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Reverse-Mode Automatic Differentiation

Reverse-mode autodiff computes gradients by sending cotangents backward through a computation graph.

published · difficulty 3/5 · 14 min read

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Suppose one scalar loss depends on millions of parameters. Do you need to run one derivative computation for each parameter to know how to change them all?

Reverse-mode automatic differentiation is the bookkeeping trick that makes the answer no.

The previous idea, computation graphs, makes dependencies visible. Reverse-mode AD adds an execution rule: during the forward pass, record the primitive operations and the intermediate values they will need later. During the reverse pass, start from the final question, "how much does the loss change if this output changes?", and walk backward through the recorded operations. Each local backward rule converts an output sensitivity into input sensitivities.

The key advantage is shape. If one scalar loss depends on many parameters, reverse mode can compute all parameter gradients in one backward sweep through the graph. Forward mode would ask, one input direction at a time, how the output changes.

The useful mental model is a tape plus a register file. The tape remembers what primitive operations ran. The registers store cotangents such as aˉ\bar a and xˉ\bar x. The model breaks if you imagine symbolic simplification: reverse mode does not expand the formula by hand; it accumulates local contributions on the graph that actually ran.

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Math

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Equation 1Inspect equation object\bar v_i = (\partial L)/(\partial v_i).Equation 2Inspect equation object\bar L = (\partial L)/(\partial L) = 1.Equation 3Inspect equation object\bar v_i \mathrel{+}= \bar v_j (\partial v_j)/(\partial v_i).

Let a differentiable computation graph produce a scalar output LL from intermediate variables v1,…,vnv_1,\dots,v_n. Reverse mode stores, for each node, an adjoint or cotangent

vˉi=∂L∂vi.\bar v_i = \frac{\partial L}{\partial v_i}.

Before the reverse sweep, initialize every non-output cotangent register to zero. Then seed the scalar output with

Lˉ=∂L∂L=1.\bar L = \frac{\partial L}{\partial L} = 1.

For scalar nodes and a local operation vj=f(vi)v_j = f(v_i), the chain rule sends sensitivity backward. When the operation producing vjv_j is processed, vˉj\bar v_j already contains all downstream contributions:

vˉi+=vˉj∂vj∂vi.\bar v_i \mathrel{+}= \bar v_j \frac{\partial v_j}{\partial v_i}.

For an operation with multiple inputs, such as vj=f(u,w)v_j=f(u,w), each input receives its own local derivative:

uˉ+=vˉj∂vj∂u,wˉ+=vˉj∂vj∂w.\bar u \mathrel{+}= \bar v_j \frac{\partial v_j}{\partial u},\qquad \bar w \mathrel{+}= \bar v_j \frac{\partial v_j}{\partial w}.

The plus-equals matters. If a value is reused by several later nodes, all downstream paths contribute to its total sensitivity. Reverse mode is therefore not symbolic simplification; it is graph-local accumulation of vector-Jacobian products.

For vector nodes, choose column-vector cotangents. If vi∈Rnv_i\in\mathbb{R}^n, vj=f(vi)∈Rmv_j=f(v_i)\in\mathbb{R}^m, vˉi∈Rn\bar v_i\in\mathbb R^n, vˉj∈Rm\bar v_j\in\mathbb R^m, and

Jji=∂vj∂vi∈Rm×n,J_{ji}=\frac{\partial v_j}{\partial v_i}\in\mathbb{R}^{m\times n},

then the reverse update is

vˉi+=JjiTvˉj.\bar v_i \mathrel{+}= J_{ji}^{\mathsf T}\bar v_j.

This is the direction contrast:

JVP: v˙j=Jjiv˙i,VJP: vˉi=JjiTvˉj.\text{JVP: }\dot v_j=J_{ji}\dot v_i,\qquad \text{VJP: }\bar v_i=J_{ji}^{\mathsf T}\bar v_j.

