Provable Convergence Bounds for Hybrid Dynamical Sampling and Optimization

Authors: Matthew X. Burns, Qingyuan Hou, Michael Huang
Primary area: probabilistic methods (Bayesian methods, variational inference, sampling, UQ, etc.)
Keywords: langevin, accelerators, sampling, optimization, diffusion, analog computing

Abstract:
Analog dynamical accelerators (DXs) are a growing sub-field in computer architecture research, offering order-of-magnitude gains in power efficiency and latency over traditional digital methods in several machine learning, optimization, and sampling tasks. However, limited-capacity accelerators require hybrid analog/digital algorithms to solve real-world problems, commonly using large-neighborhood local search (LNLS) frameworks. Unlike fully digital algorithms, hybrid LNLS has no non-asymptotic convergence guarantees and no principled hyperparameter selection schemes, particularly limiting cross-device training and inference.


In this work, we provide non-asymptotic convergence guarantees for hybrid LNLS by reducing to block Langevin Diffusion (BLD) algorithms.
Adapting tools from classical sampling theory, we prove exponential KL-divergence convergence for randomized and cyclic block selection strategies using ideal DXs. With finite device variation, we provide explicit bounds on the 2-Wasserstein bias in terms of step duration, noise strength, and function parameters. Our BLD model provides a key link between established theory and novel computing platforms, and our theoretical results provide a closed-form expression linking device variation, algorithm hyperparameters, and performance.

Decision: accept (poster)
Reviewer scores: 8, 6, 8, 8, 3

Reviews and discussion:
<thread>
creation date:
1729967699987

thread_type:
official_review

<thread_object>
creator: reviewer_kqh5

<content>
summary:
The authors analyze analog accelerators and large-neighborhood local search (LNLS) frameworks. Reducing LNLS to block Langevin Diffusion algorithms, the paper provides convergence guarantees using the tools from the classical sampling theory.

strengths:
Before I start my review, I should acknowledge that the topics of this paper, including Langevin Diffusion (BLD) algorithms, Analog dynamical accelerators, SDEs, LNLS frameworks, are very different from what I do in my research. My main field of interest is mathematical optimization.

I have no doubt that analog computations are an important direction to accelerate the current expensive digital algorithms: the topic is important and relevant. The author's attempt at approaching the issue is unusual (in a good way) and nontrivial.

weaknesses:
The main weakness is that it is challenging to read the paper. From the beginning, the authors introduce many uncommon words and terms that are very unlikely to be easily understood by most researchers from the ICLR community. I think the introduction and the background should be significantly simplified for a broad audience. For instance, the main object of interest is LNLS, but the authors do not try to explain the mathematical foundation and the background of LNLS. Figure 1 is too abstract to understand LNLS.  

Other weaknesses and questions:
1. Why do you consider Block Langevin Diffusion? Why can't we optimize w.r.t. all variables?
2. Lines 345-347: I guess there should be $||x - y||^2$ instead of $||x^2 - y^2||$
3. Assumption 5: How does the function inside the integral depends on $t$?
4. Assumption 6: In my experience, this is a very *uncommon* assumption. Also, Assumption 3 is also very uncommon.
5. Theorem 3: This theorem yields the convergence rate $\log \frac{1}{\varepsilon} + \varepsilon,$ which is $\geq 1.$ What If one wants to make the Wasserstein distance less or equal $0.001$? 

Unfortunately, reading this paper, I'm not convinced that the reduction to Langevin Diffusion algorithms can not help to improve and explain analog accelerators. At the same time, I do not have expertise in these fields, so I choose low confidence.

questions:
(see weaknesses)
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</thread>
<thread>
creation date:
1730199483538

thread_type:
official_review

<thread_object>
creator: reviewer_ycu8

<content>
summary:
This paper presents the first explicit probabilistic convergence guarantees for hybrid Langevin Noise Likelihood Sampling (LNLS) algorithms in activation sampling and optimization. The authors reduce hybrid LNLS to block sampling using continuous-time Langevin diffusion sub-samplers, analyzing randomized and cyclic block selection rules. They demonstrate that ideal accelerators converge exponentially under a log-Sobolev inequality, while finite device variation introduces bias in the Wasserstein distance. Numerical experiments on a toy Gaussian sampling problem illustrate the effects of device variation and hyperparameters.

strengths:
- The paper is clearly structured, with each theorem building on the previous results to form a coherent narrative.

