Diffusion Models as Evolutionary Algorithms.txt

Diffusion Models as Evolutionary Algorithms
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Submitted on 3 Oct 2024 (v1), last revised 4 Oct 2024 (this version, v2)]
Diffusion Models are Evolutionary Algorithms
Yanbo Zhang, Benedikt Hartl, Hananel Hazan, Michael Levin
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In a convergence of machine learning and biology, we reveal that diffusion models are evolutionary algorithms. By considering evolution as a denoising process and reversed evolution as diffusion, we mathematically demonstrate that diffusion models inherently perform evolutionary algorithms, naturally encompassing selection, mutation, and reproductive isolation. Building on this equivalence, we propose the Diffusion Evolution method: an evolutionary algorithm utilizing iterative denoising -- as originally introduced in the context of diffusion models -- to heuristically refine solutions in parameter spaces. Unlike traditional approaches, Diffusion Evolution efficiently identifies multiple optimal solutions and outperforms prominent mainstream evolutionary algorithms. Furthermore, leveraging advanced concepts from diffusion models, namely latent space diffusion and accelerated sampling, we introduce Latent Space Diffusion Evolution, which finds solutions for evolutionary tasks in high-dimensional complex parameter space while significantly reducing computational steps. This parallel between diffusion and evolution not only bridges two different fields but also opens new avenues for mutual enhancement, raising questions about open-ended evolution and potentially utilizing non-Gaussian or discrete diffusion models in the context of Diffusion Evolution.
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This paper by Yanbo Zhang, Benedikt Hartl, Hananel Hazan, and Michael Levin presents a novel perspective by positing that diffusion models, a type of generative model in machine learning, can be understood as evolutionary algorithms. The authors draw parallels between the process of evolution, which they equate with a denoising process, and diffusion, essentially the reverse of this. They mathematically demonstrate that diffusion models perform tasks akin to natural evolutionary processes���selection, mutation, and reproductive isolation.
Building on this foundational idea, the authors introduce "Diffusion Evolution," an evolutionary algorithm that employs the denoising techniques used in diffusion models to refine solutions iteratively in various parameter spaces. This method differs from traditional evolutionary algorithms in that it can efficiently find multiple optimal solutions and demonstrates superior performance compared to well-known evolutionary algorithms.
Moreover, they expand on this concept with "Latent Space Diffusion Evolution," which utilizes latent space techniques and accelerated sampling from diffusion models to solve evolutionary tasks in complex, high-dimensional parameter spaces more efficiently. This innovative approach not only shows the practical utility of blending diffusion models with evolutionary algorithms but also suggests a new area of exploration in machine learning and biology. This integration could lead to advancements in understanding open-ended evolution and the application of non-traditional diffusion model techniques, such as non-Gaussian or discrete models, in evolutionary computation.
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At least two processes in the biosphere have been recognized as capable of generalizing and driving novelty: evolution, 
a slow variational process adapting organisms across generations to their environment through natural selection (Dar- 
win, 1959; Dawkins, 2016); and learning, a faster transformational process allowing individuals to acquire knowledge 
and generalize from subjective experience during their lifetime (Kandel, 2013; Courville et al., 2006; Holland, 2000; 
Dayan & Abbott, 2001). These processes are intensively studied in distinct domains within artif i cial intelligence. Rel- 
atively recent work has started drawing parallels between the seemingly unrelated processes of evolution and learn- 
ing (Watson & Levin, 2023; Vanchurin et al., 2022; Levin, 2022; Watson et al., 2022; Kouvaris et al., 2017; Watson & 
Szathm�� ary, 2016; Watson et al., 2016; Power et al., 2015; Hinton et al., 1987; Baldwin, 2018). We here argue that in 
particular diffusion models (Sohl-Dickstein et al., 2015; Song et al., 2020b; Ho et al., 2020; Song et al., 2020a), where 
generative models trained to sample data points through incremental stochastic denoising, can be understood through 
evolutionary processes, inherently performing natural selection, mutation, and reproductive isolation. 
Theevolutionaryprocessisfundamentaltobiology, enablingspeciestoadapttochangingenvironmentsthroughmech- 
anisms like natural selection, genetic mutations, and hybridizations (Rosen, 1991; Wagner, 2015; Dawkins, 1996); this 
adaptive process introduces variations in organisms��� genetic codes over time, leading to well-adapted and diverse indi- 
viduals (Mitchell & Cheney, 2024; Levin, 2023; Gould, 2002; Dennett, 1995; Smith & Szathmary, 1997; Szathm�� ary, 
2015). Evolutionary algorithms utilize such biologically inspired variational principles to iteratively ref i ne sets of nu- 
merical parameters that encode potential solutions to often rugged objective functions (Vikhar, 2016; Golberg, 1989;

Figure 1: Evolution processes can be viewed as the inverse process of diffusion, where higher i tness populations 
(red points) have higher i nal probability density. The initially unstructured parameters are iteratively ref i ned towards 
high-f i tness regions in parameter space. 
