Venue: NeurIPS 2021 Conference

Reviews:

final_score: rating: 7: Good paper, accept
review_content: summary: The paper proposes to represent uncertainty in neural networks by augmenting neural network layers with inducing weights and then assuming that the resulting vector of weights is drawn from a matrix normal distribution which is structured so that it has fewer parameters than the original network. The parameters of the distribution are learned via variational inference and the authors present a procedure to sample efficiently from this distribution for optimizing the model parameters and for inference at test time. Experiments on classification and out of distribution detection are presented to validate the approach.
main_review: **Quality and Clarity**

The overall idea is clear but quite heavy on notation and hence may require multiple readings to process entirely. I would recommend adding a table of symbols either in the body of the paper or in the appendix to enable readers to keep track of the notation. I would also suggest modifying Figure 1 to illustrate how the method is memory efficient since that is the key selling point of this work.

**Originality and Significance**

While this is one of many approaches to representing uncertainty in deep learning, the ability to effectively represent uncertainty without increasing, and indeed sometimes decreasing, storage cost is certainly significant. My only concern is that the method appears quite sensitive to hyperparameter choices, especially from the discusiion in Section 4.2 (Lines 260-268) and associated plots and without any guidance on how to choose the hyperparameters, it might be difficult to translate the success of the method to datasets other than the ones considered in the paper.

**Strengths**

1. A new memory efficent approach and associated sampling procedure is proposed for representing uncertainties in deep learning.
2. Experiments on a range of task validate the efficacy of the approach for both prediction and out-of-distribution detection.

**Weaknesses**

1. The writing is a bit heavy on notation and the details may be hard to parse in one reading.
2. The approach appears quite sensitive to hyperparameters and it is not clear how to set the hyperparameters for new datasets.

**Queries and Suggestions**

1. Is it possible to give some guidance on choosing the hyperparameters $\sigma_{\max}$, $\lambda_{\max}$, $M$ in general or is an ablation study the only way to figure it out? If it is the latter then I would recommend highlighting that fact a bit more clearly in the paper and providing any insights you may have on reasonable ranges over which to search, based on your experience with the experiments. Exploring the efficacy of AutoML in finding the optimal hyperparameters might also be an interesting option.

2. Why does pruning FFG-W to 2% of the size of the deterministic network give the lowest ECE in Figure 7? That appears to be violating the observed trend quite significantly and is probably worth investigating or at least providing some, even speculative, thoughts on.

3. I would suggest including some comparison with other space efficient approaches for uncertainty estimation like rank-1 BNNs. Even if they are not as space efficient as this approach it would be instructive to see if the extra space they require gives any performance improvement on the accuracy/ECE metrics and if not then those approaches can be convincingly ruled out in favor of this one.

4. I also did not notice any wall-clock time numbers for the different methods. It would be beneficial to have them since it does appear that the space savings come at the cost of slightly higher time complexity. While that is not a big issue, it would be good to see how much longer the method takes on average to get a better sense of the tradeoff.


limitations_and_societal_impact: I have provided suggestions on addressing the limitations in the review above. I believe there is no potential negative societal impact of this work.
review_title: null

final_score: rating: 7: Good paper, accept
review_content: summary: The paper proposes a parameter-efficient method for uncertanity quantification in deep neural network using per layer inducing weights. The paper develops a network weight sampling scheme based on inducing weights. The proposed method employs Matheron's rule for efficient sampling. The method requires fewer parameters than even deterministic networks, while keeping good calibration and performance, at the cost of longer compute times, when few MC weight samples are used.

main_review: Originality

The paper proposes a novel idea (best to my knowledge) of using inducing points known from sparse variational GPs in the context of Bayesian neural networks.
The paper adequately cites and describes differences with related work.

Quality

Submission is technically sound and claims are well supported through detailed derivations and exhaustive experimentation. 
Methods used are appropriate and the work looks complete.
The paper is careful and honest about evaluating the strengths (smaller number of parameters) and weakness (larger computational cost) of the proposed method as well as comparison to the baseline alternatives (weight pruning).

It would be interesting to see the training times additionally to the predictions times. As far as I understand from appendix G.2, during training the method uses 1 MC sample, and this might be the regime where the method is much less effective (>10x slower). 

I am not sure whether the full weight matrix W gets materiallized during training of the FFG-U method. 
If this is the case, does it diminish all the gains in terms of memory requirement during training / testing? 

