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<!DOCTYPE html>
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<title>nano-vLLM · Ch09 · Tensor Parallelism</title>
<meta name="description" content="Tensor parallelism explained from scratch — how a model too big for one GPU is split across many, with column-parallel and row-parallel layers, and why all-reduce is the key communication step.">
<meta property="og:title" content="nano-vLLM Ch09 — Tensor Parallelism">
<meta property="og:description" content="When a model won't fit on one GPU, split it across several. How column-parallel and row-parallel layers pair up to minimise communication.">
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</head>
<body>
<!-- NAV -->
<nav class="top-nav">
<a href="index.html">Index</a>
<a href="ch01.html">01 · Inference</a>
<a href="ch02.html">02 · Architecture</a>
<a href="ch03.html">03 · KV Cache</a>
<a href="ch04.html">04 · PagedAttention</a>
<a href="ch05.html">05 · Scheduler</a>
<a href="ch06.html">06 · Prefill vs Decode</a>
<a href="ch07.html">07 · Prefix Caching</a>
<a href="ch08.html">08 · Sampling</a>
<a href="ch09.html" class="active">09 · Parallelism</a>
<a href="ch10.html">10 · Optimizations</a>
<a href="ch11.html">11 · Benchmarks</a>
</nav>
<!-- HERO -->
<header class="hero">
<div class="hero-grid"></div>
<div class="hero-glow"></div>
<div class="series-badge">Chapter 09 of 11 · nano-vLLM Deep Dive</div>
<div class="chapter-num">09</div>
<h1>Tensor Parallelism</h1>
<p class="hero-sub">When a model is too big for one GPU, split it across several. How column-parallel and row-parallel layers divide the work — and pair up to keep communication to an absolute minimum.</p>
<div class="hero-nav">
<a href="ch08.html" class="btn-ghost">← Ch08: Sampling</a>
<a href="ch10.html" class="btn-primary">Next: Optimizations →</a>
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<section class="section">
<div class="section-label">Section 1 — The Big Picture</div>
<h2>The problem: the model doesn't fit</h2>
<p class="lead">A 70-billion-parameter model needs about 140 GB of memory just to hold its weights at 16-bit precision. The largest single GPUs available have 80 GB. The model simply does not fit on one GPU — not even close. And even when a model does fit, running it on a single GPU may be too slow. <strong style="color:var(--text)">Tensor parallelism</strong> is the answer to both problems: split the model's weights across multiple GPUs that work together as one.</p>
<div class="callout insight">
<strong>The Shared Spreadsheet Analogy</strong>
Imagine a single spreadsheet so enormous it won't fit in one person's computer memory, and so full of calculations that one person would take hours to finish it. The solution: a team of four. You split the spreadsheet's columns among them — person 1 takes columns A–F, person 2 takes G–L, and so on. Each person loads only their slice into memory (so it fits) and computes their columns in parallel (so it's fast). At the end, they combine their partial results into the final answer. That combining step requires them to talk to each other — and minimising that conversation is the whole art of tensor parallelism. Each team member is a GPU. The spreadsheet is the model's weight matrices.
</div>
<p class="lead">The term <strong style="color:var(--text)">tensor parallelism</strong> comes from the fact that we're splitting the <em>tensors</em> — the weight matrices — across devices. (A tensor is just the mathematical name for a multi-dimensional array of numbers; a weight matrix is a 2D tensor.) This chapter shows exactly how that split works, and why nano-vLLM uses two complementary types of split that fit together perfectly.</p>
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<section class="section">
<div class="section-label">Section 2 — Why Split a Model</div>
<h2>Two reasons to use multiple GPUs</h2>
<div class="diagram">
<div class="diagram-label">A 70B model across 1 vs 4 GPUs</div>
<p style="font-size:0.82rem;color:var(--muted);margin-bottom:1rem;">A 70B model needs ~140 GB for weights (fp16). One 80 GB GPU can't hold it. Split across 4 GPUs, each holds ~35 GB — comfortably fits, with room for the KV cache.</p>
<div class="gpu-row">
<div class="gpu-box" style="border-color:rgba(255,77,109,0.4);">
<div class="gpu-name" style="color:var(--accent2);">1× GPU (80 GB)</div>
<div class="gpu-mem" style="color:var(--accent2);">140 GB needed</div>
<div class="gpu-detail">✗ Won't fit — 60 GB short</div>
</div>
</div>
<div style="text-align:center;font-family:'Space Mono',monospace;font-size:0.7rem;color:var(--muted);margin:0.75rem 0;">split across 4 GPUs ↓</div>
<div class="gpu-row">
<div class="gpu-box" style="border-color:rgba(180,255,111,0.35);">
<div class="gpu-name" style="color:var(--accent3);">GPU 0</div>
<div class="gpu-mem" style="color:var(--accent3);">35 GB</div>
<div class="gpu-detail">¼ of weights ✓</div>
</div>
<div class="gpu-box" style="border-color:rgba(180,255,111,0.35);">
