I'm also a visual learner and my class on dynamical systems really put a lot into perspective, particularly the parts about classifying stable/unstable/saddle points by finding eigenvectors/values of Jacobians.

A lot of optimization theory becomes intuitive once you work through a few of those and compare your understanding to arrow maps like you suggest.

At least in my undergrad multivariate real analysis class, I remember the professor arranging things to strongly suggest that the Hessian should be thought of as ∇⊗∇, and that this was the second term in a higher dimensional Taylor series, so that the third derivative term would be ∇⊗∇⊗∇ etc. Things like tensor products or even quotient spaces weren't assumed knowledge, so it wasn't explicitly covered, but I remember feeling the connection was obvious enough at the time. Then an introductory differential geometry class got into (n,m) tensors. So I'm quite sure mathematicians are fine dealing with tensors. My experience was undergrad engineering math tries to avoid even covectors though, so that will stay well clear of a coherent picture of multi-variable calculus. e.g. my engineering professors would talk of dirac δ as an infinite spike/spooky doesn't-really-exist thing that makes integrals work or whatever. My analysis professor just said δ(f) = f(0) is a linear functional.

Well for me the Hessian is the second order derivative in the special case where the co-domain is of dim 1. It's just very easy to work with...

I agree it is confusing, because starting with notation will confuse you. I personally don't like the partial derivative-first definition of those concepts, as it all sounds a bit arbitrary.

What made sense to me is to start from the definition of derivative (the best linear approximation in some sense), and then everything else is about how to represent this. vectors, matrices, etc. are all vectors in the appropriate vector space, the derivative is always the same form in a functional form, etc.

E.g. you want the derivative of f(M) ? Just write f(M+h) - f(M), and then look for the terms in h / h^2 / etc. Apply chain rules / etc. for more complicated cases. This is IMO a much better way to learn about this.

As for notation, you use vec/kronecker product for complicated cases: https://janmagnus.nl/papers/JRM093.pdf

This doesn't really help with programming, but in physics it's traditional to use up- and down-stairs indices, which makes the distinction you want very clear.

If input x has components xⁿ, and output f(x) components fᵐ, then the Jacobian is ∂ₙfᵐ which has one index upstairs and one downstairs. The derivative has a downstairs index... because x is in the denominator of d/dx, roughly? If x had units seconds, then d/dx has units per second.

Whereas if g(x) is a number, the gradient is ∂ₙg, and the Hessian is ∂ₙ∂ₙ₂g with two downstairs indices. You might call this a (0,2) tensor, while the Jaconian is (1,1). Most of the matrices in ordinary linear algebra are (1,1) tensors.

You're confusing too many things.

The Hessian is defined as the second order partial derivative of a scalar function. Therefore it will always give you a matrix.

What you're doing with the shape (m,n,n) isn't actually guaranteed at all since the output shape of an arbitrary function can be any tensor and you can apply the Hessian to each scalar value in the tensor to get another arbitrary tensor that has two dimensions more.

It's the Jacobian that is weird, since it is just a vector of gradients and therefore its partial derivative must also be a vector of Hessians.

Have you tried using Enzyme (https://enzyme.mit.edu/)? It operates on the LLVM IR, so it's available in any language that breaks down into LLVM (e.g., Julia, where I've used it for surface gradients) and it produces highly optimized AD code. Pretty cool stuff.

Sorry to actually your actually, but the derivative of a function f from a space A to a space B at the point a is a linear function Df_a from the tangent space of A at a to the tangent space of B at b = f(a).

When the spaces are Euclidean spaces then we conflate the tangent space with the space itself because they're identical.

By the way, this makes it easy to remember the chain rule formula in 1 dimension. There's only one logical thing it could be between spaces of arbitrary dimensions m, n, p: composition of linear transformations from T_a A to T_f(a) B to T_g(f(a)) C. Now let m = n = p = 1, and composition of linear transformations just becomes multiplication.

(Only half kidding)

A perhaps nicer way to look at things[0] is to hold onto your base points explicitly and say Df:: a -> (b, a -o b) = (f(p),A(p)) where f(p+v)≈f(p)+A(p)v. Then you retain the information you need to define composition Dg∘Df=D(g∘f)=(Dg._1∘Df._1, Dg(Df._1)_.2∘Df._2). i.e the chain rule.

[0] which I learned from this talk https://youtube.com/watch?v=17gfCTnw6uE

When you look at a 2D surface, you directly observe all the values on that surface.

