It's a bit snarky of me, but whenever I see some web developer or product person with a strong opinion about AI and its future, I like to ask "but can you at least tell me how gradient descent works?"
I'd like to see a future where more developers have a basic understanding of ML even if they never go on to do much of it. I think we would all benefit from being a bit more ML-literate.
I agree that understanding them is useful, but they are not abstractions much less leaky abstractions.
9 years ago, 365 points, 101 comments
I told everyone this was the best single exercise of the whole year for me. It aligns with the kind of activity that I benefit immensely but won't do by myself, so this push was just perfect.
If you are teaching, please consider this kind of assignments.
P.S. Just checked now and it's still in the syllabus :)
> “Why do we have to write the backward pass when frameworks in the real world, such as TensorFlow, compute them for you automatically?”
worries me because is structured with the same reasoning of "why we have to demonstrate we understand addition if in the real world we have calculators"
For example, Alex Graves's (great! with attention) 2013 paper "Sequence Generation with Recurrent Neural Networks" has this line:
One difficulty when training LSTM with the full gradient is that the derivatives sometimes become excessively large, leading to numerical problems. To prevent this, all the experiments in this paper clipped the derivative of the loss with respect to the network inputs to the LSTM layers (before the sigmoid and tanh functions are applied) to lie within a predefined range.
with this footnote:
In fact this technique was used in all my previous papers on LSTM, and in my publicly available LSTM code, but I forgot to mention it anywhere—mea culpa.
That said, backpropagation seems important enough to me that I once did a specialized videocourse just about PyTorch (1.x) autograd.
I understand how SGD is just taking a step proportional to the gradient and how backprop computes the partial derivative of the loss function with respect to each model weight.
But with more advanced optimizers the gradient is not really used directly. It gets per weight normalization, fudged with momentum, clipped, etc.
So really, how important is computing the exact gradient using calculus, vs just knowing the general direction to step? Would that be cheaper to calculate than full derivatives?
def clipped_error(x):
return tf.select(tf.abs(x) < 1.0,
0.5 * tf.square(x),
tf.abs(x) - 0.5) # condition, true, false
Plus his GitHub. The recently released nanochat https://github.com/karpathy/nanochat is fantastic. Having minimal, understandable and complete examples like that is invaluable for anyone who really wants to understand this stuff.
Backpropagation is a specific algorithm for computing gradients of composite functions, but even the failures that do come from composition (multiple sequential sigmoids cause exponential gradient decay) are not backpropagation specific: that's just how the gradients behave for that function, whatever algorithm you use. The remedy, of having people calculate their own backwards pass, is useful because people are _calculating their own derivatives_ for the functions, and get a chance to notice the exponents creeping in. Ask me how I know ;)
[1] Gradients being zero would not be a problem with a global optimization algorithm (which we don't use because they are impractical in high dimensions). Gradients getting very small might be dealt with by with tools like line search (if they are small in all directions) or approximate newton methods (if small in some directions but not others). Not saying those are better solutions in this context, just that optimization(+modeling) are the actually hard parts, not the way gradients are calculated.
Is this _more_ useful than the other things, which I could be learning but won't because I spent the time and effort to learn this instead?
We have a finite capacity for learning, if for no other reason then at least because we have a finite amount of time in this life, and infinite topics to learn (and there are plenty of other constraints besides time). The reason given for learning this topic, is that it has hidden failure modes which you will not be on the lookout for if you didn't know how it worked "under the hood".
Is this a good enough reason to spend time learning this rather than, say, how to model the physics of the system you're training the neural network to deal with? Tough question; maybe, maybe not. If you have time to learn both, do that, but if not, then you will have to choose which is most important. And in our education system, we do things like teach calculus but not intermediate statistics, and it would have been better to do the opposite for something like 90% of the people taking calculus.
That said, I've implemented backpropagation multiple times, it's a good way to evaluate a new language (just complex enough to reveal problems, not so complex that it takes forever).