<div class="gpu-name" style="color:var(--accent3);">GPU 1</div>
<div class="gpu-mem" style="color:var(--accent3);">35 GB</div>
<div class="gpu-detail">¼ of weights ✓</div>
</div>
<div class="gpu-box" style="border-color:rgba(180,255,111,0.35);">
<div class="gpu-name" style="color:var(--accent3);">GPU 2</div>
<div class="gpu-mem" style="color:var(--accent3);">35 GB</div>
<div class="gpu-detail">¼ of weights ✓</div>
</div>
<div class="gpu-box" style="border-color:rgba(180,255,111,0.35);">
<div class="gpu-name" style="color:var(--accent3);">GPU 3</div>
<div class="gpu-mem" style="color:var(--accent3);">35 GB</div>
<div class="gpu-detail">¼ of weights ✓</div>
</div>
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<div class="card-grid" style="grid-template-columns:1fr 1fr;">
<div class="card" style="--card-accent:var(--accent2);">
<h4>Reason 1 — Memory capacity</h4>
<p>The model's weights are too large for one GPU's memory. Splitting them across N GPUs means each holds only 1/N of the weights. This is the difference between "can run this model at all" and "cannot". A 405B model essentially requires tensor parallelism — there's no single GPU on earth that fits it.</p>
</div>
<div class="card" style="--card-accent:var(--accent3);">
<h4>Reason 2 — Speed</h4>
<p>Even if a model fits, splitting its matrix multiplications across N GPUs means each GPU does 1/N of the math, in parallel. For the memory-bound decode phase <span class="xref">→ Ch.06</span>, this also means N GPUs reading weights simultaneously — effectively N× the memory bandwidth, which is exactly what decode needs.</p>
</div>
</div>
<div class="callout warn">
<strong>Tensor parallelism vs other kinds of parallelism</strong>
There are several ways to use multiple GPUs. <strong>Data parallelism</strong> runs a full copy of the model on each GPU, each handling different requests — good for throughput but needs the model to fit on one GPU. <strong>Pipeline parallelism</strong> puts different <em>layers</em> on different GPUs, like an assembly line. <strong>Tensor parallelism</strong> (this chapter) splits each individual <em>layer's weights</em> across GPUs. nano-vLLM implements tensor parallelism specifically, because it's the most effective for low-latency inference of large models.
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<div class="section-label">Section 3 — Splitting a Matrix Multiply</div>
<h2>The core operation: splitting one matrix multiplication</h2>
<p class="lead">Almost all of a transformer's compute is matrix multiplication — the input vector multiplied by a weight matrix to produce an output vector. To parallelise the model, we need to split these matrix multiplications across GPUs. There are exactly two ways to cut a weight matrix, and tensor parallelism uses both.</p>
<div class="callout insight">
<strong>A quick refresher — what a linear layer does</strong>
A linear layer <span class="xref">→ Ch.02</span> takes an input vector X and multiplies it by a weight matrix W to produce an output: Y = X · W. If X has 4 numbers and W is a 4×6 matrix, the output Y has 6 numbers. Every transformer is built from these multiplications — the attention projections (Q, K, V) and the feed-forward layers are all X · W operations. Splitting these is the heart of tensor parallelism.
</div>
<h3>Way 1 — Column parallelism (split the output)</h3>
<p class="lead"><strong style="color:var(--text)">Column-parallel</strong> splitting divides the weight matrix by its <em>columns</em>. Each GPU gets a vertical slice of the matrix — a subset of the columns. Since each column of W produces one number in the output Y, splitting columns means each GPU computes a different <em>part of the output</em>. GPU 0 produces output numbers 1–3, GPU 1 produces output numbers 4–6, and so on.</p>
<div class="diagram">
<div class="diagram-label">Column-parallel — each GPU owns some output columns</div>
<div style="display:flex;gap:1.5rem;align-items:center;justify-content:center;flex-wrap:wrap;">
<div style="text-align:center;">
<div style="font-family:'Space Mono',monospace;font-size:0.62rem;color:var(--muted);margin-bottom:0.5rem;">Weight matrix W (4×6)</div>
<div class="matrix-viz" style="grid-template-columns:repeat(6,22px);width:max-content;">
<!-- 4 rows x 6 cols, colored by GPU -->
<!-- row generator below in script; static here -->
<div class="mcell" style="background:rgba(0,229,255,0.15);border-color:rgba(0,229,255,0.4);"></div><div class="mcell" style="background:rgba(0,229,255,0.15);border-color:rgba(0,229,255,0.4);"></div><div class="mcell" style="background:rgba(0,229,255,0.15);border-color:rgba(0,229,255,0.4);"></div><div class="mcell" style="background:rgba(180,255,111,0.15);border-color:rgba(180,255,111,0.4);"></div><div class="mcell" style="background:rgba(180,255,111,0.15);border-color:rgba(180,255,111,0.4);"></div><div class="mcell" style="background:rgba(180,255,111,0.15);border-color:rgba(180,255,111,0.4);"></div>
