Bayesian posterior inference via optimization

1 Bayesian learning rule

M. E. Khan and Rue (2024):

We show that a wide range of well-known learning algorithms from a variety of fields are all specific instances of a single learning algorithm derived from Bayesian principles. The starting point is the variational formulation by Zellner (1988), which is an extension of Eq. 1 to optimize over a well-defined candidate distribution $q(\mathit{\theta })$, and for which the minimizer $$q_{\ast }(\mathit{\theta })=\underset{q(\mathit{\theta })}{\arg min}{\mathbb{E}}_q\left[\sum _{i=1}^N\ell (y_i,f_{\mathit{\theta }}\left({\mathit{x}}_i\right))\right]+{\mathbb{D}}_{KL}[q(\mathit{\theta })\parallel p(\mathit{\theta })]$$ defines a generalized posterior (Bissiri, Holmes, and Walker 2016; Catoni 2007; T. Zhang 1999) in lack of a precise likelihood. The prior distribution is related to the regularizer, $p(\mathit{\theta })\propto \exp (-R(\mathit{\theta }))$, and ${\mathbb{D}}_{KL}[\cdot \parallel \cdot ]$ is the Kullback-Leibler Divergence (KLD). In the case where $\exp (-\ell (y_i,f_{\mathit{\theta }}\left({\mathit{x}}_i\right)))$ is proportional to the likelihood for $y_i,\mathrm{\forall }i$, then $q_{\ast }(\mathit{\theta })$ is the posterior distribution for $\mathit{\theta }$ (Zellner 1988).

The result is heavy on natural gradient and exponential families to tweak Adam to be a Bayesian posterior sampler. Probably related: Knoblauch, Jewson, and Damoulas (2022).

2 SGD as MCMC

Combining Markov Chain Monte Carlo and Stochastic Gradient Descent in the sense of using SGD to do some cheap approximation to MCMC posterior sampling. Overviews in Ma, Chen, and Fox (2015) and Mandt, Hoffman, and Blei (2017). A lot of probabilistic neural nets leverage this idea.

A related idea is estimating gradients of parameters by Monte Carlo; there is nothing necessarily Bayesian about that per se; in that case, we are doing a noisy estimate of a deterministic quantity. In this setting we are interested in the noise itself.

I have a vague memory that this argument is leveraged in Neal (1996)? Should check. For sure the version in Mandt, Hoffman, and Blei (2017) is a highly developed and modern take. Basically, they analyse the distribution near convergence as an autoregressive process:

Stochastic Gradient Descent with a constant learning rate (constant SGD) simulates a Markov chain with a stationary distribution. With this perspective, we derive several new results.

1. We show that constant SGD can be used as an approximate Bayesian posterior inference algorithm. Specifically, we show how to adjust the tuning parameters of constant SGD to best match the stationary distribution to a posterior, minimizing the Kullback-Leibler divergence between these two distributions.
2. We demonstrate that constant SGD gives rise to a new variational EM algorithm that optimizes hyperparameters in complex probabilistic models.
3. We also propose SGD with momentum for sampling and show how to adjust the damping coefficient accordingly.
4. We analyse MCMC algorithms. For Langevin Dynamics and Stochastic Gradient Fisher Scoring, we quantify the approximation errors due to finite learning rates. Finally,
5. we use the stochastic process perspective to give a short proof of why Polyak averaging is optimal. Based on this idea, we propose a scalable approximate MCMC algorithm, the Averaged Stochastic Gradient Sampler.

The article is rather beautiful. Importantly they leverage the assumption that we are sampling from approximately (log-)quadratic posterior modes, which means that we should be suspicious of the method when

1. The posterior is not quadratic, i.e. the distribution is not well approximated by a Gaussian at the mode, and
2. The same for the tails. If there are low-probability but high-importance posterior configurations such that they are not Gaussian in the tails, we should be skeptical that they will be sampled well; I have an intuition that this is a more stringent requirement, but TBH I am not sure of the exact relationship of these two conditions.

The analysis leverages gradient flow, which is a continuous limit of stochastic gradient descent.

3 Stochastic Weight Averaging

A popular recent development is the Stochastic Weight Averaging family of methods (Izmailov et al. 2018, 2020; Maddox et al. 2019; Wilson and Izmailov 2020). See Andrew G Wilson’s web page for a brief description of the sub methods, since he seems to have been involved in all of them.