I really hate what Twitter did to blogging…
> I'd like to see a future where more developers have a basic understanding of ML even if they never go on to do much of it. I think we would all benefit from being a bit more ML-literate.
Plus, I don't think a "gotcha" question like "what is gradient descent" will give you a good signal about someone if it get popularized. It probably will lead to the present-day OOP cargo cult, where everyone just memorizes whatever their lecturer/bootcamp/etc and repeats it to you without actually understanding what it does, why it's the preferred method over other strategies, etc.
There is no need for the knowledge that you propose in a world where this is solved, you will achieve more goals by utilizing higher-level tools.
I made a UI that showed how the weights and biases changed throughout the training iterations.
Perhaps, but maybe because there was more experimentation with different neural net architectures and nodes/layers back then?
Nowadays the training problems are better understood, clipping is supported by the frameworks, and it's easy to find training examples online with clipping enabled.
The problem itself didn't actually go away. ReLU (or GELU) is still the default activation for most networks, and training an LLM is apparently something of a black art. Hugging Face just released their "Smol Training Playbook: a distillation of hard earned knowledge to share exactly what it takes to train SOTA LLMs", so evidentially even in 2025 training isn't exactly a turn-key affair.
> As a developer, you just pick the best one and find good hparams for it
It would be more correct to say: "As a developer, (not researcher), whose main goal is to get a good model working — just pick a proven architecture, hyperparameters, and training loop for it."
Because just picking the best optimizer isn't enough. Some of the issues in the article come from the model design, e.g. sigmoids, relu, RNNs. And some of the issues need to be addressed in the training loop, e.g. gradient clipping isn't enabled by default in most DL frameworks.
And it should be noted that the article is addressing people on the academic / research side, who would benefit from a deeper understanding.
You've also missed the point of the article, if you're building novel model architectures you can't magic away the leakiness. You need to understand the back prop behaviours of the building blocks you use to achieve a good training run. Ignore these and what could be a good model architecture with some tweaks will either entirely fail to train or produce disappointing results.
Perhaps you're working at a level of bolting pre built models together or training existing architectures on new datasets but this course operates below that level to teach you how things actually work.
Diving through the abstraction reveals some of those.
Just because the framework you are using provides things like ReLU doesn't mean you can assume someone else has done all the work and you can just use these and expect them to work all the time. When things go wrong training a neural net you need to know where to look, and what to look for - things like exploding and vanishing gradients.
Nowadays, fine tuning LLMs is becoming quite mainstream, so even if you are not training neural nets of any kind from scratch, if you don't understand how gradients are used in the training (& fine tuning) process, then that is going to limit your ability to fully work with the technology.
Electric cars have similar complexities and limitations, for example the Bolt I owned could only go ~92MPH due to limitations in the gearing as a result of having a 1 speed gearbox. I would expect someone with a strong opinion of a car to know something as simple as the top speed.
Yes, absolutely -- a lot of ideas inspired by this have been explored in the field of optimization, and also in machine learning. The very idea of "stochastic" gradient descent using mini-batches basically a cheap (hardware compatible) approximation to the gradient for each step.
For a relatively extreme example of how we might circumvent the computational effort of backprop, see Direct Feedback Alignment: https://towardsdatascience.com/feedback-alignment-methods-7e...
Ben Recht has an interesting survey of how various learning algorithms used in reinforcement learning relate with techniques in optimization (and how they each play with the gradient in different ways): https://people.eecs.berkeley.edu/~brecht/l2c-icml2018/ (there's nothing special about RL... as far as optimization is concerned, the concepts work the same even when all the data is given up front rather than generated on-the-fly based on interactions with the environment)
Why would these things be "fudging"? Vanishing gradients (see the initial batch norm paper) are a real thing, and ensuring that the relative magnitudes are in some sense "smooth" between layers allows for an easier optimization problem.
> So really, how important is computing the exact gradient using calculus, vs just knowing the general direction to step? Would that be cheaper to calculate than full derivatives?
