Paper: Minimizing Chebyshev Risk Magically Mitigates the Perils of Overfitting (ICLR-2024)

Reviewer comment (Theory):
I could not figure out the theoretical justification for using DS in Lemma 3.1 or Corollary 3.1.1.

Author response:
Hello reviewer wkG8,
We apologize, we did not fully understand your question initially – we should go beyond the numerical side to answer your question.

Assume there are K classes and K prototypes, one for each class.  Assume that if a feature vector, v, has high cosine similarity to a prototype feature vector, p_k, from a class k, it will be classified into class k.  Likewise, for a given class and a sufficiently trained network, E[CS(v,p)] should be on the higher end of the interval [0,1] since we expect the general population of class k feature vectors to be classified correctly and similar to p_k.  

If a new example v were to deviate too far from its true prototype, it would get close to some other untrue prototype and be classified into that untrue prototype’s class (a misclassification).  Therefore, the further that v deviates from its true prototype, the risk of misclassification increases because it would be getting closer to other untrue prototypes.  By saying something mathematical about the probability of v deviating a certain distance from its true prototype, we are also saying something about its probability of misclassification.

Now, we find its best to think in terms of “distances” (really cosine distance) between vectors and prototypes when looking at the left-hand side.

The left-hand side of the inequality is | CS(v,p) – E[CS(v,p)] |.    As this difference gets larger, this means v is getting further away from its true prototype (a larger cosine distance) as compared to the rest of the feature vectors in its class.  At the same time, it is getting closer to one or more of the other (K-1) prototypes.  DS represents the average cosine distance between prototypes.  We can think about DS like a threshold distance.  If  | CS(v,p) – E[CS(v,p)] |  becomes greater than DS, then it means that v has moved a cosine distance DS away from its true prototype, but this is the average distance between prototypes from different classes!  This means that v would now very likely be close to an untrue prototype and be misclassified.  Therefore, by bounding | CS(v,p) – E[CS(v,p)] | > DS, we are bounding the probability of v transitioning from its true class prototype to an untrue class prototype and being misclassified.

Pr [ distance away from true class > misclassification threshold distance  ]   <   intra-class feature covariance /  (misclassification threshold distance^2)

We see that, for our learning method and intuitively, it is beneficial to increase the spacing between prototypes as much possible (increase DS) so that a feature vector must move further away from its true class to reach the misclassification threshold.  Increasing DS decreases the upper bound, which is our desired behavior.

Reviewer comment (Theory):
If I understand correctly, the DS part in Eq.(6) can be any positive variable. Then what is the reason for using DS here?

Author response:
If I understand correctly, the DS part in Eq.(6) can be any positive variable. Then what is the reason for using DS here?
Moreover, the authors claim to regularize DNN training by increasing DS (which is established by decreasing DS), because it leads to a smaller value of the right part of Eq.(6). However, the larger DS value leads to the looser condition from the point of view of the left part of Eq.(6).

We can put any value for 'a' into the proof of the probability bound in eqn 22, but because the left hand side | CS(v,p) - E[CS(v,p)] | only exists on the interval [0,1], any 'a' value above 1 is not useful since the probability distribution goes to zero.

Reviewer comment (Theory):
The authors claim to regularize DNN training by increasing DS (which is established by decreasing $\mathcal{L}_{CS}$), because it leads to a smaller value of the right part of Eq.(6). However, the larger DS value leads to the looser condition from the point of view of the left part of Eq.(6).

Author response:
Hi wkG8,

It seems we may have run out of discussion period - but it has been nice to interact with you and thank you for putting in so much effort.

Yes! you are right Lcs is introduced to increase DS, the spacing between class prototypes.

The key is that we don't just want the inequality to hold, we want it to compress the probability down as much as possible by lowering cov/DS^2.

To summarize our previous discussion, the left-hand side is the probability that I draw a new example feature vector v and its relative similarity to its prototype is greater than DS.  This could be True OR False with some probability.  The right hand side is telling us the upper limit on the probability of it being True.  Therefore, if we lower this upper limit, we compress the probability of the left hand side being True lower, which is what we want.  We do not want | CS(v,p) - E(CS(v,p))| > DS to be true, we prefer it to be False: | CS(v,p) - E(CS(v,p))| < DS.

Now imagine that we train 100 different ML models maybe with different loss functions (as we have done in our results).  All of these models will have different values of DS and cov() and thus different upper limits on that probability.  Our results show that if you explicitly optimize the right hand side, you can indeed lower the probability of that left hand side being True and thus have fewer misclassifications.

Reviewer comment (Theory):
In Lemma 3.1, is there an assumption that the class label of $v$ is $k$?

Author response:
In Lemma 3.1, is there an assumption that the class label of v is k?
Yes.  We propose to change the wording of lemma 3.1 to read:
“Lemma 4.3.1 Given a sufficiently trained classifier … , [a feature vector v for a class k input example],…”

Reviewer comment (Reproducibility):
In Section 5.4, I could not understand how the hyperparameters are determined in the proposed method.

Author response:
In Section 5.4, I could not understand how the hyperparameters are determined in the proposed method.
We can be more clear here.  We propose to change the wording in section 4.5.4:

“For our algorithm, we could not conduct a large hyperparameter study on (beta,gamma,zeta) for computational reasons, so we instead performed cross-entropy loss training on the first training subset for 10 epochs and adjusted the values of (beta,gamma,zeta) until the other 3 loss components Lcov, Lcs, and Lproto had similar magnitude to Lce at epoch 10.  Our goal was to ensure that after the 10 epoch warmup period the relative scales of the different losses were similar and that the full training could optimize the Chebyshev Prototype Risk (CPR) while still fitting the data.  Our results indicate that our hyperparameter choices indeed allowed the model to fit the data to 100\% training accuracy while still reducing the CPR.”

