{"id":1326,"date":"2026-10-27T22:57:00","date_gmt":"2026-10-27T22:57:00","guid":{"rendered":"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/?p=1326"},"modified":"2026-10-27T22:57:00","modified_gmt":"2026-10-27T22:57:00","slug":"persistence-pays","status":"publish","type":"post","link":"https:\/\/sites.rutgers.edu\/nawaf-bou-rabee\/persistence-pays\/","title":{"rendered":"Persistence Pays"},"content":{"rendered":"

Paper: arXiv:2610.35781<\/a>.<\/em><\/p>\n

The previous post<\/a> on this blog ended with a negative result and a list of open routes. The six standard Strang splittings of kinetic Langevin dynamics, OBABO and BAOAB among them, are not accelerated from cold starts: with step size and friction fixed from the curvature bounds and the dimension alone, no mixing time bound of order \u221a\u03ba holds over the smooth strongly convex class. Whether acceleration is possible for an implementable scheme, by warm starts, dynamic tuning, randomized integrators or Metropolis adjustment, was left open.<\/p>\n

The new preprint takes the last route, together with a warm start. It proves that Metropolis-adjusted OBABO mixes in a number of gradient evaluations of order \u221a\u03ba, up to the dimension and logarithms, provided the accept\/reject decision uses a uniform random number that is updated non-reversibly across steps, as proposed by Neal, instead of being redrawn at every step. This post describes the sampler, the identity that makes the persistent uniform effective, the two main theorems, and the structure of the proof.<\/p>\n

1. The acceleration question for adjusted samplers<\/h2>\n

Fix a target \u03bc_target(dx) \u221d exp(\u2212U(x)) dx on \u211d^d and consider kinetic Langevin dynamics,<\/p>\n

dX_t = V_t dt, dV_t = \u2212\u2207U(X_t) dt \u2212 \u03b3 V_t dt + \u221a(2\u03b3) dW_t,<\/p>\n

with invariant measure \u03bc = \u03bc_target \u2297 N(0, I_d). Throughout, m I \u2aaf \u2207\u00b2U \u2aaf L I, and \u03ba = L\/m is the condition number. With friction \u03b3 = 2\u221am the dynamics converge to \u03bc at a rate of order \u221am, independently of L: in L\u00b2(\u03bc) by Cao, Lu and Wang, and in relative entropy by Lu. Since the stability of an explicit Verlet-based discretization requires a step of order L^(\u22121\/2), a discretization that inherited this rate would reach accuracy \u03b5 after of order \u221a\u03ba log(1\/\u03b5) gradient evaluations, ignoring the dependence on d, against \u03ba log(1\/\u03b5) for overdamped samplers. This is the sense of acceleration used here, by analogy with Nesterov’s method in optimization. It is also the best one can hope for: for centered Gaussian targets, sampling to a fixed small total variation error requires in the worst case at least of order min(\u221a\u03ba, d) evaluations of U and \u2207U (Chewi, de Dios Pont, Li, Lu and Narayanan).<\/p>\n

For Metropolis-adjusted samplers, the known bounds on the full class of strongly log-concave targets were linear in \u03ba or worse. The bounds of Lee, Shen and Tian and of Wu, Schmidler and Chen for MALA depend linearly on \u03ba, and the coupling bound of Bou-Rabee and Oberd\u00f6rster for adjusted OABAO carries a factor \u03ba^(3\/2). Apers, Gribling and Szil\u00e1gyi obtained \u221a\u03ba d^(1\/4) on Gaussian targets, with a proof that relies on Gaussian structure.<\/p>\n

Question.<\/strong> Is there a Metropolis-adjusted discretization of kinetic Langevin dynamics that reaches a fixed total variation accuracy from a warm start after a number of gradient evaluations of order \u221a\u03ba, up to the dimension and logarithms, for every target in the smooth strongly log-concave class?<\/em><\/p><\/blockquote>\n

The answer given in the preprint is yes, for OBABO, under an additional Lipschitz condition on the Hessian, once the accept\/reject uniform is updated non-reversibly.<\/p>\n

2. Adjusted OBABO with a non-reversibly updated uniform<\/h2>\n

OBABO with step size h applies, in this order, a half-step of the Ornstein-Uhlenbeck flow of the velocity, a half kick by the force, a free flight, a half kick, and a second Ornstein-Uhlenbeck half-step:<\/p>\n