-  The findings of the paper are supported by clear numerical experiments.

weaknesses:
N/A

questions:
N/A
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</thread>
<thread>
creation date:
1730228684132

thread_type:
official_review

<thread_object>
creator: reviewer_o8ag

<content>
summary:
Analogue accelerators are attracting renewed interest, promising far superior power efficiency and latency compared to digital methods for problems in machine learning, optimization, and sampling.
While the theoretical understanding of analogue accelerators has evolved quickly, significant gaps remain when taking into account a fundamental practical aspect of those devices:
their limited capacity makes it necessary to solve larger problems "piece-by-piece".
That is, the device operates on a subset of the problem at a time while keeping the rest constant, progressively iterating over the entire problem.

The authors find a rich connection between this constraint and the theory of block Langevin diffusion algorithms.
With the connection to well-established theory, the authors adapt existing methods to obtain novel bounds on the performance of a class of hybrid analogue-digital algorithms and non-asymptotic guarantees for their convergence when accounting for non-ideal devices (which are inevitable in practice).

strengths:
1. The paper is well-written.
    The exposition is clear with remarkably few typos, the motivation is put clearly, the authors bring and discuss relevant limitations, and they provide some discussion after presenting design choices, results, and new ideas, in general.
    They also highlight key ideas underlying their proofs.

2. The work is decently contextualized.
    The contribution relies significantly on many existing works, which the authors appear to recognize and discuss fairly.

3. The authors are upfront and honest about the limitations of their work.

4. The general topic is interesting and timely.

5. The reduction to block Langevin diffusion seems like a natural (and, thus, promising) approach to the problem.
    I believe it should motivate several follow-up works.
    The approach also yields significantly softer assumptions compared to similar previous results.

6. The results feature valuable properties for practical applications, such as explicit constants, hyperparameter simplification, and the handling of some device variation.

weaknesses:
1. The care mentioned in strength (1) does not extend to the appendices. E.g., Appendix A would greatly benefit from some discussion about the impact and intuitions behind the choices made for the experiments.

2. The last sentences of the paragraph 051-059 ask for some substantiation, but the authors offer no references to back them up. Some further discussion could also solve this issue.

3. Despite strength (2) and my comprehension of the space constraints, I believe the paper relies too heavily on references to explain the concept.
    I do not see the reliance on previous works as a problem in general, as much of it is a side effect of the strong fruitful connection the authors made with consolidated theory.
    Still, at points such as Section 4, I felt like essential details were left to be found in the references.
    I am sure there is some curse of knowledge at play here, which is understandable, but it would be a good use of the authors' sharp eloquence to make the paper a bit more self-contained.
    I apologize for not substantiating this claim with specific examples, but it is hard to do it when my point is precisely that I did not get a good grasp of what was being presented.

4. The role of analogue-to-digital conversion is not discussed.
    While I am not sure how pertinent this is for this particular work (see question 1), ADC bottlenecks are so common in analogue computing that it should deserve at least a mention.

---
### Minor issues and suggestions

I'll only mention typographical matters because I noticed the authors were particularly zealous with that —I spotted the 2-letter `\emph` on line 269!
They did a great job, overall, and those suggestions aim to help them further improve their skills.

1. Consider numbering only the equations that are referenced in the text (rather than all of them).
2. Most colons preceding equations should be removed.
    More generally, equations are an integral part of the text, reading as sentences.
    This also means that equations should be punctuated as such (this issue affects almost all equations in this work).
    Any maths style guide would serve as reference for this. As an example, Section 13.4 of the AMS Style Guide (I'd provide a link, but this is disallowed for reviewers) mentions both issues.
3. Some `\mathrm`s and `\operatorname`s are missing.
    See, for instance, Assumption 1.
4. By eyeballing, I suspect the authors use `||` (double vertical bars) when they should use `\lVert`, `\rVert`, or `\Vert` which ensure proper spacing.
    For example, compare $||x||$ and $\lVert x \rVert$ (the latter is the correct one) or $\mathrm{D}_{\mathrm{KL}}(\mu || \pi)$ and $\mathrm{D_{KL}}(\mu \mathrel{\Vert} \pi)$ with the latter being coded as `\mathrm{D_{KL}}(\mu \mathrel{\Vert} \pi)`.
5. At 235, having the domain of $i$ specified in the definition of $\overline{B}_i$ would be helpful.
6. At 105, mentioning $\beta$ is premature.
    The sentence also references Equation 22 which is 7 pages ahead!
7. In Assumptions 3 and 4, the domain of $\delta$ (denoted by $\mathbf{D}$) is not defined.

questions:
1. In the applications familiar to me, analogue-to-digital conversion tends to be an crucial bottleneck for hybrid analogue/digital accelerators.
    This affects their accuracy, latency, power efficiency, and, most crucially, die footprint which largely determines their cost.
    ADCs are so expensive in so many ways that many applications sacrifice as much precision as possible to minimize their use.

