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Probabilistic Methods · Monte Carlo and Sampling Methods

Jie Hu, Yi-Ting Ma, Do-Young Eun

We propose a *history-driven target (HDT)* framework in Markov Chain Monte Carlo (MCMC) to improve any random walk algorithm on discrete state spaces, such as general undirected graphs, for efficient sampling from target distribution $\\boldsymbol{\\mu}$. With broad applications in network science and distributed optimization, recent innovations like the self-repellent random walk (SRRW) achieve near-zero variance by prioritizing under-sampled states through transition kernel modifications based on past visit frequencies. However, SRRW's reliance on explicit computation of transition probabilities for all neighbors at each step introduces substantial computational overhead, while its strict dependence on time-reversible Markov chains excludes advanced non-reversible MCMC methods. To overcome these limitations, instead of direct modification of transition kernel, HDT introduces a history-dependent target distribution $\\boldsymbol{\\pi}[\\mathbf{x}]$ to replace the original target $\\boldsymbol{\\mu}$ in any graph sampler, where $\\mathbf{x}$ represents the empirical measure of past visits. This design preserves lightweight implementation by requiring only local information between the current and proposed states and achieves compatibility with both reversible and non-reversible MCMC samplers, while retaining unbiased samples with target distribution $\\boldsymbol{\\mu}$ and near-zero variance performance. Extensive experiments in graph sampling demonstrate consistent performance gains, and a memory-efficient Least Recently Used (LRU) cache ensures scalability to large general graphs.

Probabilistic Methods · Monte Carlo and Sampling Methods

Jie Hu, Yi-Ting Ma, Do-Young Eun

We propose a *history-driven target (HDT)* framework in Markov Chain Monte Carlo (MCMC) to improve any random walk algorithm on discrete state spaces, such as general undirected graphs, for efficient sampling from target distribution $\\boldsymbol{\\mu}$. With broad applications in network science and distributed optimization, recent innovations like the self-repellent random walk (SRRW) achieve near-zero variance by prioritizing under-sampled states through transition kernel modifications based on past visit frequencies. However, SRRW's reliance on explicit computation of transition probabilities for all neighbors at each step introduces substantial computational overhead, while its strict dependence on time-reversible Markov chains excludes advanced non-reversible MCMC methods. To overcome these limitations, instead of direct modification of transition kernel, HDT introduces a history-dependent target distribution $\\boldsymbol{\\pi}[\\mathbf{x}]$ to replace the original target $\\boldsymbol{\\mu}$ in any graph sampler, where $\\mathbf{x}$ represents the empirical measure of past visits. This design preserves lightweight implementation by requiring only local information between the current and proposed states and achieves compatibility with both reversible and non-reversible MCMC samplers, while retaining unbiased samples with target distribution $\\boldsymbol{\\mu}$ and near-zero variance performance. Extensive experiments in graph sampling demonstrate consistent performance gains, and a memory-efficient Least Recently Used (LRU) cache ensures scalability to large general graphs.

General Machine Learning · Clustering

Mingyi Li, Michael R. Metel, Akiko Takeda

The K-means algorithm is one of the most widely studied clustering algorithms in machine learning. While extensive research has focused on its ability to achieve a globally optimal solution, there still lacks a rigorous analysis of its local optimality guarantees. In this paper, we first present conditions under which the K-means algorithm converges to a locally optimal solution. Based on this, we propose simple modifications to the K-means algorithm which ensure local optimality in both the continuous and discrete sense, with the same computational complexity as the original K-means algorithm. As the dissimilarity measure, we consider a general Bregman divergence, which is an extension of the squared Euclidean distance often used in the K-means algorithm. Numerical experiments confirm that the K-means algorithm does not always find a locally optimal solution in practice, while our proposed methods provide improved locally optimal solutions with reduced clustering loss. Our code is available at https://github.com/lmingyi/LO-K-means.

