Kati8/ICLR_2023_2025 huggingface dataset:
authors: Timothee Leleu
authors: Sam Reifenstein
reviews: 
Comments: The authors propose a novel hybrid continuous-discrete algorithm that combines deterministic continuous dynamics with Metropolis-Hastings (MH) steps for ground-state sampling in non-equilibrium dynamics.
Strengths: 1. The combination of chaotic dynamics with an MH step to ensure convergence to a target distribution represents an innovative approach to hybrid sampling.
2. The hybrid algorithm shows potential for improving ground-state sampling performance in combinatorial optimization
3. The authors address parallelization for efficient computation on GPUs
4. The discussion on combining chaotic dynamics with probabilistic methods provides a useful context for researchers working at the intersection of machine learning and statistical physics.
Weaknesses: While the authors present an interesting optimization algorithm, the clarity of the writing is a major concern. The main ideas are difficult to follow in the current presentation. I would encourage the authors to reconsider their notation and improve their writing to convey their ideas more effectively to readers.


**Major concerns**
1. The momentum and pre-conditioning typically serve different roles in optimization: momentum accumulates past gradients, and pre-conditioning captures curvature information. I do not think the connection demonstrated in Section 3.2 is trivial. A detailed explanation is needed in the main text to connect them.
2. The paper lacks theoretical guarantees regarding the convergence or performance of the proposed algorithm.
3. Since the algorithm incorporates a momentum variable, it would be more consistent to account for this momentum within the MH step (Equation (7)), rather than applying it solely to $\boldsymbol{\sigma}$.

**Other suggestions**
1. The paper lacks clear definitions for essential variables (e.g., $\boldsymbol{u}$ and $\boldsymbol{e}$ in Equation (1)).
2. The time variable $t$ is used ambiguously. It denotes continuous evolution in Equation (1) but has discrete updates in Equations (4)-(5).
3. Although the authors appear to focus on ground-state sampling, the formulation provided in Section 3.2 is more oriented toward sampling from a Gibbs measure rather than explicitly defining the ground-state sampling.
4. Ensure that all variables and abbreviations are clearly defined. For example, abbreviations such as SA, HNN, and CACm used in Table 1 should be explicitly explained in the main text.
Recommendation: 6
Reviewer confidence: 3
Soundness/correctness: 2
Clarity: 2
reviews: 
Comments: This paper presents two variants of the CAC (chaotic amplitude control) algorithm, namely CAC with momentum (CACm) and Metropolis-Hastings CAC with momentum (HMCACm). CACm is a deterministic continuous-time dynamical model for combinatorial optimization, and HMCACm is a Metropolis-Hastings adjusted version of CACm with the Boltzmann distribution as the theoretical equilibrium. MHCACm can be regarded as a unified framework that generalizes many existing methods, including simulated annealing, Hopfield neural networks, analog iterative machines, and CAC(m). Numerical results show that this method illustrates faster relaxation time on NP-hard problems. In particular, MHCACm exhibits excellent performance in sampling from easier ground states, which may be relevant to training over-parametrized neural networks.
Strengths: - State-of-the-art algorithms exploiting relaxation to a continuous state form discrete combinatorial optimization do not sample fairly from the Boltzmann distribution due to the lack of detailed balance. The MHCACm algorithm fills in this conceptual gap by adding a Metropolis-Hasting step. This new design ensures that MHCACm samples fairly from a discrete distribution while iterating over the relaxed (continuous) search space. 
- The numerical results look strong. Table 3 shows that MHCACm has a success probability higher than dSBM, another well-known Ising solver based on GPU. 
- MHCACm is well-suited for large-scale deployment on GPU because its computational bottleneck is matrix-vector multiplication.
Weaknesses: - Little theoretical justification for the effectiveness of MHCACm is provided. Specifically, it is not clear why a fair sampling strategy in the CAC framework would lead to a better performance in an optimization (ground state finding) problem. Relaxation to the ground state does not necessarily need to go through a detailed-balance algorithmic path. It would be nice to discuss how the Metropolis-Hastings step interacts with the CAC dynamics to potentially improve optimization performance.
- No empirical results for real-world optimization problems. While the performance of MHCACm has been benchmarked over the dWPE instances and GSET, these test instances are highly artificial and may not reflect the performance of the algorithm in a practical setting (e.g., quadratic assignment problems, portfolio optimization problems, etc.).
Recommendation: 6
Reviewer confidence: 4
Soundness/correctness: 3
Clarity: 3
reviews: 
Comments: The paper proposes a class of hybrid continuous-discrete algorithms by integrating continuous dynamics with Metropolis-Hastings steps. The paper also constructs a set of Ising problems with a tunable parameter to trade off between easy ground states and hard degenerate ground states, in order to experiment with the bias of different algorithms. The proposed class of algorithms are also fast solvers that achieve a great amount of acceleration on GPU due to a parallelizable structure.
Strengths: * The paper writing is nice and structured, with a comprehensive literature review and detailed problem set-up.
* The paper looks technically sound with solid mathematical proof.
* The proposed algorithm is evaluated on multiple tasks and compared with various other benchmark methods, showing competitive performance.
Weaknesses: * I am not very familiar with the literature, but seems that the tasks of ground-state sampling are not formally defined in the paper, as well as the idea of non-equilibrium dynamics.
* The connection between ground-state sampling and deep learning optimization/generalization mentioned in the paper is interesting, but the discussion is very limited. 
* For numerical experiments, the definition of TTS is hard to comprehend. Does smaller TTS indicate better algorithmic performance?
Recommendation: 6
Reviewer confidence: 2
Soundness/correctness: 3
Clarity: 3