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Input-Output Stability of Gradient Descent: A Discrete-Time Passivity-Based Approach

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Pith's one-line read Passivity-based proof shows gradient descent converges for step sizes below 2/L.

desk verdict A clean passivity-based proof of the classical GD step-size bound, with a real well-posedness gap in the loop-transformed nonlinearity that a revision should fix. read the letter →

arxiv 2411.16074 v1 pith:I6SAD2X7 submitted 2024-11-25 math.OC cs.SYeess.SY

Sepehr Moalemi , James Richard Forbes This is my paper · ORCID
classification math.OCcs.SYeess.SY MSC 93D1093C5590C25
keywords gradientdescentpassivity-basedanalysisinput-outputstabilitysector-boundedverystrictlypassivelooptransformationgainschedulingglobalconvergence
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Gradient descent iterates as $x_{k+1}=x_k-\alpha\nabla f(x_k)$, and the paper proves that for continuously differentiable functions whose gradient lies in a sector between slopes $m$ and $L$ and that have a unique minimizer, the method is input-output stable and globally convergent for every step size $\alpha\in(0,2/L)$. The proof recasts the algorithm as a feedback system: a linear controller in negative feedback with the gradient nonlinearity, then applies a loop transformation so the controller becomes passive. Because the shifted gradient is very strictly passive for this function class, the strong passivity theorem yields square-summable signals, and those signal bounds imply $\nabla f(x_k)\to 0$ and $x_k\to x^*$. The result recovers the classical step-size bound for $L$-smooth functions and extends it to a class that may include nonconvex objectives with sector-bounded gradients.

What carries the argument

The central object is the loop-transformed feedback representation of gradient descent: an LTI controller with state $\xi_{k+1}=\xi_k+\alpha u_1^k$ and output $y_1^k=\xi_k$, in negative feedback with the static memoryless nonlinearity $\Delta$. Because the original controller is strictly proper and therefore cannot be passive, a loop transformation inserts a feedthrough term $D$, yielding a modified controller $\bar{G}_{\mathrm{GD}}$ and a modified nonlinearity $\bar{\Delta}$; this transformation preserves the closed-loop dynamics. The load-bearing identities are the sector inequality from (2), which makes $\Delta$ very strictly passive, and the discrete positive-real LMI in Lemma 1, which yields the passivity condition $D\ge\alpha/2$. The strong passivity theorem then converts the dissipated energy into $\ell_2$ signal bounds, and those bounds produce convergence.

What would settle it

Run gradient descent on the paper's nonconvex example $f(x)=\frac{L-m}{4}\left(\frac{L+m}{L-m}x^2+2\sin x-2x\cos x\right)$ with $m=1$, $L=100$, using step $\alpha=1.99/L$ from any initial condition; the theorem predicts convergence of $x_k$ to $x^*$, so observing $x_k$ stay bounded away from $x^*$ or diverge for a single initial condition would falsify the claim. Alternatively, directly test the very strict passivity inequality $\langle u_2,y_2\rangle_T\ge \delta\|u_2\|_{2T}^2+\varepsilon\|y_2\|_{2T}^2$ for $\Delta$ on a long finite horizon; a violation would refute Lemma 3.

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Extended reading notes

Core claim

The central claim is that gradient descent is a negative feedback interconnection between a modified linear time-invariant controller $\bar{G}_{\mathrm{GD}}$ with feedthrough $D=\alpha/2$ and the shifted gradient nonlinearity $\Delta: u_2\mapsto \nabla f(u_2+x^*)$. Lemma 2 shows $\bar{G}_{\mathrm{GD}}$ is passive if and only if $D\ge \alpha/2$. Lemma 3 shows that for $f\in S_{m,L}$, $\Delta$ is very strictly passive with $\delta=mL/(m+L)$ and $\varepsilon=1/(m+L)$. Lemma 4 shows the positive feedback interconnection of $\Delta$ with $D$ remains very strictly passive when $D<1/L$. Combining these with the strong passivity theorem gives $y_1+Du_1\in\ell_2$ and $y_2\in\ell_2$ for $\alpha\in(0,2/L)$; Section IV.A interprets these $\ell_2$ bounds as $\nabla f(x_k)\to 0$ and $x_k-D\nabla f(x_k)\to x^*$, hence $x_k\to x^*$. For $m<L$ and $\alpha=2/L$, $\bar{\Delta}$ is only input strictly passive, so the weak passivity theorem guarantees $y_1+Du_1\in\ell_2$ but not pointwise convergence; the paper proposes the stopping criterion $\|\nabla f(x_k)+\nabla f(x_{k-1})\|_2^2<\epsilon$. It then gain-schedules the passive controller to obtain a variable-step-size variant $x_{k+1}=x_k-s_k\nabla f(s_kx_k)$ and numerically tests it on a nonconvex sector-bounded example with $m=1$, $L=100$.