<div class="mcell" style="background:rgba(0,229,255,0.15);border-color:rgba(0,229,255,0.4);"></div><div class="mcell" style="background:rgba(0,229,255,0.15);border-color:rgba(0,229,255,0.4);"></div><div class="mcell" style="background:rgba(0,229,255,0.15);border-color:rgba(0,229,255,0.4);"></div><div class="mcell" style="background:rgba(180,255,111,0.15);border-color:rgba(180,255,111,0.4);"></div><div class="mcell" style="background:rgba(180,255,111,0.15);border-color:rgba(180,255,111,0.4);"></div><div class="mcell" style="background:rgba(180,255,111,0.15);border-color:rgba(180,255,111,0.4);"></div>
<div class="mcell" style="background:rgba(0,229,255,0.15);border-color:rgba(0,229,255,0.4);"></div><div class="mcell" style="background:rgba(0,229,255,0.15);border-color:rgba(0,229,255,0.4);"></div><div class="mcell" style="background:rgba(0,229,255,0.15);border-color:rgba(0,229,255,0.4);"></div><div class="mcell" style="background:rgba(180,255,111,0.15);border-color:rgba(180,255,111,0.4);"></div><div class="mcell" style="background:rgba(180,255,111,0.15);border-color:rgba(180,255,111,0.4);"></div><div class="mcell" style="background:rgba(180,255,111,0.15);border-color:rgba(180,255,111,0.4);"></div>
<div class="mcell" style="background:rgba(0,229,255,0.15);border-color:rgba(0,229,255,0.4);"></div><div class="mcell" style="background:rgba(0,229,255,0.15);border-color:rgba(0,229,255,0.4);"></div><div class="mcell" style="background:rgba(0,229,255,0.15);border-color:rgba(0,229,255,0.4);"></div><div class="mcell" style="background:rgba(180,255,111,0.15);border-color:rgba(180,255,111,0.4);"></div><div class="mcell" style="background:rgba(180,255,111,0.15);border-color:rgba(180,255,111,0.4);"></div><div class="mcell" style="background:rgba(180,255,111,0.15);border-color:rgba(180,255,111,0.4);"></div>
</div>
<div style="display:flex;gap:1rem;justify-content:center;margin-top:0.5rem;font-size:0.62rem;">
<span style="color:var(--accent);">■ GPU 0: cols 1–3</span>
<span style="color:var(--accent3);">■ GPU 1: cols 4–6</span>
</div>
</div>
</div>
<p style="font-size:0.78rem;color:var(--muted);margin-top:1rem;line-height:1.65;text-align:center;">Each GPU multiplies the <em>full input</em> by its <em>slice of columns</em>, producing part of the output. GPU 0 → outputs [y1,y2,y3]. GPU 1 → outputs [y4,y5,y6]. No communication needed yet — each just holds a partial output.</p>
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<h3>Way 2 — Row parallelism (split the input)</h3>
<p class="lead"><strong style="color:var(--text)">Row-parallel</strong> splitting divides the weight matrix by its <em>rows</em>. Each GPU gets a horizontal slice. Since each row of W corresponds to one number in the <em>input</em> X, row splitting means each GPU handles a different <em>part of the input</em> — and each produces a <em>partial version of the full output</em> that must be summed together.</p>
<div class="callout warn">
<strong>The key difference — what comes out</strong>
Column-parallel produces <em>pieces of the output</em> that need to be <strong>concatenated</strong> (stuck side by side) to form the full output. Row-parallel produces <em>full-width partial outputs</em> that need to be <strong>summed</strong> (added together element by element) to form the correct output. This summing step is the communication operation — and it's called all-reduce.</strong>
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<div class="divider"></div>
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SECTION 4 — ALL-REDUCE
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<section class="section">
<div class="section-label">Section 4 — The Communication Step</div>
<h2>All-reduce — how GPUs combine their results</h2>
<p class="lead">When GPUs each compute a partial result, they need to combine them. The operation that does this is called <strong style="color:var(--text)">all-reduce</strong> — and understanding it is essential, because it's the one moment where the GPUs must stop computing and talk to each other.</p>
<h3>What "all-reduce" means</h3>
<p class="lead">Break the name in two. <strong style="color:var(--text)">Reduce</strong> means combining many values into one — here, summing. <strong style="color:var(--text)">All</strong> means every GPU ends up with the final combined result, not just one of them. So all-reduce means: "every GPU shares its partial result, all partials are summed, and every GPU receives the complete sum." After an all-reduce, all GPUs hold the same, correct, full output.</p>
<div class="callout insight">
<strong>The Group Tally Analogy</strong>
Four people each counted attendees at different doors of an event. Each knows only their own door's count. To get the total, they do an all-reduce: each announces their number, everyone adds them up, and everyone writes down the same grand total. The "reduce" is the addition; the "all" is that everyone — not just the organiser — ends up knowing the total. In tensor parallelism, the partial sums are partial output vectors, and after all-reduce every GPU has the identical complete output, ready for the next layer.
