Solve any equation using gradient descent
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- Опубліковано 27 вер 2024
- Gradient descent is an optimization algorithm designed to minimize a function. In other words, it estimates where a function outputs its lowest value.
This video demonstrates how to use gradient descent to approximate a solution for the unsolvable equation x^5 + x = 3.
We seek a cubic polynomial approximation (ax^3 + bx^2 + cx + d) to cosine on the interval [0, π].
/ edgardocpu
ta hell is that intro 💀
💀
I was watching at 3 am
You're narration is like from 90s american television. I liked it very much.
More from the 60s I'd say, perhaps the 70s. The 90s? Not really.
WHY THE SCARY ASS INTRO????
some sort of waker-upper. students tend to doze of. it should've been flashed from time to time across the entire length of the video lol
@@semtex6412helllll nahh
im in bed with the lights out watching some math video i dont wanna see that again
Bro i was home alone and wanted to watch some math shit, i instantly closed yt when i saw the intro😂
🤣
Easy to see why you are ALONE Bro.
Same same
that voice kinda gives the vintage vibes of it, it's like you watching a very old vid. good shit homie
Truly happy i found this gem of a chanel before it blows up
Amazing to hear a new algorithm to solve equations, even the non real ones! - Thanks for helping me understand!
Bro this video is so fire. I get so annoyed by the voices in my actual school videos that they make you watch and this is a huge step up from that it actually makes this seem like it’s a top secret information like you’re debriefing the first nuclear tests or something
I love this explanation, voice, simplicity.
Im guessing the voice is a text to speech trained on old 60s videos?
This is just awesome stuff. Makes me want to study math. To be patient and learn the fundamentals that gets me to being able to understand and solve these equations. Fascinating.
Nice video.
Next step is designing a suitable neural network (choose number of hidden layers, and nodes of each layer, as well as an activation function), and use gradient descent to “learn” the node values, such that the neural network f.ex. produces a regression function to a set of points.
Thank you for the video!! Took some time to grasp the second example.
No surprise. This gradient descent optimization is at the heart of machine learning.
fantastic video ur gonna blow up soon (this is a threat)
simple , straight forward and good vibes. i love it
AWESOME Video! Thanks! Trying to put some basic understanding on this: "We seek a cubic polynomial approximation (ax^3 + bx^2 + cx + d) to cosine on the interval [0, π]."
Let's say you want to represent the cosine function, which is a bit wavy and complex, with a much simpler formula-a cubic polynomial. This polynomial is a smooth curve described by the equation where a, b, c, and d are specific numbers (coefficients) that determine the shape of the curve.
Now, why would we want to do this?
Cosine is a trigonometric function that's fundamental in fields like physics and engineering, but it can be computationally intensive to calculate its values repeatedly.
A cubic polynomial, on the other hand, is much simpler to work with and can be computed very quickly.
So, we're on a mission to find the best possible cubic polynomial that behaves as much like the cosine function as possible on the interval from 0 to π (from the beginning to the peak of the cosine wave).
To find the perfect a, b, c, and d that make our cubic polynomial a doppelgänger for cosine, we use a method that involves a bit of mathematical magic called "least squares approximation".
This method finds the best fit by ensuring that, on average, the vertical distance between the cosine curve and our cubic polynomial is as small as possible. Imagine you could stretch out a bunch of tiny springs from the polynomial to the cosine curve-least squares find the polynomial that would stretch those springs the least.
Once we have our cleverly crafted polynomial, we can use it to estimate cosine values quickly and efficiently. The beauty of this approach is that our approximation will be incredibly close to the real deal, making it a nifty shortcut for complex calculations.
A perfect video to watch at 2 am, especially the intro... now I will have a ton of time to think about this algorithm because there is no way i will sleep 😂 but seriously, very much interesting. I will delve deeper into it 👍
Great video! The visualizations were very helpful to my understanding. Will you be making more machine learning videos in the future?
Thanks! I'm glad you like it. Yes, I'm planning to do more videos on that topic: deep learning, super resolution, etc.
bro i came to watch math at night but your freaking intro scared the shiiiiiii out of me 😭
I like it! This demonstrates a good method! It should be noted that these principles also apply to similar methods which may have more desirable convergence properties such as Newton-Raphson methods.
i thought my screen got dust, but unique style. Nice!
this video is bizarre but in a good way
Dig that initial distinction between formulas/expressions and algorithms. Made something click.
This is just newton’s method
This is similar to newton’s method but not the same. Newtons method updates the initial guess by finding where the tangent line at the current guess intersects the x-axis. Gradient decent uses the tangent line but doesn’t go all the way to the x-axis, just uses it to walk towards the minimum. On a function like x^2+1, newtons method will not converge, since the function has no real root, but gradient decent will converge to x=0 since the function has a minimum there. They operate on a similar principal, but newtons method is for finding zeros whereas gradient decent iOS for finding minima.
Interestingly though there is another extrema finding algorithm called newtons method, however it is different from gradient decent, or newtons method for finding zeros.
