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Lecture 23

9.3. Lecture 23#

Step through Mini-project IIIa to see what is needed#

Here is a summary of what you need to do for Mini-project IIIa:

  • Code for function and a standard scipy optimization.

  • Specification of statistical model and acquisition function, plotting 10 (or so) iteration and summary plots.

  • Change acquisition function to ‘LCB’. What’s that?

  • Assessing exploration vs. exploitation.

  • Adding noise.

  • Changing the GP kernel and analyzing the results.

  • Optional task: implementing BayesOpt algorithm (open the black box!)

  • Bivariate example for a plus \(\Lra\) follow details.

Now go back to Bayesian Optimization and see where these elements come from.

  1. Ingredients:

    • Objective function to optimize \(f(\thetavec) = \chi^2(\thetavec) = \sum_{i=1}^N [y_i^{\text{expt}} - y_i^{\text{th}(\thetavec)}]^2/\sigma_i^2\), which is assumed to be costly to evaluate. **Find \(\thetavec_\ast\) that minimizes \(\chi^2\).

    • Statistical model for \(f(\thetavec)\): \(p(f | \mathcal{D})\) where \(\mathcal{D}\) is the current set of evaluations of \(f\): \(\mathcal{D} = \{\thetavec_i, f(\thetavec_i)\}\). Start with \(p(f)\) as a Gaussian process GP and update via Bayes theorem (recall the GP procedure) with each additional \((\thetavec_i,f(\thetavec_i))\).

    • Acquisition function \(\mathcal{A}(\thetavec|\mathcal{D})\). Take maximum with respect to \(\thetavec\) to determine \(\theta_{i+1}\), given the current data \(\mathcal{D}\) (the full history). This balances between “exploration” (look more widely for possible global minima) and “exploitation” (refine the best currently known minimum).

  2. Step through the code

    • Univariate example: plot function and find minimum from scipy.optimize.minimum (note: requires a starting guess).

      • no noise at first

    • Create GPyOpt object with GPyOpt.methods.BayesianOptimization.

      • specify objective function \(f(\thetavec)\), domain, initial data, acquisition function, whether or not exact function.

    • run_optimization(max_iter, max_time, eps), where max_iter is the number of evaluations, max_time is the time budget, and eps is the minimum distance between \(\thetavec_i\) and \(\thetavec_{i+1}\).

      • Plot with plot_acquisition.

    • Bivariate example

      • This uses a built-in example.

      • Notice the setup for the BayesianOptimization object

        • look at choices for model_type

        • look at choices for acquisition_type

      • separate plots for mean, standard deviation (sd), acquisition function, with red dots for evaluation points.

  3. Look at options for starting samples

    • LHS and Mersenne twister (standard rng) have “holes” but the LHS projections are good.

    • Sobol sequence does not have hole.

  4. Concluding remarks: step through these.

  5. Step back to acquisition function

    • Expected improvement \(\Lra\) EI. At each \(\thetavec\) point, calculation the expectation value of \(f_{\rm min} - f(\thetavec)\), where \(f_{\rm min}\) is the lowest result so far. This is the improvement; use 0 if not improvement (\(f_{\rm min} - f(\thetavec) > 0\)).

    • Analytic evaluation of expectation value, because at a given \(\theta_\ast\) the distribution pdf for \(f\) is a Gaussian (because it is a GP) specified by \(\mu(\thetavec_\ast)\) and \(\sigma^2(\thetavec_\ast) = C(\thetavec_\ast,\thetavec_\ast)\).

    • Consider the two pieces to the integral with \(z = (f_{\rm min} - \mu)/\sigma\):

      • explorative if \(\phi(z)\) dominates \(\Lra\) prior has large uncertainty (large \(\sigma\));

      • exploitive if \(z\Phi(z)\) dominates \(\Lra\) prior has low \(\mu\).

      • With LCB you can change the relative importance of these two terms.

Bayesian neutral networks (BNNs): What is the idea?#

Think of how we might combine Bayesian ideas with neural networks (NNs) or modify NNs.

  • We use BNNs when we care about uncertainty.

  • Standard NN training via optimization is equivalent to doing maximum likelihood estimation (MLE) for the weights.

    • So point estimates only.

    • Recall issues with MLEs from “Why Bayes is Better” section.

    • General problem for NNs: susceptible to overfitting.

  • Can address overfitting (in part) by regularization \(\Lra\) don’t let weights get too big.

    • Bayesian equivalent: put priors on weights as we used naturalness priors in Mini-project I (an L2 regularization \(\Lra\) Gaussian prior pdf).

    • So now finding a MAP estimate (maximizing posterior).

  • New research on overfitting: in some cases a learning algorithm makes good predictions despite fitting to the noise in training data. This is called “benign overfitting”. See, for example, Bartlett et al., Benign Overfitting in Linear Regression. A recent explanation of this is Chatterji and Long, Foolish Crowds Support Benign Overfitting. From their abstract: “Our analysis exposes the benefit of an effect analogous to the ‘wisdom of the crowd’, except here the harm arising from fitting the noise is ameliorated by spreading it among many directions - the variance reduction arises from a foolish crowd.”

  • Bayesian way: posterior inference \(\Lra\) BNNs (start with model, update with data).

    • This is a challenge both to model and to compute.

    • Approximations such as Laplace’s method are inadequate and MCMC is computationally infeasible (because of too many parameters).

    • The alternative is variational inference, where one approximates the posterior.

    • Important for decision-making systems and smaller data situations.

  • Ordinary workflow of a neural network (supervised learning)

    • Randomly initialize the weights.

    • Given inputs, compute outputs of neurons by layers, propagate to prediction.

    • A “loss function” computes deviation of predicted output \(\hat y\) and expected \(y\).

    • The loss value is “back propagated” through layers, adjusting the weights.

  • Output of ordinary ML does not come with variability or a credibility

    • Just a point prediction - no model of the world is explicitly constructed.

    • There are weights and network topology, but no direct correlation to a statistical model.

  • BNN: Prior describes key parameters, utilized as input to the neural net. Output is used to compute likelihood with pdf. Get the posterior distribution by variational inference.

Notebooks from Christian Forssen’s course at Chalmers (updated for 2024)#

A. Bayesian neural networks

B. Variational Inference: Bayesian Neural Networks

Some notes from A:

  • Basic neural network. Bottom line goal is

    \[ p(y|\xvec, D) = \int p(y|\xvec,\wvec) p(\wvec|D) d\wvec \]

    where \(y\) is the new output, \(\xvec\) is the new input, and \(D = \{\xvec^{(i)},y^{(i)}\}\) is a given training dataset.

    • The first pdf in the integral is what the neural network gives \(\Lra\) deterministic given \(\xvec\) and \(\wvec\).

    • We marginalize over the weights.

  • We need \(p(\wvec|D) \propto p(D|\wvec)p(\wvec)\) by Bayes. Then \(p(D|\wvec) = \prod_i p(y^{(i)}| \xvec^{(i)},\wvec)\) is the likelihood.

  • So how do we calculate the marginalization integral with thousands of parameters? \(\Lra\) variational inference or VI.

  • Quick idea about VI and KL divergence.

    • \(p(\wvec|D)\) is intractable so approximate the true posterior with a proxy variational distribution \(q(\wvec|\thetavec)\), where we need to find the optimal \(\thetavec\), denoted \(\thetavec^\ast\).

      • Find \(\thetavec^\ast\) by using the Kullback-Leibler divergence.

      • Find \(\thetavec^\ast\) that maximizes \(J_{\rm ELBD}(\thetavec)\), where ELBD stands for “Evidence Lower Bound”.

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