Langevin dynamics: gradient descent plus noise

Links

• Interactive version: particles rolling in a potential you pick, with the Gibbs curve they settle onto; the density itself flowing between two wells; ULA against MALA as you turn up the step size; the shape of SGD-like noise; and annealed Langevin sampling a 2-D mixture from its score field.
• code/langevin.py prints every number on this page and regenerates the figures: python3 code/langevin.py --figures. make verify runs it, in about ten seconds.
• Paper notes that lean on this page: Achille, Mbeng & Soatto 2019 (SGD as a path integral, detailed balance, Kramers against end points); Achille, Paolini & Soatto 2020 (Eyring–Kramers and sharp minima); Achille et al. 2020, task complexity (the temperature as the price of information); Karczewski et al. 2026 (annealed Langevin along a geodesic).
• Inequalities and concentration: the KL divergence and the Donsker–Varadhan formula behind §4’s “free energy”. Optimal transport: the Fokker–Planck equation is a gradient flow in Wasserstein distance (§5).
• Standard references: Pavliotis, Stochastic Processes and Applications (2014), chapters 3–4 and 6, for everything in §1–§7; Welling & Teh, Bayesian Learning via Stochastic Gradient Langevin Dynamics (ICML 2011) for §9; Song & Ermon, Generative Modeling by Estimating Gradients of the Data Distribution (NeurIPS 2019) for §11.

In one paragraph

Gradient descent rolls a ball downhill and stops it at the bottom of whatever valley it started in. Langevin dynamics adds random kicks. At every instant the ball is pushed downhill by -\nabla U and jostled by Gaussian noise whose strength is a temperature D. Three things follow, and most of machine learning’s uses of the idea are one of them. First, the ball never stops, but its position settles into a fixed distribution, the Gibbs law p(w)\propto e^{-U(w)/D}: low places are exponentially more likely than high ones, and D sets how strongly. So if you want samples from a distribution p, run Langevin with U=-\log p and D=1 (this is SGLD, MALA and score-based sampling). Second, the ball can climb out of a valley, but the waiting time grows like e^{\text{barrier}/D} (Kramers’ law), so at low temperature the dynamics is fast inside a valley and glacial between valleys. Third, at a given temperature a wide valley holds more probability than a narrow one of the same depth, by a factor \sqrt{\text{curvature ratio}}, so noise prefers flat minima. SGD is approximately a Langevin process with D\approx\eta\sigma^2/(2B), which is how all three facts get attached to training neural networks, with a caveat about the shape of the noise (§10).

1. Two ingredients: a pull downhill and random kicks

The pull. Gradient flow, \dot w=-\nabla U(w), is gradient descent with an infinitely small step. It always goes down, and it stops at the first minimum it meets. Where it ends depends only on where it started.

The kicks. Now forget the potential and add up random steps. In each small time step \Delta t the position changes by an independent Gaussian with mean zero and variance 2D\,\Delta t. After time t there have been t/\Delta t steps, and variances of independent steps add, so the total variance is (t/\Delta t)\times 2D\Delta t=2Dt. The spread grows like \sqrt t, not like t: a random walk wanders, but slowly. The limiting process is Brownian motion (also called a Wiener process), written W_t, with W_t\sim\mathcal N(0,t). Here the kicks are \sqrt{2D}\,W_t.

The code checks both statements. With D=0.5, the variance at t=1, 4 and 16 divided by 2Dt is 1.007, 1.005, 0.987 with steps of 0.1, and 1.016, 0.991, 1.001 with steps of 0.01: the same answer whatever the step, as it should be.

Why the kick in one step is \sqrt{2D\Delta t} and not proportional to \Delta t. This is the one unfamiliar piece of arithmetic, and it is forced. If the kick in each step were 2D\,\Delta t times a standard Gaussian, the total variance after time 1 would be (1/\Delta t)(2D\Delta t)^2=4D^2\Delta t, which vanishes as the step shrinks. The code gets 0.1012, 0.0101, 0.0010 for steps 0.1, 0.01, 0.001: the noise simply disappears in the limit. For the noise to survive, its standard deviation per step has to scale like \sqrt{\Delta t}. This is also why Brownian paths are so jagged: over a short time \Delta t they move by about \sqrt{\Delta t}, which is much larger than \Delta t, so they have no velocity.

2. The Langevin equation, and how to simulate it

Put the two together:

dw=-\nabla U(w)\,dt+\sqrt{2D}\,dW_t .

