Paper-Equation Assessment

Decode four equations of the kind ML papers assume you can read at a glance — MSE, softmax, cosine similarity, and a linear/attention layer — by naming every symbol, its type, and its dimensions, then rewriting, exemplifying, and implementing each.

Paper-Equation Assessment

Below are four equations of the kind that appear, unexplained, in real ML papers. For each equation, produce a short write-up with these six parts:

Symbols & types — name every symbol and state whether it is a scalar, vector, matrix, or index, with its dimensions.
In English — rewrite the equation as one or two plain-English sentences.
Tiny example — plug in small numbers and compute the result by hand.
Pseudocode — a few lines of language-agnostic pseudocode.
NumPy — a vectorized NumPy implementation (a fenced code block).
ML purpose — one or two sentences on where and why this appears in machine learning.

Equation 1 — Mean squared error

L(\theta) = \frac{1}{n}\sum_{i=1}^{n} \left( \hat{y}_i - y_i \right)^2

Here $\hat{y}_i = f(x_i; \theta)$ is the model's prediction for example $i$ and $y_i$ is the target.

Equation 2 — Softmax

\text{softmax}(z)_i = \frac{e^{z_i}}{\sum_{j=1}^{K} e^{z_j}}

where $z \in \mathbb{R}^{K}$ is a vector of logits.

Equation 3 — Cosine similarity

\cos(\theta) = \frac{u \cdot v}{\lVert u \rVert_2 \, \lVert v \rVert_2}

for two nonzero vectors $u, v \in \mathbb{R}^{d}$ .

Equation 4 — A linear layer and attention scores

h = W x + b, \qquad S = \frac{Q K^\top}{\sqrt{d_k}}

where $W \in \mathbb{R}^{m \times d}$ , $x \in \mathbb{R}^{d}$ , $b \in \mathbb{R}^{m}$ , and for attention $Q \in \mathbb{R}^{n \times d_k}$ , $K \in \mathbb{R}^{n \times d_k}$ .

For Equation 4, pay special attention to the shapes: give the shape of $h$ , the shape of $S$ , and explain the role of the $\sqrt{d_k}$ scaling.

Marking rubric

Symbols & types — every symbol named with the correct type (scalar/vector/matrix/index) and dimensions; sum indices and their ranges identified.
Plain English — each equation restated accurately in words, capturing what is computed and over what.
Tiny example — concrete small numbers plugged in and computed correctly (e.g. MSE of a 2-example case, softmax of a length-2 or length-3 logit vector).
Pseudocode — correct, readable, language-agnostic steps that match the math.
NumPy — vectorized and correct; softmax uses the max-subtraction stability trick; cosine handles the norms correctly.
Shapes (Eq. 4) — $h$ has shape $(m,)$ and $S$ has shape $(n,n)$ ; the $\sqrt{d_k}$ scaling is explained as keeping the dot-product variance stable so softmax gradients do not vanish.
ML purpose — each equation correctly tied to its use (regression loss, converting logits to probabilities, similarity/retrieval, linear projection and attention).