Title: Hamiltonian Descent Algorithms for Optimization: Accelerated Rates via Randomized Integration Time
Venue: NeurIPS 2025
Accepted: yes
Keywords: Optimization

Hamiltonian Flow

Acceleration

Randomization
Abstract: We study the Hamiltonian flow for optimization (HF-opt), which simulates the Hamiltonian dynamics for some integration time and resets the velocity to $0$ to decrease the objective function; this is the optimization analogue of the Hamiltonian Monte Carlo algorithm for sampling. For short integration time, HF-opt has the same convergence rates as gradient descent for minimizing strongly and weakly convex functions. We show that by randomizing the integration time in HF-opt, the resulting randomized Hamiltonian flow (RHF) achieves accelerated convergence rates in continuous time, similar to the rates for accelerated gradient flow. We study a discrete-time implementation of RHF as the randomized Hamiltonian gradient descent (RHGD) algorithm. We prove that RHGD achieves the same accelerated convergence rates as Nesterov's accelerated gradient descent (AGD) for minimizing smooth strongly and weakly convex functions. We provide numerical experiments to demonstrate that RHGD is competitive with classical accelerated methods such as AGD across all settings and outperforms them in certain regimes.


Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nLet me start with a statement of (what I think is) the main result.\nThat is that there is an implementable approximation of Hamiltonian dynamics for convex optimization that has a provably accelerated convergence rate.\n\nNow for a summary. It is shown that an idealized version of Hamiltonian flow for convex optimization in discrete time achieves the convergence rate of gradient descent on the same class of functions when the step size (or integration time) is small. Inspired by the work of other authors, who have demonstrated that randomizing the integration time can speed up convergence, the authors show that a continuous-time limit of Hamiltonian flow with appropriately randomized integration times achieves accelerated convergence on convex functions. Presumably motivated by this continuous-time hint, the authors then go on to present a discretization of the continuous-time flow--that is, an implementable algorithm--and prove that the discrete-time dynamics achieves an accelerated convergence rate.\n\nSTRENGTHS AND WEAKNESSES:\nThe mathematical results of this paper are interesting, original, and relevant, and deserve to be published. The experiments need some work (they are not properly implemented or interpreted) and the whole paper could do with some rewriting here and there for clarity. I'll expand on these two points in the next section. Here, let me just say that the reason for my tepid score are the issues with the experiments. If the authors fix the experiments and the associated discussion in the experiments section, I am happy to raise my score to a firm accept.\n\nQUESTIONS:\n**Experiments.**\nThere are a few things wrong with the experiments.\nOne problem is the range on the y-axes. In the most egregious example, the y-axis goes all the way down to $10^{-77}$, in another it goes to $10^{-23}$, and in many others it goes to $10^{-15}$.\nWhat bit precision is being used in the experiments?\nIn single precision, machine epsilon is merely $10^{-7}$. In double precision that improves to something like $10^{-15}$.\nValues of $f_k-f^{\\star}$ that are smaller than machine epsilon are not sensible. At that point it is not the performance of the algorithm so much as cancellation effects that are being measured.\nThis needs to be fixed.\nThe second problem is that many of the experiments don't actually show faster convergence for the accelerated algorithm. For convex or strongly convex functions with the strong convexity constant correctly specified, we should definitely see the accelerated algorithm converging faster on average than gradient descent. The story seems to be more complicated than that, in fact it looks in Figure 1 like the two types of algorithms are converging at the same rate. I'm wondering if there is something wrong with the code, or if perhaps the vast range of the y-axis is obscuring any differentials in behavior.\nBecause of these two problems, there is also a third problem, which is that the experiments as presented simply don't support the claim in the main text (see for example, the paragraph at line 280) that RHGD converges faster than GD.\n\n**Readability**. My other main issue with this paper has to do with readability, which I'm going to break into two pieces: lack of overall narrative, and other choices in the writing.