论文检索

Probabilistic Methods/Monte Carlo and Sampling Methods

Luhuan Wu, Brian Trippe, Christian Naesseth, David Blei, John Cunningham

Diffusion models have been successful on a range of conditional generation tasks including molecular design and text-to-image generation. However, these achievements have primarily depended on task-specific conditional training or error-prone heuristic approximations. Ideally, a conditional generation method should provide exact samples for a broad range of conditional distributions without requiring task-specific training. To this end, we introduce the Twisted Diffusion Sampler, or TDS. TDS is a sequential Monte Carlo (SMC) algorithm that targets the conditional distributions of diffusion models through simulating a set of weighted particles. The main idea is to use twisting, an SMC technique that enjoys good computational efficiency, to incorporate heuristic approximations without compromising asymptotic exactness. We first find in simulation and in conditional image generation tasks that TDS provides a computational statistical trade-off, yielding more accurate approximations with many particles but with empirical improvements over heuristics with as few as two particles. We then turn to motif-scaffolding, a core task in protein design, using a TDS extension to Riemannian diffusion models; on benchmark tasks, TDS allows flexible conditioning criteria and often outperforms the state-of-the-art, conditionally trained model. Code can be found in https://github.com/blt2114/twisted_diffusion_sampler

Probabilistic Methods/Monte Carlo and Sampling Methods

Swati Padmanabhan, David Woodruff, Richard Zhang

Recent works in dimensionality reduction for regression tasks have introduced the notion of sensitivity, an estimate of the importance of a specific datapoint in a dataset, offering provable guarantees on the quality of the approximation after removing low-sensitivity datapoints via subsampling. However, fast algorithms for approximating sensitivities, which we show is equivalent to approximate regression, are known for only the $\ell_2$ setting, in which they are popularly termed leverage scores. In this work, we provide the first efficient algorithms for approximating $\ell_p$ sensitivities and other summary statistics of a given matrix. In particular, for a given $n \times d$ matrix, we compute $\alpha$-approximation to its $\ell_1$ sensitivities at the cost of $n/\alpha$ sensitivity computations. For estimating the total $\ell_p$ sensitivity (i.e. the sum of $\ell_p$ sensitivities), we provide an algorithm based on importance sampling of $\ell_p$ Lewis weights, which computes a constant factor approximation at the cost of roughly $\sqrt{d}$ sensitivity computations, with no polynomial dependence on $n$. Furthermore, we estimate the maximum $\ell_1$ sensitivity up to a $\sqrt{d}$ factor in $O(d)$ sensitivity computations. We also generalize these results to $\ell_p$ norms. Lastly, we experimentally show that for a wide class of structured matrices in real-world datasets, the total sensitivity can be quickly approximated and is significantly smaller than the theoretical prediction, demonstrating that real-world datasets have on average low intrinsic effective dimensionality.

Probabilistic Methods/Monte Carlo and Sampling Methods

Gergely Flamich

One-shot channel simulation is a fundamental data compression problem concerned with encoding a single sample from a target distribution $Q$ using a coding distribution $P$ using as few bits as possible on average. Algorithms that solve this problem find applications in neural data compression and differential privacy and can serve as a more efficient and natural alternative to quantization-based methods. Unfortunately, existing solutions are too slow or have limited applicability, preventing their widespread adaptation. In this paper, we conclusively solve one-shot channel simulation for one-dimensional problems where the target-proposal density ratio is unimodal by describing an algorithm with optimal runtime. We achieve this by constructing a rejection sampling procedure equivalent to greedily searching over the points of a Poisson process. Hence, we call our algorithm greedy Poisson rejection sampling (GPRS) and analyze the correctness and time complexity of several of its variants. Finally, we empirically verify our theorems, demonstrating that GPRS significantly outperforms the current state-of-the-art method, A* coding.

