Monte Carlo

By far the most common way to estimate an expected loss in machine learning is to draw samples, compute the loss on each one, and take the empirical average. However, sampling is not necessarily optimal. Given an MLP at initialization, we show how to estimate its expected output over Gaussian inputs without running samples through the network at all. Instead, we produce approximate representations of the distributions of activations at each layer, leveraging tools such as cumulants and Hermite expansions. We show both theoretically and empirically that for sufficiently wide networks, our estimator achieves a target mean squared error using substantially fewer FLOPs than Monte Carlo sampling. We find moreover that our methods perform particularly well at estimating the probabilities of rare events, and additionally demonstrate how they can be used for model training. Together, these findings suggest a path to producing models with a greatly reduced probability of catastrophic tail risks.

Training-free conditional diffusion provides a flexible alternative to task-specific conditional model training, but existing samplers often allocate computation inefficiently: independent guided trajectories can vary widely in quality, and additional function evaluations along a single trajectory may not recover from poor early decisions. We propose Tempered Guided Diffusion (TGD), an annealed sequential Monte Carlo framework for training-free conditional sampling with diffusion priors. TGD targets tempered posterior distributions over the clean signal, using noisy diffusion states only as auxiliary variables for proposing reconstructions and propagating particles. Particles are reweighted by incremental likelihood ratios, resampled, and propagated across noise levels, concentrating computation on trajectories plausible under both the prior and observation. Under idealized exact-reconstruction assumptions, full TGD yields a consistent particle approximation to the posterior as the number of particles grows. For expensive reconstruction tasks, Accelerated TGD (A-TGD) retains early particle exploration but prunes to a single high-likelihood trajectory partway through sampling. Experiments on a controlled two-dimensional inverse problem and image inverse problems show improved posterior approximation and favorable wall-clock speed-quality tradeoffs over independent multi-trajectory baselines.

Probabilistic values, including Shapley values and semivalues, provide a model-agnostic framework to attribute the behavior of a black-box model to data points or features, with a wide range of applications including explainable artificial intelligence and data valuation. However, their exact computation requires utility evaluations over exponentially many coalitions, making Monte Carlo approximation essential in modern machine learning applications. Existing estimators are often developed through different representation strategies, including weighted averages, self-normalized weighting, regression adjustment, and weighted least squares. Our key observation is that these seemingly distinct constructions share a common first-order expansion, in which the leading term is determined by the sampling law and a working surrogate function. This first-order representation yields an explicit expression for the leading mean squared error (MSE), which characterizes how the sampling law and the surrogate jointly determine statistical efficiency. Guided by this criterion, we propose an Efficiency-Aware Surrogate-adjusted Estimator (EASE) that directly chooses the sampling law and surrogate to minimize the first-order MSE. We demonstrate that EASE consistently outperforms existing estimators for various probabilistic values.

Automatic Emergency Braking (AEB) systems represent a safety-critical national interest, with the National Highway Traffic Safety Administration (NHTSA) Federal Motor Vehicle Safety Standard (FMVSS No. 127) requiring AEB in all new light vehicles sold in the United States by September 2029. However, production implementations frequently rely on deterministic stopping-distance or Time-to-Collision (TTC) thresholds that fail to capture uncertainty in sensing, road conditions, and vehicle dynamics. This paper presents a GPU-accelerated Monte Carlo framework for stochastic evaluation of emergency braking performance using a high-fidelity longitudinal vehicle model incorporating aerodynamic drag, road grade, brake actuator dynamics, and weight transfer effects. A one-thread-per-sample execution strategy exploits the independence of Monte Carlo rollouts, while deterministic CPU-generated sampling ensures bit-exact numerical consistency between CPU and GPU implementations. The framework is evaluated across four hardware platforms spanning development and deployment environments: two laptop GPUs (GTX 1650, RTX 5070) and two automotive-grade embedded platforms (Jetson Orin Nano, Jetson AGX Orin). Peak speedups of 54.57x are achieved while maintaining exact numerical agreement. Real-time feasibility analysis with a complete AEB timing budget (700 ms human reaction time minus 120 ms perception and 50 ms decision overhead) demonstrates that the Jetson AGX Orin can execute approximately 25,000 Monte Carlo samples within a 530 ms budget, enabling real-time probabilistic AEB evaluation as part of a complete embedded pipeline. These results establish Monte Carlo-based uncertainty evaluation as a deployable runtime component rather than an offline validation tool and provide quantitative guidance for risk-aware AEB threshold selection under the NHTSA final rule.

