10 ms·

The Matrix Calculus You Need for Deep Learning

610 points by prostoalex 8y ago

roel_v 8y ago
So I have a question somewhat related to this that I never knew where/who to ask (well actually I asked a few mathematicians at a university I work with whose answers I couldn't understand - their answers were almost as impenetrable as the Wikipedia page, and some engineering scientists who I thought would be more into 'applied math' but they didn't know. So I'm hoping some data science people reading this would better understand where I'm coming from, and be able to explain at my primitive math skills level). So, some time ago I contracted out writing some code for fitting a logistic regression onto a given set of observations. There were some specific requirements, but I think I should have been able to piece something myself together using mainstream LA libraries; some of them even hint at 'you could fit a logistic regression using these functions' but no complete examples. But I didn't understand well enough so I contracted it out. The woman who ended up writing the code used a 'Hessian' matrix to do so (she actually wrote two functions doing the same thing, one used this Hessian approach - I think the idea was that it would be faster but there wasn't a lot of time and it never got tweaked enough to make a difference). So my question - is there a layman's explanation for what a Hessian matrix is, and how it applies to fitting a logistic model? Also (with an eye to the future of my project), does it have applications for non-linear regressions? Alternatively, are there any books where this is covered? I have most standard stats/applied stats/operations research books, as well as a few like the no bullshit guide to linear algebra, but none cover this specific issue - or even how to fit a logistic regression at all, on a practical level (so not just 'conceptually you do xyz, implementation is left as an exercise for the reader').
- matheist 8y ago
  Random guess: a logistic regression is fit by maximizing a particular log likelihood. [+] You can find the maximum by (1) gradient ascent, or (2) the analogue of Newton's method in multiple dimensions [++], which involves computing the Hessian matrix. So there's your Hessian. [+] https://en.wikipedia.org/wiki/Logistic_regression#Model_fitting https://en.wikipedia.org/wiki/Logistic_regression#Model_fitt... [++] https://en.wikipedia.org/wiki/Newton%27s_method_in_optimization#Higher_dimensions https://en.wikipedia.org/wiki/Newton%27s_method_in_optimizat...
  - nerdponx 8y ago
    Good guess!
- imh 8y ago
  Look up maximum likelihood estimation [0] and look up newton's method[1]. Take the log likelihood of your data and find the parameters which maximize it. If that doesn't make sense, read up on numerical optimization. [0] https://en.wikipedia.org/wiki/Maximum_likelihood_estimation https://en.wikipedia.org/wiki/Maximum_likelihood_estimation [1] https://en.wikipedia.org/wiki/Newton%27s_method_in_optimization#Higher_dimensions https://en.wikipedia.org/wiki/Newton%27s_method_in_optimizat...
- Rainymood 8y ago
  >So my question - is there a layman's explanation for what a Hessian matrix is (I'm approaching this from a graduate level stats angle). Just like the score vector is the derivative of the ML wrt its parameters, the Hessian matrix is just the derivative of the score vector. It's just the second derivative wrt the parameters of the ML. It's the derivative of the derivative of the ML function.
- enriquto 8y ago
  The Hessian matrix is covered in just about any standard calculus book. The Hessian is a generalization of the second derivative of elementary calculus. Recall that the second derivative of a function f(x) allows to distinguish concave (f''>0) and convex (f''<0) parts of the function. If you are at a local extremum (f'=0) it allows to distinguish maxima and minima. The Hessian does the same thing, but for functions of several variables. It says in which spatial directions your function is convex or concave. At a local maximum, it is concave in all directions, and at a local minimum it is convex in all directions. At a saddle point, there will be directions where it is concave and directions where it is convex. The eigen decomposition of the hessian allows to find these directions. It is useful for function approximation because the 2nd degree coefficients of a polynomial that better fits your data are precisely the entries of the Hessian matrix (due to Taylor theorem).
- mehrdadn 8y ago
  Are you familiar with what the 2nd derivative of a function is? The Hessian is just that, when you have a function with multiple inputs. What's the second derivative of f(x,y) = x/y? Well there are four of them depending on the order in which you differentiate: f_xx = 0, f_xy = -1/y^2, f_yx = -1/y^2, f_yy = 2x/y^3. You just put these in a nice matrix and call it the Hessian matrix. So for functions of N input variables, there are N^2 second-derivatives, so the Hessian is the matrix of all those second derivatives. I'm not sure how you saw it used for fitting a logistic model, but in general knowing the concavity of a function is useful for minimizing or maximizing it (e.g. by repeatedly approximating it as a 2nd-order polynomial and minimizing that), so maybe that's what you saw.