Reviewer comment (Novelty):
The concept of minimizing intra-class variance while encouraging larger inter-class distances seems very similar to the well-observed phenomenon of neural collapse [1]. In [1], it was observed that after training for a sufficiently long time, the final layer feature embeddings collapse to class means and form a simplex ETF structure. The classifier on top also coincides with these. It was also shown to improve test performance. How does the proposed approach relate to this? I suggest including some discussion on the connection/comparisons with this.

Author response:
Thank you for pointing us towards the line of research on neural collapse phenomena, after reviewing papers in this area, it is certainly worth a paragraph discussion in our paper because we believe our basic idea behind the Chebyshev bound development contributes a different, probabilistic mathematical model to this discussion.

The main question regards whether our baseline models are in the terminal phase of training, implying that 100% of the training data has been fit and Tr(SwSb), a measure of intra-class feature covariance as a noise-to-signal ratio, has collapsed during the natural course of training (the phenomenon).  The short answer is that our baseline models have not reached the terminal phase yet and our algorithm is able to significantly reduce intra-class feature covariance compared to the baselines.

We do have data on the intra-class feature covariance (numerator of CPR) for our models at epoch 100 showing that our algorithm reduced the class average numerator by 50% compared to baselines in both weight decay settings and both datasets.  Being able to reduce these numbers implies that the intra-class feature covariance had not collapsed yet to the point we could no longer effectively reduce them.  There are two limitations to comparing directly to Papyan et. al [1].  First, we use standard flipping/cropping while they did not include data augmentation, which introduces more variation in the input space, greatly benefits test accuracy, and is more standard practice in visual tasks.  Second, our intra-class feature covariance is prototype weighted because our mathematical model required it – note how our numerator is E[pi(vi − pi)pj (vj − pj )], not the typical covariance E[(vi − pi)(vj − pj )].  Having said this, we have added the following paragraph to the paper after the CPR definition:

•	Our probabilistic model for CPR has parallels to Neural Collapse property (NC1), which is the collapse of within-class feature covariance when a neural network has been trained for a sufficient number of epochs beyond 100\% training accuracy (see \citep{Papyan_2020} for details).  Through this lens, our work shares two connections to Neural Collapse.  First, our results show that our algorithm significantly reduces intra-class feature covariance compared to baseline models within a typical duration (100 epochs) of training, which is not in the terminal phase of training – in this way our loss components accelerate the covariance effects of neural collapse, which has been shown to improve generalization in many settings and lends credence to our Chebyshev model that reducing CPR reduces overfitting \citep{Papyan_2020}.  The second connection is that we introduce a new probabilistic modeling of the similarity between an example’s features to its class mean feature vector (Lemma \ref{lem:2}), which to our knowledge has not been mathematically derived in the existing Neural Collapse discussion.  Future work could look to integrate theoretical arguments like ours into modeling Neural Collapse phenomena.

Reviewer comment (Experiments):
Can the authors verify whether the baseline numbers used for comparison are computed in the terminal phase of training where neural collapse happens (please refer to [1])? Since the proposed approach seems to have a similar motivation as this phenomenon, it would be interesting to see how much incorporating the loss helps compared to just training for a large enough time.

Author response:
The main question regards whether our baseline models are in the terminal phase of training, implying that 100% of the training data has been fit and Tr(SwSb), a measure of intra-class feature covariance as a noise-to-signal ratio, has collapsed during the natural course of training (the phenomenon).  The short answer is that our baseline models have not reached the terminal phase yet and our algorithm is able to significantly reduce intra-class feature covariance compared to the baselines.

We do have data on the intra-class feature covariance (numerator of CPR) for our models at epoch 100 showing that our algorithm reduced the class average numerator by 50% compared to baselines in both weight decay settings and both datasets.  Being able to reduce these numbers implies that the intra-class feature covariance had not collapsed yet to the point we could no longer effectively reduce them.  There are two limitations to comparing directly to Papyan et. al [1].  First, we use standard flipping/cropping while they did not include data augmentation, which introduces more variation in the input space, greatly benefits test accuracy, and is more standard practice in visual tasks.  Second, our intra-class feature covariance is prototype weighted because our mathematical model required it – note how our numerator is E[pi(vi − pi)pj (vj − pj )], not the typical covariance E[(vi − pi)(vj − pj )].  Having said this, we have added the following paragraph to the paper after the CPR definition:

•	Our probabilistic model for CPR has parallels to Neural Collapse property (NC1), which is the collapse of within-class feature covariance when a neural network has been trained for a sufficient number of epochs beyond 100\% training accuracy (see \citep{Papyan_2020} for details).  Through this lens, our work shares two connections to Neural Collapse.  First, our results show that our algorithm significantly reduces intra-class feature covariance compared to baseline models within a typical duration (100 epochs) of training, which is not in the terminal phase of training – in this way our loss components accelerate the covariance effects of neural collapse, which has been shown to improve generalization in many settings and lends credence to our Chebyshev model that reducing CPR reduces overfitting \citep{Papyan_2020}.  The second connection is that we introduce a new probabilistic modeling of the similarity between an example’s features to its class mean feature vector (Lemma \ref{lem:2}), which to our knowledge has not been mathematically derived in the existing Neural Collapse discussion.  Future work could look to integrate theoretical arguments like ours into modeling Neural Collapse phenomena.