\nWith what follows and the suggestions I make, my goal is simply to be helpful. My concerns regarding readability did not influence how I scored the paper. I am of the opinion that the quality of the written paper is not on par with the quality of the mathematical results, but I think this can be fixed without much trouble.\n\nNarrative. There isn't a narrative arc in this paper. In the summary, I wrote my impression of what I think the story might be. In the beginning of the paper the authors do a good job of identifying a gap in the literature---how to derive accelerated optimization algorithms from ideas in sampling---that serves as motivation for the main results. But after that the paper is mostly a list of theorem statements. Is the narrative I put in the summary what the authors had in mind (namely, that the fact that the continuous-time limit of randomized integration-time HMC turned out to have an accelerated rate motivated the search for a discretization that could preserve that rate)? If so, it would be worth slipping in a few sentences here and there that make it clear. This would be helpful to the reader in a paper with several algorithms and proofs of convergence. Speaking for myself, although they're cool, I'm just not sure what the place of the continuous-time results (Theorems 2 and 3) is if not to serve as motivation to search for Theorems 3 and 4.\n\nAlso along the lines of narrative, I wonder if the authors would consider promoting the discussion of lift-conserve-project schemes to the main paper? Descent lemmas are valuable in optimization. The observation that Hamiltonian dynamics gives a descent lemma for free is really nice. I think it would be worth putting that and the proof of the descent lemma in the main paper. One possibility would be to relegate some of the material in the experiments section to the appendices to make room.\n\nAnd while I'm talking about LCP schemes---the authors claim that this perspective is new. My question is, new to whom? If they mean the optimization/ML community then that is certainly plausible. But in physics the idea is folklore to the point where I don't think it even has a name. And I would be surprised if it wasn't also known in numerical analysis (but maybe it isn't!). Can the authors please clarify what they meant? Secondly, I think we should be careful not to imply (I think it's happening accidentally) that we can obtain a descent lemma for absolutely any $f$. For Hamilton's equations to be well-defined I believe we need $f$ to be differentiable at the very least.\n\nMore about the writing.\nNone of the math in this paper was new to me but it was still a difficult read. I think that part of the reason why is that the paper is absolutely teeming with acronyms. This really slows a reader down (for example, I got totally lost trying to keep RHMC, RHF, RHF-opt, RPHD, and RHGD straight). I think it would be worth dropping some of the acronyms, reserving them perhaps only for implementable algorithms such as GD and RHGD. There is a similar problem with excessive use of lemmas for well-known results such as Young's inequality. A sentence in a proof that reads ``we go from step A to step B via an application of Young's inequality'' is much faster to read than ''we go from step A to step B via Lemma 16'' and then I must flip several pages to check the statement of Lemma 16.\n\nApart from that, there are specific sentences and paragraphs that could use some rewriting for clarity. For example, no convergence result is provided for Algorithm 2 (and that's fine), but then the sentence in line 154 should be edited. Another example are the two paragraphs that begin at lines 176 and 185, respectively. We can't directly compare rates of a discrete-time dynamics with a continuous-time dynamics (like HF-opt and RHF in line 176).\n\n**Questions.**\nHF-opt is not an implementable algorithm. But having used an approximation of the Prox operator to approximate the Hamiltonian flow in the construction of RHGD, do the authors know what might happen if they did the same thing in HF-opt? Would the convergence rates of Theorem 1 be preserved?\n\nI was surprised by the discussion following the statements of Theorems 2 and 3, specifically the bit about smoothness. And are we sure that smoothness is really needed to prove a convergence rate for HF-opt? The relevant descent lemma does not require it. Could the authors please clarify what they are trying to say there?\n\n**One last thing.** Markov's inequality is quite simple to state and apply. It may be worth doing so in the paper to quantify deviations from the expected convergence rate of RHGD since the possibility of doing so is mentioned twice anyway.