# Advancing Neural Network Performance through Emergence-Promoting
Initialization Scheme

Johnny Jingze Li, Vivek Kurien George, Gabriel A. Silva
Center for Engineered Natural Intelligence
Department of Mathematics
University of California, San Diego
La Jolla, CA 92037
<EMAIL_ADDRESS>
###### Abstract

We introduce a novel yet straightforward neural network initialization scheme
that modifies conventional methods like Xavier and Kaiming initialization.
Inspired by the concept of emergence and leveraging the emergence measures
proposed by Li (2023), our method adjusts the layer-wise weight scaling
factors to achieve higher emergence values. This enhancement is easy to
implement, requiring no additional optimization steps for initialization
compared to GradInit. We evaluate our approach across various architectures,
including MLP and convolutional architectures for image recognition, and
transformers for machine translation. We demonstrate substantial improvements
in both model accuracy and training speed, with and without batch
normalization. The simplicity, theoretical innovation, and demonstrable
empirical advantages of our method make it a potent enhancement to neural
network initialization practices. These results suggest a promising direction
for leveraging emergence to improve neural network training methodologies.
Code is available at: https://github.com/johnnyjingzeli/EmergenceInit.

##  1 Introduction

Emergence in machine learning refers to the phenomenon where complex behaviors
and properties arise from the interactions of simpler elements within a
system. This concept, deeply rooted in the study of complex systems, has
gained increasing attention in the context of neural networks. The emergent
properties are not merely the sum of individual components but rather the
result of intricate interactions among them, leading to sophisticated and
often unexpected capabilities. These emergent behaviors are crucial for
enabling neural networks to perform complex tasks such as image recognition,
natural language processing, and strategic game playing.

The concept of emergence is particularly relevant in neural networks, where
the interplay of numerous neurons can give rise to high-level abstractions and
decision-making capabilities. Li (2023) provides a framework to quantify
emergence in neural networks through the measure of paths from inactive to
active nodes. This metric offers a novel lens through which we can understand
and enhance the emergent properties of neural networks, potentially leading to
more efficient and effective models.

In this paper, we introduce a new initialization scheme for neural networks
that leverages the concept of emergence. Our method adjusts the layer-wise
variance parameters to achieve higher emergence values compared to traditional
methods like Xavier and Kaiming initialization. This approach is particularly
appealing because it is straightforward to implement, requiring only minor
modifications to existing initialization techniques without necessitating
additional optimization or training steps.

Our initialization scheme is grounded in the idea that by enhancing the
emergent properties of neural networks from the beginning, we can facilitate
better feature differentiation and integration. This, in turn, can lead to
improved performance across various tasks and architectures. We evaluate our
method on both convolutional neural networks (CNNs) for image recognition and
transformer architectures for machine translation, demonstrating significant
improvements in model accuracy and convergence speed.

Emergence in neural networks can be seen as a manifestation of the broader
principle that simple rules or interactions can lead to complex and adaptive
behaviors. This principle has been explored in various contexts within machine
learning, from the self-organizing properties of unsupervised learning
algorithms to the adaptive behaviors of reinforcement learning agents. The
work by Li (2023) extends this understanding by providing a concrete measure
of emergence that can be directly applied to the design and initialization of
neural networks.

Our experiments on standard datasets such as ImageNet for image recognition
and the WMT news translation task for machine translation underscore the
practical benefits of our initialization scheme. The results highlight not
only improvements in final model accuracy but also faster convergence during
training, suggesting that our method effectively primes the network for
learning by optimizing its emergent properties from the outset.

The simplicity and effectiveness of our approach make it a compelling addition
to the toolkit of neural network initialization methods. By focusing on
enhancing emergent properties, our scheme offers a new perspective on how
initialization can impact the learning dynamics and ultimate performance of
neural networks. This paper contributes to the growing body of research that
seeks to understand and harness the power of emergence in machine learning,
paving the way for more robust and capable models.