Load-bearing premise

The loop transformation shown in Figure 3 is assumed to preserve the closed-loop dynamics of the original gradient descent update while adding the feedthrough term $D$; if this equivalence failed, the passivity-based stability conclusions would not apply to the actual algorithm.

Editorial extensions

If this is right

  • For every $f\in S_{m,L}$, including nonconvex functions with sector-bounded gradients, gradient descent is globally convergent for all $\alpha\in(0,2/L)$ without assuming a separate Lipschitz-gradient condition.
  • For $m<L$, the boundary step size $\alpha=2/L$ remains input-output stable, so the practical stopping rule based on consecutive gradients $\nabla f(x_k)+\nabla f(x_{k-1})$ being small is justified even though the iterates may oscillate without converging.
  • The gain-scheduled variant $x_{k+1}=x_k-s_k\nabla f(s_kx_k)$ is input-output stable whenever $\max_k |s_k|\in(0,\sqrt{2/L})$; for constant $s$ it reduces to standard gradient descent with step size $s^2$.
  • The analysis gives a systematic recipe for other first-order algorithms: cast the algorithm as a feedback interconnection, solve the passivity LMI for the minimal feedthrough, compare with the sector bounds, and apply the weak or strong passivity theorem.
  • The passivity framework extends stability guarantees known for strongly convex $L$-smooth functions to the larger sector-bounded nonconvex class, matching the classical convergence region of gradient descent at the $2/L$ limit.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The explicit energy parameters $\delta=mL/(m+L)$ and $\varepsilon=1/(m+L)$ suggest that convergence-rate estimates for this nonconvex class could be derived from the passivity inequalities themselves, although the paper does not provide explicit rates.
  • The same loop-transformation argument could be applied to strictly proper momentum-type updates by replacing the scalar feedthrough $D$ with a matrix feedthrough and solving the analogous LMI; this is a natural extension the paper leaves implicit.
  • For the boundary case $\alpha=2/L$, the two-dimensional example suggests that Ces\`aro averages of the iterates might still converge to $x^*$; a testable conjecture is that averaged iterates converge even when pointwise convergence fails.
  • The time-varying scheduling function $s_k$ defines a genuinely new algorithm when $s_k$ is not constant, and its convergence behavior could be tuned further, for example by scheduling $s_k$ based on the local gradient magnitude.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 5 minor

Summary. The paper presents a discrete-time passivity-based analysis of the gradient descent (GD) method for functions whose gradients satisfy a sector bound (the class S_{m,L}, which includes nonconvex functions with a unique global minimizer). The GD update is recast as a negative feedback interconnection of an LTI controller G_GD and a static nonlinearity Delta (the shifted gradient). Since G_GD is strictly proper and hence not passive, a loop transformation introduces a feedthrough term D, yielding a modified controller \bar{G}_GD and a modified nonlinearity \bar{\Delta}. Lemma 2 shows \bar{G}_GD is passive iff D >= alpha/2; Lemma 3 shows Delta is very strictly passive (VSP) with constants epsilon=1/(m+L), delta=mL/(m+L); Lemmas 4 and 5 show \bar{\Delta} is VSP for D<1/L and input strictly passive (ISP) for D=1/L when m<L. The strong passivity theorem is then invoked to conclude that for alpha in (0,2/L) the loop signals y1+Du1 and y2 lie in l2, which is used to prove global convergence to the unique minimizer. The weak passivity theorem is invoked for alpha=2/L, yielding only a weaker stability conclusion; this is illustrated with Polyak's counterexample. Finally, a gain-scheduled GD variant with variable step size is proposed and tested numerically.

