Model information as an analysis tool in deep learning
Authors: Xiao Zhang
Di Hu
Xingjian Li
Dejing Dou
Ji Wu
Venue:  (2021)

Abstract: Information-theoretic perspectives can provide an alternative dimension of analyzing the learning process and complements usual performance metrics. Recently several works proposed methods for quantifying information content in a model (which we refer to as "model information"). We demonstrate using model information as a general analysis tool to gain insight into problems that arise in deep learning. By utilizing model information in different scenarios with different control variables, we are able to adapt model information to analyze fundamental elements of learning, i.e., task, data, model, and algorithm. We provide an example in each domain that model information is used as a tool to provide new solutions to problems or to gain insight into the nature of the particular learning setting. These examples help to illustrate the versatility and potential utility of model information as an analysis tool in deep learning.

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MODEL INFORMATION AS AN ANALYSIS TOOL IN DEEP LEARNING
Anonymous authors Paper under double-blind review
ABSTRACT
Information-theoortii perspectives can provide alternative dimension of analyz- ing the learning process and complements usual performance metrics. Recently several works proposed methods for quantifying information content in a model (which we refer to as "model information"). We demonstrate using model infor- mation as general analysis tool to gain insight into problems that arise in deep learning. By utilizing model information in different scenarios with different con- trol variables, we are able adapt model information to analyze fundamental elements of learning, i.e., task, data, model, and algorithm. We provide an ex- ample in each domain that model information is used as a tool to provide new solutions to problems to gain insight into the nature of the particular learning setting. These examples help to illustrate the versatility and potential utility model information an analysis tool in deep learning.
INTRODUCTION
The ultimate goal of many deep learning research has been improving performance on specific datasets, for example, aiming for superior classification accuracy on the ILSVRC challenge. We have witnessed super-human performance on tasks in vision and language processing, but we are still far from understanding how the learning process works and whether they resemble the learning of human. This is partially due to the metric we use providing too little information on the dynamics of learning. Besides, performance on the test set as sole goal can sometimes even lead undesirable outcomes or misleading conclusions (Lipton & Steinhardt 2019).
Recently, several works propose to use the description length of model (or surrogate estimations thereof) to understand the behavior of learning. In this paper, we refer to such measures of the amount of information content in a model as model information. Blier & Ollivier (2118) first demon- strated efficiently encoding deep neural network with prequential coding technique. Zhang et al. (2020) then proposed an approximation model information and utilized model information to an- alyze the information content in task. They also used model information to explain phenomenons in transfer learning and continual learning. They showed that model information provides a different perspective than performance, directly characterizing the information transfer in the learning pro- cess. Voita & Titov (2020) used model information as a probe to analyze what kind of information present in text representation model. They claim that model information more informative and stable than performance metrics when used as probe.
Model information can provide an informational perspective learning. It can potentially help to answer questions about learning dynamics, such as how much information exists in dataset or model. or how much information transferred in learning step. Furthermore. we can also re- formulate existing problems into problems about information, for example, similarity and capacity, as we will show in this paper. Comparing model information with model performance, model infor- mation not only accounts for how good model can perform but also how fast it learns to perform well (in the sense of sample efficiency, a discussion is given by Yogatama et al. (2019)). The latter can be interpreted related the quantity of information transferred in model training.
In this paper, we try to illustrate that model information can provide framework for analyzing and understanding phenomena in deep learning. In the next section, provide general definition of model information, independent of how model information is estimated. We then unify the anal- ysis of fundamental elements of deep learning under the framework of model information. In the

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following sections, several common problems as examples to show how to use the model information framework as an analysis tool.
AN INFORMATIONAL PERSPECTIVE TO ELEMENTS OF DEEP LEARNING
To measure the amount of information in a model, Voita & Titov (2020) use the codelength of dataset subtracting the cross-entropy of the final model:
L 7.preg (y1:N (TI:N) + Σ log Pow (Vilz;))
Zhang et al. (2020) propose to estimate model information by subtracting codelength of & examples (the k examples are independent and different from the training set of size N) with initial model and final model, and show that it is more reliable than [T:
L L Oiiit Both methods share idea that model information can derived by comparing the codelength of encoding the model and data together, with the codelength of encoding the data alone.
(y1 [T1.) Lox (11. (111/(1)))))))
(2)
This is essentially edge" in model M (in Zhang et al. (2020). this is denoted by K ((M) K ((M Throughout this paper, we use D denote dataset including examples sampled from task T. and use MD to denote model M that has been trained to converge on D. Assume labels in task T have an underlying generation mechanism f: YI = f(2,, Q) (where 6 is i.i.d. noise), then the "generalizable knowledge" in T' is K (f), and generalizable knowledge in Mp is at most K(F).
approximation the Kolmogorov complexity of the "generalizable knowl-
We denote the amount generalizable information a model M contains about a dataset D as L(M, D). Then I(M D) is the "model information": the amount of generalizable information trained model MD contains about dataset D. Ideally, if learning is "perfectly efficient", then we can expect L(MD. D) = (MM)) K ((M K(f).
