Paper: A Theory of Unimodal Bias in Multimodal Learning (ICLR-2024)

Reviewer comment (Novelty):
The author claims "there is a scarce theoretical understanding of how unimodal bias arises and how it is affected by the network configuration, dataset statistics, and initialization". However, the work [1] mentioned in this paper already theoretically explored the rise of unimodal bias in a more realistic neural network setting, and their results somewhat reveal the relationship between the inferior performance of late-fusion networks with initialization and modality correlations.

Author response:
We thank the reviewer for highlighting the two related papers and would like to provide a more detailed comparison between our work and theirs.

- Huang et al studied two-layer late fusion network with smooth ReLU activation. We compare the differences between their and our problem setup in the table below. We make a stronger assumption on the activation function. However, in return, we are able to study deep networks with all three common fusion schemes trained with standard gradient descent, all of which were not covered in Huang et al.

  Actually, a main motivation of our work came from the Limitations section in Huang et al, where they point out in their Limitations section that their analysis focused on two-layer late fusion networks and immediate future directions are to study **other fusion frameworks** and **deep neural networks**.

  |                     | Huang et al                                                  | Ours                                     |
  | ------------------- | ------------------------------------------------------------ | ---------------------------------------- |
  | Network Depth       | Two-layer                                                    | **Deep**                                 |
  | Activation function | **Smoothed ReLU**                                            | Linear                                   |
  | Fusion scheme       | Late fusion                                                  | **Late, intermediate, and early fusion** |
  | Training            | Fixing the second layer, gradient descent on only the first layer | **Standard gradient descent**            |

  In terms of analytical results, Huang et al proved the occurrence of modality competition. We provide new insights into the process of this competition by studying the learning dynamics. We also derive an analytical expression to describe the influence of dataset statistics and network configurations on unimodal bias, for which Huang et al and other prior literature did not have theory-grounded answers.

Reviewer comment (Experiments):
For the data generation process, it appears that only the correlation matrices are required. Are there specific assumptions made regarding $y$ that need to be clarified?

Author response:
Thank you for this note. We have added in our main text that we assume the input covariance $\boldsymbol \Sigma$ has full rank but make no further assumption on the correlation matrices. 

  Since the gradient descent dynamics of multimodal linear networks given in Eq. (3-4) only involve the correlation matrices $\boldsymbol \Sigma$ and $\boldsymbol \Sigma_{yx}$, knowing $\boldsymbol \Sigma$ and $\boldsymbol \Sigma_{yx}$ suffices to determine the learning trajectories. We have also added an Implementation Details section at the end of Appendix.

Reviewer comment (Theory):
The author claims "we develop a theory of unimodal bias with deep multimodal linear networks." However, the frequent use of the approximation symbol ($\approx$) in the appendix when deriving mean results raises questions about the rigor and precision of the theoretical justifications provided.

Author response:
We acknowledge our frequent usage of $\approx$ in the Appendix. To control approximations more explicitly, we are re-working through our derivations to include error correction terms and will post these updates as soon as we are done.

  All our approximation signs are precisely equality when the initialization scale is taken to the limit of $0$. The approximations are also justified by our simulation results, which align well with theoretical predictions.

Reviewer comment (Experiments):
The paper presents an interesting study of unimodal bias in intermediate and late fusion contexts, yet the evidence supporting the absence of unimodal bias in early fusion remains unconvincing. The reliance on the Frobenius norm of weights as a metric for understanding unimodal bias seems reasonable, but the experiments regarding early fusion are limited to simplistic scenarios. These toy settings are insufficient to assert the absence of unimodal bias in early fusion.

Author response:
- Absence of unimodal bias in early fusion

  We would like to clarify that we only claim that unimodal bias is absent in early fusion linear networks. And we strongly agree that we cannot claim the same for other nonlinear early fusion networks. For early fusion linear networks, we studied linearly separable data without losing generality since linear networks can only learn the linear component even if the dataset contains a nonlinear task. In addition to the one example in Fig 2, we also numerically computed the time ratio for four-layer early fusion linear networks in Fig 5b-d (purple circles), which are all close to 1.