O_s(x, v) = (x, e^(\u2212\u03b3s) v + (1 \u2212 e^(\u22122\u03b3s))^(1\/2) \u03be), B_s(x, v) = (x, v \u2212 s\u2207U(x)), A_s(x, v) = (x + sv, v).<\/p>\n

The three middle steps form the velocity Verlet integrator \u03a6 = B_(h\/2) \u2218 A_h \u2218 B_(h\/2). The adjusted chain proposes \u03a6(w) from the state w after the first noise half-step, accepts the proposal or reverses the velocity, and then applies the second noise half-step. Write H(x, v) = U(x) \u2212 min U + |v|\u00b2\/2 and \u0394H(w) = H(\u03a6(w)) \u2212 H(w) for the energy error of the proposal, and h\u0304 = h\u221aL for the dimensionless step size.<\/p>\n

Writing the uniform random number of the Metropolis test as u = exp(\u2212\u2113), the test accepts exactly when \u2113 > \u0394H. The standard rule draws a fresh exponential \u2113 at every step. Neal’s non-reversibly updated uniform keeps \u2113 from step to step and updates it on acceptance: in terms of u this is u \u2190 u e^(\u0394H), and in terms of \u2113,<\/p>\n

\u2113 \u2190 \u2113 \u2212 \u0394H on acceptance, \u2113 unchanged on rejection.<\/p>\n

The variant analyzed in the preprint runs N epochs of E consecutive steps, drawing \u2113 once at the start of each epoch.<\/p>\n

Definition<\/strong> (adjusted OBABO with a non-reversibly updated uniform). One epoch from z draws<\/em> \u2113 ~ Exp(1) independently of the past and then, for k = 0, \u2026, E \u2212 1, with<\/em> w = O_(h\/2)(z): sets<\/em> (z, \u2113) \u2190 (O_(h\/2)(\u03a6(w)), \u2113 \u2212 \u0394H(w)) if<\/em> \u2113 > \u0394H(w), and<\/em> (z, \u2113) \u2190 (O_(h\/2)(S(w)), \u2113) otherwise, where<\/em> S(x, v) = (x, \u2212v) is the velocity reversal.<\/em><\/p>\n

Each step costs one gradient evaluation. Invariance follows from the auxiliary-variable involution framework of GIST, with \u2113 as the auxiliary variable: on \u211d^(2d) \u00d7 (0, \u221e), the map that sends (w, \u2113) to ((S\u2218\u03a6)(w), \u2113 \u2212 \u0394H(w)) when \u2113 > \u0394H(w), and leaves (w, \u2113) unchanged otherwise, is a volume-preserving involution that conserves the extended energy H + \u2113. So each step preserves \u03bc \u2297 Exp(1), and each epoch preserves \u03bc.<\/p>\n

\"\"
The two acceptance rules along one run on the Sonar posterior: (a) fresh threshold at every step, (b) non-reversibly updated threshold. Gray: energy error \u0394H. Black: threshold \u2113. Crosses: rejections.<\/em><\/figcaption><\/figure>\n

The figure compares the two rules along one run. With a fresh threshold the black dots are independent exponentials, and a rejection occurs whenever one falls below the gray bar. With the non-reversibly updated threshold the black line moves down by each accepted energy error and up when the error is negative. Its value before step k + 1 of an epoch is its initial value minus the sum of the energy errors accepted so far.<\/p>\n

3. Telescoping energy errors<\/h2>\n

The reason the persistent threshold helps is an exact identity for the energy error of velocity Verlet. For a step from w = (x, v), let V = v \u2212 (h\/2)\u2207U(x) be the velocity during the free flight and x_1 = x + hV the new position. Put<\/p>\n

G(x) = (h\u00b2\/8) |\u2207U(x)|\u00b2, \u03c4(w) = U(x_1) \u2212 U(x) \u2212 \u27e8\u2207U(x) + \u2207U(x_1), x_1 \u2212 x\u27e9\/2,<\/p>\n

so that \u03c4 is the error of the trapezoidal rule for U along the flight.<\/p>\n

Lemma<\/strong> (energy error of velocity Verlet). For every<\/em> w,<\/p>\n

\u0394H(w) = G(x_1) \u2212 G(x) + \u03c4(w), |\u03c4(w)| \u2264 (h\u0304\u00b2\/8) |V|\u00b2, 0 \u2264 G \u2264 (h\u0304\u00b2\/4)(U \u2212 min U).<\/p>\n