  - thomasahle 8y ago
    > I'm not sure how you saw [the Hessian] used for fitting a logistic model Probably some 2nd derivative version of gradient descent or Newton's method. https://en.wikipedia.org/wiki/Newton%27s_method_in_optimization#Higher_dimensions https://en.wikipedia.org/wiki/Newton%27s_method_in_optimizat...
    - mehrdadn 8y ago
      Yeah that's what I was referring to when I said "approximating it as a 2nd-order polynomial and minimizing that".
- TJSomething 8y ago
  Those Wikipedia pages are kind of awful for pedagogy, but they have the right equations, so I won't cover those. Say we have a curve that corresponds to how good of a fit your model is. We want to try to find the maximum on that curve. However, calculating every point of the curve is too expensive, so we want to minimize the number of points we have to check. So, we start with a guess as to the highest point on the curve and take the first and second derivatives of the curve at that point. This gives us enough to fit a parabola that approximates the curve in the neighborhood of our initial guess point. Then, it's pretty easy to solve for the highest point on the parabola. That's our new guess. Repeat that a few times until the guesses stop shifting much. If the curve is nicely shaped (i.e. smooth everywhere, only has one maximum), the guesses will converge on the highest point. This is a often faster than a similar method, gradient ascent, which relies upon only taking the first derivative. This would yield a line in the vicinity of our guess, and then we just move our guess a little bit, such that it goes up the line. This is pretty slow, since it can't just go straight to a guess of the top, and if you go too fast, then it'll blow right past the maximum. The Hessian matrix is the higher dimensional equivalent to the second derivative there and the gradient is the equivalent for the first derivative. For example, if we have a two dimensional surface in 3D, then those matrices will be 2x2 and capture the curvature of the 3D paraboloid in the vicinity of the guess. As you go up in dimensionality, they're called quadric hypersurfaces. When you're fitting a logistic regression, your hypersurface is the logarithm of the likelihood that the data you have fits the logistic curve with parameters at that point. The logarithm makes the hypersurface better behaved and makes the calculus easier. You just need the gradient and the Hessian, evaluate those at your initial guess, fit a quadric hypersurface to the guess there, pop up to the top of that hypersurface, repeat a few times, and you've got your model.
  - cimmanom 8y ago
    Thank you for a super clear explanation that was easy to grok even with no more math background than (extremely rusty) high school calculus.
    - hotwire 8y ago
      yep, these are the kinds of posts on HN that I really appreciate, not the contrarian "well akshually" kind that usually appear.
- anilakar 8y ago
  Engineering maths is usually "put your numbers in a matrix and calculate the eigenvalues and eigenvectors" regardless of the field :-)
  - MarkMMullin 8y ago
    Please make the lowest variance dimension go away to see if you can understand the problem then, and if not, rinse and repeat ?
- archgoon 8y ago
  So, when you perform a regression of any sort, what you're doing is saying "Hey, I want to find parameters X,Y,Z, etc, that make this curve best fit the data that I have". One interpretation of 'best fit' is 'minimize the mean squared error'. So regression is just a minimization problem. You're trying to find the values that minimize f(X,Y,Z...) And, well, that means that you just want to find values for X,Y,Z such that ∇f(X,Y,Z) = 0 Why? Because the derivative tracks the rate of change. If the function isn't changing much locally, you've hit either a minimum or maximum (assuming the function is smooth). Let's see how this plays out when you have a function that you're trying to minimize that only has 1 parameter. This is then just a regular old function of one variable, and we can easily visualize it. | | | x^2 + 2x \ |/ ------------ \_/| And take it's dervative | / | / | / 2x + 2 |/ | /| / | ------------ And then you'd say, oh hey I know the roots of 2x + 2 ! It's just x = -1. So x^2 + 2x would have a minimum at -1. But when dealing with many functions you may not know the functional form, or there may not be a way to solve for the roots symbolically (or you don't know the tricks to do so), you might want to try a strictly numerical approach. You say "Look, I know how to compute the original function, can't I just use that directly?" Well, one option would be to find two points on your function, one where it's positive, the other where it's negative, and then keep bisecting the interval down until you zoom in on the point where it crossed the x axis (with additional logic to handle multiple crossings). Of course, if you're function is discontinuous, or is discontinuous due to a floating point error, you're going to be hosed. This is the bisection method. However, it's not as fast as is desirable. Since each decimal point is an increase by a factor of 10 precision, and you typically want several decimal