AVOIDING UNSAFE SETS WHEN TRAINING WITH LANGEVIN DYNAMICS ADAM OBERMAN
arXiv:2607.07538v1 [cs.LG] 8 Jul 2026
Abstract. Training a model with noisy gradient descent can be idealized as overdamped Langevin dynamics on the loss landscape, and a natural safety question is to bound the probability νt (AH ) = P(Qt ∈ AH ) that the trajectory lies in a designated failure region AH . We study this for a smooth, strongly convex loss in d dimensions and a failure region separated from the minimizer by an energy gap. Three bounds emerge. At the end of training, the equilibrium mass π(AH ) is exponentially small in d, with a complementary energy-barrier rate when the noise is small. Along the trajectory, a shape-free bound q νt (AH ) ≤ π(AH ) 1 + χ20 /π(AH ) e−mt shows that the in-set probability relaxes to (twice) the static value after a burn-in time of order d, using only the global spectral gap m of the loss. A worked Ornstein-Uhlenbeck example shows this burn-in is necessary: an angular slice of the equilibrium shell can transiently swell by a factor exponential in d, even though its equilibrium mass is tiny. To rule such swelling out we introduce a local relaxation rate attached to the failure region, defined through the spectral measure of its centered indicator rather than a Dirichlet-form Rayleigh quotient. For geometrically isolated regions this rate exceeds the global one, shrinking the burn-in proportionally, and combined with a maximum-principle ceiling it caps the trajectory probability uniformly in time. The picture is that strong convexity sets how fast training relaxes, but the shape of the unsafe set decides whether the trajectory bulges through it on the way home.
1. Introduction A model trained by noisy gradient descent can be idealized as a diffusion on its loss landscape, and a basic safety question is whether the training trajectory ever enters a designated bad region of parameter space. Write Qt ∈ Rd for the parameters at training time t and AH ⊆ Rd for a failure region: a set of parameters whose induced behavior we would like the trained model to avoid. Even when training ends safely, the trajectory can pass through AH on its way to the optimum, so the object of interest is the in-set probability νt (AH ) = P(Qt ∈ AH )
at every training time t, not only at convergence. Several safety concerns share this shape. In code generation, AH is the set of parameters that emit a hidden backdoor or a known-insecure pattern, and one wants the chance that training ever lands there to be negligible. In alignment, AH is a region of misaligned or deceptive behavior that a model might drift through before settling into a benign optimum. The motivating instance for this work is the Scientist AI (SAI) Predictor safety case of [BRG+ 26], which separates an honest non-agentic Predictor from a scaffold that gates its outputs through a guardrail, and bounds the probability that a consequence-invariant training process produces a dangerous Predictor (one whose guarded deployment causes a designated harm event above a normative threshold) uniformly in t by νt (AH ) ≤ Cbad Rshell .
Here Rshell is the conditional fraction of dangerous Predictors inside a narrow loss band under the initialization, argued exponentially small on the grounds that danger requires many coordinated errors, and Cbad is the within-band enrichment factor of training, assumed bounded as a stated Date: July 9, 2026. 1
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requirement on the process. That argument treats the dynamics generating νt abstractly, as a distribution over training trajectories indexed by t; the present paper supplies those dynamics and bounds νt (AH ) directly. We model a training run as the overdamped Langevin dynamics dQt = −∇J(Qt ) dt + σ dWt ,
where J is the training loss on Rd , Wt is standard Brownian motion, and σ > 0 is a noise level set by the optimization (informed, for example, by batch size and learning rate). This paper proves an upper bound on the probability that the law of a Langevin training trajectory occupies the designated failure region AH ⊆ Rd . Under explicit smoothness, convexity, and energy-gap hypotheses on the loss J, the bound is exponentially small in d and is uniform in time, after a burn-in of order d. Two grades are proved: a shape-free version that uses only the total equilibrium mass π(AH ) and the global Poincaré constant, and a shape-aware version that uses a local relaxation rate λAH together with a maximum-principle ceiling to remove the burn-in altogether for flux-isolated sets. The Langevin idealization. This SDE is the standard continuous-time model of stochastic gradient training: at small step size, minibatch SGD on a smooth loss has Itô-SDE limits with drift −∇J and noise covariance set by the minibatch gradient covariance [LTE19, MHB17, HLLL19], and the explicitly noised SGLD algorithm of [WT11] realizes the model used here exactly. Isotropy of the noise and the continuous-time limit are idealizations; we take them as given and do not address the discretization or anisotropy gaps. The loss J has a minimizer Pn , normalized to J(Pn ) = 0. We single out a failure region AH ⊆ Rd : a set of parameters we would like the trained model to avoid. The minimizer is safe, Pn ∈ / AH , but the noise keeps Qt from ever settling exactly at Pn , so we must ask how much probability mass the trajectory places in AH . The natural hope is that if the failure region has small equilibrium mass π(AH ), then P(Qt ∈ AH ) is small for all t. This hope is false in general, and the way it fails is the organizing problem of this paper. Transient swelling. Even under the strongest possible convexity, the trajectory mass of a small set can bulge far above both its initial and its equilibrium value. The canonical demonstration is the one-dimensional Ornstein-Uhlenbeck process J(Q) = Q2 /2 with π = N (0, 1). Start the trajectory concentrated near Q = 10 and take AH = [4, 6]. Then the initial mass of AH is zero and its equilibrium mass is about 10−5 , yet at t = log 2 the law is approximately N (5, 3/4) and places roughly 0.7 of its mass in AH . The mass swells by a factor of about 105 on its way home, because AH sits directly on the transport path from the start to equilibrium. No convergence rate for KL or Wasserstein distance forbids this: those are global functionals and say nothing about a single set. Higher-dimensional swelling. Swelling is not a one-dimensional artifact. In high dimensions the equilibrium of a strongly convex loss concentrates on a thin shell, and a failure region that is an angular slice of that shell suffers the same transit swelling, now amplified by dimension. Section 7 makes this quantitative: the transient overshoot factor for shell dynamics grows exponentially in the ambient dimension unless the geometry of AH is controlled. This is the concrete reason we cannot rely on equilibrium mass alone, and it is what motivates a geometric (Cheeger-type) notion of isolation. What this paper proves. We give two complementary estimates. 1. A static mass bound on π(AH ), the probability at the end of training, which is exponentially small in the dimension d (Section 4). 2. A dynamic mass bound on P(Qt ∈ AH ) along the trajectory (Section 5), which relaxes to the static bound after a burn-in time of order d.
