I think it should be quite obvious that perceptrons are far from the smallest units that are capable of learning. They store many bytes of information, require a non-local update process, need numeric (i.e. symbolic) inputs and involve relatively complex computations. You can go much simpler. For example:
I think Karpathy's microgpt blogpost is the best in this genre in a long time, and it also includes a multi layer perceptron. It's a step up in the hierarchy, so reading both is helpful, of course.
I'm not sure if I'd like to declare a best. There are so many different approaches and I think their ability to inform is cumulative,
I like the ability of this article to do the tiny training runs in browser. It makes the point of a bias clear. Too many tutorials get sucked into the proof of zero times anything is zero. Everyone knows that. What you should show is where that mstters in the problem at hand.
3blue1brown does one of the best depictions of why we need an activation function.
Karpathy's videos are a little tougher for a beginner to grasp, but excel at solving a complete problem. I knew all of the theory behind what it takes to make micrograd before I made my own by following the video, but what you get from doing it can't be understated.
It's hard to describe but it what you learn is more of a feel than pure knowledge. It gives you a better sense of knowing when the principles apply in other circumstances.
Perhaps it's the distinction of understanding how springs and gears work, then looking at a clock and understanding how the gears and springs move the hands. There's still more needed if you want to make a clock. And that stuff is what let's you also make a wind up toy.
I can't agree more with you, It took me many days to understand the "By we need bias?"
I know maths, I know programming, but why was not clear.
I love 3blue1brown.
In the early days of machine learning (before the first AI winter), networks like this were often implemented and trained in hardware: https://en.wikipedia.org/wiki/ADALINE
That was the first thing that came to mind when I read "the smallest brain you can build". Nowadays, that "small brain" would likely be built on a breadboard using op-amps instead.
Amazing and anachronistic to see something like that from 1960. And then it makes me wonder why there wasn't more progress on neural nets being used for many things prior to the 21st century. (I haven't read the history of the AI winters but I have heard of them)
The first AI winter was largely triggered by Minsky in a book he published in 1969, which mathematically proved that single-layer perceptrons couldn't solve non-linear problems. Favorite quote: "Our intuitive judgment is that the extension [to multilayer systems] is sterile."
Yet we had the computational power to run backpropagation in the 1960s and small Transformers in the 1970s (I'm the author of both):
What was missing wasn't the raw processing power, but the ideas and algorithms themselves. Because funding and research were completely discouraged during the AI winter, neural networks research was left dormant and we lost two decades.
I wonder had we invented transformer architecture back in the 70's or 80's, if the pace of hardware innovation would have naturally slowed AI progression, and given humans decades to slowly adapt, rather than the current tidal wave (that seems to grow in size daily) bearing down on us.
> why there wasn't more progress on neural nets being used for many things prior to the 21st century
They were simply too computationally expensive to train for the limited things they could do. It wasn’t until we had the ability to train large neural networks on commodity hardware that things really took off.
The quasi-mythical memristor would be choice for bread boarding a brain. However I suppose you could train a model and then manually place fixed resistors to build the network
One day I'll write about my 1-liner physics engine...
let gravity = setInterval( _ => { if (projectile.object3D.position.y > 0) projectile.object3D.position.y \*= .99 }, 100)
Jokes aside I find that providing ridiculously short toy examples that provide the very limited foundation of a concept are extremely empowering in pedagogy. You "get" it right away because it "fits" in your mind, then you dare tinker with it and quickly see how limited it is, then get excited again. It's a powerful trick to learn more IMHO.
The IF statement is the root creator of software programming. It has the ability to compare two values against each other and branch out to blocks of instructions. So it is perceiving (reading), decision making and routing - all that which differentiate life from inanimate objects. The AI agents perform the exact same loop, by delegating the first two steps to a model.
Going further backwards, the transistor (or a PNP junction) is the hardware level enabler of the IF statement. The action (switching) driven by the current which in turn controls other switches, is the first manifestation of "observe and act" by inanimate things at the speed of electricity.