points at least, simply increasing your precision by a mere factor of 2 each time may not be good enough. This is especially true if you're doing this by hand. :) So a faster method is to start at a point, and then draw a line tangent to that point, and see where that line intersects the x-axis. Evaluate that point, and normally it'll be closer to a zero than before. repeat. This, in many cases, will converge much faster than performing bisection. This is known as the Newton-Raphson Method. Now, in order to draw a tangent line, you have to know how the function is changing at that point, so the line and the function's slopes will match, and well, that means you take the derivative. Since, however, the function which we're trying to find the root of is itself a derivative, this is now a second derivative. So it turns out Newton-Raphson generalizes upwards in dimension. So when you start off with your error function that you're trying to minimize, you take it's derivative, but now you have to track how it changes in N dimensions, so the derivative object is now a vector. Now, we're trying to minimize this vector valued function (the gradient), and set it to zero. So we take it's derivative, which, since it's vector valued, will now be a matrix, since each component can vary in N directions. So we now have a NxN matrix that tracks how everything is changing. This second order derivative is called the Hessian. And we can use it the same way (implementation left to the reader ;) ) as we did in one dimension.
- GistNoesis 8y ago
  In a general manner when you have a complex function you want to minimize to obtain your parameters. One way to proceed is to assume some initial parameters and refine them. To refine them, you locally approximate your function, get the minimum of the approximation, take a step toward this minimum and loop until convergence. At first order you can approximate locally your function as a plane (the plane that go through the point and has the same first derivative), and to minimize that you take a small step (because your approximation is only valid locally) in the direction where the plane is inclined. Alternatively you can make a better approximation of your function using higher order derivatives. So instead of approximating your function with a plane, you approximate it with a quadratic form (the multidimensional extension of the parabola). This quadratic form matches the first derivatives and also the Hessian (second derivatives in multi-dimension) of your function at your current parameters. Once you have the approximation, there are closed formula for the minimum of the quadratic form so you can directly jump closer to the result (but how it will perform will depend how close your function resemble your approximation). When to pick first order approximation or second order approximation is problem dependent, but a quick rule of thumb is second order is faster when close to the solution but consume memory quadratically with respect to the number of dimension so is only applicable when the dimension is low, or when you have problem specific simplifications like your problem being a sum of squares. In practice interesting problem are too big and first order method is all you can do. But you can also improve things a little by approximating the diagonal of the Hessian, or some low-rank approximation of the Hessian. (This is another tractable kind of approximation of your function). You can also make some probabilistic approximation of your function (only considering a few examples instead of the whole training set) and from that you can derive all "on-line" methods, but this is a story for another day.
- pX0r 8y ago
  A logistic regression model is typically used for a classification task. 'Fitting a logistic model' entails finding optimal coefficients / 'weights' of input features such that classification error is minimised. For a binary classification task, one could simply calculate mean squared error between predicted values and actual labels (as in linear regression) and then proceed to find the optimal weights iteratively using gradient descent. But the sigmoid shape of the logistic function makes gradient descent a poor choice of an optimization technique (w.r.t. lack of guarantee of finding a global optimum). A surer way to find globally optimal weights is using the Newton's method of calculating weight updates. This is a numerical optimization technique that requires one to calculate the 1st and 2nd order derivatives of the error function. The matrix that 'calculates' the 1st order derivative is called a Jacobian and the one that calculates the 2nd order derivative is called a Hessian...
sdan 8y ago
Truely amazing work. Hoped for some nice drawings but Latex did it!
Koncopd 8y ago
Absolutely best guide (for me) on these things is http://www.psi.toronto.edu/~andrew/papers/matrix_calculus_for_learning.pdf http://www.psi.toronto.edu/~andrew/papers/matrix_calculus_fo... Also "Matrix algebra" by Magnus and Abadir.
luk32 8y ago
If someone likes more lecture style explanation I can recommend 3blue1brown's material on YouTube. He explained in a pretty good an accessible way imho. I didn't learn artificial neural network stuff from there. I knew those concepts but I didn't know the matrix formalism applied to it. So this was really nice to understand why GPUs are good for this. Math-wise it was really nice watch.
