Bayesian learning

“…many well-known learning-algorithms, such as those used in optimization, deep learning, and machine learning in general, can now be derived directly following the above scheme using a single algorithm”

The One World ABC webinar today was delivered by Emtiyaz Khan (RIKEN), about the Bayesian Learning Rule, following Khan and Rue 2021 arXival on Bayesian learning. (It had a great intro featuring a video of the speaker’s daughter learning about the purpose of a ukulele in her first year!) The paper argues about a Bayesian interpretation/version of gradient descent algorithms, starting with Zellner’s (1988, the year I first met him!) identity that the posterior is solution to

when ℓ is the likelihood and π the prior. This identity can be generalised to an arbitrary loss function (also dependent on the data)  replacing the likelihood and considered for a posterior chosen within an exponential family just as variational Bayes. Ending up with a posterior adapted to this target (in the KL sense). The optimal hyperparameter or pseudo-hyperparameter of this approximation can be recovered by some gradient algorithm, recovering as well stochastic gradient and Newton’s methods. While constructing a prior out of a loss function would have pleased the late Herman Rubin, this is not the case, but rater an approach to deriving a generalised Bayes distribution within a parametric family, including mixtures of Gaussians. At some point in the talk, the uncertainty endemic to the Bayesian approach seeped back into the picture, but since most of the intuition came from machine learning, I was somewhat lost at the nature of this uncertainty.

 

 

the Bayesian learning rule [One World ABC’minar, 27 April]

The next One World ABC seminar is taking place (on-line, requiring pre-registration) on 27 April, 9:30am UK time, with Mohammad Emtiyaz Khan (RIKEN-AIP, Tokyo) speaking about the Bayesian learning rule:

We show that many machine-learning algorithms are specific instances of a single algorithm called the Bayesian learning rule. The rule, derived from Bayesian principles, yields a wide-range of algorithms from fields such as optimization, deep learning, and graphical models. This includes classical algorithms such as ridge regression, Newton’s method, and Kalman filter, as well as modern deep-learning algorithms such as stochastic-gradient descent, RMSprop, and Dropout. The key idea in deriving such algorithms is to approximate the posterior using candidate distributions estimated by using natural gradients. Different candidate distributions result in different algorithms and further approximations to natural gradients give rise to variants of those algorithms. Our work not only unifies, generalizes, and improves existing algorithms, but also helps us design new ones.

Concentration and robustness of discrepancy-based ABC [One World ABC ‘minar, 28 April]

Our next speaker at the One World ABC Seminar will be Pierre Alquier, who will talk about “Concentration and robustness of discrepancy-based ABC“, on Thursday April 28, at 9.30am UK time, with an abstract reported below.

Approximate Bayesian Computation (ABC) typically employs summary statistics to measure the discrepancy among the observed data and the synthetic data generated from each proposed value of the parameter of interest. However, finding good summary statistics (that are close to sufficiency) is non-trivial for most of the models for which ABC is needed. In this paper, we investigate the properties of ABC based on integral probability semi-metrics, including MMD and Wasserstein distances. We exhibit conditions ensuring the contraction of the approximate posterior. Moreover, we prove that MMD with an adequate kernel leads to very strong robustness properties.

approximate Bayesian inference [survey]

In connection with the special issue of Entropy I mentioned a while ago, Pierre Alquier (formerly of CREST) has written an introduction to the topic of approximate Bayesian inference that is worth advertising (and freely-available as well). Its reference list is particularly relevant. (The deadline for submissions is 21 June,)