Very. In high dimensional space, small steps can move you extremely far from a proper solution. See adversarial examples.
First of all, gradient computation with back-prop (aka reverse-mode automatic differentiation) is exact to numerical precision (except for edge-cases that are not relevant here) so it's not about the way of computing the gradient.
What Andrej is trying to tell is that when you create a model, you have freedom of design in the shape of the loss function. And that in this design what matters for learning is not so much the value of the loss function, but its slopes, and curvature (peaks and valleys).
The problematic case being flat valleys, surrounded by straight cliffs, (picture the grand canyon).
Advanced optimizers in deep learning like "Adam", are still first-order, with diagonal approximation of the curvature, which mean the optimizer in addition to the gradient it has an estimate of the scale sensitivity of each parameter independently. So the cheap thing it can reasonably do is modulate the gradient with this scale.
The length of the gradient vector, being often problematic, what optimizers would usually do was something called "line-search", which is determine the optimal step-size along this direction. But the cost of doing that is usually between 10-100 evaluation of the cost function which is often not worth the effort in the noisy stochastic context, compared to just taking a smaller step multiple times.
Higher-order optimizers necessitate that the loss function is twice differentiable, so non-linearities like relu, which are cheap to calculate can't be used.
Lower-order global optimizers don't even necessitate the gradient, which is useful when the energy-function landscape has lots of local minima, (picture an egg-box).
That said, I know Deepseek use fp32 for their gradient updates even though they use fp8 for inference. And a recent paper shows that RL+LLM training is shakier at bf16 than fp16, which would both imply that numerical precision in gradients still matters.
Non-stochastic gradient descent has to optimize over the full dataset. This doesn't matter for non-machine learning applications, because often there is no such thing as a dataset in the first place and the objective has a small fixed size. The gradient here is exact.
With stochastic gradient descent you're turning gradient descent into an online algorithm, where you process a finite subset of the dataset at a time. Obviously the gradient is no longer exact, you still have to calculate it though.
Seems like "exactness" is not that useful of a property for optimization. Also, I can't stress it enough, but calculating first order derivatives is so cheap there is no need to bother. It's roughly 2x the cost of evaluating the function in the first place.
It's second order derivatives that you want to approximate using first order derivatives. That's how BFGS and Gauss-Newton work.
How do you propose calculating the "general direction" ?
And, an example "advanced optimizer" - AdamW - absolutely uses gradients. It just does more, but not less.
For example, I believe that if we were to ask the average developer about why LLM's behave randomly, they would not be able to answer. This to me exposes a fundamental hole in their knowledge of AI. Obviously one shouldn't feel bad about not knowing the answer, but I think we'd benefit from understanding the basic mathematical and statistical underpinnings on these things.
> how important is computing the exact gradient using calculus
Normally the gradient is computed with a small "minibatch" of examples, meaning that on average over many steps the true gradient is followed, but each step individually never moves exacty along the true gradient. This noisy walk is actually quite beneficial for the final performance of the network https://arxiv.org/abs/2006.15081 , https://arxiv.org/abs/1609.04836 so much so that people started wondering what is the best way to "corrupt" this approximate gradient even more to improve performance https://arxiv.org/abs/2202.02831 (and many other works relating to SGD noise)
> vs just knowing the general direction to step
I can't find relevant papers now, but I seem to recall that the Hessian eigenvalues of the loss function decay rather quickly, which means that taking a step in most directions will not change the loss very much. That is to say, you have to know which direction to go quite precisely for an SGD-like method to work. People have been trying to visualize the loss and trajectory taken during optimization https://arxiv.org/pdf/1712.09913 , https://losslandscape.com/
We could also say AI-literate too, I suppose. I guess I just like to focus on ML generally because 1) most modern AI is possible only due to ML and 2) it’s more narrow and emphasizes the low level of how AI works.
If it's about the supply chain, understanding at least the requirements for magnets is helpful.
On way to make sure you understand all of these things is to understand the electric motor. But you could learn the separate pieces of knowledge on the fly too.