Randomized Smoothing (RS) offers formal $\ell_2$ guarantees for arbitrary base classifiers but faces two key practical bottlenecks: (i) it often relies on noise-augmented training to achieve nontrivial certificates, which increases training cost, can reduce clean accuracy, and weakens RS as a genuinely post-hoc defense; and (ii) certification is computationally expensive, typically requiring tens of thousands of noisy forward passes per input, which hinders deployment, especially on resource-constrained edge devices. To address both limitations, we propose Laplace-Bridged Smoothing (LBS), an analytic reformulation of RS that replaces high-dimensional input-space Monte Carlo (MC) sampling with efficient computations in a low-dimensional probability space. LBS preserves formal robustness guarantees without requiring noise-augmented training while substantially reducing certification burden. On CIFAR-10 and ImageNet, LBS attains stronger certified robustness than RS and reduces per-sample certification cost by nearly an order of magnitude. Notably, on NVIDIA Jetson Orin Nano and Raspberry Pi 4, LBS achieves speedups of up to $494\times$, enabling practical certified deployment on real-world edge devices. Finally, we provide theoretical justification for the analytic formulation and certificate validity of LBS.

History-dependent sampling can reduce long-run Monte Carlo variance by discouraging redundant revisits, but existing schemes typically encode history through empirical measure on finite state spaces, which is infeasible in high-dimensional discrete configuration spaces or ill-posed in continuous domains. We propose Score-Repellent Monte Carlo (SRMC) framework that summarizes trajectory history by a running average of score evaluations in $\mathbb{R}^d$, where $d$ is the dimension of the score and state representation. This history is converted into a surrogate target through an exponential score tilt, indexed with $α$ that represents the strength of repellence in controlling the magnitude of the history-based repulsion. The surrogate family is normalization-free in the standard MCMC sense, yielding a generic wrapper: at each iteration, any base kernel targeting $π$ can instead be run on the current surrogate $π_{θ_n}$ while the history is updated online. We analyze the coupled evolution of the history recursion and Monte Carlo estimators using stochastic approximation with controlled Markovian noise, establishing almost sure convergence and a joint central limit theorem. We further identify regimes in which the asymptotic covariance decreases as $α$ increases, with scaling $O(1/α)$, extending the near-zero-variance effect of finite-state history-dependent samplers to general state spaces with constant memory. Experiments on continuous targets and discrete energy-based models demonstrate improved estimator variance and mode coverage, while retaining $O(d)$ memory usage and modest per-iteration overhead.

We propose a sampling-based framework for finite-horizon trajectory and policy optimization under differentiable dynamics by casting controller design as inference. Specifically, we minimize a KL-regularized expected trajectory cost, which yields an optimal "Boltzmann-tilted" distribution over controller parameters that concentrates on low-cost solutions as temperature decreases. To sample efficiently from this sharp, potentially multimodal target, we introduce tempered sequential Monte Carlo (TSMC): an annealing scheme that adaptively reweights and resamples particles along a tempering path from a prior to the target distribution, while using Hamiltonian Monte Carlo rejuvenation to maintain diversity and exploit exact gradients obtained by differentiating through trajectory rollouts. For policy optimization, we extend TSMC via (i) a deterministic empirical approximation of the initial-state distribution and (ii) an extended-space construction that treats rollout randomness as auxiliary variables. Experiments across trajectory- and policy-optimization benchmarks show that TSMC is broadly applicable and compares favorably to state-of-the-art baselines.

