Skip to content

Latest commit

 

History

History
71 lines (43 loc) · 12.3 KB

README.md

File metadata and controls

71 lines (43 loc) · 12.3 KB

Matrix Calculus for Machine Learning and Beyond

This is the course page for an 18.063 Matrix Calculus at MIT taught in January 2025 (IAP) by Professors Alan Edelman and Steven G. Johnson.

Lectures: MWF time 11am–1pm, Jan 13–Jan 31 in room 4-370; lecture recordings to be posted (MIT only). 3 units, 2 problem sets due Jan 24 and Feb 31 — submitted electronically via Canvas, no exams. TA/grader: TBD.

Course Notes: Draft notes from IAP 2024. Other materials to be posted.

Piazza forum: Online discussions at Piazza.

Description:

We all know that calculus courses such as 18.01 and 18.02 are univariate and vector calculus, respectively. Modern applications such as machine learning and large-scale optimization require the next big step, "matrix calculus" and calculus on arbitrary vector spaces.

This class covers a coherent approach to matrix calculus showing techniques that allow you to think of a matrix holistically (not just as an array of scalars), generalize and compute derivatives of important matrix factorizations and many other complicated-looking operations, and understand how differentiation formulas must be re-imagined in large-scale computing. We will discuss reverse/adjoint/backpropagation differentiation, custom vector-Jacobian products, and how modern automatic differentiation is more computer science than calculus (it is neither symbolic formulas nor finite differences).

Prerequisites: Linear Algebra such as 18.06 and multivariate calculus such as 18.02.

Course will involve simple numerical computations using the Julia language. Ideally install it on your own computer following these instructions, but as a fallback you can run it in the cloud here: Binder

Topics:

Here are some of the planned topics:

  • Derivatives as linear operators and linear approximation on arbitrary vector spaces: beyond gradients and Jacobians.
  • Derivatives of functions with matrix inputs and/or outputs (e.g. matrix inverses and determinants). Kronecker products and matrix "vectorization".
  • Derivatives of matrix factorizations (e.g. eigenvalues/SVD) and derivatives with constraints (e.g. orthogonal matrices).
  • Multidimensional chain rules, and the significance of right-to-left ("forward") vs. left-to-right ("reverse") composition. Chain rules on computational graphs (e.g. neural networks).
  • Forward- and reverse-mode manual and automatic multivariate differentiation.
  • Adjoint methods (vJp/pullback rules) for derivatives of solutions of linear, nonlinear, and differential equations.
  • Application to nonlinear root-finding and optimization. Multidimensional Newton and steepest–descent methods.
  • Applications in engineering/scientific optimization and machine learning.
  • Second derivatives, Hessian matrices, quadratic approximations, and quasi-Newton methods.

Lecture 1 (Jan 13)

  • part 1: overview (slides)
  • part 2: derivatives as linear operators: matrix functions, gradients, product and chain rule
  • video (MIT only)

Re-thinking derivatives as linear operators: f(x+dx)-f(x)=df=f′(x)[dx] — f′ is the linear operator that gives the change df in the output from a "tiny" change dx in the inputs, to first order in dx (i.e. dropping higher-order terms). When we have a vector function f(x)∈ℝᵐ of vector inputs x∈ℝⁿ, then f'(x) is a linear operator that takes n inputs to m outputs, which we can think of as an m×n matrix called the Jacobian matrix (typically covered only superficially in 18.02).

In the same way, we can define derivatives of matrix-valued operators as linear operators on matrices. For example, f(X)=X² gives f'(X)[dX] = X dX + dX X. Or f(X) = X⁻¹ gives f'(X)[dX] = –X⁻¹ dX X⁻¹. These are perfectly good linear operators acting on matrices dX, even though they are not written in the form (Jacobian matrix)×(column vector)! (We could rewrite them in the latter form by reshaping the inputs dX and the outputs df into column vectors, more formally by choosing a basis, and we will later cover how this process can be made more elegant using Kronecker products. But for the most part it is neither necessary nor desirable to express all linear operators as Jacobian matrices in this way.)

Reviewed the (easy) derivations of the sum rule d(f+g)=df+dg and the product rule d(fg) = (df)g+f(dg), directly from the definition of f(x+dx)-f(x)=df=f′(x)[dx], dropping higher-order terms.

Discussed the chain rule for f(g(x)) (f'(x)=g'(h(x))h'(x), where this is a composition of two linear operations, performing h' then g' — g'h' ≠ h'g'!). For functions from vectors to vectors, the chain rule is simply the product of Jacobians. Moreover, as soon as you compose 3 or more functions, it can a make a huge difference whether you multiply the Jacobians from left-to-right ("reverse-mode", or "backpropagation", or "adjoint differentiation") or right-to-left ("forward-mode"). Showed, for example, that if you have many inputs but a single output (as is common in machine learning and other types of optimization problem), that it is vastly more efficient to multiply left-to-right than right-to-left, and such "backpropagation algorithms" are a key factor in the practicality of large-scale optimization.

Further reading: Draft Course Notes (link above), chapters 1 and 2. matrixcalculus.org (linked in the slides) is a fun site to play with derivatives of matrix and vector functions. The Matrix Cookbook has a lot of formulas for these derivatives, but no derivations. Some notes on vector and matrix differentiation were posted for 6.S087 from IAP 2021.

Further reading (gradients): A fancy name for a row vector is a "covector" or linear form, and the fancy version of the relationship between row and column vectors is the Riesz representation theorem, but until you get to non-Euclidean geometry you may be happier thinking of a row vector as the transpose of a column vector.

Further reading (chain rule): The terms "forward-mode" and "reverse-mode" differentiation are most prevalent in automatic differentiation (AD), which will will cover later in this course. You can find many, many articles online about backpropagation in neural networks. There are many other versions of this, e.g. in differential geometry the derivative linear operator (multiplying Jacobians and perturbations dx right-to-left) is called a pushforward, whereas multiplying a gradient row vector (covector) by a Jacobian left-to-right is called a pullback. This video on the principles of AD in Julia by Dr. Mohamed Tarek also starts with a similar left-to-right (reverse) vs right-to-left (forward) viewpoint and goes into how it translates to Julia code, and how you define custom chain-rule steps for Julia AD. In other fields, "reverse mode" is sometimes called an "adjoint method": see the notes on adjoint methods and slides from 18.335 (video).

Further reading (fancier math): the perspective of derivatives as linear operators is sometimes called a Fréchet derivative and you can find lots of very abstract (what I'm calling "fancy") presentations of this online, chock full of weird terminology whose purpose is basically to generalize the concept to weird types of vector spaces. The "little-o notation" o(δx) we're using here for "infinitesimal asymptotics" is closely related to the asymptotic notation used in computer science, but in computer science people are typically taking the limit as the argument (often called "n") becomes very large instead of very small.

Lecture 2 (Jan 15)

Further reading: Wikipedia has a useful list of properties of the matrix trace. The "matrix dot product" introduced today is also called the Frobenius inner product, and the corresponding norm ("length" of the matrix viewed as a vector) is the Frobenius norm. When you "flatten" a matrix A by stacking its columns into a single vector, the result is called vec(A), and many important linear operations on matrices can be expressed as Kronecker products. The Matrix Cookbook has lots of formulas for derivatives of matrix functions. See the notes on adjoint methods and slides from 18.335 (video).