Fokker-Planck Analysis and Invariant Laws for a Continuous-Time Stochastic Model of Adam-Type Dynamics

Abstract: We develop a continuous-time model for the long-term dynamics of adaptive stochastic optimization, focusing on bias-corrected Adam-type methods. Starting from a finite-sum setting, we identify a canonical scaling of learning rates, decay parameters, and gradient noise that yields a coupled, time-inhomogeneous stochastic differential equation for the parameters $x_t$, first-moment tracker $z_t$, and second-moment tracker $y_t$. Bias correction persists via explicit time-dependent coefficients, and the dynamics becomes asymptotically time-homogeneous. We analyze the associated Fokker-Planck equation and, under mild regularity and dissipativity assumptions on $f$, prove existence and uniqueness of invariant measures. Noise propagation is governed by $A(x)=\mathrm{Diag}(\nabla f(x))H_f(x)$. Hypoellipticity may fail on $\mathcal D_A\times\mathbb R^m\times(\mathbb R_+)^m$, where [ \mathcal D_A={x\in\mathbb R^m:\exists j,\ e_j^\top A(x)=0}\subset{x:\det A(x)=0}=\mathcal D_A^\dagger, ] and critical points of $f$ lie in $\mathcal D_A$. We show $\mathcal D_A^\dagger\neq\mathbb R^m$ and use this to prove exponential convergence of the Markov semigroup $μ0P_t$ to a unique invariant measure, uniformly in $μ_0$. The proof uses a Harris-type argument, minorization on Lyapunov sublevel sets, control constructions, and hypoellipticity on $(\mathbb R^m\setminus\mathcal D_A)\times\mathbb R^m\times(\mathbb R+)^m$. This provides a transparent continuous-time view of Adam-type dynamics.

• The paper derives a continuous-time SDE capturing bias correction, momentum, and adaptive learning rates in Adam-type optimization.
• It establishes unique invariant measures and quantifies exponential convergence through a detailed Fokker-Planck framework.
• The study develops explicit Lyapunov functions and minorization techniques to control noise propagation and address degeneracies in high dimensions.

Fokker-Planck Analysis and Invariant Laws for Continuous-Time Adam-Type Dynamics

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Introduction and Problem Setting

The paper introduces a rigorous analytical framework for understanding Adam-type adaptive optimization algorithms using continuous-time stochastic analysis. Adam, as widely used in deep learning, incorporates adaptive moment estimates with bias correction and exhibits robust empirical performance but remains poorly understood theoretically, particularly in nonconvex and stochastic regimes.

The analysis begins with the finite-sum stochastic optimization problem

$$\min_{x \in \mathbb{R}^m} f(x) = \frac{1}{n} \sum_{i=1}^n f_i(x)$$

with nonconvex, smooth $f$, in settings where computing $\nabla f(x)$ exactly is infeasible. The adaptive mechanisms of Adam-type algorithms—involving EWMA-based preconditioning and momentum—are recognized and encoded in a discrete-time dynamical system. The aim is to rigorously derive the continuous-time effective stochastic dynamics, analyze the corresponding Fokker-Planck equation, and establish the existence/uniqueness of invariant measures describing long-time statistical behavior.

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From Discrete-Time Adam to Continuous-Time SDEs

The paper carefully identifies the correct scaling limits for the algorithm parameters—learning rate, exponential decay rates, and gradient noise—to facilitate a non-trivial and mathematically tractable continuous-time model. The canonical scalings are

• $\eta = \gamma h$
• $\alpha = 1 - a h$
• $\beta = 1 - b h$ where $h \to 0$ is the time step and $a, b, \gamma > 0$ are fixed. The gradient noise is scaled as $\xi_k = \frac{\sigma}{\sqrt{h}} \zeta_k$ with standard Gaussian innovations, leading to macroscopic stochasticity in the moment tracker.

To avoid ill-posedness in the stochastic term for the adaptive variance, the authors introduce an effective closure: the coordinate-wise variance update incorporates the mean-square effect of the noise without explicit fast fluctuations, yielding

$$\lvert \partial_{x_i} f(x_k) + \xi_k^i \rvert^2 \approx (\partial_{x_i} f(x_k))^2 + \sigma^2$$

This closure is crucial for the derivation of well-posed SDE limits representative of Adam-type methods operating under realistic, batch-induced noise.

The resulting continuous-time process is a nontrivial time-inhomogeneous SDE for $f$0:

• $f$1 (parameters)
• $f$2 (bias-corrected momentum)
• $f$3 (adaptive coordinate-wise second moment)

$f$4

where $f$5 and $f$6 encode the transient bias-correction dynamics.