The paper is very math heavy and while most of the calculations looks good to me, I might have missed some errors. 

Clarity

The submission is clearly written and well organised. It adequately informs the reader.
The paper refers to complex mathematical objects and procedures (e..g Matheron's sampling, Matrix Normal closed form derivations), but aims to inform the reader through references and even explaining the basics of these concepts.

Significance

The paper proposes an interesting method for reducing the number of network parameters by a large factor. 
However, it increases the computational cost during testing and very likely also during training. 
Therefore, I expect that this method might find applicability in areas which require low memory and calibrated predictions, but the prediction time is not as important.
Nevertheless, from the theoretical standpoint, the paper develops an interesting connection between variational inference BNNs and sparse variational GPs.

limitations_and_societal_impact: N/A
review_title: null

final_score: rating: 4: Ok but not good enough - rejection
review_content: summary: This paper proposes a layer-wise variational inference scheme for Bayesian neural networks that uses "inducing weights" for each layer before up-sampling to the full (layer-wise) weight distribution. The scheme is equivalent to a hierarchical prior formulation with an analogous hierarchical variational posterior. Experiments are performed on the standard BDL benchmarks of synthetic one dimensional regression, ResNets on CIFAR-100, and out of distribution detection.
main_review: tldr: This method has a lot of promise, but the methodological writing is close to impenetrable currently, so I tend to reject it. Another round of revision to clean up the experiments and focus the writing presentation would help this paper have impact.

## Originality

*Are the tasks or methods new?* Yes, they seem to be a new version of VI for BNNs.

*Is the work a novel combination of well-known techniques?* Yes, I mostly followed along the Matheron's rule derivation, which is a nifty idea to reduce the computational complexity of prior sampling over the conditional distribution from the weights and the inducing weights.

*Is it clear how this work differs from previous contributions?* Yes, I think so. It bears a bit more resemblance to the GP style hierarchical priors and posteriors of [Karaletsos & Bui, NeurIPS, '20](https://arxiv.org/abs/2002.04033) than is mentioned in the paper. Variational matrix normal posteriors are also pretty closely related and probably deserve a bit of discussion as well (even though you do compare to them, which is helpful) [Louizos & Welling, ICML, '16](https://arxiv.org/pdf/1603.04733.pdf).

*Is related work adequately cited?* 
- See above comment plus also the probabilistic meta-hypernetwork approach of [Karaletsos et al, '18](https://arxiv.org/abs/1810.00555)
- Hypernetworks and their Bayesian counterparts are also relevant as hypernetworks can reduce the parameter count of standard networks significantly, while Bayesian hyper-networks should contain some of the same advantages of this work (better uncertainty quantification, etc...) [Ha et al, '16](https://arxiv.org/abs/1609.09106) and [Krueger et al, '17](https://arxiv.org/abs/1710.04759).

## Quality

*Is the submission technically sound?* Mostly so, a few comments:
- line 148: from my understanding the "mixture of delta posterior" is not theoretically justified due to support mismatch in KL space, which isn't an issue at all if you're not going for the bayesian interpretation (just all it ensembling), but could prove problematic if you're trying to train through that KL.
    - Thus, is it okay to drop the KL term when training?

- line 631 (but throughout): My understanding for sampling this joint would be to use Kronecker algebra to have the cholesky factors of the covariance (both defined under eq 5) and then to compute the MVM $(L_r \otimes L_c)z$ where $z$ is a standard gaussian random vector (before reshaping into a matrix). But, what you're taking is instead a non-square matrix vector product and so it's really tough to parse if that's the correct set of algebra.
    - Also, from the Kronecker approach, I think you can just write down the algebra and simplify to get out $\bar W$ and $\bar U$ (although I haven't had the chance to do so), which would simplify the presentation a good deal.
    - Eq 31 (and following): I don't follow the shapes of these matrices any more because of how the matrices are being written down, as well as how these matrix matrix products are being written down.

- line 638: "we can reduce variance ..." this statement concerns me, are you actually doing the full Matheron's rule here? or some sort of variance reduction method? How does this play into the variance based proofs in the following lines?

- line 640: why do we need to take the choleskies of these matrices? my understanding is that given any $AA' = \Sigma$ we can use $A'z$ to produce valid samples. 

- eq 36: it's a bit of a misnomer to call this a GP because its just a linear kernel (and this could be easily derived by marginalizing linear regression).