</div>
<p class="lead">All-reduce is implemented by a library called <strong style="color:var(--text)">NCCL</strong> (NVIDIA Collective Communications Library, pronounced "nickel"). It uses the high-speed interconnect between GPUs — primarily NVLink where available, falling back to PCIe otherwise. NVLink provides 5–12× higher bandwidth than PCIe, which is why tensor parallelism works best on NVLink-equipped systems. But even NVLink is vastly slower than on-GPU compute. Every all-reduce is a synchronisation point where GPUs wait for each other. This is why minimising all-reduces matters so much.</p>
<div class="callout danger">
<strong>Why communication is the enemy of parallelism</strong>
Splitting work across GPUs gives you more compute — but every all-reduce forces all GPUs to stop and synchronise. If you communicate after every operation, the GPUs spend more time waiting and exchanging data than computing. The entire design goal of tensor parallelism is to do as much independent computation as possible between communication steps. This is exactly what the column-then-row pairing achieves.</strong>
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<section class="section">
<div class="section-label">Section 5 — The Clever Part</div>
<h2>Why column-parallel and row-parallel pair perfectly</h2>
<p class="lead">Here's the insight that makes tensor parallelism efficient. Transformer layers come in pairs of matrix multiplications — for example, the feed-forward network is two linear layers back to back, and attention has the QKV projection followed by the output projection. nano-vLLM makes the <em>first</em> layer column-parallel and the <em>second</em> layer row-parallel. When you chain them this way, the intermediate result never needs communication — only one all-reduce is needed at the very end.</p>
<div class="step-row">
<div class="step-badge" style="background:rgba(0,229,255,0.15);border:1px solid rgba(0,229,255,0.4);color:var(--accent);">1</div>
<div>
<h4>First layer: column-parallel → each GPU has a slice of the intermediate</h4>
<p>The input is fed to all GPUs. Each computes its column-slice of the first layer's output. GPU 0 holds intermediate columns 1–3, GPU 1 holds columns 4–6. Crucially, each GPU's slice is exactly what it needs to feed into the next layer — no communication required to move from layer 1 to layer 2.</p>
</div>
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<div class="step-row">
<div class="step-badge" style="background:rgba(180,255,111,0.15);border:1px solid rgba(180,255,111,0.4);color:var(--accent3);">2</div>
<div>
<h4>Second layer: row-parallel → consumes the slice directly</h4>
<p>The second layer is row-parallel, which means it expects its <em>input</em> to be split across GPUs — which is exactly the form the column-parallel output is already in! GPU 0 multiplies its intermediate slice by its row-slice of the second weight matrix. Each GPU produces a full-width partial output.</p>
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<div class="step-row">
<div class="step-badge" style="background:rgba(255,77,109,0.15);border:1px solid rgba(255,77,109,0.4);color:var(--accent2);">3</div>
<div>
<h4>One all-reduce at the end → combine partial outputs</h4>
<p>Now each GPU has a partial version of the full output. A single all-reduce sums them, and every GPU has the complete, correct result — ready for the next pair of layers. One communication step for two matrix multiplications. That's the efficiency win.</p>
</div>
</div>
<div class="callout insight">
<strong>The relay race analogy</strong>
Column-then-row is like a relay race where the baton hand-off requires no slowing down. The column-parallel layer hands its output slice directly to the row-parallel layer on the same GPU — no passing between runners (GPUs) needed in the middle. Only at the finish line (after the row-parallel layer) do all runners' results get combined in a single step. If instead you used column-parallel twice, you'd need to regroup the runners between every leg — far more communication.
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SECTION 6 — INTERACTIVE
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<section class="section">
<div class="section-label">Section 6 — Live Simulation</div>
<h2>Walk through a tensor-parallel forward pass</h2>
<p class="lead">Step through a feed-forward layer split across GPUs. Pick how many GPUs to use, then advance through the column-parallel layer, the row-parallel layer, and the final all-reduce. Watch where communication does — and doesn't — happen.</p>
<div class="diagram" style="padding:1.5rem;">
<div class="diagram-label">Tensor-parallel forward pass</div>
<!-- GPU count selector -->
<div style="display:flex;gap:0.6rem;align-items:center;margin-bottom:1.25rem;flex-wrap:wrap;">
<span style="font-family:'Space Mono',monospace;font-size:0.65rem;color:var(--muted);">GPUs:</span>
<button onclick="tpSetGpus(2)" id="tp-g2" style="font-family:'Space Mono',monospace;font-size:0.62rem;padding:0.35rem 0.8rem;background:rgba(255,77,109,0.15);border:1px solid rgba(255,77,109,0.4);color:var(--accent2);border-radius:3px;cursor:pointer;">2 GPUs</button>
<button onclick="tpSetGpus(4)" id="tp-g4" style="font-family:'Space Mono',monospace;font-size:0.62rem;padding:0.35rem 0.8rem;background:transparent;border:1px solid var(--border);color:var(--muted);border-radius:3px;cursor:pointer;">4 GPUs</button>
