Quasi-Monte Carlo Variational Inference
Quasi-Monte Carlo Variational Inference
Alexander Buchholz and Florian Wenzel and Stephan Mandt
2018

Paper summary
artems
Variational Inference builds around the ELBO (Evidence Lower BOund) -- a lower bound on a marginal log-likelihood of the observed data $\log p(x) = \log \int p(x, z) dz$ (which is typically intractable). The ELBO makes use of an approximate posterior to form a lower bound:
$$
\log p(x) \ge \mathbb{E}_{q(z|x)} \log \frac{p(x, z)}{q(z|x)}
$$
# Introduction to Quasi Monte Carlo
It's assumed that both the join $p(x, z)$ (or, equivalently the likelihood $p(x|z)$ and the prior $p(z)$) and the approximate posterior $q(z|x)$ are tractable (have closed-form density and are easy to sample from). Then one can estimate the ELBO via Monte Carlo as
$$
\text{ELBO} \approx \frac{1}{N} \sum_{n=1}^N \log \frac{p(x, z_n)}{q(z_n|x)}, \quad\quad z_n \sim q(z|x)
$$
This estimate can be used in stochastic optimization, essentially stochastically maximizing the ELBO, which leads to either increasing marginal log-likelihood or decreasing the gap between the true posterior distribution $p(z|x)$ and the approximate one $q(z|x)$.
Efficiency of stochastic optimization depends on the amount of stochasticity. The bigger the variance is -- the harder it's to locate the optimum. It's well-known that in typical Monte Carlo variance scales as 1/N for a sample of size N, and hence typical error of such "approximation" has an order of $1/\sqrt{N}$
However, there are more efficient schemes to evaluate the integrals of the form of the expectation. To give you some intuition, consider
$$ \mathbb{E}_{q(z)} f(z) = \int_\mathcal{Z} f(z) q(z) dz = \int_{[0, 1]^d} f(z(u)) du $$
Here I used the fact that any random variance can be expressed as a deterministic transformation of a uniform r.v. (by application of the inverse CDF of the former r.v.), so estimating the expectation using MC essentially means sampling a bunch of uniform r.v. $u_1, \dots, u_N$ and transforming them into the corresponding $z$s. However, uniformly distributed random variables sometimes clump together and leave some areas uncovered:
https://i.imgur.com/fejsl2t.png
Low Discrepancy sequences are designed to cover the unit cube more uniformly in a sense that points are unlikely to clump and should not leave "holes" in the landscape, effectively facilitating a better exploration. The Quasi Monte Carlo then employs these sequences to evaluate the integral at, giving (a deterministic!) approximation with an error of an order $\tfrac{(\log N)^d}{N}$. If you want some randomness, there are clever randomization techniques, that give you Randomized Quasi Monte Carlo with roughly the same guarantees.
# RQMC applied to VI
Authors estimate the ELBO using samples obtained from the Randomized QMC (scrambled Sobol sequence, in particular), and show experimentally that this leads to lower gradient variance and faster convergence.
# Theoretical Properties
Authors also analyse Stochastic Gradient Descent with RQMC and prove several convergence theorems. To the best of my knowledge, this is the first work considering stochastic optimization using QMC (which is understandable given that one needs to be able to control the gradients to do so)
# Critique
The paper was a great read, and spurred a great interest in me. I find the idea of using QMC very intriguing, however in my opinion there are several problems on the road to mass-adoption
1. Authors use RQMC to get the stochastic nature of $z_n$, however that essentially changes the effective distribution of generated $z$, which should be accounted for in the ELBO, otherwise the objective they're maximizing is not an ELBO (if only asymptotically) and hence not necessary a lower bound on the marginal log-likelihood. However, finding the correct proposal density $q(z|x)$ (and successfully using it) does not seem easy as most randomization schemes give you degenerate support, and KL is not well-defined.
2. Authors have an experiment on a Bayesian Neural Network, however a very small one, there are reasons to doubt their results will translate to real ones, as the positive effect of QMC vanishes as dimension grows (because it's harder for uniform samples to clump together)
3. Standard control variates might no longer reduce the variance, further research is needed.
Quasi-Monte Carlo Variational Inference

