ADOPT: A Modified Adam Optimizer that Achieves Optimal Convergence Rate with Any β2

Concetti Chiave

The ADOPT algorithm overcomes the convergence limitations of Adam and its variants in smooth nonconvex optimization by decorrelating the second moment estimate from the current gradient and changing the order of momentum update and normalization, achieving optimal convergence rate without relying on specific hyperparameter choices or bounded noise assumptions.

Sintesi

• Bibliographic Information: Taniguchi, S., Harada, K., Minegishi, G., Oshima, Y., Jeong, S. C., Nagahara, G., Iiyama, T., Suzuki, M., Iwasawa, Y., & Matsuo, Y. (2024). ADOPT: Modified Adam Can Converge with Any β2 with the Optimal Rate. In Advances in Neural Information Processing Systems (NeurIPS 2024).

• Research Objective: This paper proposes a novel adaptive gradient method, ADOPT, to address the non-convergence issue of Adam in smooth nonconvex optimization without relying on problem-dependent hyperparameter selection or strong assumptions like bounded gradient noise.

• Methodology: The authors analyze the convergence bounds of RMSprop and Adam, identifying the correlation between the second moment estimate and the current gradient as the root cause of non-convergence. They propose ADOPT, which modifies the momentum update and normalization order in Adam to eliminate this correlation. Theoretical analysis proves ADOPT's optimal convergence rate for smooth nonconvex optimization. Experiments on a toy problem, MNIST classification, CIFAR-10 classification with ResNet-18, ImageNet classification with SwinTransformer, NVAE training for MNIST density estimation, GPT-2 pretraining, LLaMA finetuning, and deep reinforcement learning tasks validate ADOPT's superior performance and robustness compared to existing methods.

• Key Findings:

  • Adam's non-convergence stems from the correlation between the second moment estimate and the current gradient.
  • Removing this correlation and modifying the momentum update order enables convergence with any β2 without bounded noise assumptions.
  • ADOPT consistently outperforms Adam and its variants in various tasks, demonstrating faster convergence and improved performance.

• Main Conclusions: ADOPT provides a theoretically sound and practically effective solution to the long-standing convergence issue of Adam, paving the way for more reliable and efficient optimization in deep learning.

• Significance: This research significantly impacts the field of stochastic optimization by providing a more robust and efficient adaptive gradient method, potentially leading to improved training stability and performance in various machine learning applications.

• Limitations and Future Research: The analysis assumes a uniformly bounded second moment of the stochastic gradient, which could be relaxed to a bounded variance assumption in future work. Further investigation into ADOPT's behavior in other complex optimization scenarios and its application to larger-scale problems is also warranted.

Statistiche

Adam fails to converge in a simple convex optimization problem with an objective f(θ) = θ for θ ∈[−1, 1] when the gradient noise is large, even with β2 = 0.999.
In the same problem, AMSGrad exhibits slow convergence when the gradient noise is large.
In MNIST classification using a single-hidden-layer MLP with 784 hidden units, ADOPT performs slightly better than Adam, AMSGrad, and AdaShift in terms of convergence speed and final accuracy.
In CIFAR-10 classification using ResNet-18, ADOPT converges slightly faster than Adam.
In ImageNet classification using SwinTransformer, ADOPT achieves higher top-1 accuracy than AdamW and AMSGrad throughout training.
In training NVAE for MNIST density estimation, ADOPT achieves better negative log-likelihood on test data compared to Adamax.
In pretraining GPT-2 on OpenWebText with a small batch size, Adam exhibits loss spikes and fails to converge, while ADOPT trains stably.
In finetuning LLaMA-7B on 52K instruction-following data, ADOPT achieves a higher MMLU score (42.13) compared to Adam (41.2).

Citazioni

Approfondimenti chiave tratti da

ADOPT: Modified Adam Can Converge with Any $\beta_2$ with the Optimal Rate

by Shohei Tanig... alle arxiv.org 11-06-2024

https://arxiv.org/pdf/2411.02853.pdf

Domande più approfondite

How does ADOPT's performance compare to other adaptive gradient methods beyond Adam and its variants, particularly in scenarios with highly non-stationary or noisy objectives?