AVOIDING UNSAFE SETS WHEN TRAINING WITH LANGEVIN DYNAMICS
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Transient swelling in 1-D (Ornstein-Uhlenbeck) AH = [4, 6]
drift toward equilibrium
νt (AH )
vs t
density
π = N (0, 1) (= ν∞ )
t∗
∼
(peak ∼0.75) t then decays to π(AH )∼3×10−5
75% mass in AH
ν0 = δ10
π(AH ) ∼ 3×10−5
0
2
4
π(AH )
νt∗ = N (5, 3/4), t∗ = log 2
6
8
10 parameter Q
Figure 1. Transient swelling in the 1D Ornstein-Uhlenbeck process with π = N (0, 1). From the point-mass start ν0 = δ10 , the law νt passes through N (5, 3/4) at t∗ = log 2, placing about 75% of its mass in AH = [4, 6], whose equilibrium mass is π(AH ) ≈ 3 × 10−5 . The inset shows νt (AH ) peaking near t∗ and decaying to π(AH ). The dynamic bound comes in two grades. The first (Section 5) assumes only the total equilibrium mass π(AH ) and gives a clean, dimension-free relaxation rate, at the cost of a transient window during which it is uninformative: this is exactly the swelling window. The second (Section 6) assumes in addition that AH is geometrically isolated, quantified by a local relaxation rate λAH ≥ m, which shrinks the transient window by the factor m/λAH and, combined with a maximum-principle ceiling, caps the trajectory mass uniformly in time. Section 7 returns to the Ornstein-Uhlenbeck and shell examples to show both grades are sharp and to exhibit the geometries where swelling is real. 1.1. Notation. Throughout, νt denotes the law of Qt , π the stationary Gibbs measure, and AH the failure region. We use σ for the noise level (rather than an inverse temperature β; the two are related by β = 2/σ 2 ). Three scalar quantities measure how far the initial law ν0 is from π and how isolated AH is: • M := ∥ν0 /π∥∞ , the initial density ratio in L∞ . • χ20 := χ2 (ν0 ∥ π), the chi-squared divergence. • λAH , the local spectral gap of AH (Definition in Section 6). 2. Related work and positioning This section locates the paper in the surrounding literature along three axes: what is bounded, how the bound is structured, and what geometric information about the failure region is required. What is bounded. Much of the sampling literature [VW19, EHZ22, CEL+ 22, Che24, Pav14, BGL14] bounds a global divergence of the law from the target: KL, chi-squared, Rényi, or Wasserstein. The metastability literature [MS14, BEGK04, BGK05] bounds spectral gaps and exit times of metastable wells. The SGLD hitting-time analyses [RRT17, ZLC17] bound the time to first reach
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a target region. The object controlled here is none of these: it is νt (AH ), the probability of being in a fixed measurable set at a fixed time. Section 7 is included precisely because no global divergence forbids transient swelling: a global functional can be small while a single set’s mass is large. 2 How the bound p is structured. Theorem 5.1 is downstream of the standard L (π) contraction −mt 2 , which is in [Pav14, MV00, AMTU01, BGL14]. The new step is pairing ∥ut − 1∥L2 (π) ≤ χ0 e this contraction with the centered indicator 1AH − π(AH ) rather than the raw indicator. Centering replaces the second moment π(AH ) by the variance π(AH )(1 − π(AH )), which is what makes the bound informative for rare sets. The centering device is the “warm-start” trick standard in geometric sampling [LS93, LV07], applied here to a fixed measurable set in conjunction with the chi-squared contraction. The contribution is not in any single ingredient but in the synthesis: a static dimensional bound on π(AH ) (Section 4, sublevel volume plus smoothness, in the spirit of Gaussian concentration but with explicit constants), the L2 -to-set-mass conversion (Section 5), and the maximum-principle ceiling (Theorem 5.2) that closes the transient window from above.
What is asked of the geometry. The local rate λAH of Definition 1 is positioned carefully relative to two existing notions. In [ZLC17] a Cheeger constant of the target region is used to bound the SGLD hitting time of approximate stationary points; this and λAH are siblings, both measuring how isolated a set is under the dynamics, but [ZLC17] uses it on the favorable side of the dynamics (entering a good set) while Theorem 6.1 uses it on the unfavorable side (avoiding a bad set), so the resulting inequalities run in opposite directions. In the metastability literature [MS14, BEGK04, BGK05] the analogous quantity is the Eyring-Kramers rate of escape from a metastable well, which equals λAH in case (b) of the Section 6.3 taxonomy (far-tail ball). That literature does not define a comparable quantity for cases (a) and (c), the off-shell ball at the mode and the shell-slice, since neither is a basin. Definition 1 handles all three uniformly by defining λAH through the spectral measure of the centered indicator rather than through a Dirichlet-energy Rayleigh quotient, which is the technical move that makes the definition work on indicators outside the Dirichlet domain (Section 6.1). What is not improved. This paper does not strengthen the global Poincaré constant under logconcavity; the rate m from Brascamp-Lieb [BL76], from the Bakry-Émery Γ2 derivation [BE85, BGL14], and, in the non-strongly-log-concave case, from the KLS work [LV24] and the surveys of Cattiaux-Guillin [CG14], is used as a black box. Similarly, the discrete-time Langevin Monte Carlo analyses [CEL+ 22, EHZ22] give convergence under weaker functional inequalities than the chisquared bound used here (Poincaré, Latala-Oleszkiewicz, modified log-Sobolev); a future discretetime version of the present results would build on them rather than compete with them. Framing. The motivation is to bound the probability that a Langevin training trajectory occupies a designated failure region at any finite time, in a high-dimensional parameter space. This places the paper next to [RRT17, ZLC17] in motivation, with a complementary deliverable: those papers bound when training reaches a good set, while this paper bounds whether training is currently in a bad set, in two grades, shape-free (Theorem 5.1) and shape-aware (Theorem 6.1), which makes the cost of geometric ignorance explicit. The OU and shell-slice examples of Section 7 show the gap between the two grades is real and exponential in d, which is the strongest case for why a local rate is needed and not just a faster global rate. 3. Setup The dynamics and its equilibrium. With J ∈ C 2 (Rd ) and minimizer Pn , J(Pn ) = 0, the stationary distribution of the SDE is the Gibbs measure Z 1 −2J(Q)/σ2 2 , Z= e−2J(Q)/σ dQ. π(Q) = e Z Rd
AVOIDING UNSAFE SETS WHEN TRAINING WITH LANGEVIN DYNAMICS
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This captures the distribution of QT at the end of training. The law νt of Qt solves the Fokker-Planck equation
(1)
∂t νt = ∇ · (νt ∇J) +
σ2 ∆νt , 2
νt t=0 = ν0 .
