An Intrinsic Dimension Perspective of Transformers for Sequential Modeling

1. What is the space on which the ID is computed? The representation of the single tokens, or the representation of the sentence?
2. In ref https://arxiv.org/abs/2302.00294 it is observed a dependence of the ID on the depth which is non-monotonic. Even if the analysis is performed on different datasets, the monotonic behaviour seems to me at odds with the hypothesis that the first layer of a NN expand the dimension of the representation by getting rid of low-level confounders. Any explanation?
3. Can the ID really be used to guide the choice of the training set size?
4. What is the take-home-message of the observation that the ID grows with the ED? Why this observation is not obvious as it seems?

1.For experimental study, increasing depth/width of Transformer based encoder results in lower/higher ID. Do we have a trade-off between these two important hyper-parameters? The paper states that higher ID usually means lower error rate. Does this mean that a shallow and wide structure is always better than deep structure for text classification?

2.Comparing to PCA based analysis, the ID derived via TwoNN is substantially small. The paper stated that the learned representation fall in a distinct curved manifold rather than linear subspace. This might be biased since the TwoNN is non-linear analysis tool naturally.

3.In Table 5, the error rates of the model over training sets with full, long and short sequences are similar. This seems counterintuitive especially when training on short sequences. As we know, Transformer is an order-independent learning structure. For sequential learning, we need to add explicit position information (such as position embedding in Standard Transformer used in this paper). Training on short sequences will prevent its generalization ability on longer sequences. However, from Table 5, there is no significant difference between error rate especially for SST2 with diverse lengths.

4. What kind of distance is used in TwoNN for ID computation?

5. Missed related references

1. Anti-oversmoothing in deep vision transformers via the fourier domain analysis: From theory to practice. ICLR, 2022.
2. Revisiting over-smoothing in bert from the perspective of graph. ICLR, 2022.

The reviewer is open to learning about new evidences or analyses which address the “quality” and “originality” points of the “Weaknesses” section above in this review.

Clarification about which feature space is used when computing the ID. The feature space in which the authors compute the ID needs to be better defined. The datasets analyzed consist of input sentences of variable length $l_i$, embedded in $l_i \times ED$ dimensional spaces by the transformer, where ED is the embedding dimension. The authors are likely applying the TwoNN in a space with a dimension equal to ED.

1. Where do the authors define the strategy to reduce sentences of different lengths to a common ambient space where the Euclidean distances can be computed? In Sec. 3.2 the authors write that “every piece of input data is sequentially mapped in $n_l$ dimensional vector.
2. What do they mean by “piece of input data”: the tokens or the sentences? Is the meaning of hidden dimension $n_l$ different from that of embedding dimension ED? If not, why do they use different names?

Correlation between ID and accuracy The authors claim that an higher terminal feature ID correlates generally correlates with a lower classification error rate (e.g., Abstract). However, they show incomplete/contradictory evidence about this aspect. In Sec. 3.3, the claim seems valid for AG and IMDB (but the evidence is not robust for SST2).
More importantly, in Sec. 3.5 they show instead that the ID of the last layer decreases when the depth is increased, but they do not report the accuracy. Generally, increasing the transformer depth should also increase the accuracy:

3. Does the accuracy improve making the network deeper?

Clarification about intrinsic dimension estimate with PCA. The ID estimate with PCA is usually done by looking for gaps in the eigenvalue spectrum of the covariance matrix. This approach is followed by Ansuini et al., which the authors cite and also widely reported in the technical literature (see e.g. [1-3]). In Sec. 3.4 the authors instead select an arbitrary threshold corresponding to 50% or 85% of the explained variance.

4. How do they justify their approach in this case?

[1] Little et al., Multiscale geometric methods for estimating intrinsic dimension.
[2] Ceruti et al. Intrinsic Dimension Estimation: Relevant Techniques and a Benchmark Framework.
[3] Kaslovsky and Meyer, Overcoming noise, avoiding curvature: Optimal scale selection for tangent plane recovery.

Rationale of paragraphs 3.6-3.7 I have some concerns about the rationale behind paragraphs 3.6-3.7. Decreasing the dataset size deteriorates the performance (especially on the SST2 dataset). The evidence that the sentence IDs do not change with the sentence length does not seem a useful criterion for the data selection.

Minor suggestions

a. The authors have trained the transformers but do not indicate the hyperparameter setup used (number of epochs, batch size, optimizer, learning rate). They should add this information to the manuscript or in the appendix.

b. The authors should add a link to the code so that others can reproduce the experiments described in the paper. This is now a de facto mandatory requirement in this field.

See weaknesses