If moreover<\/em> \u2207\u00b2U is<\/em> L_H-Lipschitz, then<\/em> |\u03c4(w)| \u2264 (L_H\/12) h\u00b3 |V|\u00b3, and<\/em> \u03c4 \u2261 0 when<\/em> U is quadratic.<\/em><\/p>\n

The Ornstein-Uhlenbeck half-steps do not move positions. Along a path X_0, X_1, \u2026 of the unadjusted chain, the partial sums of the energy errors therefore telescope:<\/p>\n

S_n = \u0394H_0 + \u2026 + \u0394H_(n\u22121) = G(X_n) \u2212 G(X_0) + \u03c4_0 + \u2026 + \u03c4_(n\u22121).<\/p>\n

Proposition<\/strong> (acceptance of every move of an epoch). Every move of an epoch is accepted if and only if<\/em> \u2113 > max_(1\u2264n\u2264E) S_n, and on this event the adjusted epoch coincides with the unadjusted chain driven by the same noises. Consequently<\/em><\/p>\n

\u2119(a move of the epoch is rejected) = \ud835\udd3c[1 \u2212 exp(\u2212max_n S_n\u207a)] \u2264 \ud835\udd3c[min(1, (h\u0304\u00b2\/4) max_(n\u2264E) (U(X_n) \u2212 min U) + \u03a3_(k<E) |\u03c4_k|)].<\/p>\n

The leading part of the energy error enters only through a maximum over the visited positions, with no factor of E. With a fresh threshold at every step, the corresponding bound is \u03a3_(k<E) \ud835\udd3c[1 \u2212 exp(\u2212\u0394H_k\u207a)], a sum of E terms whose positive parts do not cancel.<\/p>\n

\"\"
Accumulated energy error (solid) and accumulated positive parts (dashed) along unadjusted OBABO on the Sonar posterior over one unit of time. Dotted: the median of \u2113.<\/em><\/figcaption><\/figure>\n

The figure shows both sums along one unit of time on the Sonar posterior. The accumulated signed error stays flat, well below the median of \u2113. The accumulated positive parts climb steadily and pass it. The same phenomenon is visible in the animation at the end of this post: the two chains reject the same fraction of moves, but the persistent threshold spends its rejections in bursts when it is nearly used up, and between bursts the chain runs for stretches of time without a rejection. Every rejection reverses the velocity, so a long rejection-free stretch is a stretch of ballistic motion along the slow directions of the target.<\/p>\n

For a quadratic potential \u03c4 \u2261 0, and the rule takes a simple form. On acceptance \u2113 drops by G(x_1) \u2212 G(x), and on rejection neither \u2113 nor x changes, so \u2113 + G(x) keeps its initial value \u2113_0 + G(x_0) throughout the epoch. The test \u2113 > \u0394H then reads G(x_1) < \u2113_0 + G(x_0). Within an epoch the adjusted chain is unadjusted OBABO confined to the ellipsoid {G < G(x_0) + \u2113_0}: a proposal that would leave it is rejected and the velocity reversed.<\/p>\n

4. The main result<\/h2>\n

Let l_H = L_H \/ m^(3\/2) be the scale-invariant Hessian-Lipschitz ratio, and let P denote the kernel of one epoch.<\/p>\n

Theorem 1<\/strong> (mixing of the adjusted chain). Assume<\/em> m I \u2aaf \u2207\u00b2U \u2aaf L I and that<\/em> \u2207\u00b2U is<\/em> L_H-Lipschitz. Let<\/em> \u03b5 \u2208 (0, 1) and let<\/em> \u03bd be a probability measure with<\/em> M_2 = \u2016d\u03bd\/d\u03bc\u2016_(L\u00b2(\u03bc)) < \u221e. Put<\/em> D = d + log(M_2\/\u03b5), \u039b = log(2 + \u03ba + l_H + D) and<\/em> N = \u2308log(2\/\u03b5)\/log 4\u2309. There is an absolute constant<\/em> C_0, and there are<\/em> h\u0304 and<\/em> E depending only on<\/em> \u03ba, l_H, d, M_2 and<\/em> \u03b5, such that<\/em><\/p>\n

