soVeryTired

It'd be nice to see a refutation of his arguments, or at least a recognition of them. As it stands I've just seen two ad hominems in a row and not much else.

You're talking past each other, specifically on what it means to "set interest rates".

GP is correct in that the fed funds rate (the one that makes headlines and is used for policy) isn't literally _set_ by fiat: the Fed just intervenes in the interbank money market to ensure it stays in a specified range. It has near unlimited capacity to intervene so the fed funds rate only ever strays a tiny amount outside that range, and even then it's an exceptional circumstance.

On the other hand the parent is correct in that the Fed literally sets a lot of other rates that are used for financial plumbing.

Nickles I'm sure you know this already but I'm trying to clarify so that others don't get confused by an already confusing topic.

In economics, this viewpoint is known as the "classical dichotomy" [0]. IMO it's a bit of a dangerous way of seeing things since it blinds you to some important aspects of the economy. Inflation is particularly hard on people who don't have negotiating power, and in general those people aren't doing great to begin with.

While the size of the pie is important, so is the means of distributing the pie and the assurance that pie will be served tomorrow.

[0] https://en.wikipedia.org/wiki/Classical_dichotomy

I guess the other difference between encryption and hashing is that a hash function need not be one-to-one. Many inputs can result in the same hash, though hopefully it's hard to find collisions.

So a hash function is allowed to destroy information, whereas it's pretty important that an encryption algorithm doesn't!

Meh. Imagine the set of 10k x 10k pixel RGB images. The mona lisa is in there, as are images from your worst nightmares.

But almost all of the images in that set look like grey fuzz. On average there's nothing interesting there.

Oh man. Lack of proper debugging and diffing is a _huge_ problem with jupyterlab notebooks. In several cases I've seen, data scientists don't even realise that these tools exist.

I know VS Code has been making progress recently but a good jupyter competitor with some halfway decent software development features would be a game changer.

Presumably that just shows that it's easy to fit things with inverse-power functions, rather than revealing some fundamental truth about the behaviour the equation is modelling.

It’s just a matter of the two groups having different skills. One group writes for the general case while the other specialises in corner cases.

The latter looks really impressive when it’s done well, but it’d be silly to expect someone with deep security knowledge to sit down and build a spreadsheet manager from scratch. The two skill sets are just different. There is no “best”.

I never claimed they didn't. I claimed the reserve requirements didn't constrain lending.

In most counties in the west, reserve requirements don't constrain lending. For example, Canada, the UK, and Australia have a reserve requirement of zero. Fractional reserve banking doesn't really exist anymore outside of economics textbooks.

Capital requirements are what constrain lending in the west (I think the Chinese government does try to control lending in part via a reserve requirement). For example, the "Core Tier 1 Capital Ratio" [0] is extremely important in this regard.

https://www.investopedia.com/terms/t/tier-1-capital-ratio.as...

Nice. Makes sense, thanks for the explanation

Got you. The countability argument is interesting. Thanks for the discussion, I feel like I learned something!

Not trying to be stubborn here - I just don't understand.

So we're talking about the case when tensors don’t need to be multilinear functions on a product of vector spaces.

Each element of V* is (trivially) a tensor and a linear function on V. Each element of V is (trivially) a tensor and a linear function on V*. However, not all linear functions on V* are in V. So V** is bigger than V. No problem so far.

But all elements of V** are functions on (trivial) products of vector spaces, and by definition all functions in V** are linear. So how have we misunderstood each other here?

> if your original vector space is infinite dimensional then the natural embedding into its double dual is not an isomorphism

Ok, so this feels like the crux of the matter. So how does the fact that V** is not isomorphic to V make the tensor product construction a more general concept than the linear function construction?

> Say F[x] is the vector space of polynomials in x with coefficients in the field F, and F[x,y] is the space of polynomials in the variables x, y. For example x^2 - x is an element of F[x], and yx^2 - y^2 is an element of F[x,y]. Then it’s not hard to see that as vector spaces, F[x,y] is isomorphic to (F[x] tensor F[y]). A tensor in this new space is precisely a polynomial in two variables.

I think I'm missing part of the argument. So we have a polynomial p in F[x,y]. The claim in my previous post basically says that there's always a way to associate p with a linear map that takes two elements of the dual space of F[x] and produces a real number. I don't see why that's impossible here.

That's a fair point about the tensor product being more central in algebra. But I think the 'friendliest' introduction to tensors is the multilinear function viewpoint. IMO it motivates the tensor product.