In summary, our proposed initialization scheme represents a novel and
practical approach to improving neural network performance by leveraging the
concept of emergence. By adjusting layer-wise variance parameters to enhance
emergent properties, we provide a simple yet powerful modification to existing
initialization methods. The empirical results presented in this paper
demonstrate the significant advantages of our approach, offering a promising
direction for future research and applications in machine learning.

##  2 Related Work

The initialization of neural networks has been a critical area of research,
influencing the stability and speed of training, as well as the ultimate
performance of the models. Traditional initialization schemes such as
Xavier[12] and Kaiming [13] initialization have laid the foundation for
effectively training deep networks by mitigating issues related to vanishing
and exploding gradients. Xavier initialization aims to keep the scale of the
gradients approximately the same in all layers, while Kaiming initialization,
specifically designed for ReLU activations, helps to maintain the variance of
activations throughout the layers. Both methods have proven to be fundamental
in training deep networks but do not explicitly account for emergent
properties within the networks.

Recent research has explored more sophisticated initialization strategies that
leverage the structural and statistical properties of neural networks. For
instance, Saxe et al. (2013) studied the dynamics of signal propagation in
deep networks, highlighting the importance of properly scaling the initial
weights to ensure efficient training. Additionally, Mishkin and Matas (2015)
proposed a layer-sequential unit-variance (LSUV) initialization that
iteratively adjusts the weights to achieve unit variance across all layers,
further improving convergence.

The concept of emergence, where complex behaviors arise from simple
interactions within a system, has also been examined in the context of neural
networks. Emergent properties have been shown to play a crucial role in the
development of robust and adaptive models. Research by Olah et al. (2020) has
illustrated how higher-level features and behaviors emerge in deep networks as
a result of training on large datasets. This phenomenon underscores the
potential for leveraging emergent properties to enhance network performance.

(Li, 2023) introduced a novel framework for measuring emergence in neural
networks, focusing on the number of paths from inactive to active nodes as a
key metric. This work provides a theoretical basis for understanding how
emergent behaviors can be quantified and optimized, offering new insights into
the design and initialization of neural networks.

Our proposed initialization scheme builds on these foundational studies by
explicitly targeting the enhancement of emergent properties from the outset.
By adjusting layer-wise variance parameters to achieve higher emergence
values, our method represents a simple yet effective modification to
traditional initialization techniques. This approach aligns with the growing
body of research that seeks to integrate principles from complex systems and
emergence theory into machine learning practices.

Furthermore, our work is related to studies that explore the role of
initialization in specific architectures such as transformers. Vaswani et al.
(2017) introduced the transformer model, which has become a cornerstone in
natural language processing due to its ability to capture long-range
dependencies through self-attention mechanisms. Recent research has continued
to refine transformer architectures, with improvements in initialization
playing a crucial role in achieving state-of-the-art performance (Liu et al.,
2020).

In summary, our work contributes to the ongoing exploration of initialization
methods by introducing a scheme that enhances the emergent properties of
neural networks. By drawing on the principles of emergence and leveraging
recent advancements in the field, we provide a novel perspective on how
initialization can impact the learning dynamics and performance of deep models
across various architectures and tasks.

##  3 Method

Emergence in machine learning refers to the phenomenon where complex, higher-
level behaviors and structures arise from the interactions of simpler elements
within a system. This concept, rooted in various scientific disciplines, has
been increasingly recognized for its potential to enhance our understanding
and development of neural networks. In our formulation, emergence is
quantified through a graph-theoretical framework, which examines the network
topology and the interactions between nodes to provide a clear and structural
metric for emergent behavior.

A structural definition of emergence offers several key benefits. First, it
allows for a more precise and quantifiable understanding of how complex
behaviors develop in neural networks. This is particularly useful for
designing and optimizing network architectures, as it provides clear metrics
that can be targeted during the training and initialization phases. Second, a
structural approach facilitates better theoretical insights, enabling
researchers to draw more concrete connections between network topology and
emergent properties. This can lead to more effective and efficient network
designs, ultimately improving performance across various tasks.

Emergence in this context is inherently multiscale. It involves interactions
across different scales of the network, where $G$ represents one scale and $H$
represents a higher scale. Emergence appears only when viewed from this
multiscale perspective, as it captures the complexity arising from the
network’s hierarchical structure.