Grefenstette, 1993; Holland, 1992). In another side, recent breakthroughs in deep learning have led to the development 
of diffusion models���generative algorithms that iteratively ref i ne data points to sample novel yet realistic data follow- 
ing complex target distributions: models like Stable Diffusion (Rombach et al., 2022) and Sora (Brooks et al., 2024) 
demonstrate remarkable realism and diversity in generating image and video. Notably, both evolutionary processes 
and diffusion models rely on iterative ref i nements that combine directed updates with undirected perturbations: in evo- 
lution, random genetic mutations introduce diversity while natural selection guides populations toward greater i tness, 
and in diffusion models, random noise is progressively transformed into meaningful data through learned denoising 
steps that steer samples toward the target distribution. This parallel raises fundamental questions: Are the mechanisms 
underlying evolution and diffusion models fundamentally connected? Is this similarity merely an analogy, or does it 
ref l ect a deeper mathematical duality between biological evolution and generative modeling? 
To answer these questions, we i rst examine evolution from the perspective of generative models. By considering 
populations of species in the biosphere, the variational evolution process can also be viewed as a transformation of 
distributions: the distributions of genotypes and phenotypes. Over evolutionary time scales, mutation and selection 
collectively alter the shape of these distributions. Similarly, many biologically inspired evolutionary algorithms can 
be understood in the same way: they optimize an objective function by maintaining- and iteratively changing a large 
population���s distribution. In fact, this concept is central to most generative models: the transformation of distri- 
butions. Variational Autoencoders (VAEs) (Kingma, 2013), Generative Adversarial Networks (GANs) (Goodfellow 
et al., 2014), and diffusion models are all trained to transform simple distributions, typically standard Gaussian distri- 
butions, into complex distributions, where the samples represent meaningful images, videos, or audio, etc. 
On the other hand, diffusion models can also be viewed from an evolutionary perspective. As a generative model, 
diffusion models transform Gaussian distributions in an iterative manner into complex, structured data-points that 
resemble the training data distribution. During the training phase, the data points are corrupted by adding noise, and 
the model is trained to predict this added noise to reverse the process. In the sampling phase, starting with Gaussian- 
distributed data points, the model iteratively denoises to incrementally ref i ne the data point samples. By considering 
noise-free samples as the desired outcome, such a directed denoising can be interpreted as directed selection, with each 
step introducing slight noise, akin to mutations. Together, this resembles an evolutionary process (Fields & Levin, 
2020), where evolution is formulated as a combination of deterministic dynamics and stochastic mutations within the 
framework of non-equilibrium thermodynamics (Ao, 2005). This aligns with recent ideas that interpret the genome 
as a latent space parameterization of a multi-scale generative morphogenetic process, rather than a direct blueprint of 
an organism (Mitchell & Cheney, 2024; Hartl et al., 2024; Levin, 2023; Gould, 2002). If one were to revert the time 
direction of an evolutionary process, the evolved population of potentially highly correlated high-f i tness solutions will 
dissolve gradually, i.e., step by step and thus akin to the forward process in diffusion models, into the respectively 
chosen initial distribution, typically Gaussian noise, see Figure 1.
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The passage you provided delves into the intriguing parallels between biological evolution and machine learning processes, particularly diffusion models, suggesting a deeper connection between these two domains that extends beyond mere analogy to potentially a mathematical duality.
In biology, evolution operates through mechanisms like natural selection, genetic mutations, and hybridizations, creating a dynamic environment where species continually adapt and diversify. Evolutionary algorithms in computational sciences mirror this process by using similar principles to iteratively refine numerical parameters within complex objective functions, capturing the essence of biological adaptation in algorithmic form.
The text also discusses the concept of diffusion models in machine learning, which are a class of generative models that incrementally refine data points through stochastic denoising steps to produce realistic and complex data outputs, akin to natural selection and mutation in evolution. This iterative refinement, whether in evolutionary biology or in diffusion models, involves a combination of directed updates (guided changes towards a goal, like fitness in evolution or accuracy in data generation) and undirected perturbations (random mutations or noise introduction).
By conceptualizing evolution as a transformation of distributions over time ��� in genotypes and phenotypes ��� and comparing it to how diffusion models transform Gaussian noise into structured data points, the text proposes a framework where both processes are seen as analogous transformations of distributions. This raises fundamental questions about whether the underlying mechanisms of evolution and generative modeling are not just similar but are connected through deeper mathematical structures, potentially viewing the genome itself as a type of latent space that encodes a generative process.
This interdisciplinary insight could redefine how we perceive the relationship between biological processes and artificial intelligence, suggesting that methods used in one field could be adapted to advance the other, offering a richer understanding of both evolutionary biology and machine learning.
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Driven by this intuition, we conduct a thorough investigation into the connections between diffusion models and evo- 
lutionary algorithms, discovering that these seemingly disparate concepts share the same mathematical foundations. 
This insight leads to a novel approach, the Diffusion Evolution algorithm, which directly utilizes the framework of 
diffusion models to perform evolutionary optimization. This can be obtained by inverting the diffusion process with 
2 
Diffusion Models are Evolutionary AlgorithmsA PREPRINT 
the Bayesian method. Our analytical study of Diffusion Evolution reveals promising parallels to biological evolution, 
naturally incorporating concepts such as mutation, hybridization, and even reproductive isolation. 