- tw1010 8y ago
  It's amazing what a rich-get-richer effect products and content that really manage to solve problems in a high quality way get in comments sections around the web. (E.g. 3B1B.)
  - closetCS 8y ago
    Is that such a bad thing? The web has allowed the sharing of high quality content across the world for little to no cost, but has also created so much noise for the average user that they have little to no hope of finding the high quality content on their own. Comment sections across the web fix this problem by promoting producers that offer a superior product. This encourages everyone to make better content.
  - shrimp_emoji 8y ago
    Yep. And 3b1b seems to agree: https://youtu.be/VcgJro0sTiM https://youtu.be/VcgJro0sTiM
  - Twisol 8y ago
    I'm not really sure why you're being downvoted. You're right that 3B1B really does manage to solve a problem in a high-quality way, and it's amazing how much of an effect that really has on people, especially considering the relatively niche topics that 3B1B goes into. (You'd think that his "Essence of" series would be more popular, but the one-off problem analyses have ridiculously more views in general.) In some ways, it is a "rich-get-richer" effect. But creators like 3B1B expend a lot of time and resources to do what they do, and the word of mouth he gets is an acknowledgement that the work he does is worth the money and views we provide.
cimmanom 8y ago
The explanatory style here is excellent. They explain clearly in 3 paragraphs what I was unable to get a good handle on in a semester of college math (due to awful teaching styles and textbooks), namely why partial derivatives are calculated the way they are.
qiqitori 8y ago
This book had a really gentle explanation of the required calculus: https://www.amazon.com/Make-Your-Own-Neural-Network-ebook/dp/B01EER4Z4G https://www.amazon.com/Make-Your-Own-Neural-Network-ebook/dp... It also builds some really simple Python code to create a simple (non-"deep", i.e. with just one hidden layer) neural network capable of recognizing human-drawn digits with good accuracy.
calebh 8y ago
There's a tool now that can do matrix calculus: http://www.matrixcalculus.org/ http://www.matrixcalculus.org/
bag531 8y ago
This is really useful. Matrix calculus was one of the hardest parts of my machine learning course at university simply because it had never been covered by any of the many math classes I'd taken.
dm7 8y ago
an accessible and concise refreshment of material
hypersoar 8y ago
Does anyone know of good resources for studying machine learning or data science given a strong mathematical background? I'm transitioning careers from pure math research into industry. I know very little about machine learning, but I know the crap out of linear algebra and real analysis (and other, less relevant fields). It'd be great to have some sources that leverage that without assuming much prior CS knowledge.
- srean 8y ago
  Given your math bent expecially analysis, I would recommend you start with Luc Devroye, follow it up with Vapnik. These are a bit dated but still very relevant. After these two go with Alex Smola and Bernard Scholkopf. The books that have been suggested to you so far aren't bad but with high fluff to meat ratio, not meant for a reader like you. If you have function analysis in your bag you will feel right at home with the books I suggested. Deep NN are much in the news, but on the math side of the practice of DNN, there is not much to it apart from chain rule for differentiation.
  - poster123 8y ago
    Thanks for the recommendations. Devroye wrote several books, and I assume the one being mentioned is "A Probabilistic Theory of Pattern Recognition".
    - srean 8y ago
      Yes that's the one I had in mind. His other books are also very good, but are more specialized. You may also enjoy graycat's comments on HN. He know his math but is contrarian about machine learning but its good to have that view point.
  - poster123 8y ago
    The book "A Probabilistic Theory of Pattern Recognition" by Devroye, Györfi and Lugosi is available from the site of Gyorfi at http://www.szit.bme.hu/~gyorfi/pbook.pdf http://www.szit.bme.hu/~gyorfi/pbook.pdf .
- 8y ago
andrepd 8y ago
Too extensive and not focusing enough on the core concepts. Teaching mathematics is about making the underlying structure as clear as possible. It's a very hard thing to do effectively.
jonnydubowsky 8y ago
This thread is a glowing reminder of how effective and friendly the HN community can be, when good will is extended in both the question and the answer. Thank you to all who contributed as the comments have provided me (and many others I'm sure) with a clear and concise map of the matrix calculus full of helpful resources. My 4th of July beach reading is now complete.
stablemap 8y ago
Previous discussion: https://news.ycombinator.com/item?id=16267178 https://news.ycombinator.com/item?id=16267178
- dang 8y ago
  Hmm that definitely qualifies this one as a dupe, but the discussion is unusually substantive so I guess we'll leave it up.