Significance. If the technical issue identified below is repaired, the paper provides a clean, self-contained passivity derivation of the classical step-size bound 2/L for a class of possibly nonconvex functions. The modular structure (controller passivity condition plus nonlinearity VSP property) is conceptually appealing and may be applicable to other first-order methods. The paper also gives a useful comparison with existing IQC and dissipativity approaches, and the proposed gain-scheduled variant is an interesting extension. The derivations in Lemmas 2-5 are correct as algebraic/inequality statements, and the numerical experiments are carefully done. The main concern is that the proof of the central theorem invokes the strong passivity theorem for an interconnection that is not shown to be well-posed as an operator from l2e to l2e.

major comments (1)
  1. The proof of Theorem 3 applies the strong passivity theorem (Theorem 2) to the feedback interconnection of \bar{G}_GD and \bar{\Delta}, but the paper does not establish that \bar{\Delta}, defined as the positive feedback interconnection of Delta and D1, is a single-valued operator from l2e to l2e. Lemma 4 only proves an algebraic inequality: whenever y=Delta(u) and \bar{u}=u-Dy, the inequality (15) holds. For f in S_{m,L}, the equation y=nabla f(\bar{u}+Dy+x*) can have multiple solutions. A scalar example is f(x)=x^2+(1-cos(x^2))/2, for which f'(x)=x(2+sin(x^2)) lies in the sector [1,3] with unique minimizer x*=0; taking D=0.3<1/3 and \bar{u}=1 gives at least two distinct solutions y. Thus \bar{\Delta} is set-valued, and Theorem 2, stated for operators, cannot be invoked as written. The closed-loop GD trajectory is well-posed because the D terms cancel in the specific interconnection, but this cancellation is not proved, and the proof of the main input-output stability claim therefore has a load-bearing gap. Please add a rigorous well-posedness argument for the closed-loop interconnection of Figure 3, or provide a direct proof of the l2 conclusions using the passivity inequalities (without relying on \bar{\Delta} being an operator).
minor comments (5)
  1. The statement says the step size satisfies alpha in (0,2/L), but the proof and the subsequent discussion in Section IV.A concern alpha=2/L. The statement should read alpha=2/L (or alpha in (0,2/L] with the endpoint resolved through the weak passivity theorem).
  2. There is a notation inconsistency in the claim 'y1+Du1 in l2'. The state-space realization of \bar{G}_GD is given with output y1 = C xi + D u1, so the modified controller output already includes the term D u1. Clarify whether 'y1' denotes the original output xi (in which case y1+Du1 is the modified output) or the modified controller output (in which case the '+Du1' is redundant).
  3. The equations shown inside the block for \bar{G}_GD in Figure 3 appear to be the original controller equations (y_k^1 = xi_k), which contradicts the stated realization (A,B,C,D)=(1,alpha,1,D1). Make the figure consistent with the text or clarify that the modified output is y_k^1 + D u_k^1.
  4. The implication from (20) and (21) to x_k -> x* is correct but terse; it would be clearer to write x_k = (x_k - D nabla f(x_k)) + D nabla f(x_k) and then take limits.
  5. The condition for the gain-scheduled method, max_{k in T} |s_k| in (0, sqrt(2/L)], is not fully derived. Specify whether the endpoint uses Lemma 5 (ISP) rather than Lemma 4, and state whether \bar{r}_2=0 is required, as in Theorem 3(ii).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: all stability conditions are derived from stated sector-bound assumptions and external passivity theorems, with no fitted parameters or self-citation chain.

full rationale

The derivation is self-contained. The paper assumes f ∈ S_{m,L}, meaning ∇f satisfies the sector-bound inequality (2), and then derives the very strict passivity of the shifted gradient ∆ in Lemma 3 by summing that inequality over a truncated interval. The controller passivity condition in Lemma 2 is obtained from the standard positive-real LMI with minimal realization (A,B,C,D) = (1, α1, 1, D1), giving D ≥ α/2. Lemma 4 combines these two algebraic inequalities, completing squares to obtain the explicit bound D < 1/L, and Theorem 3 simply intersects D ≥ α/2 with D < 1/L to reach α < 2/L. No parameter is fitted to the conclusion, and no result is renamed: the convergence step in Section IV.A derives x_k → x* from y2 ∈ ℓ2 and y1 + Du1 ∈ ℓ2, which are consequences of the strong passivity theorem applied to the transformed loop. The cited passivity theorems are external standard results, and the cited optimization results are prior independent work, not self-citations of the present authors. The only noteworthy concern is a possible well-posedness gap in Lemma 4 when ∇f is non-monotone, because the paper proves an algebraic passivity inequality for arΔ without proving that the implicit feedback equation defines a unique operator; that is a correctness risk, not a circularity, since the VSP inequality itself is not assumed as the conclusion. Accordingly, no circular step is identified.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The analysis rests on the sector-bound assumption defining S_{m,L}, the passivity theorems, and the loop-transformation equivalence that is stated but not formally proved. No free parameters are fitted to data; the step size and scheduling function are algorithm design choices, and the sector bounds m and L are inputs from the function class definition.