Remark (consequence invariance). The dynamics depends on the loss landscape only through J: the drift is −∇J, the noise is independent of everything, and ν0 is assumed independent of any auxiliary objective. Replacing any function the procedure does not see by an arbitrary other function leaves the trajectory law unchanged. This rules out by construction the mesa-optimization [HvMM+ 19] and reward-hacking [SHKK22] failure modes, in which a training procedure ostensibly minimizing J implicitly steers toward a hidden objective. The contrast is the implicit-bias literature for discrete-time SGD [BD21, SDBD21]: finite-step discretization introduces implicit regularizers (such as ∥∇J∥2 /2) on top of J, whereas the continuous-time SDE studied here does not. Convexity and smoothness. Most results assume the two-sided Hessian envelope m I ⪯ ∇2 J(Q) ⪯ L I
for all Q ∈ Rd ,
0 < m ≤ L,
which integrates to the quadratic envelope m L ∥Q − Pn ∥2 ≤ J(Q) ≤ ∥Q − Pn ∥2 . 2 2 The upper bound (L-smoothness) alone suffices for the static dimensional bound of Section 4; the lower bound (m-strong convexity) is what supplies the spectral gap used in Sections 5 and 6. The failure region. We take AH = {Q ∈ Rd : aH (Q) > α} for a measurable alarm function aH ≥ 0 with aH (Pn ) = 0, and we assume a monotone link to the loss, J(Q) ≥ ψ(aH (Q)),
for a strictly increasing ψ : [0, ∞) → [0, ∞) with ψ(0) = 0. By monotonicity AH ⊆ {J ≥ ψ(α)}: triggering the alarm costs at least ψ(α) in loss. This energy gap is what makes AH rare under π. Chi-squared divergence of the start. When the initial law matters we record Z 2 Z 2 ν0 ν0 2 2 − 1 dπ = dQ − 1. χ0 := χ (ν0 ∥ π) = π π 4. Static mass bound: probability at the end of training The stationary measure π describes QT for T large, so π(AH ) is the probability of ending training inside the failure region. We show it is exponentially small in d. The mechanism is dimensional: π spreads over a large volume, and the rare, high-loss set AH captures exponentially little of it. This section uses only L-smoothness and a volume bound on AH .
Dimensionless variables. Define the natural length scale, the dimensionless energy variable, and the dimensionless gap πσ 2 2t 2ψ(α) ℓ2 := , s := 2 , α̂ := . L σ σ2
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Sublevel-volume bound. Let V (t) := {Q ∈ AH : J(Q) ≤ t} be the Lebesgue volume of the part of AH below loss level t. Assume V (t) ≤ Φ(t) for some non-decreasing Φ with Φ(t) = 0 for t < ψ(α), and set the dimensionless sublevel volume Φ̂(s) := Φ(sσ 2 /2) ℓd , normalized by the volume cell ℓd . It satisfies Φ̂(s) = 0 for s < α̂.
Proposition 4.1 (dimensionless static mass bound). Under ∇2 J ⪯ L I and J(Pn ) = 0, Z ∞ π(AH ) ≤ Φ̂(s) e−s ds = ES∼Exp(1) Φ̂(S). 0
R∞ 2 2 Proof. Write π(AH ) = N/Z. The layer-cake identity e−2J/σ = (2/σ 2 ) 0 1{J≤t} e−2t/σ dt and Fubini give Z ∞ Z ∞ Z ∞ 2 2 −2t/σ 2 −2t/σ 2 d Φ̂(s) e−s ds, V (t) e dt ≤ 2 Φ(t) e dt = ℓ N = 2 σ 0 σ 0 0
the last equality being the substitution s = 2t/σ 2 . The smoothness envelope J(Q) ≤ L2 ∥Q − Pn ∥2 2 2 2 gives e−2J/σ ≥ e−L∥Q−Pn ∥ /σ , so 2 d/2 Z πσ −L∥Q−Pn ∥2 /σ 2 Z ≥ e dQ = = ℓd . L Rd □
Divide.
Theorem 4.2 (exponential-in-d static mass bound). Suppose the sublevel volume has the canonical polynomial form Φ(t) = C (t − ψ(α))η+ for some C > 0, η > 0. Define the dimensionless rate and prefactor 2 2 η πσ σ A := log e−α̂ , , K := C Γ(η + 1) L 2
with α̂ = 2ψ(α)/σ 2 . When σ is large enough that πσ 2 > L (equivalently A > 0), π(AH ) ≤ K e−Ad/2 .
For fixed σ, L, and ψ(α), the static mass decays exponentially in the ambient dimension d at rate 2 A/2 > 0, with an additional energy-gap suppression e−α̂ = e−2ψ(α)/σ in the prefactor. Proof. Apply Theorem 4.1 with Φ̂(s) = Ĉ (s − α̂)η+ and substitute u = s − α̂: Z ∞ Z ∞ Ĉ (s − α̂)η+ e−s ds = Ĉ e−α̂ uη e−u du = Ĉ Γ(η + 1) e−α̂ . 0
0
From the dimensionless variables Ĉ = C (σ 2 /2)η /ℓd and ℓd = (πσ 2 /L)d/2 = eAd/2 . gives π(AH
) ≤ K e−Ad/2 .
Substituting □
4.1. The small-σ (Arrhenius) regime and the combined bound. Theorem 4.2 is informative only when πσ 2 > L, that is, when the noise is large or the dimension high. In the opposite regime a complementary barrier argument applies, √ using strong convexity. Since π is (2m/σ 2 )-strongly log-concave p it has Gaussian tails of width σ/ m around Pn , while smoothness places AH at radius at least 2ψ(α)/L from Pn . Gaussian concentration then gives, when ψ(α) > dLσ 2 /(4m), 2(m/L) ψ(α) π(AH ) ≲ exp − , σ2
AVOIDING UNSAFE SETS WHEN TRAINING WITH LANGEVIN DYNAMICS
volume cell
threshold
7
ℓd = (πσ 2 /L)d/2 : π spreads over this
AH
{J = ψ(α)}
subset of
{J ≥ ψ(α)}
ψ(α) p 2ψ(α)/L
energy gap dist
≥
Pn
Figure 2. The level-set cap. The equilibrium π spreads over the volume cell ℓd = (πσ 2 /L)d/2 (dashed circle), p while the failure region AH sits past the threshold {J = ψ(α)} at distance ≥ 2ψ(α)/L from Pn . Both effects suppress π(AH ): volume spreading captures a factor e−Ad/2 , the energy gap contributes the Arrhenius factor 2 e−2ψ/σ . the bare Arrhenius rate 2ψ(α)/σ 2 shaved by the condition number m/L ∈ (0, 1]. Folding the two bounds together through min(e−a , e−b ) = e− max(a,b) and defining the effective dimensional rate 4m ψ(α) , Aeff := max A, L d σ2 gives the combined static bound
π(AH ) ≤ e−Aeff d/2 .