610 points by prostoalex 8y ago

So I have a question somewhat related to this that I never knew where/who to ask (well actually I asked a few mathematicians at a university I work with whose answers I couldn't understand - their answers were almost as impenetrable as the Wikipedia page, and some engineering scientists who I thought would be more into 'applied math' but they didn't know. So I'm hoping some data science people reading this would better understand where I'm coming from, and be able to explain at my primitive math skills level). So, some time ago I contracted out writing some code for fitting a logistic regression onto a given set of observations. There were some specific requirements, but I think I should have been able to piece something myself together using mainstream LA libraries; some of them even hint at 'you could fit a logistic regression using these functions' but no complete examples. But I didn't understand well enough so I contracted it out. The woman who ended up writing the code used a 'Hessian' matrix to do so (she actually wrote two functions doing the same thing, one used this Hessian approach - I think the idea was that it would be faster but there wasn't a lot of time and it never got tweaked enough to make a difference). So my question - is there a layman's explanation for what a Hessian matrix is, and how it applies to fitting a logistic model? Also (with an eye to the future of my project), does it have applications for non-linear regressions? Alternatively, are there any books where this is covered? I have most standard stats/applied stats/operations research books, as well as a few like the no bullshit guide to linear algebra, but none cover this specific issue - or even how to fit a logistic regression at all, on a practical level (so not just 'conceptually you do xyz, implementation is left as an exercise for the reader').

