SlowGuess/NeurIPS_2023_OCR huggingface dataset:
CIFAR-10	CIFAR-100	ImageNet
MI-based	Ranking	MI-based	Ranking	MI-based	Ranking
K	|S|=0	Ψ	τ	K	|S|=0	Ψ	τ	K	|S|=0	Ψ	τ
CP	0.38	0.59	0.71	0.83	0.49	0.53	0.80	0.80	0.75	0.75	0.59	0.92
DP	0.38	0.59	0.80	0.84	0.42	0.47	0.75	0.76	0.81	0.81	0.60	0.94
D.2 Training error
Tables 17 and 18 show the training error values (the average, minimum, median, and the maximum) associated with the empirical results reported in Section 3.1 and 3.2, respectively.
Table 17: Training error of the models analyzed in Section 3.1. The four numbers represent the average, minimum, median, and maximum values  $(\%)$ .
CIFAR-10	CIFAR-100	ImageNet
Figure 1,3,6	0.4	0.0	0.0	8.8	4.7	0.0	0.1	53.6	24.9	3.0	23.5	56.6
Figure 2	3.3	0.0	2.6	10.9	18.2	0.2	13.6	56.3	44.5	16.8	45.6	65.3
Figure 5,7	0.5	0.0	0.0	10.1	4.0	0.0	0.0	54.5	16.3	0.1	11.3	59.0
Table 18: Training error of the models analyzed in Section 3.2. The four numbers represent the average, minimum, median, and maximum values  $(\%)$ .
Case#1	Case#2	Case#3	Case#4	Case#5
10.0	8.8	10.1	11.6	4.5	2.1	4.5	6.7	0.06	0.02	0.04	0.17	0.04	0.01	0.03	0.11	0.0	0.0	0.0	0.0
Case#6	Case#7	Case#8	Case#9	Case#10
0.7	0.3	0.8	1.1	0.1	0.1	0.1	0.2	4.8	3.9	4.9	5.5	2.7	1.9	2.9	3.3	2.8	0.5	1.8	6.5
D.3 More on inconsistency and disagreement
Inconsistency takes how strongly the models disagree on each data point into account while disagreement ignores it. That is, the information disagreement receives on each data point is binary (whether the classification decisions of two models agree or disagree) while the information inconsistency receives is continuous and more complex. On the one hand, this means that inconsistency could use information ignored by disagreement and thus it could behave quite differently from disagreement as seen in our empirical study. On the other hand, it should be useful also to consider the situation where inconsistency and disagreement are highly correlated since in this case our theoretical results can be regarded as providing a theoretical basis for the correlation of not only inconsistency but also disagreement with generalization gap though indirectly.
To simplify the discussion towards this end, let us introduce a slight variation of Theorem 2.1, which uses 1-norm instead of the KL-divergence since disagreement is related to 1-norm as noted in Section 2.
Proposition D.1 (1-norm variant of Theorem 2.1). Using the notation of Section 2, define 1-norm inconsistency  $\mathcal{C}_{1,P}$  and 1-norm instability  $S_{1,P}$  which use the squared 1-norm of the difference in place of the KL-divergence as follows.
$$
\mathcal {C} _ {1, P} = \mathbb {E} _ {Z _ {n}} \mathbb {E} _ {\Theta , \Theta^ {\prime} \sim \Theta_ {P | Z _ {n}}} \mathbb {E} _ {X} \| f (\Theta , X) - f (\Theta^ {\prime}, X) \| _ {1} ^ {2} \quad (1 - \text {n o r m i n c o s i s t e n c y})
$$
$$
\mathcal {S} _ {1, P} = \mathbb {E} _ {Z _ {n}, Z _ {n} ^ {\prime}} \mathbb {E} _ {X} \| \bar {f} _ {P | Z _ {n}} (X) - \bar {f} _ {P | Z _ {n} ^ {\prime}} (X) \| _ {1} ^ {2} \quad (1 \text {- n o r m i n s t a b i l i t y})
$$
Then with the same assumptions and definitions as in Theorem 2.1, we have
$$
\mathbb {E} _ {Z _ {n}} \mathbb {E} _ {\Theta \sim \Theta_ {P | Z _ {n}}} \left[ \Phi_ {\mathbf {Z}} (\Theta) - \Phi (\Theta , Z _ {n}) \right] \leq \inf  _ {\lambda > 0} \left[ \frac {\gamma^ {2}}{2} \psi (\lambda) \lambda \left(\mathcal {C} _ {1, P} + \mathcal {S} _ {1, P}\right) + \frac {\mathcal {I} _ {P}}{\lambda n} \right].
