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Even for non-smooth or discontinuous objectives, I'd still reach for methods that use gradient-like information over zeroth-order methods. For non-smooth objectives, Clarke-generalized subdifferentials have been pretty effective outside of ML, and have been used in automatic differentiation contexts at least 10-15 years ago. A carelessly quick literature search suggested conservative gradients, too.

For discontinuous objectives, I know there's been work on using envelope approximations, but the little I'm aware of in that work was in low-dimensional settings where the structure of the discontinuity was known explicitly. On the other extreme, lack of continuity comes up all the time in infinite-dimensional, PDE-constrained optimization, and some methods rely on tangent cones or various generalized notions of subdifferentiability (e.g., Mordukhovich, Bouligand) to demonstrate convergence. Admittedly, that work was somewhat outside my area of expertise, so I may be getting the details there slightly wrong, but the broad point stands that even in those settings, some directional information can be obtained and used profitably without resorting to zeroth-order methods.

It's still important to support research in this direction cause us meatbags cost a LOT less energy to train than GPTs even if you assume that it takes 30 years to train a PhD. Our current training methods honestly leave a lot to be desired. We almost certainly dont do full backpropagation.

I think it's fun back-of-the-napkin sometimes to compare meat to matmuls but ultimately fallacious to its core, making it an intellectual tarpit.

It's much harder to argue with the math and empirical results.

Energy cost may be a pointless comparison since we don't eat electricity, but it's very reasonable to look at sample efficiency and ask what we have there that our theories miss.

PHD inference doesn't scale, though.

I’m not sure that’s true. We have literal billions of years of training.

Human "pre-training" is vastly underappreciated.

Hell, animals are an even better example. Many animals pop out of the womb and start walking and eating and acting just like an adult!

One could argue that humans are more adaptable to novel environments

If you wanna get philosophical then the decades of research that humans did is also AI "pre-training". And all of that is built on knowledge that humans have spent our whole existence amassing so maybe all of human "pre-training" is also AI pre-training.

A fun but silly exercise.

I'm sure we don't literally do backpropagation, but differential equations that settle toward stationary states show up a lot in biology

You need to factor in the cost of training all those PhDs that never end up producing much of interest though you don't get to pick the best afterwards and claim all it took was to train him/her. Plus, once trained, it's productive only for 6 subjective hours per day (including weekends and holidays). Might have costed gazillions to train a frontier LLM but it works for millions of hours/day.

Kind of like random projections. Pre ChatGPT there would every now and then come a paper that states something along the lines of: instantiate a large randomly populated matrix, multiply by the input, and win! It kinda makes sense because you "stretch out" the space and separation boundaries become easier, but I have never seen it fully utilised in production in any meaningful way. Anyone remember the DANs (Deep Averaging Networks)?

This is used quite often in SoTA quantization algorithms, which definitely count as production

Interesting, thanks for the share!

Polymarket is banned in my country, but I would really like to bet on the other side. Just like second order can provide much information than first order, the computation cost forbids it. There could be a way that is easier to compute/parallalize that overpowers the benifit of gradient. I don't think the current methods like EA/Dust is the solution either, though, they are just imitating gradient.

Polymarket is available via VPN, but I advise not betting on such things.

Random selection seems to favor post training

https://arxiv.org/pdf/2603.12228

If it's cheaper then I guess it will might get practical. Also more likely that mind uses simpler techniques more similar to these.

I wonder though if someone tested mutating training objective though, like keeping original loss/goal and somehow defining loss differently and then comparing against original. This intuitively feels like how mind tries to handle difficult tasks.

It is like Monte Carlo integration and the curse of dimensionality. Yes, you can use this for basically anything but it pays off only if the object you are trying to integrate is multidimensional, otherwise traditional methods outperforms

"A strict complexity gap between gradient-based and derivative-free Lipschitz convex optimization" has been known for decades. It's covered in standard textbooks like Nesterov's "Introductory lectures on convex optimization". That AI result is about establishing the gap in a fairly niche accuracy regime.

Thanks for the correction, I had a feeling that should be true but the headline was at top of search results and I'm not close enough to the field to know off the top of my head.

Isn't the 'we need gradients' a foregone conclusion?

Gradients can be calculated numerically, meaning that any method that samples the cost function and makes optimization decisions based on that can actually compute gradients if it needs to.

> Derivative-free optimization can be useful for genuinely discontinuous objectives [1], but common neural network objectives are smooth and/or Lipschitz.

There is even an analog to the continuous derivative for discrete binary functions, called "Boolean variation": https://proceedings.neurips.cc/paper_files/paper/2024/hash/7...

Like for derivatives, there is a chain rule for Boolean variations, so you can use something like backpropagation, but without needing any expensive floating point math. Though I don't think this has been used much so far. There must be some other downside.

And yet the best learning algorithm we have is derivative-free!

Perhaps the optimizer is worse but the architecture or objective function is better.

Imagine how good it would be with derivatives

Yes, SVMs and Random Forests still rule the many worlds of classification problems often not in the spotlight.

Support Vector Machines involve solving a large QP optimization problem... which is often done by running a variant of gradient descent (or one of the second- or quasi-second-order optimization algorithms, which involves finding the Hessian as well as the gradient)

I was under the impression that solving QP optimization problems (subject to constraints) was largely performed by SMO.

Back when I was a researcher, I had a classification problem where the default random forest classifier built into scikitlearn completely dominated the neural method. The best numbers we got were something like a ~30% improvement from baseline for the neural network to ~95% for the random forest.

The disparity was so large that I was certain I must have made a mistake and I spent a few hours debugging, and then a few hours more trying different neural architectures.

Turns out this is just a super common experience for anyone in NNs who would also try the more established learning algorithms.

Bayes Optimal Classifier?