While the paper primarily focuses on comparing ADOPT with Adam and its variants (AMSGrad, AdaShift), its theoretical properties suggest potential advantages for scenarios with highly non-stationary or noisy objectives. Here's a breakdown:

Robustness to Noise: ADOPT's convergence analysis relies on the bounded second moment assumption (Assumption 2.5), which is weaker than the bounded gradient assumption (Assumption 2.6) used by AMSGrad. This suggests ADOPT might be more robust to high gradient noise, a common characteristic of non-stationary objectives. The toy example with varying k values in the paper provides some evidence for this, where ADOPT outperforms AMSGrad as noise increases.
Handling Non-Stationarity: ADOPT's use of the previous second moment estimate (vt-1) for both momentum update and normalization helps decouple the current gradient's influence. This could be beneficial in non-stationary settings where the optimal step direction changes frequently. By not overly emphasizing the current, potentially noisy, gradient, ADOPT might exhibit more stable convergence behavior.
However, further empirical investigation is crucial to validate these potential advantages. Comparing ADOPT with other adaptive methods like:

AdaGrad: Known for its robustness in sparse gradients and suitability for non-stationary objectives.
RMSprop: Shares similarities with ADOPT in using a moving average of squared gradients.
Algorithms with Variance Reduction: Techniques like SVRG or SAGA could be combined with ADOPT to further enhance its performance in noisy settings.
Evaluating ADOPT on tasks known for high non-stationarity, such as reinforcement learning with dynamically changing environments or online learning with concept drift, would provide valuable insights.

Could the benefits of ADOPT's decorrelation and momentum update order be incorporated into other optimization algorithms beyond the Adam family?

It's certainly possible to explore incorporating ADOPT's core ideas into other optimization algorithms.

Decorrelation: The principle of decorrelating the current gradient from the second moment estimate can be applied to any algorithm that relies on such estimates. For instance, in RMSprop, instead of using vt = β2vt-1 + (1-β2)gt⊙gt, one could explore vt = β2vt-1 + (1-β2)E[gt⊙gt], where the expectation is approximated using past gradients.
Momentum Update Order: The idea of normalizing the gradient before incorporating it into the momentum term could potentially benefit momentum-based methods beyond Adam. For example, in classical momentum SGD, instead of mt = μmt-1 + gt, one could investigate mt = μmt-1 + gt / √vt-1 + ε, where vt-1 is a suitable second moment estimate.
However, directly transferring these modifications might not always guarantee improvement. The effectiveness of these techniques is intertwined with the specific update rules and convergence properties of each optimization algorithm. Careful theoretical analysis and empirical validation would be necessary to assess the benefits and potential drawbacks in each case.

What are the potential implications of ADOPT's improved convergence properties for the development of more efficient and robust training procedures for large-scale deep learning models, especially in resource-constrained settings?

ADOPT's improved convergence properties, particularly its robustness to hyperparameter choices and noise, hold promising implications for large-scale deep learning, especially in resource-constrained environments:

Reduced Hyperparameter Tuning: Training large models demands extensive computational resources. ADOPT's ability to converge well with a wider range of β2 values could translate to significant savings in hyperparameter search time and costs. This is particularly valuable in resource-constrained settings where extensive tuning is infeasible.
Faster Training: While the paper focuses on theoretical convergence rates, the empirical results on ImageNet and language modeling hint at potentially faster training with ADOPT. Faster convergence implies reaching a satisfactory performance level with fewer training epochs, directly reducing computational expenses and time-to-solution.
Improved Stability: Large-scale training is prone to instability due to noisy gradients from large datasets or mini-batch sizes. ADOPT's robustness to noise, as suggested by its theoretical analysis, could lead to more stable training processes, reducing the risk of divergence or requiring costly restarts.
These advantages could be particularly impactful in scenarios like:

Federated Learning: Training on decentralized data often involves higher noise levels. ADOPT's noise robustness could be crucial for efficient and stable federated learning.
Edge Computing: Deploying models on devices with limited resources necessitates efficient training procedures. ADOPT's potential for faster and more robust training aligns well with these constraints.
However, it's essential to acknowledge that these implications are based on the current results and further research is needed to fully realize ADOPT's potential in large-scale settings. Investigating its scalability to massive datasets and model sizes, along with its performance on diverse hardware platforms, will be crucial for widespread adoption in resource-constrained deep learning.