THEORY-035: Deep networks generalise because the diffusion phase of SGD compresses each layer's information about the input

Source

• Shwartz-Ziv & Tishby (2017), LIT-324, §§2.3–2.4, 3.2, 3.4, 3.5 and 3.8, as read in NOTE-298. The account’s origin.
• Saxe, Bansal, Dapello, Advani, Kolchinsky, Tracey & Cox (2018), LIT-372, §§2–5 and Apps. B–E and I, as read in NOTE-320. The independent test.
• Tishby, Pereira & Bialek (1999), LIT-338, the IB functional it applies, as read in NOTE-300.

What was actually shown

The account. LIT-324 trains small tanh networks on a 12-bit synthetic rule whose joint distribution is known exactly. It estimates each layer’s information by binning every neuron into 30 bins (§3.2). Over training, I(T;Y) rises within a few hundred epochs, and then I(X;T) falls over thousands (§3.4). The bend coincides with a fall in per-layer gradient SNR, from drift to diffusion, at about 350 epochs (§3.5). The paper reads the diffusion phase as entropy-maximising relaxation under the training-error constraint. It then asserts that “compression by noise” explains the absence of overfitting: “we believe” (§1), “If our findings hold…” (§4). The causal chain from diffusion to compression to generalisation is deferred to “a rigorous analysis … elsewhere” (NOTE-298, C3, C5).

Why the record rejects it. LIT-372 reruns LIT-324’s code and changes one thing at a time. Each link in the account fails as a general statement (NOTE-320, C1–C6).

1. The phases depend on the nonlinearity and the estimator. The tanh replication reproduces LIT-324’s fitting-then-compression paths. ReLU, softplus and linear networks do not compress, on the 12-bit task and on MNIST. That holds under binning, a KDE bound and a Kraskov entropy estimate (Figs. 1, 8, 12). Re-binning the same tanh run, with edges evenly spaced in net input, removes the compression in most layers (App. C, Fig. 14). The later phase is tanh units saturating as their weights grow, which a fixed-width binning reads as lower entropy (Fig. 2, App. E). Weight norms rise through it.
2. The measured quantity belongs to the estimator. For a deterministic network, I(h;X) is infinite, so any finite value comes from noise or binning the analyst adds (App. C, eqs. 17–20). On a finite input set with binning, I(X;T) = H(T) exactly (eqs. 1–3). “Compression” then means only that the binned activity takes fewer distinct values. The same function, reparametrised across two layers, gets a different I(T;X) (eq. 21). LIT-324 itself concedes the root of this in §2.4: for deterministic maps, information “is insensitive to the complexity of the function”. It also runs two unreconciled noise models, binning (§3.2) and “sigmoid outputs as probabilities” (§2.3).
3. Compression and generalisation dissociate. All four combinations occur (§3, Figs. 1, 3, 4). Tanh on the full task compresses and generalises. ReLU and deep linear networks generalise without compressing. A linear network overtrains without compressing. Tanh on 30% of the data compresses and overfits. That last cell matches LIT-324’s own small-sample result. With 5% of the data, compression lowers I(T;Y), and the paper attributes the overfitting “largely” to compression (§3.4).
4. Gradient noise does not cause the compression. Full-batch gradient descent gives “robust compression in tanh networks”, as SGD does. It also gives the same information dynamics in ReLU and linear networks (§4, Figs. 5, 19). Saxe et al. also point out a category error in the argument. A maximum-entropy distribution over weights describes variation across runs, while H(X|T) is about inputs for one set of weights. “There is no general reason” one draw maximises it (§4).
5. “Layers lie on the IB bound” is a fit. β is chosen per layer to minimise the divergence between the layer’s encoder and the IB encoder built from the layer’s own decoder (eq. 12). The claim says only that some β makes each layer nearly IB-consistent (NOTE-298, C4). LIT-372 does not test this claim or LIT-324’s depth claim. This point is the record’s own reading of LIT-324.

What this does not say

• That the IB method is wrong. LIT-338 poses a source-coding problem with a known p(x,y) and proves the form of its optimum (Thm 4). It says nothing about SGD, finite samples or generalisation (fn. 1). Nothing here touches it.
• That the tanh two-phase observation is false. With tanh units and this binning, the information paths are what LIT-324 reports, consistent across 50 runs. LIT-372 reproduces them with the original code. What fails is their generality and their reading.
• That the gradient-SNR transition is spurious. The drift-to-diffusion change is measured from gradient statistics, not from information estimates, and LIT-372 confirms it. It also shows the change is generic (App. I, Figs. 9–10, 20–21). It occurs in ReLU nets, on MNIST, under full-batch GD, and in a 1-1-1 linear network that cannot compress. Its explanation is elementary: the mean gradient goes to zero near a minimum while the per-example spread stays finite. Earlier work described the same two phases (Murata 1998; Chee & Toulis 2017). It is also the precursor of McCandlish’s noise scale (LIT-348), which cites LIT-324. It survives this rejection as a property of approaching a minimum, not as evidence for compression.
• That no representation discards input information. In a linear network with task-irrelevant inputs, information about the irrelevant subspace rises and then falls, while total I(X;T) rises (LIT-372, §5, Fig. 6). That happens during fitting, not in a later phase. It is measured under an imposed noise σ²_MI = 1.
• That no information quantity bears on generalisation. Input–output information does bound expected generalisation error (LIT-347), but that quantity is I(S;W), not I(X;T). Sefidgaran et al. (LIT-245, Theorem 4) bound the generalisation error of an encoder–decoder learner by a third quantity, the KL divergence of the encoder’s train-and-ghost latents to a prior symmetric under label-preserving permutations (NOTE-742), filed as THEORY-238. That too is not I(X;T).
• Anything beyond small settings. LIT-372 tests the 12-bit task, one MNIST MLP and linear networks. It does not test convolutional or transformer networks. Nor does it test anisotropic inputs, which its authors flag as open (NOTE-320).

Connections

• LIT-226 (CEB) inherits the IB programme. NOTE-300 shows algebraically that its objective is eq. (15) of LIT-338 reparametrised. Fischer’s §4 argues that compressed models generalise better, citing Shamir et al.’s bound, but measures compression by its own variational bounds on a stochastic encoder, not by binning a deterministic network, and makes no claim about a compression phase of SGD (NOTE-754). It neither supports nor bears on the account rejected here.
• LIT-240 (Vera et al.) takes up “the IB term controls generalisation” formally, with a bound in I(X;U) that holds for finite alphabets. It is Unavailable: no copy could be obtained, so it is unread. LIT-245 (Sefidgaran et al.) takes the claim up to replace it. It argues that the existing I(X;U) bounds, Vera et al.’s among them, are vacuous or restricted to finite alphabets, and that I(X;U) is infinite for deterministic continuous encoders, which agrees with LIT-372, App. C. It does not prove that I(X;U) cannot bound generalisation (NOTE-742).
• None of the later phases of training reported elsewhere in the record is a measured fall in I(X;T). See LIT-345 (NOTE-283), LIT-371 (NOTE-319), LIT-369 (NOTE-317) and LIT-370 (NOTE-318).