Read it as a recipe for one short time step dt: move downhill by \nabla U\,dt, then add a Gaussian kick of variance 2D\,dt in every coordinate. Physicists write the same thing as \dot w=-\nabla U+\sqrt{2D}\,\xi(t) with “white noise” \xi, which is the derivative of W that does not really exist; the dW form avoids pretending it does.

Simulating it means taking that recipe with a finite step \eta (the Euler–Maruyama scheme):

w_{k+1}=w_k-\eta\,\nabla U(w_k)+\sqrt{2D\eta}\;\xi_k,\qquad \xi_k\sim\mathcal N(0,I)\ \text{independent}.

That is gradient descent with step \eta plus a Gaussian of standard deviation \sqrt{2D\eta} added after every step. Nothing more.

The simplest example: a quadratic bowl. With U=\frac h2w^2 the equation is dw=-hw\,dt+\sqrt{2D}\,dW, the Ornstein–Uhlenbeck (OU) process. Its average follows gradient flow, \mathbb E w(t)=w_0e^{-ht}, and its spread grows from zero and levels off at \operatorname{Var}w=\frac Dh(1-e^{-2ht})\to\frac Dh: the pull and the kicks balance (right panel above). The code: with h=2, D=0.2 and step 0.01 the stationary variance is 0.1010, against D/h=0.1000. The extra 1% is the finite-step bias, exactly 1/(1-h\eta/2) (§9).

Equipartition. In a bowl with many directions of different curvature h_i, each direction ends up with variance D/h_i: wide in flat directions, narrow in stiff ones. But the average height in each direction is \mathbb E[\frac12h_iw_i^2]=\frac12h_i\cdot\frac D{h_i}=\frac D2, the same for all of them. With 50 directions of curvature from 0.1 to 10 and D=0.2, the code finds 0.1027 in the flattest and 0.0999 in the stiffest (both D/2=0.1), and a total of 5.025 against kD/2=5. So a Langevin particle at the bottom of a quadratic minimum sits, on average, kD/2 above it, whatever the curvatures, while its spread is \sqrt{D/h_i}: 1.414 in the flattest direction and 0.141 in the stiffest. Keep this in mind for SGD: the loss plateaus above the minimum by an amount proportional to the temperature and the number of directions.

3. Where the particles end up: the Gibbs law

Run many particles for a long time and histogram their positions. The histogram stops changing and settles on

\pi(w)=\frac1Z\,e^{-U(w)/D},\qquad Z=\int e^{-U(w)/D}\,dw .

In words: a point that is \Delta U higher than another is e^{-\Delta U/D} times less likely. Temperature is the exchange rate between height and improbability. At low D almost all the mass sits in the lowest valley; at high D it spreads almost evenly. Z just makes the total probability 1.

Where it comes from (one dimension, line by line). Think of p(w,t) as a fluid. Two things move it:

• the drift carries probability downhill with velocity -U'(w), a flow of -U'(w)\,p;
• diffusion moves probability from crowded to sparse places, a flow of -D\,p'(w) (more crowding to the right pushes mass to the left). This is Fick’s law, the same rule as heat spreading in a bar.

The total flow is the probability current J=-U'p-Dp', and probability is conserved, so the density changes by the amount flowing in minus the amount flowing out: \partial_tp=-\partial_wJ. At equilibrium nothing changes. In one dimension, with nothing escaping to infinity, that means J=0 everywhere:

-U'\pi-D\pi'=0\quad\Longrightarrow\quad\frac{\pi'}{\pi}=-\frac{U'}D\quad\Longrightarrow\quad\log\pi=-\frac UD+\text{const}.

The downhill pull and the outward spreading cancel point by point. For the OU bowl this gives \pi\propto e^{-hw^2/2D}, a Gaussian of variance D/h, as found above.

The code checks it on a tilted double well, U=(w^2-1)^2+0.25w, whose left well is 0.5 deeper. The Gibbs probability of the left well is 0.993 at D=0.1, 0.823 at D=0.3 and 0.613 at D=1. A Langevin simulation at D=0.4 (3000 particles, started uniformly, 200 time units) puts 0.757 of its time in the left well against the Gibbs value 0.756. Its histogram is within total-variation distance 0.007 of the curve.

A word on why this is useful. To sample from a density p you would normally need its normalising constant, which is an intractable integral for anything interesting. Langevin with U=-\log p and D=1 has \pi=p, and it only ever uses \nabla U=-\nabla\log p, in which the constant has disappeared. That gradient is called the score. Everything in §9 and §11 builds on this.