Applications/Chemistry and Drug Discovery

Pedro O. Pinheiro, Joshua Rackers, Joseph Kleinhenz, Michael Maser, Omar Mahmood, Andrew Watkins, Stephen Ra, Vishnu Sresht, Saeed Saremi

We propose a new score-based approach to generate 3D molecules represented as atomic densities on regular grids.First, we train a denoising neural network that learns to map from a smooth distribution of noisy molecules to the distribution of real molecules.Then, we follow the _neural empirical Bayes_ framework [Saremi and Hyvarinen, 2019] and generate molecules in two steps: (i) sample noisy density grids from a smooth distribution via underdamped Langevin Markov chain Monte Carlo, and (ii) recover the "clean" molecule by denoising the noisy grid with a single step.Our method, _VoxMol_, generates molecules in a fundamentally different way than the current state of the art (ie, diffusion models applied to atom point clouds). It differs in terms of the data representation, the noise model, the network architecture and the generative modeling algorithm.Our experiments show that VoxMol captures the distribution of drug-like molecules better than state of the art, while being faster to generate samples.

Probabilistic Methods/Monte Carlo and Sampling Methods

Congye Wang, Ye Chen, Heishiro Kanagawa, Chris Oates

Stein discrepancies have emerged as a powerful tool for retrospective improvement of Markov chain Monte Carlo output. However, the question of how to design Markov chains that are well-suited to such post-processing has yet to be addressed. This paper studies Stein importance sampling, in which weights are assigned to the states visited by a $\Pi$-invariant Markov chain to obtain a consistent approximation of $P$, the intended target. Surprisingly, the optimal choice of $\Pi$ is not identical to the target $P$; we therefore propose an explicit construction for $\Pi$ based on a novel variational argument. Explicit conditions for convergence of Stein $\Pi$-Importance Sampling are established. For $\approx 70$% of tasks in the PosteriorDB benchmark, a significant improvement over the analogous post-processing of $P$-invariant Markov chains is reported.

Reinforcement Learning/Planning

Michael Painter, Mohamed Baioumy, Nick Hawes, Bruno Lacerda

Monte-Carlo Tree Search (MCTS) methods, such as Upper Confidence Bound applied to Trees (UCT), are instrumental to automated planning techniques. However, UCT can be slow to explore an optimal action when it initially appears inferior to other actions. Maximum ENtropy Tree-Search (MENTS) incorporates the maximum entropy principle into an MCTS approach, utilising Boltzmann policies to sample actions, naturally encouraging more exploration. In this paper, we highlight a major limitation of MENTS: optimal actions for the maximum entropy objective do not necessarily correspond to optimal actions for the original objective. We introduce two algorithms, Boltzmann Tree Search (BTS) and Decaying ENtropy Tree-Search (DENTS), that address these limitations and preserve the benefits of Boltzmann policies, such as allowing actions to be sampled faster by using the Alias method. Our empirical analysis shows that our algorithms show consistent high performance across several benchmark domains, including the game of Go.

Probabilistic Methods/Bayesian Models and Methods

Zhe Zeng, Guy Van den Broeck

Bayesian neural networks (BNNs) provide a formalism to quantify and calibrate uncertainty in deep learning. Current inference approaches for BNNs often resort to few-sample estimation for scalability, which can harm predictive performance, while its alternatives tend to be computationally prohibitively expensive. We tackle this challenge by revealing a previously unseen connection between inference on BNNs and volume computation problems. With this observation, we introduce a novel collapsed inference scheme that performs Bayesian model averaging using collapsed samples. It improves over a Monte-Carlo sample by limiting sampling to a subset of the network weights while pairing it with some closed-form conditional distribution over the rest. A collapsed sample represents uncountably many models drawn from the approximate posterior and thus yields higher sample efficiency. Further, we show that the marginalization of a collapsed sample can be solved analytically and efficiently despite the non-linearity of neural networks by leveraging existing volume computation solvers. Our proposed use of collapsed samples achieves a balance between scalability and accuracy. On various regression and classification tasks, our collapsed Bayesian deep learning approach demonstrates significant improvements over existing methods and sets a new state of the art in terms of uncertainty estimation as well as predictive performance.