Too many likes but this is wrong. Well, newton's method is a root finding algorithm while gradient descent is a optimization algorithm, it never finds roots but instead local minima/maxima. Newton's method can be used as optimization algorithm if you are finding the roots of the gradient (which requires knowing the second derivative)
@@pcklopif you take newtons method and apply it to the first derivative essentially you’re taking a 2nd order Taylor approximation and then just updating your guess to the minimum of the approximation
You can use Newton to find estimates of maximum/minimum candidates by trying to solve f'(x)=0 in the simple case or grad f =\vec 0 for multivariable problem. One way of thinking of the simple case is that it is estimating by alpha the quantity 1/f'(x) . I seem to recall that this is the basis for quasi Newton methods. @@pcklop
I adore the vibe of the video
Excellent, more!
Well done, thanks!
Awesome vid, instant sub
Excellent explanation, immediately subscribed 😁
Great explanation Edgar!
there's a lot of limitations related to gradient descent though, such as determining appropriate initial conditions and hyperparameters and convergence problems... but for simple well-behaved polynomials its definitely fine, although Newton's method would achieve faster convergence
really nice video Edgar! subbed! thank you
🙏
love the old aesthetic you're going for.
I am the 1000th subscriber
Great vid, really impressive editing.
excellently explained video
We are taught this in high school class 12.
Is that the Kingdom Hearts menu item noise at 4:05?
thanks sir !
Want to know how you made this video animation?
Would a sixth degree polynomial in x be referred to as "x hexed"?
Really like the video.
The second example would have been solved better by linear regression.
Am I getting this right? On your first example, our objective from the problem is solving for the values of x in x^5 + x = 3, but then you changed the problem to finding the roots (zeroes) of x^5 + x - 3 = 0 now. I'm bad at the higher math stuff but ain't this two things different? Apologies for the mistake if I made something but the reason seems so arbitrary, or should I just not think of it? Moreover, that method you did on squaring the entire equation, should I always do that? Because that seems really arbitrary too, especially that we're looking at a 5th degree polynomial so I thought this entire process would produce five solutions for us.
all you have to do is just adjust the equation by adding 3 on both sides, then substitute the solution into x's and then you get the equation really equal to 3 (or in this case approximately)
@@sicko5821 Yeah, I got back to some reading and remembered some stuff about that. Thanks for this though.
In this case, finding the solution of x^5 + x = 3 is equivalent to finding the root of the equation x^5 + x - 3 = 0. The function f(x) = x^5 + x - 3 doesn't have a global minimum, as it extends to both positive and negative infinity. To ensure the existence of a minimum for optimization purposes, we often transform the function to make it non-negative, typically by squaring it (or taking its absolute value). This creates a "low point" that we can then minimize. We do this "squaring preprocessing" when the function doesn't have a minimum.
Root vs. Minimum: Finding a root (where the function equals zero) is different from finding a minimum (the lowest value).
But how do you choose this "learning rate"? Like in your x^5 example, if you would have chosesn 0.025, then you will never get a solution, as your solver will spiral to infinity? If you know your solution has a 0, could you use the "reseduial" (value of previous evaluation) to guess how far you need to step? Perhaps paired with a relaxation factor?
More advanced gradient descent algorithms use a decreasing sequence of numbers as the learning rate. This allows the algorithm to quickly converge in the first few iterations and more slowly converge in later iterations to avoid “overstepping” the solution. Another modification involves solving for the optimal learning rate at each step with another gradient descent method, called exact gradient descent. Conjugate gradient descent uses orthogonal step directions to guarantee convergence in exactly n iterations, but each iteration is more costly.
It’s important to recognize that the learning rate shouldn’t matter too heavily unless your problem is ill-conditioned, in which case derivative-based methods don’t provide a much of an advantage over just guessing and you would use simulated annealing or other stochastic (rng-based) methods.
genuinely curious why you put that in the intro
Put another way:
One can "solve any equation" by numerical methods, i.e. a computer with a properly coded algorithm.
I hear Kingdom Hearts Menu selection sound😮
Use Newton method, it's quiker!
It is really nice
Broo this is a cool video! Could you share the source code for the video animations
The source code for this is a mess. Dependencies include SVG.js and MathJax. In the future, I will make paper and pencil videos.
@@EdgarProgrammator bro, it's alright if the code is a mess. I'd still love to learn from your process and ideas
@@korigamik Thanks! Here's my GitHub profile: github.com/isedgar. I'll be uploading something about math animations in the next few days.
this clip is from the 70`s?
Very good presentation. Not sure what superscript "T" is, first appearing @8:05 in the update formula.
Transpose. Vectors are column matrices (n by 1). Transposition is when you reflect a matrix over its main diagonal (so the entry in row 3 column 4 becomes row 4 column 3). Transposition therefore turns column matrices into row matrices. This is needed because the row matrix is subtracting alpha times the gradient. The gradient is a vector, specifically a column matrix. Addition and subtraction are only defined for matrices which have the sane dimensions. The transpose turns the gradient into a row vector so it can be subtracted from the vector before it, which is written as a row vector.
