Your Non-Convex Control Never Left the Quadratic: A Resolution Ratio for Auditing Optimizer Validation Experiments

1. Motivation and provenance

This paper began as a referee check. While reviewing the Recensorium submission ap_ppr_ycewd54ym50324pbh5vz ("Constant-Step Adam Does Not Reach Stationarity"), I noticed that its Section 7 reported, for the non-convex objective f(x) = log cosh x at noise scale s = 1e-2, the values SGD 5.14e-7, RMSProp 6.39e-5, Adam 4.27e-5 — identical to three significant figures to the values it reported for the quadratic f(x) = x^2/2 at the same noise scale. My first hypothesis was duplicated numbers. I implemented log cosh independently (gradient tanh x, averaging tanh(x)^2) and ran it: the numbers are real and reproduce exactly. The identity is not an error. It is a consequence of the fact that the iterates never leave the region where log cosh is indistinguishable from a quadratic. Tracking the largest |x| visited during the averaging window gives 0.010 at s = 1e-2 and only 0.283 at s = 1.0, and tanh(x) agrees with x to a fraction of a percent throughout. The paper's stated justification for the choice — "whose curvature vanishes in the tails" — describes a region the run never visits. The control was vacuous, and three of the five prior reviews on that paper had credited it as evidence.

That paper is not unusual and this is not a complaint about it; its analytical content is correct and its code reproduces exactly, which is why the failure was detectable at all. The general pattern is: an author validates a claim on objective g, then reruns on a "harder" objective f and reports that the result persists. The rerun is evidence only to the extent that f and g differ where the optimizer actually goes. When they do not, agreement is guaranteed a priori and its report is uninformative. My purpose here is to turn that from a matter of referee intuition into a number an author can compute before running, and a referee can demand after.

Everything below is either proved or measured by code included in Section 9. I have run no experiment other than the simulations described, and I claim no result on real networks or in high dimension; Section 8 states what is and is not established.

2. Setting

I work in the one-dimensional constant-step setting used by the motivating paper, because it is the minimal setting in which the phenomenon appears and it makes the diagnostic checkable end to end. Let g be a baseline objective and f a perturbed one. The stochastic oracle returns the gradient plus Rademacher noise xi_t = +/- s with probability 1/2, independent across t. The two optimizers are

SGD: x_{t+1} = x_t - eta (grad(x_t) + xi_t) RMSProp: v_t = beta2 v_{t-1} + (1-beta2)(grad(x_t)+xi_t)^2, vhat_t = v_t/(1-beta2^t), x_{t+1} = x_t - eta (grad(x_t)+xi_t)/(sqrt(vhat_t) + eps)

with eta = 0.01, beta2 = 0.999, eps = 1e-8, x_1 = 0.5, T = 2e5 steps, and the first quarter discarded as burn-in. The measured quantity is the stationarity floor, the time-averaged expected squared gradient Floor(f) = (1/|W|) sum_{t in W} f'(x_t)^2 over the averaging window W. This is the quantity the motivating paper reports and it is the quantity a "does the effect persist on a harder objective" control compares.

Two derived quantities matter. The probed radius R is the largest |x| attained during W (I also report the 0.999 quantile, which is within 25% of it in every run). The perturbation is Delta(x) = f'(x) - g'(x).

3. A coupling bound: why the trajectories are pinned together

The reason the log-cosh control returned the quadratic's answer is not a coincidence of averaging. The whole trajectory is pinned.

Theorem 1 (synchronous coupling). Let g' be L-smooth and mu-strongly monotone on an interval I (that is, mu <= g'' <= L on I), and let sup_{x in I} |f'(x) - g'(x)| <= delta. Run the two SGD recursions driven by the same noise realisation, x_{t+1} = x_t - eta(f'(x_t) + xi_t), y_{t+1} = y_t - eta(g'(y_t) + xi_t), from a common start x_1 = y_1, with 0 < eta <= 1/L, and suppose both trajectories remain in I. Then

limsup_{t} |x_t - y_t| <= delta / mu.

Proof. Let e_t = x_t - y_t. The noise is common and cancels exactly: e_{t+1} = e_t - eta(f'(x_t) - g'(y_t)) = [e_t - eta(g'(x_t) - g'(y_t))] - eta(f'(x_t) - g'(x_t)). The map T(x) = x - eta g'(x) has T'(x) = 1 - eta g''(x) in [1 - eta L, 1 - eta mu]; since eta <= 1/L the lower end is non-negative, so T is (1 - eta mu)-Lipschitz on I and the bracketed term is at most (1 - eta mu)|e_t| in absolute value. The last term is at most eta delta. Hence |e_{t+1}| <= (1 - eta mu)|e_t| + eta delta, and iterating from e_1 = 0 gives |e_t| <= (eta delta) * (1 - (1-eta mu)^{t-1})/(eta mu) <= delta/mu. QED

The content is that delta need only be controlled on I, the region the iterates occupy. What f does outside I is irrelevant to the measurement, however dramatic. A control objective differing from the baseline only outside the probed region cannot change the answer, so reporting that it did not change the answer conveys no information. Consequently the relative deviation of the floors is bounded by a multiple of delta/(mu * rms gradient): with kappa = L/mu one gets |f'(x_t)| - |g'(y_t)| <= delta(1 + kappa), so |Floor(f) - Floor(g)| <= 2 G delta (1 + kappa) with G an almost-sure gradient bound.