matheist 8y ago
Random guess: a logistic regression is fit by maximizing a particular log likelihood. [+] You can find the maximum by (1) gradient ascent, or (2) the analogue of Newton's method in multiple dimensions [++], which involves computing the Hessian matrix. So there's your Hessian. [+] https://en.wikipedia.org/wiki/Logistic_regression#Model_fitting https://en.wikipedia.org/wiki/Logistic_regression#Model_fitt... [++] https://en.wikipedia.org/wiki/Newton%27s_method_in_optimization#Higher_dimensions https://en.wikipedia.org/wiki/Newton%27s_method_in_optimizat...
- nerdponx 8y ago
  Good guess!
imh 8y ago
Look up maximum likelihood estimation [0] and look up newton's method[1]. Take the log likelihood of your data and find the parameters which maximize it. If that doesn't make sense, read up on numerical optimization. [0] https://en.wikipedia.org/wiki/Maximum_likelihood_estimation https://en.wikipedia.org/wiki/Maximum_likelihood_estimation [1] https://en.wikipedia.org/wiki/Newton%27s_method_in_optimization#Higher_dimensions https://en.wikipedia.org/wiki/Newton%27s_method_in_optimizat...
Rainymood 8y ago
>So my question - is there a layman's explanation for what a Hessian matrix is (I'm approaching this from a graduate level stats angle). Just like the score vector is the derivative of the ML wrt its parameters, the Hessian matrix is just the derivative of the score vector. It's just the second derivative wrt the parameters of the ML. It's the derivative of the derivative of the ML function.
enriquto 8y ago
The Hessian matrix is covered in just about any standard calculus book. The Hessian is a generalization of the second derivative of elementary calculus. Recall that the second derivative of a function f(x) allows to distinguish concave (f''>0) and convex (f''<0) parts of the function. If you are at a local extremum (f'=0) it allows to distinguish maxima and minima. The Hessian does the same thing, but for functions of several variables. It says in which spatial directions your function is convex or concave. At a local maximum, it is concave in all directions, and at a local minimum it is convex in all directions. At a saddle point, there will be directions where it is concave and directions where it is convex. The eigen decomposition of the hessian allows to find these directions. It is useful for function approximation because the 2nd degree coefficients of a polynomial that better fits your data are precisely the entries of the Hessian matrix (due to Taylor theorem).
mehrdadn 8y ago
Are you familiar with what the 2nd derivative of a function is? The Hessian is just that, when you have a function with multiple inputs. What's the second derivative of f(x,y) = x/y? Well there are four of them depending on the order in which you differentiate: f_xx = 0, f_xy = -1/y^2, f_yx = -1/y^2, f_yy = 2x/y^3. You just put these in a nice matrix and call it the Hessian matrix. So for functions of N input variables, there are N^2 second-derivatives, so the Hessian is the matrix of all those second derivatives. I'm not sure how you saw it used for fitting a logistic model, but in general knowing the concavity of a function is useful for minimizing or maximizing it (e.g. by repeatedly approximating it as a 2nd-order polynomial and minimizing that), so maybe that's what you saw.
- thomasahle 8y ago
  > I'm not sure how you saw [the Hessian] used for fitting a logistic model Probably some 2nd derivative version of gradient descent or Newton's method. https://en.wikipedia.org/wiki/Newton%27s_method_in_optimization#Higher_dimensions https://en.wikipedia.org/wiki/Newton%27s_method_in_optimizat...
  - mehrdadn 8y ago
    Yeah that's what I was referring to when I said "approximating it as a 2nd-order polynomial and minimizing that".
TJSomething 8y ago
Those Wikipedia pages are kind of awful for pedagogy, but they have the right equations, so I won't cover those. Say we have a curve that corresponds to how good of a fit your model is. We want to try to find the maximum on that curve. However, calculating every point of the curve is too expensive, so we want to minimize the number of points we have to check. So, we start with a guess as to the highest point on the curve and take the first and second derivatives of the curve at that point. This gives us enough to fit a parabola that approximates the curve in the neighborhood of our initial guess point. Then, it's pretty easy to solve for the highest point on the parabola. That's our new guess. Repeat that a few times until the guesses stop shifting much. If the curve is nicely shaped (i.e. smooth everywhere, only has one maximum), the guesses will converge on the highest point. This is a often faster than a similar method, gradient ascent, which relies upon only taking the first derivative. This would yield a line in the vicinity of our guess, and then we just move our guess a little bit, such that it goes up the line. This is pretty slow, since it can't just go straight to a guess of the top, and if you go too fast, then it'll blow right past the maximum. The Hessian matrix is the higher dimensional equivalent to the second derivative there and the gradient is the equivalent for the first derivative. For example, if we have a two dimensional surface in 3D, then those matrices will be 2x2 and capture the curvature of the 3D paraboloid in the vicinity of the guess. As you go up in dimensionality, they're called quadric hypersurfaces. When you're fitting a logistic regression, your hypersurface is the logarithm of the likelihood that the data you have fits the logistic curve with parameters at that point. The logarithm makes the hypersurface better behaved and makes the calculus easier. You just need the gradient and the Hessian, evaluate those at your initial guess, fit a quadric hypersurface to the guess there, pop up to the top of that hypersurface, repeat a few times, and you've got your model.
- cimmanom 8y ago
  Thank you for a super clear explanation that was easy to grok even with no more math background than (extremely rusty) high school calculus.
  - hotwire 8y ago
    yep, these are the kinds of posts on HN that I really appreciate, not the contrarian "well akshually" kind that usually appear.
anilakar 8y ago
Engineering maths is usually "put your numbers in a matrix and calculate the eigenvalues and eigenvectors" regardless of the field :-)
- MarkMMullin 8y ago
  Please make the lowest variance dimension go away to see if you can understand the problem then, and if not, rinse and repeat ?
archgoon 8y ago