4 Actually overparameterization biases us to a good GP approximation

Such is the hypotheses of Mingard et al. (2021); see also overparameterization for a discussion of the general idea.

Deep neural networks (DNNs) generalise remarkably well in the overparameterised regime, suggesting a strong inductive bias towards functions with low generalisation error. We empirically investigate this bias by calculating, for a range of architectures and datasets, the probability $P_{\text{SGD }}(f\mid S)$ that an overparameterised DNN, trained with stochastic gradient descent (SGD) or one of its variants, converges on a function $f$ consistent with a training set $S$. We also use Gaussian processes to estimate the Bayesian posterior probability $P_{\mathrm{B}}(f\mid S)$ that the DNN expresses $f$ upon random sampling of its parameters, conditioned on $S$.

Our main findings are that $P_{\text{SGD }}(f\mid S)$ correlates remarkably well with $P_{\mathrm{B}}(f\mid S)$ and that $P_{\mathrm{B}}(f\mid S)$ is strongly biased towards low-error and low complexity functions. These results imply that strong inductive bias in the parameter-function map (which determines $P_{\mathrm{B}}(f\mid S)$ ), rather than a special property of SGD, is the primary explanation for why DNNs generalise so well in the overparameterised regime.

While our results suggest that the Bayesian posterior $P_{\mathrm{B}}(f\mid S)$ is the first order determinant of $P_{\mathrm{SGD}}(f\mid S)$, there remain second order differences that are sensitive to hyperparameter tuning. A function probability picture, based on $P_{\mathrm{SGD}}(f\mid S)$ and/or $P_{\mathrm{B}}(f\mid S)$, can shed light on the way that variations in architecture or hyperparameter settings such as batch size, learning rate, and optimiser choice, affect DNN performance.

5 Stochastic Gradient Langevin MCMC

“a Markov Chain reminiscent of noisy gradient descent” (Welling and Teh 2011) extending vanilla Langevin dynamics.

6 Stein Variational GD

Perhaps related? An ensemble method. See Stein VGD.

7 SG Hamiltonian Monte Carlo

This, surprisingly, works, I am told? T. Chen, Fox, and Guestrin (2014).

8 SG thermostats

Some kind of variance control using auxiliary variables? See Ding et al. (2014).

9 SG Fisher scoring

See Ahn, Korattikara, and Welling (2012). I assume there is a connection to MC gradients via the score trick?.

10 Non-likelihood based methods

See Generalized Variational Inference.

11 Incoming

(M. Khan et al. 2018; Osawa et al. 2019; G. Zhang et al. 2018).

• Bayesian Neural Nets and how to effectively train them with Stochastic Gradient Markov Chain Monte Carlo | Simen Eide

Knoblauch, Jewson, and Damoulas (2019):

We advocate an optimization-centric view on and introduce a novel generalization of Bayesian inference. Our inspiration is the representation of Bayes’ rule as infinite-dimensional optimization problem (Csiszár 1975; Donsker and Varadhan 1975; Zellner 1988). First, we use it to prove an optimality result of standard Variational Inference (VI): Under the proposed view, the standard Evidence Lower Bound (ELBO) maximizing VI posterior is preferable to alternative approximations of the Bayesian posterior. Next, we argue for generalizing standard Bayesian inference. The need for this arises in situations of severe misalignment between reality and three assumptions underlying standard Bayesian inference: (1) Well-specified priors, (2) well-specified likelihoods, (3) the availability of infinite computing power. Our generalization addresses these shortcomings with three arguments and is called the Rule of Three (RoT). We derive it axiomatically and recover existing posteriors as special cases, including the Bayesian posterior and its approximation by standard VI. In contrast, approximations based on alternative ELBO-like objectives violate the axioms. Finally, we study a special case of the RoT that we call Generalized Variational Inference (GVI). GVI posteriors are a large and tractable family of belief distributions specified by three arguments: A loss, a divergence and a variational family. GVI posteriors have appealing properties, including consistency and an interpretation as approximate ELBO. The last part of the paper explores some attractive applications of GVI in popular machine learning models, including robustness and more appropriate marginals. After deriving black box inference schemes for GVI posteriors, their predictive performance is investigated on Bayesian Neural Networks and Deep Gaussian Processes, where GVI can comprehensively improve upon existing methods.

• Team ApproxBayes

12 References