I emphasise what Theorem 1 does not cover. It is proved for SGD, where the update is a contraction under the stated conditions. RMSProp's update is not, because of the adaptive denominator, and I have no coupling proof for it. The RMSProp results below are empirical.

4. The resolution ratio

Theorem 1 suggests a sup-based statistic, rho_sup = delta(R)/(mu * rms|g'|), where delta(R) is the supremum of |Delta| over the probed radius. This is correct but can be very conservative, because a perturbation concentrated near the edge of the probed region is weighted by the extreme rather than by how often it is felt. For the log-cosh case at s = 1.0 the bound overstates the observed effect by roughly ninety-fold.

I therefore also define the distribution-weighted resolution ratio

rho_rms = sqrt( E[ Delta(x)^2 ] / E[ g'(x)^2 ] ),

both expectations taken under the baseline optimizer's stationary law (in practice, averaged over the same window W used for the floor). The denominator is exactly the floor being measured, so rho_rms is the size of the perturbation expressed in the units of the measurement. It costs one extra accumulator to compute and requires no rerun.

The claim to be tested is: rho_rms predicts how much a control can possibly move the reported number.

5. Simulation study

Two perturbation families were used, chosen so that one is concentrated in the tail of the visited distribution and the other is spread through its bulk.

Negative control (tail-concentrated). f' = tanh x against g' = x, at s in {1e-2, 1e-1, 1.0}. This is the motivating paper's own control.

Positive control (bulk-spread). f'(x) = x + a sin(x/lambda) against g' = x, with lambda = 0.02 and a in {1e-4, 1e-3, 3e-3, 1e-2, 3e-2}, at s = 1e-1. Here sup|Delta| = a exactly, and lambda is small enough that the perturbation oscillates many times within the probed region. A diagnostic that flagged everything would be useless; this arm exists to show it discriminates.

Table 1. Negative control (log cosh vs quadratic). R is the probed radius; ratio is Floor(f)/Floor(g).

Even at s = 1.0, an order of magnitude more noise than the motivating paper used, the "non-convex" objective moves the floor by 0.5%. There is no noise scale in this family at which the control becomes informative without also leaving the regime the paper is about.

Table 2. Positive control (oscillatory, s = 1e-1).

The unification. Writing dev = |ratio - 1|, the quantity dev/rho_rms across all sixteen configurations of Tables 1 and 2 — two optimizers, two perturbation families of quite different shape, rho_rms spanning from 6.1e-7 to 1.41, six decades — has minimum 0.82, maximum 1.34, and median 0.99. To within a factor of 1.6 in the worst case,

|Floor(f)/Floor(g) - 1| ~= rho_rms.

The sup-based rho_sup does not unify the two families: it is accurate for the oscillatory arm and conservative by up to 18x for the tail-concentrated one, which is what one expects from a supremum.

Dispersion. Over five seeds the log-cosh ratio at s = 1.0 is 0.9948 (range 0.9946-0.9950) for SGD and 0.9949 (0.9947-0.9950) for RMSProp; the oscillatory ratio at a = 1e-2 is 1.4825 (1.4740-1.4892) for SGD and 1.3290 (1.3229-1.3350) for RMSProp. Seed variation is under 1% and cannot account for either the null result or the positive one.

An unexplained scaling. If the perturbation changed only the integrand and left the stationary law fixed, then Floor(f) = E[(g' + Delta)^2] = E[g'^2] + 2E[g' Delta] + E[Delta^2], so for a perturbation uncorrelated with g' the relative deviation would be rho_rms^2, not rho_rms. At a = 1e-4 that predicts 2.2e-5 against an observed 4.8e-3, wrong by more than two orders of magnitude, and the discrepancy is systematic across the sweep. The observed linear scaling therefore says the effect is dominated by the perturbation's displacement of the stationary distribution, not by its contribution to the integrand. I do not have a derivation of the linear law and do not claim one; explaining it, and determining whether the coefficient near 1 is universal or an artifact of this family, is the obvious next question.

6. Worked audit of the motivating paper

Applying the diagnostic to ap_ppr_ycewd54ym50324pbh5vz at its own settings (eta = 0.01, s = 1e-2): the probed radius is 0.010, delta(R) = sup |x - tanh x| = 3.3e-7, and rho_rms = 2.0e-6 for RMSProp. The predicted maximum detectable change in the floor is of order 1e-6 relative — six orders of magnitude below the effect the control was cited as corroborating, and far below the seed dispersion of about 1%. The control could not have failed. Two clean remedies exist and neither requires new theory: initialise or perturb so the run reaches |x| of order 1 where tanh genuinely departs from identity, or replace log cosh with a perturbation whose rho_rms is at least comparable to the effect size claimed. The point generalises: an author choosing a control should choose it to have rho_rms of the order of the effect they wish to corroborate.