The more you understand the fundamentals of what you're talking about, the more likely you are to have genuine insight because you can connect more aspects of the problem and understand more of the "why".
TL;DR it depends, but it almost always helps.
"Computers are good at maths" is normally a pretty obvious statement... but many things we take for granted from analytical mathematics, is quite difficult to actually implement in a computer. So there is a mountain of clever algorithms hiding behind some of the seemingly most obvious library operations.
One of the best courses I've ever had.
Back propagation is reverse mode auto differentiation. They are the same thing.
And for those who don't understand what back propagation is, it is just an efficient method to calculate the gradient for all parameters.
All you need is:
- Basic understanding of how a Markov chain can generate text (generating each word using corpus statistics on the previous few words).
- Understanding that you can then replace the Markov chain with a neural model which gives you more context length and more flexibility (words are now in a continuous space so you don't need to find literally the same words, you can exploit synonyms, similarity, etc., plus massive training data also helps).
- Finally, you add the instruction tuning (among all the plausible continuations the model could choose, teach it to prefer the ones human prefer - e.g. answering a question rather than continuing with a list of similar questions. You give the model cookies or slaps so it learns to prefer the answers humans prefer).
- But the core is still like in the Markov chain (generating each word using corpus statistics on the previous words).
I often give dissemination talks on LLMs to the general public and I have the feeling that with this mental model, you can basically know everything a lay user needs to know about how they work (you can explain things like hallucinations, stochastic nature, relevance of training data, relevance of instruction tuning, dispelling myths like "they always choose the most likely word", etc.) without any calculus at all; although of course this is subjective and maybe some people will think that explaining it in this way is heresy.
The point is that you can't abstract away the details of back propagation (which involve computing gradients) under some circumstances. For example, when we are using gradient descend. Maybe in other circumstances (global optimization algorithm) it wouldn't be an issue, but the leaky abstraction idea isn't that the abstraction is always an issue.
(Right now, back propagation is virtually the only way to calculate gradients in deep learning)
https://youtu.be/lXUZvyajciY?si=vbqKDOOY7l-491Ka&t=7028
Not too many details on timeline - just that he's working on it.
Then again, it might have been the corporate stuff that burned him out rather than the engineering.
> Yesterday I was browsing for a Deep Q Learning implementation in TensorFlow (to see how others deal with computing the numpy equivalent of Q[:, a], where a is an integer vector — turns out this trivial operation is not supported in TF). Anyway, I searched “dqn tensorflow”, clicked the first link, and found the core code. Here is an excerpt:
Notice how it's "browse" and "search" not just "I asked chatgpt". Notice how it made him notice a bug
I respect Karpathy’s contributions to the field, but often I find his writing and speaking to be more than imprecise — it is sloppy in the sense that it overreaches and butchers key distinctions. This may sound harsh, but at his level, one is held to a higher standard.
Later I understood that they don’t need to understand LLMs, and they don’t care how they work. Rather they need to believe and buy into them.
They’re more interested in science fiction discussions — how would we organize a society where all work is done by intelligent machines — than what kinds of tasks are LLMs good at today and why.
Sure, instead of "the problem with backpropagation is that it's a leaky abstraction" he could have written "the problem with not learning how back propagation works and just learning how to call a framework is that backpropagation is a leaky abstraction". But that would be a terrible sub-heading for an introductory-level article for an undergraduate audience, and also unnecessary because he already said that in the introduction.
Secondly, the article is from 2016, ChatGPT didn’t exist back then
> I think congrats again to OpenAI for cooking with GPT-5 Pro. This is the third time I've struggled on something complex/gnarly for an hour on and off with CC, then 5 Pro goes off for 10 minutes and comes back with code that works out of the box. I had CC read the 5 Pro version and it wrote up 2 paragraphs admiring it (very wholesome). If you're not giving it your hardest problems you're probably missing out.
And the issue you mention in the last paragraph is very relevant, since the scenario is plausible, so it is something we definitely should be discussing.
