Provable Convergence and Limitations of Geometric Tempering for Langevin Dynamics

Authors: Omar Chehab, Anna Korba, Austin J Stromme, Adrien Vacher
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
Keywords: Sampling, Langevin, Annealing

Abstract:
Geometric tempering is a popular approach to sampling from challenging multi-modal probability distributions by instead sampling from a sequence of distributions which interpolate, using the geometric mean, between an easier proposal distribution and the target distribution. In this paper, we theoretically investigate the soundness of this approach when the sampling algorithm is Langevin dynamics, proving both upper and lower bounds. Our upper bounds are the first analysis in the literature under functional inequalities. They assert the convergence of tempered Langevin in continuous and discrete-time, and their minimization leads to closed-form optimal tempering schedules for some pairs of proposal and target distributions. Our lower bounds demonstrate a simple case where the geometric tempering takes exponential time, and further reveal that the geometric tempering can suffer from poor functional inequalities and slow convergence, even when the target distribution is well-conditioned. Overall, our results indicate that the geometric tempering may not help, and can even be harmful for convergence.

Decision: accept (poster)
Reviewer scores: 6, 6, 8, 8

Reviews and discussion:
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creation date:
1730639870041

thread_type:
official_review

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creator: reviewer_mtxb

<content>
summary:
This work analyzes the convergence rate of Langevin dynamics with geometric tempering (LD-GT), a modification of the Langevin dynamics which attempts to follow the geometric path between a proposal distribution $\nu$ (e.g. a standard Gaussian) and the target distribution $\pi$. 
More precisely, LD-GT with tempering schedule $(\lambda_k)_k \subset [0,1]$ is

$$X_0 \sim \nu, ~~~~ X_{k+1} = X_k + h \nabla \log \mu_k(X_k) + \sqrt{2 h} \epsilon_k,$$

where $\mu_k \propto \nu^{1-\lambda_k} \pi^{\lambda_k}$ is the geometric path (and $\epsilon_k$ are independent Gaussians).

(Per the authors' account of the literature,) LD-GT was proposed since the 1990s, and one of the motivations is the intuition that sampling progressively from the path $\mu_k$ is easier than sampling directly from the target $\pi$, especially if $\pi$ is multi-modal. 

This work's contributions are two-fold:
- Precise convergence guarantees for LD-GT under certain common assumptions on the proposal $\nu$ and the target $\pi$: Poincare inequality (PI), log-Sobolev inequality (LSI), strong log-concavity.\
To this aim, a key sub-question is to estimate the PI or LSI constant of the path $(\mu_k)_k$. Besides showing how to best utilize well-known upper bounds, the authors also identify cases where these constants are surprisingly poor, leading to the next item.
- This work provides evidence that the original intuition motivating LD-GT is wrong, by exhibiting cases where LD-GT must converge very slowly (regardless of the choice of schedule). Remarkably this can even happen for well-conditioned and uni-modal targets $\pi$, for which vanilla LD can be expected to converge fast.

strengths:
This paper addresses a natural question on the convergence of a sampling algorithm. The positive results (first item in "Summary") are of theoretical interest, as they address the technically difficult question of optimizing the upper bounds w.r.t the temperature schedule. The negative findings are surprising: namely, adding geometric tempering may actually slow down Langevin dynamics. This new insight is significant for both theory and practice.

The presentation is very clear and "flows" very nicely. All the technical claims are correct are far as I checked.

weaknesses:
No substantial weaknesses, but the negative results of this paper naturally lead to a question which is not addressed nor mentioned in this paper, see "Questions" below.