I'm not familiar with the infinite-dimensional case. Do you have an example of a tensor that isn't a multilinear function on a product of vector spaces? I'd be interested to refine my understanding here.

Take a look at my definition of the "mathematician's tensor" in the top-level comments. The rank of a tensor is the number of row- and column-vectors you need to feed the tensor to get it to return a real number. AFAIK it's not related to 'rank' in the sense of the rank-nullity theorem.

To move from the mathematician's definition to the ML definition, pick a basis for your row and colum vectors. Now if you want the (i, j, ...) element of the multidimensional array, gather the vectors with one in the i-th position and zero elsewhere, the j-th position and zero elsewhere, etc. Then feed them to the tensor in that order.

It's easiest to see how it works with row vectors (rank (0,1)), column vectors (rank (1, 0)) and matrices (rank (1, 1)) and work from there.

I think one of the issues that makes tensors difficult to understand is that different disciplines use the word in different ways. I'm going to generalize here a bit about how each group uses the term, but I suspect not everyone will agree with me. Apologies in advance!

For machine-learning types, a tensor is a multidimensaional array of numbers that admits certain operations (addition, subtraction, hadamard product, etc) [0]

For mathematicians, a tensor is a function F. We pass the function an ordered tuple of N vectors and M covectors (or, to keep things simple, N column vectors and M row vectors), and the function returns a scalar. The function F needs to be linear in each of the N vectors and M covectors. In this view, a matrix is a tensor with N=1, M=1. The operations used by machine learning types arise naturally once you crank the mathematical handle a little bit.

For physicists, a tensor is what mathematicians would refer to as a tensor field. So they take a space X (think R^N), and for each point in the space they associate a mathematician's tensor F as defined above. The properties of the function F are permitted to change from point to point. The Riemannian metric tensor is a classic example of 'tensor' according to this usage.

[0] https://machinelearningmastery.com/introduction-to-tensors-f...

I don’t think it’s a question of ‘belief’. AOC is logically independent of the other ZF axioms: you may find it aesthetically ugly (a viewpoint I’m sympathetic to). But you can’t disbelieve it any more than you can disbelieve Euclid’s parallel postulate.

I think you mean Monte Carlo Tree Search rather (MCTS) rather than MCMC?

But this is part of the author's point: where are all the production reinforcement learning models?

Can you clarify what you mean by 'non-differentiable outcomes'?

That’s in a vacuum, in a straight line. In fibre optic cable, light travels around two thirds of that speed.

It's laughable given the current state of computer vision. CV has made enormous progress over the past decade, but it's still brittle and probably shouldn't be relied on in a life-or-death situation

Has the author done any serious work in computer vision? Imagenet is one thing, but there are many many problems that need to be solved before a vision-based vehicle can run reliably. I worked at a self-driving company for a while, and they struggled with issues like multiple target tracking and reasoning about occluded vehicles.

And in a sense, vision is the easy part. There are whole areas like prediction and motion planning where the current state of the art isn't really up to scratch.

Wikipedia says Strassen's algorithm is useful for n > 100 and is implemented in BLAS [0]. AFAIK Coppersmith-Winograd is more of a theoretical advance.

Edit: ah, we're probably talking about the same thing. Strassen was 1970.

[0] https://en.wikipedia.org/wiki/Matrix_multiplication_algorith...

They're counting the additions needed to add the matricies, not to multiply them. The point being that matrix addition is order n^2 so even if you need to do a large number of matrix additions in some intermediate calculation, they don't hurt you as n gets large if you're trying to speed up matrix multiplication.

But it would change the times people have their curtains closed.

(Not that I think it's a particularly good argument. And no, I'm not in the pocket of Big Curtain)

Decent quants know about the shortcomings of Markowitz allocation. The mean is more or less impossible to measure, and covariances shift over time and are awkward to measure concurrently. So the art is in specifying the parameters. I've used zero mean and block-diagonal covariance in asset allocation before, which isn't far off 1/N allocation.

The issue with Kelly betting is the same as with Markowitz: knowing the parameters of the problem. If you overstate your probability of win by mis-estimating, you'll get ruined. For what it's worth, the version of Kelly betting I'm familiar with is for Bernoulli trials. I think the Gaussian version of the Kelly criterion essentially reduces to Markowitz.

Side note: kelly betting isn't new! Daniel Bernoulli knew about it back in the 1700s. Kelly just connected the formula to ideas in information theory.

“When winter rolls around, the gorillas simply freeze to death.”