- 3.2 and 3.3: which version is actually implemented?
    - Do the variables $U_c$ play a role in the Matheron's rule based approach?
    - If not, are they in the implementation anywhere or are they just an object of theoretical interest due to their connection to the deep GP literature?
    - line 305: the scale of the KL term is probably important, and suggests that there's probably a missing constant or something there. After all, the scale determines if you're doing proper Bayesian inference or a generalized version (which isn't practically important, just you should be clear).

*Are claims well supported (e.g., by theoretical analysis or experimental results)?* 
- The experimental work demonstrates that the hierarchical VI scheme does outperform SVI, but it definitely doesn't seem to outperform either ensembling approach or really a deterministic NN, so the claims of "significant improvement" are not really well supported.

*Are the methods used appropriate?* 
- I'm not a fan of the comparisons and think that some other reduced space parameterization technique really ought to be compared to (even if the memory overheads are somewhat higher). 
    - From what I can tell, MIMO networks [Havasi et al, ICLR, '20](https://arxiv.org/pdf/2010.06610.pdf), rank-1 BNNs [Dusenberry et a l, ICML, '20](https://arxiv.org/abs/2005.07186), and subspace inference [Izmailov et al, '19](https://arxiv.org/abs/1907.07504) are probably also relevant comparisons. At least one of these would be very helpful (probably MIMO or rank-1 BNNs b/c I think that they are using the same architectures that you are).
    - This is especially needed because FFG-W (aka SVI) is such a weak baseline and then your approach is often dominated by your own baseline of Ensemble-U.
        - Even the claim of "not including these baselines [because they increase the parameter count]" (lines 223-225) is insincere because FFG-W doubles the parameter count as well.
    - A comparison to the standard Bayesian approaches for BDL is also missing [VOGN, SGMCMC, SWAG, GG-MC, etc.], but may be less relevant if the authors wish to focus solely on the parameter counting aspects. But these should still probably be cited.

- Similarly, a comparison to a post facto pruning method (down to 25% of the weights) would also be helpful. Potentially, something like [Savarese et al, '20](https://arxiv.org/pdf/1912.04427v4.pdf) would be a good comparison or simple magnitude based pruning.
    - Pruning represents a similar reduction in model capacity that the proposed approach does as well.
    - Fig. 7 is a nice attempt but a deterministic comparison should be performed.
    - One could always just imagine doing post facto pruning, and then a cheap and cheerful bayesian approach on retraining in the sparse space, say SWAG ([Maddox et al, NeurIPS, '19](https://arxiv.org/abs/1902.02476)).

- An alternative comparison (and probably more relevant) would probably be the sparsity based SGMCMC procedure of [Deng et al, NeurIPS, '19](https://proceedings.neurips.cc/paper/2019/file/2d71b2ae158c7c5912cc0bbde2bb9d95-Paper.pdf), which is both designed for producing sparsity in deep networks and for producing strong uncertainty calibration.

*Please note that I'm not suggesting the authors compare to all of these, please just pick one or two in the rebuttal (but cite the possibility of the rest). Preferably the SGMCMC one and/or deterministic pruning.*

*Is this a complete piece of work or work in progress?* Complete work.

*Are the authors careful and honest about evaluating both the strengths and weaknesses of their work?* No weaknesses are discussed from my reading. The obvious one is what happens if $U$ is too small of a vector / matrix, in which the model will have too little capacity from the approximation algorithm. This might be seen in Fig. 4 right but I'm not sure.

## Significance

*Are the results important?* Possibly.

*Are others (researchers or practitioners) likely to use the ideas or build on them?* Probably, this is a popular field right now.

*Does the submission address a difficult task in a better way than previous work?* While the approach beats SVI, that's a very low bar. Both ensemble based approaches (one of which is your own baseline) seem to outperform it.

*Does it advance the state of the art in a demonstrable way?* Empirically maybe not, in terms of practical sampling mechanisms for matrix normals, probably so.

*Does it provide unique data, unique conclusions about existing data, or a unique theoretical or experimental approach?* A unique theoretical approach, same experiments as many papers in this space.

## Discussion

- The uncertainty bands in Fig. 3 are pretty nice overall.

- Fig 4 looks like a mixed bag, the accuracy of FFG-U (which I take to be your method) is reasonably worse in terms of accuracy than even a standard deterministic NN (although better calibrated).

