Imagine for a second that you were tasked with
finding the absolute perfect settings for an
incredibly complex system. Oh yeah, like standing
in front of a massive control board. Exactly.
You're tweaking dials, you're flipping switches,
trying to squeeze out the best possible performance,
but there is a massive paralyzing catch. Right,
the cost. Right. Every single guess you make,
every time you turn a dial just a fraction of
a millimeter to see what happens, it costs an
immense amount of time. or a staggering amount
of money, or just a colossal drain on computing
power. You basically can't just try everything.
The stakes are way too high to just guess and
check. So how do you find the absolute best answer
with the fewest possible attempts? Well, welcome
to today's deep dive. Today we are cracking open
the mathematical engine that solves this exact
problem. It's a fascinating one, too. It really
is. It's a sequential design strategy called
Bayesian optimization. And we are pulling from
a massive comprehensive source document on the
subject to really understand how it works. Our
mission today is basically to figure out how
machines and, you know, the scientists who program
them, make these highly efficient optimal choices
when they are flying completely blind. And the
stakes for understanding this framework? couldn't
be higher for you or really anyone following
modern technology. Absolutely. If you look at
the visual backdrop behind me today, these shifting
interconnected star charts that kind of dissolve
into complex math equations, that really represents
the unseen landscape we're navigating here. It's
a great visual for it. Because with the 21st
century explosion of artificial intelligence,
this specific mathematical framework has quietly
become the secret engine behind the scenes. It's
everywhere. It is the invisible hand guiding
everything from the training of massive neural
networks to the design of entirely new physical
materials in a laboratory. It's literally the
science of making the most out of the unknown.
Okay, let's unpack this. Before we get into how
this technology evolved to practically run the
modern AI world, we first need to understand
the fundamental problem it was built to solve
in the first place. Right, the core issue. Yeah,
this concept of the black box function. The black
box is the perfect starting point. So Bayesian
optimization is deployed when you have an objective
function that is continuous, but its internal
structure is completely, completely hidden from
you. Like you literally cannot see the gears
turning inside. Exactly. And furthermore, it's
incredibly difficult or computationally expensive
to evaluate. All you can do is observe the output
for a given input. So you feed the box a specific
set of numbers, and then you just wait. You wait,
sometimes for hours or even days, and you see
what single number comes out the other side.
And importantly, you cannot evaluate its derivatives.
Meaning, you don't get those handy mathematical
shortcuts that tell you which direction is up
or down the slope of the function. Right. You
can't just follow the curve. Without derivatives,
you have no local compass. And as a practical
limitation to keep in mind, the math underlying
this framework generally works best when you
are dealing with 20 dimensions or fewer. Which,
just to be clear, trying to optimize something
across 20 different intersecting dimensions is
still incredibly complex to visualize. Oh, absolutely.
It's mind -bending. But I think I have a way
for you, the listener, to picture what this black
box really feels like. Imagine you are trying
to find the absolute highest peak in a vast,
rugged mountain range. OK, I like this. but you're
completely blindfolded. You can't see the peaks
in the distance. You can't see the valleys below
you. The only way you can know your current altitude
is by stopping, setting up a massive camp, and
taking an incredibly expensive time -consuming
GPS reading. Right, because you can't just walk
in a neat little grid taking readings every five
feet. Exactly. You'd run out of expedition money
and time almost immediately. You need a much
smarter strategy to guess where the highest peak
might be based on just the very few scattered
readings you've already taken. That analogy captures
the tension perfectly. You can't wander aimlessly,
and you certainly can't afford to be exhaustive.
So what's the move? This is where the Bayesian
strategy steps in to save the day. Because the
true landscape of that mountain range, the function
itself is unknown. We treat it mathematically
as a random function. And what we do first is
place a prior. over it. A prior. So mapping that
back to the mountain analogy, that would be our
initial pre -existing belief about what the mountain
range probably looks like before we even take
a single step. That is the essence of it, yeah.
Like assuming mountains generally slope upward
or that peaks might cluster together. Right.
The prior captures our initial mathematical beliefs
about the behavior of the function before we've
taken any of those costly measurements. Then
you take a reading. You get some actual data.
Yes. You gather your first piece of real data
in math terms, a function. evaluation and based
on that concrete new data you update your prior
to form what the framework calls a posterior
distribution. So the posterior is basically our
updated slightly less blind map of the landscape.
Exactly. It takes our assumptions and bends them
around the cold hard facts of the GPS reading
we just took. It represents our newly updated
understanding of the world. Every single time
you take a costly measurement, your posterior
distribution shifts, warps, and refines itself.
It's adapting. Yes. You aren't just looking for
the peak. You are mathematically mapping out
the uncertainty itself. You're zeroing in on
where the highest peak is most likely to be hiding
in the dark. Now that we understand the core
goal, this idea of intelligently mapping an unseen
landscape while blindfolded, I really want to
look at how mathematicians actually crack the
code on this. The history is pretty fascinating.
