We introduce Adam, an algorithm for first-order gradient-based optimization of stochastic objective functions, based on adaptive estimates of lower-order moments. The method is straightforward to implement, is computationally efficient, has little memory requirements, is invariant to diagonal rescaling of the gradients, and is well suited for problems that are large in terms of data and/or parameters. The method is also appropriate for non-stationary objectives and problems with very noisy and/or sparse gradients. The hyper-parameters have intuitive interpretations and typically require little tuning. Some connections to related algorithms, on which Adam was inspired, are discussed. We also analyze the theoretical convergence properties of the algorithm and provide a regret bound on the convergence rate that is comparable to the best known results under the online convex optimization framework. Empirical results demonstrate that Adam works well in practice and compares favorably to other stochastic optimization methods. Finally, we discuss AdaMax, a variant of Adam based on the infinity norm.
Adam is a first-order gradient-based optimization algorithm designed for large-scale and high-dimensional problems.
It adapts learning rates using estimates of the first and second moments of gradients, enhancing its efficiency and versatility.
The paper provides a rigorous theoretical analysis, showing Adam's convergence properties and establishing a regret bound.
Empirical evidence supports Adam's effectiveness, demonstrating superior performance in various machine learning tasks compared to other optimizers.
The publication under review presents Adam, a first-order gradient-based optimization algorithm specifically designed to handle large-scale problems involving high-dimensional parameter spaces and substantial amounts of data. The novelty of Adam lies in its adaptive moment estimation, which calculates individual learning rates for different parameters through estimates of the first and second moments of the gradients. With properties such as computational efficiency, low memory requirements, resistance to the diagonal rescaling of gradients, and suitability for non-stationary objectives, Adam is positioned as a versatile solution for an array of machine learning challenges.
Adam displays its robust design by integrating features from two eminent algorithms: AdaGrad and RMSProp. The AdaGrad component efficiently deals with sparse gradients, while RMSProp contributes to handling non-stationary objectives. One of Adam's key characteristics is the update rule's invariance to gradient rescaling, a bounded step size governed by the selected hyperparameter, and an inherent mechanism that performs a form of step size annealing, all of which contribute to its stability.
At its core, Adam maintains exponential moving averages of both the gradient and its square, which are ambitious estimates of the first (mean) and uncentered variance, respectively. Nevertheless, these estimates initially suffer from bias towards zero, particularly when the moving averages are initiated and decay rates are high. The paper introduces a method for bias correction, proving essential and noteworthy in the algorithm's overall performance effectiveness.
The authors conduct a thorough theoretical analysis, demonstrating Adam's convergence properties and establishing a regret bound. Under specific conditions, Adam achieves a regret of O(√T), aligning with the finest known results for this type of convex online learning problems. Notably, these achievements extend to scenarios with sparse gradients, when the regret can be even more tightly bounded.
Empirical assessments across different models and datasets corroborate the theoretical findings. In logistic regression tasks and multi-layer neural networks, including those with dropout regularization, Adam consistently shows comparable or superior performance against competing optimization methods. Additionally, in convolutional neural networks (CNNs), Adam adapts learning rates for different layers, providing a compelling alternative to manually fine-tuning learning rates as required by methods like SGD.
The strength of Adam is exemplified in both its theoretical foundation and practical applicability. This optimization tool addresses several challenges in the stochastic optimization landscape. By effectively handling sparse gradients, adapting to non-stationary objectives, and adjusting learning rates individually across high-dimensional parameters, Adam proves to be robust and well-suited for various non-convex optimization issues prevalent in machine learning today.