# Session 1 — From a line to a learned line > **Where we're starting:** you know basic 2D math (lines, equations) and what > a function is. That's it. Everything else is built here. > > Companion to the live teaching session. Read time: ~15 minutes. ## What we're standing on - A **function** = a rule that takes an input and gives back an output. Same input → same output, every time. - A **line** = `y = mx + b`. Slope `m`, intercept `b`. Give me `x`, I give you `y`. That's enough to build everything from. --- ## The flip In math class, you usually see questions like: > *"Given y = 2x + 3, what's y when x = 5?"* You're **told** the line — `m` and `b` are handed to you. You plug in `x = 5`, get `y = 13`. Done. But that's not how the world works. In real life, you almost never know the line in advance. You see **data** — examples of inputs paired with outputs — and you have to figure out the rule yourself. Concrete example. You collect 5 houses: ``` sqft | price ($K) -------|------------ 1000 | 250 1500 | 350 2000 | 420 2500 | 510 3000 | 600 ``` You strongly suspect: *price increases roughly linearly with sqft.* So you guess the rule has the shape: ``` price = m · sqft + b ``` There IS some `m` and some `b` that makes this approximately true. You don't know what they are yet. Plot the data: ``` price 600 | • 500 | • 400 | • 300 | • 250 | • +-------------------------- sqft 1000 1500 2000 2500 3000 ``` By eye, you can almost *see* the line going through those points. Maybe `m ≈ 0.18`, `b ≈ 75`. Doesn't matter — the point is **a good line exists**, and the job is to find it. That job — finding the right `m` and `b` from data — is **learning**. --- ## Same equation, two completely different questions This is the conceptual centerpiece. Same equation `y = mx + b`. Two opposite ways to use it: | | Math class | Machine learning | | ------------------ | ----------------------------- | -------------------------------------- | | **What you have:** | `m`, `b`, and one `x` | A pile of `(x, y)` pairs | | **What you want:** | The `y` that matches that `x` | Good values of `m` and `b` | | **The work:** | Plug in. Compute. | Search. Try values. Adjust. Try again. | Same equation. Different question. Everything else in this field — every neural net, every LLM, every image model — is a fancier version of this exact flip. The function gets more complicated. The data gets more complicated. But the question stays the same: *given examples, find the rule.* --- ## Sidebar: what's "fixed" vs what "changes" during learning This trips everyone up the first time. The four symbols in `y = mx + b` play two very different roles: | Symbol | Role | Where it comes from | Does it change during training? | | ------ | ------------- | --------------------- | ------------------------------- | | `x` | input | given by the data | **no** — frozen | | `y` | output | given by the data | **no** — frozen | | `m` | **parameter** | started as a guess | **yes** — search variable | | `b` | **parameter** | started as a guess | **yes** — search variable | Learning = the `(x, y)` pairs sit still, and we move `m` and `b` around until predictions land near the actual y's. When someone says "a 7-billion parameter model," that 7B is the count of `m`-and-`b`-like things. The inputs aren't part of "the model." **The model IS its parameters.** --- ## So what's a neural network? The simplest possible neural network is `y = mx + b`. The smallest unit (called a **neuron**) is literally a line. We only change two things, both cosmetic: | Math-class name | ML name | Same thing? | | --------------- | ---------- | -------------- | | slope `m` | weight `w` | yes, identical | | intercept `b` | bias `b` | yes, identical | So `y = mx + b` and `y = wx + b` are the same equation. ML people just renamed it. Why "neuron"? Mostly historical baggage — 1950s researchers loosely analogized multiply-and-add to brain cells "weighing" incoming signals. **Don't take the brain metaphor seriously.** A neuron is multiply-and-add. That's it. When you hear "neural network," picture **a line being fit to points.** Everything more elaborate is built by stacking and twisting this base case. --- ## What's NOT in this picture yet Deliberately left out — named without teaching: - **More than one input** (price depends on sqft AND bedrooms): just `price = w₁·sqft + w₂·bedrooms + b`. Still a "line" in higher dimensions. → future session. - **More than one output**: multiple lines in parallel. → future session. - **The data isn't actually linear** (price grows fast, then plateaus): one line can't capture this, which is why "non-linearities" exist. → future session. - **How do you actually search for `m` and `b`?** Gradient descent. → next session. --- ## Self-check Don't peek. Answer in your own words: 1. What does it mean to "learn" `m` and `b`? 2. If I hand you `y = 3x + 1`, is that a trained model or untrained? Why? 3. If you had just ONE data point — `(sqft=1000, price=250)` — could you find `m` and `b`? Why or why not? Question 3 has a punchline: **with N parameters, you generally need at least N data points to pin them down.** Modern LLMs have hundreds of billions of parameters — and accordingly need oceans of data. That's the data race. --- ## Where this connects - Single neuron = **linear regression** (200 years old; Gauss & Legendre, early 1800s). The "neural network" framing is mostly historical branding on top of "stack many linear regressions with non-linear bumps between them." See `journal.md` Entry 1. - The dense, vectors-and-matrices version of this material lives in `archive/session-01-neural-net-basics-dense.md` — same ideas, faster pace, assumes more math fluency. - Next: **Session 2** — how the search for `m` and `b` actually works (gradient descent, done properly).