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1 . Huiqi Deng∗, Qihan Ren∗, Hao Zhang, Quanshi Zhang†
Shanghai Jiao Tong University
* Equal contribution.
† corresponding author.
2 . Background and Motivation
Traditional models Deep neural networks (DNNs)
Performance
DNNs
Traditional model
Amount of data
Why superior performance? Representation capacities
A common hypothesis
3 . Background and Motivation
• How to study the representation capacities of DNNs ?
parameter
complexity
overfitting
Representation
capacity Generalization
ability
DNNs Adversarial
+ =
robustness
Panda Gibbon
✓ Reflect the representation capacity
from a certain perspective
......
4 . Motivation
• Unlike previous studies, we focus on the following questions
⚫ Any common tendencies of DNNs in representing concepts, i.e.,
which types of concepts are (un)likely to be encoded in DNNs?
⚫ Does a DNN encode similar visual concepts to human beings
for image classification?
5 . Conclusions
Types of concepts ? Complexity of interactions
⚫ Any common tendencies of DNNs in representing concepts, i.e.,
which types of concepts are (un)likely to be encoded in DNNs?
✓ Simple interactions × Middle-complex interactions
✓ Complex interactions
⚫ Does a DNN encode similar visual concepts to human beings
for image classification?
visual concepts encoded in a DNN ≠ human beings
6 . Interactions
• Interactions and interaction concepts
independently
DNN
inference
Intetaction utility
interact 0.1
trigger the importance of i
and changed by j
variable i variables j Interaction concept (head)
The inference of DNN:
× considers input variables working independently.
✓ encodes the interaction between input variables to form an interaction concept for inference.
7 . Multi-order interactions
• Complexity of interaction concepts
variable i variable i
interact interact
variable j variable j
A few large amounts of
contexts S contexts S
Simple interactions Complex interactions
• Multi-order interactions to represent complexity
The importance of i
changed by j
the importance of i the importance of i
(when j is present) (when j is absent)
Multi-order
interaction
order
8 . Multi-order interactions
• Complexity of interaction concepts
variable i variable i
interact interact
variable j variable j
A few large amounts
contexts S of contexts S
Simple interactions Complex interactions
• Multi-order interactions to represent complexity
A simple collaboration
A small m (low-order)
between a few variables
A complex collaboration
A larger m (high-order)
between massive variables
9 . Output vs Multi-order interactions
• Efficiency axiom of multi-order interactions
Efficiency axiom. the network output can be decomposed into utilities
of multi-order interactions of differnent orders (i.e., interaction
concepts of different complexities).
Output independent utilities Overall Utilities of multi-order intearctions
small m large m
medium m
low-order middle-order high-order
(simple) (middle-complex) (complex)
➢ Therefore, interaction concepts can be exactly categorized into concepts of
low-order (simple), middle-order (middle-complex), and high-order (complex).
10 . Discovering the bottleneck
• The relative interaction strength of the m-th order
Representation bottleneck: A DNN usually encodes strong low-order and
strong high-order interactions, but encodes weak middle-order interactions.
𝑱(𝒎)
• The representation bottleneck phenomenon is widely shared by different
DNN architectures trained on different datasets.
11 . Bottleneck → Cognition gap
Cognition gap. i.e., DNNs and humans encode different types of interaction
patterns for inference.
context: a few patches context: middle number context: massive patches
of patches
• Whether humans/DNNs can extract new information from additional new
patches under contexts of different sizes.
12 . Explaining the bottleneck
• Proof: the change of network weights can be decomposed into the
sum of gradients of multi-order interactions w.r.t. weights.
the strength of learning
the m-order interactions.
much higher when the order m is small or large
Training strength
Bottleneck
much lower when the order m is medium
13 . Explaining the bottleneck
• Verification of the theory
Simulate
Theoretical training strength Empirical interaction strength
(in Theorem 1) (in real applications)
Simulations of the distributions based on curves on ImageNet.
14 . Train DNNs encoding specific orders of interactions
• Can we force the DNN to encode interactions of specific orders ?
We prove that the mainly encodes interactions of orders.
15 . Train DNNs encoding specific orders of interactions
Encourage specific orders of interactions:
Penalize specific orders of interactions:
Total loss:
In experiments, we found that the two
losses usually could encourage/penalize
interactions of the -th orders.
16 . Investigating representation capacities
We investigate the representation capacities of four types of DNNs
• Normal DNN: normally trained DNN
• Low-order DNN: penalize high-order interactions.
• Middle-order DNN: encourage middle-order interactions.
• High-order DNN: penalize low-order interactions.
17 . Investigating representation capacities
Part I: Classification accuracy
➢ The four types of DNNs achieved similar accuracies
➢ Middle-order interactions can also provide discriminative information
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18 . Investigating representation capacities
Part II: Adversarial robustness
➢ High-order interactions are vulnerable to adversarial attacks
➢ Low-order interactions are more robust to adversarial attacks
Low-order Middle-order High-order
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19 . Investigating representation capacities
Part III: Bag-of-words vs. structural representations.
random
masking
masked
surrounding
masking
masked
➢ The high-order DNN encodes more structural information.
High-order DNN High-order DNN
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20 . Investigating representation capacities
Part III: Bag-of-words vs. structural representations.
random
masking
masked
surrounding
masking
masked
➢ The low-order DNN prefers bag-of-words representation.
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21 . Conclusions
• Discover a representation bottleneck phenomenon of DNNs.
• Theoretically explain the representation bottleneck.
• Propose losses to force DNNs to encode interactions of specific orders.
• Investigate the representation capacities of low-order, middle-order,
and high-order DNNs.
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