Alexander Buchholz and Florian Wenzel and Stephan Mandt

arXiv e-Print archive - 2018 via Local arXiv

Keywords: stat.ML, cs.LG

**First published:** 2018/07/04 (7 months ago)

**Abstract:** Many machine learning problems involve Monte Carlo gradient estimators. As a
prominent example, we focus on Monte Carlo variational inference (MCVI) in this
paper. The performance of MCVI crucially depends on the variance of its
stochastic gradients. We propose variance reduction by means of Quasi-Monte
Carlo (QMC) sampling. QMC replaces N i.i.d. samples from a uniform probability
distribution by a deterministic sequence of samples of length N. This sequence
covers the underlying random variable space more evenly than i.i.d. draws,
reducing the variance of the gradient estimator. With our novel approach, both
the score function and the reparameterization gradient estimators lead to much
faster convergence. We also propose a new algorithm for Monte Carlo objectives,
where we operate with a constant learning rate and increase the number of QMC
samples per iteration. We prove that this way, our algorithm can converge
asymptotically at a faster rate than SGD. We furthermore provide theoretical
guarantees on QMC for Monte Carlo objectives that go beyond MCVI, and support
our findings by several experiments on large-scale data sets from various
domains.
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Alexander Buchholz and Florian Wenzel and Stephan Mandt

arXiv e-Print archive - 2018 via Local arXiv

Keywords: stat.ML, cs.LG

[link]
Variational Inference builds around the ELBO (Evidence Lower BOund) -- a lower bound on a marginal log-likelihood of the observed data $\log p(x) = \log \int p(x, z) dz$ (which is typically intractable). The ELBO makes use of an approximate posterior to form a lower bound: $$ \log p(x) \ge \mathbb{E}_{q(z|x)} \log \frac{p(x, z)}{q(z|x)} $$ # Introduction to Quasi Monte Carlo It's assumed that both the join $p(x, z)$ (or, equivalently the likelihood $p(x|z)$ and the prior $p(z)$) and the approximate posterior $q(z|x)$ are tractable (have closed-form density and are easy to sample from). Then one can estimate the ELBO via Monte Carlo as $$ \text{ELBO} \approx \frac{1}{N} \sum_{n=1}^N \log \frac{p(x, z_n)}{q(z_n|x)}, \quad\quad z_n \sim q(z|x) $$ This estimate can be used in stochastic optimization, essentially stochastically maximizing the ELBO, which leads to either increasing marginal log-likelihood or decreasing the gap between the true posterior distribution $p(z|x)$ and the approximate one $q(z|x)$. Efficiency of stochastic optimization depends on the amount of stochasticity. The bigger the variance is -- the harder it's to locate the optimum. It's well-known that in typical Monte Carlo variance scales as 1/N for a sample of size N, and hence typical error of such "approximation" has an order of $1/\sqrt{N}$ However, there are more efficient schemes to evaluate the integrals of the form of the expectation. To give you some intuition, consider $$ \mathbb{E}_{q(z)} f(z) = \int_\mathcal{Z} f(z) q(z) dz = \int_{[0, 1]^d} f(z(u)) du $$ Here I used the fact that any random variance can be expressed as a deterministic transformation of a uniform r.v. (by application of the inverse CDF of the former r.v.), so estimating the expectation using MC essentially means sampling a bunch of uniform r.v. $u_1, \dots, u_N$ and transforming them into the corresponding $z$s. However, uniformly distributed random variables sometimes clump together and leave some areas uncovered: https://i.imgur.com/fejsl2t.png Low Discrepancy sequences are designed to cover the unit cube more uniformly in a sense that points are unlikely to clump and should not leave "holes" in the landscape, effectively facilitating a better exploration. The Quasi Monte Carlo then employs these sequences to evaluate the integral at, giving (a deterministic!) approximation with an error of an order $\tfrac{(\log N)^d}{N}$. If you want some randomness, there are clever randomization techniques, that give you Randomized Quasi Monte Carlo with roughly the same guarantees. # RQMC applied to VI Authors estimate the ELBO using samples obtained from the Randomized QMC (scrambled Sobol sequence, in particular), and show experimentally that this leads to lower gradient variance and faster convergence. # Theoretical Properties Authors also analyse Stochastic Gradient Descent with RQMC and prove several convergence theorems. To the best of my knowledge, this is the first work considering stochastic optimization using QMC (which is understandable given that one needs to be able to control the gradients to do so) # Critique The paper was a great read, and spurred a great interest in me. I find the idea of using QMC very intriguing, however in my opinion there are several problems on the road to mass-adoption 1. Authors use RQMC to get the stochastic nature of $z_n$, however that essentially changes the effective distribution of generated $z$, which should be accounted for in the ELBO, otherwise the objective they're maximizing is not an ELBO (if only asymptotically) and hence not necessary a lower bound on the marginal log-likelihood. However, finding the correct proposal density $q(z|x)$ (and successfully using it) does not seem easy as most randomization schemes give you degenerate support, and KL is not well-defined. 2. Authors have an experiment on a Bayesian Neural Network, however a very small one, there are reasons to doubt their results will translate to real ones, as the positive effect of QMC vanishes as dimension grows (because it's harder for uniform samples to clump together) 3. Standard control variates might no longer reduce the variance, further research is needed. |

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