Thisequivalenceprovidesanewwayofimprovingevolutionaryalgorithmsandhasthepotentialtounifydevelopments 
in both i elds. By mimicking biological evolution, evolutionary algorithms have shown promising results in numeri- 
cal optimization, particularly for tasks that cannot be effectively trained using gradient-based methods (Wang et al., 
2024; Goodfellow et al., 2014). These algorithms thus excel in exploring complex, rugged search spaces and i nding 
globally optimal or near-optimal solutions (Ho & Salimans, 2022; Hansen, 2016; Hansen & Ostermeier, 2001; Sehnke 
et al., 2010). While the biosphere exhibits extreme diversity in lifeforms of life, many evolutionary strategies, such as 
CMA-ES (Hansen & Ostermeier, 2001), and PEPG (Sehnke et al., 2010), struggle to i nd diverse solutions (Lehman & 
Stanley, 2011). However, our Diffusion Evolution Algorithm offers a new approach. By naturally incorporating muta- 
tion, hybridization, and reproductive isolation, our algorithm can discover diverse solutions, mirroring the diversity of 
the biosphere, rather than converging on a single solution as is often the case with traditional methods. Since this par- 
allel between diffusion and evolution exists naturally and not imposed by our design, the two i elds ��� diffusion models 
and evolutionary computing ��� can mutually benef i t from each other. For example, we demonstrate that the concept of 
latent diffusion (Rombach et al., 2022) and accelerated sampling (Nichol & Dhariwal, 2021) can signif i cantly improve 
the performance of our Diffusion Evolution algorithm. 
In the following sections, we will i rst review evolutionary strategies and diffusion models, introduce the mathematical 
connection between diffusion and evolution, and propose the Diffusion Evolution algorithm. Then, we will quanti- 
tatively compare our algorithm to conventional evolutionary strategies, demonstrating its capability to i nd multiple 
solutions, solve complex evolutionary tasks, and incorporate developments from diffusion model literature. Finally, 
the emerging connections between the derived algorithm and evolution will be discussed, along with the potentials 
of this i nding and the limitations of our algorithm. Codes are available on Github2and the package can be installed 
via pip install diffevo.
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The excerpt discusses a significant breakthrough in computational science, where the authors have established mathematical connections between diffusion models and evolutionary algorithms. This connection has culminated in the creation of a novel algorithm termed "Diffusion Evolution," which leverages the principles of diffusion models to enhance evolutionary optimization.
Key Insights from the Diffusion Evolution Algorithm:
Mathematical Foundations:
The authors have identified that diffusion models and evolutionary algorithms share underlying mathematical principles. By reversing the diffusion process using Bayesian methods, they can apply these models directly to evolutionary optimization.
Biological Analogies:
The Diffusion Evolution algorithm naturally integrates biological concepts such as mutation, hybridization, and reproductive isolation���core elements of natural evolution. This integration suggests that the algorithm not only optimizes but does so in a way that mirrors the biological processes that result in diversity and adaptation.
Improving Evolutionary Algorithms:
Traditional evolutionary algorithms are adept at navigating complex, rugged search spaces to find globally optimal or near-optimal solutions. However, they often fall short in generating a diversity of solutions. The Diffusion Evolution algorithm, by contrast, is designed to discover a variety of solutions, reflecting the diversity found in nature. This is particularly advantageous for tasks where diversity is beneficial or required.
Technological Synergies:
The implementation of advanced diffusion model techniques like latent diffusion and accelerated sampling within the Diffusion Evolution algorithm demonstrates significant performance improvements. These techniques help the algorithm not only in finding diverse solutions but also in doing so more efficiently.
Broader Implications and Availability:
The potential of the Diffusion Evolution algorithm to unify and advance both fields of diffusion models and evolutionary computing is substantial. The availability of this algorithm as open-source code on GitHub and its installable package via pip further democratizes access to this advanced tool, allowing a wider community of researchers and practitioners to explore and expand upon this innovative approach.
Future Directions:
The paper promises to delve deeper into these insights, offering a quantitative comparison of the Diffusion Evolution algorithm against conventional evolutionary strategies. This comparison will likely highlight the algorithm's strengths in handling complex evolutionary tasks and finding multiple optimal solutions. Additionally, the discussion will extend to the broader implications of these findings, exploring both the potential benefits and the limitations of the Diffusion Evolution algorithm.
This approach represents a significant step forward in computational methods, potentially leading to more robust, efficient, and diverse optimization strategies that better mimic and leverage the complexities of natural processes.
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2.1Evolutionary Algorithms 
The principles of evolution extend far beyond biology, offering exceptional utility in addressing complex systems 
across various domains. The key components of this process ��� imperfect replication with heredity and i tness-based 
selection ��� are suff i ciently general to i nd applications in diverse i elds. In computer and data science, for instance, 
evolutionary algorithms play a crucial role in optimization (Vikhar, 2016; Grefenstette, 1993; Golberg, 1989; Holland, 
1992). These heuristic numerical techniques, such as CMA-ES (Hansen & Ostermeier, 2001) and PEPG (Sehnke 
et al., 2010), maintain and optimize a population of genotypic parameters over successive generations through opera- 
tions inspired by biological evolution, such as selection of the i ttest, reproduction, genetic crossover, and mutations. 
