Memory Math

Last verified: 2026-07-13 — refresh when a new size-class anchor is validated or optimizer/dtype defaults change. Use it to give an agent explicit responsibilities, steps and constraints.

Editorial note: curated source snapshot published by Collider.club under the MIT License. Source attribution is preserved in the front matter.

Source snapshot

Last verified: 2026-07-13 — refresh when a new size-class anchor is validated or optimizer/dtype defaults change.

Memory Math

A worksheet for estimating whether a model size-class, method, and batch/pack combination fits available memory before a run. This is planning math, not a guarantee — leave headroom rather than sizing to the byte. Base models are never named here; every example is labeled by size class only (for example, "8B-class LoRA bf16"). See model-catalog.md for which actual model to use at a given size class.

The Four Terms

Total footprint ≈ weights + optimizer states + gradients + activations, plus a near-zero term for LoRA/QLoRA adapters. Work each term from parameter count and dtype, then sum.

1. Weights

params × bytes/param, by dtype:

This term dominates for full fine-tuning, and the calculation (params × bytes/param) is the same formula regardless of method — but the dtype, and so the result, is not: bf16 LoRA loads weights at 2 bytes/param while int4 QLoRA loads the same parameter count at 0.5 bytes/param, a 4x gap. Reuse the formula across methods; never reuse the resulting weight-memory number from one method's dtype for another's.

2. Optimizer states

Full fine-tuning carries optimizer state for every trainable parameter; LoRA and QLoRA carry it only for the adapter parameters, which is why this term is negligible for them regardless of base model size.

8-bit AdamW roughly quarters this term versus the fp32 variant for any run that isn't LoRA/QLoRA- adapter-only, where it's already negligible.

3. Gradients

Same dtype as compute precision — typically bf16, so 2 bytes/param — and, like optimizer state, only for trainable parameters. Full fine-tuning pays this for every weight; LoRA and QLoRA pay it only for the adapter, since frozen base weights never accumulate a gradient.

4. Activations

The hardest term to pin to a single number — it scales with batch size, sequence/packing length, and architecture, not just parameter count. Two levers matter more than exact estimation:

• Gradient checkpointing trades recompute for memory: expect roughly 30% savings on this term versus no checkpointing, at the cost of a recompute pass per checkpointed segment.
• Packing/sequence length is a more direct lever than batch size for this term.

5. LoRA/QLoRA adapter overhead

A rank-r adapter on a linear layer adds r × (in + out) parameters — A is r×in and B is out×r, so together they contribute r·in + r·out. At normal rank sizes (1–32 for RL, up to ~256 for SFT-at-scale), this is a small fraction of a percent of base model size — round it to zero in the worksheet unless an unusually high rank is in play.

Worked Examples

8B-class LoRA, bf16

Weights dominate; optimizer state and gradients are adapter-only and small.

params = 8e9
weights_gb = params * 2 / 1e9   # bf16, step 1
adapter_gb = 0.2                # step 5, negligible
total_gb = weights_gb + adapter_gb  # + activations
print(f"{total_gb:.0f}GB before activations")

Weights alone land around 16GB — the reference point for "an 8B-class model fits comfortably on a single high-memory GPU in bf16 LoRA."

8B-class QLoRA

Same parameter count, quantized weights:

params = 8e9
weights_gb = params * 0.5 / 1e9  # int4 NF4, step 1
adapter_gb = 0.2                 # step 5, negligible
total_gb = weights_gb + adapter_gb  # + activations
print(f"{total_gb:.0f}GB before activations")

Weights land around 4GB — roughly a 4x reduction versus bf16 LoRA, which is why QLoRA is the method that buys headroom for larger batch size or longer packing at the same size class, not just a way to fit bigger models.

70B-class QLoRA (≈40GB anchor)

params = 70e9
weights_gb = params * 0.5 / 1e9  # int4 NF4, step 1
adapter_gb = 0.5                 # step 5, negligible
total_gb = weights_gb + adapter_gb  # + activations
print(f"{total_gb:.0f}GB before activations")

The idealized formula lands weights at ≈35GB (decimal GB, weights only); treat ≈40GB as the real-world anchor once quantization metadata (NF4 double-quant constants) and runtime overhead are included — the reference point for "a 70B-class model is reachable via QLoRA, not bf16," where bf16 weights alone (≈140GB) would already exceed most single-device budgets before optimizer state, gradients, or activations are added. A plan estimating far above the ≈40GB anchor for the same size class is a signal to recheck dtype and method, not just add headroom.

Using These Numbers

1. Pick the size class and method from model-catalog.md.
2. Sum weights + optimizer + gradients from the tables above for that combination.
3. Add activations, applying the ~30% gradient- checkpointing saving if it's enabled.
4. Compare against the closest worked example or anchor above rather than trusting the estimate in isolation — a plan far off an anchor for the same size class and method is a signal to recheck inputs before assuming the hardware won't work.

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License

MIT License — Copyright (c) 2026 Collider.club. Full text: LICENSE · https://opensource.org/licenses/MIT