Probabilistic Methods/Monte Carlo and Sampling Methods

Gergely Flamich, Stratis Markou, José Miguel Hernández-Lobato

Relative entropy coding (REC) algorithms encode a sample from a target distribution $Q$ using a proposal distribution $P$ using as few bits as possible. Unlike entropy coding, REC does not assume discrete distributions and require quantisation.As such, it can be naturally integrated into communication pipelines such as learnt compression and differentially private federated learning. Unfortunately, despite their practical benefits, REC algorithms have not seen widespread application, due to their prohibitively slow runtimes or restrictive assumptions. In this paper, we make progress towards addressing these issues. We introduce Greedy Rejection Coding (GRC), which generalises the rejection sampling-based algorithm of Harsha et al. (2007) to arbitrary probability spaces and partitioning schemes. We first show that GRC terminates almost surely and returns unbiased samples from $Q$, and then focus on two variants of GRC, namely GRCS and GRCD. We show that for continuous $Q$ and $P$ over $\mathbb{R}$ with unimodal $dQ/dP$, the expected runtime of GRCS is upper bounded by $\beta D_{KL}(Q||P) + \mathcal{O}(1)$ where $\beta \approx 4.82$, and its expected codelength is optimal. This makes GRCS the first REC algorithm with guaranteed optimal runtime for this class of distributions, up to the multiplicative constant $\beta$. This significantly improves upon the previous state-of-the-art method, A* coding (Flamich et al., 2022). Under the same assumptions, we experimentally observe and conjecture that the expected runtime and codelength of GRCD are upper bounded by $D_{KL}(Q||P) + \mathcal{O}(1)$. Finally, we evaluate GRC in a compression pipeline with variational autoencoders on MNIST, and show that a modified training objective and a codelength-compression method can further improve compression efficiency.

Optimization

Yueming LYU

Black-box optimization has gained great attention for its success in recent applications. However, scaling up to high-dimensional problems with good query efficiency remains challenging. This paper proposes a novel Rank-1 Lattice Targeted Sampling (RLTS) technique to address this issue. Our RLTS benefits from random rank-1 lattice Quasi-Monte Carlo, which enables us to perform fast local exact Gaussian processes (GP) training and inference with $O(n \log n)$ complexity w.r.t. $n$ batch samples. Furthermore, we developed a fast coordinate searching method with $O(n \log n)$ time complexity for fast targeted sampling. The fast computation enables us to plug our RLTS into the sampling phase of stochastic optimization methods. This improves the query efficiency while scaling up to higher dimensional problems than Bayesian optimization. Moreover, to construct rank-1 lattices efficiently, we proposed a closed-form construction. Extensive experiments on challenging benchmark test functions and black-box prompt fine-tuning for large language models demonstrate the query efficiency of our RLTS technique.

Reinforcement Learning

Yangqing Fu, Ming Sun, Buqing Nie, Yue Gao

Monte Carlo Tree Search (MCTS) algorithms such as AlphaGo and MuZero have achieved superhuman performance in many challenging tasks. However, the computational complexity of MCTS-based algorithms is influenced by the size of the search space. To address this issue, we propose a novel probability tree state abstraction (PTSA) algorithm to improve the search efficiency of MCTS. A general tree state abstraction with path transitivity is defined. In addition, the probability tree state abstraction is proposed for fewer mistakes during the aggregation step. Furthermore, the theoretical guarantees of the transitivity and aggregation error bound are justified. To evaluate the effectiveness of the PTSA algorithm, we integrate it with state-of-the-art MCTS-based algorithms, such as Sampled MuZero and Gumbel MuZero. Experimental results on different tasks demonstrate that our method can accelerate the training process of state-of-the-art algorithms with 10%-45% search space reduction.

