Notation
The single notation used across every course of Applied AI Mastery: vectors, matrices, probability, learning and reinforcement learning, with the macro that writes each symbol.
One notation runs through every course, so a symbol learned in linear algebra still means the same thing in reinforcement learning. Lessons write symbols with the macros below, and tools/check_lessons.py rejects ad-hoc alternatives.
Where the field has a strong convention we follow it: Sutton and Barto for reinforcement learning, and the common usage of Mathematics for Machine Learning and Understanding Deep Learning for linear algebra and learning.
General
| Meaning | Symbol | Write | Notes |
|---|---|---|---|
| Scalar | \(a, b, c, x\) | a |
Lower-case italic |
| Vector | \(\vect{v}, \vect{x}\) | \vect{v} |
Lower-case bold. A column unless stated |
| Matrix | \(\mat{A}, \mat{W}\) | \mat{A} |
Upper-case bold |
| Entry of a vector | \(v_i\) | v_i |
Italic, not bold. Indices start at 1 in maths, 0 in code |
| Entry of a matrix | \(a_{ij}\) | a_{ij} |
Row \(i\), column \(j\) |
| Real numbers | \(\R\), \(\R^n\) | \R, \R^n |
|
| Transpose | \(\mat{A}\T\) | \mat{A}\T |
|
| Norm | \(\norm{\vect{v}}\) | \norm{\vect{v}} |
Euclidean unless subscripted |
| Absolute value | \(\abs{x}\) | \abs{x} |
|
| Standard basis of the plane | \(\ihat, \jhat\) | \ihat, \jhat |
\(\ihat = (1, 0)\), \(\jhat = (0, 1)\) |
| Standard basis of \(\R^n\) | \(\vect{e}_1, \dots, \vect{e}_n\) | \vect{e}_i |
|
| A map from \(A\) to \(B\) | \(T : A \to B\) | T : A \to B |
|
| Partial derivative | \(\pd{L}{w}\) | \pd{L}{w} |
|
| Gradient | \(\nabla L\) | \nabla L |
A vector with the shape of its argument |
| Arg max, arg min | \(\argmax_a\), \(\argmin_w\) | \argmax_a |
In running text a vector may be written as a tuple, \((x, y)\). In displayed equations that involve matrices it is written as a column.
Complex numbers
| Meaning | Symbol | Notes |
|---|---|---|
| The imaginary unit | \(i\) | \(i^2 = -1\). Code writes it 1j |
| A complex number | \(z = a + bi\) | A second one is \(u\), because \(w\) is a weight |
| Real and imaginary parts | \(\operatorname{Re} z\), \(\operatorname{Im} z\) | |
| Modulus and argument | \(\abs{z}\), \(\arg z\) | Length and angle of \(z\) in the plane |
| Conjugate | \(z^*\) | Not an overbar, which is kept for means |
Probability
| Meaning | Symbol | Write |
|---|---|---|
| Probability of an event | \(\Prob(A)\) | \Prob(A) |
| Expectation | \(\E[X]\) | \E[X] |
| Expectation under a policy or distribution | \(\E_\pi[\,\cdot\,]\) | \E_\pi[...] |
| Random variable | \(X, S_t, R_t\) | Upper-case italic |
| A value it takes | \(x, s, r\) | Lower-case italic |
| Conditional probability | \(p(y \mid x)\) | p(y \mid x) |
Learning
| Meaning | Symbol | Notes |
|---|---|---|
| Number of examples | \(n\) | |
| Example \(i\) | \((x_i, y_i)\) | Input and target. The target is also called the label, above all in classification |
| Number of features | \(d\) | An input with several features is a vector \(\vect{x}\) in \(\R^d\) |
| Prediction | \(\hat{y}_i\) | |
| Weight, bias | \(w\), \(b\) | \(\vect{w}\) when there are several weights |
| All the parameters of a model | \(\theta\) | Used when a statement holds for any model, whatever its parameters are |
| Decision threshold | \(c\) | Predict the positive class when the score is at least \(c\). A lesson that uses “above \(c\)” says so: the two differ only at a tie |
| Threshold on a probability | \(\tau\) | Greek tau. Predict the positive class when the predicted probability is at least \(\tau\). For logistic regression this is the cut \(c = \ln\frac{\tau}{1 - \tau}\) on the score. Not used for anything else |
| Loss | \(L\) | \(L(w, b)\) as a function of the parameters |
| Learning rate | \(\eta\) | Greek eta. Code calls it lr |
| Step counter | \(t\) | \(w_t\) is the weight after \(t\) updates |
| Mini-batch | \(B\) | A set of example indices, of size \(\abs{B}\) |
Reinforcement learning
Following Sutton and Barto, second edition.
| Meaning | Symbol | Notes |
|---|---|---|
| State, action, reward at time \(t\) | \(S_t, A_t, R_t\) | \(R_{t+1}\) is the reward that follows \(A_t\) |
| Values they take | \(s, a, r, s'\) | \(s'\) is the next state |
| Dynamics | \(p(s', r \mid s, a)\) | Probability of next state and reward |
| Policy | \(\pi(a \mid s)\) | Probability of action \(a\) in state \(s\) |
| Discount rate | \(\gamma\) | \(0 \le \gamma \le 1\) |
| Return | \(G_t\) | \(R_{t+1} + \gamma R_{t+2} + \gamma^2 R_{t+3} + \cdots\) |
| State-value function | \(v_\pi(s)\) | Expected return from \(s\) under \(\pi\) |
| Action-value function | \(q_\pi(s, a)\) | Expected return after taking \(a\) in \(s\), then following \(\pi\) |
| Optimal values | \(v_*(s)\), \(q_*(s, a)\) | |
| An estimate | \(V(s)\), \(Q(s, a)\) | Upper-case for a table being learned |
| Bellman operators | \(T_\pi\), \(T_*\) | \((T_\pi v)(s)\) is one backup of the table \(v\) under \(\pi\); \(T_*\) takes the best action |
| Max norm | \(\norm{v}_\infty\) | The largest absolute entry of a value table |
| Transition matrix and reward vector under a policy | \(\mat{P}_\pi\), \(\vect{r}_\pi\) | So that \(\vect{v}_\pi = \vect{r}_\pi + \gamma \mat{P}_\pi \vect{v}_\pi\) |
Colour in equations
A colour means the same thing in an equation as in a figure. Use it sparingly, to tie a term to the part of the picture it describes.
| Role | Write | Shows as |
|---|---|---|
| Data and inputs | \cdata{x} |
\(\cdata{x}\) |
| Model output | \cmodel{\hat{y}} |
\(\cmodel{\hat{y}}\) |
| Ground truth | \ctruth{y} |
\(\ctruth{y}\) |
| Gradients and updates | \cgrad{\nabla L} |
\(\cgrad{\nabla L}\) |
| Loss and error | \closs{L} |
\(\closs{L}\) |
| Parameters | \cparam{w} |
\(\cparam{w}\) |
| Second class | \csecond{x'} |
\(\csecond{x'}\) |
In a sentence, the matching spans are [the data]{.c-data}, [the prediction]{.c-model} and so on.
The macros live in site/macros.html. To add a symbol, add it there and to this page in the same change.