$$
Sketch of proof. Using the triangle inequality of norms and Jensen's inequality, replace inequality (2) of the proof of Theorem 2.1
$$
\mathbb {E} _ {Z _ {n}} \mathbb {E} _ {\Theta \sim \Theta_ {P \mid Z _ {n}}} \mathbb {E} _ {X} \| f (\Theta , X) - \bar {f} _ {P} (X) \| _ {1} ^ {2} \leq 4 \left(\mathcal {C} _ {P} + \mathcal {S} _ {P}\right) \tag {2}
$$
with the following,
$$
\begin{array}{l} \mathbb {E} _ {\mathcal {Z} _ {n}} \mathbb {E} _ {\Theta \sim \Theta_ {P | Z _ {n}}} \mathbb {E} _ {X} \| f (\Theta , X) - \bar {f} _ {P} (X) \| _ {1} ^ {2} \\ \leq 2 \mathbb {E} _ {Z _ {n}} \mathbb {E} _ {\Theta \sim \Theta_ {P | Z _ {n}}} \mathbb {E} _ {X} \left[ \| f (\Theta , X) - \bar {f} _ {P | Z _ {n}} (X) \| _ {1} ^ {2} + \| \bar {f} _ {P | Z _ {n}} (X) - \bar {f} _ {P} (X) \| _ {1} ^ {2} \right] \\ \leq 2 \mathbb {E} _ {Z _ {n}} \mathbb {E} _ {\Theta , \Theta^ {\prime} \sim \Theta_ {P | Z _ {n}}} \mathbb {E} _ {X} \| f (\Theta , X) - f (\Theta^ {\prime}, X) \| _ {1} ^ {2} + 2 \mathbb {E} _ {Z _ {n}} \mathbb {E} _ {Z _ {n} ^ {\prime}} \mathbb {E} _ {X} \| \bar {f} _ {P | Z _ {n}} (X) - \bar {f} _ {P | Z _ {n} ^ {\prime}} (X) \| _ {1} ^ {2} \\ = 2 \left(\mathcal {C} _ {1, P} + \mathcal {S} _ {1, P}\right). \\ \end{array}
$$
Now suppose that with the models of interest, the confidence level of model outputs is always high so that the highest probability estimate is always near 1, i.e., for any model  $\theta$  of interest, we have  $1 - \max_{i} f(x, \theta)[i] < \epsilon$  for a positive constant  $\epsilon$  such that  $\epsilon \approx 0$  on any unseen data point  $x$ . Let  $c(\theta, x)$  be the classification decision of  $\theta$  on data point  $x$  as in Section 2:  $c(\theta, x) = \arg \max_{i} f(\theta, x)[i]$ . Then it is easy to show that we have
$$
\frac {1}{2} \| f (\theta , x) - f \left(\theta^ {\prime}, x\right) \| _ {1} \approx \mathbb {I} [ c (\theta , x) \neq c \left(\theta^ {\prime}, x\right) ]. \tag {8}
$$
Disagreement measured for shared training data can be expressed as
$$
\mathbb {E} _ {Z _ {n}} \mathbb {E} _ {\Theta , \Theta^ {\prime} \sim \Theta_ {P \mid Z _ {n}}} \mathbb {E} _ {X} \mathbb {I} [ c (\Theta , X) \neq c \left(\Theta^ {\prime}, X\right) ]. \tag {9}
$$
Comparing (9) with the definition of  $\mathcal{C}_{1,P}$  above and considering (8), it is clear that under this high-confidence condition, disagreement (9) and 1-norm inconsistency  $\mathcal{C}_{1,P}$  should be highly correlated; therefore, under this condition, Proposition D.1 suggests the relation of disagreement (measured for shared training data) to generalization gap indirectly through  $\mathcal{C}_{1,P}$ .
While this paper focused on the KL-divergence-based inconsistency motivated by the use of the KL-divergence by the existing algorithm for encouraging consistency, the proposition above suggests that 1-norm-based inconsistency might also be useful. We have conducted limited experiments in this regard and observed mixed results. In the settings of Appendix D.1, the correlation scores of 1-norm inconsistency with respect to generalization gap are generally either similar or slightly better, which is promising. As for consistency encouragement during training, we have not seen a clear advantage of using 1-norm inconsistency penalty over using the KL-divergence inconsistency penalty as is done in this paper, and more experiments would be required to understand its advantage/disadvantage.
In our view, however, for the purpose of encouraging consistency during training, KL-divergence inconsistency studied in this paper is more desirable than 1-norm inconsistency in at least three ways. First, minimization of the KL-divergence inconsistency penalty is equivalent to minimization of the standard cross-entropy loss with soft labels provided by the other model; therefore, with the KL-divergence penalty, the training objective can be regarded as a weighted average of two cross-entropy loss terms, which are in the same range (while 1-norm inconsistency is not). This makes tuning of the weight for the penalty more intuitive and easier. Second, optimization of the standard cross-entropy loss term with the KL-divergence inconsistency penalty has an interpretation of functional gradient optimization, as shown in [19]. The last (but not least) point is that optimization may be easier with KL-divergence inconsistency, which is smooth, than with 1-norm inconsistency, which is not smooth.
Related to the last point, disagreement, which involves arg max in the definition, cannot be easily integrated into the training objective, and this is a crucial difference between inconsistency and disagreement from the algorithmic viewpoint.
Finally, we believe that for improving deep neural network training, it is useful to study the connection between generalization and discrepancies of model outputs in general including instability, inconsistency, and disagreement, and we hope that this work contributes to progress in this direction.