Mechanical equivalents existed ofcourse - speed of a governer which controls the flow of fuel which in turn controls the speed of the governer.
> Going further backwards, the transistor (or a PNP junction) is the hardware level enabler of the IF statement. The action (switching) (..)
Back up a bit please! Analog computing is a thing. And it isn't even new - not by a long shot.
There are good reasons why practically all computing today is the digital kind. But electronic 'equivalents' of neural nets is one area where analog might make sense. Adding inputs can be as simple as a bunch of resistors + a transistor. Even on modern silicon nodes, that might be a more efficient setup than digital inputs, N-bit adders/multipliers etc. Not saying that's the case, and AI hardware should be based on analog circuitry. But it could be, and perhaps found to be practical.
> But electronic 'equivalents' of neural nets is one area where analog might make sense.
That's an interesting idea, but could the weights be transferred to different hardware and still work? If not, that would be a significant limitation, even if it were preferable in some cases.
An artificial neuron needs values to compare (the sum of weighted inputs). You can add values with transistors of course, but you need more than a dozen just to do simple addition. The activation function could be a simple binary comparison (e.g. between a weight and a threshold), but it’s usually more complicated.
Artificial neurons are significantly more complex that single transistors, and even a minimal hardwired circuit to implement just one neuron requires quite a number of transistors.
So, what if, we build a stack/set of transistors in same shape as a trained model? It would eliminate most of the software stack too and should run very fast. No memory/gpu required, the chip acts as both storage and processing device, purpose built to be physical model of a trained model.
There’s lots of things you can do in hardware that could be done in software but cost. FPGA should have solved this long ago, but apparently the guys who own the IP want to make it as hard as possible to use it …
The alternative IF expression or statement, in the form introduced by John McCarthy in 1958 (which he used in early 1959 to define his version of AND, OR and NOT), is one of the possible primitives for computation.
There exist several equivalent sets of primitive operations. While the sets containing only NAND or only NOR, or both AND and NOT or both OR and NOT are more notorious, these logical operations are more abstract and they do not indicate precisely a hardware implementation, i.e. there are many distinct hardware methods to make such logical gates.
Other sets of primitive operations map directly to hardware devices, e.g. the sets of primitive operations composed of maximum and complement or of minimum and complement map directly to a hardware implementation using rectifier diodes and inverting amplifiers (which can be made with either semiconductor devices or with vacuum tubes, or also with pneumatic or hydraulic devices).
Other sets of primitive operations are obtained by replacing the maximum or minimum circuits with series or parallel connections of switches, like in the CMOS logic that is nowadays dominant.
The alternative IF expression corresponds in hardware to a 2-way multiplexer, which, together with the 2 constant functions "0" and "1" (a.k.a. "false" and "true" or "low voltage" and "high voltage"), is sufficient for a complete set of computational primitives.
Besides those mentioned above, the main remaining variant for a complete set of computational primitives consists of an analog (possibly weighted) adder and an analog comparator, which had been used in the so-called RTL circuits (resistor-transistor logic) and which also corresponds to perceptrons. RTL had been used in some early integrated circuits, before being replaced by DTL and TTL circuits (which are based on minimum and complement functions).
In hardware, e.g. in RTL circuits, a combined analog adder+comparator can be made with a single high-gain amplifying device, together with a set of weighting resistors and a bias resistor. RTL circuits can implement complex logic with fewer devices (e.g. they can implement a neural network in the analog domain), but they were replaced during the sixties of the past century with DTL, then TTL, because those were faster (in RTL, the resistors limit the charging currents for input capacitors and parasitic capacitors, which slows down the logical transitions) and the fact that they needed more devices was not important, due to the quick increase in circuit density.
This reminds me of a book, whose title I forget, about creating a compact set of knowledge that will enable us to "rebuild civilization from scratch" in case of a future/fictional post-collapse scenario. I sometimes wonder, given the immense complexity and global supply-chain dependencies of our computing stack, whether we could bootstrap from first principles something of equivalent power and expressivity, but orders of magnitude simpler.
