Interactive lab · stochastic stability · 1-D cocycle · start here: stability tutorial · 2-D sharp–tangent lab →

The shape of SGD iterates Watch the quasi-stationary distribution of a scalar stochastic gradient map across stability edges

$$X_{t+1}=(1-\eta h_t)\,X_t+\alpha X_t^{3}-\beta X_t^{5}+\sigma\xi_t$$

Curvature \(h_t\) is redrawn each step from a chosen law with mean \(\mu\) and variance \(v\). Drag \(\eta\) across the mean-square edge \(\eta_{\mathrm{MS}}\), the Lyapunov edge \(\eta_L\), and the empirical bimodality threshold \(\hat\eta_H\) to see the iterate cloud go from a spike at \(0\) to a two-lobed distribution.

What each coefficient does
\(\eta\) — step size

Primary control. Below \(\eta_{\mathrm{MS}}\) the origin is mean-square stable; past \(\eta_L\) pathwise linear stability is lost; near/above \(\hat\eta_H\) central mass depletes and two lobes appear.

\(\mu,\,v\) — curvature law

Mean and variance of \(h_t\). Larger \(v\) shrinks \(\eta_{\mathrm{MS}}=2\mu/(\mu^2+v)\) and widens the gap to \(\eta_L\). Keep \(v/\mu^2\) small for Gaussian if you want clean lobes.

\(\alpha\) — cubic / quartic map coeff.

Coefficient of \(X_t^3\) in the map (not multiplied by \(\eta\)). \(\alpha>0\) drives the flip / period-2 lobes about \(0\); \(\alpha<0\) flips that cubic drive.

\(\beta\) — quintic / sextic map coeff.

Coefficient in front of \(-X_t^5\). \(\beta>0\) confines large amplitudes; \(\beta<0\) anti-confines (can explode). Tune with \(\alpha\) so lobes stay finite.

\(\sigma\) — additive noise

Scale of \(\xi_t\sim\mathcal{N}(0,1)\). Default \(0\) is fine for the bimodal presets; a small \(\sigma>0\) softens lobes and helps when the cloud collapses to \(0\).

Law of \(h_t\)

Gaussian / Uniform / Bernoulli / \(\chi^2\) (matched \(\mu,v\)), deterministic \(h\equiv\mu\), or rare spike: typically small \(h\), rarely huge \(h\), with fixed mean \(\mu\).

Rare spike \(\varepsilon,\,h_{\mathrm{low}}\)

\(P(h=h_{\mathrm{high}})=\varepsilon\), else \(h_{\mathrm{low}}\), with \(h_{\mathrm{high}}\) chosen so \(\mathbb{E}[h]=\mu\). Can be bimodal with \(\eta\ll 2/\mu\) while \(h\equiv\mu\) stays at \(0\). Often lag-1 \(>0\) (well-hopping), not flip period-2.

Tip. Start from the Gaussian default preset, then move only \(\eta\). Watch the Fig. 1 strip: spike at \(0\) → broadening → two lobes about \(\pm 1\).
Deterministic mode. All edges collapse to \(2/\mu\). Additive noise \(\sigma\) broadens the period-2 orbit into two stochastic lobes (Thm. 1–3).
Rare-spike mode. \(\eta_{\mathrm{MS}}\) and \(\eta_L\) collapse near \(2/h_{\mathrm{high}}\ll 2/\mu\). You can sit past those rare-event edges yet far below \(2/\mu\): two lobes appear, while the mean-field control \(h\equiv\mu\) at the same \(\eta\) stays at \(0\). Lag-1 is often positive (well-hopping), not classic flip period-2.

Controls

Curvature

Sets the scale of the edge: roughly \(\eta\sim 2/\mu\).

Matched across laws. Small \(v\) ⇒ \(\eta_{\mathrm{MS}}\approx\eta_L\approx 2/\mu\).

Nonlinearity

Map term \(+\alpha X^3\). Positive ⇒ period-2 drive about \(0\).

Map term \(-\beta X^5\). Sign/magnitude set confinement.

Step size

drag across \(\eta_{\mathrm{MS}}\to\eta_L\to\hat\eta_H\)

Simulation

Across the edges

Same map and curvature; only the step size changes. Spike, then tails after Lyapunov, then two lobes.

Quasi-stationary iterate distribution

histogram of the particle cloud after burn-in

density \(p(X)\) \(\eta_{\mathrm{MS}}\) \(2/\mu\) \(\eta_L\) \(\hat\eta_H\)

Sample trajectory

One particle. True period-2 about \(0\) alternates sign (green / orange markers on the last steps).