Yeah, because we have this great theoretical
concept. But in the early 1960s, an American
applied mathematician named Harold J. Kushner
started laying the actual groundwork. Kushner's
work in 1964 was a massive leap forward. He published
a paper proposing a new method for locating the
maximum point of an arbitrary multi -peak curve
in a noisy environment. Imagine a roller coaster
track, hidden in dense fog, where your altimeter
occasionally gives you slightly wrong readings.
Exactly, and he was trying to find the highest
point on that track. Now, he wasn't explicitly
calling it Bayesian optimization just yet, but
he provided the crucial theoretical foundation
for making decisions under that kind of mathematical
fog. But the framework really takes its modern
shape a bit later. In 1978, thanks to a Lithuanian
scientist named Jonas Malkus, he published a
paper discussing how to use Bayesian methods
specifically to seek extreme values under uncertainty.
and he proposed a mechanism called the expected
improvement principle. Yes, the EI principle.
This seems like the moment the light bulb really
went on for the field. What's fascinating here
is that Mockus's expected improvement principle
isn't just a historical footnote, it is literally
still one of the core sampling strategies driving
algorithms today. Really? Still today? Oh yeah.
Before Marcus, you might just look for any point
that might be higher than your current spot.
Expected improvement actually calculates a magnitude
of the potential gain. So it doesn't just ask,
will this next step be higher? Right. It asks,
how much higher could it possibly be, multiplied
by the probability that it actually is higher?
It fundamentally cracked the code on how to balance
the risk of exploring new areas versus optimizing
the function efficiently based on what you already
know. And then the timeline jumps again to 1998.
Donald R. Jones and his colleagues took Mockus's
work and introduced something called the Gaussian
process, while elaborating heavily on that expected
improvement principle. They really brought it
into a new era. They started bringing this out
of pure theoretical math and into computer science
and engineering. But wait, I have a question
about this. Sure, what is it? If they are bringing
this highly complex, constantly updating topographical
map into computer science in the late 90s, wouldn't
the computers of 1998 basically catch on fire
trying to process this? Oh yeah. I mean, the
computing power back then must have been a massive
bottleneck for something this math heavy. You're
totally right. The computational complexity of
Bayesian optimization was absolutely a wall they
slammed into. The processing power available
in 1998 just couldn't handle the sheer weight
of the math for anything beyond relatively simple
problems. It was an incredibly elegant theory
locked inside sluggish, incapable hardware. It
was a sports car stuck in endless traffic. It
just had to wait for the roads to clear. And
those roads clear dramatically in the 21st century.
They really did. With the sudden massive rise
of artificial artificial intelligence, deep learning,
and bionic robots, Bayesian optimization finally
had the horsepower it required. Suddenly we had
the massive parallel compute power necessary
to run these complex posterior updates in real
time. Exactly. And it rapidly became a vital
foundational tool for what the industry calls
hyperparameter tuning. Major tech players like
Google, Meta, OpenAI began actively embedding
Bayesian optimization into their deep learning
frameworks. Just to drastically improve their
search efficiency. Right. Because training a
massive AI model is the ultimate blindfolded
walk in the mountains. Every single time they
test a new configuration for a neural network,
it takes huge amounts of server time. Thousands
of dollars in electricity. And days of waiting.
They desperately needed Mockus and Kushner's
math to find the best AI settings without going
bankrupt running endless trial and error tests.
It's crazy to think about, but the theoretical
mathematicians of the 1960s and 70s build the
exact perfect engine that the AI pioneers of
the 2000s didn't even know they needed yet. It's
a beautiful full circle moment in computer science.
So with companies like OpenAI utilizing this
today to build the tools we use, we need to look
under the hood of how it actually operates in
practice. Let's do it. We know we have this updated
posterior distribution or actively updating map
of the mountain, but how does the system actually
use that map to decide where to place the very
next guess? That's the million dollar question.
Right. If I'm on the mountain, How does the system
tell me whether I should take a helicopter to
a completely uncharted dark side of the mound
range, which is exploration, or just set up camp
right next to the highest peak I've already found
and check a few feet to the left, which is exploitation?
This raises an important question, and it is
the exact tension at the heart of the entire
system. Exploration versus exploitation. Okay.
The mechanism that makes this highly critical
decision is called an acquisition function. You'll
sometimes see it referred to as an infill sampling
criterion in the literature. Acquisition function,
got it. These are strict mathematical rules that
use your current updated map to determine the
absolute best coordinates for your very next
blindfolded jump. It's the decision engine. It
looks at the map and points the finger. And the
nuance here is brilliant because there isn't
just one single way to point that finger. You
choose different acquisition functions depending
on your specific appetite for risk. Like what?