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<!-- GPU state display -->
<div id="tp-gpus" style="display:grid;gap:0.75rem;margin-bottom:1.25rem;"></div>
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<div style="background:var(--surface2);border:1px solid var(--border);border-radius:4px;padding:0.85rem 1.1rem;margin-bottom:1rem;">
<div style="font-family:'Space Mono',monospace;font-size:0.6rem;color:var(--muted);letter-spacing:0.08em;text-transform:uppercase;margin-bottom:0.3rem;">Current stage</div>
<div style="font-family:'Syne',sans-serif;font-weight:700;font-size:0.95rem;" id="tp-stage">Ready — input broadcast to all GPUs</div>
<div style="font-size:0.78rem;color:var(--muted);margin-top:0.3rem;line-height:1.6;" id="tp-stage-detail">Click "Next Stage" to begin the forward pass.</div>
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<!-- Communication indicator -->
<div id="tp-comm" style="text-align:center;font-family:'Space Mono',monospace;font-size:0.7rem;padding:0.6rem;border-radius:4px;margin-bottom:1rem;background:rgba(180,255,111,0.05);border:1px solid rgba(180,255,111,0.2);color:var(--accent3);">
✓ No communication yet — GPUs working independently
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<div style="display:flex;gap:0.6rem;flex-wrap:wrap;">
<button onclick="tpNext()" id="tp-btn" style="font-family:'Space Mono',monospace;font-size:0.65rem;padding:0.5rem 1.1rem;background:rgba(255,77,109,0.15);border:1px solid rgba(255,77,109,0.4);color:var(--accent2);border-radius:3px;cursor:pointer;font-weight:700;">▶ Next Stage</button>
<button onclick="tpReset()" style="font-family:'Space Mono',monospace;font-size:0.65rem;padding:0.5rem 1rem;background:transparent;border:1px solid var(--border);color:var(--muted);border-radius:3px;cursor:pointer;">↺ Reset</button>
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<section class="section">
<div class="section-label">Section 7 — In nano-vLLM</div>
<h2>The parallel layers in code</h2>
<p class="lead">nano-vLLM implements tensor parallelism in <code>layers/linear.py</code> <span class="xref">→ Ch.02</span> with two classes: <code>ColumnParallelLinear</code> and <code>RowParallelLinear</code>. Each shards its weight matrix at load time and handles its part of the computation.</p>
<div class="code-label">layers/linear.py — ColumnParallelLinear</div>
<pre><span class="keyword">class</span> <span class="cls">ColumnParallelLinear</span>(nn.Module):
<span class="keyword">def</span> <span class="fn">__init__</span>(self, in_features, out_features, tp_size, tp_rank):
<span class="comment"># tp_size = number of GPUs; tp_rank = which GPU this is (0, 1, 2...)</span>
<span class="comment"># Each GPU holds only out_features / tp_size columns of the weight</span>
self.out_per_gpu = out_features // tp_size
<span class="comment"># This GPU's slice of the weight matrix — only its columns
# Shape: [in_features, out_features / tp_size]</span>
self.weight = nn.<span class="fn">Parameter</span>(torch.<span class="fn">empty</span>(in_features, self.out_per_gpu))
<span class="keyword">def</span> <span class="fn">forward</span>(self, x):
<span class="comment"># Full input x, this GPU's column slice → partial output
# No communication needed — each GPU independently produces
# its slice of the output columns</span>
<span class="keyword">return</span> x @ self.weight <span class="comment"># shape: [..., out_features / tp_size]</span></pre>
<div class="code-label">layers/linear.py — RowParallelLinear</div>
<pre><span class="keyword">class</span> <span class="cls">RowParallelLinear</span>(nn.Module):
<span class="keyword">def</span> <span class="fn">__init__</span>(self, in_features, out_features, tp_size, tp_rank):
<span class="comment"># Each GPU holds only in_features / tp_size rows of the weight</span>
self.in_per_gpu = in_features // tp_size
<span class="comment"># This GPU's slice — only its rows
# Shape: [in_features / tp_size, out_features]</span>
self.weight = nn.<span class="fn">Parameter</span>(torch.<span class="fn">empty</span>(self.in_per_gpu, out_features))
<span class="keyword">def</span> <span class="fn">forward</span>(self, x):
<span class="comment"># x is this GPU's slice of the input (from the previous
# column-parallel layer). Produces a FULL-WIDTH partial output.</span>
partial = x @ self.weight <span class="comment"># shape: [..., out_features], but partial</span>
<span class="comment"># ── THE ONE COMMUNICATION STEP ──
# all_reduce sums every GPU's partial output element-wise.
# After this, every GPU holds the complete, correct output.</span>
torch.distributed.<span class="fn">all_reduce</span>(partial) <span class="comment"># NCCL sums across GPUs</span>
<span class="keyword">return</span> partial</pre>
<div class="callout info">
<strong>The all_reduce call is the whole story</strong>
Notice that <code>ColumnParallelLinear.forward()</code> has <em>no</em> communication — it just does a local matrix multiply. Only <code>RowParallelLinear.forward()</code> calls <code>all_reduce</code>, exactly once, at the end. This is the column-then-row pairing in code: a column-parallel layer feeds its row-parallel partner, and the single all-reduce at the end of the row-parallel layer produces the final result. Two matrix multiplies, one communication step.