Probabilistic Methods/Monte Carlo and Sampling Methods

Ali Younis, Erik Sudderth

Particle filters flexibly represent multiple posterior modes nonparametrically, via a collection of weighted samples, but have classically been applied to tracking problems with known dynamics and observation likelihoods. Such generative models may be inaccurate or unavailable for high-dimensional observations like images. We instead leverage training data to discriminatively learn particle-based representations of uncertainty in latent object states, conditioned on arbitrary observations via deep neural network encoders. While prior discriminative particle filters have used heuristic relaxations of discrete particle resampling, or biased learning by truncating gradients at resampling steps, we achieve unbiased and low-variance gradient estimates by representing posteriors as continuous mixture densities. Our theory and experiments expose dramatic failures of existing reparameterization-based estimators for mixture gradients, an issue we address via an importance-sampling gradient estimator. Unlike standard recurrent neural networks, our mixture density particle filter represents multimodal uncertainty in continuous latent states, improving accuracy and robustness. On a range of challenging tracking and robot localization problems, our approach achieves dramatic improvements in accuracy, will also showing much greater stability across multiple training runs.

Probabilistic Methods/Monte Carlo and Sampling Methods

Weijian Luo, Boya Zhang, Zhihua Zhang

Efficiently sampling from un-normalized target distributions is a fundamental problem in scientific computing and machine learning. Traditional approaches such as Markov Chain Monte Carlo (MCMC) guarantee asymptotically unbiased samples from such distributions but suffer from computational inefficiency, particularly when dealing with high-dimensional targets, as they require numerous iterations to generate a batch of samples. In this paper, we introduce an efficient and scalable neural implicit sampler that overcomes these limitations. The implicit sampler can generate large batches of samples with low computational costs by leveraging a neural transformation that directly maps easily sampled latent vectors to target samples without the need for iterative procedures. To train the neural implicit samplers, we introduce two novel methods: the KL training method and the Fisher training method. The former method minimizes the Kullback-Leibler divergence, while the latter minimizes the Fisher divergence between the sampler and the target distributions. By employing the two training methods, we effectively optimize the neural implicit samplers to learn and generate from the desired target distribution. To demonstrate the effectiveness, efficiency, and scalability of our proposed samplers, we evaluate them on three sampling benchmarks with different scales. These benchmarks include sampling from 2D targets, Bayesian inference, and sampling from high-dimensional energy-based models (EBMs). Notably, in the experiment involving high-dimensional EBMs, our sampler produces samples that are comparable to those generated by MCMC-based methods while being more than 100 times more efficient, showcasing the efficiency of our neural sampler. Besides the theoretical contributions and strong empirical performances, the proposed neural samplers and corresponding training methods will shed light on further research on developing efficient samplers for various applications beyond the ones explored in this study.

Probabilistic Methods/Structure Learning

Enrico Giudice, Jack Kuipers, Giusi Moffa

Gaussian Process Networks (GPNs) are a class of directed graphical models which employ Gaussian processes as priors for the conditional expectation of each variable given its parents in the network. The model allows the description of continuous joint distributions in a compact but flexible manner with minimal parametric assumptions on the dependencies between variables. Bayesian structure learning of GPNs requires computing the posterior over graphs of the network and is computationally infeasible even in low dimensions. This work implements Monte Carlo and Markov Chain Monte Carlo methods to sample from the posterior distribution of network structures. As such, the approach follows the Bayesian paradigm, comparing models via their marginal likelihood and computing the posterior probability of the GPN features. Simulation studies show that our method outperforms state-of-the-art algorithms in recovering the graphical structure of the network and provides an accurate approximation of its posterior distribution.

