Figure 2: Testlog-perplexityofa Transformer -Bigmodelon WMT'14 en!fr, whentrainingwith batchsizesof 384 (left) and 768 (right).

Our method retains the benefits of per-parameter adaptivity while allowing significantly larger models and batch sizes.

We propose SMMF (Square-Matricized Momentum Factorization), a memory-efficient optimizer that reduces the memory requirement of the widely used adaptive learning rate optimizers, such as Adam, by up to 96%. SMMF enables flexible and efficient factorization of an arbitrary rank (shape) of the first and second momentum tensors during optimization, based on the proposed square-matricization and one-time single matrix factorization. From this, it becomes effectively applicable to any rank (shape) of momentum tensors, i.e., bias, matrix, and any rank-d tensors, prevalent in various deep model architectures, such as CNNs (high rank) and Transformers (low rank), in contrast to existing memory-efficient optimizers that applies only to a particular (rank-2) momentum tensor, e.g., linear layers. We conduct a regret bound analysis of SMMF, which shows that it converges similarly to non-memory-efficient adaptive learning rate optimizers, such as AdamNC, providing a theoretical basis for its competitive optimization capability. In our experiment, SMMF takes up to 96% less memory compared to state-of-the-art memory efficient optimizers, e.g., Adafactor, CAME, and SM3, while achieving comparable model performance on various CNN and Transformer tasks.

Training large-scale neural networks in vision, and multimodal domains demands substantial memory resources, primarily due to the storage of optimizer states. While LoRA, a popular parameter-efficient method, reduces memory usage, it often suffers from suboptimal performance due to the constraints of low-rank updates. Low-rank gradient projection methods (e.g., GaLore, Flora) reduce optimizer memory by projecting gradients and moment estimates into low-rank spaces via singular value decomposition or random projection. However, they fail to account for inter-projection correlation, causing performance degradation, and their projection strategies often incur high computational costs. In this paper, we present COAP (Correlation-Aware Gradient Projection), a memory-efficient method that minimizes computational overhead while maintaining training performance. Evaluated across various vision, language, and multimodal tasks, COAP outperforms existing methods in both training speed and model performance. For LLaMA-1B, it reduces optimizer memory by 61% with only 2% additional time cost, achieving the same PPL as AdamW. With 8-bit quantization, COAP cuts optimizer memory by 81% and achieves 4x speedup over GaLore for LLaVA-v1.5-7B fine-tuning, while delivering higher accuracy.

There is growing evidence of the effectiveness of Shampoo, a higher-order preconditioning method, over Adam in deep learning optimization tasks. However, Shampoo's drawbacks include additional hyperparameters and computational overhead when compared to Adam, which only updates running averages of first- and second-moment quantities. This work establishes a formal connection between Shampoo (implemented with the 1/2 power) and Adafactor -- a memory-efficient approximation of Adam -- showing that Shampoo is equivalent to running Adafactor in the eigenbasis of Shampoo's preconditioner. This insight leads to the design of a simpler and computationally efficient algorithm: $\textbf{S}$hampo$\textbf{O}$ with $\textbf{A}$dam in the $\textbf{P}$reconditioner's eigenbasis (SOAP). With regards to improving Shampoo's computational efficiency, the most straightforward approach would be to simply compute Shampoo's eigendecomposition less frequently. Unfortunately, as our empirical results show, this leads to performance degradation that worsens with this frequency. SOAP mitigates this degradation by continually updating the running average of the second moment, just as Adam does, but in the current (slowly changing) coordinate basis. Furthermore, since SOAP is equivalent to running Adam in a rotated space, it introduces only one additional hyperparameter (the preconditioning frequency) compared to Adam. We empirically evaluate SOAP on language model pre-training with 360m and 660m sized models. In the large batch regime, SOAP reduces the number of iterations by over 40% and wall clock time by over 35% compared to AdamW, with approximately 20% improvements in both metrics compared to Shampoo. An implementation of SOAP is available at https://github.com/nikhilvyas/SOAP.

Training language models becomes increasingly expensive with scale, prompting numerous attempts to improve optimization efficiency. Despite these efforts, the Adam optimizer remains the most widely used, due to a prevailing view that it is the most effective approach. We aim to compare several optimization algorithms, including SGD, Adafactor, Adam, and Lion, in the context of autoregressive language modeling across a range of model sizes, hyperparameters, and architecture variants. Our findings indicate that, except for SGD, these algorithms all perform comparably both in their optimal performance and also in terms of how they fare across a wide range of hyperparameter choices. Our results suggest to practitioners that the choice of optimizer can be guided by practical considerations like memory constraints and ease of implementation, as no single algorithm emerged as a clear winner in terms of performance or stability to hyperparameter misspecification. Given our findings, we further dissect these approaches, examining two simplified versions of Adam: a) signed momentum (Signum) which we see recovers both the performance and hyperparameter stability of Adam and b) Adalayer, a layerwise variant of Adam which we introduce to study Adam's preconditioning. Examining Adalayer leads us to the conclusion that the largest impact of Adam's preconditioning is restricted to the last layer and LayerNorm parameters, and, perhaps surprisingly, the remaining layers can be trained with SGD.

This paper establishes a mathematical foundation for the Adam optimizer, elucidating its connection to natural gradient descent through Riemannian and information geometry. We rigorously analyze the diagonal empirical Fisher information matrix (FIM) in Adam, clarifying all detailed approximations and advocating for the use of log probability functions as loss, which should be based on discrete distributions, due to the limitations of empirical FIM. Our analysis uncovers flaws in the original Adam algorithm, leading to proposed corrections such as enhanced momentum calculations, adjusted bias corrections, adaptive epsilon, and gradient clipping. We refine the weight decay term based on our theoretical framework. Our modified algorithm, Fisher Adam (FAdam), demonstrates superior performance across diverse domains including LLM, ASR, and VQ-VAE, achieving state-of-the-art results in ASR.

Optimization algorithms play an indisputable role in the remarkable development of AI, especially in the realm of modern deep learning. In recent years, the emergence of breakthroughs in architectural innovation [3], as well as practical applications [37], has further promoted the necessity for embracing efficient training paradigms, which encompass optimization algorithms striking a balance between performance and manageable memory costs. Stochastic gradient descent (SGD) is widely regarded as the standard algorithm for training deep learning models, supported by extensive theoretical foundations [31, 32, 34, 43]. However, it requires thorough tuning of hyperparameters and frequently exhibits undesirable convergence rates when applied to many contemporary architectures [10, 36, 40]. Meanwhile, adaptive gradient methods such as Adam [17], AdaGrad [12], AMSGrad [29], etc., can adjust the learning rate for each parameter throughout the optimization process by utilizing cumulative second-order statistics.