In this paper, we investigate what can achieved with a model information measure L. We found that model information deep learning. This covers the fundamental elements of machine learning: task, data, model, and algorithm. In Table [] information can be used to perform analysis of the corresponding problem.
powerful tool for analyzing and understanding many phenomenons in
we list problems as examples for each domain and summarize how model
Table 1: Model information as an analysis tool in deep learning
Element
Relevant problem
Use of model information
Task
Task difficulty
L(Mp; D)
Data
Domain similarity
L(Mp,e D2)
Model (structure)
Model capacity
max L(M,, D)
(parameter)
Ablation study
L(MB D)
Algorithm
Knowledge distillation
L(MD+Dr: D)
As we shall see, model information can enable attacking some of the above problems from neat and theoretically-sound perspective. Table i not an exhaustive list but just examples of what one can do with model information. the following sections, detail the application of model information to perform analysis of each problem and explain how it can lead to useful insights.
For following experiments in this paper, we use to estimate model information and set the en- coding set size k=10000, unless otherwise stated.
3 TASK: DIFFICULTY
The success of deep learning is marked by its superior performance at solving difficult tasks, like large-scale visual classification and question answering. For evaluating the capability of algorithms,
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Figure 2: Injecting three kinds of noise on MNIST.
it is important to understand the difficulty of the tasks and the datasets we have. Human definitions of difficulty are mostly subjective, for example, difficulty scoring, task performance, and time needed to complete the task (lonescu et al. 2016).
Several metrics have been proposed for measuring the difficulty of a task. Many focus on the com- plexity of the data. For images, difficulty is linked with image quality, objectiveness, and clutter-ness (oonesu et al. 2016). For text, the relevant factors include perplexity, reading ease, and diversity (Collins al. 2018). For general real-valued data, can measure the distribution of feature value, class separability, and the geometry of class manifolds (Ho & Basu 2002). There are also other factors that affect the difficulty of task, for example, class balance and class ambiguity.
As pointed out by Collins et al. (2018), these characteristics all describe some aspect of the task, and difficulty cannot be described by any one of them alone. Complexity measures of input data can sometimes have little impact on task difficulty. An important line of work defines the difficulty of classification problem by the complexity of the optimal decision boundary (Ho & Basu 2002). The complexity of the boundary can be characterized by its Kolmogorov complexity or its minimum description length. This coincidences with the idea of the model information. While for simple tasks, one can directly characterize the geometry of the decision boundary, for complex tasks that use neural network to model the decision boundary, the description length can only be approximated.
With description length, the difficulty of task can be interpreted as the amount of information needed to solve the task. This amount of information is at most K (J), which the input-output relationship of the task. We measure the model information L ((MD, D) of a trained model. and use as a measure of the information content in dataset D. In Figure [ we use five 10-class classification tasks to illustrate the idea. The results show that the datasets vary greatly in information content: For classifying digits, SVHN (Netzer et al.] (011) requires more information than MNIST (LeCun et al., 1998) or EMNIST (Cohen et al. (017). meaning classifying digits in street view is more difficult than in standardized MNIST images. CIFAR-10 (Krizhevsky & Hinton 2009) is richer in information content and therefore more difficult, as natural objects are more complex than digits.
complexity of
There is also general trend that the more difficultt the task, the lower the performance and the confidence of the model. However, performance and confidence are not only determined by task complexity, they are also affected by noise present in the task.
Next we want to disentangle the two factors: task complexity (measured by information) and noise. We apply three kinds of noise on MNIST represent different forms of noise present in a dataset (Figure 2 There are two kinds of image noise: missing pixels, where small random block of pixels is missing the image, missing objects, where large portion the digit is missing, and one form of label noise: massart noise, where small ratio of labels are replaced by random labels. To see how different kinds noise affect model behavior, we plot model information, model accuracy, and confidence in Figure
What we observe is that information complexity and noise are two independent dimensions that affect performance and require more information to correctly classify the digits. This is because one must learn more features corresponding to each digit, as any of the features could be absent at any time. However, the
task. Pixels missing from the image will make the task more complex
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Figure 3: Task information, model performance, and model confidence, with different kinds and varying levels of noise.
model can still perform well after learning enough information about the task. Missing objects and wrong labels, the other hand, are pure noise that does not affect the information complexity of the task. They effectively make some of the examples uninformative and simply confusing, which makes the model less confident in its predictions.