There's a wide variety of computational primitives, including lambda calculus, combinators, cellular automata, rewriting systems. Perhaps some are more practical to implement in hardware, particularly the kind of DIY electronics or analog machines that can also be put together from scratch. It might look like a whole building of mechanical switches, powered by a water wheel ("watermill"), for example.
I think if civilization collapses they'll have other priorities, and very little benefit to get from the effort required for such simpler computing, for a good while...
I suppose people will have more important things to work on, like growing food and fighting off roving bands of bandits, than building primitive analog calculators/computers.
I wish that the tutorial went just one more step. It presents a one dimensional perceptron. But most perceptrons are multi-input.
Adapting the article's 1D perceptron to three-input, for example:
import random
learning_rate = 0.1
EPOCHS = 50
NUM_INPUTS = 3
weights = [random.uniform(-1, 1) for _ in range(NUM_INPUTS)]
bias = random.uniform(-1, 1)
data = []
for _ in range(100):
inputs = [random.uniform(-1, 1) for _ in range(NUM_INPUTS)]
result = sum(inputs) > 0
data.append((inputs, result))
for epoch in range(EPOCHS):
for inputs, result in data:
weighted_sum = bias
for i in range(NUM_INPUTS):
weighted_sum += inputs[i] * weights[i]
prediction = weighted_sum > 0
if prediction != result:
error = int(result) - int(prediction)
for i in range(NUM_INPUTS):
weights[i] += learning_rate * error * inputs[i]
bias += learning_rate * error
print(f"Final weights: {[round(w, 3) for w in weights]}")
print(f"Final bias: {round(bias, 3)}")
Is this something that could be scaled up and used, for instance, to recognize features in images?
Or to put it another way are there any local only tools that can be trained on my own set of images to automatically tag new images? Tools that do not already have built in classes of image.
I take a lot of photographs and it would be handy to reduce the drudgery of tagging them so to say broadly what the subject was so that they are easier to find later.
If you want to learn the fundamentals of ML I recommend a book, such as Deep Learning: Foundations and Concepts by Chris Bishop. If you insist on staying online, one option is https://course.fast.ai/
If you don't know ML I don't think you're going to learn much through ad hoc demos.
>I wish I was smart enough to hold it all together.
I used to have a wife, but they took her in the divorce!
The human mind isn't very good at correlating its contents[0]. You can "know" something for years without realizing its implications.
The human mind traverses its knowledge like a man with a small flashlight in total darkness. Our beam of attention is small and narrow, so you need to put the right things in it, or the magic doesn't happen.
This has important implications for learning. I don't know what they are though.
Probably something like, "you can know something without knowing what it means." You haven't connected it to the things it's supposed to be connected to yet. I don't know how to fix that though. (Something involving the Feynman technique, maybe?)
I didn't know Bishop had released a new textbook. I will have to take a look at it. I wasn't the biggest fan of his Pattern Recognition book as I found it overly dense. I much preferred the Murphy and Alpaydin books.
I still find his pattern recognition book useful and informative. It may be dense, but some of us consider that a positive for 'reference' literature. That book was one of very few that still holds up well fr when it was published - truly in on of the last "dark ages" of ML.
I think those down voting you are perhaps overly eager. I upvoted. Grab "Deep Learning" - you'll find it useful, imteresting, and likely less 'dense' in the negative sense!
Appreciate your comment. I skimmed the online version and it covers all the 2010s era developments all the way to Transformers which is enough to earn it a spot on my bookshelf.
> Grab "Deep Learning" - you'll find it useful, imteresting, and likely less 'dense' in the negative sense!
Hey, I am sorry if you felt that way, I watched https://www.youtube.com/watch?v=l-9ALe3U-Fg and got inspiration,
I found your content to be similar to that video as well.
I don't understand these concepts as deeply as you can, but I have tried to make it simple and toy like.
I have to say, you have really good content, if I use any of your resources in my future blogs, surely I will give acknowledgement to you or any other creator!