I think that's more because he's trying to write to an audience who isn't hardcore deep into ML already, so he simplifies a lot, sometimes to the detriment of accuracy.
At this point I see him more as a "ML educator" than "ML practitioner" or "ML researcher", and as far as I know, he's moving in that direction on purpose, and I have no qualms with it overall, he seems good at educating.
But I think shifting the mindset of what the purpose of his writings are maybe help understand why sometimes it feels imprecise.
If you live in a world of horse carriages, you can be thinking about what the world of cars is going to be like, even if you don't fully understand what fuel mix is the most efficient or what material one should use for a piston in a four-stroke.
(Thinking about it, would that necessarily be a bad thing...)
Eureka runs LLM101n, which is teaching software for pedagogic symbiosis.
The question here is whether the details are important for the major issues, or whether they can be abstracted away with a vague understanding. To what extent abstracting away is okay depends greatly on the individual case. Abstractions can work over a large area or for a long time, but then suddenly collapse and fail.
The calculator, which has always delivered sufficiently accurate results, can produce nonsense when one approaches the limits of its numerical representation or combines numbers with very different levels of precision. This can be seen, for example, when one rearranges commutative operations; due to rounding problems, it suddenly delivers completely different results.
The 2008 financial crisis was based, among other things, on models that treated certain market risks as independent of one another. Risk could then be spread by splitting and recombining portfolios. However, this only worked as long as the interdependence of the different portfolios was actually quite small. An entire industry, with the exception of a few astute individuals, had abstracted away this interdependence, acted on this basis, and ultimately failed.
As individuals, however, we are completely dependent on these abstractions. Our entire lives are permeated by things whose functioning we simply have to rely on without truly understanding them. Ultimately, it is the nature of modern, specialized societies that this process continues and becomes even more differentiated.
But somewhere there should be people who work at the limits of detailed abstractions and are concerned with researching and evaluating the real complexity hidden behind them, and thus correcting the abstraction if necessary, sending this new knowledge upstream.
The role of an expert is to operate with less abstraction and more detail in her oder his field of expertise than a non-expert -- and the more so, the better an expert she or he is.
He's just test driving LLMs, nothing more.
Nobody's asking this core question in podcasts. "How much and how exactly are you using LLMs in your daily flow?"
I'm guessing it's like actors not wanting to watch their own movies.
Imagine if you were using single layer perceptrons without understanding seperability and going "just a few more tweaks and it will approximate XOR!"
And his _examples_ are about gradients, but nowhere does he distinguish between backpropagation, a (part of) an algorithm for automatic differentiation and the gradients themselves. None of the issues are due to BP returning incorrect gradients (it totally could, for example, lose too much precision, but it doesn't).
Feynman was right that "If you can't build it, you don't understand it", but of course not everyone needs or wants to fully understand how an LLM works. However, regarding an LLM as a magic black box seems a bit extreme if you are a technologist and hope to understand where the technology is heading.
I guess we are in an era of vibe-coded disposable "fast tech" (cf fast fashion), so maybe it only matters what can it do today, if playing with or applying towards this end it is all you care about, but this seems a rather blinkered view.
And in fact this is true of any tool, you don’t have to know exactly how to build them but any craftsman has a good understanding how the tool works internally. LLMs are not a screw or a pen, they are more akin to an engine, you have to know their subtleties if you build a car. And even screws have to be understood structurally in advanced usage. Not understanding the tool is maybe true only for hobbyists.
There are things that you just can’t expect from current LLMs that people routinely expect from them.
They start out projects with those expectations. And that’s fine. But they don’t always learn from the outcomes of those projects.
> In other words, it is easy to fall into the trap of abstracting away the learning process — believing that you can simply stack arbitrary layers together and backprop will “magically make them work” on your data.
Then follows this with multiple clear examples of exactly what he is talking about.