The goal is to gradually adapt the parameters of the entire population so individual genotypic samples, or short indi- 
viduals, perform well when evaluated against an objective- or i tness function. These algorithms harness the dynamics 
of evolutionary biology to discover optimal or near-optimal solutions within vast, complex, and otherwise intractable 
parameter spaces. The evaluated numerical i tness score of an individual correlates with its probability of survival and 
reproduction, ensuring that individuals with superior traits have a greater chance of passing their genetic information 
to the next generation, thus driving the evolutionary process toward more optimal solutions. Such approaches are par- 
ticularly valuable when heuristic solutions are needed to explore extensive combinatorial and permutation landscapes. 
Some evolutionary algorithms operate with discrete, others with continuous sets of parameters. Here, we focus on the 
latter since discrete tasks can be seen as a subcategory of continuous tasks. Typically, the structure of the parameter 
space is apriori unknown. Thus, the initial population is often sampled from a standard normal distribution. As 
explained above, this initially random population is successively adapted and ref i ned, generation by generation, to 
perform well on an arbitrary objective function. Thus, initially randomized parameters are successively varied by 
evolutionary algorithms into sets of potentially highly structured parameters that perform well on the specif i ed task, 
eventually (and hopefully) solving the designated problem by optimizing the objective function. Thus, evolutionary 
algorithms can be understood as generative models that use heuristic information about already explored regions of 
the parameter space (at least from the previous generation) to sample potentially better-adapted offspring individuals 
for the next generation (c.f., CMA-ES (Hansen et al., 2003), etc.).
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This section details the foundational principles and functioning of evolutionary algorithms (EAs), a significant branch of computer science and optimization theory that leverages evolutionary principles such as selection, reproduction, genetic crossover, and mutations to solve complex optimization problems.
Key Concepts of Evolutionary Algorithms:
Imperfect Replication with Heredity and Fitness-based Selection:
Imperfect Replication:
In EAs, individuals in a population are subjected to random changes or mutations, mirroring biological evolution's imperfect replication of genetic material.
Heredity:
Traits of individuals (solutions) can be passed down to the next generation, ensuring continuity and the potential to refine good traits over time.
Fitness-based Selection:
Individuals are selected for reproduction based on their fitness, which is typically measured by how well they perform against a predefined objective function. This ensures that better-performing individuals are more likely to contribute to the gene pool of the next generation.
Optimization in Complex Parameter Spaces:
EAs are particularly effective in navigating large, complex, and rugged parameter spaces where traditional gradient-based optimization methods might struggle. By simulating the process of natural selection, EAs incrementally improve the solutions, adapting the population towards optimal or near-optimal parameters.
Generative Model Perspective:
Viewing EAs as generative models highlights their capability to transform an initial random distribution of parameters (typically Gaussian) into a structured set of parameters that perform well on a specific task. This transformation is achieved through iterative selection and mutation processes, refining the population based on the information gathered from previous generations.
Application Across Continuous and Discrete Tasks:
While the discussion primarily focuses on EAs that operate with continuous parameters, the underlying principles also apply to discrete parameter sets. The adaptability of EAs to both types of parameters showcases their versatility and broad applicability across different types of optimization problems.
Heuristic Sampling and Solution Exploration:
EAs use heuristic information from previously explored regions of the parameter space to guide the generation of new individuals. This approach allows EAs to "learn" from past iterations and potentially skip over less promising areas of the search space, making them efficient for exploring extensive combinatorial landscapes.
Significance and Utility:
EAs are indispensable for problems where the parameter space is vast and not well-understood beforehand. Their ability to iteratively refine and adapt a population of solutions makes them particularly useful for tasks where precise analytical solutions are infeasible or where the landscape of possible solutions is too complex for direct enumeration. This adaptability extends their utility beyond theoretical applications, making them practical tools in fields like robotics, artificial intelligence, and complex system design, where dynamic solution adaptation is crucial.
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2.2Diffusion Models 
Diffusion models, such as denoising diffusion probabilistic models (DDPM) (Ho et al., 2020) and denoising diffusion 
implicit models (DDIM) (Song et al., 2020a), have shown promising generative capabilities in image, video, and even 
neural network parameters (Wang et al., 2024). Similar to other generative approaches such as GANs, VAEs, and 
l ow-based models (Dinh et al., 2016; Chen et al., 2019), diffusion models transform a simple distribution, often a 
Gaussian, into a more complex distribution that captures the characteristics of the training data. Diffusion models 
achieve this, in contrast to other techniques, via iterative denoising steps, progressively transforming noisy data into 
less noisy (Raya & Ambrogioni, 2024), more coherent representations (Sohl-Dickstein et al., 2015). 