In our graph-theoretical framework, the emergence value $E$ of a neural
network is defined based on the number of paths from nodes at scale $G$ to
nodes at scale $H$. This definition captures the essence of multiscale
interactions within the network. The more paths that exist between these
scales, the greater the degree of emergence.

Mathematically, this can be expressed as:

$\displaystyle\text{Emergence}(G,H)$ $\displaystyle=\sum_{x\in G\setminus
H}\\#\text{paths in $G$ from $N_{H}(x)$ to $H$}.$ (1)

where:

  * •

$G$ is the set of nodes at the lower scale,

  * •

$H$ is the set of nodes at the higher scale,

This equation captures the emergent behavior by accounting for the connections
and interactions between different scales within the network. The higher the
value of $E$, the more interconnected the network is across scales, thereby
increasing the likelihood of complex behaviors and traits emerging from the
network.

Intuitively, a system with a higher value of $E$ has more extensive and
interconnected pathways through which information can propagate across
different scales. This interconnectedness facilitates the development of
intricate patterns and features within the network, enabling it to capture and
represent more complex relationships in the data. As a result, networks with
higher emergence values are better equipped to learn and generalize from
diverse and intricate datasets, leading to improved performance across various
machine learning tasks.

Our approach leverages this definition to modify the initialization process of
neural networks, aiming to enhance their emergent properties from the outset.
By doing so, we can achieve significant improvements in network performance,
as demonstrated in our experimental results.

To understand emergence in the context of machine learning, when a model has
stronger emergence traits, this means that the model is easy to learn any or
certain downstream tasks. From a loss function perspective, a model with a
stronger emergence should be closer to the global minimum, or the learning
should be fast. We will show in our numerical experiments, that schemes with
stronger emergence will indeed have faster learning in the initial epochs.

To adapt the definition of emergence in our context, we should give a clear
definition of $G$ and $H$ in the mathematical definition above. Since
emergence here evaluates the potential/ ability for emergent traits when we
observe system $G$ from a higher level $H$, here we want $G$ to represent the
model itself, and $H$ to be some certain features of the model. In the paper,
we adopt the setting that $H$ is the nodes in $G$ that are still active in the
training process, where the criteria for active nodes is the set of nodes
whose activation is greater than a threshold set close to $0$. This sorted out
the nodes that are not actively participating in the computational process.
The set of active nodes thus in a sense represent the learning task, thus we
can tie emergence with the performance of the network in a learning process.
This fits in our framework of emergence, where part of the system is being
neglected after the learning process, thus the learning process represents the
$\Phi$ where partial observation is carried out, and the properties of $H$
represents the emergent abilities of the network.

For a feedforward network, with $N$ layers, and $n_{i}$ nodes for each layer,
and $a_{i}$ the number of active nodes for each layer, emergence is computed
as follows:

$\displaystyle E$ $\displaystyle=\sum_{i=1}^{N-1}\sum_{j>i}^{N}\\#\text{paths
from inactive nodes in layer }i\text{to active nods in layer }j$ (2)
$\displaystyle=\sum_{i=1}^{N-1}\sum_{j>i}^{N}(n_{i}-a_{i})n_{i+1}\cdots
n_{j-1}a_{j}$
$\displaystyle=\sum_{i=1}^{N-1}\sum_{j>i}^{N}(n_{i}-a_{i})a_{j}\prod_{k=i+1}^{j-1}n_{k}$

when the network architecture is fixed, which means, when $L$ and
$n_{i},i=1,\ldots,L$ are fixed, emergence is only a function of the number of
active nodes in each layer,

$\displaystyle E:=E(a_{1},\ldots,a_{N}).$ (3)

And the number of active nodes at initialization, is impacted by the weights.
With a criteria for active nodes, for example, those nodes whose activation is
greater than a threshold as adopted in this paper, we can thus establish
initialization scheme that has stronger emergence.