What are our options? Yeah. Well, for instance,
you have the probability of improvement. This
function just looks for any guaranteed step up
no matter how small. It's highly exploitative.
Then you have Marcus's expected improvement,
which we discussed. Right. Trying to balance
the actual size of the potential gain against
the risk. Exactly. Finding the biggest possible
lead, not just a tiny inch upward. Oh, and you
also have Bayesian expected losses. Okay. Wow.
Yeah, and then you have something like the upper
confidence bound or UCB This one is fascinating
because it specifically rewards ignorance. Wait
rewards ignorance. How does a math equation reward
ignorance? It's clever UCB looks at the map and
intentionally calculates the uncertainty of a
region. It essentially says We have absolutely
no idea what is over in that dark corner of the
map. Because our uncertainty is so high, the
potential for a massive undiscovered peak existing
there is technically also high. So it forces
you to go look. It aggressively forces exploration
so you don't get stuck on a tiny foothill thinking
it's Mount Everest just because you haven't looked
anywhere else. And there are others too, like
Thompson sampling, which uses randomized sampling
from the posterior to naturally balance that
exploration exploitation trade -off. And how
do they actually maximize these functions? Is
there a specific technique? Yeah, they are typically
maximized using numerical optimization techniques.
Things like Newton's method or the BFGS algorithm
are very common here. Wait, if we keep taking
GPS readings on this mountain and we are using
this proxy model we mentioned earlier, the Gaussian
process, to constantly redraw this smooth, perfect
3D topographical map, won't our map eventually
become a problem? What do you mean? Won't it
become so insanely detailed and massive that
it takes longer to read the map than to just
climb the mountain? Like if you feed a Gaussian
process a million data points, doesn't it just
completely buckle under its own weight? That
is a fantastic point, and yes, that is the exact
limitation of the standard approach. I knew it.
When there is a massive amount of data, the mathematical
training of a Gaussian process becomes brutally
slow. The computational cost skyrockets because
it is trying to maintain a continuous, perfectly
smoothed mathematical landscape connecting every
single point to every other point. Which makes
it really difficult for standard Bayesian optimization
to work well in highly complex fields, right?
Oh, yeah, like drug development or sprawling
medical experiments where the data sets aren't
just large, they are gargantuan. So what do they
do if the Gaussian map gets too heavy to unfold?
Do they just abandon the Bayesian approach entirely?
Not at all. They just swap out the proxy model
for something lighter. The sources highlight
a really clever computationally cheaper alternative
called the Parsons tree estimator or TPE. How
does the Parsons tree bypass that data overload?
How does it draw the map differently? Instead
of trying to maintain that perfect continuous
topographical landscape, the Parsons Tree estimator
essentially stops trying to draw the mountain
and instead just sorts your data into buckets.
Yeah, it constructs two separate distributions.
It looks at all your past GPS readings and says,
let's put the top 10 % of our highest readings
into bucket A, the high performers, and let's
throw the remaining 90 % into bucket B, the low
performers. So it's just splitting the data into
good and bad rather than mapping every single
inch. Right. And by simply comparing the mathematical
likelihood of those two distinct buckets, it
can highly efficiently find the location that
maximizes expected improvement. Oh, that's smart.
It asks where on the map is bucket A most likely
to occur compared to bucket B. It finds the next
best spot without getting bogged down by the
sheer crushing weight of calculating a smooth
curve through all the data combined. It's a brilliant
shortcut. Here's where it gets really interesting.
We have these pristine mathematical models. We
have Gaussian processes laying down smooth fabric.
We have Parson Tree estimators sorting data into
neat buckets. But the real world is rarely pristine.
Never pristine. Exactly. What happens when the
GPS reader breaks or the wind blows the climber
off the mountain? How does Bayesian optimization
survive outside the vacuum of pure theory? That
brings us to a subfield known as exotic Bayesian
optimization. Exotic Bayesian optimization. It
sounds like a math problem wearing a Hawaiian
shirt. It's a great term because standard optimization
assumes that every single point you want to test
is relatively easy to evaluate and that the answers
you get back are clean, static, and reliable.
But the real world isn't like that. Great. Exotic
problems occur when the real world injects chaos
into the system. Chaos like what? Give me an
example. Imagine there is inherent unavoidable
noise in the data you are collecting. Your sensors
are faulty. Or maybe you are running parallel
evaluations, like sending out five blindfolded
climbers at the exact same time, and their results
are arriving back to base camp entirely out of
order. Oh, that would be a nightmare. Or what
if the quality of your evaluations relies on
a strict trade -off between the difficulty of
the test and the accuracy of the result? Sometimes
there are random shifting environmental conditions
throwing off the baseline. Or what if the evaluation
suddenly requires calculating partial derivatives
on the fly? Exactly. When the clean assumptions
of the standard model completely break down,
the optimization becomes exotic. And despite
all that real -world messiness, the applications
for this framework are just everywhere. Once
you know what it is, you see it hiding in every
industry. It's not just a theoretical exercise.