4. Two wells: depth against width

Which valley holds more probability? Integrate e^{-U/D} over each. Near a minimum w_i with height U_i and curvature h_i, the integrand is a Gaussian bump of height e^{-U_i/D} and width \sqrt{D/h_i} (this is the Laplace approximation), so

P(\text{well }i)\propto e^{-U_i/D}\sqrt{\frac{2\pi D}{h_i}} =\exp\Big(-\frac1D\Big[U_i+\frac D2\log h_i\Big]\Big)\times\text{const}.

The bracket is the well’s free energy F_i=U_i+\frac D2\log h_i (in k dimensions, \frac D2\log\det H_i). Depth counts through U_i, narrowness through \log h_i, and the temperature sets the exchange rate between them. At low D the deepest well wins. At high D the widest one does.

The code builds a well that is 0.3 deeper but 25 times stiffer than its neighbour:

The narrow well loses its majority at D=0.192. The free-energy rule predicts the crossover where 0.3=\frac D2\log25, at D=0.186. Counting depth alone, the narrow well would keep its majority at every temperature. This is the precise sense of “noise prefers flat minima”: not that noise pushes the particle out of sharp minima faster (that is the barrier’s job, §6), but that at equilibrium a flat valley simply has more room.

The same potential shows that where the mass ends up and how soon it gets there are separate questions. The narrow well sits 2.96 below the ridge between the wells. Particles dropped into it take 5448 time units on average to leave at D=0.3, although at equilibrium only 0.362 of the mass belongs there; at D=0.8 they leave in 17.1. The first question is the Gibbs law. The second is Kramers’ law (§6).

5. How the density moves: the Fokker–Planck equation

§3 used the current J only at equilibrium. The same bookkeeping gives the full time evolution of the density, the Fokker–Planck equation:

\partial_tp=\nabla\cdot\big(p\,\nabla U\big)+D\,\nabla^2p .

The first term is the fluid being carried downhill, the second is it spreading out. (The Langevin equation describes one particle’s random path; Fokker–Planck describes the deterministic evolution of the cloud of all possible paths. They are two views of the same process.) A second reading, for readers who know optimal transport: Fokker–Planck is gradient descent, in Wasserstein distance, on the free-energy functional \int U\,p+D\int p\log p. Its minimiser is the Gibbs law (Jordan, Kinderlehrer & Otto, 1998).

The code solves it exactly on a grid for the tilted double well at D=0.25, starting from a spike at w=+1 in the shallower right well:

The equilibrium value is 0.866. Within t\approx0.3 the spike has relaxed to the local shape of the right well. Getting the mass into the deeper well takes a hundred times longer.

Two time scales. The density relaxes as a sum of modes, each decaying like e^{-\lambda t}. The rates \lambda are the eigenvalues of the Fokker–Planck operator. On the grid they are 0 (the equilibrium itself), then \lambda_1=0.0416, then 3.34, 5.61, and upward. The sanity check on a quadratic of curvature 8 gives 7.997, 15.994, 23.99, against the exact OU rates 8, 16, 24.

• \lambda_1 is the hopping rate between the wells. The exact mean passage times (§6) are 187.4 (left to right, uphill) and 28.94 (right to left), and 1/187.4+1/28.94=0.0399, within 4% of \lambda_1.
• Everything after \lambda_1 is of the order of the curvatures: U''=8.73 and 7.22 at the two minima, \lvert U''\rvert=3.95 at the saddle. That is a gap of 80 between the slow rate and the next. At D=0.05 the fast rates sharpen: 3.77 belongs to a mode sitting on the saddle, and 7.12 lies next to the right well’s curvature, 7.22.

So in practice the density first equilibrates inside each valley and then, much later, shares mass between valleys. That separation is what makes “which basin did training end up in” a meaningful question at low temperature.

6. Waiting to cross a barrier: Kramers’ law

How long until a particle in one well reaches the other? In one dimension there is an exact formula for the mean first-passage time from a to b>a (with nothing escaping to the left):

\tau(a\to b)=\frac1D\int_a^b e^{U(y)/D}\left[\int_{-\infty}^{y}e^{-U(z)/D}\,dz\right]dy .