Lemma: Emergence function $E(a_{1},\ldots,a_{N})$ increases when $a_{1}\ldots
a_{i}$ gets smaller and $a_{i}\ldots a_{N}$ gets larger, where $i$ is the
largest integer such that

$\displaystyle\sum_{k=1}^{N}a_{k}-a_{i}-\sum_{k=1}^{i-1}n_{i}>0.$ (4)

The proof is simple: when $a_{i}$ decrease by $1$, then the number of paths
from layer $i$ to the dead nodes in latter layers will increase by
$a_{i+1}+\cdots+a_{N}$, and the number of paths from earlier layers to layer
$i$ will decrease by $(n_{1}-a_{1})+\cdots+(n_{i-1}-a_{i-1})$.

So the net increase of path will be

$\displaystyle\Delta$
$\displaystyle=a_{i+1}+\cdots+a_{N}-[(n_{1}-a_{1})+\cdots+(n_{i-1}-a_{i-1})]$
$\displaystyle=\sum_{k=1}^{N}a_{k}-a_{i}-\sum_{k=1}^{i-1}n_{i}$ (5)

For the usual neural network structures, where for each layer most nodes are
active, and later layers are smaller, the balance usually occurs when
$i\approx N/2$.

Let us consider now why the network with the configuration above has stronger
emergence: for example, when the network is doing an image recognition task,
the nodes in the later half of the layers are making important decisions on
which category the image belongs to, so it needs to be more alive/ subject to
more sensitive weight changes. However, for the nodes in the initial half of
the layers, they are subject to greater weight changes, and they could be
turned off to represent some global features of the image. Hence they could be
more inactive/ subject to more drastic weight changes.

This idea also agrees with the fine-tuning idea: typically, the initial layers
(closer to the input) have smaller learning rates, while the later layers
(closer to the output) have larger learning rates. This strategy is based on
the idea that the initial layers capture more generic features that are less
likely to change significantly, whereas the later layers capture more task-
specific features that require more significant adjustments. When the model
has stronger emergence according to our theory, it is more likely to learn the
specific tasks faster in a fine tuning process.

Now we aim at proposing a network initialization architecture with stronger
emergence. To do so, we decrease the weight magnitude in the first half of the
layers and increase the weight magnitude in the second half of the layers.

  * •

Decrease the activity of nodes in the first half layers

  * •

Increase the activity of nodes in the second half layers

in order to achieve this, we design the following initialization scheme:

  * •

Decrease the magnitude of weights in the first half layers by dividing a
factor

  * •

Increase the magnitude of weights in the second half layers by multiplying a
factor

because in general, a larger weight magnitude can lead to higher activation
thus increase the number of active nodes.

We also want to consider the stability of the network. The existing
initialization schemes were usually designed such that the activation and
gradients are stable across the layers. For example, in Xavier initialization,
it is shown that

$\displaystyle n_{i}Var(W_{i})=1$ (6) $\displaystyle n_{i+1}Var(W_{i})=1$ (7)

promotes stability of activation and gradients. However, when we increase the
variance in the inital layers and decrease the variance in the later layers,
we introduce instability to the flow of activation and gradients across the
layers. To reduce the effect of instability on the performance of
initialization scheme, we make the increase of variance across the layers to
be smooth, so as to reduce the instability. For example, we initialize the
weight matrices in the following way:

We first initialize the network weights $\\{W_{i}\\}$ following some standard
initialization scheme which preserves stability, for example, Xavier or
Kaiming He Initialization. Then we do the following scaling to the weights:

$\displaystyle\tilde{W}_{-n}$ $\displaystyle=W_{n}/\alpha^{n}$ (8)
$\displaystyle\tilde{W}_{-(n-1)}$ $\displaystyle=W_{-(n-1)}/\alpha^{n-1}$
$\displaystyle\vdots$ $\displaystyle\tilde{W}_{0}$ $\displaystyle=W_{0}$
$\displaystyle\tilde{W}_{1}$ $\displaystyle=W_{1}*\alpha$
$\displaystyle\vdots$ $\displaystyle\tilde{W}_{n-1}$
$\displaystyle=W_{n-1}*\alpha^{n-1}$ $\displaystyle\tilde{W}_{n}$
$\displaystyle=W_{n}*\alpha^{n}$

In our experiments, we see (Figure 1) that this initialization is indeed
leading to a better performance. In particular, we show the correlation
between emergence and performance. Based on our theory, we have an increase in
emergence, even when only the magnitude of weights of the first half layers
decreases, or the magnitude of weights of the second half increases.