For a scalar loss L:Rp→RL:\mathbb R^p\to\mathbb R, one forward evaluation records the needed values, and one reverse sweep gives the full gradient

∇θL=(∂L∂θ1,…,∂L∂θp)T\nabla_\theta L = \left(\frac{\partial L}{\partial \theta_1},\dots,\frac{\partial L}{\partial \theta_p}\right)^{\mathsf T}

assuming primitive backward rules are available. The cost is memory for saved forward values on a tape, or extra compute if some values are recomputed. An autodiff engine automates this bookkeeping by recording primitive operations during the forward pass and executing their local backward rules in reverse topological order.

These equations assume the recorded primitives are differentiable at the saved forward values. For nonsmooth primitives, an implementation must choose a convention, use a subgradient, or report that the derivative is undefined. If the program has control flow, reverse mode differentiates the branch that actually ran.

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Code

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Code witness 1import math x, y = 2.0, 3.0 # Forward graph: # a = x * y # b = sin(a) # L = a + b a = x * y b...python
import math

x, y = 2.0, 3.0

# Forward graph:
# a = x * y
# b = sin(a)
# L = a + b
a = x * y
b = math.sin(a)
L = a + b

# Reverse-mode table: each bar_* stores dL/d(node).
# Start with empty cotangent registers, then seed the output.
bar_L = 0.0
bar_a = 0.0
bar_b = 0.0
bar_x = 0.0
bar_y = 0.0

bar_L = 1.0

# Read the tape backward.
# L = a + b sends one unit of sensitivity to both inputs.
bar_a += bar_L * 1.0
bar_b += bar_L * 1.0

# b = sin(a) contributes another path back into a.
bar_a += bar_b * math.cos(a)

# a = x * y sends sensitivity to both inputs.
bar_x += bar_a * y
bar_y += bar_a * x

print("L:", round(L, 4))
print("dL/dx:", round(bar_x, 4))
print("dL/dy:", round(bar_y, 4))

The code initializes every cotangent register and then uses += for local contributions. The reused node aa receives two contributions: directly through L=a+bL=a+b, and indirectly through b=sin⁡(a)b=\sin(a). With x=2x=2 and y=3y=3, the output is approximately L=5.7206L=5.7206, ∂L/∂x=5.8805\partial L/\partial x=5.8805, and ∂L/∂y=3.9203\partial L/\partial y=3.9203.

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Live Concept Demo

Explore Reverse-Mode Automatic Differentiation

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difficulty 3/5undergraduatecode-aligned
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01Choose lensTrace a quantity
02ObserveDemo state pending
03GroundName the equation, invariant, or control that explains it.
04CarryNext: Backpropagation

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Use the sliders to change xx and yy, then compare the three phases.

Forward tape mode records the primitive operations and the saved values that local backward rules will need. Reverse sweep mode reads the same tape backward, starts from Lˉ=1\bar L=1, and fills the cotangent registers. Cost shape mode highlights the main reason reverse mode matters for deep learning: when many inputs feed one scalar loss, one reverse sweep gives the whole gradient vector.

Try the second preset after making a prediction. It changes the product regime so the same tape can expose a different cotangent-accumulation pattern.

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4/4 sections ready

Concept: Reverse-Mode Automatic Differentiation

What is the smallest example that makes Reverse-Mode Automatic Differentiation click without losing the math?

BeforeComputation GraphsNow4/4 sections readyTryManipulate one control and predict the visible change.NextBackpropagation
Object contextCalculus
StateCalculus
1 source attachedChecking local snapshot
concept:calculus/reverse-mode-autodiff
Math3 equation objects\bar v_i = (\partial L)/(\partial v_i).CodeReverse-Mode Automatic Differentiation code witness 1x, y = 2.0, 3.0DemoReverse-Mode Automatic Differentiation predictionManipulate one control and predict the visible change.
ConceptLearner lens

Reverse-Mode Automatic Differentiation

What is the smallest example that makes Reverse-Mode Automatic Differentiation click without losing the math?

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4/4 sections ready
Carry inComputation Graphs

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Work hereReverse-Mode Automatic Differentiation

Reverse-mode autodiff computes gradients by sending cotangents backward through a computation graph.