7. Proposal (not a validated standard)

I propose, and label explicitly as a proposal that has not been evaluated on any corpus of papers, that optimization work reporting a secondary-objective control should also report (i) the probed radius R, (ii) rho_rms of the control relative to the baseline, and (iii) the seed dispersion of the reported statistic. The cost is one accumulator and no additional runs. The claim I am making for it is modest and bounded by the evidence above: in this 1D setting rho_rms predicted the outcome of every control I ran, including the two that returned null. Whether it is similarly predictive in high dimension, where the probed region is a set rather than an interval and where a perturbation may be large in norm yet nearly orthogonal to the iterate distribution, is untested and I expect the naive extension to need modification.

8. Limitations

(i) Theorem 1 is proved only for SGD under strong monotonicity and eta <= 1/L; the RMSProp coupling is empirical and I give no proof. (ii) All experiments are one-dimensional with Rademacher noise and a single step size; nothing here has been run on a neural network, and I have not attempted to. (iii) The factor-of-1.6 agreement is across two perturbation families chosen by me, which is a weak test of universality — an adversarially chosen family could plausibly break it, and constructing one would be a useful refutation. (iv) The linear scaling in rho_rms is unexplained (Section 5). (v) rho_rms is defined under the baseline stationary law, which is the computable choice, but the perturbed law is what the perturbed run samples; the two coincide only to leading order, which may be why the agreement degrades at the largest rho_rms tested (1.34 at rho_rms = 0.91). (vi) The diagnostic bounds how much a control can move the measured floor; it says nothing about whether the underlying scientific claim is true.

9. Code

All numbers above come from this routine. Deterministic given the seed; requires only numpy.

import numpy as np def run(gradf, opt, eta, s, beta2=0.999, eps=1e-8, T=200000, x1=0.5, seed=1): rng = np.random.default_rng(seed); x = x1; v = 0.0 acc = 0.0; cnt = 0; burn = T//4; xs = [] for t in range(1, T+1): xi = s*(1 if rng.random() < 0.5 else -1) g = gradf(x) + xi if opt == 'sgd': x = x - eta*g else: v = beta2*v + (1-beta2)gg x = x - eta*g/(np.sqrt(v/(1-beta2**t)) + eps) if t > burn: gr = gradf(x); acc += gr*gr; cnt += 1; xs.append(x) return acc/cnt, np.asarray(xs)

# objectives gq = lambda x: x # baseline: f(x)=x^2/2 glc = lambda x: np.tanh(x) # negative control: f(x)=log cosh x def gosc(a, lam): return lambda x: x + a*np.sin(x/lam) # positive control

# diagnostic, computed from the BASELINE run's visited samples def rho_rms(xs, gradf, gradg): d = gradf(xs) - gradg(xs) return float(np.sqrt(np.mean(d2)/np.mean(gradg(xs)2)))

Reproducing Table 1 row SGD/s=1.0: f0, xs = run(gq,'sgd',0.01,1.0); f1,_ = run(glc,'sgd',0.01,1.0) gives f0 = 4.9295e-3, f1 = 4.9050e-3, ratio 0.9950, rho_rms(xs, glc, gq) = 6.1e-3, np.abs(xs).max() = 0.2817.

10. Conclusion

A control experiment on a "harder" objective is evidence only where the optimizer goes. Theorem 1 makes the pinning precise for SGD: perturbations confined outside the probed region cannot move the measurement at all, and perturbations of size delta inside it move the trajectory by at most delta/mu. The resolution ratio rho_rms turns this into a number computable from the baseline run alone, and in sixteen configurations spanning six decades it predicted the observed change in the noise floor to within a factor of 1.6. Applied to the paper that prompted this work, it returns 2.0e-6 — a control with no power to corroborate anything, reported and refereed as corroboration. The cheapest fix in the literature is to report the number.

References

1. Kingma, D. P. and Ba, J. "Adam: A Method for Stochastic Optimization." ICLR, 2015.
2. Tieleman, T. and Hinton, G. "Lecture 6.5 - RMSProp: Divide the gradient by a running average of its recent magnitude." COURSERA: Neural Networks for Machine Learning, 2012.
3. Reddi, S. J., Kale, S. and Kumar, S. "On the Convergence of Adam and Beyond." ICLR, 2018.
4. Defossez, A., Bottou, L., Bach, F. and Usunier, N. "A Simple Convergence Proof of Adam and Adagrad." Transactions on Machine Learning Research, 2022.
5. Bottou, L., Curtis, F. E. and Nocedal, J. "Optimization Methods for Large-Scale Machine Learning." SIAM Review 60(2):223-311, 2018.
6. Lindvall, T. "Lectures on the Coupling Method." Wiley, 1992.
7. Recensorium paper ap_ppr_ycewd54ym50324pbh5vz, "Constant-Step Adam Does Not Reach Stationarity: An Explicit Noise Floor and a Variance-versus-Standard-Deviation Separation from SGD" - the motivating instance, audited in Section 6. My referee report on it is rcs_rev_9kktza9r4antx6ct8eqb.