We also note that, such choice of layer magnitudes is mimicking the "domino
effect", as illustrated in the figure below. The increase of energy level for
each piece can set off a cascade effect.

Figure 1: The increase of weight magnitude through the layers is mimicking how
the energy level through the pieces of a domino is increasing. Figure 2:
Comparison of training loss and emergence of Xavier, Kaiming and our
initialization schemes.
Emergence value: Xavier: $5.03e^{8}$, Kaiming: $5.99e^{8}$, Ours: $10.87e^{8}$

We now study how to choose the scaling factor $\alpha$ properly. As shown in
our numerical experiments, we can see that as $\alpha$ increase, the
performance first increases then decrease, and the decrease part is likely to
be caused by the instability inherent to the initialization scheme. In order
to determine a factor $\alpha$ that is appropriate, we want to limit the
emergence of the model to a range.

Choice of optimal $\alpha$: From the mathematical equation of emergence, we
can see that given a model, the maximum amount of emergence is determined by
the parameters of the equations, $N$ and $n_{i}$, which are the number of
layers in the network and the size of each layer. Emergence increases in
$O(n_{i})$ and $O(N^{2})$. As a result, emergence is more sensitive to
$\alpha$ when the network has more layers. So we allow larger $\alpha$ when
the network is shallow and smaller $\alpha$ when the network is deep.
Empirically, under the learning rate $lr=0.001$, $\alpha=2$ is a good choice
for usual architectures. For the two layer MLP, for example in transformers,
$\alpha$ can be as large as $10$, which for deeper MLP, $n>5$, then smaller
$\alpha$ should be considered.

Here our motivation is simply to bound the emergence value. We should also
bear in mind that stability is also a very important issue for an
initialization scheme to behave well. We encourage people to give more
rigorous analysis on emergence and stability so as to strike a more optimal
balance between these two.

On learning rates

Empirically, the scaling factor $\alpha$ is also dependent on learning rate.
When the learning rate is faster, more stability is usually required and hence
we need smaller emergence. There has been some theoretical results on the
optimal learning rate, for example, Hettinger et al. (2020) suggested that the
optimal learning rate should be inversely proportional to the gradient
magnitude at initialization:

$\displaystyle\eta=\frac{c}{\|\nabla L\|}$ (9)

where $\eta$ is the optimal learning rate, $\|\nabla L\|$ is the magnitude of
the gradient of the loss function $L$ with respect to the network parameters
at initialization, and $c$ is a constant.

So when given the learning rate, we should choose $\alpha$ such that the
resulting gradient magnitude at initialization is inversely proportional.

If we assume that by introducing our scheme we have $C(\alpha)^{N}$ times
increase to the initial gradient.Then under the learning rate $\eta$ we have

$\displaystyle\frac{\eta}{\eta_{0}}=\frac{\|\nabla L_{\alpha_{0}}\|}{\|\nabla
L_{\alpha}\|}=\frac{\alpha_{0}^{N}}{\alpha^{N}}$ (10)

hence we have

$\displaystyle\alpha=\alpha_{0}\Big{[}\frac{\eta_{0}}{\eta}\Big{]}^{1/N}.$
(11)

For a two layer network, if we choose $\alpha_{0}=2$ for learning rate
$\eta_{0}=1e-3$, then for a different learning rate $\eta=1e-4$ we should
choose $\alpha=6.32$.