A rigorous weak convergence proof is established: the discrete iterates under the considered scaling converge in law (on compact intervals bounded away from $f$7) to the unique strong solution of the above SDE system. The singularity at $f$8 is explained as an inevitable consequence of bias-correction transient dynamics.

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Long-Time Dynamics and Invariant Measures

Upon letting $f$9, the bias-correction coefficients stabilize $\nabla f(x)$0 and the system becomes time-homogeneous: $\nabla f(x)$1 The primary analytical focus is then on the long-term statistical equilibrium characterized by invariant measures.

The existence, uniqueness, and exponential convergence to a unique invariant measure $\nabla f(x)$2 on $\nabla f(x)$3 are rigorously established under global smoothness and dissipativity assumptions on $\nabla f(x)$4. The approach involves:

• Construction of suitable Lyapunov functions, yielding geometric ergodicity.
• Quantitative minorization on compact sets, realized via explicit control-theoretic arguments.
• Careful analysis of the generator's hypoelliptic structure, with the stochasticity entering only the $\nabla f(x)$5 (momentum) variables, and drift terms coupling all variables.

Critically, noise propagation depends on the matrix

$\nabla f(x)$6

where $\nabla f(x)$7 is the Hessian. Hypoellipticity may fail along the set $\nabla f(x)$8, essentially corresponding to points where some row of $\nabla f(x)$9 vanishes—the set containing all coordinate-wise critical points. The argument overcomes the absence of global hypoellipticity by constructing skeleton (controlled ODE) paths into strictly hypoelliptic regions, leveraging the structure of $\eta = \gamma h$0 and the objective's geometry.

Ergodicity holds under the mild topological condition $\eta = \gamma h$1, with $\eta = \gamma h$2. This ensures that there exists an open set of parameters where all degrees of freedom are coupled to the noise, facilitating exponential mixing for the Markov semigroup.

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Fokker-Planck Equation, Regularity, and Support

The infinitesimal generator $\eta = \gamma h$3 and the associated Fokker-Planck (forward Kolmogorov) equation for densities $\eta = \gamma h$4 are derived, exhibiting a hypoelliptic diffusive structure. The presence of degenerate diffusion (only in $\eta = \gamma h$5), with all other variables being coupled via the drift, induces the need for careful hypoelliptic and control-theoretic analysis. Calculations reveal that the bracket-generating property depends on the invertibility and nondegeneracy of $\eta = \gamma h$6; thus, classical Hörmander theory applies locally.

The downstream effect is that, while densities of the process exist and are smooth off the degenerate set, invariant distributions do not generally admit a closed-form (no Gibbs measure structure). They reflect complicated correlations induced by the adaptive and bias-corrected dynamics.

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Numerical and Theoretical Implications

The existence of a unique exponentially attractive invariant measure $\eta = \gamma h$7 gives a precise description of long-run statistical behavior for Adam-type methods, including in nonconvex and high-dimensional regimes, going beyond the existing deterministic or convex analyses.

• Theoretically, this clarifies the dissipative mechanism—nonconvexity and critical point geometry only matter via the structure of $\eta = \gamma h$8. The precise propagation of noise plays a determining role in stationary exploration of parameter space, influencing escape from saddle points and "flat" regions in the loss landscape.
• Practically, it justifies modeling Adam-type training dynamics using continuous-time SDEs to understand steady-state behaviors such as sample path fluctuations, updates statistics, and adaptive learning rates.
• The analysis opens the door to conditional-Gaussian or Hermite-Galerkin expansion approaches for efficient approximation of the invariant law, especially given the Ornstein-Uhlenbeck-like structure for the fast variable $\eta = \gamma h$9.

Notably, the framework highlights that the effectiveness of Adam-type methods is formally justified only in parameter regimes where sufficient noise propagation is ensured, and that the bias-correction mechanism modulates the transient but not the asymptotic regime.

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Conclusion

This work delivers the first comprehensive stochastic and PDE-based analysis of Adam-type dynamics in continuous time, extending the ergodic theory for stochastic gradient-based algorithms to modern adaptive optimizers. It establishes strong notions of ergodicity under weak assumptions via complex hypoelliptic structure analysis, control-theoretic arguments, and carefully constructed Lyapunov function techniques. The approach paves the way for finer-grained, non-asymptotic analysis of stochastic adaptive optimization procedures and offers pathways for modeling heavy-tailed and structured noise, nonlinear loss landscapes, and further algorithmic variants.

Reference:

"Fokker-Planck Analysis and Invariant Laws for a Continuous-Time Stochastic Model of Adam-Type Dynamics" (2604.00840)