You get whatever you assign 0 to. You can combine it with other simple building blocks like f(x) = ~x to do large calculations to answer more complicated questions.
I know you're joking, but if you really wanted to, you can if you have a network of these, since you can build NAND gates from perceptrons. If you have NAND gates, then you can build any other gates from these and then you can build a computer [1].
https://medium.com/@VictorBanev/the-simplest-learning-machin...
This is a description of a 5-line algorithm that learns and stores approximate probability of an event using just 1 byte of persistent memory.
https://karpathy.github.io/2026/02/12/microgpt/
I like the ability of this article to do the tiny training runs in browser. It makes the point of a bias clear. Too many tutorials get sucked into the proof of zero times anything is zero. Everyone knows that. What you should show is where that mstters in the problem at hand.
3blue1brown does one of the best depictions of why we need an activation function.
Karpathy's videos are a little tougher for a beginner to grasp, but excel at solving a complete problem. I knew all of the theory behind what it takes to make micrograd before I made my own by following the video, but what you get from doing it can't be understated.
It's hard to describe but it what you learn is more of a feel than pure knowledge. It gives you a better sense of knowing when the principles apply in other circumstances.
Perhaps it's the distinction of understanding how springs and gears work, then looking at a clock and understanding how the gears and springs move the hands. There's still more needed if you want to make a clock. And that stuff is what let's you also make a wind up toy.
That was the first thing that came to mind when I read "the smallest brain you can build". Nowadays, that "small brain" would likely be built on a breadboard using op-amps instead.
Yet we had the computational power to run backpropagation in the 1960s and small Transformers in the 1970s (I'm the author of both):
https://github.com/dbrll/Xortran (backprop on IBM 1130, 60s)
https://github.com/dbrll/ATTN-11 (Transformer on PDP-11, 70s)
What was missing wasn't the raw processing power, but the ideas and algorithms themselves. Because funding and research were completely discouraged during the AI winter, neural networks research was left dormant and we lost two decades.
They were simply too computationally expensive to train for the limited things they could do. It wasn’t until we had the ability to train large neural networks on commodity hardware that things really took off.
Tl;dr - compute was the bottleneck.
I am not associated with this channel/video, just love it. I’ve shared it here before.
[0] https://youtu.be/glWvwvhZkQ8?si=XjcwWWy43305tl6O
Going further backwards, the transistor (or a PNP junction) is the hardware level enabler of the IF statement. The action (switching) driven by the current which in turn controls other switches, is the first manifestation of "observe and act" by inanimate things at the speed of electricity.
Mechanical equivalents existed ofcourse - speed of a governer which controls the flow of fuel which in turn controls the speed of the governer.
Back up a bit please! Analog computing is a thing. And it isn't even new - not by a long shot.
There are good reasons why practically all computing today is the digital kind. But electronic 'equivalents' of neural nets is one area where analog might make sense. Adding inputs can be as simple as a bunch of resistors + a transistor. Even on modern silicon nodes, that might be a more efficient setup than digital inputs, N-bit adders/multipliers etc. Not saying that's the case, and AI hardware should be based on analog circuitry. But it could be, and perhaps found to be practical.
That's an interesting idea, but could the weights be transferred to different hardware and still work? If not, that would be a significant limitation, even if it were preferable in some cases.
Artificial neurons are significantly more complex that single transistors, and even a minimal hardwired circuit to implement just one neuron requires quite a number of transistors.
It's the same thing as stimulus, response.
Unchanging in response to circumstances is static.
Changing in the absence of circumstances is randomness.
The conditional is all that remains. Changing in response to circumstances
(Arguably, unchanging in the absence of circumstances completes the truth table, but it's a whole lot of nothing)
Try it, it's llama 3.1 8B at 16000 tokens per second.
chatjimmy.ai https://taalas.com/the-path-to-ubiquitous-ai/
There exist several equivalent sets of primitive operations. While the sets containing only NAND or only NOR, or both AND and NOT or both OR and NOT are more notorious, these logical operations are more abstract and they do not indicate precisely a hardware implementation, i.e. there are many distinct hardware methods to make such logical gates.