Diffusion models have two phases: diffusion and denoising. In the diffusion phase, we are blending original data 
points with some extent of Gaussian noise. Specif i cally, let x0be the original data point and xTbe the fully distorted 
data, then the process of diffusion can be represented as: 
xt= ����� tx0+ ���1 ��� �� t��, (1) 
where the amount of total noise �� ��� N(0,I) added to the data x0at time step t ��� [0,T] is controlled by ��tthat 
is 
monotonously decreasing from ��0= 1 to ��T��� 0. Thus, while x0represents 
the original data, xTwill consist entirely 
of Gaussian noise. To restore such diffused data, a predictive model, typically a neural network ����with parameter ��, is 
trained to predict the added total noise given xtand time step t. Thus, diffusion models can be trained by minimizing 
the loss function: L = Ex0���pdata,�����N(0,I)�������(�����tx0+ ���1 ��� �� t��,t) ��� �����2, 
(2) 
where pdatais the distribution of training data. So, conventionally, diffusion models are understood as predicting the 
added noise during the diffusion process. 
In the denoising phase, starting with a noisy pattern, the trained models are used to iteratively remove the predicted 
noise from current data: from xT��� N(0,I), iteratively ref i ne to xT���1,xT���2, 
..., until x0. In the DDIM framework, 
this sampling process is given by: 
xt���1= ����� t���1 ?x 
t��� ���1 ��� �� t����(xt,t) 
����� t ? 
+ q 1 ��� ��t���1��� ��2 
t����(xt,t) + ��tw, (3) 
where ��tcontrols the amount of noise w ��� N(0,I) added during the denoising phase. Notably, the schedule of ��t 
and ��twill both affect the denoising process and can be chosen based on our needs under the DDIM framework.
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Overview of Diffusion Models
Diffusion models, particularly denoising diffusion probabilistic models (DDPM) and denoising diffusion implicit models (DDIM), represent a class of generative models that have demonstrated significant capabilities in generating complex data outputs such as images, videos, and even neural network parameters.
How Diffusion Models Work
1. Diffusion Phase:
Initial Process:
The model begins with a data point
x
0
x_0
x
0
���
(the original data) and progressively adds Gaussian noise over a series of time steps
t
t
t
, leading to a fully distorted data point
x
T
x_T
x
T
���
that essentially consists of random noise.
Mathematical Representation:
This process can be expressed mathematically as:
x
t
=
��
t
x
0
+
1
���
��
t
��
x_t = \sqrt{\alpha_t} x_0 + \sqrt{1 - \alpha_t} \epsilon
x
t
���
=
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t
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x
0
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+
1
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��
t
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Here,
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N
(
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I
)
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��
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(
0
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I
)
is the Gaussian noise, and
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t
\alpha_t
��
t
���
is a coefficient that decreases monotonically from 1 to a value close to 0 across the diffusion steps, controlling the extent of noise added.
2. Training Phase:
Noise Prediction:
A neural network
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\epsilon_\theta
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��
���
with parameters
��
\theta
��
is trained to predict the noise added at each diffusion step. This is achieved by minimizing the loss function:
L
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x
0
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p
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a
t
a
,
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N
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)
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t
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,
t
)
���
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2
L = \mathbb{E}_{x_0 \sim p_{data}, \epsilon \sim N(0, I)} \|\epsilon_\theta(\sqrt{\alpha_t}x_0 + \sqrt{1 - \alpha_t}\epsilon, t) - \epsilon\|^2
L
=
E
x
0
���
���
p
d
a
t
a
���
,
��
���
N
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,
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���
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2
Goal of Training:
The training essentially teaches the model to reverse the noise addition process by predicting the exact noise that was added during the diffusion phase.
3. Denoising Phase:
Noise Removal:
Starting with the noisy data
x
T
x_T
x
T
���
, the model uses the trained
��
��
\epsilon_\theta
��
��
���
to iteratively remove the predicted noise, effectively reversing the diffusion process step-by-step until it reaches
x
0
x_0
x
0
���
.
DDIM Sampling Process:
In the DDIM framework, this denoising is detailed by:
x
t
���
1
=
��
t
���
1
(
x
t
���
1
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t
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(
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)
+
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(
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+
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w
x_{t-1} = \sqrt{\alpha_{t-1}} \left(x_t - \sqrt{1 - \alpha_t} \epsilon_\theta(x_t, t)\right) + \sqrt{1 - \alpha_{t-1} - \sigma_t^2} \epsilon_\theta(x_t, t) + \sigma_t w
x
t
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1
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=
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t
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1
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(
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w
where
��
t
\sigma_t
��
t
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controls additional noise
w
���
N
(
0
,
I
)
w \sim N(0, I)
w
���
N
(
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,
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)
added during denoising, and both
��
t
\alpha_t
��
t
���
and
��
t
\sigma_t
��
t
���
can be adjusted to fine-tune the denoising process.
Key Characteristics
Iterative Transformation:
Similar to how biological evolution operates through small, iterative changes, diffusion models transform a simple Gaussian distribution into a complex data distribution by iteratively refining the data through a series of small, noise-adjusting steps.
Generative Power:
These models rival other generative approaches like GANs and VAEs by achieving high-quality generative results without directly competing frameworks (e.g., discriminator in GANs).