Minor comments on the presentation:
- use citep instead of citet on lines 119, 233, 254, 468
- correct typos and/or grammar on lines 271, 326, 420, 496, 1389, 1494, 1838
- justify the fact that chi^2, KL > TV rather than say it "of course" holds (line 493)
- line 988 contains the proof of Corollary 13, not 17
- add details on the argument on line 1824 (I could not reconstruct it using Cauchy-Schwarz, only Jensen)
- use different markers for each curve in Figure 2
- consider using a log scale or showing less iterations in Figure 3
- consider including the example of section 4.1 in Figure 4, in addition to Figure 1 (which shows only $\lambda \in \{0, 0.45, 1\}$)

questions:
The theoretical results in this work suggest that geometric tempering may not help the convergence of Langevin dynamics. Yet tempering is a strategy that is used in practice (per the authors' presentation of the literature). In practice, is tempering observed to lead to improved performance compared to vanilla Langevin dynamics? If yes, is there any intuitive reason why?

Minor questions:
- Would the conclusions, and the analysis techniques, of this paper still apply if one takes $\nu$ to be the Lebesgue measure instead of a probability measure?
- In practice, is the proposal $\nu$ always taken to be a Gaussian? Is it sometimes taken to be the Lebesgue measure? Or multi-modal?
- Is the $\frac{1}{2\alpha_\pi t}$ rate in Proposition 7 (line 433) classical? If so please give a reference.
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<thread>
creation date:
1730706121902

thread_type:
official_review

<thread_object>
creator: reviewer_pomc

<content>
summary:
This paper offers a thorough study of geometric tempering combined with a Langevin MCMC scheme. In particular, a general theory is given which characterizes the error induced by said dynamics for arbitrary tempering schemes. Negative results are then given for the efficacy of tempering schemes (over the naive Langevin dynamics) both in terms of the intermediate distributions' log-Sobolev constants, as well as the worst-case convergence rate, although some regimes where the tempering is beneficial are highlighted.

strengths:
The main positive result on geometric tempering (Theorem 3) seems quite thorough (in that it comprises every reasonable regime of interest) and about as good as one could hope for in this context.

The negative example is very intuitive and is a worthwhile inclusion into the paper. It offers a good characterization about why one might be skeptical about the occasional poor performance of these schemes in practice, and gives good intuition about the heart of the problem (the appearance of multimodality).

The inclusion of more concrete lower bounds is also insightful.

weaknesses:
It would be more helpful if the paper offered more positive examples of instances where the tempering can improve over vanilla Langevin by at least polynomial factors; in particular, a comparative bound would be helpful in Propositions 6, 7.

It would also be good if the paper could explore further the areas where tempering has a provable benefit over Langevin, especially in cases of multimodality where the algorithm would likely be used.

questions:
The following suggestions relate to minor areas of the paper:

In Figure 2, should we not be scaling the $y$-axis logarithmically for a more reasonable demonstration?

The comment after (3) is strange. Probably, you mean to take $kh = t$ for a fixed choice of $t \in \mathbb R$, and then some schedule $h = t/K$ for a set of integers $K$, rather than what is written.

There is a spacing issue in Line 190~191.

Line 240: Lebesgue -> Lebesgue measure.

Line 425: where obtain-> where we obtain

It is a bit strange to cite Durmus 2019 for the Langevin rate in the str. convex + smooth setting, compared to earlier work such as [1].

Durmus, Alain, and Eric Moulines. "High-dimensional Bayesian inference via the unadjusted Langevin algorithm." (2019): 2854-2882.
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</thread>
<thread>
creation date:
1730765106105

thread_type:
official_review

<thread_object>
creator: reviewer_8hqm

<content>
summary:
This paper presents a theoretical analysis of geometric tempering when applied to Langevin dynamics, a popular sampling method in machine learning and statistics. Geometric tempering is a technique that attempts to improve sampling from complex multi-modal distributions by sampling from a sequence of intermediate distributions that interpolate between an easy-to-sample proposal distribution and the target distribution. The authors provide the first convergence analysis under functional inequalities, proving both upper and lower bounds for tempered Langevin dynamics in continuous and discrete time. They also derive optimal tempering schedules for certain pairs of proposal and target distributions.