It reads: the inner integral is how much equilibrium probability sits behind the point y; the outer factor e^{U(y)/D} is large where the potential is high, so the integral is dominated by the highest point on the way, the saddle. For a barrier of height \Delta E above the start, at low temperature this becomes Kramers’ law:

\tau\approx\frac{2\pi}{\sqrt{U''(a)\,\lvert U''(s)\rvert}}\;e^{\Delta E/D},

with a the bottom of the starting well and s the saddle. The exponential is the Arrhenius factor (the chance of one attempt making it over the top). The prefactor measures how often attempts happen, through the curvature of the well and of the barrier top. Note what is not in it: the depth of the destination. Once over the saddle the particle rolls down quickly, however far down it goes.

On the symmetric well U=(w^2-1)^2 (barrier 1):

Kramers is an asymptotic formula, exact as D\to0. Its error here grows roughly in proportion to D, to 25% at D=0.5, where the barrier is only twice the temperature. Simulations with 4000 particles give 22.95\pm0.35 at D=0.35 and 10.42\pm0.15 at D=0.5, on the exact values. The slope of \ln\tau against 1/D between D=0.1 and 0.2 is 0.988, the barrier height. Halving the temperature squares the waiting time.

In several dimensions the same holds with determinants (the Eyring–Kramers or Langer formula): \tau\approx\frac{2\pi}{\lvert\lambda_s\rvert}\sqrt{\lvert\det H_s\rvert/\det H_a}\,e^{\Delta E/D}, with \lambda_s the one negative eigenvalue of the Hessian at the saddle. The determinant ratio is the free-energy correction of §4 applied to the well and to the saddle’s cross-section. A wide mountain pass is easier to find than a narrow one.

7. Reversibility, and when it fails

Film a Langevin particle at equilibrium and play the film backwards. For a gradient drift and constant isotropic noise you cannot tell the difference: every transition x\to y happens exactly as often as y\to x. This is detailed balance,

\pi(x)\,p_t(y\mid x)=\pi(y)\,p_t(x\mid y)\qquad\Longrightarrow\qquad \frac{p_t(y\mid x)}{p_t(x\mid y)}=e^{-[U(y)-U(x)]/D}\ \text{ at every }t .

It is the same statement as J=0 in §3, and it is why the downhill/uphill passage times in §5 differ by exactly the equilibrium odds. The Achille 2019 notes use it to split a transition probability into a symmetric “reachability” part and an end-point part. Two ways to break it, both relevant to SGD:

• A drift that is not a gradient. Add a rotation to the pull, dw=-(w-\Omega Jw)\,dt+\sqrt{2D}\,dW with J a 90° rotation and \Omega=2. The equilibrium snapshot is unchanged: the code solves for the stationary covariance and gets exactly DI, the Gibbs law of \frac12\lVert w\rVert^2. But the film now runs in circles. The lagged covariances \mathbb E[w_1(t+s)w_2(t)]=-0.510 and \mathbb E[w_2(t+s)w_1(t)]=+0.510 at s=0.5 would be equal for a reversible process. A snapshot alone cannot tell you the dynamics is reversible.
• Noise that is not isotropic. Keep the gradient drift on a quadratic loss with curvatures 10 and 0.1, but give the noise a covariance proportional to the Hessian, 2cH, as SGD’s noise roughly is (§10). The stationary covariance \Sigma solves H\Sigma+\Sigma H=2cH, whose solution is \Sigma=cI: a round cloud, not the Gibbs law. With isotropic noise of the same total size the spreads are D/h: variances 0.5 and 0.005. With Hessian-shaped noise they are 0.0099 in both directions.

The second case changes what “flat minima are preferred” means. With isotropic noise, the average excess loss is kD/2 at any minimum (equipartition, §2), and flatness helps only through the free-energy volume of §4. With Hessian-shaped noise, the excess loss is c\operatorname{tr}H/2. At a minimum 10 times sharper it grows from 0.050 to 0.500 in the code, while the isotropic value stays at 0.050. Sharpness then costs loss directly, not only probability mass.

8. The weight of a whole path

The Gaussian kicks also give a probability to an entire trajectory. Chop time into steps. The chance of a particular sequence of positions is the product of the Gaussian densities of the kicks it needed, \exp\big(-\lVert\Delta w+\nabla U\Delta t\rVert^2/(4D\Delta t)\big) per step. In the limit:

\text{weight}[w(\cdot)]\;\propto\;\exp\Big(-\frac1{4D}\int\lVert\dot w+\nabla U(w)\rVert^2\,dt\Big),

the Onsager–Machlup (or, dropping a correction of order D, Freidlin–Wentzell) action. A path that moves exactly as the gradient pushes it needs no kicks at all and costs nothing. Every deviation costs its squared size, divided by the temperature.