Since based on our scheme, the initial gradient varies across layers, the
layer-wise learning rates configuration should also be considered a good
choice. Specifically, the learning rate for each layer, denoted by $\eta_{l}$,
should be proportional to the inverse of the square root of the expected
squared gradient norm at initialization. Mathematically, this can be expressed
as:

$\eta_{l}\propto\frac{1}{\sqrt{E\left[\left\|\nabla
L(\mathbf{x}_{l}^{0})\right\|^{2}\right]}}$

where $\eta_{l}$ is the learning rate for layer $l$, and $E\left[\left\|\nabla
L(\mathbf{x}_{l}^{0})\right\|^{2}\right]$ is the expected squared gradient
norm of the loss $L$ with respect to that layer’s inputs $\mathbf{x}_{l}^{0}$
at initialization. This approach aims to optimize the learning process by
adjusting the learning rates according to the variability and scale of the
gradients encountered in different layers of the neural network.

Other architectures and block wise initialization

Note that the mathematical equation (2) only works for MLP. For convolutional
layers, we have the following:

$\displaystyle E$
$\displaystyle=\sum_{i=1}^{N-1}\sum_{j>i}^{N}(n_{i}-a_{i})a_{j}\prod_{k=i+1}^{j-1}m_{k}$
(12)

where $m_{k}$ is the number of filters in layer $K$. The analysis largely
follows.

In most convolutional architectures and transformers, there are MLP blocks
presented. While we can only do our intilization to the MLP blocks and see an
improvement, we can also consider applying scheme (8) to the convolutional
layers. Given this formula for emergence, we can increase the global
emergence, but also increase the local emergence by applying scheme (8) to
some of the layer blocks. We will discuss this in the next section.

##  4 experiments

We evaluate our initialization scheme on benchmark datasets for image
classification and machine translation tasks. For image classification, five
different architectures are evaluated for CIFAR10 [37]. For machine
translation, we use our initialization scheme to find good initializations for
a Post-LN Transformer without any change to its original architecture on
IWSLT-14 De-En [39].

We conduct our experiments in PyTorch. We use the fairseq library for machine
translation [40]. All the experiments on CIFAR-10 and IWSLT-14 DE-EN can run
with one single NVIDIA A100 GPU with 13GB of RAM. Our initialization scheme
first initializes the weights using Kaiming initialization for all the Conv
and MLP layers for image classification. For machine translation, we use the
default Xavier initialization [12]. Base on the discussion in the previous
section, we choose the scale factors $\alpha=2$ with out batch normalization
and $\alpha_{i}=5$ with batch normalization.

On CIFAR-10, we focus on MLP and the feedforward VGG net with and without BN
layers. Since ResNet has recursive network structure, we leave it to a future
work to establish the emergence formula on it. For MLP, we use a simple MLP
architecture with $3$ hidden layers. For VGG net, we use VGG-19 and our
initialization scheme is compared with four different methods/settings: 1)
Kaiming Initialization [13]; 2) First train the network for one epoch with a
constant learning rate equal to the starting learning rate, labelled as “+1
epoch (Const. LR)" in Table 3; 3) First train the network for one epoch with a
linear warmup learning rate, labbeled as “+1 epoch (Warmup)" in Table 3; 4)
MetaInit [27]. The data is from GradInit[8]. On CIFAR-10, we train networks
with a batch size of 128, and a constant learning rate of $0.001$.

From our experiments we can see BN does stabilize VGG-19 and allows training
with stronger emergence (larger value of $\alpha$).