N E \u2264 C_0 \u039b^(3\/2) (\u221a(\u03baD) + \u221a(l_H) D^(3\/4)) log(2\/\u03b5)<\/p>\n

and<\/em> \u2016\u03bd P^N \u2212 \u03bc\u2016_TV \u2264 \u03b5.<\/p>\n

A Gaussian centered within distance L^(\u22121\/2) of the minimizer, with covariance (2L)^(\u22121) I, satisfies M_2 \u2264 e (2\u03ba)^(d\/2), and such a point is found from any x_0 by Nesterov’s accelerated gradient method in O(\u221a\u03ba log(2 + \u03baL|x_0 \u2212 x_*|\u00b2)) gradient evaluations. From this start the bound becomes<\/p>\n

\u00d5(\u221a(\u03bad) + \u221a(L_H) (d\/m)^(3\/4)),<\/p>\n

where \u00d5 hides logarithmic factors. The first term dominates when l_H \u2264 \u03ba\/\u221ad. This does not require U to be a small perturbation of a quadratic: it allows the Hessian to change by order L over a distance \u221a(d\/m), which is the largest scale of the position marginal.<\/p>\n

The \u221a\u03ba dependence is necessary for this algorithm, already on Gaussian targets and from a warm start.<\/p>\n

Proposition<\/strong> (sharpness in \u03ba). Let<\/em> d \u2265 2 and<\/em> U(x) = (m\/2) x_1\u00b2 + (L\/2)(x_2\u00b2 + \u2026 + x_d\u00b2). Consider adjusted OBABO with any friction<\/em> \u03b3 > 0 and any step size with<\/em> h\u0304 < 2, using either a fresh uniform at every step or the non-reversibly updated uniform with any<\/em> E. Start from<\/em> \u03bc(\u00b7 | x_1 \u2264 0), whose density with respect to<\/em> \u03bc is at most 2. Then the position after<\/em> n steps is at total variation distance greater than 1\/4 from the position marginal of<\/em> \u03bc whenever<\/em><\/p>\n

n < (\u221a(2\u03ba)\/(4h\u0304)) (1 + h\u0304\u2076\/64)^((d\u22121)\/2),<\/p>\n

and hence, for every<\/em> h\u0304 \u2208 (0, 2), whenever<\/em> n < (\u221a2\/8) \u221a\u03ba d^(1\/6).<\/p>\n

The proof combines a conductance bound for the half-space {x_1 \u2264 0} with an exact Gaussian integral that bounds the probability of an accepted crossing. At the step size of Theorem 1, which on this Gaussian is h\u0304 \u224d (D\u039b)^(\u22121\/2), the first condition requires n of order \u221a(\u03baD\u039b), so there the term \u221a(\u03baD) is sharp up to logarithms. Over all step sizes, the dimension dependence on this family lies between d^(1\/6) and d^(1\/2).<\/p>\n

5. Entropy decay of unadjusted OBABO<\/h2>\n

The proof of Theorem 1 rests on a contraction estimate for the unadjusted chain. Let K be the OBABO kernel, r = e^(\u2212\u221am h) the velocity contraction factor of a noise half-step, and<\/p>\n

\u03bb = r\u00b2\u221am \/ (4(16 + r)) \u2265 (\u221am\/68) e^(\u22122\u221am h).<\/p>\n

Theorem 2<\/strong> (relative entropy decay of unadjusted OBABO). Assume<\/em> m I \u2aaf \u2207\u00b2U \u2aaf L I, that<\/em> \u2207\u00b2U is<\/em> L_H-Lipschitz,<\/em> \u03b3 = 2\u221am and<\/em> \u221am h \u2264 3\/16. Then for every<\/em> z \u2208 \u211d^(2d) and<\/em> n \u2265 1,<\/p>\n

KL(\u03b4_z K^(n+1) \u2016 \u03bc) \u2264 (17\/15) e^(\u2212\u03bbnh) [\ud835\udd3c_z H(Z_1) + d log(\u03ba \/ (2e h\u0304\u00b2 (1 \u2212 \u221am h)))]
\n + (16\/15) [(h\u0304\u00b2\/4) \ud835\udd3c_z (U(X_(n+1)) \u2212 min U) + sup_(1\u2264k\u2264n) ((L_H h\u00b2\/(12\u03bb)) \ud835\udd3c_z|V_k|\u00b3 + (221\/(16r)) h\u0304\u00b2 \ud835\udd3c_z|V_k|\u00b2)].<\/p>\n

The rate involves no \u03ba. The upper Hessian bound enters only through the factor h\u0304\u00b2 of the bias, and L_H only through the third moment of the velocity. This is the discrete counterpart of Lu’s sharp entropic hypocoercivity estimate for the continuous dynamics. Lu’s functional is the relative entropy plus a multiple of the transport corrector<\/p>\n