In the large-σ or large-d regime, Aeff = A and spreading does the work. In the small-σ regime σ 2 ≪ 4mψ(α)/(dL), Aeff d/2 = 2mψ(α)/(Lσ 2 ) and the energy barrier does the work. The two regimes meet at the crossover Ad/2 ≈ 2mψ(α)/(Lσ 2 ), where the bound is least tight, and this crossover coincides with the burn-in scale of Section 5. 5. Dynamic mass control along the trajectory The static bound is a statement at stationarity (t → ∞). For finite t we control νt (AH ) = P(Qt ∈ AH ) directly. The main result of this section assumes only the total mass π(AH ), together with strong convexity and a finite initial chi-squared divergence. It is the dimension-free, shape-free bound. Its one weakness, a transient window in which it is uninformative, is exactly the swelling phenomenon of the introduction, and it is shortened in Section 6 by adding geometric information.
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5.1. The common machinery. All bounds in this section and the next track the density ratio νt ut := . π Substituting νt = ut π into equation (1) and using the identity ∇π = − σ22 π ∇J (the chain rule 2 applied to π ∝ e−2J/σ ), the time-independent factors cancel and ut solves the backward Kolmogorov equation σ2 ∂t ut = Lut , L= ∆ − ∇J · ∇, 2 where L is the Langevin generator. Integration by parts against π shows L is self-adjoint in L2 (π) with Dirichlet form Z σ2 E(f ) = −⟨f, Lf ⟩π = |∇f |2 dπ, 2 and generates a conservative Markov semigroup Pt = etL with Pt 1 = 1. These are the standard generator, reversibility, and Dirichlet-form facts for the Langevin diffusion; see Pavliotis [Pav14, Ch. 4] and [BGL14, Sec. 1.6 and 4.2]. Every bound below converts a statement about ut − 1 into a statement about the set mass by pairing against the indicator 1AH . 5.2. The chi-squared spectral-gap bound. Theorem 5.1 (mass control along the trajectory). Assume m I ⪯ ∇2 J and let ν0 have χ20 < ∞. Then for every measurable AH ⊆ Rd and every t ≥ 0, (2)
q P(Qt ∈ AH ) ≤ π(AH ) 1 + χ20 /π(AH ) e−mt .
The same argument gives the symmetric lower bound νt (AH ) ≥ π(AH ) − νt (AH ) → π(AH ) at exponential rate m.
p χ20 π(AH ) e−mt , so
Proof. The argument runs in six steps. The first three are the standard machinery of reversible diffusions, for which we refer to Pavliotis [Pav14, Ch. 4]; the last three are the explicit estimate. Step 1 (change of variables). As recorded in Section 5.1, ut = νt /π solves the backward equation ∂t ut = Lut , obtained by substituting νt = ut π into equation (1) and cancelling the stationary part using ∇π = − σ22 π∇J ([Pav14, Ch. 4]). Step 2 (self-adjointness). Integration by parts against π makes L self-adjoint in L2 (π) with 2 R Dirichlet form E(f ) = σ2 |∇f |2 dπ, and Pt = etL is a Markov semigroup with Pt 1 = 1 (Section 5.1; [Pav14, Ch. 4]). 2 Step 3 (Poincaré inequality). Brascamp-Lieb applied to π ∝ e−2J/σ , whose potential has Hessian I, gives ∇2 (2J/σ 2 ) ⪰ 2m σ2 1 Varπ (f ) ≤ E(f ), m so the spectral gap of −L on L2 (π) is at least m (see [BL76], the Bakry-Émery Γ2 derivation [BE85], [BGL14, Sec. 4.8-4.9], or the textbook statement [Pav14, Ch. 4]). The two factors of 2/σ 2 , one from the log-concavity constant and one from the Dirichlet form, cancel, which is why the rate is m and not 2m/σ 2 . Step 4 (L2 contraction). For any centered g ∈ L2 (π), differentiate the squared norm along the R semigroup R and use, in order, self-adjointness, the Dirichlet form, and Step 3 (the centering Pt g dπ = g dπ = 0 is preserved, so Poincaré applies to Pt g): d ∥Pt g∥2L2 (π) = 2⟨Pt g, LPt g⟩π = −2 E(Pt g) ≤ −2m ∥Pt g∥2L2 (π) . dt Grönwall’s inequality gives ∥Pt g∥L2 (π) ≤ e−mt ∥g∥L2 (π) .
AVOIDING UNSAFE SETS WHEN TRAINING WITH LANGEVIN DYNAMICS
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R Step 5 (applyR to u0 − 1). The constant 1 is2 fixed by R L, so ut − 1 = Pt (u0 − 1). It is centered, (u0 − 1) dπ = ν0 dQ − 1 = 0, with ∥u0 − 1∥L2 (π) = (ν0 /π − 1)2 dπ = χ20 . Step 4 gives
q ∥ut − 1∥L2 (π) ≤ e−mt ∥u0 − 1∥L2 (π) = χ20 e−mt . R Step 6 (from L2 decay to set mass). Since (ut − 1) dπ = 0 we may subtract the constant π(AH ) inside the pairing for free, replacing the indicator by the centered indicator: Z νt (AH ) − π(AH ) = (ut − 1) dπ = ⟨ut − 1, 1AH − π(AH )⟩π .
(3)
AH
Centering tightens Cauchy-Schwarz: the second factor becomes the variance of a Bernoulli(π(AH )) variable, ∥1AH − π(AH )∥2L2 (π) = π(AH )(1 − π(AH )) ≤ π(AH ), rather than the second moment π(AH ), which for a rare set is dramatically smaller. With equation (3), q p νt (AH ) − π(AH ) ≤ ∥ut − 1∥L2 (π) π(AH )(1 − π(AH )) ≤ χ20 π(AH ) e−mt .