DATA: DOMAIN SIMILARITY
Understanding the data is fundamental element of understanding learning. Labeled data for deep learning comes from many sources and domains, and people seek to train models that can adapt well new domain, generalize unseen domains, or be independent Lopez-Paz 2020 However, few attempts to understand the data before using them to train the model. Similarity fundamental concept for understanding the relationship between different data. In deep learning, we are more concerned about similarity semantic level. Semantically similar images can sometimes differ very much the RGB space.
domains (Gulrajani &
The informational similarity measure proposed by Lin (1998) measures the similarity between A and B by the ratio the amount of information needed to describe the commonality between A and B and the amount information needed to describe A together with B:
sim A, B) = Iccommon (A, B))
K(fB) K(falfA)
(3)
T(description A, B))
K(FA.JB)
is the Jaccard similarity between and B measured with information. A property of the in- formational similarity measure is universality: it does not depend on particular representation (or modeling) of A and B. We can use model information to estimate the information terms in Bh. and turn the similarity measure into (4): (using approximations L(MA, z K(fA) and L(MA, B) K(fB) - K(fBlfA)) see discussion on the symmetricity of S in Appendix A.3
L(MA, B)
S(A, B) L(MA, A) + L(MB, B) L(MA, B)
Another benefit of information similarity is that the similarity can be measured with respect one side, resulting in a unidirectional similarity measure:
Mcoomon
L(Ma, B)
Sund (A, B) [ddesripptin
(5)
L(M B)
Note that generally Suni(A, B) * Sumk (B, A). The unidirectional similarity measure is useful for depicting the relationship between A and B. For example, if Sumi B) < 1 and gumi (B, 1, then one can tell that A is a subset of B.
We perform experiments et al. 2010) and Office-Home (Venkateswara et al.) 2017). They each have 3 and 4 different do- mains for imgges of the same set of classes (Figure 4). We calculate information similarity S and unidirectional information similarity gumi for each pair domains in each dataset. The baseline for comparison is the distance of first-order and second-order statistics in the feature space
two commonly-used domain adaptation datasets: Office-31 (Saenko
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amszon
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Figure Image domains Office-31 and Effice-Home.
produced by ResNet-556 (He et al., 2016b) pretrained ImageNet (Russakovsky et al. 2015) (see Appendix for details). This is inspired by the Maximum Mean Discrepancy (MMD) (Gretton et al.) 2006) method for comparing data distributions. In domain adaptation literature, one often aims to minimize such distances to match domains (Pan et al. 2009).
As illustrated Figure information similarity measure give similarity score between and 1. which is intuitive and comparable across datasets. S and Sums largely agree with feature distances, but the latter not comparable across datasets. For instance, dslr and webcam in Office-31 are much more similar than any other pair in the two datasets, which is reflected in but not in feature distance. is also more faithful within dataset for corresponding to vi- sual similarities. Unidirectional information similarity Suni gives extra information, for instance, Suxi (amazon, dslr) > Sumi (dsir, amazon) shows that amazon contains more distinct informa- tion than dslr. This be explained that each category in amazon has images for 90 different objects, while each category in dsir only has images for objects.
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Figure 5: Pairwise domain similarity on Office-31 (top) and Office- Home (bottom). S and Srmms are information similarities, Dj and D2 are first-order and second-orde feature distances.
MODEL STRUCTURE: CAPACITY
Neural networks are powerful learners that can scale to learn extremely large datasets given enough neurons and layers. It largely remains mystery how to qualify the true capacity of neural network. Capacity can be defined different fashion, for example, [Collins et al.) (2017) measure capacity by the number of bits of data the network can remember, while Baldi & Vershynin (2019) define capacity as the number of distinct functions a network can represent. We can define the information capacity of network M as the maximum amount of information model can hold for given task T. directly using model information to measure:
(MM) = max L(Mp, D).
(6)
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To illustrate the information capacity of model, experiment with models from small to large (4 configurations of ResNet-11: large, standard, small, and riny. See Appendix for details), and datasets of varying complexity (subsets of TinyImageNed" with with 50, 100, 150 and 200 classes.). We measure the model information of every model trained on each task, and plot results in Figure
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Figure 6: Measuring capacity using information. (a) the amount of information a model can store is capped by its capacity. (b) increase in capacity is correlated with an increase in the number of parameters. (c) task has an inherent amount of information content.
The first thing we notice in Figure ( a that with increasing complexity of the task, model infor- mation saturates certain point for each model. This displays the capacity of model (in dotted lines). Larger models can store more information and saturate later at larger tasks. Figure plots the increase of information capacity w.r.t. the increase in the number of parameters. Information capacity roughly increases with the number of parameters in the model, which agrees with the ob- servation in Collins al. (2017).
The same observation also apply to datasets: each dataset has a definite amount of information content. Larger models learn more information from the dataset, but can hardly learn more than this amount. This is shown in Figure [|c, and dataset information is indicated by the dotted lines.
6 MODEL PARAMETER: ABLATION
Next turn to the analysis of parameters within network. common way understand com- ponents in a network to perform ablations: remove some units or structural components from the network, and compare the performance before and after ablation (Meyes al. 2019). A larger performance drop signifies larger importance of the ablated component for performing on the task.
However, there several caveats this approach: firstly, ablations cannot reveal the true tribution of a component "in vivo." If network layer is removed and the model re-trained, the functionalities of the layer can get substituted other layers. In this case, ablations fail reveal the contribution of the layer in the original network. Secondly, ablation cannot be performed on components vital to model performance, for example, residue connections are removed from very deep CNN, the network can fail to train. It would be unreasonable to therefore conclude that residue connections contribute 100% to the model.