- My overarching takeaway from the experiments is that the performance across all experiments goes *deep ensembles > ensembles-U > FFG-U > FFG-W* which suggests that FFG-U only outperforms FFG-W, but the hierarchical prior structure has some sort of performance limitations (as standard deep ensembles tend outperform ensembles over $U$).
    - This limits the practical usefulness of the results to be only that the approach reduces the parameter count, but then further comparisons to pruning would be helpful.
    - This ends up being hidden, as a result, but looking at the uncertainty quantification results in Tables 10+, it seems like generically the deterministic NN outperforms FFG-U and ensemble-U in terms of accuracy, but loses in terms of ECE, thus continuing to weaken the empirical evidence.
        - This type of result really shouldn't be hidden deep in the appendix.

- At its core, one of the weaknesses of SVI for NNs is that it tends to significantly underfit the data (see for example [Blier & Ollivier, NeurIPS, '18](https://papers.nips.cc/paper/2018/file/3b712de48137572f3849aabd5666a4e3-Paper.pdf)), hence, why the variance is clipped practically (lines 240-246), yet, your approach still seems like it'll underfit (given empirically that you still have to clamp the variances and the improved ECE, which is often an indicator of the amount of posterior variance).
    - Should your approach get around this under-fitting issue or is its success solely due to the increased amount of structure?
    - What happens if you don't clamp the variance in your approach?


### Questions

- More details on the *Ensemble-U* baseline would be very helpful. Is this just an ensemble of five models with the extra regularizer of $p(\text{vec}(W) | \text{vec}(U))$ as the prior?
    - If that's not quite what it is, then that should be the baseline.
    - Also, see the comment on line 148 above.

- What does the reduction in parameter count mean practically? Is there actually a memory reduction or is this negated by the requirement of forming the full layerwise matrices $W$ via sampling using $U$?
    - Does it induce some sort of sparsity in weight space or something along those lines?

- What is the say "accuracy vs model size" tradeoff required on CIFAR-10 / 100? It's hard to parse from the two small points in Fig. 4.

- Why does ECE skyrocket in Fig 7 as more pruning is performed? 
    - This seems to be the case in Fig 5 bottom right for small $M$.

- What is training time like? In Fig. 4 you mention prediction time? It seems like it'll be a few times slower due to the extended Matheron's rule comments.

- Why should magnitude based pruning not change the probabilistic model in FFG-U?

- Overall, what happens if you ignore the layer-wise approximation and do the reduction on the full space (e.g. $\text{vec}(W)$ refers to the vector of all parameters of the NN)? Does the approach still work?

### Writing Comments

- Fig 4 left is too small and detailed, had to zoom into 400% to get it to be visible, as well as to be able to parse taht there's actually two separate architectures in there. I'd suggest displaying only one architecture and the two inference types to make this clearer.
    - Can this be shown on a log scale as well? Is there ever a "break-even point" where with enough samples, the Matheron's rule approach is as fast, so to speak?

- line 657: mean + noise isn't helpful.

- Personally, I'd suggest replacing most of the tables with plots using the x axis as the skew intensity level. The tables are really tough to parse as nothing is bolded or anything.

- throughout: define "push-forward distribution" is this just the transformed (matrix) normal distribution after some sort of linear transformation?
limitations_and_societal_impact: maybe suggest why smaller neural networks are better or worse for society?
review_title: null

final_score: rating: 6: Marginally above the acceptance threshold
review_content: summary: This paper proposes a method for reducing the storage cost of BNNs. The idea is to apply the inducing point method and fast Matheron's sampling rule, commonly used in GPs, to standard variational BNNs. Experiments show that the resulting BNNs stay competitive to standard BNNs while enjoying up to 75% reduction in parameter size.
main_review: **Pros:**

This paper is generally well written, though not without blemishes (see below). The method is solid and can be seen as a "natural" counterpart of the standard inducing weight method in function space (GPs) for weight space (BNNs).


**Cons:**

* While the proposed method significantly reduces the storage cost of BNNs, it introduces a large overhead during prediction. E.g. the proposed BNNs are around twice as slow as the vanilla counterparts for the commonly-used number of MC samples (e.g. 10, see Tab. 4). This makes me wonder whether the trade-off is worth it at all, considering ResNet-50 (the architecture that the authors use) for ImageNet only requires just under 100MB of storage space. In contrast, prediction latency has a real-world, tangible effect in any real-time predictive system, e.g. in self-driving cars. I hope to see a discussion from the authors during the rebuttal period.

