Learning moves inside the network
Rosenblatt’s perceptron made random association activity adaptive. Reinforcement changed a response disposition, and class structure could make an unseen member recover the same source-set bias as trained members. Yet the useful intermediate organisation was still largely supplied by the architecture and its fixed sampling statistics. Paper 0048 asks a more pointed question. If a network has hidden layers whose desired states are never specified by the task, can error at the output teach those layers what to detect?
The difficulty is easy to miss because the output has a target. For an input-output pair, the teacher can say which output units should be on and off. Direct connections can be adjusted according to the discrepancy. A hidden unit has no such label. The task does not say “unit seven should represent generation” or “unit two should detect asymmetry”. Deciding when a hidden unit should become active amounts to deciding what internal feature it should represent. A general learning method must derive that decision from the output objective.
Rumelhart, Hinton and Williams describe a solution through differential sensitivity. Run the network forwards and record every unit’s activity. Compare actual and desired outputs. Then move backwards, asking how a small change at each unit would change total error. The chain rule decomposes this global question into messages along connections. Once a unit has an error derivative, every incoming weight has a derivative too.
The title’s “learning representations” is therefore not ornamental. The procedure is valuable because hidden units come to encode task regularities that were not named in the input or output code. In the family-tree experiment, internal activity discovers nationality, generation and branch-like structure. In the symmetry experiment, paired positions acquire organised weights. The network is not handed these predicates. They are useful coordinates in weight space because they reduce the supervised output error.
Historically, the safe claim is precise. This short Nature letter clearly described, demonstrated and popularised back-propagation for learning useful internal representations. It should not be called the sole invention of reverse-mode differentiation or every earlier neural use of error propagation. The contribution here is the lucid multilayer learning procedure joined to concrete representation-learning demonstrations.
A differentiable composition, one layer at a time
The analysed network has input units at the bottom, one or more intermediate layers and output units at the top. Connections inside a layer or from a higher layer to a lower one are forbidden in the simplest feed-forward form, though a connection may skip an intermediate layer. Units within a layer can be computed in parallel; layers must be computed sequentially because a higher state depends on lower states.
For a unit j, total input xj is a weighted linear combination of lower outputs. The paper writes wji for the weight from lower unit i to unit j. A permanently active extra input supplies the bias. A positive bias weight lowers the effective threshold; a negative one raises it.
yj = 1 / (1 + exp(−xj)) (2)
The logistic function maps the unbounded total input to a smooth activity between zero and one. Its derivative is y(1−y), a fact that lets the backward calculation reuse the stored forward output. The authors stress that these exact functions are not mandatory. The useful requirements are a bounded derivative and a linear combination before the nonlinearity, which keeps each weight’s local derivative simple.
Differentiability changes the status of a hidden unit. A hard threshold only reports whether it crossed a boundary; an infinitesimal change is zero almost everywhere and undefined at the boundary. A smooth unit reports how responsive it is at its current operating point. That local slope is the coupling between an arriving error sensitivity and the total input that produced the activity.
Causal probe 1 · Why save forward activities?
The backward pass needs both the lower unit’s activity and the receiving unit’s logistic slope. Recomputing them is possible, but the 1986 procedure describes retaining the forward states so each derivative can be evaluated against the exact activity that caused the current output.
Worked example 1 · A bias as an always-on input
Two lower units have outputs 0.7 and 0.4, with weights 0.5 and −0.2. An always-on bias input has weight 0.1. The total is x=0.7×0.5+0.4×(−0.2)+1×0.1=0.37. The logistic output is about 0.591. Nothing special has to be added to the derivative machinery: the bias is simply a connection whose presynaptic activity is always one.
Error is a surface, not an instruction
For each training case c, the network produces output yj,c where the desired value is dj,c. The article sums half the squared difference over all output units and all cases. The factor one-half cancels the two introduced by differentiating a square. The result is a scalar error for the entire finite training set.
A scalar loss ranks weight settings, but it does not by itself say how to move. The gradient supplies a direction in weight space: one partial derivative per parameter. Its component ∂E/∂w tells how rapidly error would change under a small positive change of that weight while other coordinates are held fixed. Moving a sufficiently small distance in the opposite direction reduces the local first-order approximation to error.
