Finite-Time Nonsmooth Convergence of Signum and Muon

Can momentum by itself repair sign-based optimization when the objective is nonsmooth?

A short overview. Full technical note: Finite-Time Nonsmooth Convergence of Signum and Muon.

Why This Question Matters

Sign-based optimizers replace each gradient coordinate by its sign. SignSGD made this idea a basic primitive for compressed nonconvex optimization [Bernstein et al., 2018]. The same sign map is attractive for communication-efficient distributed learning and is naturally robust to low-precision training, because it discards magnitude information and keeps only direction at the coordinate level [Bernstein et al., 2019; Gupta et al., 2015; Yu et al., 2026a]. This sign viewpoint also helps explain the behavior of Adam-like methods [Kingma and Ba, 2014; Balles and Hennig, 2018; Kunstner et al., 2023].

Momentum is a natural candidate for fixing the bias of sign updates. In smooth problems, momentum can stabilize sign directions and improve SignSGD-type guarantees [Sun et al., 2023; Jiang et al., 2025; Yu et al., 2026b; Tao et al., 2026]. The nonsmooth setting is harsher. ReLU activations, max operations, routing layers, clipping, and quantization introduce kinks where smooth descent inequalities disappear [Nair and Hinton, 2010; Shazeer et al., 2017; Gupta et al., 2015]. Even deterministic SignSGD can fail on simple convex Lipschitz objectives [Karimireddy et al., 2019]. Existing finite-time remedies use mechanisms beyond plain momentum, such as error feedback or structural stochasticity [Karimireddy et al., 2019; Yu et al., 2026a]. In other words, they repair sign descent by adding algorithmic components that are not present in bare Signum.

Muon makes the question more timely. It is a matrix sign optimizer for neural-network hidden layers [Jordan et al., 2024], and the polar factor of a diagonal matrix reduces to the entrywise sign. As a result, diagonal Signum counterexamples can often be lifted to diagonal Muon dynamics [Parshakova et al., 2026].

Can momentum by itself restore finite-time nonsmooth convergence for sign-based methods?

The Setup

The central object is Signum with a Euclidean projection onto a closed convex domain \(\mathcal X\):

\[ m_{t+1}=\beta_t m_t+(1-\beta_t)g_t,\qquad x_{t+1}=\Pi_{\mathcal X}\!\left(x_t-\eta_t\operatorname{sign}(m_{t+1})\right). \]

The unconstrained case is \(\mathcal X=\mathbb R^d\). This single update separates two sources of difficulty. In constrained problems, projection can create geometry-driven failures. In unconstrained problems, any failure must come from the objective and the sign-momentum recursion itself.

Muon uses the matrix sign, or polar factor, in place of the coordinatewise sign. If \(A=U\Sigma V^\top\), then \(\operatorname{Sign}(A)=UV^\top\). On a diagonal matrix \(A=\operatorname{Diag}(a_1,\ldots,a_d)\), this becomes

\[ \operatorname{Sign}(A)=\operatorname{Diag}(\operatorname{sign}(a_1),\ldots,\operatorname{sign}(a_d)). \]
Lemma: Diagonal Muon-to-SignSGD/Signum reduction.

If a Muon iterate, momentum matrix, and stochastic or deterministic oracle remain diagonal, then the diagonal entries follow the corresponding coordinatewise sign recursion. Without momentum this is the SignSGD recursion, and with momentum it is the Signum recursion. Therefore, any SignSGD or Signum counterexample on diagonal coordinates can be embedded as a diagonal Muon counterexample when the construction preserves diagonality.

In online convex optimization (OCO), an algorithm chooses \(x_t\in\mathcal X\), then observes a convex loss \(\ell_t\). The performance target is sublinear regret [Zinkevich, 2003]:

\[ \operatorname{Regret}_T =\sum_{t=0}^{T-1}\ell_t(x_t) -\min_{x\in\mathcal X}\sum_{t=0}^{T-1}\ell_t(x) =o(T). \]

For unconstrained nonsmooth stationarity, the measure is a Goldstein-type gap [Cutkosky et al., 2023]:

\[ G_\delta(x)= \operatorname{dist}\!\left( 0,\operatorname{conv}\bigcup_{y\in B_\infty(x,\delta)}\partial f(y) \right). \]

