Gradient Checkpointing and Activation Recomputation

The Problem

Training a transformer stores, for each layer, the inputs to every op that is differentiated in backward: the attention inputs, the Q/K/V projections, the softmax output, the FFN inputs, the norm outputs, and the residual stream. For a layer with hidden size d, sequence length L, batch B, this is on the order of 12 * B * L * d floats per layer.

For d=8192, L=8192, B=1, that's about 805M values, or 1.6 GB/layer in BF16. A 64-layer model is about 103 GB of activations, and that's before you multiply by microbatch size, before you add attention-softmax intermediates (L^2 per head), and before you factor tensor-parallel partial copies.

The two-sided bill: BF16 weights plus optimizer state might fit in 80GB, but activations push you past. Gradient checkpointing (aka activation recomputation) is the standard fix. Drop most activations; redo the forward during backward to get them back. Cost: extra FLOPs. Benefit: memory drops by the ratio of checkpoint segments to total layers.

Done naively, checkpointing costs roughly 33% more forward-pass FLOPs per step. Done well — selective checkpointing per the "smart selection" of Korthikanti et al. — you save 5x memory for under 5% FLOP overhead. And with FP8 matmuls, FSDP offload, and expert-parallel MoE this really matters: you can't afford either the memory or the wasted compute.

The Concept

What Backward Actually Needs

output = layer(input). Backward wants grad_input and grad_params. To compute them it needs:

• input (to compute grad_params = input.T @ grad_output for linear layers)
• some activation derivative intermediates (the derivative of ReLU/GELU/softmax depends on the activation value)

The forward pass stores these automatically in the autograd graph. Every tensor.retain_grad() and every op that needs its input retains a reference.

Naive Full Checkpointing

Split the network into N segments. During forward, store only the input to each segment. When backward needs intermediates, rerun the segment's forward pass to materialize them, then differentiate.

Example: 32-layer transformer split into 32 segments of 1 layer each.

• Memory: 32 layer-inputs (small) vs 32 * (activation volume per layer) (huge).
• Extra compute: 1 extra forward per segment, i.e., ~33% more forward FLOPs total (since backward is 2x forward, full step becomes 1 + 1 + 2 = 4 units instead of 1 + 2 = 3).

This is the original Chen et al. 2016 recipe: one checkpoint every sqrt(L) layers to balance memory and compute. For $L=64$, that's 8 checkpoints.

Selective Checkpointing (Korthikanti 2022)

Not all activations cost the same. The attention softmax output is B*L*L*heads and grows quadratically with sequence length. The FFN hidden activation is B*L*4d and grows linearly. For long sequences the softmax dominates.

Selective checkpointing keeps the cheap-to-store activations (linear projections, residuals) and recomputes only the expensive ones (attention). You pay minimal FLOPs to recompute but save the $O(L^2)$ memory.

Megatron-Core implements this as "selective" activation recomputation. Used in most 2024+ frontier training runs.

Offload

Alternative to recompute: ship activations to CPU RAM between forward and backward. Requires PCIe bandwidth; beneficial when idle bandwidth exceeds the cost of rematerialization. Mixed strategies are common: checkpoint some layers, offload others.

FSDP2 ships offload as a first-class option. Offload shines when GPU is bottlenecked on memory but CPU-GPU transfer has headroom.

Recompute Cost Model

Per-step FLOPs with naive checkpointing every k layers out of L:

flops_fwd_normal = L * f_layer
flops_bwd_normal = 2 * L * f_layer
flops_total_normal = 3 * L * f_layer

flops_fwd_ckpt = L * f_layer
flops_recompute = L * f_layer  # one extra forward per layer in the segment
flops_bwd_ckpt = 2 * L * f_layer
flops_total_ckpt = 4 * L * f_layer
overhead = 4 / 3 - 1 = 0.33 = 33%

With selective checkpointing you recompute only the attention kernel, not the whole layer:

flops_recompute_selective = L * f_attention ~= L * f_layer * 0.15
overhead_selective = (3 + 0.15) / 3 - 1 = 0.05 = 5%

Memory Savings Model

Activation volume per layer: A. For L layers, total activation memory: L * A.