For a loss-function, the value at each point must be computed.

You can compute them all and "look at" the surface and just directly choose the lowest - that is called a grid search.

For high dimensions, there's just way too many "points" to compute.

You're thinking of situations where you are able to see a whole object at once. If you were dealing with an object too large to see all of, you'd have to start making decisions about how to explore it.

Evaluating a function using a densely spaced grid and plotting it does work. This is brute-force search. You will see the global minima immediately in the way you describe, provided your grid is dense enough to capture all local variation.

It's just that when the function is implemented on the computer, evaluating so many points takes a long time, and using a more sophisticated optimization algorithm that exploits information like the gradient is almost always faster. In physical reality all the points already exist, so if they can be observed cheaply the brute force approach works well.

Edit: Your question was good. Asking superficially-naive questions like that is often a fruitful starting point for coming up with new tricks to solve seemingly-intractable problems.

People here are giving you mathematical answers which is what you are asking for, but I want to challenge your intuition here.

In construction, grading a site for building is a whole process involving surveying. If you dropped a person on a random patch of earth that hasn't previously been levelled and gave them no tools, it would be a significant challenge for that person to level the ground correctly.

What I'm saying is, your intuition that "I can look around me and find the minimum of anything" is almost certainly wrong, unless you have a superpower that no other person has.

When you look at, for instance, a bowl, or even one of those egg carton mattress things, and you want to find the global minimum, you are looking at a surface which is 2 dimensions in and 1 dimension out. It's easy enough for your brain to process several thousand points and say ok the bottom of the bowl is right here.

When a computer has a surface which is 2 dimensions in and 1 dimension out, you can actually just do the same thing. Check like 100 values in the x/y directions and you only have to check like 10000 values. A computer can do that easy peasy.

When a computer does ML with a deep neural network, you don't have 2 dimensions in and 1 dimension out. You have thousands to millions of dimensions in and thousands to millions of dimensions out. If you have 100000 inputs, and you check 1000 values for each input, the total number of combinations is 1000^100000. Then remember that you also have 100000 outputs. You ain't doin' that much math. You ain't.

So we need fancy stuff like Jacobians and backtracking.

Many practical optimisation problems are less like "let's go hiking and climb a literal hill which we can see in front of us" and more like "find the best design in this space of possible designs that maximises some objective"

Here are some alternative example problems, that are a lot more high dimensional, and also where the dimensions are not spatial dimensions so your eyes give you absolutely no benefit.

(a) Your objective is to find a recipe that produces a maximally tasty meal, using the ingredients you have in your kitchen cupboard. To sample one point in recipe-space, you need to (1) devise a recipe, (2) prep and cook a candidate meal following the recipe, and (3) evaluate the candidate recipe, say by serving it to a bunch of your friends and family. That gets you one sample point. Maybe there are 1 trillion possible "recipes" you could make. Are you going to brute-force cook and serve them all to find a meal that maximises tastiness, or is there a more efficient way that requires fewer plan recipe->prep&cook->serve->evaluate cycles?

(b) Your objective is to find the most efficient design of a bridge, that can support the required load and stresses, while minimising the construction cost.

Your eyes compute gradients, as part of the shitton of visual processing your brain does to get an estimate of where the local and global minima are.

It is not perfect though, see the many optical illusions.

But we follow gradients all the time, consciously or not. You know you are at the bottom of the hole when all the paths go up for instance.

Without looking up the answer (because someone has already computed this for you), how would you find the highest geographic point (highest elevation) in your country?

well, first of all ... you can't. and it is very easy to come up with all sorts of (not even special) cases where you simply couldn't for literally obvious reasons. what you are imagining is some sort of stereoscopic ray tracing. that is anyway much more compute intensive then calculating a derivative.

What if you're trying to find the minimum of something that you can't see? Or what if the differences are so small that you can't perceive them with your eyes even though you can see?

The inputs you can process visually are of trivial size even for naive algorithms, and probably also simple instances. I certainly can’t find global minima in 2d for any even slightly adversarial function.

You're ignoring all the calculations that go on unconsciously that realize your conscious experience of "immediately" apprehending the global minima.

Your visual cortex is a massively parallel processor.

touch and sight sense essentially the same...the difference is in the magnitudes involved

Ok, but a good HN post should explain what is correct, so those who don't know can learn.

> (...) if the function was a straight line

There are a bit more to go from local linearization to a complete view of a derivative but it's not exactly incorrect.