C(p) = \ud835\udd3c_p \u27e8v, x \u2212 T(x)\u27e9,<\/p>\n

where T is the Brenier map from the position marginal of p to the position marginal of \u03bc. Its proof does not use an upper bound on the Hessian. Every step of it has an exact counterpart for OBABO except one: in the splitting the force acts impulsively at the two ends of a free flight, and the resulting mismatch in the time derivative of the corrector is the error of the trapezoidal rule applied to the differential entropy of the position marginal along the flight.<\/p>\n

6. An entropy inequality for free transport<\/h2>\n

That trapezoidal error is controlled by the following inequality, which may be of independent interest.<\/p>\n

Proposition<\/strong> (entropy inequality for free transport). Let<\/em> X, V be random vectors in<\/em> \u211d^d, let<\/em> g be a nondegenerate Gaussian density on<\/em> \u211d^d, and put<\/em><\/p>\n

\u2130_g(t) = KL( Law(X + tV, V) \u2016 Law(X + tV) \u2297 g ).<\/p>\n

If<\/em> \u2130_g(0) and<\/em> \u2130_g(1) are finite, then<\/em> \u2130_g(t) \u2264 \u2130_g(0) + \u2130_g(1) for all<\/em> t \u2208 [0, 1].<\/p>\n

The quantity \u2130_g(t) measures how far the velocity is from the reference Gaussian g, conditionally on the position, at time t of the free flight. The proposition says it cannot exceed the sum of its values at the two ends. The paper gives two proofs. The first adds Gaussian noise of variance t(1 \u2212 t) to the intermediate position X + tV = (1 \u2212 t)X + t(X + V), which gives the marginal at time t of a Brownian bridge from X to X + V, and applies strong subadditivity of entropy to the triple formed by the two endpoints and this point; the logarithmic terms in t(1 \u2212 t) cancel. It assumes finite second moments and finite differential entropy. The second is a dual argument that needs only the finiteness of \u2130_g(0) and \u2130_g(1).<\/p>\n

The inequality is stated for a general pair (X, V) and does not see the potential. Combined with the exact one-step identities for the noise half-steps and the kicks, it gives a one-step decay of entropy plus a multiple of the corrector, and summation over steps gives Theorem 2.<\/p>\n

7. From entropy decay to mixing<\/h2>\n

Theorem 2 controls the unadjusted chain; the adjusted chain is handled by a coupling of whole epochs.<\/p>\n

Lemma<\/strong> (maximal endpoint coupling). There is a probability kernel<\/em> (z, z\u0303) \u21a6 \u0393_(z,z\u0303) to pairs of noise blocks for one epoch, with both marginals i.i.d. standard Gaussian, such that the unadjusted chains from<\/em> z and<\/em> z\u0303 driven by the two blocks differ at the end of the epoch with probability exactly<\/em> \u2016\u03b4_z K^E \u2212 \u03b4_(z\u0303) K^E\u2016_TV.<\/p>\n

Run two copies of the adjusted chain, one from the initial distribution \u03bd and one from the target distribution \u03bc, use the same \u2113 for both, and drive each epoch by noise blocks drawn from \u0393. On the event that neither copy rejects a move, both coincide with their unadjusted chains, by the acceptance proposition of \u00a73. So the copies differ at the end of the epoch with probability at most the total variation distance between the unadjusted endpoint laws plus the two rejection probabilities. The coupling is not adapted to the filtration of the chains: it is built on whole epochs, from the endpoints back to the noises. This is necessary. For Gaussian targets the meeting probability of any coupling of two copies of OBABO that acts step by step decays strictly more slowly than their total variation distance, as shown in joint work with Sonja Cox and Roy Schieven.<\/p>\n

Proposition<\/strong> (coupling). Let<\/em> \ud835\udc9f be a measurable set of states and<\/em><\/p>\n

q = sup_(z, z\u0303 \u2208 \ud835\udc9f) ( \u2016\u03b4_z K^E \u2212 \u03b4_(z\u0303) K^E\u2016_TV + \u03b2(z) + \u03b2(z\u0303) ),<\/p>\n

where<\/em> \u03b2(z) is the probability that an epoch from<\/em> z rejects some move. Then for every<\/em> N \u2265 1,<\/p>\n