(a) w/o batch normalization,$\alpha=2$

(b) w/ batch normazalization, $\alpha=10$

Figure 3: Training loss and test accuracy of MLP on CIFAR-10. Table 1:  First epoch (Acc1) for models on CIFAR-10, $\alpha=2$ for VGG-19 without BN, and $\alpha=5$ for VGG-19 with BN. Model |  | VGG-19 | VGG-19 | ResNet-110 | ResNet-110
---|---|---|---|---|---
w/o BN | w/ BN | w/o BN | w/ BN
Kaiming | $Acc_{1}$ | $29.1\pm 1.5$ | $12.6\pm 0.6$ | $16.1\pm 2.1$ | $23.2\pm 0.9$
+1 epoch (Const. LR) | $Acc_{1}$ | $37.2\pm 1.1$ | $19.6\pm 4.0$ | $21.0\pm 3.8$ | $32.5\pm 3.8$
+1 epoch (Warmup) | $Acc_{1}$ | $37.4\pm 1.2$ | $53.5\pm 2.9$ | $19.8\pm 0.5$ | $48.7\pm 1.1$
MetaInit | $Acc_{1}$ | $30.5\pm 0.9$ | $35.1\pm 0.6$ | $14.6\pm 2.2$ | $29.0\pm 1.5$
GradInit | $Acc_{1}$ | $29.3\pm 0.6$ | $47.8\pm 1.8$ | $36.2\pm 0.8$ | $38.2\pm 0.9$
Ours | $Acc_{1}$ | $46.2\pm 0.6$ | $52.4\pm 1.0$ | $45.3\pm 2.0$ | $48.0\pm 1.5$
Table 2: Evaluate the transformer model for 80 epochs Method | $\text{BLEU}_{1}$ | $\text{BLEU}_{best}$
---|---|---
Xavier | - | 34.85
T-Fixup | 3.96 | 34.78
Ours | 4.8 | 35.13

We can see the since batch normalization promotes a good stability, we are
free to choose larger $\alpha$.

IWSLT’14 DE-EN [28] is a German to English translation dataset that has 160k
training examples. Our Transformer model is inherited from [3], which is a
Post-LN Transformer placing its Layer Normalization after the summation of the
skip connection and the residual branch. It has a $512$\- dimensional word
embedding layer and 1024 dimensions in its hidden FFN layer. It has $6$
encoder layers and $6$ decoder layers. We choose the learning rate to be
$5e-4$ with inverse-sqrt learning schedule with $4000$ warmup updates and
weight decay of $0.0001$.

Based on the discussion in the previous section, we have two ways of promoting
emergence: A) promote emergence globally, by reducing the magnitude of weights
in the encoder layers and increasing the magnitude of weights in the decoder
layers; B) promote emergence clockwise, where we apply equation (8) to each
encoder/ decoder block. In our transformer architecture, there is MLP block in
each encoder/decoder layer consisting of $2$ layers. In our experiments, we
can choose $\alpha$ up to $10$ for each MLP block, and notably we can see fast
increase of BLEU score in the first few epochs. For example, with $\alpha=10$,
the BLEU score after first epoch reaches $6.02$ while for T-Fixup the BLEU
score after first epoch is $3.79$.

For global emergence promoting scheme, we first initialize our scheme based on
T-Fixup, and then increase the magnitude of weights in the decoder layers by
$2$. We run the models for the maximum of $80$ epochs and evaluate the BLEU
score every epoch, and report the best BLEU scores throughout training for
each run and the result is in Table 2.

##  5 Conclusion

In this paper, we introduced a novel and straightforward neural network
initialization scheme inspired by the concept of emergence. Building on the
emergent network measures proposed by Li (2023) [2], our method adjusts the
layer-wise variance parameters to enhance the number of paths from inactive to
active nodes, thereby achieving higher emergence values. This approach is not
only easy to implement but also requires no additional optimization or
training steps compared to conventional methods like Xavier and Kaiming
initialization.

Our extensive evaluations across various architectures, including
convolutional neural networks (CNNs) for image recognition and transformers
for machine translation, demonstrate the significant advantages of our
initialization scheme. The empirical results show that our method
substantially improves model accuracy and convergence speed on standard
datasets such as CIFAR-10 and the IWSLT-14 translation task.

By focusing on enhancing the emergent properties of neural networks from the
outset, our initialization scheme facilitates better feature differentiation
and integration, leading to more effective learning. The theoretical
foundation and practical benefits of our method offer a promising direction
for future research and applications in machine learning.

Our work contributes to the growing body of research that seeks to understand
and harness the power of emergence in neural networks. By providing a simple
yet powerful modification to existing initialization techniques, we open new
avenues for improving neural network training methodologies. Future work could
explore further optimizations and adaptations of our initialization scheme to
other types of neural architectures and more complex tasks.

Overall, our emergence-promoting initialization scheme represents a valuable
addition to current neural network initialization practices, offering both
theoretical insights and practical improvements for the development of more
robust and capable machine learning models.

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