Other sets of primitive operations map directly to hardware devices, e.g. the sets of primitive operations composed of maximum and complement or of minimum and complement map directly to a hardware implementation using rectifier diodes and inverting amplifiers (which can be made with either semiconductor devices or with vacuum tubes, or also with pneumatic or hydraulic devices).
Other sets of primitive operations are obtained by replacing the maximum or minimum circuits with series or parallel connections of switches, like in the CMOS logic that is nowadays dominant.
The alternative IF expression corresponds in hardware to a 2-way multiplexer, which, together with the 2 constant functions "0" and "1" (a.k.a. "false" and "true" or "low voltage" and "high voltage"), is sufficient for a complete set of computational primitives.
Besides those mentioned above, the main remaining variant for a complete set of computational primitives consists of an analog (possibly weighted) adder and an analog comparator, which had been used in the so-called RTL circuits (resistor-transistor logic) and which also corresponds to perceptrons. RTL had been used in some early integrated circuits, before being replaced by DTL and TTL circuits (which are based on minimum and complement functions).
In hardware, e.g. in RTL circuits, a combined analog adder+comparator can be made with a single high-gain amplifying device, together with a set of weighting resistors and a bias resistor. RTL circuits can implement complex logic with fewer devices (e.g. they can implement a neural network in the analog domain), but they were replaced during the sixties of the past century with DTL, then TTL, because those were faster (in RTL, the resistors limit the charging currents for input capacitors and parasitic capacitors, which slows down the logical transitions) and the fact that they needed more devices was not important, due to the quick increase in circuit density.
There's a wide variety of computational primitives, including lambda calculus, combinators, cellular automata, rewriting systems. Perhaps some are more practical to implement in hardware, particularly the kind of DIY electronics or analog machines that can also be put together from scratch. It might look like a whole building of mechanical switches, powered by a water wheel ("watermill"), for example.
A transistor (driven to saturation) is a much better model.
I played with similar approach in JavaScript and built a NanoNeuron https://github.com/trekhleb/nano-neuron (it is more verbose than Python though)
Or to put it another way are there any local only tools that can be trained on my own set of images to automatically tag new images? Tools that do not already have built in classes of image.
I take a lot of photographs and it would be handy to reduce the drudgery of tagging them so to say broadly what the subject was so that they are easier to find later.
> In 1958, a researcher named Frank Rosenblatt built a machine *he called* the perceptron.
> It was *inspired* by a single brain cell, a neuron.
I included a remark about how time travellers would find Rosenblatt a better target than Miles Dyson.
I was never quite sure on how close, or over, the line that was on appropriateness. It was definitely thought provoking.
and its an a great learning resource - which is arguably more important :-)
If you don't know ML I don't think you're going to learn much through ad hoc demos.
I used to have a wife, but they took her in the divorce!
The human mind isn't very good at correlating its contents[0]. You can "know" something for years without realizing its implications.
The human mind traverses its knowledge like a man with a small flashlight in total darkness. Our beam of attention is small and narrow, so you need to put the right things in it, or the magic doesn't happen.
This has important implications for learning. I don't know what they are though.
Probably something like, "you can know something without knowing what it means." You haven't connected it to the things it's supposed to be connected to yet. I don't know how to fix that though. (Something involving the Feynman technique, maybe?)
[0] H.P. Lovecraft quote - https://www.goodreads.com/quotes/193944-the-most-merciful-th...
(before numarray and numpy merged!)
EDIT: His son is co-author?
I think those down voting you are perhaps overly eager. I upvoted. Grab "Deep Learning" - you'll find it useful, imteresting, and likely less 'dense' in the negative sense!
> Grab "Deep Learning" - you'll find it useful, imteresting, and likely less 'dense' in the negative sense!
Absolutely! I just ordered it and it's enroute :)
f(x) = 0.
[1] https://www.nand2tetris.org/
https://www.dandwiki.com/wiki/Doom_(5e_Campaign_Setting)