So, when you perform a regression of any sort, what you're doing is saying "Hey, I want to find parameters X,Y,Z, etc, that make this curve best fit the data that I have". One interpretation of 'best fit' is 'minimize the mean squared error'. So regression is just a minimization problem. You're trying to find the values that minimize f(X,Y,Z...) And, well, that means that you just want to find values for X,Y,Z such that ∇f(X,Y,Z) = 0 Why? Because the derivative tracks the rate of change. If the function isn't changing much locally, you've hit either a minimum or maximum (assuming the function is smooth). Let's see how this plays out when you have a function that you're trying to minimize that only has 1 parameter. This is then just a regular old function of one variable, and we can easily visualize it. | | | x^2 + 2x \ |/ ------------ \_/| And take it's dervative | / | / | / 2x + 2 |/ | /| / | ------------ And then you'd say, oh hey I know the roots of 2x + 2 ! It's just x = -1. So x^2 + 2x would have a minimum at -1. But when dealing with many functions you may not know the functional form, or there may not be a way to solve for the roots symbolically (or you don't know the tricks to do so), you might want to try a strictly numerical approach. You say "Look, I know how to compute the original function, can't I just use that directly?" Well, one option would be to find two points on your function, one where it's positive, the other where it's negative, and then keep bisecting the interval down until you zoom in on the point where it crossed the x axis (with additional logic to handle multiple crossings). Of course, if you're function is discontinuous, or is discontinuous due to a floating point error, you're going to be hosed. This is the bisection method. However, it's not as fast as is desirable. Since each decimal point is an increase by a factor of 10 precision, and you typically want several decimal points at least, simply increasing your precision by a mere factor of 2 each time may not be good enough. This is especially true if you're doing this by hand. :) So a faster method is to start at a point, and then draw a line tangent to that point, and see where that line intersects the x-axis. Evaluate that point, and normally it'll be closer to a zero than before. repeat. This, in many cases, will converge much faster than performing bisection. This is known as the Newton-Raphson Method. Now, in order to draw a tangent line, you have to know how the function is changing at that point, so the line and the function's slopes will match, and well, that means you take the derivative. Since, however, the function which we're trying to find the root of is itself a derivative, this is now a second derivative. So it turns out Newton-Raphson generalizes upwards in dimension. So when you start off with your error function that you're trying to minimize, you take it's derivative, but now you have to track how it changes in N dimensions, so the derivative object is now a vector. Now, we're trying to minimize this vector valued function (the gradient), and set it to zero. So we take it's derivative, which, since it's vector valued, will now be a matrix, since each component can vary in N directions. So we now have a NxN matrix that tracks how everything is changing. This second order derivative is called the Hessian. And we can use it the same way (implementation left to the reader ;) ) as we did in one dimension.
GistNoesis 8y ago
In a general manner when you have a complex function you want to minimize to obtain your parameters. One way to proceed is to assume some initial parameters and refine them. To refine them, you locally approximate your function, get the minimum of the approximation, take a step toward this minimum and loop until convergence. At first order you can approximate locally your function as a plane (the plane that go through the point and has the same first derivative), and to minimize that you take a small step (because your approximation is only valid locally) in the direction where the plane is inclined. Alternatively you can make a better approximation of your function using higher order derivatives. So instead of approximating your function with a plane, you approximate it with a quadratic form (the multidimensional extension of the parabola). This quadratic form matches the first derivatives and also the Hessian (second derivatives in multi-dimension) of your function at your current parameters. Once you have the approximation, there are closed formula for the minimum of the quadratic form so you can directly jump closer to the result (but how it will perform will depend how close your function resemble your approximation). When to pick first order approximation or second order approximation is problem dependent, but a quick rule of thumb is second order is faster when close to the solution but consume memory quadratically with respect to the number of dimension so is only applicable when the dimension is low, or when you have problem specific simplifications like your problem being a sum of squares. In practice interesting problem are too big and first order method is all you can do. But you can also improve things a little by approximating the diagonal of the Hessian, or some low-rank approximation of the Hessian. (This is another tractable kind of approximation of your function). You can also make some probabilistic approximation of your function (only considering a few examples instead of the whole training set) and from that you can derive all "on-line" methods, but this is a story for another day.
pX0r 8y ago
A logistic regression model is typically used for a classification task. 'Fitting a logistic model' entails finding optimal coefficients / 'weights' of input features such that classification error is minimised. For a binary classification task, one could simply calculate mean squared error between predicted values and actual labels (as in linear regression) and then proceed to find the optimal weights iteratively using gradient descent. But the sigmoid shape of the logistic function makes gradient descent a poor choice of an optimization technique (w.r.t. lack of guarantee of finding a global optimum). A surer way to find globally optimal weights is using the Newton's method of calculating weight updates. This is a numerical optimization technique that requires one to calculate the 1st and 2nd order derivatives of the error function. The matrix that 'calculates' the 1st order derivative is called a Jacobian and the one that calculates the 2nd order derivative is called a Hessian...

