I get your point, but I don't think your nit-pick is useful in this case.

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)

Yes. No need to be apologetic or timid about it — it’s not a nit to push back against a flawed conceptual framing.

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.

I was slightly surprised that my colleagues, who are extremely invested in capabilities of LLMs, didn’t show any interest in Karpathy’s communication on the subject when I recommended it to them.

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.

And to all the LLM heads here, this is his work process:

> 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

The difference in understanding (for me and how my brain works) between reading the paper in what appears to be a future or past alien language & doing a minimal paper / code example is massive.

I did this in highschool from some online textbook in plain Java. I recall implementing matrix multiplication myself being the hardest part.

I made a UI that showed how the weights and biases changed throughout the training iterations.

Is that paper publicly available?

> It seems to me that in 2016 people did (have to) play a lot more tricks with the backpropagation than today

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.

Two thoughts:

> 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/

> 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?

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)

>computing the exact gradient using calculus

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).

> But with more advanced optimizers the gradient is not really used directly. It gets per weight normalization, fudged with momentum, clipped, etc.

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.

Calculus isn't that complicated, at least not what's done in backprop.

How do you propose calculating the "general direction" ?

And, an example "advanced optimizer" - AdamW - absolutely uses gradients. It just does more, but not less.

That's an interesting idea, it sounds similar to the principles behind low precision models like BitNet (where each weight is +-1 or 0).

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.

You don't need exact gradients, since gradient descent is self-correcting (which can make it hard to find gradient calculation bugs!). One approach using inexact gradients is to use predicted "synthetic gradients" which avoids needing to wait for backward pass for weight updates.

It is possible to compute the approximate gradient (direction to step) without using the formulas: we can change the value of each parameter individually, compute the loss, set the values of all parameters in such a way that the loss is minimized, and then repeat. This means, however, that we have to do number-of-parameters forward passes for one optimization step, which is very expensive. With formulas, we can compute all these values in one backward pass.

All first order methods use the gradient or Jacobian of a function. Calculating the first order derivatives is really cheap.

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.

Agree, better title for this post; the fact that back prop is a leaky abstraction is a reason one should understand it and know how to do the mechanics by hand to truly experience it and develop understanding and intuition. Software / code abstracting away even more of the process leaves it open to magical thinking. I had to do hand calculations and convolutions in my Deep Learning graduate school course.

The counter-argument would be that you can make excellent arguments for why we should understand what the compiler is doing, understand what the transistors are doing (e.g. to understand the limitations and risks of overclocking), understand how sort algorithms work (doing sort algorithms manually was at one time a typical interview question). It's not that these things aren't useful, but whether or not that learning would be useful is not the right question.

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).

From the perspective of the university, the students are being trained to become researchers, not engineers.

> Any leakiness in BackProp is addressed by researchers who introduce new optimizers

> 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.

The problem isn't with backprop itself or the optimizer - it's potentially in (the dervatives of) the functions you are building the neural net out of, such as the Sigmoid and ReLU examples that Karpathy gave.

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.

It's for a CS course at Stanford not a PyTorch boot camp. It seems reasonable to expect some level of academic rigour and need to learn and demonstrate understanding of the fundamentals. If researchers aren't learning the fundamentals in courses like these where are they learning them?

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.

The problem with your reasoning is you never tackle your "unknown unknowns". You just assume they are "known unknowns".

Diving through the abstraction reveals some of those.

If we don't subtract from the second branch, there will be a discontinuity around x = 1, so the derivative will not be well-defined. Also the value of the loss will jump at this value, which will make it hard to inspect the errors, for one thing.

You do that to make things smoother when plotted. You could in theory add some crazy stairstep that adds a hundred to the middle part. It would make your loss curves spike and increase towards convergence but then those spikes are just visual artifacts from doing weird discontinuous nonsense with yoru loss.

square roots are expensive

He gives an update in the Dwarkesh interview:

https://youtu.be/lXUZvyajciY?si=vbqKDOOY7l-491Ka&t=7028

Not too many details on timeline - just that he's working on it.

He does have a history of abandoning projects. OpenAI. Tesla. OpenAI again...

Then again, it might have been the corporate stuff that burned him out rather than the engineering.

I'm wondering: how can understanding gradient descent help in building AI systems on top of LLMs? To mee it feels like the skills of building "AI" are almost orthogonal to skills of building on top of "AI"

Impossible requirement. The inherent quality of abstractions is to allow us to get more done without understanding everything. We dont write raw assembly for the same reason, you dont make fire by rubbing sticks, you dont go hunting for food in the woods, 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.

so if you want to have a strong opinion on electric cars you need to be able to explain how an electric engine works right?

  > 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.

Yes. Pretraining and fine-tuning use standard Adam optimizers (usually with weight-decay). Reinforcement learning has been the odd-man out historically, but these days almost all RL algorithms also use backprop and gradient descent.

Are LLMs still trained by (variants of) stochastic GRADIENT descent? AFAIK what used to be called "backprop" is nowadays known as "automatic differentiation". It's widely used in PyTorch, JAX etc

article is from 2016

He has a new blog at https://karpathy.bearblog.dev/blog/