A framework for adaptive MCMC targeting multimodal distributions

Pompe, Emilia; Holmes, Chris; Łatuszyński, Krzysztof

doi:10.1214/19-aos1916

Cited by 35 publications

(32 citation statements)

References 54 publications

(93 reference statements)

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“…Whereas one typically trains a generative model from a dataset of positive examples, a GFlowNet is trained to match a given energy function and convert it into a sampler, which we view as a generative policy because the composite object s is constructed through a sequence of steps. This is similar to what MCMC methods achieve but does not require a lengthy stochastic search in the space of such objects and avoids the modemixing intractability challenge of MCMC methods (Jasra et al, 2005;Bengio et al, 2013;Pompe et al, 2020). GFlowNets exchange that intractability for the challenge of amortized training of the generative policy.…”

Section: Introductionmentioning

confidence: 66%

“…Although these methods guarantee that asymptotically (in the length of the chain) we obtain samples drawn from the correct distribution, there is an important set of distributions for which finite chains are unlikely to provide enough diversity of the modes of the distribution. This is known as the mode-mixing problem (Jasra et al, 2005;Bengio et al, 2013;Pompe et al, 2020): the chances of going from one mode to a neighboring one may become exponentially small (and thus require exponentially long chains) if the modes are separated by a long sequence of low-probability configurations. This can be alleviated by burning more computation (sampling longer chains) but becomes exponentially unsustainable with increased mode separation.…”

Section: Contrast With Monte-carlo Markov Chain Methodsmentioning

confidence: 99%

“…For example, Bengio et al (2021) build a GFlowNet that constructs a molecule via a small sequence of actions, each of which adds an atom or a molecular substructure to an existing molecule represented by a graph, starting from an empty graph. Only one such configuration needs to be considered, in contrast with MCMC methods, which require potentially very long chains of such configurations, and suffer from the challenge of mode-mixing (Jasra et al, 2005;Bengio et al, 2013;Pompe et al, 2020), which can take time exponentially long in the distance between modes. In GFlowNets, this computational challenge is avoided but the computational demand is converted to that of training the GFlowNet.…”

Section: Gflownets As An Alternative To Mcmc Samplingmentioning

confidence: 99%

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GFlowNet Foundations

Bengio¹,

Salem²,

Tristan³

et al. 2021

Preprint

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Generative Flow Networks (GFlowNets) have been introduced as a method to sample a diverse set of candidates in an active learning context, with a training objective that makes them approximately sample in proportion to a given reward function. In this paper, we show a number of additional theoretical properties of GFlowNets. They can be used to estimate joint probability distributions and the corresponding marginal distributions where some variables are unspecified and, of particular interest, can represent distributions over composite objects like sets and graphs. GFlowNets amortize the work typically done by computationally expensive MCMC methods in a single but trained generative pass. They could also be used to estimate partition functions and free energies, conditional probabilities of supersets (supergraphs) given a subset (subgraph), as well as marginal distributions over all supersets (supergraphs) of a given set (graph). We introduce variations enabling the estimation of entropy and mutual information, sampling from a Pareto frontier, connections to reward-maximizing policies, and extensions to stochastic environments, continuous actions and modular energy functions.

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Section: Introductionmentioning

confidence: 66%

Section: Contrast With Monte-carlo Markov Chain Methodsmentioning

confidence: 99%

Section: Gflownets As An Alternative To Mcmc Samplingmentioning

confidence: 99%

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GFlowNet Foundations

Bengio¹,

Salem²,

Tristan³

et al. 2021

Preprint

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“…The first collection of approaches uses "mode jumping" proposals, e.g. Tjelmeland and Hegstad [2001], Ibrahim [2009], Liu et al [2000], Tjelmeland and Eidsvik [2004] and Pompe et al [2020]. There is typically no explicit state space augmentation with such methods.…”

Section: Alps Motivationmentioning

confidence: 99%

Annealed Leap-Point Sampler for Multimodal Target Distributions

Tawn¹,

Moores²,

Roberts³

2021

Preprint

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In Bayesian statistics, exploring multimodal posterior distribution poses major challenges for existing techniques such as Markov Chain Monte Carlo (MCMC). These problems are exacerbated in high-dimensional settings where MCMC methods typically rely upon localised proposal mechanisms. This paper introduces the Annealed Leap-Point Sampler (ALPS), which augments the target distribution state space with modified annealed (cooled) target distributions, in contrast to traditional approaches which have employed tempering. The temperature of the coldest state is chosen such that its corresponding annealed target density can be sufficiently well-approximated by a Laplace approximation. As a result, a Gaussian mixture independence Metropolis-Hastings sampler can perform mode-jumping proposals even in high-dimensional problems. The ability of this procedure to mode hop at this super-cold state is then filtered through to the target state using a sequence of tempered targets in a similar way to that in parallel tempering methods. ALPS also incorporates the best aspects of current gold-standard approaches to multimodal sampling in high-dimensional contexts. A theoretical analysis of the ALPS approach in high dimensions is given, providing practitioners with a gauge on the optimal setup as well as the scalability of the algorithm. For a d-dimensional problem the it is shown that the coldest inverse temperature level required for the ALPS only needs to be linear in the dimension, O(d), and this means that for a collection of multimodal problems the algorithmic cost is polynomial, O d 3 . ALPS is illustrated on a complex multimodal posterior distribution that arises from a seemingly-unrelated regression (SUR) model of longitudinal data from U.S. manufacturing firms.

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“…Many methods for accelerating the exploration of MCMC algorithms use prior knowledge of the target. For example, known mode locations may be used to design global moves for the sampler Andricioaei et al (2001); Pompe et al (2020); Lan et al (2014); Sminchisescu and Welling (2007); Sminchisescu et al (2003); Tjelmeland and Hegstad (2001), or moves may be guided by the known derivatives of a differentiable target density Lan et al (2014); Tak et al (2018).…”

Section: Related Workmentioning

confidence: 99%

A Metropolis-class sampler for targets with non-convex support

2021

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We aim to improve upon the exploration of the general-purpose random walk Metropolis algorithm when the target has non-convex support $$A\subset {\mathbb {R}}^d$$ A ⊂ R d , by reusing proposals in $$A^c$$ A c which would otherwise be rejected. The algorithm is Metropolis-class and under standard conditions the chain satisfies a strong law of large numbers and central limit theorem. Theoretical and numerical evidence of improved performance relative to random walk Metropolis are provided. Issues of implementation are discussed and numerical examples, including applications to global optimisation and rare event sampling, are presented.

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A framework for adaptive MCMC targeting multimodal distributions

Abstract: Please refer to published version for the most recent bibliographic citation information. If a published version is known of, the repository item page linked to above, will contain details on accessing it.

Cited by 35 publications

References 54 publications

GFlowNet Foundations

GFlowNet Foundations

Annealed Leap-Point Sampler for Multimodal Target Distributions

A Metropolis-class sampler for targets with non-convex support

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