assumptions (4)
  • domain assumption The objective function f belongs to S_{m,L}, i.e., continuously differentiable, has a unique global minimizer x* with grad f(x*) = 0, and satisfies the sector-bound inequality (2) for all x.
    This inequality, taken from [9, Proposition 5], is the sole property of the objective used to prove Delta is VSP in Lemma 3. It defines the class of functions for which the results hold.
  • ad hoc to paper The loop transformation in Figure 3 preserves the closed-loop dynamics of the original GD method while making the modified controller Gbar have a feedthrough term D.
    The paper asserts this transformation without a formal proof; the equivalence is the structural step that connects the passivity analysis to the actual GD algorithm.
  • standard math Discrete-time passivity theorems (Theorem 1 weak, Theorem 2 strong) from Desoer-Vidyasagar [13] are valid for the systems considered.
    The paper uses these theorems to conclude l2 stability of the feedback interconnection; they are cited as known results.
  • standard math Discrete positive real lemma (Lemma 1) from Hitz-Anderson [14] characterizes passivity of LTI systems.
    Used in the proof of Lemma 2 to derive the condition D >= alpha/2 for the modified controller to be passive.

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Pith. "Pith review of Input-Output Stability of Gradient Descent: A Discrete-Time Passivity-Based Approach." pith.science (2026). https://pith.science/paper/I6SAD2X7

@misc{pith2026241116074,
  author       = {Pith},
  title        = {Pith review of: Input-Output Stability of Gradient Descent: A Discrete-Time Passivity-Based Approach},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I6SAD2X7}},
  note         = {Machine review of arXiv:2411.16074}
}
read the original abstract

This paper presents a discrete-time passivity-based analysis of the gradient descent method for a class of functions with sector-bounded gradients. Using a loop transformation, it is shown that the gradient descent method can be interpreted as a passive controller in negative feedback with a very strictly passive system. The passivity theorem is then used to guarantee input-output stability, as well as the global convergence, of the gradient descent method. Furthermore, provided that the lower and upper sector bounds are not equal, the input-output stability of the gradient descent method is guaranteed using the weak passivity theorem for a larger choice of step size. Finally, to demonstrate the utility of this passivity-based analysis, a new variation of the gradient descent method with variable step size is proposed by gain-scheduling the input and output of the gradient.

Figures

Figures reproduced from arXiv: 2411.16074 by the authors.

Figure 1
Figure 1. The negative feedback interconnection of two systems [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 4
Figure 4. A discrete-time system ∆¯ , composed of the positive feedback interconnection of a VSP system ∆, and the feedthrough term D1. From the Schur complement lemma, M ⪯ 0 if and only if αP − 1 = 0, α 2 P − 2D1 ⪯ 0. (7) From (7), it follows that P = 1 α and D ≥ α/2, for α ∈ R>0 . Finally, Lemma 1 in tandem with Remark 1 concludes that G¯ GD is passive if and only if D ≥ α/2, for α ∈ R>0 . Next, it is shown that for f ∈ Sm,… view at source ↗
Figure 3
Figure 3. Loop transformation of the negative feedback interconnection [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (4 more)
Figure 2
Figure 2. Figure 2: 2) Using the minimal realization of the controller, deter￾mine the lower bound on the feedthrough term D by solving the linear matrix inequality (LMI) in (1). 3) Determine the bounds on the algorithm’s step size(s) that guarantee the feedthrough term D is within the bo…
Figure 6
Figure 6. Figure 6: The function f ∈ Sm,L defined in (25) for m = 1 and L = 100. This function is non-convex but has a sector-bounded gradient and a unique global minimizer at x ∗ = 0. and only if D¯ ≥ [PITH_FULL_IMAGE:figures/full_fig_p005_6.png]
Figure 5
Figure 5. Figure 5: Two gain-scheduling architectures resulting in the same [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 7
Figure 7. Figure 7: Comparison of number of iterations needed to achieve [PITH_FULL_IMAGE:figures/full_fig_p006_7.png]

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