Lyapunov Constructions for System Interconnections Arising from Adaptation in Some Optimization Methods

Abstract: Interconnected systems have been widely studied, with a focus on interconnected systems whose subsystems are solely input-to-state stable (ISS) or passive systems. The focus of this work is on the interconnected systems that appear in adaptive gradient methods. In adaptive gradient methods, one subsystem seeks to move parameters of a cost function towards a minimizer while the other subsystem works to estimate some derivative information about the cost function to help determine the direction of the parameter update. This work studies instances of such interconnected systems and gives various Lyapunov function constructions for them using different techniques. In doing so, adaptive gradient optimizers are proven to be globally asymptotically stable (GAS), and the methods for constructing the Lyapunov functions that certify this are presented.

• The paper proves global asymptotic stability for adaptive gradient interconnections under assumptions including a unique minimizer, a single critical point, radial unboundedness, and continuous derivative estimates.
• It develops three Lyapunov constructions—a cost-referenced max function, a scalar error-coordinate function, and an integral-transform function—to provide explicit stability certificates beyond generic small-gain results.
• The analysis applies to continuous-time RMSProp and AdaHessian, while identifying limits involving nonconvex objectives, bounded adaptive-gain integrals, discontinuous derivative bounds, and extensions to stochastic or momentum-based methods.

Problem setting and motivation

Adaptive gradient optimizers such as RMSprop and AdaHessian can be written in continuous time as an interconnection of two subsystems: a parameter-update subsystem $\dot{\vartheta} = -K(\varphi)\nabla J(\vartheta)$ driven by a positive definite adaptive gain $K(\varphi)$, and an auxiliary low-pass filter $$\dot{\varphi} = \omega_\ell(q(\vartheta) - \varphi)$$ that estimates derivative information of the cost $J$ (typically squared gradient or squared Hessian diagonal entries). Both subsystems are ISS, so the optimizer is an ISS-ISS interconnection whose stability is a property of the coupling rather than either subsystem alone. While small-gain theorems and max-/sum-type Lyapunov constructions from the general interconnected-systems literature apply here (2608.16851), the ISS gains are rarely available in closed form, and the resulting Lyapunov functions are frequently non-elementary. The paper's contribution is to exploit problem structure—properties of $J$ itself—to obtain GAS certificates with simpler, more explicit Lyapunov functions.

Under four assumptions (unique minimizer at the origin, single critical point, radial unboundedness of $J$ via $$\alpha_{\vartheta,1} \leq J \leq \alpha_{\vartheta,2}$$, and continuity of $q$), the auxiliary subsystem is ISS with gain $\rho_q$, while the parameter subsystem satisfies $$\dot{J} \leq -W(\vartheta,\varphi) = -\lambda_{\min}(K(\varphi))\lvert\nabla J\rvert^2$$; notably, its ISS gain can be taken arbitrarily small because the dissipation is expressed through $K(\varphi)$0 rather than $K(\varphi)$1. This arbitrariness already yields GAS by small-gain composition, but the authors emphasize that a Lyapunov function—not merely the stability conclusion—is needed for forward-invariant sublevel sets and $K(\varphi)$2 bounds supporting robustness and parameter-selection analyses.

Baseline construction: max-type Lyapunov function

Theorem 1 establishes that system under Assumptions 1–4 is GAS, certified by

$K(\varphi)$3

The key structural insight is that the switching bound argument uses $K(\varphi)$4 rather than $K(\varphi)$5; since $K(\varphi)$6 decreases strictly along trajectories, the auxiliary state's admissible envelope shrinks automatically. Unlike the max-formulations of Jiang–Mareels–Wang, which switch between subsystem Lyapunov functions, this max activates $K(\varphi)$7 only above a cost-dependent threshold. Because $K(\varphi)$8 is nonsmooth on the switching surface, the proof proceeds via upper-right Dini derivatives in three regions, obtaining $K(\varphi)$9 where $$\dot{\varphi} = \omega_\ell(q(\vartheta) - \varphi)$$0 is positive definite but only lower semicontinuous—which suffices for uniform asymptotic stability via a comparison-lemma argument with explicit settling-time bound $$\dot{\varphi} = \omega_\ell(q(\vartheta) - \varphi)$$1. The scalar example's $$\dot{\varphi} = \omega_\ell(q(\vartheta) - \varphi)$$2 instantiates this construction.