The distinction between loss and gradient is essential. A large error does not imply a large useful update if the network is locally insensitive. A small error can coexist with important derivatives if several residuals cancel in the aggregate value. Back-propagation is not the act of subtracting the output error from every weight. It is an efficient calculation of the structured derivative through a composed system.
Worked example 2 · Residual, loss and direction
One output is y=0.8 when d=1. The residual is −0.2 and the case loss is ½(−0.2)²=0.02. The derivative with respect to y is −0.2: raising y would reduce error. The derivative with respect to total input is −0.2×0.8×0.2=−0.032. The logistic slope has converted an output-space instruction into an input-space sensitivity.
Causal probe 2 · Why does saturation slow learning?
When y is near zero or one, y(1−y) is small. Even a sizeable output residual is multiplied by a weak local slope. The unit’s total input can change substantially while its output barely moves, so the error derivative sent to its inputs becomes small.
The first derivative is visible at the supervised boundary
At an output unit, the desired state is known. Differentiating squared error gives ∂E/∂yj=yj−dj. This residual has a sign: a positive value says the unit is too active; a negative one says it is not active enough. It also has a magnitude in output coordinates.
The unit is controlled through total input x, not directly through y. The chain rule therefore multiplies the output residual by dy/dx. For the logistic function, dy/dx=y(1−y). The resulting quantity, often called a delta in later notation, measures how error changes when the unit’s total input changes.
∂E/∂xj = (∂E/∂yj) yj(1−yj) (5)
This decomposition separates two causes. The residual belongs to the task and the current prediction. The slope belongs to the unit’s current operating state. The same residual sent through two output units can yield different total-input sensitivities because one is near its steep midpoint and the other is saturated.
Worked example 3 · Same residual, different slope
Two units are each 0.1 below target. Unit A is at y=0.5, whose slope is 0.25, so its input sensitivity magnitude is 0.025. Unit B is at y=0.95, whose slope is 0.0475, so its magnitude is 0.00475. The task discrepancy is identical, but Unit B’s current nonlinearity makes its incoming parameters less able to change the output.
A connection inherits activity from below and blame from above
Total input xj contains the product yiwji. Its derivative with respect to that weight is simply yi. Chain this local fact to the receiving unit’s sensitivity and the weight derivative becomes ∂E/∂xj multiplied by presynaptic activity.
The rule has an appealing local form once the backward sensitivity has arrived. A silent lower unit makes no case-wise contribution to the weight gradient because changing its outgoing weight cannot alter the current total input. A strongly active lower unit scales the sensitivity. The hard part is not the final multiplication. It is producing the correct ∂E/∂x at every hidden destination.
For a bias weight, yi=1, so its derivative is the receiving delta. Biases learn by the same rule as other connections. This uniformity is one reason to represent thresholds as weights from a constant unit.
Causal probe 3 · Why does an inactive input receive no update for one case?
If yi=0, changing wji does not change the product yiwji on that case. The derivative is therefore zero even if the receiving unit has a large error signal. Other cases where the input is active may still update the weight.
Worked example 4 · One connection’s derivative
A receiving unit has δ=−0.032 and its lower unit is active at 0.6. The weight derivative is −0.0192. With learning rate ε=0.5 and no momentum, Δw=−ε∂E/∂w=+0.0096. Increasing the weight raises the receiving input, which is the locally correct direction because the output was below target.
Every downstream route returns a weighted consequence
A hidden unit has no direct target, so its error derivative must be assembled from the units it influences. If hidden output yi feeds several higher units j, a small change in yi changes each higher total input by its connection weight wji. Each route contributes the higher input sensitivity multiplied by that weight. The contributions add.
The hidden unit then multiplies this derivative by its own yi(1−yi) to obtain ∂E/∂xi. Its incoming weights use equation 6. Repeating the operation computes derivatives for successively earlier layers. “Back” names the order of derivative evaluation, not a reversal of the forward activations or a second prediction.
The same forward weight reappears because that weight quantified the causal sensitivity of the downstream total input to the hidden output. If yi affects xj through wji, then dxj/dyi=wji. Exact back-propagation therefore uses transposed forward weights to carry sensitivities. This mathematical reuse later becomes central to debates about biological plausibility.