Main Results at a Glance

Finite-time status for Signum. Diagonal Muon inherits the listed counterexamples whenever the diagonal reduction applies.
Model Domain Schedule Status
Constrained OCO \(\mathcal X\subset\mathbb R^2\), compact convex Any \(\eta_t\), any \(\beta_t\in[0,1]\) Linear regret is possible.
Unconstrained deterministic \(\mathbb R^2\) Fixed \(\beta<1\) Finite-time failure examples exist.
Residual scale \(\mathbb R^2\) Fixed \(\beta<1\) The known diagonal obstruction shrinks like \(1-\beta\).
Adapted stochastic oracle \(\mathbb R^2\) Fixed \(\beta<1\) A conditionally unbiased oracle can freeze progress.
Coupled high momentum \(\mathbb R^d\) \(1-\beta_t=\Theta(\eta_t)\) Finite-time nonsmooth theory remains open.

Projection Can Defeat Any Schedule

The cleanest negative result is geometric. For every stepsize schedule \((\eta_t)\) and every momentum schedule \((\beta_t)\), one can construct a two-dimensional compact convex set \(\mathcal X\), a repeated linear loss, and an initialization such that projected Signum has linear regret.

Theorem: constrained OCO failure.

For any \(\eta_t\ge 0\), any \(\beta_t\in[0,1]\), and \(m_0=0\), there exist a compact full-dimensional \(\mathcal X\subset\mathbb R^2\), a repeated linear loss \(\ell_t\equiv\ell\), and an initialization \(x_0\in\mathcal X\) such that projected Signum has \(\operatorname{Regret}_T=\Omega(T)\). The same construction transfers to constrained diagonal Muon through the diagonal reduction above.

Projected Signum OCO counterexample A thin rectangle in z and w coordinates. Signum moves vertically and projection keeps z fixed, while the minimizer lies at a larger z coordinate. progress coordinate z thin coordinate w z = -1 z = 1 start x0 minimizer x* projected sign step loss wants larger z the z-coordinate stays fixed, so regret grows linearly
Figure 1. In the constrained OCO counterexample, it is clearest to look at the rotated coordinates (z,w). The sign update changes only the thin coordinate w. Projection clips it back into the feasible set and leaves the progress coordinate z stuck at its initial value.

The construction uses a rotated rectangle. The signed step points along a direction that, after Euclidean projection, changes only the wrong coordinate. The coordinate that would reduce the loss stays fixed, so regret accumulates linearly. This gives a complete negative answer for arbitrary compact constrained OCO domains, and the same construction transfers to constrained diagonal Muon.

Fixed Momentum Fails Without Projection

Projection is not the whole story. In the unconstrained exact-oracle model, fixed momentum still fails on convex Lipschitz functions. The basic family is

\[ f_{a,b,C}(u,v)=|a u+v|+b|u+C|. \]
Fixed beta deterministic Signum two-cycle Two points on the diagonal alternate across the kink line. The momentum sign chooses the diagonal direction back and forth, creating a period-two orbit. u v 0 diagonal trajectory kink au + v = 0 u = -C x+ x- Signum step constant stepsize momentum sign flips with the kink
Figure 2. For constant stepsize, the fixed-beta construction can force an exact two-cycle between the two diagonal points x+ and x-. The iterate crosses the kink au + v = 0, and the biased momentum keeps choosing the diagonal sign direction that sends the next point back across the kink.

Along a diagonal trajectory, one subgradient coordinate alternates with the nonsmooth kink while another coordinate carries a persistent bias. For any fixed \(\beta<1\), parameters can be chosen so that momentum keeps selecting the wrong signed direction.

Proposition: fixed-\(\beta\) deterministic failure.

Fix \(\beta\in[0,1)\). For suitable \(a,b,C\) in \(f_{a,b,C}\), Signum can be forced into a period-two orbit under constant stepsizes, can converge to a nonstationary point under summable stepsizes, and can converge to a nonstationary kink under strictly decreasing nonsummable stepsizes. In each case, for some \(\delta>0\), the Goldstein gap satisfies \(G_\delta(x_t)\ge c_\beta>0\) along the constructed trajectory.