    In this light, how are those aspects relevant to your work?
    Do the experiments take them into account?
    Do previous works on the topic address them?


2. As the authors say, performing experiments with Gaussian distributions allows for closed-form solutions for the 2-Wasserstein distance. Yet, even though the plots from Figure 2 display a $y$-axis with units, it is hard to reason quantitatively reason in terms of $W_2$. Could you provide some general guidance for that? I mean, is a $W_2$ of 1 large? I understand this can be problem-dependent, but some general guidance would be helpful.
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</thread>
<thread>
creation date:
1730568782164

thread_type:
official_review

<thread_object>
creator: reviewer_zah3

<content>
summary:
I am not engaged in research related to this problem, so I am unable to provide an
objective evaluation on this topic. Please disregard my review comments.

strengths:
N/A

weaknesses:
N/A

questions:
1. In this paper, the authors assume that a vector can be decomposed using tensor products or Kronecker products. However, this decomposition does not span the entire Hilbert space, which implies that the conclusions presented in the paper lack generality.

2. All equations lack punctuation and should be corrected.
</content>
</thread_object>

</thread>
<thread>
creation date:
1730703760455

thread_type:
official_review

<thread_object>
creator: reviewer_wnfd

<content>
summary:
This paper concerns analysis of hybrid large neighborhood local search (LNLS) frameworks, in which the authors provide non-asymptotic convergence guarantees for this framework. In particular, an exponential non-asymptotic bound is obtained for the KL divergence of DXs employing two different strategies (randomized and cyclic block) and a bias bound on the 2-Wasserstein distance is established for finite device variation. Numerical experiments supporting the theoretical results developed.

strengths:
This is an interesting paper and I believe the contributions are novel. The authors provide a good literature review and contextualization of their results with respect to the past literature. Moreover, the authors did a good job in identifying the limitations of their work. 

The paper is objective and its contributions are clearly identified.

I did not have time to review all proofs in detail.

weaknesses:
-I believe that a discussion on the performance differences between Random and Cyclic block approaches would be good for clarification (see questions).

questions:
- Can the authors provide further examples of distributions that would satisfy the LSI? How realistic is that assumption in applications?

- Random and Cyclic block approaches seem to produce similar outcomes. What is the motivation to choosing one over another? Is there any intuition on which one should I choose based upon my application?

- In Figure 2 (e), why doesn't the curve associated with \delta=0 match the ideal curve?
</content>
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</thread>
<thread>
creation date:
1732125018496

thread_type:
official_comment

<thread_object>
creator: authors

<content>
title:
Revision Uploaded

comment:
Dear Reviewers and AC,

We have uploaded a revised manuscript in an attempt to address reviewer concerns and clarify our presentation. New text is marked in blue to assist with the reviewing process. 

The primary content changes are:
- Simplification of some overly technical/reference-heavy portions of the Introduction, Background, and Numerical Experiment sections (in line with comments from **Reviewers o8aG and kqh5**)
- A simplification of our assumptions for Theorem 3. Specifically, we removed the exponential integrability assumption as we realized it was unecessary, and could be replaced by a simpler dissipativity assumption (stemming from a comment by **Reviewer kqh5**).
- Added examples of LSI distributions with associated references to the background section (**Reviewer wnfD**)
- Added further discussion of our experimental methods to Appendix A (**Reviewer o8aG**)

Moreover, we have made a myriad of small typographical/presentation fixes, including properly punctuating our equations, fixing typos, and clarifying figures.

We hope that these changes address the Reviewers' concerns, and we look forward to any further feedback.
</content>
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</thread>


Metareview:
metareview:
Summary:
This paper presents a comprehensive analysis of hybrid large neighborhood local search (LNLS) frameworks in the context of analog accelerators, providing non-asymptotic convergence guarantees. The authors reduce LNLS to block sampling using continuous-time Langevin diffusion sub-samplers and analyze both randomized and cyclic block selection approaches. The work establishes exponential convergence under log-Sobolev inequality conditions for ideal accelerators while quantifying bias from finite device variation in the Wasserstein distance.

Main strengths:

- Novel theoretical analysis bridging LNLS with block Langevin diffusion algorithms
- Clear presentation of convergence guarantees with explicit constants
- Thorough experimental validation of theoretical claims
- Practical relevance for analog accelerator implementations
- Careful consideration of real-world constraints like device capacity limitations

Main weaknesses:

- Some sections rely heavily on references without sufficient self-contained explanation
- Technical presentation may be challenging for readers unfamiliar with both sampling theory and analog computing
- Limited discussion of ADC (analog-to-digital conversion) implications
- Appendices could benefit from more detailed experimental discussion