Two alternative constructions under additional structure

For the scalar case ($$\dot{\varphi} = \omega_\ell(q(\vartheta) - \varphi)$$3, $$\dot{\varphi} = \omega_\ell(q(\vartheta) - \varphi)$$4), Proposition 1 gives a smooth Lyapunov function in error coordinates $$\dot{\varphi} = \omega_\ell(q(\vartheta) - \varphi)$$5:

$$\dot{\varphi} = \omega_\ell(q(\vartheta) - \varphi)$$6

The sign-controlled integral term compensates the cross-coupling $$\dot{\varphi} = \omega_\ell(q(\vartheta) - \varphi)$$7 in the error dynamics, exploiting $$\dot{\varphi} = \omega_\ell(q(\vartheta) - \varphi)$$8 from Assumptions 1–2 and the bound $$\dot{\varphi} = \omega_\ell(q(\vartheta) - \varphi)$$9. This differs from value-domain scaling à la Ito: it operates in error coordinates rather than scaling a supplied Lyapunov value.

For the multivariable case, Proposition 2 requires a lower bound $J$0 and constructs

$J$1

with $J$2 encoding growth of the gradient along level sets. The scalings inside these integral transforms are selected directly from $J$3 and its derivative growth, not from subsystem supply rates as in Ito's state-dependent scaling framework. The resulting derivative bound is negative definite provided the antiderivative of $J$4 diverges—a genuine restriction discussed below.

Application to RMSProp and AdaHessian

Both RMSprop and AdaHessian fit the abstraction ($J$5 diagonal with entries $J$6; $J$7 equal to squared gradients or squared Hessian diagonals, respectively), yielding corollaries that each optimizer acting on any $J$8 satisfying Assumptions 1–3 is GAS at the origin. For AdaHessian on a quadratic cost, all three constructions apply; the naive sum-of-squares works there, but the paper notes that $J$9 does not generalize to arbitrary $J$0, whereas the systematic constructions do.

The integral-transform approach has a demonstrated limitation: for the system $J$1, $J$2 satisfies the eigenvalue bound but has bounded integral, so no radially unbounded $J$3 exists—even though the max- and scalar-error constructions still certify GAS via negative definite $J$4 and $J$5. The authors concede this construction therefore fails to be fully generalizable across adaptive gradient systems, though they expect applicability "for most systems an analyst is studying."

Limitations and open questions

Several caveats are stated plainly in the paper. The assumptions exclude costs with multiple critical points or non-radially-unbounded $J$6, so saddle-point-rich landscapes typical in neural network training fall outside the analysis. The settling-time characterization relies on $J$7 being positive definite yet discontinuous, giving qualitative (not exponential) convergence guarantees. The integral construction requires both the pointwise bound on $J$8 and divergence of $J$9; moreover, closed-form expressions for the antiderivative may not exist even when the integral diverges (e.g., for $J$0), limiting usability of level sets. Whether the constructions extend to vector-valued error coordinates, momentum-augmented variants (excluded here for AdaHessian), or stochastic discretizations remains open.

Conclusion

This work proves GAS for continuous-time adaptive gradient interconnections and supplies three Lyapunov constructions—max-type over a cost-referenced switching surface, scalar error-coordinate compensation, and sum-type integral transforms—whose scalings derive from properties of the cost function rather than subsystem dissipation rates. The constructions yield more elementary Lyapunov functions than generic interconnected-system tools, though their scope is bounded by the assumptions on $J$1 and by integrability conditions on the adaptive-gain lower bound.