sdan 8y ago

Truely amazing work. Hoped for some nice drawings but Latex did it!

Koncopd 8y ago

Absolutely best guide (for me) on these things is http://www.psi.toronto.edu/~andrew/papers/matrix_calculus_for_learning.pdf http://www.psi.toronto.edu/~andrew/papers/matrix_calculus_fo... Also "Matrix algebra" by Magnus and Abadir.

luk32 8y ago

If someone likes more lecture style explanation I can recommend 3blue1brown's material on YouTube. He explained in a pretty good an accessible way imho. I didn't learn artificial neural network stuff from there. I knew those concepts but I didn't know the matrix formalism applied to it. So this was really nice to understand why GPUs are good for this. Math-wise it was really nice watch.

tw1010 8y ago
It's amazing what a rich-get-richer effect products and content that really manage to solve problems in a high quality way get in comments sections around the web. (E.g. 3B1B.)
- closetCS 8y ago
  Is that such a bad thing? The web has allowed the sharing of high quality content across the world for little to no cost, but has also created so much noise for the average user that they have little to no hope of finding the high quality content on their own. Comment sections across the web fix this problem by promoting producers that offer a superior product. This encourages everyone to make better content.
- shrimp_emoji 8y ago
  Yep. And 3b1b seems to agree: https://youtu.be/VcgJro0sTiM https://youtu.be/VcgJro0sTiM
- Twisol 8y ago
  I'm not really sure why you're being downvoted. You're right that 3B1B really does manage to solve a problem in a high-quality way, and it's amazing how much of an effect that really has on people, especially considering the relatively niche topics that 3B1B goes into. (You'd think that his "Essence of" series would be more popular, but the one-off problem analyses have ridiculously more views in general.) In some ways, it is a "rich-get-richer" effect. But creators like 3B1B expend a lot of time and resources to do what they do, and the word of mouth he gets is an acknowledgement that the work he does is worth the money and views we provide.

cimmanom 8y ago

The explanatory style here is excellent. They explain clearly in 3 paragraphs what I was unable to get a good handle on in a semester of college math (due to awful teaching styles and textbooks), namely why partial derivatives are calculated the way they are.

qiqitori 8y ago

This book had a really gentle explanation of the required calculus: https://www.amazon.com/Make-Your-Own-Neural-Network-ebook/dp/B01EER4Z4G https://www.amazon.com/Make-Your-Own-Neural-Network-ebook/dp... It also builds some really simple Python code to create a simple (non-"deep", i.e. with just one hidden layer) neural network capable of recognizing human-drawn digits with good accuracy.

calebh 8y ago

There's a tool now that can do matrix calculus: http://www.matrixcalculus.org/ http://www.matrixcalculus.org/

bag531 8y ago

This is really useful. Matrix calculus was one of the hardest parts of my machine learning course at university simply because it had never been covered by any of the many math classes I'd taken.

dm7 8y ago

an accessible and concise refreshment of material

hypersoar 8y ago

Does anyone know of good resources for studying machine learning or data science given a strong mathematical background? I'm transitioning careers from pure math research into industry. I know very little about machine learning, but I know the crap out of linear algebra and real analysis (and other, less relevant fields). It'd be great to have some sources that leverage that without assuming much prior CS knowledge.

srean 8y ago
Given your math bent expecially analysis, I would recommend you start with Luc Devroye, follow it up with Vapnik. These are a bit dated but still very relevant. After these two go with Alex Smola and Bernard Scholkopf. The books that have been suggested to you so far aren't bad but with high fluff to meat ratio, not meant for a reader like you. If you have function analysis in your bag you will feel right at home with the books I suggested. Deep NN are much in the news, but on the math side of the practice of DNN, there is not much to it apart from chain rule for differentiation.
- poster123 8y ago
  Thanks for the recommendations. Devroye wrote several books, and I assume the one being mentioned is "A Probabilistic Theory of Pattern Recognition".
  - srean 8y ago
    Yes that's the one I had in mind. His other books are also very good, but are more specialized. You may also enjoy graycat's comments on HN. He know his math but is contrarian about machine learning but its good to have that view point.
- poster123 8y ago
  The book "A Probabilistic Theory of Pattern Recognition" by Devroye, Györfi and Lugosi is available from the site of Gyorfi at http://www.szit.bme.hu/~gyorfi/pbook.pdf http://www.szit.bme.hu/~gyorfi/pbook.pdf .
8y ago

andrepd 8y ago

Too extensive and not focusing enough on the core concepts. Teaching mathematics is about making the underlying structure as clear as possible. It's a very hard thing to do effectively.

jonnydubowsky 8y ago

This thread is a glowing reminder of how effective and friendly the HN community can be, when good will is extended in both the question and the answer. Thank you to all who contributed as the comments have provided me (and many others I'm sure) with a clear and concise map of the matrix calculus full of helpful resources. My 4th of July beach reading is now complete.

stablemap 8y ago

Previous discussion: https://news.ycombinator.com/item?id=16267178 https://news.ycombinator.com/item?id=16267178

dang 8y ago
Hmm that definitely qualifies this one as a dupe, but the discussion is unusually substantive so I guess we'll leave it up.