Credit can cancel. Suppose one downstream route says increasing a hidden unit would reduce error while another says it would increase error. Their weighted derivatives add algebraically. A near-zero result may mean the unit is irrelevant, saturated or caught between opposing responsibilities. Inspecting only the final gradient cannot distinguish those explanations without the route-level terms.
Causal probe 4 · Why add the routes?
A small hidden change affects every downstream recipient simultaneously. The total differential of error is the sum of those first-order effects. Choosing only the largest route would calculate the derivative of a different system in which the other routes had been removed.
Worked example 5 · Two routes disagree
A hidden unit feeds Output A with weight 0.8 and δA=−0.05, contributing −0.04. It feeds Output B with weight −0.3 and δB=−0.04, contributing +0.012. The derivative with respect to hidden output is −0.028. If the hidden activity is 0.5, its logistic slope is 0.25 and hidden delta is −0.007. Both consequences matter, but the first dominates.
Batch accumulation, momentum and broken symmetry
The paper describes two ways to use weight derivatives. One can update after every case, avoiding separate derivative memory, or accumulate derivatives over the full set and update once per sweep. The reported experiments use accumulated gradients. These alternatives follow different trajectories even when they optimise the same total error because interim changes alter later derivatives.
Plain descent changes each weight by −ε∂E/∂w, where ε is the learning-rate factor. The accelerated rule adds α times the preceding change. The parameter α, between zero and one, controls how much earlier gradients contribute to the current velocity. Momentum can smooth alternating derivatives across a narrow direction and preserve motion along a consistently descending direction. It can also overshoot if the scale is poorly chosen.
Δw(t) = −ε ∂E/∂w(t) + αΔw(t−1) (9)
Small random starting weights have another role: they break symmetry. Two hidden units with identical incoming weights, outgoing weights and biases produce identical activities. They receive identical derivatives and remain clones forever under deterministic updates. A small asymmetry gives them different sensitivities, allowing one to specialise in a feature the other does not.
Random initialisation is not a guarantee of a good solution. It makes distinct solutions reachable by preventing an invariant redundancy. The error surface can still contain local minima, plateaux and ill-conditioned directions. Rumelhart and colleagues explicitly say global convergence is not guaranteed. They report that serious poor local minima were rare in their experience and suggest extra connections introduce weight-space dimensions that provide routes around barriers. That is a practical observation and intuition, not a theorem.
Worked example 6 · Identical twins cannot divide the task
Hidden units H₁ and H₂ begin with the same two input weights, bias and outgoing weight. Every input gives them the same activation. Their downstream contributions and returned deltas match, so each corresponding gradient matches. After one update they are still identical; induction gives the same conclusion for every update. Perturb one incoming weight by 0.01 and the invariant is broken.
Causal probe 5 · Why can extra connections help without proving anything?
Additional parameters can create a path around a barrier visible in a restricted subspace, but they also create new saddles, flat directions and equivalent solutions. More dimensions change geometry; they do not certify that a particular descent path reaches a global minimum.
Two hidden units discover mirror symmetry
A six-bit vector is symmetrical when positions equally distant from the centre match. No individual input bit supplies evidence for that global property. Direct input-output connections only add independent contributions from positions; they cannot express the required comparisons. An intermediate layer is essential for the network considered in the paper.
Figure 1 shows a network trained on all 64 possible binary input vectors. It used two hidden units and one output unit, small initial random weights uniformly distributed between −0.3 and 0.3, accumulated gradients, ε=0.1 and α=0.9. Training required 1,425 sweeps. The displayed solution gives paired positions equal-magnitude, opposite-sign weights into a hidden unit, with an organised 1:2:4 ratio across pairs.
Each half-pattern thereby produces a distinctive weighted sum. For a symmetric input, the matching contribution from the opposite half cancels it, so both hidden units receive zero net input from the bits. Their negative biases keep them off, while the output unit’s positive bias makes the symmetry response on. For an asymmetric input, the two half-codes differ; one hidden unit receives positive evidence and suppresses the output.
The learnt hidden activities do not name “left half” and “right half” in the input or output vocabulary. They implement a comparison code useful for the task. The exact printed weights are one elegant outcome, not proof that every run reaches the same factorisation or needs exactly 1,425 sweeps.