Konrad Mundinger, Max Zimmer, Aldo Kiem, Christoph Spiegel, Sebastian Pokutta

We demonstrate how neural networks can drive mathematical discovery through a case study of the Hadwiger-Nelson problem, a long-standing open problem at the intersection of discrete geometry and extremal combinatorics that is concerned with coloring the plane while avoiding monochromatic unit-distance pairs. Using neural networks as approximators, we reformulate this mixed discrete-continuous geometric coloring problem with hard constraints as an optimization task with a probabilistic, differentiable loss function. This enables gradient-based exploration of admissible configurations that most significantly led to the discovery of two novel six-colorings, providing the first improvement in thirty years to the off-diagonal variant of the original problem (Mundinger et al., 2024a). Here, we establish the underlying machine learning approach used to obtain these results and demonstrate its broader applicability through additional numerical insights.

Yiming Qin, Manuel Madeira, Dorina Thanou, Pascal Frossard

Graph generative models are essential across diverse scientific domains by capturing complex distributions over relational data. Among them, graph diffusion models achieve superior performance but face inefficient sampling and limited flexibility due to the tight coupling between training and sampling stages. We introduce DeFoG, a novel graph generative framework that disentangles sampling from training, enabling a broader design space for more effective and efficient model optimization. DeFoG employs a discrete flow-matching formulation that respects the inherent symmetries of graphs. We theoretically ground this disentangled formulation by explicitly relating the training loss to the sampling algorithm and showing that DeFoG faithfully replicates the ground truth graph distribution. Building on these foundations, we thoroughly investigate DeFoG's design space and propose novel sampling methods that significantly enhance performance and reduce the required number of refinement steps. Extensive experiments demonstrate state-of-the-art performance across synthetic, molecular, and digital pathology datasets, covering both unconditional and conditional generation settings. It also outperforms most diffusion-based models with just 5–10\% of their sampling steps.

Sibylle Marcotte, Rémi Gribonval, Gabriel Peyré

While conservation laws in gradient flow training dynamics are well understood for (mostly shallow) ReLU and linear networks, their study remains largely unexplored for more practical architectures. For this, we first show that basic building blocks such as ReLU (or linear) shallow networks, with or without convolution, have easily expressed conservation laws, and no more than the known ones. In the case of a single attention layer, we also completely describe all conservation laws, and we show that residual blocks have the same conservation laws as the same block without a skip connection. We then introduce the notion of conservation laws that depend only on *a subset* of parameters (corresponding e.g. to a pair of consecutive layers, to a residual block, or to an attention layer). We demonstrate that the characterization of such laws can be reduced to the analysis of the corresponding building block in isolation. Finally, we examine how these newly discovered conservation principles, initially established in the continuous gradient flow regime, persist under discrete optimization dynamics, particularly in the context of Stochastic Gradient Descent (SGD).

Social Aspects · Security

Yu-Wei Chen, Raghu Pasupathy, Jordan A Awan

This work identifies the first privacy-aware stratified sampling scheme that minimizes the variance for general private mean estimation under the Laplace, Discrete Laplace (DLap) and Truncated-Uniform-Laplace (TuLap) mechanisms within the framework of differential privacy (DP). We view stratified sampling as a subsampling operation, which amplifies the privacy guarantee; however, to have the same final privacy guarantee for each group, different nominal privacy budgets need to be used depending on the subsampling rate. Ignoring the effect of DP, traditional stratified sampling strategies risk significant variance inflation. We phrase our optimal survey design as an optimization problem, where we determine the optimal subsampling sizes for each group with the goal of minimizing the variance of the resulting estimator. We establish strong convexity of the variance objective, propose an efficient algorithm to identify the integer-optimal design, and offer insights on the structure of the optimal design.