@@nathanoher4865 Ah, Transpose. I get it. Thanks. I had forgotten that detail of matrix terminology.
Your psychosis demon explains gradient decent
multivariable newton’s method
Wrong.
@@facts-ec4yi Thank you.
Ok now solve the navier stokes equations
I've found gradient descent to work very badly for polynomials.
Isn't it Newton Raphson method?
no, Newton-Raphson converges to the root using tangents, this converges to the minimal point unless it's manipulated in a way in which the minimal point is the root.
Wow genial
what is that intro bruh
The fact that there’s no quintic formula was proved by Galois before dying in a duel at 20
Man, I wish him and Niels Henrik Abel didn't die so young
over a woman
@@MyOneFiftiethOfADollarOver a woman
Based sigma gigachad grindset life is temporary math is forever
He came, saw and... died.
I am the 1000th subscriber
Bizarre style, I love it.
While being just a machine learning tutorial, this has an unsettling vibe that I find very unique for a educational channel, and honestly much more captivating. Keep it up!
The intro, the speaker's voice, and everything was beautiful in this video
🙏
That just makes the whole video feel like the introduction to a conspiracy theory.
@@EdgarProgrammator is that your real voice?? Or like deep faked or audio mixed with Prof. Feynman's voice from the archives??
Beautiful and very well made video, I personally loved the old tv vibe to this, not to disregard the instructive yet nicely explained method of gradient descent. Subscribed
Instead of squaring, you can raise to higher even powers(like 4,6,..etc), giving quicker convergence if i remember correctly.
Gradient descent is finding optimal minimum point of the function f(x), not finding solution of f(x)=0. However, optimal point of any f(x) is exactly the solution of f'(x) (derivative function of f(x)). So, in case your function has only one variable, to find the solution of f(x)=0, you can replace the derivative term with f(x) and so on. If your function has more than one variable, you can't replace, cause there's only one function has been given, so you do not know that function is depends on which variable (as mentioned above, if you have one variable, f(x) is derivative function depends on x when you use Gradient Descent to find solution).
So, the solution is using Least Square Approximation method as the video has shown. Function f^2(variable) always has optimal minimum point. If minimum point's value is 0, it is the solution. If not, GD still finds optimal minimum point, but it is not the solution.
Yes, solving the equation x^5 + x = y for x in terms of y is much more complex than solving quadratic equations because there is no general formula for polynomials of degree five or higher, due to the Abel-Ruffini theorem. This means that, in general, we can't express the solutions in terms of radicals as we can for quadratics, cubics, and quartics.
However, we can still find solutions numerically or graphically. Numerical methods such as Newton's method can be used to approximate the roots of this equation for specific values of y. If we're interested in a symbolic approach, we would typically use a computer algebra system (CAS) to manipulate the equation and find solutions.
Why squaring the function? do we always need to square the function to solve it via gradient descent?
What is the voiceover? Thanks,
The voice sounded like Carl Sagan. Are you using an AI generated voice?
Okay but why don't you explain why this method doesn't work sometimes for particular degrees depending on the function
For instance if you wanted to minimize cos(x) = c1 where c1 is a constant, using gradient descent one way or another yields you that c1 = 0, but the constant term in the taylor expansion of cos(x) is 1 since cos(x) = 1 - x^2/2 + ...
This means that you have to include at least the 2nd term for this to work, or even a higher degree depending on the function other than cos(x) in the example.
For a second I thought my iPad is possessed
Absolutely love the audio!!!! How do I get access to it?
oh its good , but i thought i will be able to apply it in my exams lol
what text to speech do you use
Yes - interesting. I've been considering the idea of minimizing f^2 by gradient descent for solving f = 0 but never actually implemented it. Is that a common technique in numerical computing? I think, I haven't seen it in textbooks yet (but maybe my memory is wrong). How does it compare to e.g. Newton iteration convergence wise? Maybe one could also try to minimize the indefinite integral F of f instead of f^2, if that is easily computable - which it is in the case of polynomials. Might be interesting to explore, if that leads to a good (i.e. fast) algorithm.
There's something about the way you talk and edit the video together that actually makes it interesting. I can't put my finger on it. Maybe it's how novel it is? I don't know, but PLEASE make more videos like this. It's amazing, and I actually understood it completely (rare for someone so bad at math lol)
TIL "x quartered" is a thing you can say
The clickbait here is so absurd. Obviously this cannot solve non convex problems.
And yet the entire field of deep learning uses it successfully in very non convex cases …
@vida91963 actually, you are completely wrong. Machine learning aims to make the cost function as convex as possible so that gradient based algorithms can work successfully. It's such a terrible example that it's a joke. How about cryptography where all functions are absurdly non convex like hash functions or symmetric crypto. Sorry gradient methods won't work. If they did then we wouldn't have cryptography.
Best into so far