Convergence Rates for the Adam Optimizer: When Adaptive Methods Follow Unexpected Trajectories

Introduction

The Adam optimizer, introduced by Kingma and Ba in 2014, has become the default workhorse for training deep neural networks across virtually every domain of artificial intelligence. Its combination of momentum acceleration and adaptive learning rates has proven empirically superior to vanilla stochastic gradient descent in countless applications. Yet, for a decade, a striking theoretical gap persisted: despite its ubiquity, no convergence proof existed for Adam even in the simplest nontrivial setting of strongly convex quadratic optimization problems. This is not merely a technical omission; it reflects the fundamental complexity introduced by adaptive moment estimation.

In their recent work, "Convergence rates for the Adam optimizer" (arXiv:2407.21078v1), Steffen Dereich and Arnulf Jentzen resolve this open problem by establishing optimal convergence rates for Adam on a broad class of stochastic optimization problems. Their analysis reveals something far more nuanced than a standard optimization guarantee. Rather than converging to critical points of the objective function (the zeros of the gradient), Adam converges to zeros of a distinct Adam vector field, an object that captures the macroscopic behavior of the algorithm but differs fundamentally from the negative gradient direction. This finding challenges conventional intuition about adaptive optimization and provides the first rigorous mathematical foundation for understanding where Adam actually leads us.

The Adam Vector Field and Non-Convergence to Critical Points

The central conceptual contribution of Dereich and Jentzen's analysis is the introduction of the Adam vector field, a vector-valued function that accurately describes the limiting behavior of the Adam iteration. In classical stochastic gradient descent, the optimization process follows (in expectation) the negative gradient of the objective function, and under standard assumptions, converges to critical points where the gradient vanishes. One might intuitively expect Adam, as a variant of SGD, to exhibit similar behavior, perhaps with modified convergence rates due to its adaptive nature.

However, the authors prove that this intuition is incorrect. The Adam optimizer typically does not converge to zeros of the gradient of the objective function. Instead, it converges to zeros of the Adam vector field, which constitutes a different set of points in parameter space. This distinction is not merely academic; it implies that Adam exhibits an intrinsic bias that persists even in the limit of infinite training time. The algorithm solves a different problem than the one originally posed, converging to a point that minimizes not the original objective, but rather a modified objective implicitly defined by the Adam vector field.

This phenomenon arises from the interaction between the first moment estimate (momentum) and the second moment estimate (adaptive scaling) in Adam's update rules. The exponential moving averages of gradients and squared gradients create a dynamical system whose fixed points differ from the stationary points of the underlying loss landscape. The authors formalize this by viewing the Adam iteration as a non-expansive mapping on an appropriately defined Adam sequence space, allowing them to characterize the limiting behavior through the zeros of this newly defined vector field.

Quantitative Convergence Rates and the Bias-Variance Tradeoff

Dereich and Jentzen establish their results through a careful analysis of quadratic stochastic optimization problems, where the objective takes the form of minimizing $\mathbb{E}[\ell(\theta, U)]$ with $\ell$ being the standard mean squared error loss $\ell(\theta, u) = |\theta - u|^2$. Even in this strongly convex setting, which is trivial for standard SGD, the analysis requires sophisticated machinery involving ordinary differential equation approximations and perturbation analysis.

Their main result, Theorem 1.1, establishes three key properties for the Adam optimizer with damping parameters $\alpha, \beta \in [0,1)$ satisfying $\alpha < \sqrt{\beta}$, step sizes $\gamma_n$ decreasing to zero with $(\gamma_n - \gamma_{n+1})/\gamma_n^2 \to 0$, and mini-batch size $M$:

1. Almost sure convergence: The Adam iterates $\theta_n$ converge almost surely to a random limit point $\vartheta_M$ as $n \to \infty$.

2. Optimal convergence rate: The $L^p$ distance between the iterates and the limit satisfies $\mathbb{E}[|\theta_n - \vartheta_M|^p]^{1/p} \leq \eta \sqrt{\gamma_n}$ for some constant $\eta$. This $O(\sqrt{\gamma_n})$ rate is optimal for stochastic approximation methods, matching the rates achievable by standard SGD.

3. Bias characterization: The distance between the random limit point $\vartheta_M$ and the true global minimizer $\mathbb{E}[U(1)]$ satisfies $|\vartheta_M - \mathbb{E}[U(1)]| \leq \eta M^{-1}$.