Deep Learning · Algorithms

Lorenzo Magnino, Yuchen Zhu, Mathieu Lauriere

Optimal stopping is a fundamental problem in optimization with applications in risk management, finance, robotics, and machine learning. We extend the standard framework to a multi-agent setting, named multi-agent optimal stopping (MAOS), where agents cooperate to make optimal stopping decisions in a finite-space, discrete-time environment. Since solving MAOS becomes computationally prohibitive as the number of agents is very large, we study the mean-field optimal stopping (MFOS) problem, obtained as the number of agents tends to infinity. We establish that MFOS provides a good approximation to MAOS and prove a dynamic programming principle (DPP) based on mean-field control theory. We then propose two deep learning approaches: one that learns optimal stopping decisions by simulating full trajectories and another that leverages the DPP to compute the value function and to learn the optimal stopping rule using backward induction. Both methods train neural networks to approximate optimal stopping policies. We demonstrate the effectiveness and the scalability of our work through numerical experiments on 6 different problems in spatial dimension up to 300. To the best of our knowledge, this is the first work to formalize and computationally solve MFOS in discrete time and finite space, opening new directions for scalable MAOS methods.

Deep Learning · Large Language Models

Valentyn Boreiko, Alexander Panfilov, Václav Voráček, Matthias Hein, Jonas Geiping

A plethora of jailbreaking attacks have been proposed to obtain harmful responses from safety-tuned LLMs. These methods largely succeed in coercing the target output in their original settings, but their attacks vary substantially in fluency and computational effort. In this work, we propose a unified threat model for the principled comparison of these methods.Our threat model checks if a given jailbreak is likely to occur in the distribution of text. For this, we build an N-gram language model on 1T tokens, which, unlike model-based perplexity, allows for an LLM-agnostic, nonparametric, and inherently interpretable evaluation. We adapt popular attacks to this threat model, and, for the first time, benchmark these attacks on equal footing with it. After an extensive comparison, we find attack success rates against safety-tuned modern models to be lower than previously presented and that attacks based on discrete optimization significantly outperform recent LLM-based attacks. Being inherently interpretable, our threat model allows for a comprehensive analysis and comparison of jailbreak attacks. We find that effective attacks exploit and abuse infrequent bigrams, either selecting the ones absent from real-world text or rare ones, e.g., specific to Reddit or code datasets.

Applications · Everything Else

Konrad Mundinger, Max Zimmer, Aldo Kiem, Christoph Spiegel, Sebastian Pokutta

We demonstrate how neural networks can drive mathematical discovery through a case study of the Hadwiger-Nelson problem, a long-standing open problem at the intersection of discrete geometry and extremal combinatorics that is concerned with coloring the plane while avoiding monochromatic unit-distance pairs. Using neural networks as approximators, we reformulate this mixed discrete-continuous geometric coloring problem with hard constraints as an optimization task with a probabilistic, differentiable loss function. This enables gradient-based exploration of admissible configurations that most significantly led to the discovery of two novel six-colorings, providing the first improvement in thirty years to the off-diagonal variant of the original problem (Mundinger et al., 2024a). Here, we establish the underlying machine learning approach used to obtain these results and demonstrate its broader applicability through additional numerical insights.

Optimization

Lisa Jin, Jianhao Ma, Zechun Liu, Andrey Gromov, Aaron Defazio, Lin Xiao

We develop a novel optimization method for quantization-aware training (QAT). Specifically, we show that convex, piecewise-affine regularization (PAR) can effectively induce neural network weights to cluster towards discrete values. We minimize PAR-regularized loss functions using an aggregate proximal stochastic gradient method (AProx) and prove that it enjoys last-iterate convergence. Our approach provides an interpretation of the straight-through estimator (STE), a widely used heuristic for QAT, as the asymptotic form of PARQ. We conduct experiments to demonstrate that PARQ obtains competitive performance on convolution- and transformer-based vision tasks.