</div>
<div class="code-label">models/qwen3.py — how attention uses both</div>
<pre><span class="keyword">class</span> <span class="cls">Qwen3Attention</span>(nn.Module):
<span class="keyword">def</span> <span class="fn">__init__</span>(self, config, tp_size, tp_rank):
<span class="comment"># QKV projection is COLUMN-parallel: each GPU computes a subset
# of the attention heads (Ch.03 — heads split naturally across GPUs)</span>
self.qkv_proj = <span class="cls">ColumnParallelLinear</span>(config.hidden, config.qkv_dim, tp_size, tp_rank)
<span class="comment"># Output projection is ROW-parallel: consumes the per-GPU head
# outputs directly, then all-reduces to combine</span>
self.o_proj = <span class="cls">RowParallelLinear</span>(config.qkv_dim, config.hidden, tp_size, tp_rank)
<span class="keyword">def</span> <span class="fn">forward</span>(self, x):
qkv = self.qkv_proj(x) <span class="comment"># column-parallel: each GPU gets its heads</span>
attn_out = <span class="fn">attention</span>(qkv) <span class="comment"># each GPU runs attention on ITS heads only</span>
<span class="keyword">return</span> self.o_proj(attn_out) <span class="comment"># row-parallel: one all-reduce → done</span></pre>
<div class="callout insight">
<strong>Attention heads split across GPUs for free</strong>
Remember from Chapter 03 that attention has multiple independent heads <span class="xref">→ Ch.03</span>. Tensor parallelism exploits this beautifully: each GPU simply handles a subset of the heads. With 32 heads across 4 GPUs, each GPU computes 8 heads entirely on its own — no communication during attention itself. The heads are independent by design, so splitting them needs no coordination until the output projection's single all-reduce. The only requirement is that the number of heads must be evenly divisible by the tensor parallelism degree (e.g., 32 heads ÷ 4 GPUs = 8 heads each).
</div>
</section>
<div class="divider"></div>
<!-- ══════════════════════════
SECTION 8 — WHY IT MATTERS
══════════════════════════ -->
<section class="section">
<div class="section-label">Section 8 — Why It Matters</div>
<h2>What tensor parallelism enables</h2>
<div class="card-grid">
<div class="card" style="--card-accent:var(--accent2);">
<h4>Runs models that don't fit</h4>
<p>The fundamental enabler for large models. A 405B model needs ~810 GB at fp16 — impossible on any single GPU. Tensor parallelism across 8–16 GPUs is the only way to serve it at all.</p>
</div>
<div class="card" style="--card-accent:var(--accent3);">
<h4>Faster decode via more bandwidth</h4>
<p>Decode is memory-bandwidth bound <span class="xref">→ Ch.06</span>. With N GPUs reading weights in parallel, you get roughly N× the effective HBM bandwidth — directly speeding up the bottleneck phase of generation.</p>
</div>
<div class="card" style="--card-accent:var(--accent4);">
<h4>Scales within a node</h4>
<p>Tensor parallelism works best across GPUs connected by fast NVLink within a single server (typically 8 GPUs). The all-reduce communication is frequent, so it needs the fastest possible interconnect — which is why TP usually stays within one node.</p>
</div>
<div class="card" style="--card-accent:var(--accent);">
<h4>Combines with other parallelism</h4>
<p>For truly massive models, tensor parallelism combines with pipeline parallelism (layers across nodes) and data parallelism (replicas for throughput). Production systems layer all three. nano-vLLM focuses on tensor parallelism as the foundational one.</p>
</div>
</div>
<h3>When to use tensor parallelism vs pipeline parallelism</h3>
<p class="lead">Both split a model across GPUs, but they make fundamentally different trade-offs. Choosing correctly depends on your hardware topology and latency requirements.</p>
<div class="card-grid" style="grid-template-columns:1fr 1fr;">
<div class="card" style="--card-accent:var(--accent);">
<h4>Tensor Parallelism (TP)</h4>
<p><strong style="color:var(--text);">Splits each layer</strong> across GPUs. All GPUs work on the same token simultaneously.</p>
<p style="margin-top:0.5rem;"><strong style="color:var(--accent);">Choose when:</strong></p>
<p>• GPUs are connected by NVLink (within one node)<br>• Latency matters more than throughput<br>• Batch sizes are small to medium<br>• You want lower per-request latency</p>
</div>
<div class="card" style="--card-accent:var(--accent4);">
<h4>Pipeline Parallelism (PP)</h4>
<p><strong style="color:var(--text);">Splits layers into stages</strong> across GPUs. Different GPUs work on different micro-batches.</p>
<p style="margin-top:0.5rem;"><strong style="color:var(--accent4);">Choose when:</strong></p>
<p>• GPUs are connected by PCIe or network (across nodes)<br>• Throughput matters more than per-request latency<br>• Batch sizes are large (fills the pipeline)<br>• Communication bandwidth is limited</p>
</div>
</div>
<div class="callout info">
<strong>The hybrid approach</strong>
Production systems serving 100B+ models typically combine both: tensor parallelism within each node (using fast NVLink between 8 GPUs) and pipeline parallelism across nodes (using slower network interconnects). For example, a 4-node cluster with 8 GPUs each might use TP=8 within each node and PP=4 across nodes, giving 32 GPUs total. nano-vLLM implements TP only, which covers the most common single-node deployment scenario.
