure; (2) assign a single but good learning rate to each parameter block. We then provide one simple way to find good learning rates and propose Adam-mini. Empirically, we verify that Adam-mini performs on par or better than AdamW on various language models sized from 39M to 13B for pre-training, supervised fine-tuning, and RLHF. The reduced memory footprint of Adam-mini also alleviates communication overheads among GPUs, thereby increasing throughput. For instance, Adam-mini achieves $49.6$% higher throughput than AdamW when pre-training Llama 2-7B on $2\times$ A800-80GB GPUs, which saves 33% wall-clock time for pre-training.

Reinforcement Learning from Human Feedback (RLHF) is key to aligning Large Language Models (LLMs), typically paired with the Proximal Policy Optimization (PPO) algorithm. While PPO is a powerful method designed for general reinforcement learning tasks, it is overly sophisticated for LLMs, leading to laborious hyper-parameter tuning and significant computation burdens. To make RLHF efficient, we present ReMax, which leverages 3 properties of RLHF: fast simulation, deterministic transitions, and trajectory-level rewards. These properties are not exploited in PPO, making it less suitable for RLHF. Building on the renowned REINFORCE algorithm, ReMax does not require training an additional value model as in PPO and is further enhanced with a new variance reduction technique. ReMax offers several benefits over PPO: it is simpler to implement, eliminates more than 4 hyper-parameters in PPO, reduces GPU memory usage, and shortens training time. ReMax can save about 46% GPU memory than PPO when training a 7B model and enables training on A800-80GB GPUs without the memory-saving offloading technique needed by PPO. Applying ReMax to a Mistral-7B model resulted in a 94.78% win rate on the AlpacaEval leaderboard and a 7.739 score on MT-bench, setting a new SOTA for open-source 7B models. These results show the effectiveness of ReMax while addressing the limitations of PPO in LLMs.

Adam is widely adopted in practical applications due to its fast convergence. However, its theoretical analysis is still far from satisfactory. Existing convergence analyses for Adam rely on the bounded smoothness assumption, referred to as the L-smooth condition. Unfortunately, this assumption does not hold for many deep learning tasks. Moreover, we believe that this assumption obscures the true benefit of Adam, as the algorithm can adapt its update magnitude according to local smoothness. This important feature of Adam becomes irrelevant when assuming globally bounded smoothness. This paper studies the convergence of randomly reshuffled Adam (RR Adam) with diminishing learning rate, which is the major version of Adam adopted in deep learning tasks. We present the first convergence analysis of RR Adam without the bounded smoothness assumption. We demonstrate that RR Adam can maintain its convergence properties when smoothness is linearly bounded by the gradient norm, referred to as the (L0, L1)-smooth condition. We further compare Adam to SGD when both methods use diminishing learning rate. We refine the existing lower bound of SGD and show that SGD can be slower than Adam. To our knowledge, this is the first time that Adam and SGD are rigorously compared in the same setting and the advantage of Adam is revealed.

SGD performs worse than Adam by a significant margin on Transformers, but the reason remains unclear. In this work, we provide an explanation through the lens of Hessian: (i) Transformers are "heterogeneous'': the Hessian spectrum across parameter blocks vary dramatically, a phenomenon we call "block heterogeneity"; (ii) Heterogeneity hampers SGD: SGD performs worse than Adam on problems with block heterogeneity. To validate (i) and (ii), we check various Transformers, CNNs, MLPs, and quadratic problems, and find that SGD can perform on par with Adam on problems without block heterogeneity, but performs worse than Adam when the heterogeneity exists. Our initial theoretical analysis indicates that SGD performs worse because it applies one single learning rate to all blocks, which cannot handle the heterogeneity among blocks. This limitation could be ameliorated if we use coordinate-wise learning rates, as designed in Adam.