Worked example 7 · Cancellation reveals symmetry
Encode the left three bits with values 1, 2 and 4. Give their mirrored partners weights −1, −2 and −4. For 101|101, the weighted sum is (1+0+4)−(1+0+4)=0. For 101|100 it is 5−1=4. A pair of oppositely oriented hidden units can detect either sign of the mismatch and turn the output off.
Two family trees share one internal geometry
The second task stores facts from two isomorphic family trees, one English and one Italian. A fact is a triple: person 1, relationship, person 2. Twelve relations include parent, spouse, child, sibling, aunt, uncle, niece and nephew forms. The input activates one unit for person 1 and one for relationship; the correct output is the person 2 unit that completes the proposition.
The network was trained on 100 of 104 possible triples and correctly completed the four withheld cases. Its lower groups learned distributed codes for people and relations, a central layer combined them, and later units selected the answer. Figure 4 displays learnt person weights. One hidden feature mainly distinguishes the English from the Italian family, another encodes generation, and another distinguishes branches.
The important regularity is isomorphism. An English person and the Italian person in the corresponding family position acquire similar representations even though the input code gives every person a separate unit. The network can share structural knowledge from one tree with the other because its hidden features describe roles rather than merely memorising names.
Training used 1,500 sweeps. For the first twenty, ε=0.005 and α=0.5; thereafter ε=0.01 and α=0.9. After each weight change, weights decayed by 0.2 per cent. To prevent unbounded weights merely pushing logistic outputs closer to exact zero or one, an output counted as correct with zero error above 0.8 for on and below 0.2 for off. These details matter: the demonstration includes scheduling, momentum, regularisation and a tolerance convention, not just the chain rule.
Four withheld triples are persuasive evidence that the learnt code captured some shared structure in this small domain. They are not an open-ended test of systematic relational reasoning. The training set covers almost the entire finite space, and the two trees were constructed to be isomorphic. The result demonstrates representation reuse under favourable structure.
Causal probe 6 · What makes an Italian role look like an English role?
Names never overlap, but their relation patterns do. Corresponding people participate in matching parent, child, spouse and sibling triples. Gradients reward hidden directions that reduce error across many cases, so a role feature can serve both trees more efficiently than two unrelated memorisation codes.
Worked example 8 · Transfer through a role code
If Colin and Alfonso occupy corresponding positions, facts involving their parents, siblings and aunts push their distributed person vectors towards similar relation-sensitive features. Withholding one “aunt of” triple can still be solvable because the hidden code learnt the family position from the remaining edges and reused the relation transformation learnt elsewhere.
Unroll time, retain states, combine shared-weight gradients
The main derivation concerns layered feed-forward networks, but Figure 5 shows how to map a synchronous iterative network onto a layered one. Three iterations become three copies of the unit layer, one per time step. An edge across time becomes a connection between successive copies. The forward recurrence is now an acyclic computation graph over a finite horizon.
Two complications follow. First, backward calculation needs intermediate outputs, so the iterative system must retain the activity history for each unit. Second, copied edges may represent the same recurrent parameter. Treating each time copy as an independent weight would train a different model. The paper says to average ∂E/∂w across corresponding weights and apply the change to their shared value.
In modern language this is a compact precursor of back-propagation through time, but the article does not use that full later vocabulary or study long-horizon gradient decay and explosion. It says the construction can train iterative searches or sequential structures. A faithful reading keeps the finite unrolling and shared-gradient logic while avoiding results not reported in the letter.
Worked example 9 · One recurrent weight used three times
Unroll a recurrent edge over three steps. Its copied derivatives are 0.04, −0.01 and 0.03. If the implementation sums contributions, the shared parameter gradient is 0.06; if it follows the paper’s averaging convention, it is 0.02. Updating three independent copies would break the recurrent model’s shared-weight constraint.
Useful representations do not settle optimisation or biology
The authors name the most obvious optimisation drawback: the error surface may contain local minima, so descent is not guaranteed to find a global minimum. Their experience across tasks was that networks rarely became stuck in poor local minima substantially worse than the global solution. They suggest networks with slightly more connections have extra weight-space dimensions that provide routes around barriers present in a smaller subspace.
This is neither a convergence theorem nor a claim that local minima never matter. The reported demonstrations use particular architectures, schedules and small datasets. They do not report multiple-seed distributions, wall-clock time or comparisons to a modern optimisation baseline. One must separate the exact gradient derivation from contingent properties of a gradient trajectory.