strengths:
Perhaps surprisingly, the paper's findings are largely negative regarding the effectiveness of geometric tempering. The authors demonstrate that geometric tempering can actually worsen functional inequalities exponentially, even when both the proposal and target distributions have favorable properties. Through theoretical analysis, they show a simple bimodal case where geometric tempering takes exponential time to converge. More strikingly, they prove that similar poor convergence results can occur even with unimodal target distributions that have good functional inequalities. These results suggest that geometric tempering may not only fail to help with convergence but could actually be harmful in some cases, challenging the conventional wisdom about its utility.

weaknesses:
In this paper they consider targets of the form $\nu^{1 - \lambda} \pi^{\lambda}$, where $\nu$ is called the proposal. In many other prior works, the targets are of the form $\pi^{\lambda}$, which corresponds to $\nu$ being an improper uniform distribution. This seems to be the main source of the largely negative results provided in this paper. Could the authors clarify the reason for considering target the above form?

questions:
Please see question abobe
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<thread>
creation date:
1731027160769

thread_type:
official_review

<thread_object>
creator: reviewer_exbp

<content>
summary:
This work studies the convergence guarantee of geometric tempering for the Langevin diffusion and its time-discretization the Langevin algorithm. The authors prove a convergence rate under a general tempering schedule, demonstrating dependency on the isoperimetry of the intermediate probability measures, in particular their log-Sobolev constant. While this constant can be suitably controlled when both measures are strongly log-concave, the authors show that even when both proposal and target densities are unimodal, intermediate measures can suffer from a poor log-Sobolev constant that scales exponentially with the distance between the modes of proposal and target measures.

strengths:
While I am not an expert in annealing or tempering algorithms for sampling, it seems that this is the first paper that proves the convergence of geometric tempering for the Langevin diffusion using functional inequalities, which is interesting. The negative results also provide a good example of why tempering may not work in practice, despite both proposal and target measures having suitable isoperimetry.

weaknesses:
* The related work section could be better structured. For example, breaking into multiple paragraphs and adding paragraph titles could help with readability and following the discussion.
* The lower bound examples hold in dimension 1, and show exponentially bad dependence on the distance between modes. While these bounds are interesting, it is not very intuitive to me why it would be natural for the distance between modes to grow in fixed dimensions. On the other hand, in a high-dimensional setting, it is more intuitive that $m$ grows with square root of dimension. Could it be straightforward to (perhaps only intuitively) extend the lower bounds to high-dimensional settings?

questions:
* I believe for $KL(p_0, \mu_0)$ to disappear in Corollary 5, one needs to set $\lambda_0 = 0$. In that case, it would not be possible to choose $\lambda_t = 1$ for all $t > 0$ in a continuous manner.

* If all $\lambda_i$s are very close to 1 in Theorem 9, we are effectively running vanilla Langevin. In that case, why should we have exponential convergence time?

* The vanilla Langevin analysis only requires the log-Sobolev inequality and smoothness for discretization. Why do we additionally need dissipativity of proposal and target measures here?

    * In fact, the Langevin algorithm is known to convergence under extremely mild conditions, namely a weak Poincaré inequality (which holds for all locally bounded potentials, although without explicit control on the constant) and smoothness of the gradients, see e.g. [1] and references therein. Are there major challenges for obtaining convergence guarantees under (weak) Poincaré inequalities for the tempered Langevin algorithm? 

* Is there a sense in which one can choose optimal proposal distributions $\nu$ when we only know some information about $\pi$?

* I believe a summation over $i$ is missing in Equation (12).

* Some typos:
    * Line 152 missing absolute continuity before “... and $+\infty$ otherwise”.
    * Line 175: potential -> potentials
    * Line 119, 233, 253, 467: missing parentheses in citation
    * Line 238: satisfy -> satisfies
    * Line 336, 339, 383: section … -> Section …
    * Line 420: are unknown -> is unknown
    * Line 425: where we obtain
    * A typo in Line 439 makes the sentence unreadable.




