> This is total nonsense...
If you assert that it is differentiable the concept looses all meaning.
Citation please.
Yes Lipschitz functions need not be differentiable but when they are that bound holds and is very useful. Using that bound is bread and butter in convex analysis of differentiable convex functions.
The article links to the Wikipedia page, which gets it right.
"An everywhere differentiable function g : R → R is Lipschitz continuous (with K = sup |g′(x)|) if and only if it has a bounded first derivative"
Lipschitz continuity for differentiable functions is just having a bounded derivative. The Lipschitz property suddenly becomes uninteresting as it just falls out of the assumptions, the interesting fact which allows you to use non-differentiable functions is that not assuming differentiability, but assuming Lipschitz continuity, is enough.
>Using that bound is bread and butter in convex analysis of differentiable convex functions.
It also is the bread and butter in Analysis of PDEs, but it is the bread and butter because Lipschitz continuity is a weaker property than differentiability. In the context of the article you want to talk about non-differnetiable functions, e.g. max, which you couldn't if you assumed differentiability.
The reason this is important because choosing Lipschitz over differentiable is what makes all this work.
My point is that the article is using the Lipschitz property to get its results. This makes it unnecessary and even wrong to introduce Lipschitz continuity only for functions with a derivative. Especially since the article actually uses functions which do not have a derivative.
For what it's worth, what you originally said was: "This is total nonsense." The points you're making are valid, but it isn't "total nonsense". Something not being exactly factually correct doesn't mean that it's "total nonsense". Different publications adhere to different levels of rigor. Just because it doesn't meet your own personal standard doesn't make it nonsense for the target audience of the article.
this is a standard thing in "mature" areas of math and it's absolutely the opposite of what's good for the student (all of the machinery being hidden in the definition instead of developed in the theorem's proof).
EDIT: if you hate "a monad is a monoid in the category of endofunctors" then you also hate "definitions should be hard and theorems easy".
My first instinct is to agree, but I'm not sure actually. What I really want when learning a new area of math is the full motivation for the tricky definition, taking as much time as needed to follow the dead ends of easier but worse definitions. Then I get the whole picture. IMO the motivation is the key thing for students, not the definition being easy.
Though maybe the way this course would work is in fact by proceeding through a series of easy but explicitly flawed definitions, and proving both real results and nonsense from them, so you see why the real definition is justified.
> What I really want when learning a new area of math is the full motivation for the tricky definition
yes so then you want proofs that actually exercise real machinery instead of playing the shell game of "an X is a Y and a Y is a Z, and has ABC properties, there for X has ABC properties"; you want a proof that goes through the process of using properties ABC to build Y from Z and X from Y (or something akin to that).
definitions aren't for people learning math, they're for people using math ie practising professional mathematicians that are proving more theorems; Hausdorff didn't invent "Hausdorff spaces", he used/worked with various properties of topological spaces and then when the next person came along and needed to right another paper on top, that person invented "Hausdorff space".
It’s interesting that you use the “a monad is a monoid in the category of endofunctors” example. That kind of statement is definitely hard to parse when you’re trying to learn concepts.
However, the more I’ve learned about category theory, the more I’ve understood it as a way of defining what things are and what properties follow from those definitions.
Like, a monad really doesn’t have meaning beyond “monoid in the category of endofunctors”. The same is true for monoids and endofunctors: it’s all about the properties of those objects.
In the context of programming, we can impose all kinds of meaning, but the definitions and laws are really what makes it all work when you piece it together.
I guess my approach is to suffer through it until some understanding is gleaned, which admittedly isn’t very satisfying or easy haha.
It is the exploration and enumeration of the possible rhythms that led to the discovery of Fibonacci sequence and binary representation in around 200 BC.
"Very well. Let's have a love poem, lyrical, pastoral, and expressed in the language of pure mathematics. Tensor algebra mainly, with a little topology and higher calculus, if need be. But with feeling, you understand, and in the cybernetic spirit."
"Love and tensor algebra? Have you taken leave of your senses?" Trurl began, but stopped, for his electronic bard was already declaiming:
Are their big barriers to entry to become a booky ?
One would assume it would be quite attractive to become one if he enjoys gambling. One gets to participate in the action and make money no matter which horse wins -- as long as they manage the edge right.