Here we propose to use model information to measure the contribution of a network component. The information I in network component can be measure in the following fashion (where M
I(c) L(MD, D) I(MM, D).
To calculate the information in e, we reset all parameters within this component to their initialized value before training. Then we measure the model information of this ablated model ME. The difference information Mp and M, the information content in the parameters of component In effect, ablated, the model structure is kept intact.
performing information ablation: only information stored in parameters is
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Figure 7: Information ablation. (a) Structure a variant of ResNet model in (He et al. 201 (b) Information ablation by each block in the model. (c) Information ablation by different types of convolution layers. "bn" stands for the bottleneck layer.
Information ablation helps uncover how the information in each component contributes to the whole network, or in other words, how information in a network is stored by its components. In ResNet model
Figure we perform information ablation of series of residue blocks, and each block contains several convolutional layers. Figure b shows the information in each block, as well the information per model parameter ("densityy") ) There are several observations: blocks in the middle (block2 5) contribute the most information. blocks near input have larger information density, and blocks that reduce spatial resolution (with '/2' in notation) contains more information than blocks that preserves resolution. The last phenomenon is likely because convolution layers in downsampling blocks do more job of combining smaller object parts into larger parts, thus having more knowledge about the constitution of objects.
example. The model consists
Figure (c) illustrates something unique the information ablation method: this time we measure information contribution by four different kinds of convolution layers in each and every block. We found out that the majority information resides in the 3x3 bottleneck layer, which where most spatial feature transformations take place. Convolution layers the residue path contain surpris- ingly little information, despite having roughly the same number of parameters as the layers before and after the bottleneck. This indicates that residue layers serve more of structural functionality to ease training, rather than doing substantial information processing.
ALGORITHM: DISTILLATION
Knowledge distillation is method to transfer knowledge from larger teacher models to smaller student models, which can result in better performing compact models (Hinton et al. 2015). Student models are trained with output probabilities of a teacher model in addition to the true labels. It is speculated that the student model can benefit from "dark knowledge" in teacher's predictions. "dark knowledge" means the presence of extra information, with proper tools, we verify the existence of extra information provided by the teacher model and quantify how much information is transferred in distillation.
We could measure the information transferred from teacher to student by the following formula:
I = L(MD+D,, D) L(MD, D).
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Next iteration:
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Figure 8: (a) illustration of the training process in synced distillation. indicates training proce- dures with information flow the arrow directions. (b) information gain of students from different teachers. (c) information gain of the student when trained with different distillation coefficient a.
Dr is teacher's predictions on dataset D. However L(M D) cannot be directly measured with online coding because Dr already contains much information about the labels, and the student model having access to Dr can encode D with minimal effort by merely looking at Dr. We propose a method called synced distillation to calculate online codelength of dataset D with the presence of teacher model.
The process online coding gradually increases the size of the dataset D and updates the model accordingly. In synced distillation, update the teacher and the student model in synchronization (Figure a)): in each iteration, we train the teacher with subset D.. then using D, and teacher's predictions train the student via distillation. The student model is then used encode 1/+11. In the end, the student will be the same as training with conventional distillation. But during the process, the teacher never leaks any information about future examples the student. This enables the student to generate valid encoding of D, which can then be used to calculate model information.
As pointed out in Zhang et al. 2020), model information can help illustrate different aspect of learning dynamics than model performance. For the simple 5-layer CNN model that we use as student, distillation with different teachers and distillation coefficient a yields similar performance (Figure Øb and c). However, from measuring the information gain 7, ResNet-56 and ResNet-101 transfer more information to the student than ResNer-20. This helps the student reach lower cross- entropy with fewer examples, although the final performance is similar. A larger distillation coeffi- cient also increases information transfer. This shows that as the weighting of the KL-divergence term increase in the student's loss function, the student learns more from the teacher. Observations such as this can help analyze and design better algorithms or understand why algorithms like knowledge distillation lead performance gain (Yim al. 2017).
DISCUSSION
Currently, methods for estimating the description length of neural network models are still quite preliminary, lacking through analysis and guarantees of their efficiency. Using prequential coding to estimate model information or Kolmogorov complexity also introduces dependency architecture and training procedure, for example, dropout, batch normalization, and SGD optimizer will all affect the model information estimations. Because prequential codelength is always larger
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than Kolmogorov complexity, can optimize the training hyperparameters to achieve lower codelength as possible, which makes the estimation tighter.