Dividing the upper inequality by π(AH ) yields (⋆); keeping the sign gives the symmetric lower bound. □
The centering device in Step 6 is the “warm-start” trick standard in the sampling literature ([LS93], [LV07], [VW19], [CEL+ 22]). The analytic-PDE route to the same L2 decay is in [MV00], [AMTU01], and a sampling-oriented exposition is in [Che24]. Explicit χ20 for a Gaussian start. The constant χ20 is finite under mild conditions. If ν0 = N (Pn , σ02 I) and mI ⪯ ∇2 J ⪯ LI, then writing the requirement that the start be no wider than π allows, one has the closed-form envelope 2 −d/2 −d/2 σ0 σ02 σ2 σ2 2 2 2 χ0 + 1 ≤ 2 − , s := , s := , − + 2m 2L s2− s2+ finite precisely when σ02 < 2s2+ = σ 2 /L. In particular log χ20 = O(d), so χ20 contributes at most an O(d) term to the burn-in below. (The clean Gaussian-Gaussian case J = a2 ∥Q − Pn ∥2 gives equality, and χ20 = 0 when σ02 = σ 2 /(2a), i.e. the start equals π.) 5.3. The maximum-principle ceiling. If the initial law has a bounded density ratio, a second, purely uniform-in-time bound is available, and it complements (⋆) at small t. Proposition 5.2 (maximum-principle ceiling). If M := ∥ν0 /π∥∞ < ∞, then for every measurable AH and all t ≥ 0, νt (AH ) ≤ M π(AH ). Proof. Since Pt is positivity-preserving with Pt 1 = 1, applying it to u0 R≥ 0 gives 0 ≤ ut = Pt u0 ≤ ∥u0 ∥∞ = M pointwise π-a.e. Integrating against π over AH , νt (AH ) = AH ut dπ ≤ M π(AH ). □ R Combining Theorem 5.2 with Theorem Theorem 5.1, and using ∥u0 − 1∥2L2 (π) ≤ ∥u0 ∥∞ u0 dπ − p 1 = M − 1, gives the two-sided sandwich with σt := (M − 1) π(AH )(1 − π(AH )) e−mt , π(AH ) − σt ≤ νt (AH ) ≤ min M π(AH ), π(AH ) + σt . 1 H) The two upper bounds cross at t⋆ = 2m log (M1−π(A −1)π(AH ) : the ceiling M π(AH ) is tighter for t ≤ t⋆ , the relaxation bound for t ≥ t⋆ . The ceiling requires M < ∞, which fails for a point-mass start, and this failure is not cosmetic: it is exactly the case where swelling is real (Section 7).
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5.4. Burn-in time and the combined safety bound. The bound (⋆) is informative once its transient term is at most order one: q 1 χ20 χ20 /π(AH ) e−mt ≤ 1 ⇐⇒ t ≥ log . 2m π(AH ) Below this threshold the bound can exceed 1 and is vacuous; this is the swelling window. Substituting the static bound π(AH ) ≤ K e−Ad/2 from Theorem Theorem 4.2 gives the explicit burn-in time 1 χ2 A t⋆ := log 0 + d. 2m K 4m So t⋆ grows linearly in d at rate A/(4m), plus a d-independent offset set by χ20 and K. For t ≥ t⋆ , combining (⋆) with the static bound gives the combined safety bound P(Qt ∈ AH ) ≤ 2 π(AH ) ≤ 2K e−Ad/2 ,
exponentially small in d at rate A/2. In words: after a burn-in of order d, the Langevin trajectory inherits the dimensionally suppressed safety of the stationary law. 6. Geometric isolation: shortening the burn-in Theorem 5.1 uses the global Poincaré constant m and only the total mass π(AH ). It pays for this generality with the burn-in window, during which it cannot exclude swelling. The window is genuine for sets that lie on the transport path from ν0 to π (Section 7), but for many failure regions of interest it is an artifact of discarding the geometry of AH . This section restores that geometry through a local relaxation rate λAH ≥ m. The resulting bound relaxes at this faster, set-dependent rate, shortening the burn-in by the factor m/λAH , and combined with the maximum-principle ceiling it caps the trajectory mass uniformly in time. 6.1. A pinned-down local relaxation rate. The informal idea is that mass cannot enter AH faster than it can cross the boundary ∂AH , so a well-isolated set should relax faster than the global rate m. Making this precise requires care, because the obvious candidate fails: the Rayleigh quotient of the centered indicator ϕAH := 1AH − π(AH ) is Z σ2 E(ϕAH ) , E(ϕAH ) = |∇1AH |2 dπ = +∞, Varπ (ϕAH ) 2 since ∇1AH is the surface measure on ∂AH . An indicator is not in the domain of the Dirichlet form, so a “restricted Poincaré constant” over step functions is vacuous. The right object is not the Dirichlet energy of the indicator but the rate at which the semigroup actually relaxes it. Definition 1 (local relaxation rate). For measurable AH with 0 < π(AH ) < 1, let ϕAH = 1AH − π(AH ) ∈ L2 (π), which is centered and has ∥ϕAH ∥2L2 (π) = π(AH )(1 − π(AH )). The local relaxation rate of AH is the best exponential rate at which the semigroup contracts this centered indicator, λAH := sup λ ≥ 0 : ∥Pt ϕAH ∥L2 (π) ≤ ∥ϕAH ∥L2 (π) e−λt for all t ≥ 0 .
Equivalently, by the spectral theorem for the self-adjoint operator −L, with µϕ (dλ) := d⟨Eλ ϕAH , ϕAH ⟩π the scalar spectral measure of ϕAH , λAH = inf supp(µϕ ). Because ϕAH ⊥ 1 and the Poincaré inequality of Theorem 5.1 places the spectrum of −L on the nonconstant subspace in [m, ∞), we have supp(µϕ ) ⊆ [m, ∞), hence λAH ≥ m.
The rate is exactly m when ϕAH overlaps the slowest mode of −L, and strictly larger when that overlap vanishes. This is the spectral content of “isolation”: an isolated set is one whose indicator is orthogonal to the slow modes.
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Conductance as the geometric diagnostic. The quantity that reads off isolation from the geometry of AH is the conductance under the Langevin diffusion, with diffusion coefficient σ 2 /2, R σ2 2 ∂AH π dS . hAH := min π(AH ), π(AcH )
Say AH is flux-isolated if hAH is bounded below independently of d. A small conductance forces a slow-mode overlap and hence λAH close to m, while for the isolated families of Section 6.3 one verifies λAH > m directly by symmetry or barrier comparison. We do not assert a universal one-line Cheeger lower bound λAH ≥ h2AH /2 for this spectral-measure quantity; the conductance is used as the geometric diagnostic, and the lower bound is established per family. 6.2. The flux-isolation theorem. Theorem 6.1 (mass control at the local rate). Assume mI ⪯ ∇2 J and χ20 < ∞. Then for every measurable AH with 0 < π(AH ) < 1 and every t ≥ 0, q q χ20 π(AH ) (1 − π(AH )) e−λAH t ≤ χ20 π(AH ) e−λAH t , νt (AH ) − π(AH ) ≤ p equivalently νt (AH ) ≤ π(AH ) 1 + χ20 /π(AH ) e−λAH t . If in addition M := ∥ν0 /π∥∞ < ∞, then combining with Theorem 5.2, q νt (AH ) ≤ min M π(AH ), π(AH ) + χ20 π(AH ) e−λAH t .