Deploying large language models to heterogeneous hardware is often constrained by memory, not compute. We introduce MCAP (Monte Carlo Activation Profiling), a load-time per-layer importance estimator that enables dynamic precision and memory placement decisions on the target device. MCAP produces a lightweight per-layer signal that drives both precision dispatch (W4A8 vs. W4A16) and residency tier (GPU, RAM, SSD), allowing a single set of weights to operate across diverse memory budgets. Our system, NVE, achieves 1.5-1.8x higher decode throughput than llama-cpp Q4_0 on NVIDIA T4 and enables models to run in memory regimes previously infeasible without modifying weights.

The standard Monte Carlo estimator $\widehat{I}_N^{\mathrm{MC}}$ of $\int fdω$ relies on independent samples from $ω$ and has variance of order $1/N$. Replacing the samples with a determinantal point process (DPP), a repulsive distribution, makes the estimator consistent, with variance rates that depend on how the DPP is adapted to $f$ and $ω$. We examine two existing DPP-based estimators: one by Bardenet & Hardy (2020) with a rate of $\mathcal{O}(N^{-(1+1/d)})$ for smooth $f$, but relying on a fixed DPP. The other, by Ermakov & Zolotukhin (1960), is unbiased with rate of order $1/N$, like Monte Carlo, but its DPP is tailored to $f$. We revisit these estimators, generalize them to continuous settings, and provide sampling algorithms.

Standard approaches to goal-conditioned reinforcement learning (GCRL) that rely on temporal-difference learning can be unstable and sample-inefficient due to bootstrapping. While recent work has explored contrastive and supervised formulations to improve stability, we present a probabilistic alternative, called survival value learning (SVL), that reframes GCRL as a survival learning problem by modeling the time-to-goal from each state as a probability distribution. This structured distributional Monte Carlo perspective yields a closed-form identity that expresses the goal-conditioned value function as a discounted sum of survival probabilities, enabling value estimation via a hazard model trained via maximum likelihood on both event and right-censored trajectories. We introduce three practical value estimators, including finite-horizon truncation and two binned infinite-horizon approximations to capture long-horizon objectives. Experiments on offline GCRL benchmarks show that SVL combined with hierarchical actors matches or surpasses strong hierarchical TD and Monte Carlo baselines, excelling on complex, long-horizon tasks. Webpage and Code: https://simple-robotics.github.io/publications/survival-value-learning/

Coherent systems are representative of many practical applications, ranging from infrastructure networks to supply chains. Probabilistic evaluation of such systems remains challenging, however, because existing decomposition-based methods scale poorly as the number of components grows. To address this limitation, this study proposes the Reference-state System Reliability (RSR) method. Like existing approaches, RSR characterises the boundary between different system states using reference states in the component-state space. Where it departs from these methods is in how the state space is explored: rather than using reference states to decompose the space into disjoint hypercubes, RSR uses them to classify Monte Carlo samples, making computational cost significantly less sensitive to the number of reference states. To make this classification efficient, samples and reference states are stored as matrices and compared using batched matrix operations, allowing RSR to exploit the advances in high-throughput matrix computing driven by modern machine learning. We demonstrate that RSR evaluates the system-state probability of a graph with 119 nodes and 295 edges within 10~seconds, highlighting its potential for real-time risk assessment of large-scale systems. We further show that RSR scales to problems involving hundreds of thousands of reference states -- well beyond the reach of existing methods -- and extends naturally to multi-state systems. Nevertheless, when the number of boundary reference states grows exceedingly large, RSR's convergence slows down, a limitation shared with existing reference-state-based approaches that motivates future research into learning-based representations of system-state boundaries.