Application Flexibility:
The ability to generate a variety of complex outputs makes diffusion models highly versatile and applicable across different domains, including creative industries and neural network design.
This sophisticated mechanism of operation positions diffusion models as powerful tools in the realm of artificial intelligence, capable of creating highly realistic and diverse outputs that closely mimic the distribution of the training data.
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3Diffusion Models are Evolutionary Algorithms 
Similar to the relationship between energy and probability in statistical physics, evolutionary tasks can be connected 
to generative tasks by mapping i tness to probability density: higher i tness correspond to higher probability density. 
Thus, given a i tness function f : Rn��� R, we can choose a mapping g to transform f into a probability density 
function p(x) = g[f(x)]. When aligning the denoising process in a diffusion model with evolution, we want x0to 
follow this density function, i.e., p(x0= x) = g[f(x)]. This requires an alternative view of diffusion models (Song 
et al., 2020a): diffusion models are directly predicting the original data samples from noisy versions of those samples 
at each time step. Given the diffusion process xt= ����� tx0+ ���1 ��� �� 
t��, we can easily express x0in terms of the 
noise ��, and vise versa: x0= xt��� ���1 ��� �� t�� 
����� t 
, and �� = xt��� ����� tx0 
���1 ��� �� 
t 
.(4) 
In diffusion models, the error �� between x0and xtis estimated by a neural network, i.e., �� �� = ����(xt,t). Thus, 
Equation 4 provides an estimation �� x0for x0when replacing �� with �� ��. Hence, the sampling process of DDIM (Song 
et al., 2020a) in Equation 3 can be written as: 
xt���1= ����� t���1�� x0+ q 1 ��� ��t���1��� ��2 
t�� �� + ��tw. (5) 
Since the denoising step in diffusion models requires an estimation of x0, we need to derive it from sample xtand the 
corresponding i tness f(xt). The estimation of x0can be expressed as a conditional probability p(x0= x|xt). 
Using 
Bayes��� theorem and p(x0= x) = g[f(x)] yields: 
p(x0= x|xt) 
= p(xt|x0= x)p(x0= x) 
p(xt) 
= p(xt|x)g[f(x)] 
p(xt)

Figure 2: (a) Diffusion Evolution on a two-peak i tness landscape: Populations near the two black crosses have higher 
i tness. For each individual xt(black star), its target �� x0(red dots) is estimated by a weighted average of its neighbors 
(c.f., dots within the blue disks, respectively); larger dot-size indicates higher i tness. The individual then moves a 
small step forward to the next generation (orange star). As evolution proceeds, the neighbor range decreases, making 
the process increasingly sensitive to local neighbors, thereby enabling global competition originally, while ���zooming 
in��� eventually to balance between optimization and diversity. (b) By mapping the population to a 1-D space (dashed 
lines in (a)), we track the progress of Diffusion Evolution. As evolution progresses, both the individuals (gray) and 
their estimated origins (red) move closer to the targets (vertical dashed lines), with the estimated origins advancing 
faster. 
Here, p(xt|x0= x) can be computed easily by N(xt; ����� tx,1 ��� ��t) 
given the design of the diffusion process, i.e., 
xt= ����� tx0+ ���1 ��� �� t��. Since deep-learning-based diffusion models are trained using mean squared error loss, the 
x0estimated by xtshould be the weighted average of the sample x. Hence, the estimation function of x0becomes: 
�� x0(xt,��,t) = X 
x���peval(x) p(x0= x|xt)x 
= X 
x���peval(x) 
g[f(x)]N(xt; ����� tx,1 ��� ��t) 
p(xt) 
x,(7) 
where pevalis the evaluation sample on which we compute the i tness score, here given by the current population 
Xt= (x(1) 
t ,x(2) 
t ,...,x(N) 
t ) of N individuals. Equation 7 has three weight terms: The i rst term g[f(x)] assigns larger 
weights to high i tness samples. For each individual sample xt, the second Gaussian term N(xt; ����� tx,1�����t) 
makes 
each individual only sensitive to local neighbors of evaluation samples. The third term p(xt) is a normalization term. 
Hence, �� x0can be simplif i ed to: 
�� x0(xt,��,t) = 
1 
Z X 
x���Xt g[f(x)]N(xt; ����� tx,1 ��� ��t)x, 
(8) 
where Z is the normalization term: 
Z = p(xt) = X 
x���Xt 
g[f(x)]N(xt; ����� tx,1 ��� ��t). (9) 
When substituting Equation 8 into Equation 4 we can express �� �� as: 
�� ��(xt,��,t) = xt��� ����� t�� x0(xt,��,t) 
���1 ��� �� 
t 
,(10) 
and by substituting Equations 8 and 10 into Equation 5, we derive the Diffusion Evolution algorithm: an evolutionary 
optimization procedure based on iterative error correction akin to diffusion models but without relying on neural networks at all, see psuedocode in Algorithm 1. When inversely denoising, i.e., evolving from time T to 0, while 
increasing ��t, the Gaussian term will initially have a high variance, allowing global exploration at i rst. As the 
evolution progresses, the variance decreases giving lower weight to distant populations, leads to local optimization 
(exploitation). This locality avoids global competition and thus allows the algorithm to maintain multiple solutions 
and balance exploration and exploitation. Hence, the denoising process of diffusion models can be understood in an 
evolutionary manner: �� x0represents an estimated high i tness parameter target. In contrast, xtcan be considered as 
diffused from high-f i tness points. The i rst two parts in the Equation 5, i.e., ����� t���1�� x0+ p1 
��� �� t���1��� ��2t�� ��, guide the 
individuals towards high i tness targets in small steps. The last part of Equation 5, ��tw, is an integral part of diffusion 
models, perturbing the parameters in our approach similarly to random mutations. 