This is Theorem 5.1 with the global rate m replaced by the local rate λAH ≥ m. For a flux1 isolated set the burn-in time shrinks from 2m log(χ20 /π(AH )) to 2λ1A log(χ20 /π(AH )), a saving by H the factor m/λAH , and the min above caps the trajectory mass at M π(AH ) for all t, so no swelling above that ceiling ever occurs. Proof. Write ϕ := ϕAH = 1AH −π(AH ), so ∥ϕ∥2L2 (π) = π(AH )(1−π(AH )). Since ut −1 = Pt (u0 −1) R and (ut − 1) dπ = 0, centering the indicator and using self-adjointness of Pt in L2 (π), νt (AH ) − π(AH ) = ⟨ut − 1, ϕ⟩π = ⟨Pt (u0 − 1), ϕ⟩π = ⟨u0 − 1, Pt ϕ⟩π .
By the spectral representation in Definition 1, with supp(µϕ ) ⊆ [λAH , ∞), Z Z −2λAH t 2 −2λt ∥Pt ϕ∥L2 (π) = e dµϕ (λ) ≤ e dµϕ (λ) = e−2λAH t ∥ϕ∥2L2 (π) . p
χ20 , and ∥ϕ∥2L2 (π) = π(AH )(1 − π(AH )) ≤ π(AH ) give q νt (AH ) − π(AH ) ≤ ∥u0 − 1∥L2 (π) ∥Pt ϕ∥L2 (π) ≤ χ20 π(AH )(1 − π(AH )) e−λAH t .
Cauchy-Schwarz, ∥u0 − 1∥L2 (π) =
Dividing the upper inequality by π(AH ) gives the multiplicative form. The min statement adjoins the maximum-principle ceiling of Theorem 5.2. □ Remark (why not the linear prefactor). The qualitatively stronger statement |νt (AH ) − π(AH )| ≤ (M − 1)π(AH ) e−λAH t , with prefactor linear in π(AH ) and hence genuinely free of any burn-in, would follow from an L1 (π) contraction ∥Pt ϕ∥L1 (π) ≲ ∥ϕ∥L1 (π) e−λAH t paired against ∥u0 − 1∥∞ ≤ M − 1. The semigroup is an L1 (π) contraction but not at a positive rate in general; obtaining the exponential L1 rate requires hypercontractivity (available here from the Bakry-Émery log-Sobolev inequality) to lift L1 to L2 after a fixed lag, which yields the global log-Sobolev rate rather than the local rate λAH . We therefore state the rigorous L2 bound and obtain the uniformin-time control by intersecting it with the maximum-principle ceiling, as in the min above.
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(c) shell-slice √ equilibrium shell: π lives on ∥Q∥ ∼ d
(b) far-tail ball
Pn
(a) o-shell ball
Figure 3. High-dimensional taxonomy of failure regions for the quadratic √ loss 1 2 J(Q) = 2 ∥Q∥ , whose equilibrium π concentrates on the shell ∥Q∥ ∼ d. (a) Off-shell ball at the mode: entropic moat, flux-isolated, λAH ≥ 2m. (b) Far-tail ball: energetic moat (Arrhenius barrier), flux-isolated. (c) Shell-slice: mass flows freely along the shell, λAH → 0, not flux-isolated, only the ceiling M π(AH ) survives. Theorem 6.1 applies to (a) and (b), is vacuous for (c). 2 Remark (the off-shell ball gains a factor). For J(Q) = m 2 ∥Q∥ the nonconstant eigenfunctions of −L are Hermite polynomials, with the slowest modes (eigenvalue m) the linear coordinates Qi . R For a ball centered at the mode, AH = {∥Q∥ < r}, symmetry gives ⟨ϕAH , Qi ⟩π = AH Qi dπ = 0 for every i, so ϕAH is orthogonal to the entire eigenvalue-m subspace and λAH ≥ 2m. A shell-slice, by contrast, has nonzero overlap with the linear modes, so λAH = m and Theorem 6.1 reduces to Theorem 5.1: no improvement, consistent with the swelling of Section 7.
6.3. High-dimensional taxonomy. The payoff of λAH is in the high-dimensional geometry of a strongly log-concave π, where the mass√concentrates on a thin shell. For the prototype J(Q) = 1 2 d d with O(1) width. Three families of small-π-mass failure 2 ∥Q∥ on R , π concentrates on ∥Q∥ ≈ regions behave very differently. √ • Off-shell ball at the mode, AH = {∥Q∥ < r} with r ≪ d. The mode has lowest loss, but vanishing surface area faces the bulk: an entropic moat. Here π(AH ) ∼ e−d/2 while the boundary flux is comparably small, so hAH and hence λAH are bounded below. Flux-isolated; Theorem 6.1 applies. √ • Far-tail ball, AH = {∥Q − Q0 ∥ < r} with ∥Q0 ∥ ≫ d. A genuine energy barrier of height ∼ ∥Q0 ∥2 /2: hitting times grow exponentially (Eyring-Kramers) and λAH inherits the barrier-induced gap. Flux-isolated; Theorem 6.1 applies.