Full checkpoint (segment size 1): store only L * input_volume (~`L * 1/10 A for a standard transformer). Saves ~9 * L * A * 1/10`.

Checkpoint every k layers: store L/k * A plus k-1 layers' worth within the active segment.

At k = sqrt(L), memory and recompute cost both scale with sqrt(L) — the optimal tradeoff for uniform-cost layers.

When Not to Checkpoint

• The innermost layers of a pipeline stage already in-flight. They have to finish anyway.
• The first and last layers if they dominate the stage's compute (rare in transformers).
• Attention kernels already using FlashAttention — Flash already recomputes the softmax fast, so additional layer-level checkpointing adds little on top.

Implementation Patterns

1. Function wrapper: wrap a segment in torch.utils.checkpoint.checkpoint(fn, input). PyTorch stores only input, recomputes everything else on backward.

2. Decorator-based: label layers as checkpointable; the trainer decides at config time which segments get wrapped.

3. Manual explicit recompute: write the backward pass yourself, calling a custom recompute_forward that duplicates the forward with the stored input.

All three give the same functional result. Wrappers are the standard idiom.

Interaction with TP / PP / FP8

• Tensor parallel: checkpoint inputs must be gathered or rescattered on recompute; handle the communication cost.
• Pipeline parallel: typical pattern is to checkpoint each pipeline-stage's forward so reverse-order microbatches can reuse activation memory.
• FP8 recompute: amax histories updated during recompute must match the original forward's, or the FP8 scale drifts. Most frameworks snapshot the scale.

activation-recompute

Build It

Step 1: A Toy Model With Segments

import numpy as np


def linear_forward(x, w, b):
    return x @ w + b


def relu(x):
    return np.maximum(x, 0)


def layer_forward(x, w1, b1, w2, b2):
    h = relu(linear_forward(x, w1, b1))
    return linear_forward(h, w2, b2)


def model_forward(x, params):
    activations = [x]
    h = x
    for w1, b1, w2, b2 in params:
        h = layer_forward(h, w1, b1, w2, b2)
        activations.append(h)
    return h, activations

Step 2: Naive Backward Needing All Activations

def model_backward(grad_output, activations, params):
    grads = [None] * len(params)
    g = grad_output
    for i in range(len(params) - 1, -1, -1):
        w1, b1, w2, b2 = params[i]
        x_in = activations[i]
        h_pre = linear_forward(x_in, w1, b1)
        h = relu(h_pre)
        gh = g @ w2.T
        gw2 = h.T @ g
        gb2 = g.sum(axis=0)
        g_pre = gh * (h_pre > 0)
        gx = g_pre @ w1.T
        gw1 = x_in.T @ g_pre
        gb1 = g_pre.sum(axis=0)
        grads[i] = (gw1, gb1, gw2, gb2)
        g = gx
    return g, grads

Step 3: Checkpoint-Every-k Memory

def model_forward_checkpointed(x, params, k=4):
    saved_inputs = [x]
    h = x
    for i, (w1, b1, w2, b2) in enumerate(params):
        h = layer_forward(h, w1, b1, w2, b2)
        if (i + 1) % k == 0:
            saved_inputs.append(h)
    return h, saved_inputs


def model_backward_checkpointed(grad_output, saved_inputs, params, k=4):
    grads = [None] * len(params)
    g = grad_output
    segments = [(j * k, min((j + 1) * k, len(params))) for j in range(len(saved_inputs))]
    for seg_idx in range(len(saved_inputs) - 1, -1, -1):
        start, end = segments[seg_idx]
        if start >= end:
            continue
        x_in = saved_inputs[seg_idx]
        _, seg_acts = model_forward(x_in, params[start:end])
        g, seg_grads = model_backward(g, seg_acts, params[start:end])
        for j, gr in enumerate(seg_grads):
            grads[start + j] = gr
    return g, grads