Applications · Chemistry, Physics, and Earth Sciences

Keyue Qiu, Yuxuan Song, Jie Yu, Hongbo Ma, Ziyao Cao, Zhilong Zhang, Yushuai Wu, Mingyue Zheng, Hao Zhou, Wei-Ying Ma

Structure-based molecule optimization (SBMO) aims to optimize molecules with both continuous coordinates and discrete types against protein targets.A promising direction is to exert gradient guidance on generative models given its remarkable success in images, but it is challenging to guide discrete data and risks inconsistencies between modalities.To this end, we leverage a continuous and differentiable space derived through Bayesian inference, presenting Molecule Joint Optimization (MolJO), the gradient-based SBMO framework that facilitates joint guidance signals across different modalities while preserving SE(3)-equivariance.We introduce a novel backward correction strategy that optimizes within a sliding window of the past histories, allowing for a seamless trade-off between explore-and-exploit during optimization.MolJO achieves state-of-the-art performance on CrossDocked2020 benchmark (Success Rate 51.3\%, Vina Dock -9.05 and SA 0.78), more than 4x improvement in Success Rate compared to the gradient-based counterpart, and 2x ``Me-Better'' Ratio as much as 3D baselines. Furthermore, we extend MolJO to a wide range of optimization settings, including multi-objective optimization and challenging tasks in drug design such as R-group optimization and scaffold hopping, further underscoring its versatility.Code is available at https://github.com/AlgoMole/MolCRAFT.

Optimization · Discrete and Combinatorial Optimization

Han Li, Fei Liu, Zhi Zheng, Yu Zhang, Zhenkun Wang

Vehicle routing problems (VRPs) are significant combinatorial optimization problems (COPs) holding substantial practical importance. Recently, neural combinatorial optimization (NCO), which involves training deep learning models on extensive data to learn vehicle routing heuristics, has emerged as a promising approach due to its efficiency and the reduced need for manual algorithm design. However, applying NCO across diverse real-world scenarios with various constraints necessitates cross-problem capabilities. Current cross-problem NCO methods for VRPs typically employ a constraint-unaware model, limiting their cross-problem performance. Furthermore, they rely solely on global connectivity, which fails to focus on key nodes and leads to inefficient representation learning. This paper introduces a \underline{C}onstraint-\underline{A}ware \underline{D}ual-\underline{A}ttention Model (CaDA), designed to address these limitations. CaDA incorporates a constraint prompt that efficiently represents different problem variants. Additionally, it features a dual-attention mechanism with a global branch for capturing broader graph-wide information and a sparse branch that selectively focuses on the key node connections. We comprehensively evaluate our model on 16 different VRPs and compare its performance against existing cross-problem VRP solvers. CaDA achieves state-of-the-art results across all tested VRPs. Our ablation study confirms that each component contributes to its cross-problem learning performance. The source code for CaDA is publicly available at \url{https://github.com/CIAM-Group/CaDA}.

Optimization · Discrete and Combinatorial Optimization

Zhi Zheng, Zhuoliang Xie, Zhenkun Wang, Bryan Hooi

Handcrafting heuristics for solving complex optimization tasks (e.g., route planning and task allocation) is a common practice but requires extensive domain knowledge. Recently, Large Language Model (LLM)-based automatic heuristic design (AHD) methods have shown promise in generating high-quality heuristics without manual interventions. Existing LLM-based AHD methods employ a population to maintain a fixed number of top-performing LLM-generated heuristics and introduce evolutionary computation (EC) to iteratively enhance the population. However, these population-based procedures cannot fully develop the potential of each heuristic and are prone to converge into local optima. To more comprehensively explore the space of heuristics, this paper proposes to use Monte Carlo Tree Search (MCTS) for LLM-based heuristic evolution. The proposed MCTS-AHD method organizes all LLM-generated heuristics in a tree structure and can better develop the potential of temporarily underperforming heuristics. In experiments, MCTS-AHD delivers significantly higher-quality heuristics on various complex tasks. Our code is available.