Algorithm 1 Diffusion Evolution 
Require: Population size N, parameter dimension D, i tness function f, density mapping function g, total evolution 
steps T, diffusion schedule �� and noise schedule ��. 
Ensure: ��0��� 1,��T��� 0,��i> ��i+1,0 < ��i< ���1 ��� �� i���1 
1:[x(1) 
T ,x(2) 
T ,...,x(N) 
T ] ��� N(0,IN��D)��� 
Initialize population 
2:for t ��� [T,T ��� 1,...,2] do 
3:���i ��� [1,N] : Qi��� g[f(x(i) 
t )]��� Fitness are cached to avoid repeated evaluations 
4:for i ��� [1,2,..,N] do 
5:Z ��� 
N X 
j=1 QjN(x(i) 
t ; ����� tx 
(j) 
t ,1 ��� ��t) 
6:�� x0��� 
1 
Z 
N X 
j=1 QjN(x(i) 
t ; ����� tx 
(j) 
t ,1 ��� ��t)x(j) 
t 
7:w ��� N(0,ID) 
8:x(i) 
t���1��� ����� t���1�� x0+ p1 
��� �� t���1��� ��2t 
x(i) 
t ��� ����� t�� x0 
���1 ��� �� 
t 
+ ��tw 
9:end for 
10:end for 
Figure 2(a) demonstrates the detailed evolution process of a multi-target i tness landscape with two optimal points (see 
exact i tness function in Appendix A.1). Each individual estimates high i tness parameter targets and moves toward 
the target along with random mutations. The high i tness parameter targets �� x0are estimated based on their neighbors��� 
i tness scores (neighbors are shown in blue disks, with radius proportional to ���1 ��� �� t/�����t). The estimated targets �� x0 
typically move faster than the individuals while the individuals are successively ref i ned in small denoising steps in the 
direction of the estimated target, see Figure 2(b). Although �� x0often have higher i tness, they exhibit lower diversity, 
hence they are used as a goal of individuals instead of the i nal solutions. This difference also provides l exibility in 
balancing between more greedy and more diverse strategies.
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This detailed description of the "Diffusion Evolution" algorithm illustrates a sophisticated blend of concepts from both diffusion models and evolutionary algorithms, aiming to harness the strengths of both approaches in an innovative optimization procedure. Here���s a breakdown of the key elements and functionalities of the Diffusion Evolution algorithm:
Integration of Diffusion and Evolutionary Concepts:
Fitness and Probability Mapping
:
The algorithm begins by mapping a fitness function
f
:
R
n
���
R
f: \mathbb{R}^n \rightarrow \mathbb{R}
f
:
R
n
���
R
into a probability density function
p
(
x
)
p(x)
p
(
x
)
, where higher fitness corresponds to higher probability density. This mapping is pivotal because it allows the algorithm to treat the optimization landscape as a probability distribution, facilitating a natural integration of evolutionary and diffusion strategies.
Reversing the Diffusion Process
:
By adapting the denoising phase of diffusion models, the algorithm estimates
x
0
x_0
x
0
���
from
x
t
x_t
x
t
���
using the relationship defined in the diffusion models. This mirrors the evolutionary concept of selecting for higher fitness, as
x
0
x_0
x
0
���
represents a less noisy (or higher fitness) version of
x
t
x_t
x
t
���
.
Iterative Refinement
:
The main operational phase of the algorithm involves iterative error correction that does not rely on neural networks, focusing instead on direct estimations from noisy data points to their cleaner versions. Each iteration involves recalculating fitness, adjusting noise levels, and refining individual data points to gradually approach optimal solutions.
Algorithmic Steps:
Initialization
:
The population is initialized with parameters sampled from a Gaussian distribution, representing a wide exploration of the solution space.
Evolutionary Dynamics
:
For each iteration, the algorithm:
Computes the fitness of each individual.
Estimates the "less noisy" version of each data point
x
0
x_0
x
0
���
based on its current state
x
t
x_t
x
t
���
and the calculated fitness, effectively using the diffusion model���s framework to estimate a target state that corresponds to higher fitness.
Updates each individual by moving it towards its estimated target, incorporating a controlled amount of noise to maintain diversity and prevent premature convergence.
Balance Between Exploration and Exploitation
:
Early in the process, higher noise levels allow for global exploration of the fitness landscape. As the process advances, the noise is gradually reduced, increasing the weight given to local optimizations and enabling fine-tuning towards the fitness peaks.