Step 4: Cost Model

def checkpoint_cost(n_layers, segment_size, flops_per_layer=1.0):
    fwd = n_layers * flops_per_layer
    recompute = n_layers * flops_per_layer
    bwd = 2 * n_layers * flops_per_layer
    return {
        "fwd": fwd,
        "recompute": recompute,
        "bwd": bwd,
        "total": fwd + recompute + bwd,
        "overhead_vs_no_ckpt": (fwd + recompute + bwd) / (fwd + bwd) - 1.0,
    }


def selective_checkpoint_cost(n_layers, attention_fraction=0.15,
                              flops_per_layer=1.0):
    fwd = n_layers * flops_per_layer
    recompute = n_layers * attention_fraction * flops_per_layer
    bwd = 2 * n_layers * flops_per_layer
    return {
        "fwd": fwd,
        "recompute": recompute,
        "bwd": bwd,
        "total": fwd + recompute + bwd,
        "overhead_vs_no_ckpt": (fwd + recompute + bwd) / (fwd + bwd) - 1.0,
    }

Step 5: Memory Estimator

def activation_memory_mb(n_layers, hidden=8192, seq=8192,
                        batch=1, bytes_per_value=2):
    per_layer = 12 * batch * seq * hidden * bytes_per_value
    return n_layers * per_layer / 1e6


def memory_after_checkpoint(n_layers, segment_size, hidden=8192,
                           seq=8192, batch=1, bytes_per_value=2):
    n_seg = max(1, n_layers // segment_size)
    saved = (n_seg + segment_size) * 1 * batch * seq * hidden * bytes_per_value
    return saved / 1e6

Step 6: Optimal Segment Size

def optimal_segment(n_layers):
    return int(round(np.sqrt(n_layers)))

Step 7: Selective Checkpoint Decision

def should_recompute(layer_type, activation_bytes, recompute_flops_ratio):
    if layer_type == "attention" and activation_bytes > 100 * 1e6:
        return True
    if layer_type == "ffn" and activation_bytes > 500 * 1e6:
        return recompute_flops_ratio < 0.1
    return False

Use It

• torch.utils.checkpoint: from torch.utils.checkpoint import checkpoint — the canonical wrapper in PyTorch. Wraps a function; stores only inputs, recomputes on backward.
• Megatron-Core activation recomputation: supports selective, full, and block modes. Standard in 2024+ frontier training.
• FSDP2 offload: module.to_empty(device="cpu") with offload_policy in FSDP2 shards activations to CPU instead of recomputing.
• DeepSpeed ZeRO-Offload: CPU offload for optimizer states and activations, complementing checkpointing.

Ship It

This lesson produces outputs/prompt-activation-recompute-policy.md — a prompt that takes your model config (layers, hidden, seq, batch) and available GPU memory and emits a per-layer recompute policy (none / selective / full / offload).

Exercises

1. Verify correctness. Run model_forward + model_backward (full activations) vs model_forward_checkpointed + model_backward_checkpointed (segments). Parameter gradients must be identical to machine precision.

2. Sweep segment size k from 1 to L. Plot FLOP overhead and memory. Find the knee of the curve.

3. Implement selective checkpointing: store the attention-module input but not its intermediates. Measure the FLOP overhead vs full-layer checkpointing for a 32-layer model at seq=8192.

4. Add offload. Save segment inputs to a simulated "CPU buffer" (a separate list). Measure "PCIe bandwidth" as bytes/time and find the breakeven point between offload and recompute.

5. Benchmark a real PyTorch transformer with and without torch.utils.checkpoint. Measure memory (via torch.cuda.max_memory_allocated) and step time.

Key Terms

Further Reading

• Chen et al., 2016 -- "Training Deep Nets with Sublinear Memory Cost" -- the original paper that formalized gradient checkpointing
• Korthikanti et al., 2022 -- "Reducing Activation Recomputation in Large Transformer Models" -- selective activation recomputation and the formal cost analysis
• Pudipeddi et al., 2020 -- "Training Large Neural Networks with Constant Memory using a New Execution Algorithm" -- alternative constant-memory approach via reverse-mode rematerialization
• Ren et al., 2021 -- "ZeRO-Offload: Democratizing Billion-Scale Model Training" -- activation offload at scale
• PyTorch torch.utils.checkpoint docs -- the standard API
• Megatron Bridge activation recomputation documentation -- selective, full, and block modes