\u2016\u03bd P^N \u2212 \u03bc\u2016_TV \u2264 q^N + N \u03bc(\ud835\udc9f^c) + M_2 (N \u03bc(\ud835\udc9f^c))^(1\/2).<\/p>\n

This is the localization idea of the earlier work with Oberd\u00f6rster, in a form where the contraction is a total variation bound at the end of an epoch instead of a one-step Wasserstein contraction. No bound on the excursions of the adjusted chain is needed: under \u03bc each epoch starts with law \u03bc, so a union bound over epochs controls the probability of starting outside \ud835\udc9f, and the Cauchy-Schwarz inequality transfers this to the warm start.<\/p>\n

To prove Theorem 1, take \ud835\udc9f = {H \u2264 R} with R of order D. For every z \u2208 \ud835\udc9f it then suffices that (i) the unadjusted epoch from z ends within a fixed small total variation distance of \u03bc, and (ii) the adjusted epoch from z is rejection-free except with small probability. For (i), Theorem 2 gives relative entropy at most a constant after a physical time of order \u039b\/\u221am, that is E \u224d \u039b\u221a\u03ba\/h\u0304 steps, and Pinsker’s inequality converts this to total variation. For (ii), the acceptance proposition bounds the rejection probability by h\u0304\u00b2 times the maximal energy along the epoch plus the accumulated \u03c4, which is small for h\u0304 \u224d D^(\u22121\/2). Then q \u2264 1\/4, and N \u224d log(1\/\u03b5) epochs suffice.<\/p>\n

Both steps need moments of the unadjusted chain over an epoch, including third moments of the velocity. Below a velocity v_* \u224d \u03ba\/(l_H h\u0304\u00b2), friction dominates the energy error of a Verlet step, and a Lyapunov function, the energy with a small twist \u27e8x, v\u27e9, gives exponential moments for the chain stopped when it leaves this region. Above v_* there is no control uniform over the class. The argument of Theorem 2 is therefore applied to a comparison chain that reverses the velocity instead of making a flight faster than v_*. It coincides with the unadjusted chain except on an event whose probability is at most E times a quantity exponentially small in v_*\u00b2, and the second term of Theorem 1 comes from the smaller step that the L_H terms force.<\/p>\n

8. Bayesian logistic regression<\/h2>\n

The paper compares the two acceptance rules on Bayesian logistic regression posteriors with a Gaussian prior for the breast cancer and Sonar data sets, in the setup of Chopin and Ridgway. In the coordinates used there, m = 1, and the prior scale sets the condition number. The chain uses \u03b3 = 2 and redraws \u2113 once per unit of time.<\/p>\n

At a step based on the global bound L, h = h\u0304\/\u221aL with the practical value h\u0304 = 2\/\u221a(d + 2), both chains reject fewer than 0.2% of the moves for condition numbers from 11 to 1.9\u00b710\u2074, and the integrated autocorrelation time in units of time does not grow with \u03ba. At these rejection rates the two rules give nearly the same chain.<\/p>\n

The global bound L exceeds the largest Hessian eigenvalue L\u0302 at the mode by a factor 1.4 to 52, so this step is conservative. With h = 1\/\u221aL\u0302, outside the parameter range of Theorem 1, both chains reject about 7% to 13% of the moves, and the condition number at the mode \u03ba\u0302 ranges from 9.5 to 9.5\u00b710\u2074 (breast cancer) and from 8 to 5.6\u00b710\u2074 (Sonar). The curvature varies along the chain, so the chains do not see a nearly Gaussian target.<\/p>\n

\"\"
Gradient evaluations per effective sample against \u03ba\u0302 for the (a) breast cancer and (b) Sonar posteriors.<\/em><\/figcaption><\/figure>\n

The integrated autocorrelation time of the chain with the non-reversibly updated uniform stays between about 4 and 8 units of time over these ranges, while that of the fresh-uniform chain grows from about 5 to 64 and 74. Least-squares fits of the logarithm of the cost against log \u03ba\u0302 give slopes 0.54 \u00b1 0.01 (non-reversibly updated) and 0.76 \u00b1 0.02 (fresh) for breast cancer, and 0.50 \u00b1 0.02 and 0.76 \u00b1 0.05 for Sonar. The measured cost of the persistent chain grows like the square root of the condition number at the mode, beyond what Theorem 1 proves on this family, where the Hessian-Lipschitz constant grows like \u03ba^(3\/2) and the second term of the theorem gives \u03ba^(3\/4).<\/p>\n

9. Scope and open problems<\/h2>\n