General Machine Learning · Representation Learning

Xiaoyu Li, Zhao Song, Shenghao Xie

The Fourier transform is a fundamental tool in computer science and signal processing. In particular, when the signal is sparse in the frequency domain---having only $k$ distinct frequencies---sparse Fourier transform (SFT) algorithms can recover the signal in a sublinear time (proportional to the sparsity $k$). Most prior research focused on SFT for discrete signals, designing both randomized and deterministic algorithms for one-dimensional and high-dimensional discrete signals. However, SFT for continuous signals (i.e., $x^*(t)=\sum_{j=1}^k v_j e^{2\pi \mathbf{i} f_j t}$ for $t\in [0,T]$) is a more challenging task. The discrete SFT algorithms are not directly applicable to continuous signals due to the sparsity blow-up from the discretization. Prior to this work, there is a randomized algorithm that achieves an $\ell_2$ recovery guarantee in $\widetilde{O}(k\cdot \mathrm{polylog}(F/\eta))$ time, where $F$ is the band-limit of the frequencies and $\eta$ is the frequency gap.Nevertheless, whether we can solve this problem without using randomness remains open. In this work, we address this gap and introducethe first sublinear-time deterministic sparse Fourier transform algorithm in the continuous setting. Specifically, our algorithm uses $\widetilde{O}(k^2 \cdot \mathrm{polylog}(F/\eta))$ samples and $\widetilde{O}(k^2 \cdot \mathrm{polylog}(F/\eta))$ time to reconstruct the on-grid signal with arbitrary noise that satisfies a mild condition. This is the optimal recovery guarantee that can be achieved by any deterministic approach.

Theory · Reinforcement Learning and Planning

Qian Qi

We establish a continuous-time framework for analyzing Deep Q-Networks (DQNs) via stochastic control and Forward-Backward Stochastic Differential Equations (FBSDEs). Considering a continuous-time Markov Decision Process (MDP) driven by a square-integrable martingale, we analyze DQN approximation properties. We show that DQNs can approximate the optimal Q-function on compact sets with arbitrary accuracy and high probability, leveraging residual network approximation theorems and large deviation bounds for the state-action process. We then analyze the convergence of a general Q-learning algorithm for training DQNs in this setting, adapting stochastic approximation theorems. Our analysis emphasizes the interplay between DQN layer count, time discretization, and the role of viscosity solutions (primarily for the value function $V^*$) in addressing potential non-smoothness of the optimal Q-function. This work bridges deep reinforcement learning and stochastic control, offering insights into DQNs in continuous-time settings, relevant for applications with physical systems or high-frequency data.

Optimization · Discrete and Combinatorial Optimization

Zijun Liao, Jinbiao Chen, Debing Wang, Zizhen Zhang, Jiahai Wang

Neural Combinatorial Optimization (NCO) has emerged as a promising approach for NP-hard problems. However, prevailing RL-based methods suffer from low sample efficiency due to sparse rewards and underused solutions. We propose *Best-anchored and Objective-guided Preference Optimization (BOPO)*, a training paradigm that leverages solution preferences via objective values. It introduces: (1) a best-anchored preference pair construction for better explore and exploit solutions, and (2) an objective-guided pairwise loss function that adaptively scales gradients via objective differences, removing reliance on reward models or reference policies. Experiments on Job-shop Scheduling Problem (JSP), Traveling Salesman Problem (TSP), and Flexible Job-shop Scheduling Problem (FJSP) show BOPO outperforms state-of-the-art neural methods, reducing optimality gaps impressively with efficient inference. BOPO is architecture-agnostic, enabling seamless integration with existing NCO models, and establishes preference optimization as a principled framework for combinatorial optimization.