The final paragraph is equally direct about biology. In its current form the learning procedure is not a plausible model of learning in brains. Exact back-propagation requires a backward calculation linked to forward weights and stored intermediate activities. The demonstrations show that gradient descent in weight space can construct interesting internal representations. They motivate looking for biologically plausible ways of approximating such descent; they do not claim that nervous tissue implements the presented algorithm exactly.
The model also requires supervised desired outputs. The family-tree system is told the correct person; the symmetry system is told the correct class. Back-propagation allocates credit under that objective. It does not create the objective, decide which cases deserve labels or supply a theory of consciousness. Representation is shaped by the loss, architecture, cases, initialisation and optimisation schedule together.
Causal probe 7 · Why is exact weight reuse biologically contentious?
The backward message from a downstream unit is multiplied by the forward connection’s value. A biological implementation would need some mechanism by which the appropriate reciprocal or transported quantity is available to the learning process. The mathematics is exact; the physical delivery mechanism is not supplied by the letter.
Evidence boundary
The primary is a four-page Nature letter. It contains a derivation, two principal learning demonstrations, a recurrent unrolling argument and explicit limitations. It does not establish sole historical priority, global convergence, large-scale generalisation, biological implementation or conscious representation. Numerical laboratories in this companion audit the equations and mechanisms; they are not replications of the reported training runs.
Compare the backward calculation with independent perturbations
The laboratory uses a two-input, two-hidden, one-output logistic network. It computes all nine analytic gradients, then independently perturbs each parameter on both sides to form a centred finite-difference estimate. The controls expose a normal gradient, one descent step, a severed hidden-to-output route and identical hidden twins. Three exact audits run for every profile.
Four cards for rebuilding back-propagation
Forward card
Store each weighted total x and activity y. The backward pass evaluates derivatives at the states that produced the current output.
Delta card
At an output: residual × local slope. At a hidden unit: sum of weighted downstream deltas × local slope.
Weight card
Connection gradient = receiving delta × sending activity. Bias is the same rule with sending activity fixed at one.
Boundary card
An exact gradient does not guarantee a global optimum, a reusable representation beyond the task, biological plausibility or consciousness.
The entire algorithm in seven dependencies
1 · Compose a differentiable forward map
Weighted sums feed bounded nonlinear activities, layer by layer.
2 · Compare actual and desired outputs
Aggregate half-squared residuals across output units and cases.
3 · Differentiate the supervised boundary
The output derivative begins as y−d and crosses the logistic slope.
4 · Send consequences backwards
A hidden output derivative is the sum of downstream deltas multiplied by its outgoing weights.
5 · Cross each local nonlinearity
Multiply by y(1−y) to turn an output sensitivity into a total-input sensitivity.
6 · Differentiate every incoming connection
Multiply the receiving delta by presynaptic activity.
7 · Follow a declared optimiser
Accumulate by case or sweep, descend with ε, optionally carry velocity with α, and retain shared-parameter constraints.
How to audit a representation-learning claim
- Name the supervised signal. Which outputs have desired values, and who supplies them?
- Write the forward graph. Include skipped layers, shared weights, biases and recurrence.
- Record intermediate states. Derivatives must correspond to the actual forward computation.
- Verify gradients independently. Compare analytic values with centred finite differences on a small deterministic case.
- Separate derivative from optimiser. Learning rate, batching, momentum and decay are trajectory choices.
- Break symmetry deliberately. Confirm that hidden units do not begin as permanent clones.
- Inspect the learnt code. Ask which task regularities are captured and which distinctions disappear.
- Hold out structural cases. Report genuinely withheld combinations separately from trained-case fit.
- Respect parameter sharing. Combine derivatives from every use of a recurrent or tied weight.
- State optimisation uncertainty. A low loss found once is not a global-minimum guarantee.
- Grade biological claims. Mathematical efficiency and physical plausibility require different evidence.
From statistical separability to end-to-end feature construction
Paper 0046 supplied a logic for fixed neural nets. Paper 0047 added reinforcement to random association populations and showed how environmental class structure could become a response bias. Paper 0048 sends learning deeper. Intermediate units no longer require hand-fixed feature roles or desired states. Their usefulness is inferred from how their activity changes a supervised output objective.