Probabilistic Methods/Monte Carlo and Sampling Methods

Xiang Cheng, Bohan Wang, Jingzhao Zhang, Yusong Zhu

MCMC algorithms offer empirically efficient tools for sampling from a target distribution $\pi(x) \propto \exp(-V(x))$. However, on the theory side, MCMC algorithms suffer from slow mixing rate when $\pi(x)$ is non-log-concave. Our work examines this gap and shows that when Poincar\'e-style inequality holds on a subset $\mathcal{X}$ of the state space, the conditional distribution of MCMC iterates over $\mathcal{X}$ mixes fast to the true conditional distribution. This fast mixing guarantee can hold in cases when global mixing is provably slow. We formalize the statement and quantify the conditional mixing rate. We further show that conditional mixing can have interesting implications for sampling from mixtures of Gaussians, parameter estimation for Gaussian mixture models, and Gibbs-sampling with well-connected local minima.

Probabilistic Methods/Monte Carlo and Sampling Methods

Oren Mangoubi, Nisheeth K. Vishnoi

Given a Lipschitz or smooth convex function $f:K \to \mathbb{R}^d$ for a bounded polytope $K:=${ $\theta \in \mathbb{R}^d: A\theta \leq b$}, where $A\in \mathbb{R}^{m\times d}$ and $b \in \mathbb{R}^m$, we consider the problem of sampling from the log-concave distribution $\pi(\theta) \propto e^{-f(\theta)}$ constrained to $K$. Interest in this problem derives from its applications to Bayesian inference and differential privacy. We present a generalization of the Dikin walk to this setting that requires at most $O((md + d L^2 R^2) \times md^{\omega-1} \log(\frac{w}{\delta}))$ arithmetic operations to sample from $\pi$ within error $\delta>0$ in the total variation distance from a $w$-warm start. Here $L$ is the Lipschitz constant of $f$, $K$ is contained in a ball of radius $R$ and contains a ball of smaller radius $r$, and $\omega \approx 2.37$ is the matrix-multiplication constant. This improves on the running time of prior works for a range of structured settings important for the aforementioned inference and privacy applications. Technically, we depart from previous Dikin walks by adding a soft-threshold regularizer derived from the Lipschitz or smoothness properties of $f$ to a barrier function for $K$ that allows our version of the Dikin walk to propose updates that have a high Metropolis acceptance ratio for $f$, while at the same time remaining inside the polytope $K$.

Applications/Physics

Michael Scherbela, Leon Gerard, Philipp Grohs

Obtaining accurate solutions to the Schrödinger equation is the key challenge in computational quantum chemistry. Deep-learning-based Variational Monte Carlo (DL-VMC) has recently outperformed conventional approaches in terms of accuracy, but only at large computational cost.Whereas in many domains models are trained once and subsequently applied for inference, accurate DL-VMC so far requires a full optimization for every new problem instance, consuming thousands of GPUhs even for small molecules.We instead propose a DL-VMC model which has been pre-trained using self-supervised wavefunction optimization on a large and chemically diverse set of molecules. Applying this model to new molecules without any optimization, yields wavefunctions and absolute energies that outperform established methods such as CCSD(T)-2Z.To obtain accurate relative energies, only few fine-tuning steps of this base model are required.We accomplish this with a fully end-to-end machine-learned model, consisting of an improved geometry embedding architecture and an existing SE(3)-equivariant model to represent molecular orbitals. Combining this architecture with continuous sampling of geometries, we improve zero-shot accuracy by two orders of magnitude compared to the state of the art.We extensively evaluate the accuracy, scalability and limitations of our base model on a wide variety of test systems.

Reinforcement Learning/Deep RL

Janaka Brahmanage, Jiajing LING, Akshat Kumar

Action-constrained reinforcement learning (ACRL) is a popular approach for solving safety-critical and resource-allocation related decision making problems. A major challenge in ACRL is to ensure agent taking a valid action satisfying constraints in each RL step. Commonly used approach of using a projection layer on top of the policy network requires solving an optimization program which can result in longer training time, slow convergence, and zero gradient problem. To address this, first we use a normalizing flow model to learn an invertible, differentiable mapping between the feasible action space and the support of a simple distribution on a latent variable, such as Gaussian. Second, learning the flow model requires sampling from the feasible action space, which is also challenging. We develop multiple methods, based on Hamiltonian Monte-Carlo and probabilistic sentential decision diagrams for such action sampling for convex and non-convex constraints. Third, we integrate the learned normalizing flow with the DDPG algorithm. By design, a well-trained normalizing flow will transform policy output into a valid action without requiring an optimization solver. Empirically, our approach results in significantly fewer constraint violations (upto an order-of-magnitude for several instances) and is multiple times faster on a variety of continuous control tasks.