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 5, "CLARITY": 2, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-26"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper presents new connections between optimization and Hamiltonian dynamics. Specifically the authors study the Hamiltonian version of the gradient flow dynamics. They show that for periodic resetting of the velocity, the exact integration yields convergence rates that match deterministic gradient descent, in both weakly and strongly convex/PL condition cases. Aspiring to improve these rates to match those of accelerated gradient descent (AGD), the authors consider the idea of randomizing the reset/integration time. Typically they propose a non-homogeneous Poisson process to govern the resets. Within each period, they further disco ties the Hamiltonian dynamics to obtain and implementable algorithm (RHGD), which has expected convergence rates matching AGD. I am quadratic opt simulation and logistic regression problems, the performance of RHGD is similar to other accelerated methods and more robust in some circumstances.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. In my view this is a high-quality submission. The results are technically sound and seem mature. Although I did not read the appendix in detail, the claims are well supported by extensive proofs. The paper is also written well, well organized, and clearly articulates the main ideas and results.\n2. To my knowledge the results of this paper and quite significant and interesting. The most similar previous results are Wang [2024], who use Chebyshev polynomials to set the reset times, and achieve results only on strongly convex quadratic functions. This paper presents a significant leap forward, with a new idea of randomized reset times, and results that hold across a much wider class of problems -- problems where the PL condition holds and weakly convex problems. Furthermore, the results significantly strengthen the connections between sampling and optimization, both of which are core topics in machine learning.\n3. To the best of my knowledge, there are no overlapping works in the literature and this work appears to present original new insights, which I think will be valuable.\n\nWeaknesses:\n1. My reading of this paper is quite positive and I don't have many strong weaknesses. However I will mention that I would appreciate if the authors could provide more insight/intuition on *why* the idea of randomized reset/integration times following a Poisson process actually leads to acceleration. I understand the \"real\" details are in the proof, but to strengthen the impact of this work some further intuition in the main body (which I could not find) would be quite helpful.\n2. I furthermore have several questions below, which I don't inherently view as weaknesses per se, but I would appreciate to see addressed.\n\nQUESTIONS:\nMajor:\n1. In the paragraph \"Approximate Poisson process.\" an acceptance-rejection scheme is described. However, I can think of alternative approach that first samples a path/trace of the Poisson process for $K$ steps. Indeed, this can be done totally independently of the dynamics I think. This gives a sequence of inter-arrival times. Then, within each subinterval of that sequence, you can discretize the HF dynamics. Will this approach also work? If not, shat is the drawback of this approach compared to yours?\n2. You propose an implicit (backward Euler) integrator. Is explicit Euler not stable here? Can you elaborate?\n3. You mention \"Nevertheless, convergence in expectation can still imply high-probability bounds...via Markov’s inequality.\" However, traditionally in optimization, Markov is not enough and you need a light-tail assumption and stronger concentration inequalities to get high-probability bounds. I think this is firmly beyond the scope of this paper, but I would still appreciate further comment (or correction) on this brief comment you made.\n4. All of your quadratic instances in Section 5.1 have minimizer $x^\\ast = 0$. This is a bit non-standard (usually a linear term is added), but perhaps WLOG. Can you elaborate on this choice?\n\nMinor:\n1. Is the phrase ``$\\alpha$-dominated graident\" very common in the literature? I more commonly have seen it referred to as the Polyak-Lojasiewicz (PL) Condition, and I might suggest that nomenclature.\n2. Why is the constraint $\\alpha \\leq 1$ needed? Most common in the literature is to simply assume $0 \\leq \\alpha \\leq L$.\n3. Although they are defined in the notation section, are big O and big Theta ever actually used in the paper? I could not find them. (And in fact, I really appreciate how your bounds are explicit!)\n4. I think $HF_{\\eta}(x, y)$ is first used in Algorithm 1 and needs a proper definition before that.\n5. Why does Theorem 1 need $L$-smoothness but Theorem 2 as well as results for the exact traditional gradient flow do not?