The target audience was people building and training neural networks (such as his CS231n students), so I think it's safe to assume they knew what backprop and gradients are, especially since he made them code gradients by hand, which is what they were complaining about!
And what gives you that confidence? A few AI nerds already claimed that in the 80s.
We're currently exploring what LLMs can do. There is no indication that any further fundamental breakthrough is around the corner. Everybody is currently squeezing the same stone.
For example, things like "AI" image and video generation are amazing, as are things like AlphaGo and AlphaFold, but none of these have anything to do with LLMs, and the only technology they share with LLMs is machine learning and neural nets. If you lump these together with LLMs, calling them all "AI", then you'll come to the wrong conclusion that all of these non-LLM advances indicate that "AI" is rapidly advancing and therefore LLMs (also being "AI") will do too ...
Even if you leave aside things like AlphaGo, and just focus on LLMs, and other future technology that may take all our jobs, then using terms like "AI" and "AGI" are still confusing and misleading. It's easy to fall into the mindset that "AGI" is just better "AI", and that since LLMs are "AI", AGI is just better LLMs, and is around the corner because "AI" is advancing rapidly ...
In reality LLMs are, like AlphaFold, something highly specific - they are auto-regressive next-word predictor language models (just as a statement of fact, and how they are trained, not a put-down), based on the Transformer architecture.
The technology that could replace humans for most jobs in the future isn't going to be a better language model - a better auto-regressive next-word predictor - but will need to be something much more brain like. The architecture itself doesn't have to be brain-like, but in order to deliver brain-like functionality it will probably need to include another half-dozen "Transformer-level" architectural/algorithmic breakthroughs including things like continual learning, which will likely turn the whole current LLM training and deployment paradigm on it's head.
Again, just focusing on LLMs, and LLM-based agents, regarding them as a black-box technology, it's easy to be misled into thinking that advances in capability are broadly advancing, and will rise all ships, when in reality progress is much more narrow. Headlines about LLMs achievement in math and competitive programming, touted as evidence of reasoning, do NOT imply that LLM reasoning is broadly advancing, but you need to get under the hood and understand RL training goals to realize why that is not necessarily the case. The correctness of most business and real-world reasoning is not as easy to check as is marking a math problem as correct or not, yet that capability is what RL training depends on.
I could go on .. LLM-based agents are also blurring the lines of what "AI" can do, and again if treated as a black box will also misinform as to what is actually progressing and what is not. Thousands of bright people are indeed working on improving LLM-adjacent low-hanging fruit like this, but it'd be illogical to conclude that this is somehow helping to create next-generation brain-like architectures that will take away our jobs.
He's doing a capability check in this video (for the general audience, which is good of course), not attacking a hard problem in ML domain.
Despite this tweet: https://x.com/karpathy/status/1964020416139448359 , I've never seen him citing an LLM helped him out in ML work.
If he did not believe in the capability of these models, he would be doing something else with his time.
That's like saying, well, given how fast bicycles make us, so much closer to horse speed, I wonder if we can tweak this a little to move faster than any animal can run. But cars needed more technological breakthroughs, even though some aspects of them used insights gained from tweaking bicycles.
The Math Olympiad results are impressive, but at the end of the day is just this same next word prediction, but in this case fine tuned by additional LLM training on solutions to math problems, teaching the LLM which next word predictions (i.e. output) will add up to solution steps that lead to correct problem solutions in the training data. Due to the logical nature of math, the reasoning/solution steps that worked for training data problems will often work for new problems it is then tested on (Math Olympiad), but most reasoning outside of logical domains like math and programming isn't so clear cut, so this approach of training on reasoning examples isn't necessarily going to help LLMs get better at reasoning on more useful real-world problems.
That's because you, as you admit in the next sentence, have almost no understanding of how they work.
Your reasoning is on the same level as someone in the 1950s thinking ubiquitous flying cars are just a few years away. Or fusion power, for that matter.
In your defense, that seems to be about the average level of engagement with this technology, even on this website.
Truth be told, a whole lot of things are more important than copyright law.