Probabilistic Methods/Monte Carlo and Sampling Methods

Yuling Yao, Justin Domke

To check the accuracy of Bayesian computations, it is common to use rank-based simulation-based calibration (SBC). However, SBC has drawbacks: The test statistic is somewhat ad-hoc, interactions are difficult to examine, multiple testing is a challenge, and the resulting p-value is not a divergence metric. We propose to replace the marginal rank test with a flexible classification approach that learns test statistics from data. This measure typically has a higher statistical power than the SBC test and returns an interpretable divergence measure of miscalibration, computed from classification accuracy. This approach can be used with different data generating processes to address simulation-based inference or traditional inference methods like Markov chain Monte Carlo or variational inference. We illustrate an automated implementation using neural networks and statistically-inspired features, and validate the method with numerical and real data experiments.

Deep Learning/Sequential Models, Time series

Zihao Zhou, Rose Yu

Learning continuous-time point processes is essential to many discrete event forecasting tasks. However, integration poses a major challenge, particularly for spatiotemporal point processes (STPPs), as it involves calculating the likelihood through triple integrals over space and time. Existing methods for integrating STPP either assume a parametric form of the intensity function, which lacks flexibility; or approximating the intensity with Monte Carlo sampling, which introduces numerical errors. Recent work by Omi et al. proposes a dual network approach for efficient integration of flexible intensity function. However, their method only focuses on the 1D temporal point process. In this paper, we introduce a novel paradigm: `Auto-STPP` (Automatic Integration for Spatiotemporal Neural Point Processes) that extends the dual network approach to 3D STPP. While previous work provides a foundation, its direct extension overly restricts the intensity function and leads to computational challenges. In response, we introduce a decomposable parametrization for the integral network using ProdNet. This approach, leveraging the product of simplified univariate graphs, effectively sidesteps the computational complexities inherent in multivariate computational graphs. We prove the consistency of `Auto-STPP` and validate it on synthetic data and benchmark real-world datasets. `Auto-STPP` shows a significant advantage in recovering complex intensity functions from irregular spatiotemporal events, particularly when the intensity is sharply localized. Our code is open-source at https://github.com/Rose-STL-Lab/AutoSTPP.

Probabilistic Methods/Monte Carlo and Sampling Methods

Omar Chehab, Aapo Hyvarinen, Andrej Risteski

Recent research has developed several Monte Carlo methods for estimating the normalization constant (partition function) based on the idea of annealing. This means sampling successively from a path of distributions which interpolate between a tractable "proposal" distribution and the unnormalized "target" distribution. Prominent estimators in this family include annealed importance sampling and annealed noise-contrastive estimation (NCE). Such methods hinge on a number of design choices: which estimator to use, which path of distributions to use and whether to use a path at all; so far, there is no definitive theory on which choices are efficient. Here, we evaluate each design choice by the asymptotic estimation error it produces. First, we show that using NCE is more efficient than the importance sampling estimator, but in the limit of infinitesimal path steps, the difference vanishes. Second, we find that using the geometric path brings down the estimation error from an exponential to a polynomial function of the parameter distance between the target and proposal distributions. Third, we find that the arithmetic path, while rarely used, can offer optimality properties over the universally-used geometric path. In fact, in a particular limit, the optimal path is arithmetic. Based on this theory, we finally propose a two-step estimator to approximate the optimal path in an efficient way.