  It is also important to note that the learning curves for different modalities are not exactly the same in early fusion. By absence, we intend to mean that we see no conspicuous unimodal phase during training. Since early fusion linear networks learn both modalities during one transitional period, the time lag is bounded by the duration of the transitional period, which approaches zero when the initialization scale is taken to the limit of $0$. In comparison, the time lag is much longer in late fusion linear networks since two modalities are learned during two separated transitional periods with a unimodal phase in between. We can revise the wording "absent" if it is the word that particularly misleads readers.

Reviewer comment (Experiments):
All experiments are conducted with linearly separable data. The inclusion of experiments with XOR data, where early fusion fails to perform effectively, further casts doubt on the claims for early fusion. Considering that the main focus of the paper was to investigate the interplay between unimodal bias, network configuration, and dataset statistics, the scope and depth of the study become critical. When this research is contrasted with prior studies that have identified unimodal bias across a diverse range of datasets, the robustness of the current findings for complex, real-world scenarios appears uncertain.

Author response:
- Uncertain robustness for real-world scenarios

  Our results provide a fundamental understanding of unimodal bias in deep linear networks, and in this specific setting, earlier fusion does shorten the unimodal phase. However, this analysis is only one part of a more complete picture: in nonlinear settings, several pre-fusion layers may be necessary to implement nonlinear transformations. This is why we include experiments on XOR, in order to show the additional factors that enter in nonlinear settings. We hope our work provides practitioners with some theoretical intuitions and theorists with a prerequisite to study unimodal bias in more complex scenarios.

  We note that all theoretical approaches at present study simplified settings to obtain greater tractability, and this work is one step toward a more complete treatment. Our intuition and qualitative conclusions can still apply to complex scenarios. For instance, the idea that a shallower fusion layer enables modalities to cooperate more reciprocally and alleviates unimodal bias still holds in complex multimodal networks. This is because it remains true that the gradient descent updates for each layer share more common terms when the fusion layer is shallower.

  We have also found our results consistent with empirical findings while containing more details for our specific setup. To give examples, multimodal networks were found to utilize different modalities at different speeds during training in [1,2], which is consistent with our time ratio results. [3] found that balancing the dataset mitigates unimodal bias, which is consistent with our conclusion that smaller disparities in input-output correlations reduce unimodal bias. 

  [1] Wu, et al. "Characterizing and overcoming the greedy nature of learning in multi-modal deep neural networks." ICML. 2022.

  [2] Wang, et al. "What makes training multi-modal classification networks hard?." CVPR. 2020.

  [3] Goyal, et al. "Making the v in vqa matter: Elevating the role of image understanding in visual question answering." CVPR. 2017.

Reviewer comment (Writing):
In equation 2, did you intend to use W^{tot} alongside y?.

Author response:
- Notation in Eq 2

  We checked that we do intend to write down $\boldsymbol W^{\text{tot}}{\boldsymbol x}$. We re-write Eq 2 with more details below:
  $$
  \hat y (\boldsymbol x; \boldsymbol W) = \prod _{j=L_f+1}^L \boldsymbol W^j \left( \prod _{i=1}^{L_f} \boldsymbol W^i_A \boldsymbol x_A + \prod _{i=1}^{L_f} \boldsymbol W^i_B \boldsymbol x_B \right) 
  \equiv \boldsymbol W^{\text{tot}}_A \boldsymbol x_A + \boldsymbol W^{\text{tot}}_B \boldsymbol x_B 
  \equiv \boldsymbol W^{\text{tot}} \boldsymbol x  ,
  $$
  where
  $$
  \boldsymbol W^{\text{tot}}_A \equiv  \prod _{j=L_f+1}^L \boldsymbol W^j \prod _{i=1}^{L_f} \boldsymbol W^i_A ,
  $$

  $$
  \boldsymbol W^{\text{tot}}_B \equiv \prod _{j=L_f+1}^L \boldsymbol W^j \prod _{i=1}^{L_f} \boldsymbol W^i_B  ,
  $$

  $$
  \boldsymbol W^{\text{tot}} \equiv \left[ \boldsymbol W^{\text{tot}}_A, \boldsymbol W^{\text{tot}}_B \right] .
  $$