Carry outBackpropagation

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Test the linkManipulate one control and predict the visible change.Then continue to Backpropagation

After The First Pass

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ConceptReverse-Mode Automatic DifferentiationCalculus
01MechanismInspect motionSee how the idea changes form before adding more explanation.02VisualChoose a cueBind one visible mark to one mathematical role.03SourcesCheck boundary1 source link attached to the object.04ClaimAudit evidenceTreat the page claim as something to verify.05PracticeAttempt transferWrite before feedback so the model has something to repair.06RoomAsk groundedAttach the question to the same exact object.

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Reverse-mode autodiff computes gradients by sending cotangents backward through a computation graph.

Demo notes open01 / Intuition
Editorial autodiff illustration of a forward tape and reverse cotangent sweep through computation nodes.
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Visual Inquiry

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Reverse-mode autodiff computes gradients by sending cotangents backward through a computation graph.

4/4 stages readyDemo notes connected
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Object - ConceptReverse-Mode Automatic DifferentiationQuestion

What is the smallest example that makes Reverse-Mode Automatic Differentiation click without losing the math?

concept:calculus/reverse-mode-autodiff
Boundary

sources: baydin-2018-ad-survey

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selected object source · paper · 2018Automatic differentiation in machine learning: a surveyBaydin et al.
Located CF editorial boundary

Grounds reverse mode as the efficient way to compute gradients of scalar losses with many parameters.

Used here as

Baydin et al. describe reverse mode as running code forward to populate intermediate variables and record graph dependencies, then propagating adjoints backward. Their example shows incre...

Caveat

Checks reverse-mode bookkeeping for one executed differentiable computation: forward tape/saved values, then reverse cotangent sweep with primitive backward rules. Not checkpointing, reco...

Open source

Claim Review

Reverse-mode autodiff computes gradients by sending cotangents backward through a computation graph.

Object - ConceptReverse-Mode Automatic DifferentiationQuestion

What is the smallest example that makes Reverse-Mode Automatic Differentiation click without losing the math?

concept:calculus/reverse-mode-autodiff
Boundary

sources: baydin-2018-ad-survey

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1 CF editorial source-scope review recorded

Publisher-side editorial review is not independent replication. Claims without it still need exact source-support review. 1 reference and 3 local witnesses are available for inspection.

Reverse-mode AD records a forward evaluation trace of primitive operations and saved values, seeds a scalar output cotangent with 1, then sweeps backward applying local VJPs and += accumulation so one reverse pass computes the full gradient of one scalar loss with respect to many inputs.
Registered CF editorial boundary
Automatic differentiation in machine learning: a surveyGrounds reverse mode as the efficient way to compute gradients of scalar losses with many parameters.
Used here as

Baydin et al. describe reverse mode as running code forward to populate intermediate variables and record graph dependencies, then propagating adjoints backward. Their example shows incremental adjoint accum...

Local witness
source-span-baydin-2018-ad-surveymath-object-1math-object-2code-witness-1interactive-demo
Equation 1
vˉi=∂L∂vi.\bar v_i = \frac{\partial L}{\partial v_i}.
Equation 2
Lˉ=∂L∂L=1.\bar L = \frac{\partial L}{\partial L} = 1.
Caveat

Checks reverse-mode bookkeeping for one executed differentiable computation: forward tape/saved values, then reverse cotangent sweep with primitive backward rules. Not checkpointing, recomputation, nonsmooth...

Review stateCF editorial source-scope reviewClaim metadata: source checkedPublisher-side editorial review only; not independent replication. Check caveats and exact source scope.

Checked Baydin et al. 3.2: reverse mode runs code forward to populate intermediates and record dependencies, then propagates adjoints backward. The example starts from output adjoint 1, accumulates reused-variable cotangents from downstream paths, and gets both input derivatives in one reverse pass. Baydin says for f:R^n->R one reverse application computes the full gradient. Local witnesses match tape values, bar L=1, += pullbacks, VJP/J^T notation, and scalar-loss shape.