Federated learning effectively addresses issues such as data privacy by collaborating across participating devices to train global models. However, factors such as network topology and computing power of devices can affect its training or communication process in complex network environments. Computing and network convergence (CNC) of sixth-generation (6G) networks, a new network architecture and paradigm with computing-measurable, perceptible, distributable, dispatchable, and manageable capabilities, can effectively support federated learning training and improve its communication efficiency. By guiding the participating devices’ training in federated learning based on business requirements, resource load, network conditions, and computing power of devices, CNC can reach this goal. In this paper, to improve the communication efficiency of federated learning in complex networks, we study the communication efficiency optimization methods of federated learning for CNC of 6G networks that give decisions on the training process for different network conditions and computing power of participating devices. The simulations address two architectures that exist for devices in federated learning and arrange devices to participate in training based on arithmetic power while achieving optimization of communication efficiency in the process of transferring model parameters. The results show that the methods we proposed can cope well with complex network situations, effectively balance the delay distribution of participating devices for local training, improve the communication efficiency during the transfer of model parameters, and improve the resource utilization in the network.

Ever since \citet{reddi2019convergence} pointed out the divergence issue of Adam, many new variants have been designed to obtain convergence. However, vanilla Adam remains exceptionally popular and it works well in practice. Why is there a gap between theory and practice? We point out there is a mismatch between the settings of theory and practice: \citet{reddi2019convergence} pick the problem after picking the hyperparameters of Adam, i.e., $(\beta_1,\beta_2)$; while practical applications often fix the problem first and then tune $(\beta_1,\beta_2)$. Due to this observation, we conjecture that the empirical convergence can be theoretically justified, only if we change the order of picking the problem and hyperparameter. In this work, we confirm this conjecture. We prove that, when the 2nd-order momentum parameter $\beta_2$ is large and 1st-order momentum parameter $\beta_1 < \sqrt{\beta_2}<1$, Adam converges to the neighborhood of critical points. The size of the neighborhood is propositional to the variance of stochastic gradients. Under an extra condition (strong growth condition), Adam converges to critical points. It is worth mentioning that our results cover a wide range of hyperparameters: as $\beta_2$ increases, our convergence result can cover any $\beta_1 \in [0,1)$ including $\beta_1=0.9$, which is the default setting in deep learning libraries. To our knowledge, this is the first result showing that Adam can converge {\it without any modification} on its update rules. Further, our analysis does not require assumptions of bounded gradients or bounded 2nd-order momentum. When $\beta_2$ is small, we further point out a large region of $(\beta_1,\beta_2)$ combinations where Adam can diverge to infinity. Our divergence result considers the same setting (fixing the optimization problem ahead) as our convergence result, indicating that there is a phase transition from divergence to convergence when increasing $\beta_2$. These positive and negative results provide suggestions on how to tune Adam hyperparameters: for instance, when Adam does not work well, we suggest tuning up $\beta_2$ and trying $\beta_1< \sqrt{\beta_2}$.

Modern neural networks are often quite wide, causing large memory and computation costs. It is thus of great interest to train a narrower network. However, training narrow neural nets remains a challenging task. We ask two theoretical questions: Can narrow networks have as strong expressivity as wide ones? If so, does the loss function exhibit a benign optimization landscape? In this work, we provide partially affirmative answers to both questions for 1-hidden-layer networks with fewer than $n$ (sample size) neurons when the activation is smooth. First, we prove that as long as the width $m \geq 2n/d$ (where $d$ is the input dimension), its expressivity is strong, i.e., there exists at least one global minimizer with zero training loss. Second, we identify a nice local region with no local-min or saddle points. Nevertheless, it is not clear whether gradient descent can stay in this nice region. Third, we consider a constrained optimization formulation where the feasible region is the nice local region, and prove that every KKT point is a nearly global minimizer. It is expected that projected gradient methods converge to KKT points under mild technical conditions, but we leave the rigorous convergence analysis to future work. Thorough numerical results show that projected gradient methods on this constrained formulation significantly outperform SGD for training narrow neural nets.