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√ • Thin shell-slice, an angular wedge of the equilibrium shell ∥Q∥ ≈ d. Equilibrium mass is small, but mass transports freely along the shell with no barrier, so hAH → 0 and λAH → 0. Not flux-isolated; Theorem 6.1 is vacuous and only the ceiling M π(AH ) survives. The slogan: π(AH ) small is not enough to keep νt (AH ) small for all t; one also needs λAH bounded below. The shell-slice is the high-dimensional version of the swelling example, quantified next. 7. Sharpness and worked example: Ornstein-Uhlenbeck and the shell This section makes the swelling phenomenon quantitative for the exactly solvable OrnsteinUhlenbeck (OU) case. It shows that the burn-in window of Section 5 and the flux-isolation hypothesis of Section 6 are both necessary, and it exhibits the precise geometry, an angular slice of the equilibrium shell, where the transient overshoot is exponential in d. 1P Take the quadratic loss J(Q) = 2 i λi Q2i with 0 < λ1 ≤ · · · ≤ λd , so the dynamics decouples into coordinatewise OU processes and π = N (0, Σ) with Σ = diag(σ 2 /(2λi )). Start from an isotropic Gaussian ν0 = N (0, σ02 I). Two quantities organize the analysis: the transient overshoot factor Cbad :=
νt⋆ (AH ) ν0 (AH )
at the worst time t⋆ , and the absolute mass νt (AH ) itself. 7.1. Isotropic case. Let J(Q) = a2 ∥Q∥2 and let AH be confined to a level-set shell at radius B ⋆ . With the single dimensionless ratio u :=
(B ⋆ )2 d σ02
(squared shell radius over squared initial typical radius),
a direct Gaussian computation gives d Cbad = exp ϕ(u) , 2
ϕ(u) := u − 1 − log u ≥ 0,
with equality ϕ(u) = 0 only at u = 1. So the overshoot is exponential in d at rate 12 ϕ(u) whenever the shell radius is mismatched to the initial scale. This is the high-dimensional swelling: a set of tiny mass can gain an eΘ(d) factor in transit. The saving grace in the isotropic case is that νt stays isotropic, so its mass on any shell set is bounded by the angular fraction of the shell, νt (AH ) ≤ Rshell
(angular fraction of AH on S d−1 ).
If Rshell is itself exponentially small in d, the overshoot is absorbed before it reaches absolute mass. This is exactly the dichotomy of Section 6: the shell-slice has small π(AH ) (equilibrium reason) but is not flux-isolated (transit reason), so the burn-in window is real and Cbad measures its depth. 7.2. Anisotropic case. With distinct λi , the coordinates run at rates 2λi . Writing ρ0,i = λi σ02 and the per-coordinate ratios ui = Q20,i /σ02 , the overshoot factors out: " d # 1X Cbad ≤ exp ϕ(ui ) . 2 i=1
Equality requires the per-coordinate peak times to align, which generically fails once the λi spread, so the effective dimension deff = #{i : ui far from 1} is governed by the geometry of AH relative to the initial scale. Anisotropy also destroys the uniform angular density, weakening the angular bound to d Y σmax (t) νt (AH ) ≤ Rshell · , σi (t) i=1
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where p σ (t) is the time-t standard deviation in coordinate i. The prefactor is 1 at t = 0 and grows Q i to i λi /λ1 as t → ∞, exponential in d for a spread spectrum.
7.3. Bounded condition number. The clean controllable case is a bounded condition number √ K := λd /λ1 together with an initial scale chosen in range, σ02 ∈ [1/λd , 1/λ 2 1 ] (in the σ = √ 2 normalization). Then σi (t) stays in [1/λd , 1/λ1 ] for all i and t, so σmax /σi ≤ K and the angular bound becomes uniform in time, νt (AH ) ≤ Rshell · K d/2
for all t ≥ 0.
Three regimes: K = 1 recovers the isotropic bound; K − 1 = O(1/d) keeps K d/2 = O(1); a fixed K > 1 gives exponential growth at rate 12 log K. The bound is useful when Rshell decays faster than √ K d/2 grows, that is for AH of angular radius below arcsin(1/ K) on S d−1 . 7.4. Combined statement and where dimension enters. Putting the transit and angular bounds together, Q νt (AH ) ≤ min Cbad · ν0 (AH ), Rshell · i σmax (t)/σi (t) .
P Two independent sources of exponential-in-d growth appear: the overshoot Cbad = exp[ 21 i ϕ(ui )], Q controlled by the geometry of AH through the ui ; and the angular amplification i σmax /σi ≤ K d/2 , controlled by the spectral spread of J. Avoiding both requires either placing AH in a low-deff subspace with the initial law matched to stationarity in the complementary directions, or a bounded K with σ0 in range and AH angularly thin. This is the constructive counterpart of the flux-isolation hypothesis of Theorem 6.1. 7.5. The one-dimensional swelling example. The simplest instance, recovering the introduction: J(Q) = Q2 /2 on R with σ 2 = 2, so π = N (0, 1) and m = 1. Start at ν0 = δ10 and take AH = [4, 6]. Then ν0 (AH ) = 0, π(AH ) ≈ 10−5 , but νt = N (10e−t , 1 − e−2t ), so at t = log 2 the law is N (5, 3/4) and νt (AH ) ≈ 0.7, an overshoot of about 105 . Here M = ∞ (point mass), so the ceiling Theorem 5.2 is vacuous, and AH is the one-dimensional analogue of a shell-slice on the transport path, so λAH is not bounded below and Theorem 6.1 does not apply. Both protective hypotheses fail at once, which is why this is the canonical demonstration that strong convexity controls the relaxation rate but not the transient peak. 8. Summary The picture has three layers. At stationarity, the failure region has exponentially-small-in-d mass, π(AH ) ≤ Ke−Ad/2 (Theorem 4.2), with a complementary Arrhenius rate in the small-noise regime. Along the trajectory, a shape-free bound (Theorem 5.1) relaxes to twice this static mass after a burn-in of order d, using only the total mass and the global gap m. Adding geometric information, a local relaxation rate λAH ≥ m (Definition Definition 1, Theorem 6.1) replaces the global rate by a faster set-dependent one for flux-isolated sets, shortening the burn-in by the factor m/λAH ; combined with the maximum-principle ceiling it caps the trajectory mass uniformly in time. The two grades of dynamic bound are the two ends of one tradeoff. Total-mass control is universal and dimension-free but transiently uninformative; shape-aware control shortens the transient and, via the ceiling, caps the mass uniformly, but requires verifying that AH is geometrically isolated and that the start has a bounded density ratio. The worked OU example (Section 7) shows the gap between them is real: an angular shell-slice has tiny equilibrium mass yet an eΘ(d) transient overshoot, and only the flux-isolation hypothesis rules it out. Strong convexity sets the relaxation rate; the geometry of AH decides whether there is a bulge before relaxation wins.
