Gradient-free Hamiltonian Monte Carlo with Efficient Kernel Exponential Families

By Heiko Strathmann, Dino Sejdinovic, Samuel Livingstone, Zoltán Szabó, Arthur Gretton. 2015

78 citations recorded by the source corpus.

We propose Kernel Hamiltonian Monte Carlo (KMC), a gradient-free adaptive MCMC algorithm based on Hamiltonian Monte Carlo (HMC). On target densities where classical HMC is not an option due to intractable gradients, KMC adaptively learns the target's gradient structure by fitting an exponential family model in a Reproducing Kernel Hilbert Space. Computational costs are reduced by two novel efficient approximations to this gradient. While being asymptotically exact, KMC mimics HMC in terms of sampling efficiency, and offers substantial mixing improvements over state-of-the-art gradient free samplers. We support our claims with experimental studies on both toy and real-world applications, including Approximate Bayesian Computation and exact-approximate MCMC.

Subfields: Numerical Analysis, Statistics and Probability, Probabilistic & Bayesian Methods

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Citations in the corpus

Generalizing Hamiltonian Monte Carlo with Neural Networks (2017, background): “, 2014) and for approximating the gradient of the energy (Strathmann et al., 2015).”
Kernel Mean Embedding of Distributions: A Review and Beyonds (2016, background): “Strathmann et al. (2015) introduces a kernel Hamiltonian Monte Carlo (KHMC) to address the case where one cannot run a traditional Hamiltonian Monte Carlo (HMC) since the target is doubly intractable, i.e., gradients of the energy functions are not available.”
A Kernel Test of Goodness of Fit (2016, methodology): “, 2014), and a recent approximation to this model using random Fourier features (Strathmann et al., 2015).”
Optimal Rates for Random Fourier Features (2015, neutral): “To this end, we considerk as in (1) and defineha := cos(πa2 + ·), a ∈ N (in other wordsh0 = cos, h1 = − sin, h2 = − cos, h3 = sin andha = hamod4).”
Density Estimation in Infinite Dimensional Exponential Families (2013, methodology): “Inspired by this preliminary empirical investigation, recently, the proposed estimator has been explored in two concrete applications of gradient-free adaptive MCMC sampler (Strathmann et al., 2015) and graphical model structure learning (Sun et al.”
Sliced Score Matching: A Scalable Approach to Density and Score Estimation (2019, background): “Beyond training unnormalized models, sliced score matching can also be naturally adapted as an objective for estimating the score function of a data generating distribution (Sasaki et al., 2014; Strathmann et al., 2015) by training a score function model parameterized by deep neural networks.”
Efficient Learning of Generative Models via Finite-Difference Score Matching (2020, methodology): “SM methods can be also used in score estimation [35, 54, 61].”
Learning deep kernels for exponential family densities (2018, methodology): “(Strathmann et al., 2015), and learning discriminative features (Janzamin et al.”
Kernel Exponential Family Estimation via Doubly Dual Embedding (2018, methodology): “Finally, we compare the MMD between the generated samples from the learned model to the training data with the current state-of-the-art method for KEF (Sutherland et al., 2017), which performs better than KDE and other alternatives (Strathmann et al., 2015).”
A Spectral Approach to Gradient Estimation for Implicit Distributions (2018, background): “Though KMC is carefully designed, its gradients are estimated by ﬁrst ﬁtting a kernel exponential family as a surrogate and then taking derivatives through it, while SSGE is arguably more direct.”
Robust generalised Bayesian inference for intractable likelihoods (2021, unknown): “Finite rank approximations have previously been considered for frequentist learning of kernel exponential families in Strathmann et al. (2015) and Sutherland et al. (2018).”
Gradient Estimators for Implicit Models (2017, methodology): “[34] and [40] derived a parametric solution by first approximating the log density up to a constant with the RBF kernel K(x,x′) := exp [ − 1 2σ2 ||x− x ||(2)2 ] (i.”