This changes the design boundary. A multilayer architecture can be trained as one composed system because the chain rule allocates sensitivity through every layer. The forward representation and the output rule are adjusted together. Later papers will ask how broad the resulting function class is, how convolution exploits spatial structure, how depth and residual paths affect optimisation, and how scale changes representation. Those advances depend on the credit-assignment hinge made vivid here.
For agency research, back-propagation shows that internal variables can acquire instrumental meaning relative to an objective without receiving semantic labels. That is a mechanism for functional representation, not a guarantee of truth, causal grounding or aligned purpose. For consciousness research, distributed hidden structure can support flexible behaviour while leaving phenomenal status untouched. A learnt coordinate is evidence of internal organisation, not evidence that the system experiences what the coordinate tracks.
What is derived, demonstrated and merely suggested
| Layer | Supported claim | Boundary |
|---|---|---|
| Mathematics | Equations 1–9 give exact derivatives for the stated layered differentiable network and squared error. | No global convergence follows. |
| Symmetry task | Two hidden units learnt a structured solution over all 64 six-bit inputs after the reported training run. | One run does not determine every solution or trajectory. |
| Family task | A distributed hidden code completed four triples withheld from 104 possibilities across isomorphic trees. | This is not open-domain relational reasoning. |
| Recurrence | Finite synchronous iteration can be unrolled, with states retained and shared gradients combined. | Long-horizon gradient pathologies are not studied. |
| Biology | Interesting internal representations motivate a search for plausible neural gradient mechanisms. | The presented procedure is explicitly called implausible as a brain-learning model. |
What this companion adds
This companion rebuilds the four-page letter without flattening back-propagation into a slogan. Ten paper-specific animated figures follow the missing hidden target, forward composition, error geometry, output delta, connection derivative, hidden credit junction, momentum and symmetry breaking, mirror detector, family-tree representation and recurrent unrolling.
The executable gradient laboratory checks all nine analytic derivatives against centred finite differences, proves a small negative-gradient step reduces the chosen error, and severs one downstream route to expose its hidden-credit contribution. The teaching sequence preserves the exact experimental counts, schedules and tolerances from the primary while marking every authored reconstruction and every limit.
Two comprehensive passes
Scholarly and editorial review. All four scan pages were rendered and inspected. Equations 1–9, Figures 1–5, the 64-pattern symmetry run, the 100/104 family-tree split, schedules, decay, output tolerances, recurrent sharing, local-minimum discussion and final biological caveat were checked against printed pages 533–536. The companion avoids sole-invention language and does not import results from the longer PDP chapter.
Executable and publication review. Every laboratory profile must pass analytic-versus-numerical gradient agreement, loss descent and severed-route controls. Responsive rendering at 1440, 768 and 390 pixels, accessible SVG names, reduced-motion behaviour, duplicate identifiers, keyboard-labelled controls, substantive length, placeholder text and source/public identity are checked before production publication.
References and source trail
- David E. Rumelhart, Geoffrey E. Hinton and Ronald J. Williams, “Learning representations by back-propagating errors,” Nature 323, 533–536 (1986). DOI record; publisher record; complete scan used for page verification.
- David E. Rumelhart, Geoffrey E. Hinton and Ronald J. Williams, “Learning internal representations by error propagation,” in Parallel Distributed Processing: Explorations in the Microstructure of Cognition, vol. 1, 318–362 (MIT Press, 1986). MIT Press record. Listed as the longer companion cited by the Nature letter; its extra material is not silently attributed to the letter.
- Frank Rosenblatt, Principles of Neurodynamics: Perceptrons and the Theory of Brain Mechanisms (Spartan Books, 1962). Archive record. The earlier perceptron-convergence contrast cited by the primary.
- Marvin Minsky and Seymour Papert, Perceptrons: An Introduction to Computational Geometry (MIT Press, 1969). Publisher record. The limitation reference cited by the primary.
- Paul J. Werbos, “Applications of advances in nonlinear sensitivity analysis,” in System Modeling and Optimization (1982). DOI record. A priority-relevant predecessor used only to constrain historical wording.
- Timothy P. Lillicrap et al., “Backpropagation and the brain,” Nature Reviews Neuroscience 21, 335–346 (2020). DOI record. A later review of biological credit-assignment proposals, not evidence for the 1986 experiments.