When applying model information perform analysis, to vary only one variable at a time. Because ((M, D) is a function of both the model and the dataset, when analyzing models, the dataset needs to be fixed for codelengths to be comparable. Similarly when datasets are the center of interest the model needs to be fixed. This also introduces dependency. For instance, theoretically we can define difficulty by [J),, but empirically when estimating K (J) using L(Mp, D), dependency on model architecture becomes inevitable. One can choose any adequate model architecture for training on the task, and fix that compare the difficulty of tasks.
did in this paper, it is also necessary
Nonetheless, we demonstrate that model information could be a powerful analysis tool in deep learning. To summarize, the key observations from our experiments are:
Model information allows us to quantify properties such as difficulty, capacity, and simi- larity, especially for complex data and deep models, in consistent fashion.
Model information can help analyze and understand learning in deep networks by showing how information is transferred and stored.
Such a tool widely applicable to many kinds of problems because does not depend on specific models or learning algorithms.
We hope problems discussed in this work serve as examples of the versatility of an informational perspective in investigating neural network learning.
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Yann LeCun, Léon Bottou, Yoshua Bengio, and Patrick Haffner. Gradient-based learning applied to document recognition. Proceedings of the IEEE, 86(11) (6611):2277-23244 1998.
Dekang Lin. An information-thhorrett definition of similarity. In Jude W. Shavlik (ed.), Proceedings of the Fifieenth International Conference on Machine Learning (ICMI. 1998), Madison, Wiscon- sin, USA, July 24-27, 1998, pp. 296-304. Morgan Kaufmann, 1998.
Zachary C. Lipton and Jacob Steinhardt. Troubling trends in machine learning scholarship. ACM Queue, 17(1):80, 2019.
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Yuval Netzer, Tao Wang, Adam Coates, Alessandro Bissacco, Bo Wu, and Andrew Y Ng. Reading digits in natural images with unsupervised feature learning. NIPS Workshop on Deep Learning and Unsupervised Feature Learning, 2011.
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Olga Russakovsky, Jia Deng, Hao Su, Jonathan Krause, Sanjeev Satheesh, Sean Ma, Zhiheng Huang, Andrej Karpathy, Aditya Khosla, Michael Bernstein, Alexander C. Berg, and Li Fei-Fei. ImageNet Large Scale Visual Recognition Challenge. International Journal of Computer Vision (IJCV), 115(3):211-255 2015.
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10

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Under review as a conference paper at ICLR 2021
A EXPERIMENTS
A.1 GENERAL EXPERIMENT SETTINGS
For experiments in this paper. we use 2 to estimate model information and set the encoding set size k=10000 (except for experiments on Office-31 and Office-Home, see below). All models are trained using Adam optimizer and early-stopping on the validation set.
A.2 DIFFICULTY
Statistics of noise injection experiments given in Table 2 We trained ResNet-56 models on MNIST with different kinds of injected noise. Model confidence is defined as the margin between the probability of the top and the second class predicted by the model:
con f'nennece = E (P,, (2) Pea (x:)
(9)
C1 arg max pk (xx)
(10)
C2 arg max pæ: (x) k
(11)
Table 2: Model statistics on MNIST with image noise (missing pixels)
Noise level
0
0.05
0.1
0.2
L(Mp, D)
1559
1718
2120
2719
Accuracy
0.996
0.992
0,988
0.981
Confidence
0.994
0.989
0.985
0.985
Table 3: Model statistics MNIST with image noise (missing objects)
Noise level
0
0.05
0.1
0.2
L(Mp, D)
1559
1489
1560
1296
Accuracy
0,996
0.953
0.909
0.865
Confidence
0.994
0.946
0.899
0.848
Table 4: Model statisties MNIST with label noise (massart noise)
Noise level
0.05
0.1
0.2
L(Mp, D)
1559
1833
1735
1605
Accuracy 0.996
0.949
0.908
0.867
Confidence
0.994
0.973
0.941
0.908
A.3. DOMAIN SIMILARITY
We use ResNet-18 model pre-trained on ImageNet as the initial model, then finetune on source dataset S and measure L(Ms. T') on target dataset Pairwise codelength representing L(Mr, T)
L(Ms. T) are given in Table 6 Diagonal values are L(Mr, T).
L(Mr, T)
L(Ms, T') can be directly calculated using (
L(Mr, 1) L(Ms, L mieg (21:) L preg (11)/111))
(12)
This assumes that information from ImageNet is known and is not included in calculating similari- ties. It is possible to use random models initialization, but as the number of examples is too small for some domains in Office-31, we found that using pre-trained models gives more stable estimation of codelength.