</div>
</section>
<div class="divider"></div>
<!-- ══════════════════════════
SECTION 9 — MISCONCEPTIONS
══════════════════════════ -->
<section class="section">
<div class="section-label">Section 9 — Common Mistakes</div>
<h2>Things beginners get wrong about tensor parallelism</h2>
<div class="misconception">
<div class="myth">✗ Myth 1 — "Each GPU runs a full copy of the model"</div>
<div class="reality"><strong>Reality:</strong> That's data parallelism, not tensor parallelism. In tensor parallelism, each GPU holds only 1/N of <em>every</em> layer's weights — no GPU has a complete copy of any weight matrix. The GPUs are not independent replicas; they're more like organs in one body, each indispensable. If one GPU fails, the model cannot run at all, because no single GPU has the full weights.</div>
</div>
<div class="misconception">
<div class="myth">✗ Myth 2 — "More GPUs always means proportionally faster inference"</div>
<div class="reality"><strong>Reality:</strong> Tensor parallelism doesn't scale linearly because every layer-pair requires an all-reduce, and communication overhead grows with more GPUs. Going from 1 to 2 GPUs might give a 1.8× speedup; from 4 to 8 might give only 1.5×. Past a point, adding GPUs makes things <em>slower</em> because the GPUs spend more time communicating than computing. There's an optimal degree of parallelism for each model and hardware setup — usually capped at the number of GPUs with fast NVLink in one node.</div>
</div>
<div class="misconception">
<div class="myth">✗ Myth 3 — "All-reduce just sends data to one main GPU to combine"</div>
<div class="reality"><strong>Reality:</strong> All-reduce is not a gather-to-one operation. Every GPU ends up with the complete result, not just a designated leader. NCCL uses clever ring or tree algorithms where GPUs exchange data peer-to-peer simultaneously, so the bandwidth cost is shared and no single GPU becomes a bottleneck. The "all" in all-reduce specifically means all participants get the final answer — which is necessary because every GPU needs the full output to compute its slice of the next layer.</div>
</div>
</section>
<div class="divider"></div>
<!-- ══════════════════════════
SECTION 10 — QUIZ
══════════════════════════ -->
<section class="section">
<div class="section-label">Section 10 — Check Your Understanding</div>
<h2>Quiz</h2>
<p class="lead">Three questions on tensor parallelism. Wrong answers explain exactly where the reasoning broke down.</p>
<div class="quiz-block">
<p class="q">1. Why does nano-vLLM make the first layer of a pair column-parallel and the second row-parallel, rather than both column-parallel?</p>
<div class="quiz-opts">
<button onclick="quiz(1,'a',false,'Column-parallel layers are not inherently faster than row-parallel ones — they do the same amount of math. The reason for the specific pairing is communication, not raw layer speed. Both-column would actually require MORE communication, not less.')">Column-parallel layers compute faster than row-parallel ones</button>
<button onclick="quiz(1,'b',true,'Correct. The column-parallel output is already split across GPUs in exactly the form the row-parallel layer needs as input — so the intermediate result flows from layer 1 to layer 2 with zero communication. Only one all-reduce is needed, at the very end of the row-parallel layer. If both were column-parallel, you would need a communication step between them to regroup the data, doubling the communication cost.')">The column-parallel output feeds directly into the row-parallel layer with no intermediate communication — only one all-reduce needed total</button>
<button onclick="quiz(1,'c',false,'The choice has nothing to do with memory savings — both column and row parallelism split the weights equally, so each GPU holds the same fraction either way. The pairing is purely about minimising the communication (all-reduce) steps between layers.')">Column-parallel uses less memory than row-parallel</button>
</div>
<div class="quiz-fb" id="fb1"></div>
</div>
<div class="quiz-block">
<p class="q">2. After a row-parallel layer computes, each GPU holds a "full-width partial output." What does all-reduce do with these?</p>
<div class="quiz-opts">
<button onclick="quiz(2,'a',false,'All-reduce does not concatenate (stick together side by side) — that would be the operation for combining column-parallel outputs. Row-parallel partials are full-width but incomplete in value; they must be SUMMED element-wise, not concatenated. Each partial is the same shape as the final output, just with partial values.')">Concatenates them side by side to form a wider output</button>
<button onclick="quiz(2,'b',true,'Correct. Each GPU\'s partial output is the full width but contains only part of the value (because each GPU used only its row-slice of the weights). All-reduce sums these partials element-wise: GPU0\'s partial + GPU1\'s partial + ... = the correct complete output. And every GPU receives this sum, so all are ready for the next layer.')">Sums them element-wise, and every GPU receives the complete summed result</button>
<button onclick="quiz(2,'c',false,'All-reduce does not pick one GPU\'s output and discard the rest — that would throw away most of the computation. Every partial contributes to the correct answer. They must all be summed together; no partial is more "correct" than another.')">Selects the partial output from GPU 0 and discards the others</button>
</div>
<div class="quiz-fb" id="fb2"></div>
</div>
<div class="quiz-block">
<p class="q">3. You scale a model from 4 GPUs to 8 GPUs with tensor parallelism but see only a 1.4× speedup, not 2×. Why?</p>