* The paper can be hard to read at times. I think this is because of how dense the presentation of the paper is. Given how dense this paper is already, it is thus not good to see that the authors still tried to squeeze every little bit of space by using e.g. text style symbols (e.g. \textstyle\sum) in display equations. I think it would be much better presentation-wise if the authors move some least important sections to the appendix, and then be more judicious with whitespace.


**Minor comments/suggestions:**

* Please *do not* just copy-and-paste bibtex entries from Google Scholar. Many references are either inaccurate or inconsistent. For instance, Hendrycks & Dietterich, Kingma & Ba are in ICLR; "A tutorial on sparse gaussian processes ..." -> "A tutorial on sparse **G**aussian processes"; etc.
* L.36, L.41: please be consistent with the em dashes.
* Starting from L.57: The usage of the boldface in math should be consistent. The matrices $\mathbf{X}, \mathbf{Y}$ are in bold, but $W_l$ among other quantities are not.
* L.60: Hinton and Van Camp, 1993 should be cited.
* L.65: Missing iterates over $i, j$ in the definition of $\phi$
* L.78: "This approach returns ..." -> "This approach yields ..."
* L.78-79: Break the sentence up
* L.81: "... so can share parameters" what does it mean exactly?
* L.87-88: Too dense with the inline math, please use display math
* L.94-95: Inline math hard to read. Consider adding some words in between the two $\widetilde{p}(\mathbf{v})$'s
* L.134-135: More efficient compared to what complexity?
* Fig. 1: Use $\left( ... \right)$, esp. for the middle term consisting of green and white vertical bars
* L.175: "... the non-linearity ..." -> "... a non-linearity ..."
* L.184: What does "d" on top of \sim mean?
* Fig. 3: Need Ensemble-W result


---

**Post-rebuttal**

I increased my score. Nevertheless, I still think that the presentation of the paper needs to be improved:

* The application of parameter-efficient-but-slower BNNs should be discussed more explicitly in the introduction. This will help motivate the method more. Also, it is instructive to show the parameter efficiency of the method on larger networks (that take more than 100MB of storage).
* The exposition of the method should be improved more---there should be a balance between information content and density.
limitations_and_societal_impact: I might have missed it, but I could not find any explicit discussion regarding the limitations of the method. This is especially crucial due to the first point in my "Cons" above.
review_title: null

Meta-review: This paper comes up with a variational inference strategy for learning a factored weight matrix, that they can then sample over efficiently using Matheron’s rule.  They need <25% of the parameters of the original deep ensemble but are competitive across metrics.  Four reviews were given for this paper with scores 6, 7, 4, 7.  Thus there was a general consensus to accept with one reviewer arguing for a reject.  In general the reviewers found the paper technically strong, novel and important in terms of the problem it addresses.

One of the major criticisms of the paper is that it is too dense and the reader is overloaded with technical background.  All reviewers seem to agree with this point (i.e. in discussion: “I wholly agree .. that the presentation of this paper is its major weakness”), but differ quite significantly in how important they think this is.  Some of the reviewers applaud the authors for striving for completeness, but seem to have found it a little excessive.  In particular, the reviewers didn't think presenting both the function space and weight space perspective was necessary.  Thus the authors are strongly recommended to move one of these to the appendix to make space for other parts of the paper.

There’s some criticism of the priors used - the reviewer felt like the empirical comparison wasn’t apples to apples because of the priors were different between their method and the baselines.  This seems reasonable, but the choice of prior seems core to the proposed method (needs to be decomposable due to Matheron’s rule).  Thus it would seem acceptable to not require the authors to change the prior of their method.

The final criticism is that the method is slower than the original ensemble, so sparsity doesn’t help - though there is still (in theory) considerable memory savings.  Again, the reviewers differed in their assessment of how important this issue is.  

The majority felt that this is a strong paper. Considering that the major cited weakness is in the presentation of the technical background, the recommended decision is to accept the paper with the ask that the authors rework the technical background to make it more accessible.  In particular, the reviewers recommended that the function space perspective could be moved to the appendix in the camera ready.  This would free up space to make the paper less mathematically dense and allow more space for other parts of the paper.

Decision: Accept (Poster)