11

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Under review as a conference paper at ICLR 2021
Also, because the smallest domain in Office-31 and Office-Home only have 399 and 1942 training examples respectively, we use about half the dataset, i.e., k=200 for Office-31 and 1=1000 for Office-Home to estimate L in (
Theoretically, the similarity measure in anddd is symmetric:
S(A, B) = I(commonl A, B))
K(fB) K(fBlfA) K(fA) K(fAlf)) S(B,A) {(fA,fB)
I[description(A4. (A,
f(fA.BB)
which is because ((fa) + K(falfa) K(fB) + K(falfe) K(fA.fB) Empirically, when
using L(MA, B) as an approximation of K(ʃB) Kf fa), usually S(A, B) + S(B, A) because
L(MA, B) and ((MB, A) doe not equal exactly. In this paper report empirical results using (4
Table 5: L(Mr, T) L(Ms 7) on domains in Office-31 31
Source
Target
amazon
dslr
webcam
amazon dslr webcam
263.0
124.7
106.3
139.4
318.4
26.6
153.9
6.6
316.6
Table L(Mr,T) T) L(Ms, T') on domains in Office-Home
Source
Target
art
clipart
product
real_world
art
2776.4
1076.2
764.8
451.9
clipart
2001.2
1630.3
756.8
617.5
product
2047.6
1144.8
1480.2
466.0
real_world
1609.2
879.2
411.9
1420.2
The first-order and second-order feature distance d1 and d2 defined as:
dj [E]]f(()) Ef] | [[)) - [ss]]
(13)
d2 =
(14)
where fa(z) and fz(z) are the representations of x produced by model trained on domain A and B. F stands for Frobenius norm.
A.4 CAPACITY
To show the capacity of a model, we need to saturate a model by information in the training set. Therefore choose to use ResNet-11, 1, the smallest of ResNet configurations, and four different layer width in ResNet-11, 1, as follows: Large: [16, 48, 96), Standard: [16, 32, 64), Small: [16, 24, 32). Tiny: [16, 16, 24]. For example, [16, 48, 96] means the layer width is 16, 48, and 96 in the first, second, and the third residue block. The datasets we use are subsets of Tiny-ImageNet, each containing 50, 150, and 200 classes. The number of examples in each dataset fixed 12500.
L(Mp, D) for models of different size, and datasets of different complexity, are listed in Table 7
Table 7: L(Mp, D) for model and dataset of different size
Dataset
Model size
Large
Standard
Small
Tiny
50 class
7687.0
7140.0
7358.2
6970.1
100 class
8907.8
8496.2
8343.9
7492.2
150 class
9540.7
9322.1
8086.6
7777.0
200 class
10264.0
9410.7
8170.3
7448.8
Number of parameters
264657
124433
49905
29457
12

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Under review as a conference paper at ICLR 2021
A.5 ABLATION
We train a ResNet-20 model (referred to as MD) on CIFAR-1O, reset the parameters of some net- work component e (this ablated model is referred MD) and measure the difference of model information by:
I(c) L(MD, D) LIMB. D)
(15)
L (VI.k((TER) L (MaakTi))
(16)
Ablation results for different network layers are shown in Table
Table 8: Information different kinds convolutional layers
Model component
[(c)
# param
per param
conv Ixl (before bottleneck)
1985
32000
0.062
conv3x3 (bottleneck)
4243
96768
0.044
conv Ixl (after bottleneck)
1231
43008
0.029
conv 1xl (residue)
467
41984
0.011
Table Information / in different residue blocks
Model component
[(c)
param in c
per param
conv3x3
379
432
0.877
block0
751
4704
0.160
block1
485
4544
0.107
block2
2088
23808
0.088
block3
1030
17792
0.058
block4
2137
94720
0.023
block5
1256
70400
0.018
fc
167
2570
0.065
all
5900
219482
0.027
A.6 DISTILLATION
In distillation experiments, we use LeNet-5 as student and ResNet-20,, ResNet-56, and ResNet-101 teachers. The loss when training student model given in (17).
L = ==akD + (1 acC
(17)
LKD = -2- Σ (1
(18)
(Pa(F)/T
(k(x) Σ, E)PP(T/
(19)
We also experiment with different coefficient o, to control the influence teacher models. Results shown in Table I0 We observe that the student's information gain plateaus after 0.2, which kind of agrees with the conventional choose of a = 0.1 in knowledge distillation.