This decomposition reveals a fundamental bias-variance structure in Adam's convergence. The step size $\gamma_n$ controls the variance of the optimization process (how quickly the iterates settle), while the mini-batch size $M$ controls the bias (how far the settled point is from the true optimum). Notably, the bias decays inversely with the mini-batch size, suggesting that practitioners using Adam with small batches may converge to points systematically offset from the true minimizer, regardless of how long they train.

Technical Innovations and Methodological Framework

The authors' proof strategy represents a significant methodological advance for analyzing adaptive optimization methods. Rather than treating Adam as a perturbed gradient flow, they develop an ODE based error analysis that views the Adam dynamics through the lens of the Adam vector field. This approach allows them to handle the nonlinearity introduced by the adaptive learning rates, which depend on the history of squared gradients through the second moment estimate.

A key technical condition is the requirement that $\alpha < \sqrt{\beta}$, which ensures that the momentum term does not dominate the adaptive scaling term excessively. This inequality, often satisfied by default hyperparameter choices ($\beta_1 = 0.9, \beta_2 = 0.999$ in standard notation), ensures the stability of the Adam vector field. The step size condition, $\lim_{n\to\infty} (\gamma_n - \gamma_{n+1})/\gamma_n^2 = 0$, is satisfied by standard decay schedules such as $\gamma_n = c/n$ or $\gamma_n = c/\sqrt{n}$, and is necessary to prevent the learning rate from decaying too abruptly.

The authors also introduce the concept of the Adam sequence space, a functional analytic framework in which the Adam iteration becomes a non-expansive mapping. This perspective allows them to apply fixed-point theory and contraction mapping arguments to establish the existence and uniqueness of limits, providing tools that extend beyond Adam to other adaptive methods such as RMSprop, Adagrad, and Adadelta.

Your Take: Implications for Theory and Practice

Beyond the technical achievement of proving convergence rates, this work carries significant implications for how we understand adaptive optimization. The demonstration that Adam converges to zeros of the Adam vector field rather than true critical points suggests that practitioners have been solving implicitly regularized problems all along. The "Adam vector field" acts as a modification of the original optimization landscape, potentially explaining why Adam often finds different solutions than SGD in overparameterized neural networks, with different generalization properties.

The bias term of order $M^{-1}$ is particularly noteworthy. In modern deep learning, there is a trend toward using smaller batch sizes for better generalization, yet this analysis suggests that small batches introduce a systematic bias in the adaptive optimization trajectory that does not vanish with increased training time. Only by increasing the mini-batch size can one reduce the distance between the Adam limit point and the true optimum. This creates a tension between statistical efficiency (small batches) and optimization accuracy (convergence to the correct point), a tradeoff that is quantified here for the first time.

Furthermore, the framework developed by Dereich and Jentzen opens the door to analyzing other adaptive methods through their respective vector fields. One might conjecture that RMSprop converges to its own vector field zeros, and that the differences between these algorithms can be characterized by the geometries of their respective vector fields. This could lead to a taxonomy of adaptive optimizers based not on their algorithmic descriptions, but on the implicit biases they introduce into the optimization process.

Limitations and Open Questions

The analysis is currently limited to strongly convex quadratic problems, a setting far removed from the non-convex, high-dimensional loss landscapes of deep neural networks. Whether the Adam vector field concept extends to general non-convex optimization remains an open question. In particular, the behavior of Adam near saddle points and the characterization of the Adam vector field in the presence of flat minima or sharp minima in deep networks require further investigation.

Additionally, the paper focuses on the standard Adam formulation with constant hyperparameters $\alpha$ and $\beta$. In practice, practitioners often use warmup schedules, learning rate annealing, or decoupled weight decay (AdamW). How these modifications alter the Adam vector field and whether they mitigate or exacerbate the bias toward non-critical points remains to be analyzed.

Conclusion

Dereich and Jentzen's work represents a foundational step in the mathematical understanding of adaptive optimization. By establishing that the Adam optimizer converges to zeros of the Adam vector field rather than critical points of the objective, they provide a new lens through which to view the algorithm's empirical behavior. The optimal convergence rates they establish, combined with the explicit characterization of the bias introduced by finite mini-batch sizes, offer practical guidance for hyperparameter selection.

The most profound contribution may be the conceptual shift from viewing Adam as an approximate gradient descent to understanding it as a distinct dynamical system with its own geometric structure. As the field moves toward more complex optimizers and training regimes, the framework of optimizer-specific vector fields may become essential for predicting and controlling the behavior of adaptive algorithms. Future work extending these results to non-convex settings and characterizing the generalization properties of Adam's implicit bias promises to bridge the gap between the theoretical guarantees established here and the empirical realities of deep learning practice.