This mechanism produces a period-two orbit for constant stepsizes, convergence to a nonstationary point for summable stepsizes, and convergence to a nonstationary kink for decreasing nonsummable stepsizes. The same exact-gradient mechanism also yields a fixed-coefficient Lion obstruction in the standard parameter regime [Chen et al., 2023].

High Momentum Is Not Settled

The phrase residual scale refers to the size of the stationarity gap left by the known diagonal counterexample after the objective is normalized to have bounded Lipschitz constant. The deterministic diagonal obstruction is strong for each fixed \(\beta<1\), but it weakens as \(\beta\) approaches one. Under a fixed Lipschitz normalization, the local Goldstein gap generated by this construction can shrink like \(1-\beta\).

Proposition: residual scale of the diagonal obstruction.

The same two-cycle mechanism can keep Signum away from stationarity, but its normalized Goldstein lower bound is only of order \(\Omega(1-\beta)\). Thus the construction rules out every fixed \(\beta<1\), while leaving open schedules where \(\beta\) approaches one with the horizon or with the stepsize.

This is why fixed-\(\beta\) failures do not automatically rule out horizon-dependent choices such as

\[ \beta = 1-\Theta(1/T), \]

or coupled high-momentum schedules of the form

\[ 1-\beta_t=\Theta(\eta_t). \]

This distinction is important. The known diagonal examples show that fixed momentum is not enough, but they do not close the most plausible momentum-only route to finite-time nonsmooth convergence.

Stochasticity Can Make the Sign Problem Worse

A tempting hope is that stochastic gradients might average out the bias. Conditional unbiasedness alone is not enough.

For every fixed \(\beta<1\), there is an adapted conditionally unbiased stochastic oracle that freezes the true progress coordinate forever while maintaining a positive Goldstein gap. The oracle injects zero-mean noise in a direction that is flat for the objective, but the distribution is adapted so that the current signed momentum follows the flat direction instead of making progress.

Theorem: stochastic shield.

Let \(e=(1,1)\) be the true progress direction and \(h=(1,-1)\) be a flat direction. For an objective with local subgradient \(-ce\), construct stochastic gradients

\[ g_t=-ce+\xi_t h, \]

where the scalar noise \(\xi_t\) is adapted to the past and satisfies \(\mathbb E[\xi_t\mid\mathcal F_t]=0\). For every fixed \(\beta<1\), \(\xi_t\) can be chosen so that \(\langle e,x_t\rangle\) never changes, while \(G_\delta(x_t)\) stays bounded away from zero.

The key pitfall is that

\[ \mathbb E[\zeta_t\mid\mathcal F_t]=0 \]

does not imply that \(\zeta_t\) is harmless after applying \(\operatorname{sign}(m_{t+1})\), because \(m_{t+1}\) already contains the current noisy sample. Here \(\zeta_t=\xi_t h\) is the zero-mean noise vector. In the shield construction, the current noise can be positively aligned with the signed update direction. The moment bound deteriorates like \((1-\beta)^{-1}\), so this does not settle i.i.d. data models or uniformly bounded high-momentum regimes.

What Remains Open

The most important remaining regime is coupled high momentum:

\[ \beta_t=1-\tau\eta_t. \]

This keeps the spatial memory length roughly fixed while sending the fixed-\(\beta\) residual scale to zero. Existing asymptotic stability results suggest that related inertial stochastic subgradient methods can converge under additional assumptions [Xiao et al., 2023], but a finite-time Goldstein guarantee for Signum is still missing.

Several sharper questions remain:

Takeaway

Momentum alone does not universally fix nonsmooth sign descent. Constrained OCO has a clean projection-based obstruction. Unconstrained fixed-momentum Signum has deterministic convex counterexamples. Adapted stochastic oracles can freeze progress even when they are conditionally unbiased.

What survives is a narrower and more interesting frontier: coupled high momentum, stronger stochastic assumptions, and a sharper understanding of when diagonal Signum pathologies become Muon pathologies.

References

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BibTeX

@misc{yu2026finite,
    title = {Finite-Time Nonsmooth Convergence of Signum and Muon},
    author = {Yu, Dingzhi and Liu, Yuxing and Pan, Rui and Zhang, Lijun and You, Yang and Zou, Difan},
    year = {2026},
    url = {https://www.lamda.nju.edu.cn/yudz/pdf/Finite-Time_Nonsmooth_Convergence_of_Signum_and_Muon.pdf}
}