13

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Under review as a conference paper at ICLR 2021
Table 10: Distillation experiments with different teacher model
Teacher
Teacher accuracy
Student accuracy
None
82.7
ResNet-20
87.07
84.0
452.6
ResNet-56
88.78
83.8
1246.2
ResNet- 101
89.69
84.2
1277.7
Table 11: Distillation experiments with different coefficient a
T
Accuracy
1
0
4
82.72
0.0
0.05
4
83.93
938.1
0.1
4
83.73
1040.9
0.2
4
83.85
1178.9
0.3
4
83.77
1188.6
0.5
4
83.64
1214.4
0.7
4
83.83
1246.2
0.9
83.94 4
1177.5
14


Reviews:
[{"review_id": "jmzXj92LlxD", "reviewer": "AnonReviewer3", "summary": "", "questions": "", "limitations": "", "rating": 6, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The central concept of this paper is model information, a description length of a discriminative model. The authors advocate usage of model information for analyzing several aspects in deep learning. In particular, they show how model information can be used to judge about difficulty of supervised tasks, domain similarity, model capacity, roles of different network components, and knowledge distillation. All of these are important topics and are relevant to the ICLR community. While most of the definitions and interpretations seem intuitive and valid, there are a few concerns and questions, and in some cases, it is hard to decide whether the conclusions of the interpretations should be trusted.\n\n**Model information**\n- The term *model information* is vague and may create associations with other quantities like the Shannon mutual information model has about the training dataset, $I(\\theta : D)$. I suggest to clarify early that by model information it is meant some kind of description length of the discriminative model $p_\\theta(Y \\mid X)$.\n- As model information is not a well-defined concept, it is not appropriate to say \"an approximation of model information\". Better to say \"an instance/a variant/a definition of model information\".\n- As all experiments are done using the instance of model information defined by Zhang et al. [1] (called Information Transfer), the conclusions may not hold for other instances of model information. For example, if one defines model information with the prequential coding of Bleir and Ollivier [2], then increasing the number of examples with noisy labels will make the task more difficult. \n- Please clarify whether the $k$ examples in equation (2) are from the training set $D$ for not.\n\n**Task difficulty**   \nI really like using model information for assessing task difficulty. The only concern is that the task difficulty depends on the network and the training algorithm. Additionally, I suggest to plot task difficulty versus the number of training examples. Will we see that it plateaus after enough number of training examples are given? How would task difficulty behave in the small data regime?\n\n**Domain similarity**\n- What do union and intersection signs mean for datasets in equation (4)? If they mean the same as in equation (3), you can just repeat the same notation.\n- Please explain how the right-hand-side of equation (4) fits into the framework of equation (3).\n- When reading that $S^\\text{uni}(A, B)$ is asymmetric, one might imply $S(A,B)$ is symmetric, which is not true. Please clarify that both measures are unidirectional.\n- \"For example, if $S^\\text{uni}(A, B) < 1$ and $S^\\text{uni}(B, A) = 1$, then one can tell that A is a subset of B.\" Please explain why is this sentence true?\n- The definitions of domain similarity of equations (4) and (5) seem a little bit arbitrary. Why should they be defined like that? How do these domain similarity measures compare to other measures, such as domain similarity computed by Task2Vec [3].\n- For easier interpretation of Fig. 5, you can use the same color map in all subplots.\n\n**Model capacity**\n- As I understand model capacity defined in equation (6) depends on the task. Why shouldn't one take supremum over tasks too?\n- In the experiments of Fig. 6, do all datasets have the same size?\n- The straight lines in Fig. 6c don't approximate the curves well. To have more points on the horizontal axis you can consider Resnet-[18/35/50]-k networks and vary $k$.\n\n**Ablation**\n- How do we know that quantity defined in equation (7) truly captures how important a component is and that we should trust the conclusions of experiments presented in Fig. 7?\n\n**Minor notes**\n- To make the paper more self-contained, please mention in the main text which definition of model information is used in the experiments. The same applies for model confidence.\n- In eq. (7), $M_D^{\\bar{c}}$ should be defined beforehand.\n\n*References*  \n[````1] Xiao Zhang, Xingjian Li, Dejing Dou, and Ji Wu. Measuring information transfer in neural networks. arXiv preprint arXiv:2009.07624, 2020.\n[2] Leonard Blier and Yann Ollivier. The description length of deep learning models. NeurIPS 2018.\n[3] Achille, Alessandro, et al. \"Task2vec: Task embedding for meta-learning.\" Proceedings of the IEEE International Conference on Computer Vision. 2019.\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "XC0GcjYVC_b", "reviewer": "AnonReviewer1", "summary": "", "questions": "", "limitations": "", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The authors propose a statistic from information theory as a tool to analyze several aspects in machine learning: data, model, etc. \nuse the codelength of a dataset subtracting the cross-entropy of the final model. The results are interesting, but slightly fall short of novelty, and do not lead to any significant implication/improvement to the existing machine learning pipeline/model. \n\nClarity&Originality&Significance: \n\nBelow I will take Section 3 as a specific example to elaborate.\nIn Section 3, the authors try to use L(M_D,D) to \"define\" task complexity, and argues that noise and task complexity are two independent directions. \n\nIt seems what the authors try to define, is simply the mutual information between X and Y, where {x_i,y_i} is drawn from the joint distribution of (X,Y). The quantity L(M_D,D) is an approximation (lower bound) of the mutual information. This is a commonly known fact, which makes the results fall short of novelty.\n\nWhen the authors define the approximation as the task complexity, it becomes dependent on both model and the dataset, not task alone. Therefore, L(M_D,D) will be strongly dependent on how one trains the model, and the size of dataset. For example, with/without dropout, L(M_D,D) will differ significantly. \n\nAs a result, it is difficult to get useful take-away from the results of Section 3, such as Figure 3. \n\nIn addition, it is unclear what the implications would be if we have an approximation of mutual information between X and Y. Could we use it to improve the learning of the task?