Reviewer: codex+oracle; reviewed 2026-05-07

Practice · Reverse-Mode Automatic Differentiation

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Reverse-mode autodiff computes gradients by sending cotangents backward through a computation graph.

Concept · Current object

Reverse-Mode Automatic Differentiation

Source boundary: sources: baydin-2018-ad-survey

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Calculus

concept:calculus/reverse-mode-autodiff
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    Selected object routeAsk from this object; carry one invariant back.sources: baydin-2018-ad-survey
    1. ObjectConceptReverse-Mode Automatic Differentiation
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    3. WitnessCompare codeReverse-Mode Automatic Differentiation code witness 1
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    ConceptReverse-Mode Automatic DifferentiationCalculus
    Code witness comparisonReverse-Mode Automatic Differentiation code witness 1x, y = 2.0, 3.0Prediction before revealReverse-Mode Automatic Differentiation predictionManipulate one control and predict the visible change.
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    conceptCalculus

    Reverse-Mode Automatic Differentiation

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    I am working in Continuous Function's research reading room. Object: concept - Reverse-Mode Automatic Differentiation Object key: concept:calculus/reverse-mode-autodiff Context: Calculus Anchor id: concept/concept-notebook/calculus/reverse-mode-autodiff Open question: What is the smallest example that makes Reverse-Mode Automatic Differentiation click without losing the math? Evidence to inspect: - Source ids to inspect: baydin-2018-ad-survey - Definition, prerequisite, and contrast concept links - The equation or code witness that makes the concept operational - One demo state that shows the invariant instead of a slogan Deterministic role lenses for this object: - Boundary: fixed perspectives, not people, community contributions, or independent review - Source-checking summary: Treat this as a mechanism object: connect the definition to one equation, code witness, or demo before broadening the discussion. - Proposed experiment: Ask the learner to perturb one representation, then check whether the same invariant survives in math, code, and demo. - Teach/transfer move: Turn the mechanism into one sentence that predicts a neighboring concept. - Assumptions: - Source ids baydin-2018-ad-survey must support the exact object, not just the surrounding topic. - The stable content-object key lets local drafts, prompts, and route memory attach without changing the source page. - The concept explanation is local atlas prose until checked against its math, code, and source support. - Prerequisite gaps should become a repair route, not a reason to leave the object vague. - Role-lens requests: - Learner: ask for "Ask what would make "Reverse-Mode Automatic Differentiation" feel predictable rather than familiar." | assumption: Source ids baydin-2018-ad-survey must support the exact object, not just the surrounding topic. | next action: The learner can state the mechanism in their own words - Researcher: ask for "Source ids to inspect: baydin-2018-ad-survey" | assumption: The stable content-object key lets local drafts, prompts, and route memory attach without changing the source page. | next action: The learner can name the prerequisite that would repair confusion - Experimenter: ask for "Choose one variable or condition to perturb before asking for an explanation." | assumption: The concept explanation is local atlas prose until checked against its math, code, and source support. | next action: The learner can predict how the mechanism changes under one perturbation - Professor: ask for "Find the smallest transferable rule a learner could reuse without the AI." | assumption: Prerequisite gaps should become a repair route, not a reason to leave the object vague. | next action: Teach or transfer: Turn the mechanism into one sentence that predicts a neighboring concept. What would resolve this: - The learner can state the mechanism in their own words - The learner can name the prerequisite that would repair confusion - The learner can predict how the mechanism changes under one perturbation Answer as a careful research tutor: stay source-grounded, separate verified evidence from assumptions, name the relevant math objects, and end with one next action. Current deterministic role lens for this object: - Role lens: Learner - Evidence request: Ask what would make "Reverse-Mode Automatic Differentiation" feel predictable rather than familiar. - Assumption to keep visible: Source ids baydin-2018-ad-survey must support the exact object, not just the surrounding topic. - Proposed experiment: Ask the learner to perturb one representation, then check whether the same invariant survives in math, code, and demo. - Next action: The learner can state the mechanism in their own words

    concept/concept-notebook/calculus/reverse-mode-autodiff concept:calculus/reverse-mode-autodiff