90 minutes of deck, 20 for the notebook. One example all the way: the reactor.

Coffee roasting works too: green, roasted, burnt.

We built the curve from the kinetics only to play the lab and check the answer. The optimizer never sees it.

The curves are sketches of first-order kinetics, A -> B -> C, not data. Saeman's paper is behind the ACS paywall.

From my seminar, as in Lecture 13.

The green diamond: where the uncertainty is largest.

How do you draw one of these random curves from the GP, in practice? Evaluate the GP on a grid of temperatures (here 241 of them). That gives a mean vector mu and a covariance matrix K. Factor K = L L^T (Cholesky). Draw a vector z of standard normal random numbers. The curve is mu + L z. In BoTorch the same thing is gp.posterior(T_grid).rsample().

Play. Prior first: same belief everywhere. Then 30, 50, 110, 80, 65 °C. The sample curves agree where we measured.

Where is Mockus in Shahriari et al. (2016)? The review spells him Močkus, with a caron, and names him only in the reference list. Page 17 of the PDF: "The term Bayesian optimization was coined in the seventies [115]". Reference [115] is J. Močkus, V. Tiesis and A. Žilinskas, "The application of Bayesian methods for seeking the extremum", 1978. The link opens that page.

The four panels are the first pass of the worked example: two old runs, EI peaks at 54 °C, run it, refit.

Step through slowly. Step 6: the jump to 140 °C, where the GP is unsure. After that it never goes back there.

What is kappa, the number on the slider? It is the weight on uncertainty in the upper confidence bound, mu(x) + kappa sigma(x): how many standard deviations of optimism we add to the GP mean. Kappa = 0: the score is just the mean, so the next run goes right next to the best point so far (58 °C). As kappa grows, the uncertainty term takes over and the pick moves to where the band is widest.

Slide κ from 0 up. At 0 it sits next to 54 °C. Around 2 it moves out. At 5 it goes where the band is widest.

What are the Greek letters? Xi (ξ) is the margin. It is in z. In plain terms, the smallest gain that counts: with xi = 0.01, a temperature counts as an improvement only if it beats 0.236 by more than 0.01. A larger xi ignores tiny gains right next to f*, so the search moves away from it. We use xi = 0. Capital and lowercase phi (Φ, φ) are NOT in z. They take z as input in PI and EI: Φ is the standard normal cumulative distribution, φ its density. They are defined on the next two slides.

PI in plain words. At 60 °C the GP does not give one yield. It gives a bell of possible yields, centred at 0.242, spread 0.046. The best yield so far is 0.236. PI is the share of the bell above 0.236: 55%. A 55% chance that 60 °C beats the best so far. Phi(z) is the same number reached through z. Shifting and rescaling the bell to a standard one turns the share above 0.236 into the share left of z = 0.13, and Phi(0.13) = 0.55. The weakness: a point right next to f*, with a narrow bell, gets PI near 1 for a gain of almost nothing. That is why EI weighs each gain by its size.

What is the gain f minus f*? If we ran the experiment at 60 °C and the yield came out f, the gain over the best so far would be f minus f*. If f is below f*, we gain nothing: the gain is 0, not negative. Bottom panel: the gain for every possible yield. Top panel: how likely each yield is. EI averages the gain over the bell: 0.022 at 60 °C. Each card lights up its term in the equation as it appears.

Why LogEI was created. Far from the data, EI is so small that the computer rounds it to exactly 0 (underflow; below about 1e-324 in double precision). Here, after 12 experiments, that happens at 12% of the temperatures (red). Where EI is exactly 0, its slope is 0 too, so the optimizer that follows the slope has no direction. Log EI is about -1000 there instead: a normal number with a slope. The log does not move the peak, so the next experiment is the same. Ament et al. (2023): EI's values "vanish numerically in many regions".

What is kappa exactly, besides the knob? How many standard deviations of optimism. UCB scores each temperature by the top of its band, mu + kappa sigma. Kappa = 2 is the top of the 95% band. What is beta in BoTorch? UpperConfidenceBound(gp, beta) computes mu + sqrt(beta) sigma, so beta = kappa squared: beta = 4 means kappa = 2. Not on this slide on purpose: BoTorch comes later. It is in the notes' BoTorch section.

The figure redraws the idea of the paper's Figs. 8 to 11 on our own function: the prediction alone only refines the local peak near the cluster; EI is largest in the gap. Jones (2001) is the follow-up taxonomy.

Is "automatic differentiation" true for BoTorch? Yes. Checked in the BoTorch 0.18.1 source: optimize_acqf calls gen_candidates_scipy, which runs SciPy's L-BFGS-B, and the gradients come from torch.autograd.grad. The GP is a formula, so the acquisition has exact gradients. Only the experiment itself has none.

Play. Watch Nelder-Mead and L-BFGS-B sit in the cold corner. Random search sprays points everywhere.

BO comes with no guarantee of finding the global optimum. It explores the whole range, which helps it avoid getting stuck, but nothing promises the best point. We do not call it "global".

The Shields et al. quote: the paper spells it "decisionmaking", one word. The slide writes "decision making".

Online experiments, in plain words. Facebook tuned two systems. A ranking system: the order in which items are shown to users. And its server software: the compiler flags, settings of the compiler that change how fast the servers run. Each try is a live test on real traffic, so every result is noisy (Letham et al., 2019).

Constraints, in plain words. The best yield may need a temperature the reactor cannot take, for example a hot spot above its limit. We can only find out by running the experiment. So a second GP learns that limit from the same experiments, and the acquisition is multiplied by the probability of staying within it. BO then proposes only runs likely to be safe.

The 70 °C limit and the rough model are made-up examples on the reactor. The batch is qLogEI on the GP after three experiments. The Pareto front is computed from the kinetics.

What is x_{n+1} = argmax_x sigma(x) doing? It reads: the next experiment is the input where the GP's uncertainty sigma(x) is largest, the widest point of the band. No mean in it at all: active learning does not care where the yield is high, only where the model is unsure.