Do you mean the person at the tracks taking odds or a business doing online bookmaking.
Being a bookie would be fun if you like maths, are street smart, somewhat not highly ethical but you are disciplined and can control your emotions. Similar to poker I guess. And are knowledgeable about the sport with respect to odds not trivia.
I doubt a typical gambler would enjoy it. Does a "foodie" love working in a lab testing flour samples?
Thorpe is a dyed in wool information theory, probability theory academic who happens to be a super cool hacker at heart *.
Among his many accomplishments are, him obtaining a more general version of Black-Scholes model, independently and before Black, Scholes and Mertens had derived theirs. You may recall that got them the Nobel prize.
Managing money and breaking Las Vegas casinos were one of his, side, and mostly academic, entertainments.
There is a nice story of him visiting Shannon to discuss an information theory research paper that he, Thorpe, was writing. The meeting had taken a lot effort to schedule -- Shannon was a busy man, already a celebrity. But then they ended up over-extending their scheduled meeting, discussing, brainstorming with playful eagerness, different mathematical methods to break several popular gambling games, starting with Blackjack. Turns out Shannon was a sucker for such entertaining hacks. (This would be one of the first formulations of card counting [0]) Thorpe and his students would then try these out in Vegas.
*Taking things apart logically, to probe where things break, to understand how they work, that's as quintessentially an exercise in hacking as you can get.
Prof Thorpe wrote about and proved his strategies along with characterizing the assumptions under which they would and had worked.
Finally, you are you and then there is Prof. Thorpe.
One thing I took note of was that he was a chemistry nerd in school.
It seems like the subject of nerd culture changed from waaay back when he was a kid being chemistry (you could make bombs), to electronics such as radio (you could do ham radio), to computers and then programming (video games).
Nowadays you won't find anyone who can tell you how chemistry works, and a few old guys will tell you how to solder your own electronics, but everyone will be able to tell you how to code a website.
Was also perhaps career focused on latest tech?? For example my granddad did structural engineering, my father studied chemical engineering, I did electronic engineering
I once ended up participating in rock, paper scissors strategy competition on a whim.
At that time I was quite tired of polishing and repolishing a paper draft at my university. (I write very poorly). And there it was, an announcement of this fun competition. The deadline was just an hour away.
I had no time to implement any respectable algorithm myself.
So, all my submission did was take the history of plays of my opponent, up to the present point in the ongoing joust, and extend it by three possible completions. I would compress each of the three resulting strings. Whichever completion gave the shortest compressed final string, I assumed that to be my opponent's predicted play. I played whatever beat that predicted play.
This did not win the competition, but survived the knockout for many rounds, beyond my expectation. If I remember correctly, I had used plain old zip. A compressor that converges faster would have been better.
In essence my strategy was similar to a Markov chain predictor. It would have been exact match had I used a Markov model based compressor.
The number of rounds of play against one's opponent was not long. Had it been so, Lempel-Ziv being an universal compressor, would have been hard to beat.
Of course we know from game theoretic analysis what the Nash strategy is for this game, but where is the fun in playing that ?
One might think that a transformer based player would be an excellent strategy, not necessarily. One needs to learn potentially non-stationary strategies of the opponent from very few examples - (near) zero-shot online learning is required.
If I had more time, I would have tried context-weighted trees - universal learners for stochastic strings. The failure mode would be non-ergodic plays by the opponent, assuming that, it too is another stochastic parrot.
Probably not. Most writing is self-congratulatory garbage. The best writing I've ever read was not meant to be fancy but got right to the point from the heart, by people who couldn't care less what others thought about them. Your writing in this comment was fine and easy to read.
I am still very bad as far as writing research papers go. That's one area my advisor's feedback helped a lot. Anything I wrote would need multiple passes.
My writing's bad, partly because of bad grammar and partly because once the cool results of the research is done, my heart is really not into writing it. It becomes a chore.
EDIT: I see no reason why your comment got down-voted.
Ignore the downvotes. They don't really mean anything on HN, and points fluctuate wildly over the course of 24-48 hours in both directions. I have a theory that HN literally adds random up/down votes to comments periodically just to mess with us.
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