Speculative decoding (SD) accelerates language model inference by drafting tokens from a cheap proposal model and verifying them against an expensive target model via rejection sampling. Because rejection truncates the draft block at the first error, throughput degrades when draft and target diverge. Rather than rejecting draft tokens outright, we propose to reweight them. To this end, we introduce sequential Monte Carlo speculative decoding (SMC-SD), which replaces token-level rejection with importance-weighted resampling over a population of draft particles. SMC-SD is a principled approximate inference scheme that trades exactness for additional speed, while preserving theoretical bounds on its per-step approximation error. Because LLM inference is memory bandwidth-bound, the arithmetic needed to draft particles and to score them in parallel comes nearly for free -- SMC-SD uses idle compute to turn verification into a vectorized, fixed-size operation with no rollback. Empirically, SMC-SD achieves 2.36x speed-up over speculative decoding and a 5.2x speed-up over autoregressive decoding, while remaining within 3% of the target model's accuracy on reasoning, instruction-following, and coding benchmarks.

In online clustering problems, there is often a large amount of uncertainty over possible cluster assignments that cannot be resolved until more data are observed. This difficulty is compounded when clusters follow complex distributions, as is the case with text data. Sequential Monte Carlo (SMC) methods give a natural way of representing and updating this uncertainty over time, but have prohibitive memory requirements for large-scale problems. We propose a novel SMC algorithm that decomposes clustering problems into approximately independent subproblems, allowing a more compact representation of the algorithm state. Our approach is motivated by the knowledge base construction problem, and we show that our method is able to accurately and efficiently solve clustering problems in this setting and others where traditional SMC struggles.

Gibbs measures, such as Coulomb gases, are popular in modelling systems of interacting particles. Recently, we proposed to use Gibbs measures as randomized numerical integration algorithms with respect to a target measure $π$ on $\mathbb R^d$, following the heuristics that repulsiveness between particles should help reduce integration errors. A major issue in this approach is to tune the interaction kernel and confining potential of the Gibbs measure, so that the equilibrium measure of the system is the target distribution $π$. Doing so usually requires another Monte Carlo approximation of the \emph{potential}, i.e. the integral of the interaction kernel with respect to $π$. Using the methodology of large deviations from Garcia--Zelada (2019), we show that a random approximation of the potential preserves the fast large deviation principle that guarantees the proposed integration algorithm to outperform independent or Markov quadratures. For non-singular interaction kernels, we make minimal assumptions on this random approximation, which can be the result of a computationally cheap Monte Carlo preprocessing. For the Coulomb interaction kernel, we need the approximation to be based on another Gibbs measure, and we prove in passing a control on the uniform convergence of the approximation of the potential.

Kernel herding belongs to a family of deterministic quadratures that seek to minimize the maximum mean discrepancy (MMD), that is, the worst-case integration error over a reproducing kernel Hilbert space (RKHS). These MMD minimization procedures come with strong experimental support, but comparatively less theoretical footing. In particular, apart from recent progress in distribution compression, little has been proved in favor of an improvement of MMD minimization over classical Monte Carlo quadrature when the RKHS is infinite-dimensional. In this paper, we study a joint probability distribution over quadrature nodes, a tailored Gibbs distribution, whose support intuitively tends to concentrate around MMD minimizers as a temperature parameter is decreased. Our main contribution is to prove that drawing integration nodes from our distribution does outperform i.i.d Monte Carlo. While our bounds on the worst-case integration error feature the same rate as i.i.d. Monte Carlo, we do obtain a tighter concentration inequality as the temperature parameter decreases. This means smaller confidence intervals as the number of quadrature nodes increases. While arguably a first step, our results demonstrate that the mathematical toolbox developed around Gibbs measures can help understand to what extent kernel herding and its variants improve on computationally cheaper methods. There remains the issue of sampling from our Gibbs distribution. In our numerical experiments, we demonstrate that a simple MCMC chain already yields approximate samples that lead to improved confidence intervals around the target integrals, as supported by our theoretical results.