\n\nIn authors' response, I hope the authors can clarify the dependence of their metric on model, dataset, besides the task. Also, please clarify how it relates to mutual information, whether there is any related work that studies the approximation of mutual information between X and Y, and their relationship with the current work. Also, it would be interesting to know if there are any potential application for the proposed descriptive statistic on task complexity.\n", "extra_scores": {}, "extra_text": {}}, {"review_id": "Q8JPfwVIy7J", "reviewer": "AnonReviewer2", "summary": "", "questions": "", "limitations": "", "rating": 4, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "The paper examines different use-cases of a quantity proposed in prior works which is said to capture the model information. It shows that this quantity behaves as expected overall. Quantifying the amount of information a deep neural network is a very interesting question for the community with both theoretical and practical appeal (from an analytical perspective but also from a model selection perspective, for example).\n\nOverall, I’m scoring the paper with a reject. While the model information seemingly behaves as expected, the weaknesses in the exposition, as well as potential mistakes and a lack of rigour in the paper, will require careful editing.\n\nQuantifying model information can be an important tool. This paper sets out to show that quantities defined by Voita & Titov (2020) and Zhang et al (2020), respectively, seem meaningful and behave as one would expect.\n\nHowever, the examples lack falsification and depth. \n\nThe structure of the ResNet in 7(a) is either wrong or non-standard: The residual connection is usually just an identity function: `out = in + block(in)`. Is this intentional? And if it is, what does this tell us about regular ResNets then?\n\nSimilarly, for the model ablation, it is not clear why resetting the parameters to their initial parameters is the appropriate method of ablation.\n\nAnother potential issue is that the definitions in Section 4 pretend that $L(M_A, B) = L(M_B, A)$ when it is clearly not. See also [Xu 2020](https://arxiv.org/abs/2002.10689) for example.\n\nThis reviewer would wish for a more in-depth treatment of the various examples in order to be thoroughly convinced.\n\n### Minor concerns\n\nFigure 1 uses a line plot, even though the x-axis refers to unordered categories.\n\n---\n### Rebuttal\n\nI thank the authors for their detailed reply. I still consider the contribution to be too high-level and to cover too much ground without going into sufficient depth. I am not sure I can follow the argument about Kolmogorov complexity. Its chain rule is also only equal up to a logarithmic factor. I will keep my score the same.", "extra_scores": {}, "extra_text": {}}, {"review_id": "6WZhE8bFdYj", "reviewer": "AnonReviewer4", "summary": "", "questions": "", "limitations": "", "rating": 4, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "", "summary_of_recommendation": "", "technical_quality": "", "clarity_of_presentation": "", "potential_impact": "", "robotics_focus": "", "quality_of_limitations": "", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "\nThis paper discusses the use of model information to assess properties of deep learning models and problems. In particular, it illustrates how model information may be used to capture the difficulty of a task, the degree of similarity between domains, the capacity of a model \n\nThe idea of using a quantity related to codelength to assess properties of a learning model is interesting; indeed quantifying the properties chosen by the authors (task complexity, domain similarity) would be certainly useful. However, I am dubious whether the assessment using model information would be reliable.\n\nTo my understanding model information evaluates the amount of information a trained model M_D contains about dataset D as a difference in codelength of the data and codelenghth of model+data (as explained in Section 2). Critically, it seems to me that this quantity depends both on the data AND the model. It is therefore a particular measure for a specific model (or at most a family of models) and a dataset. (Incidentally, although guessable, it would be proper to define all the symbols in the equations).\n\nIn Section 3, it is not clear to me how this measure may be used to evaluate, for instance, the difficulty of a task: how is M_D chosen? Is there an underlying claim that this measure would be independent from the choice of model? What is the sensitivity of information measure to the choice of the model? I am also perplexed by the results of the simulations: it seems that dropping samples or scrambling labels makes the task easier; this, though, is quite counterintuitive as it would suggest that to simplify the problem we could literally drop samples or scramble labels (I guess, but I may be wrong, that this outcome is due to the fact that \"difficulty\" is not evaluated in terms of generalization).\n\nSimilarly, in Section 4, it seems to me that a discussion on M_A and M_B is lacking. It is stated that a property of informational similarity is its independence from particular representations/modelling. This seems to me not to hold for model information where we have to rely on M_A and M_B.\n\nIn Section 5, how is a given task T defined? This particularly important as it defines the set over which the maximization is computed and with respect to which capacity is defined. My intuition is that computing this quantity on a small set of tasks would return at best a lower bound.\n\nI have some doubts also in Section 7, on Equation (8). It would seem to that the model information of the student should be relative to a different dataset, one enriched with the output probabilities generated by the teacher. Why is it not?", "extra_scores": {}, "extra_text": {}}]

Author rebuttal:

Meta-review: This work presented a broad set of interesting applications of model information toward understanding task difficulty, domain similarity, and more. However, reviewers were concerned around the validity and rigor of the conclusions. Going into more depth in a subset of the areas presented would strengthen the paper, as would further discussions and experiments around the limitations of model information with regards to specific models and dataset sizes (as you have begun to discuss in Section 8). Additionally, reviewers found the updated paper with connections to Kolmogorov complexity interesting, but reviewers wanted a more formal treatment and analysis of the relationship. 
Decision: Reject