Two consequences, checked on the tilted double well:

• Downhill along gradient descent is free. From the saddle into the left well, D\timesaction is 5\times10^{-16}.
• Uphill costs exactly the climb. The cheapest way up is gradient descent played backwards (\dot w=+\nabla U), and its D\timesaction is \int\lVert\nabla U\rVert^2dt=\Delta U. The code gets 1.2616 for a climb of 1.2616. So the weight of the likeliest climb is e^{-\Delta U/D}, the Arrhenius factor again. Any other route costs more: the best straight climb at constant speed has 1.3288.

This is the most intuitive way to see Kramers’ law. Leaving a well means climbing to its lowest pass, and climbing costs e^{-\Delta E/D} in probability no matter how you do it. The Achille 2019 notes develop this into the paper’s “static” and “reachability” factors.

9. Sampling with Langevin: ULA, MALA, SGLD

The unadjusted Langevin algorithm (ULA) is Euler–Maruyama with U=-\log p and D=1: w\leftarrow w+\eta\nabla\log p(w)+\sqrt{2\eta}\,\xi. The continuous process has \pi=p exactly. The finite step does not. For a standard normal target the update is w\leftarrow(1-\eta)w+\sqrt{2\eta}\,\xi. Its stationary variance v solves v=(1-\eta)^2v+2\eta:

v=\frac{2\eta}{1-(1-\eta)^2}=\frac1{1-\eta/2}.

The samples are too spread out, by 2.6% at \eta=0.05, a factor 2 at \eta=1 and 20 at \eta=1.9, and at \eta\ge2 the chain diverges. The code’s samples match the formula to within Monte Carlo error at every step size (1.029, 1.114, 1.331, 1.998, 4.000, 20.06).

MALA (Metropolis-adjusted Langevin) uses the Langevin step as a proposal and accepts it with the Metropolis–Hastings probability, which corrects the bias exactly. Its variance is 1.00 at every step size tested. The price is rejections: the acceptance rate falls from 0.998 at \eta=0.05 to 0.784 at \eta=1 and 0.525 at \eta=1.9.

SGLD (stochastic gradient Langevin dynamics, Welling & Teh 2011) is ULA with the gradient of the log-posterior estimated on a mini-batch, which is how Bayesian inference is scaled to neural networks. The mini-batch error adds its own noise. Per step it has variance \eta^2g^2, against the injected 2\eta, so it raises the effective temperature to 1+\eta g^2/2. The ratio shrinks with the step, which is why SGLD’s step size must decay. On the posterior of a mean from N=1000 observations with batches of 10 (g^2\approx9.8\times10^4), the stationary variance over the true posterior variance is:

A simulation at 10^{-4} gives 6.17 (the exact value for this linear model is 6.20), with the mean right at the posterior mean. At a practical step size, SGLD samples a much hotter posterior than the one written down. This is one ingredient of the “cold posterior” debate in Bayesian deep learning.

10. SGD as (approximately) Langevin

A mini-batch gradient is the full gradient plus an error with mean zero and covariance \Sigma_g/B. So an SGD step is a gradient-descent step plus a kick:

w_{k+1}=w_k-\eta\nabla L(w_k)-\eta\,\big(\nabla\hat L_B-\nabla L\big).

Read the step count as time, t=k\eta. The kick has variance \eta^2\sigma^2/B per coordinate (for \Sigma_g=\sigma^2I), and Langevin’s kick over time \eta has 2D\eta. Matching them,

D\approx\frac{\eta\,\sigma^2}{2B}.

On the toy loss \frac12(w-z_i)^2 with \eta=0.05 the stationary variance of w (which should equal D, curvature 1) is 0.01421, 0.00361, 0.00089 for B=4, 16, 64, against 0.01391, 0.00348, 0.00087. The 2–4% excess is ULA’s 1/(1-\eta/2). So learning rate over batch size is a temperature, which is the intuition behind scaling the learning rate with the batch size (to keep D fixed) and behind “large batches find sharper minima”.

The approximation has three well-known weak spots, and each changes a conclusion above.

