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The geometry of forgetting in continual learning

Companion note to the draft Interference and retention in continual learning. Draft PDF

When a network learns tasks in sequence it tends to suffer catastrophic forgetting: weights move to satisfy the new objective and, in doing so, dismantle the features that served the old one. The usual responses are engineering fixes. Experience replay retrains on stored samples; elastic regularisation penalises motion in directions the Fisher information marks as important. Both work. Neither says what forgetting is.

Forgetting is intuitively read as geometric interference between task directions. Fix the features (the frozen-feature or linearised regime — a strong assumption, discussed below) and let Δ be the weight update produced by learning Task B. The loss increase on Task A is then exactly a quadratic form:

F(Δ)=12 Δ⊤ΣA ΔF(\Delta) = \tfrac{1}{2}\, \Delta^{\top} \Sigma_A\, \Delta

Here ΣA is the feature covariance of Task A. Where ΣA has large eigenvalues, Task A's predictions depend sharply on those coordinates, and any component of Δ lying along them is expensive. Where ΣA has an exact null space, motion is free and forgetting = zero. Between those extremes, the cost of an update is set by the spectrum of ΣA.

Where this breaks. The identity above holds when features are frozen and the loss is locally quadratic. Under end-to-end training the features themselves move, ΣA is no longer a constant, and F becomes a first-order approximation whose error grows with the size of the update. It is a lens, not a theorem about deep networks.

Interference-gated functional allocation (IGFA)

If forgetting is interference, task projection in space helps visually the problem. IGFA measures the alignment between a proposed update and the dominant directions of earlier tasks. Below a threshold τ the directions are effectively independent, the update passes untouched, and capacity is shared. Above it, IGFA projects the update onto the orthogonal complement of the protected subspace, keeping the descent direction it can afford and discarding the rest.

There is no such thing as a free lunch:

  • The projection discards part of the gradient. Task B learns more slowly, and in the limit of many protected tasks the free subspace shrinks toward nothing, giving us a capacity ceiling.
  • The gate is a hard threshold, so behaviour is discontinuous at τ. Whether that matters depends on how alignment is distributed in practice; on the sequences tested it does not, but that is an empirical claim about those sequences.

Interactive · Vector alignment

Rotate the proposed Task B update ΔB. Interference energy rises as it aligns with Task A's dominant direction. Above the gate threshold, IGFA projects the update onto the orthogonal complement. Shorter arrow > less progress > far less damage.

45°
Cosine alignment
0.71
Interference, raw
52%
Interference, after IGFA
2%
Retained step length
71%

IGFA gate (τ = 0.40)

Projecting — Task A protected

Model: ΣA = diag(1.00, 0.05), F(θ) = λ₁cos²θ + λ₂sin²θ, normalised to λ₁. The subspace has finite width, so interference does not reach exactly zero at θ = 90° (null space). Illustrative, not fitted to data.

Task A's dominant subspace shown as an ellipse, with the proposed Task B update and its IGFA-projected version. Task A subspace Task A ΔB IGFA

At θ = 0 the update lies straight along Task A's high-variance direction and interference is maximal. Rotating away, the raw interference falls as cos²θ, but the gate does not release until alignment drops below τ. In the gated region the projected update survives with length |Δ|·sinθ. So at θ = 20° the controller keeps roughly a third of the step, and near the threshold it keeps most of it.

What the gate buys on a real benchmark

The widget above is schematic. The figure below is not: it plots the measured accuracy–forgetting trade-off from the draft's Rotated-Digits benchmark (domain-incremental, five rotations, frozen backbone, mean ± 95% CI over five seeds).

Scatter plot of average accuracy versus average forgetting for five continual-learning methods on Rotated-Digits. IGFA reaches accuracy 0.771 with forgetting 0.002; only replay is more accurate, at the cost of a stored data buffer. 0.00 0.05 0.10 0.15 0.20 average forgetting (lower is better) 0.65 0.70 0.75 0.80 average accuracy Naive EWC OGD/GPM Replay (buffer) IGFA
Measured, not schematic: Rotated-Digits domain-incremental results, mean ± 95% CI over five seeds (Table 2 of the draft). IGFA reaches the highest accuracy among replay- and Fisher-free methods (0.771 ± 0.030) with forgetting near zero (0.002 ± 0.027). Only replay scores higher (0.795 ± 0.012), and it pays for that with a stored data buffer.