\n\nLIMITATIONS:\nyes", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 4, "CLARITY": 4, "IMPACT": 4, "ORIGINALITY": 4, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper studies the Hamiltonian flow for optimization (HF-opt). The authors show that HF-opt has the same convergence rates as gradient descent for minimizing strongly and weakly convex functions. The authors also propose the randomized Hamiltonian flow (RHF) by randomizing the integration time, which achieves accelerated convergence rates in continuous time. This paper also studies a discrete-time implementation of RHF as the randomized Hamiltonian gradient descent (RHGD) algorithm, which achieves the same accelerated convergence rates as Nesterov’s accelerated gradient descent (AGD) for minimizing smooth strongly and weakly convex functions.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n1. This paper propses the randomized Hamiltonian flow (RHF) for optimization and stablishes its accelerated convergence rates of $O(exp(-\\sqrt{a/5}t))$ for $a$-strongly convex functions and $O(1/t^2)$ for weakly convex functions. \n\n2. This paper proposes the randomized Hamiltonian gradient descent (RHGD), a discretization of RHF. Under L-smoothness assumption, The authors establish the $\\widetilde O(\\sqrt{L/a})$ complexity for a-strongly convex functions and $O(\\sqrt{L/\\epsilon})$ for weakly convex functions. These complexities matches the optimal accelerated rates of AGD. \n\n3. The RHGD is a new algorithmic formulation of acceleration, expanding the family of accelerated algorithms with a new member. It may advance the understanding of acceleration mechanism from the view of Hamiltonian flow.\n\n4. The theory is solid and the proofs are organized well.\n\n5. The paper is well-written.\n\nWeaknesses:\n\n1. The RHGD algorithm in Algorithm 3 needs two gradient computations at each iteration. In most machine learning applications, gradient computation constitutes the most time-consuming operation. This makes each step of Algorithm 3 two times slower than the standard AGD. Especially, in Figures 1 and 2 which uses the iteration as the X-axis, RHGD performs similar to AGD, making RHGD almost two times slower than AGD when using time as the X-axis.\n\n2. The convergence rate of RHGD holds in expectation. The authors argue that  convergence in expectation can still imply high-probability bounds for via Markov’s inequality. However, Markov’s inequality makes the algorithm $1/\\gamma$ (or $log\\frac{1}{\\gamma}$) times slower when the convergence holds with probability at least $1-\\gamma$.\n\nIn conclusion, this paper is well-written. The theory is solid. The Hamiltonian flow perspective of acceleration seems interesting. However, it seems slower than AGD and lacks practical utility.\n\nQUESTIONS:\n1. Theorems 2 and 3 establish the convergence rate in continuous time without smoothness assumption. For the discrete time case, Theorems 4 and 5 need the L-smooth assumption. What is role of L-smooth when discretizing a continuous system?\n\n2. Why can randomness accelerate Hamiltonian flow? What is the intuition?\n\n3. Can the authors elucidate the physical principles underlying the descrete-time Algorithm 3? For example, what does each of the four steps on lines 3-6 mean? There are many accelerated algorithms developed from the continuous-to-descrete view, providing an intuitive physical interpretation would significantly advance our understanding of acceleration mechanism.\n\nLIMITATIONS:\nThe proposed descrete RHGD algorithm needs two gradient computations at each iteration, which may make RHGD two times slower than the standard AGD.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper proposes to add a downward jump term to the velocity to the Hamiltonian flow such that at random Poisson times (which can be non-homogeneous), the velocity is reset to zero. By randomizing the integration time, the paper shows that the resulting randomized Hamiltonian flows achieves accelerated convergence rates in continuous time, similar to those for accelerated gradient flow. The paper also studies the discrete-time algorithm based on randomized Hamiltonian flows, and obtained theoretical guarantees for expected suboptimality,  which enjoy the same convergence rates as Nesterov's accelerated gradient descent in both strongly convex and weakly convex settings.  Numerical results are also provided to show the efficiency of the proposed algorithm.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n\n(1) The continuous-time and discrete-time methods proposed in this paper are novel to the best of my knowledge. Interestingly, the proposed method can achieve the same accelerated convergence rates as Nesterov's accelerated method. \n\n(2) The paper is very well written, and the analysis seems to be rigorous.