• The noise is not isotropic. Near a minimum the gradient covariance is close to the Fisher information, which is close to the Hessian. §7 shows what that does: the stationary cloud is round, not D H^{-1}, and detailed balance with respect to e^{-L/D} fails.
• The noise is not constant. \Sigma_g depends on w. With state-dependent noise the stationary law depends on how the SDE is read (Itô or Stratonovich), and neither need be a Gibbs law.
• The noise may not be Gaussian. Mini-batch gradients can be heavy-tailed. Whether that matters for deep networks is debated; with heavy tails, escapes happen by jumps and Kramers’ e^{\Delta E/D} gives way to polynomial waiting times.

The Achille notes use the isotropic idealisation, and state where it enters.

11. Score-based sampling and diffusion models

Langevin needs only the score \nabla\log p. If a network s_\theta(x)\approx\nabla_x\log p(x) has been trained on data (score matching), then x\leftarrow x+\eta\,s_\theta(x)+\sqrt{2\eta}\,\xi generates new samples. Song & Ermon (2019) point out two problems, and one fix covers both.

• The score is only learned where the data are. Between the modes of p there are no training points, so the score there is unreliable.
• Mixing between modes is a Kramers problem. For p=0.8\,\mathcal N(-4,1)+0.2\,\mathcal N(4,1) the barrier of -\log p between the modes is 6.39 above the small mode. At temperature 1 that means crossings take of order e^{6.4}\approx600 time units, and a run of 20 time units effectively never crosses. Started uniformly on [-8,8], plain Langevin (with the exact score) puts 0.524 of its samples in the left mode. Each particle stays in whichever basin it started in, so the mode weights are wrong although every sample sits in a correct-looking mode.

Annealed Langevin runs the same sampler on smoothed versions of the target first: p_\sigma=p*\mathcal N(0,\sigma^2), the data plus Gaussian noise of size \sigma. At \sigma=10 the two modes have merged into one broad hump, so particles move freely and share themselves out in the right proportions. As \sigma decreases through 3 and 1 to 0, the modes separate after the mass has been divided. With \sigma from 10 down to 0.1 (then 0), 200 steps at each level, the code recovers a left-mode weight of 0.788 against the true 0.8. The smoothed scores are also easier to learn, because noise spreads training data into the gaps.

This is the bridge to diffusion models. Their generative process runs a reverse-time SDE, dx=[f(x,t)-g(t)^2\nabla\log p_t(x)]dt+g(t)\,d\bar W, which is Langevin-like dynamics whose target p_t slides continuously from pure noise to the data. The annealing above is a discrete version of that slide. The Karczewski et al. notes use annealed Langevin in exactly this role, along a path between two low-energy states.

12. A note on momentum (underdamped Langevin)

Everything above is overdamped: the particle has no inertia, and the kick changes its position directly. The underdamped version gives it a velocity v with friction \gamma: dw=v\,dt, dv=-\nabla U\,dt-\gamma v\,dt+\sqrt{2\gamma D}\,dW. Its equilibrium is e^{-(U(w)+\frac12\lVert v\rVert^2)/D}, so the position still follows the Gibbs law and the velocity is an independent Gaussian. With large friction it reduces to the overdamped equation. It is the continuous version of SGD with momentum, and of Hamiltonian Monte Carlo with partial momentum refresh, and it explores long flat valleys faster because it does not have to diffuse across them.

Questions and doubts

• “SGD is Langevin” is a model, not a theorem. The derivation in §10 needs small steps, Gaussian noise and, for the Gibbs law, isotropic and constant noise. Real SGD violates all three to some degree. The robust part is the scaling D\propto\eta/B. The Gibbs form of the equilibrium is the fragile part.
• Which “flat minima” argument? Three different mechanisms get the same name. The free-energy volume factor \sqrt{\det H} (§4) is equilibrium probability. Faster escape from minima with low barriers (§6) is kinetics. The excess loss c\operatorname{tr}H/2 under Hessian-shaped noise (§7) is a loss penalty. They can disagree about which minimum is “preferred”, and papers often switch between them.
• Kramers’ prefactor matters less than it looks, but not nothing. At a barrier of only two to four times D it is off by 15–25% (§6). The exponent, which carries the barrier, is what the simulations pin down, and it is what the curvature arguments of the papers tend to overlook.
• Itô or Stratonovich? With constant noise the two readings agree and nothing on this page depends on the choice. With state-dependent noise (SGD) they differ by a drift term, and the “right” one is whichever limit the discrete algorithm actually converges to, which for SGD is Itô.

Cheat sheet