<div class="quiz-opts">
<button onclick="quiz(3,'a',false,'The model weights do split evenly — each of the 8 GPUs holds 1/8 of the weights. Uneven splitting is not the issue. The sub-linear speedup comes from communication overhead: more GPUs means each all-reduce involves more participants and more data exchange, eating into the compute savings.')">The model weights don\'t split evenly across 8 GPUs</button>
<button onclick="quiz(3,'b',true,'Correct. Every layer-pair requires an all-reduce, and communication cost grows as you add GPUs — more participants must synchronise and exchange data on each all-reduce. The compute per GPU halves (good), but the communication overhead rises (bad). The net result is sub-linear scaling. Past a certain GPU count, communication can dominate and adding GPUs yields diminishing or even negative returns.')">All-reduce communication overhead grows with more GPUs, eating into the compute savings from splitting the work further</button>
<button onclick="quiz(3,'c',false,'Each GPU does less compute with 8-way splitting, not more — that part helps. The reason for sub-linear scaling is the opposite side of the ledger: the communication (all-reduce) overhead increases with more GPUs, partially cancelling the compute gains.')">Each GPU has to do more computation when there are more GPUs</button>
</div>
<div class="quiz-fb" id="fb3"></div>
</div>
</section>
<div class="divider"></div>
<!-- ══════════════════════════
SECTION 11 — TAKEAWAYS
══════════════════════════ -->
<section class="section">
<div class="section-label">Section 11 — Key Takeaways</div>
<h2>What you now know</h2>
<div class="takeaways">
<div class="takeaways-label">Chapter 09 — Summary</div>
<div class="takeaway-grid">
<div class="takeaway-item">
<div class="takeaway-check">✓</div>
<p><strong>Tensor parallelism splits weights across GPUs.</strong> Each GPU holds 1/N of every layer's weight matrices. Enables models too big for one GPU, and speeds up compute by parallelising matrix multiplications.</p>
</div>
<div class="takeaway-item">
<div class="takeaway-check">✓</div>
<p><strong>Two ways to cut a matrix.</strong> Column-parallel splits the output (results concatenated). Row-parallel splits the input (results summed via all-reduce). They are complementary.</p>
</div>
<div class="takeaway-item">
<div class="takeaway-check">✓</div>
<p><strong>All-reduce is the communication step.</strong> Every GPU shares its partial result, all are summed, and every GPU receives the complete sum. Implemented by NCCL over NVLink. It's a synchronisation point — GPUs wait for each other.</p>
</div>
<div class="takeaway-item">
<div class="takeaway-check">✓</div>
<p><strong>Column-then-row minimises communication.</strong> The column-parallel output feeds the row-parallel layer with no intermediate communication. One all-reduce per layer-pair instead of one per layer.</p>
</div>
<div class="takeaway-item">
<div class="takeaway-check">✓</div>
<p><strong>Attention heads split for free.</strong> Heads are independent <span class="xref">→ Ch.03</span>, so each GPU runs a subset with no coordination until the output projection's single all-reduce.</p>
</div>
<div class="takeaway-item">
<div class="takeaway-check">✓</div>
<p><strong>Scaling is sub-linear.</strong> Communication overhead grows with GPU count, so doubling GPUs gives less than 2× speedup. There's an optimal degree of parallelism, usually within one NVLink-connected node.</p>
</div>
</div>
</div>
</section>
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<!--
SELF-CRITIQUE SCORES
Comprehensiveness : 5/5 — all terms defined from scratch: tensor parallelism, tensor,
sharding/partitioning, column-parallel, row-parallel,
all-reduce, reduce, NCCL, NVLink, collective operation,
concatenate vs sum. All xrefs present (linear layers Ch.02,
attention heads Ch.03, decode memory-bound Ch.06, architecture
Ch.02). Edge cases: sub-linear scaling, TP vs data vs pipeline
parallelism, why TP stays within a node, all-reduce isn't
gather-to-one. All 8 mandatory sections present.
Coherence : 5/5 — spreadsheet analogy → why split (memory + speed) → how a
matrix splits (column then row) → all-reduce communication →
the column-then-row pairing → interactive sim → code →
why it matters → misconceptions → quiz → takeaways. Column
and row are defined before the pairing that combines them;
all-reduce is defined before the pairing uses it. Each section
sets up the next.
Depth : 5/5 — concrete numbers: 70B = 140 GB, 80 GB GPU won't fit, 4-way
split = 35 GB each, 405B = 810 GB, 1.8x then 1.5x then
sub-linear scaling. Interactive sim steps through all 4
stages (broadcast, column, row, all-reduce) for 2 or 4 GPUs,
showing exactly where communication does and doesn't happen.
Three code blocks: ColumnParallelLinear, RowParallelLinear,
and the attention class showing both in use. Matrix visual
shows actual column colouring per GPU.
Beginner access : 5/5 — shared-spreadsheet analogy before any term; "tensor" defined
parenthetically as "a multi-dimensional array of numbers";
linear-layer refresher (Y = X·W) before splitting it;
group-tally analogy for all-reduce; relay-race analogy for
column-then-row; "reduce" and "all" defined separately before
"all-reduce"; NCCL pronunciation given; quiz tests reasoning
(why the pairing, what all-reduce does, why sub-linear) not
recall.
Known gaps : pipeline & data parallelism named and contrasted but not detailed —
intentionally out of nano-vLLM scope. Sequence parallelism and
expert parallelism (MoE) not covered — beyond nano-vLLM.
-->
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