---
[1] A. Mousavi-Hosseini, T. Farghly, Y. He, K. Balasubramanian, M. A. Erdogdu. "Towards a Complete Analysis of Langevin Monte Carlo: Beyond Poincaré Inequality". COLT 2023.
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creation date:
1732050699958

thread_type:
official_comment

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creator: authors

<content>
title:
Response to all reviewers

comment:
We thank all four reviewers for their positive feedback. In this general reply, we address a question that was
shared among reviewers. The remaining comments
are addressed in detail in the individual replies. 


**When does tempering outperform Langevin? (reviewers pomC and mTXb) ---** 
Reviewer pomC asked us to identify settings where
tempering outperforms Langevin. And reviewer mTXb asked
how our results fit with the fact that
tempering is often used successfully in practice.
For both of these questions, we refer to our upper bounds,
Theorems 1 and 3. These results gives rates of convergence
for tempered Langevin dynamics in terms of the inverse log-Sobolev constants
$\alpha_t$ along the tempering path. While it is true that
our lower bound on the log-Sobolev constant in Theorem 4
rules out the possibility of tempering generically improving the log-Sobolev
constant, we emphasize that Theorem 4 only shows
one poorly conditioned example, specific to a pair of proposal and target distributions.
In particular, in any given example,
it may happen that the intermediate log-Sobolev contants $\alpha_t$ 
are significantly better than those of the target $\pi$ (this is, essentially,
the core intuition behind tempering).
In such a case, tempering can be expected to converge more quickly
than vanilla Langevin.

To gain some intuition for why this can happen, 
imagine $\pi$ is a bimodal distribution, $\nu$ is a wide distribution
roughly uniformly spread over the modes,
and the initial-time distribution $p_0$ is concentrated in one mode.
In general, the log-Sobolev constant $\mu_t$ should scale
as the height of the energy barrier between the modes: this is the hill
that a particle must cross to move from one mode to the other.
But, when $\lambda_t$ is small, the target $\mu_t \approx \pi^{\lambda_t}$,
so, in particular,
the height of the energy barrier is significantly smaller.
Thus, the log-Sobolev constants of $\mu_t$ should be significantly better
than those of the target, early in the tempering.

Given such control of the log-Sobolev constants, we could
plug it in to Theorem 1 to obtain a rate with which could
then improve on vanilla Langevin. 
Rigorously obtaining such control is a fascinating open question which builds upon
the theory developed in this work;
we expect that this will delicately depend on the details
of the specific example in consideration. Generally speaking,
the control of log-Sobolev constants for mixtures and other multimodal distribution
is a challenging area of ongoing research,
see [1, 2] for some recent
work
in this direction. 
To finally answer reviewer mTXb's question: 
situations where tempering performs well in practice
can be explained within our theory
as situations where the intermediate log-Sobolev constants
are significantly better than those of the target.
And to respond to reviewer pomC: 
our upper bounds provide a general framework, through 
the log-Sobolev constants, to understand when
tempering does and does not improve on vanilla Langevin.
Although Theorem 4 (and our other lower bounds) rules out generic good behavior of the tempering,
there still exist
situations (such as in Figure 2) in practice where the behavior is better than that
of vanilla Langevin,
and our results
provide a solid foundation for future research into this phenomenon.

[1] Chen, Hong-Bin, Sinho Chewi, and Jonathan Niles-Weed. ``Dimension-free log-Sobolev inequalities for mixture distributions." Journal of Functional Analysis 281.11 (2021): 109236.

[2] Schlichting, André. ``Poincaré and log–sobolev inequalities for mixtures." Entropy 21.1 (2019): 89.
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Metareview:
metareview:
The authors analyze the convergence of tempered Langevin dynamics. Most interestingly, the authors derive lower bounds where tempering leads to exponentially poor convergence. This result is quite novel as agreed upon by all reviewers. On top of this, the authors also provide the first upper bound results on convergence under functional inequality, which is already a nice result in itself. 

No significant criticisms were raised during the review process, and both reviewers with a score of 6 have agreed with acceptance, albeit not good enough to raise the scores to 8. Therefore, I believe this paper is welcomed contribution to the sampling literature and a clear accept.