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References [AMTU01]
Anton Arnold, Peter Markowich, Giuseppe Toscani, and Andreas Unterreiter. On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker–Planck type equations. Communications in Partial Differential Equations, 26(1–2):43–100, 2001. [BD21] David G. T. Barrett and Benoit Dherin. Implicit gradient regularization. In International Conference on Learning Representations (ICLR 2021), 2021. [BE85] Dominique Bakry and Michel Émery. Diffusions hypercontractives. In Jacques Azéma and Marc Yor, editors, Séminaire de Probabilités XIX 1983/84, volume 1123 of Lecture Notes in Mathematics, pages 177–206. Springer, Berlin, Heidelberg, 1985. [BEGK04] Anton Bovier, Michael Eckhoff, Véronique Gayrard, and Markus Klein. Metastability in reversible diffusion processes I: Sharp asymptotics for capacities and exit times. Journal of the European Mathematical Society, 6(4):399–424, 2004. [BGK05] Anton Bovier, Véronique Gayrard, and Markus Klein. Metastability in reversible diffusion processes II: Precise asymptotics for small eigenvalues. Journal of the European Mathematical Society, 7(1):69–99, 2005. [BGL14] Dominique Bakry, Ivan Gentil, and Michel Ledoux. Analysis and Geometry of Markov Diffusion Operators, volume 348 of Grundlehren der mathematischen Wissenschaften. Springer, Cham, 2014. [BL76] Herm Jan Brascamp and Elliott H. Lieb. On extensions of the Brunn–Minkowski and Prékopa–Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. Journal of Functional Analysis, 22(4):366–389, 1976. [BRG+ 26] Yoshua Bengio, Oliver Richardson, Tomáš Gavenčiak, Michael Cohen, Rory Svarc, Damiano Fornasiere, Gael Gendron, David Hyland, Aton Kamanda, Adam Oberman, Francis Rhys Ward, Anna Gavenčiak, Jacob Livingston Slosser, Vincent Mai, Iulian Serban, and Joumana Ghosn. Safety from honesty in a disinterested AI predictor. arXiv:2606.29657 [cs.AI], 2026. https://arxiv.org/abs/2606.29657. [CEL+ 22] Sinho Chewi, Murat A. Erdogdu, Mufan Bill Li, Ruoqi Shen, and Matthew S. Zhang. Analysis of Langevin Monte Carlo from Poincaré to log-Sobolev. In Proceedings of the 35th Conference on Learning Theory (COLT 2022), volume 178 of Proceedings of Machine Learning Research, pages 1–2. PMLR, 2022. [CG14] Patrick Cattiaux and Arnaud Guillin. Semi log-concave Markov diffusions. Séminaire de Probabilités, XLVI:231–292, 2014. Surveys the Poincare constant under (semi) log-concavity, a standing assumption of Theorem 4.1. [Che24] Sinho Chewi. Log-Concave Sampling. 2024. Book draft, available at https://chewisinho.github.io/ main.pdf. [EHZ22] Murat A. Erdogdu, Rasa Hosseinzadeh, and Shunshi Zhang. Convergence of Langevin Monte Carlo in chi-squared and Rényi divergence. In Proceedings of the 25th International Conference on Artificial Intelligence and Statistics (AISTATS 2022), volume 151 of Proceedings of Machine Learning Research, pages 8151–8175. PMLR, 2022. [HLLL19] Wenqing Hu, Chris Junchi Li, Lei Li, and Jian-Guo Liu. On the diffusion approximation of nonconvex stochastic gradient descent. Annals of Mathematical Sciences and Applications, 4(1):3–32, 2019. [HvMM+ 19] Evan Hubinger, Chris van Merwijk, Vladimir Mikulik, Joar Skalse, and Scott Garrabrant. Risks from learned optimization in advanced machine learning systems. Technical report, Machine Intelligence Research Institute, 2019. [LS93] László Lovász and Miklós Simonovits. Random walks in a convex body and an improved volume algorithm. Random Structures & Algorithms, 4(4):359–412, 1993. [LTE19] Qianxiao Li, Cheng Tai, and Weinan E. Stochastic modified equations and dynamics of stochastic gradient algorithms I: Mathematical foundations. Journal of Machine Learning Research, 20(40):1–47, 2019. [LV07] László Lovász and Santosh Vempala. The geometry of logconcave functions and sampling algorithms. Random Structures & Algorithms, 30(3):307–358, 2007. [LV24] Yin Tat Lee and Santosh S. Vempala. Eldan’s stochastic localization and the KLS conjecture: Isoperimetry, concentration and mixing. Annals of Mathematics, 199(3):1043–1092, 2024. [MHB17] Stephan Mandt, Matthew D. Hoffman, and David M. Blei. Stochastic gradient descent as approximate Bayesian inference. Journal of Machine Learning Research, 18(134):1–35, 2017. [MS14] Georg Menz and André Schlichting. Poincaré and logarithmic Sobolev inequalities by decomposition of the energy landscape. The Annals of Probability, 42(5):1809–1884, 2014. [MV00] Peter A. Markowich and Cédric Villani. On the trend to equilibrium for the Fokker–Planck equation: an interplay between physics and functional analysis. Matemática Contemporânea, 19:1–29, 2000.
16
ADAM OBERMAN
[Pav14] [RRT17]
[SDBD21]
[SHKK22]
[VW19]
[WT11]
[ZLC17]
Grigorios A. Pavliotis. Stochastic Processes and Applications: Diffusion Processes, the Fokker–Planck and Langevin Equations, volume 60 of Texts in Applied Mathematics. Springer, New York, 2014. Maxim Raginsky, Alexander Rakhlin, and Matus Telgarsky. Non-convex learning via stochastic gradient Langevin dynamics: A nonasymptotic analysis. In Proceedings of the 30th Conference on Learning Theory (COLT 2017), volume 65 of Proceedings of Machine Learning Research, pages 1674–1703. PMLR, 2017. Samuel L. Smith, Benoit Dherin, David G. T. Barrett, and Soham De. On the origin of implicit regularization in stochastic gradient descent. In International Conference on Learning Representations (ICLR 2021), 2021. Joar Skalse, Nikolaus H. R. Howe, Dmitrii Krasheninnikov, and David Krueger. Defining and characterizing reward hacking. In Advances in Neural Information Processing Systems 35 (NeurIPS 2022), 2022. Santosh S. Vempala and Andre Wibisono. Rapid convergence of the unadjusted Langevin algorithm: Isoperimetry suffices. In Advances in Neural Information Processing Systems 32 (NeurIPS 2019), pages 8094–8106, 2019. Max Welling and Yee Whye Teh. Bayesian learning via stochastic gradient Langevin dynamics. In Proceedings of the 28th International Conference on Machine Learning (ICML 2011), pages 681–688, 2011. Yuchen Zhang, Percy Liang, and Moses Charikar. A hitting time analysis of stochastic gradient Langevin dynamics. In Proceedings of the 30th Conference on Learning Theory (COLT 2017), volume 65 of Proceedings of Machine Learning Research, pages 1980–2022. PMLR, 2017.
(A. Oberman) LawZero and Mila – Quebec AI Institute and McGill University Email address: [email protected]