Applications · Health / Medicine

Sophia Tang, Yinuo Zhang, Pranam Chatterjee, PhD

We present **PepTune**, a multi-objective discrete diffusion model for simultaneous generation and optimization of therapeutic peptide SMILES. Built on the Masked Discrete Language Model (MDLM) framework, PepTune ensures valid peptide structures with a novel bond-dependent masking schedule and invalid loss function. To guide the diffusion process, we introduce **Monte Carlo Tree Guidance (MCTG)**, an inference-time multi-objective guidance algorithm that balances exploration and exploitation to iteratively refine Pareto-optimal sequences. MCTG integrates classifier-based rewards with search-tree expansion, overcoming gradient estimation challenges and data sparsity. Using PepTune, we generate diverse, chemically-modified peptides simultaneously optimized for multiple therapeutic properties, including target binding affinity, membrane permeability, solubility, hemolysis, and non-fouling for various disease-relevant targets. In total, our results demonstrate that MCTG for masked discrete diffusion is a powerful and modular approach for multi-objective sequence design in discrete state spaces.

Optimization · Discrete and Combinatorial Optimization

Yichuan Deng, Xiaoyu Li, Zhao Song, OMRI WEINSTEIN

A recent work by [Larsen, SODA 2023] introduced a faster combinatorial alternative to Bansal's SDP algorithm for finding a coloring $x \in \\{-1, 1\\}^n$ that approximately minimizes the discrepancy $\mathrm{disc}(A, x) := \\| A x \\|_{\infty}$ of a real-valued $m \times n$ matrix $A$. Larsen's algorithm runs in $\widetilde{O}(mn^2)$ time compared to Bansal's $\widetilde{O}(mn^{4.5})$-time algorithm, with a slightly weaker logarithmic approximation ratio in terms of the hereditary discrepancy of $A$ [Bansal, FOCS 2010]. We present a combinatorial $\widetilde{O}(\mathrm{nnz}(A) + n^3)$-time algorithm with the same approximation guarantee as Larsen's, optimal for tall matrices where $m = \mathrm{poly}(n)$. Using a more intricate analysis and fast matrix multiplication, we further achieve a runtime of $\widetilde{O}(\mathrm{nnz}(A) + n^{2.53})$, breaking the cubic barrier for square matrices and surpassing the limitations of linear-programming approaches [Eldan and Singh, RS\&A 2018]. Our algorithm relies on two key ideas: (i) a new sketching technique for finding a projection matrix with a short $\ell_2$-basis using implicit leverage-score sampling, and (ii) a data structure for efficiently implementing the iterative Edge-Walk partial-coloring algorithm [Lovett and Meka, SICOMP 2015], and using an alternative analysis to enable ``lazy'' batch updates with low-rank corrections. Our results nearly close the computational gap between real-valued and binary matrices, for which input-sparsity time coloring was recently obtained by [Jain, Sah and Sawhney, SODA 2023].

Applications · Chemistry, Physics, and Earth Sciences

Seul Lee, Karsten Kreis, Srimukh Veccham, Meng Liu, Danny Reidenbach, Yuxing Peng, Saee Paliwal, Weili Nie, Arash Vahdat

Drug discovery is a complex process that involves multiple stages and tasks. However, existing molecular generative models can only tackle some of these tasks. We present *Generalist Molecular generative model* (GenMol), a versatile framework that uses only a *single* discrete diffusion model to handle diverse drug discovery scenarios. GenMol generates Sequential Attachment-based Fragment Embedding (SAFE) sequences through non-autoregressive bidirectional parallel decoding, thereby allowing the utilization of a molecular context that does not rely on the specific token ordering while having better sampling efficiency. GenMol uses fragments as basic building blocks for molecules and introduces *fragment remasking*, a strategy that optimizes molecules by regenerating masked fragments, enabling effective exploration of chemical space. We further propose *molecular context guidance* (MCG), a guidance method tailored for masked discrete diffusion of GenMol. GenMol significantly outperforms the previous GPT-based model in *de novo* generation and fragment-constrained generation, and achieves state-of-the-art performance in goal-directed hit generation and lead optimization. These results demonstrate that GenMol can tackle a wide range of drug discovery tasks, providing a unified and versatile approach for molecular design.