It is a testament to how versatile the core philosophy
really is. I mean, we're talking about learning
to rank. algorithms for search engines. Yes,
and visual design by crowds. Which is fascinating,
using this math to optimize aesthetics based
on messy subjective human feedback. And then
there's sensor networks, reinforcement learning.
Even experimental particle physics uses this
to design the actual physical geometry of particle
detectors. It's in chemistry material design.
There's a specific mention in the sources of
using Bayesian optimization to develop ultra
-high specific strength carbon nanolattices.
Literally finding the perfect microscopic structure
for new super materials. But if we connect this
to the bigger picture, one of the most compelling
and visual applications is in modern robotics.
Oh, specifically, automatic gait optimization.
Getting a robot to walk without falling over.
Exactly. Helping a bionic robot learn to walk
under extreme uncertainty. The robot is the blindfolded
climber. It doesn't know the exact friction of
the laboratory floor. It doesn't know the exact
weight distribution of its own limbs while they
are in motion. It has to try a step. observe
the result, which is almost always violently
falling over, and then update its posterior distribution
with that failure. And then use an acquisition
function to decide exactly how much power to
send to its knee and ankle joints for the very
next step. It is literally learning to walk through
Bayesian inference. Every single fall makes the
map clearer. That is incredible. But there was
one specific deep dive example in the material
that I found highly relatable. because it's a
technology you and I use almost every single
day. Facial recognition. Yes. The optimization
of the histogram of oriented gradients, or the
HOG algorithm. Right. For those unfamiliar, HOG
is a really popular feature extraction method
for computer vision. It essentially looks at
an image and tries to find the edges and outlines
of an object like a human face by analyzing the
gradients of light to dark pixels. But its accuracy
relies heavily on how its internal parameters
are set. And optimizing those parameters manually
is notoriously frustrating. Yeah. Like, how many
pixels should it group together? What's the threshold
for an edge? It's a classic black box problem.
You tweak a setting, run 1 ,000 images through
it, and see if it recognized the faces better
or worse. It would take forever. It does. But
scientists proposed a novel approach using that
lighter, less expensive proxy model we talked
about earlier. The tree -structured Parson estimator,
the TPE. Yes, the bucket method. They applied
the TPE bucket method to the facial recognition
software. By using that specific Bayesian technique,
they were able to simultaneously optimize both
the complex internal parameters of the HOG algorithm
and the size of the images being processed. That's
huge. Because TPE could handle the complex overlapping
variables without buckling under the computational
weight, it drastically pushed the accuracy of
the facial recognition forward. And the source
notes that this optimized approach has massive
potential to be adapted for all kinds of other
computer vision applications, right? Exactly.
Pushing the boundaries of handcrafted parameter
-based algorithms far beyond what human trial
and error could ever achieve. So what does this
all mean? We pull back from the carbon analogies,
the falling robots, and the computer vision algorithms.
The ultimate takeaway for you listening to this
is that Bayesian optimization is the premier
framework for making decisions under uncertainty.
It really is. It mathematically proves that you
do not need to know everything to make the best
possible choice. You don't need to take a million
GPS readings. You just need to learn with ruthless
efficiency from the very few choices you can
make constantly updating your map of the world.
That is the absolute essence of it. It's about
maximizing the value of your own ignorance. And
honestly, looking at how elegantly this math
updates its beliefs, it raises a final somewhat
provocative thought for you to mull over. Oh,
I'm ready. Where are we taking this? Since Bayesian
optimization is fundamentally about mathematically
updating our beliefs, turning our initial priors
into updated posteriors based strictly on the
arrival of new evidence, could we theoretically
use this to map Human stubbornness. Whoa. What
do you mean by that? Mapping stubbornness. Think
about it. We all operate with our own internal
personal acquisition functions. Every single
day we are faced with the choice to either explore
new unknown ideas that might challenge us or
exploit the comfortable known beliefs we already
hold. Imagine if we could actually measure that
mathematically. Imagine if we could calculate
your personal posterior distribution to see exactly
where you fail to adapt to new data. Where do
you ignore the GPS reading? Simply because you
prefer the warm comfort of exploitation over
the intellectual growth and the inherent risk
of exploration. Exactly. So the real question
is, are you constantly updating your map when
new facts arrive? Or are you just sitting in
the dark on the side of the mountain, refusing
to take the blindfold off because you like the
spot you're sitting in? It's a tough question
to answer. That is a wild, brilliant thought
to leave on. Thank you for joining us on this
deep dive, keep updating your priors, keep exploring
the dark parts of the map, and we will catch
you next time.