\n\n(3) Numerical results are promising.\n\nWeaknesses:\n\n(1) It would be great if the author(s) can provide some guidance when it is better to use Nesterov's accelerated method and when it is better to use your proposed method, especially because it works under expectations only. \n\n(2) In practice, stochastic gradient is often used, and it would be interesting to investigate the sensitivity to gradient noise for the proposed method. But I understand that this is beyond the scope of the paper, and should be left as a future research direction.\n\nQUESTIONS:\n(1) In your equation (RHF), I think you should write $Y_{t-}dN_{t}$ instead to be more rigorous.\n\n(2) The Poisson intensity $\\gamma(t)$ is constant for strongly convex setting and decreasing for weakly convex setting. That reminds me of choosing constant stepsize for strongly convex setting and decreasing stepsize for Nesterov's accelerated gradient descent. I am wondering if there are any connections here.\n\n(3) It would be helpful if the author(s) can add some discussions (if not already) the intuitions why the proposed method can accelerate. Also, $\\gamma(t)$ is a function of time in general, and that is indeed the choice in the weakly convex setting. But $\\gamma(t)$ is independent of the Poisson arrivals in the model. I am wondering what is the intuition to choose $\\gamma(t)$ this way, instead of for example choosing $\\gamma(t)=\\phi(t-\\tau_{i})$ between two Poisson arrivals $\\tau_{i}$ and $\\tau_{i+1}$, i.e. reset the Poisson intensity after the velocity hits zero.\n\n(4) It seems that you missed quite a few references when you discuss underdamped Langevin algorithms, for example, Chau and Rasonyi (2022) Stochastic Gradient Hamiltonian Monte Carlo for Non-convex Learning, Dalalyan and Riou-Durand (2020) On sampling from a log-concave density using kinetic Langevin diffusions, Gao et al. (2022) Global Convergence of Stochastic Gradient Hamiltonian Monte Carlo for Nonconvex Stochastic Optimization.\n\n(5) In the reference Andre Wibisono, Ashia C Wilson, and Michael I Jordan, proceedings should be Proceedings.\n\nLIMITATIONS:\nYes.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 4, "DATE": "2025-07-02"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper proposes an optimization method inspired by Hamiltonian Monte Carlo (HMC), which follows the system’s equations of motion. At random intervals, the momentum is reset to zero, causing the state to converge to a (local) minimum rather than oscillating around it. The approach aligns with recent Hamiltonian-based optimization techniques, such as [Wang 2024]. To the best of my understanding, the primary difference is that in [Wang 2024], momentum truncation occurs after fixed evolution times, whereas in this work, those truncation times are sampled at random.\n\nThe authors provide theoretical convergence rate bounds which, as far as I can tell, are comparable to those established in prior work on related algorithms. Given these similarities, I find it difficult to clearly assess the novelty of the proposed method relative to the existing literature.\n\nSTRENGTHS AND WEAKNESSES:\nStrengths:\n1. The paper is well-written. The notations are clear, and the formal derivations are rigorous. \n2. In practice, (on the toy examples that they provide), their algorithm is comparable to continuized AGD (CAGD).\n3. Theoretical bounds on the convergence rate are proved. \n4. [After rebuttal] An Interesting and deep relation between the proposed optimizer and HMC is established. \n\nWeaknesses:\n1. The significance of the contributions, compared with similar methods such as \"Frictionless Hamiltonian Descent and Coordinate Hamiltonian Descent for strongly convex quadratic problems and beyond\" (https://arxiv.org/pdf/2402.13988) is either not clarified or the contributions are minimal. \n2. No comparison is made with any existing Hamiltonian-descent-based optimization algorithms, such as [Wang 2024]. \n3. The comparison is only carried out on toy models.\n\nQUESTIONS:\n1. As you mentioned, most existing Hamiltonian-based optimization methods use a damping term for dissipation energy. You highlight that your works (along with [Wang 24]) is not so. My question is that why should that be an important feature? Since, as far as I can say, setting the momentum to zero (either at random or fixed intervals) in effect plays the role of friction and damping, causing the state to converge to a (local) minimum rather than oscillating around it. Can you comment on this?\n\nLIMITATIONS:\nOne limitation is that the established bounds hold in expectation due tot he randomness of the algorithm which is specified in the paper.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 2, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}