Reviewer comment (Presentation):
The latter part of the caption for figure 2, particularly the description following parts a-c, is somewhat confusing and could benefit from a more straightforward explanation.

Author response:
- Caption of Fig 2

  We apologize for inducing confusion and have changed the order of the caption for Fig 2. In Fig 2b and 2d, we plot the loss and total weight trajectories with respect to time (i.e., epoch) for a 2D simple case. The loss trajectory is the training mean square error loss. The total weights are $\boldsymbol W^{\text{tot}}_A, \boldsymbol W^{\text{tot}}_B$ as defined in Eq 2 and clarified above. Since we are assuming scalars $x_A, x_B$ in Fig 2, $\boldsymbol W^{\text{tot}}_A, \boldsymbol W^{\text{tot}}_B$ are also scalars and thus can be directly plotted.

  Please let us know if this is still unclear or we got the question wrong.

Reviewer comment (Presentation):
Can the authors provide more details with respect to Fig 3e)?

Author response:
- Caption of Fig 3e

  We realize it can be confusing that we plotted the time ratio twice in Fig 3b and 3e and want to re-try explaining.

  Lines in Fig 3b and the entire Fig 3e are plotted with theoretical predictions. Specifically, the theoretical prediction here is a 2D corollary to the general time ratio in late fusion linear networks given in Eq 8. By plugging the standard 2D covariance matrix $\boldsymbol\Sigma = \left[ \sigma_A^2 , \rho \sigma_A \sigma_B ; \rho \sigma_B \sigma_A , \sigma_B^2 \right]$ and target output $y = x_A + x_B$ into the Eq 8, we obtain the time ratio in the special case of 2D inputs:
  $$
  \frac{t_B}{t_A} = 1 + \frac{\frac{\sigma_A^2}{\sigma_B^2}-1}{1-\rho^2} .
  $$
  We plot Fig 3b to show that our theoretical predictions align closely with simulations. We plot Fig 3e to show how the theoretical predictions vary with continuous values of correlation $\rho$ and variance ratio $\sigma_A/\sigma_B$. We are open to the removal Fig 3e if most readers find it redundant.

  The rationale for assuming 2D inputs in Fig 3 is that metrics for variances inside a modality and correlations across modalities are well-defined in 2x2 covariance matrices, which allows us to systematically sweep the variance and correlation parameters. Furthermore, we have found that covariance matrices with higher dimensions do not give rise to any qualitatively different behavior as long as the input dimension is not larger than the number of training samples.

Reviewer comment (Evaluation):
“Two-layer early fusion ReLU networks do not learn XOR features and can even fail to learn this task” – can the authors comment and provide more reasoning about this observation with respect to the XOR experiment?

Author response:
- More comments on the XOR experiment

  We thank the reviewer for their interest in our XOR experiment. We have added videos in the supplementary material and extended the Appendix H.2 on the XOR experiment. The videos show the time evolution of features corresponding to Fig 11.
  
  Late fusion ReLU networks learn the XOR and linear task better because the XOR branch of the network forms the XOR features in the two-dimensional space without being interfered by the linear branch. The features in the XOR branch preserve its independence from the linear branch as how the weights are independently initialized. However, early fusion ReLU networks favor particular feature directions in the three-dimensional space that correlate the most with the target output. The favorable feature directions tilt more towards the direction of the linear modality and are more prone to let the network be stuck in local minimum when the variance in the linear modality is larger.