Outcome and Diversity
:
The algorithm aims not just to find optimal solutions but also to maintain a diversity of solutions. This is crucial in complex optimization tasks where multiple valid solutions exist, and preserving diversity can provide a broader range of options or insights.
Visualization and Practical Implications:
The process is visualized in a schematic where individual trajectories in the fitness landscape are tracked, showing how each individual estimates high fitness targets and adjusts accordingly. This visualization emphasizes the algorithm's ability to adaptively refine solutions, drawing parallels to natural evolutionary processes.
The "Diffusion Evolution" algorithm represents a significant theoretical advancement by blending generative and evolutionary techniques. It leverages the conceptual strengths of both fields to address complex optimization problems more effectively, offering a novel tool that could potentially lead to breakthroughs in areas where traditional methods struggle, particularly in dealing with high-dimensional and multimodal fitness landscapes.
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4.2Latent Space Diffusion Evolution 
050100150200250300350400450500 
time steps 
2 
4 
6 
8 
10 
generation 
(a) evolution process 
2.55.07.510.0 
generation 
10 
100 
300 
500 
reward 
(b) reward comparison 
DiffEvo 
latent DiffEvo 
latent DiffEvo (high-d) 
CMA-ES 
max reward 101 
z1 
1.0 
0.5 
0.0 
0.5 
1.0 
z2 
(c) latent space comparison 202 
x 
(d) cart-pole system 
Figure 4: (a) Evolution process of cart-pole tasks: The horizontal axis shows survival time, and the vertical axis 
represents generations. Each point indicates an individual���s state (pole angle, cart shift) at their i nal survival. As the 
evolution progresses, more systems survive longer and achieve higher rewards. (b) Compared to the original Diffusion 
Evolution (blue), the Latent Space Diffusion Evolution method (red) signif i cantly improves performance, while the 
CMA-ES method (gray) fails to i nd any solutions in given generations. This latent method can even be applied to 
high-dimensional spaces (orange), with dimensions as high as 17,410. Each experiment is repeated 100 times, with 
medians (solid lines) and ranges (25% to 75% quantile) shown as shaded areas. (c) Projecting the parameters of 
individuals into a latent space visualize their diversity. The same projection is used for all results (except for the high- 
dimensional experiment, which has a different original dimension). This indicates enhanced diversity with the latent 
method. (d) The cart-pole system consists of a pole hinged to the cart. And the controller balances the pole by moving 
the cart left or right. 8 
Diffusion Models are Evolutionary AlgorithmsA PREPRINT 
Here, we apply the Diffusion Evolution method to reinforcement learning tasks (Sutton & Barto, 1998) to train neural 
networks for controlling the cart-pole system (Barto et al., 1983). This system has a cart with a hinged pole, and the 
objective is to keep the pole vertical as long as possible by moving the cart sideways while not exceeding a certain 
range, see Figure 4(d). The game is terminated if the pole-angle exceeds ��12���or the cart position exceeds ��2.4. 
Thus, longer duration yield higher i tness. We use a two-layer neural network of 58 parameters to control the cart, 
with inputs being the current position, velocity, pole angle, and pole angular velocity. The output of the neural network 
determines whether to move left or right. See more details about the neural network in Appendix A.5.1. The task is 
considered solved when a i tness score (cumulative reward) of 500 is reached consistently over several episodes. 
Deploying our original Diffusion Evolution method to this problem results in poor performance and lack of diversity, 
see Figure 4(b-c). To address this issue, we propose Latent Space Diffusion Evolution: inspired by the latent space dif- 
fusion model (Rombach et al., 2022), we map individual parameters into a lower-dimensional latent space in which we 
perform the Diffusion Evolution Algorithm. This approach signif i cantly improves performance and restores diversity. 
ThekeyinsightcomesfromtheGaussianterminEquation8forestimatingtheoriginalpoints �� x0: thedistancebetween 
parameters increases with higher dimensions, making the evolution more local and slower. Moreover, the parameter 
or genotype space may have dimensions that don���t effectively impact i tness, known as sloppiness (Gutenkunst et al., 
2007). Assigning random values to these dimensions often doesn���t affect i tness, similar to genetic drift or neutral 
genes, suggesting the true high-f i tness genotype distribution is lower-dimensional. The straightforward approach is 
directly denoising in a lower-dimensional latent space z and estimating high-quality targets z0via: 
�� z0(zt,��,t) = X 
z p(z|zt)z 
= 
1 
Z X 
x���peval(x) g[f���(z)]N(zt; ����� tz,1 ��� ��t)z. 
(11) 
However, this approach requires a decoder and a new i tness function f���for z, which can be challenging to obtain. To 
circumvent this, we approximate the latent diffusion by using the latent space only to calculate the distance between 
individuals. While we don���t know the exact distribution of x apriori, a random projection can often preserve the 
distance relationships between populations, as suggested by the Johnson-Lindenstrauss lemma (Johnson, 1984). To 
do this, we change Equation 11 to: 
�� x0(xt,��,t) = X 
x���peval(x) p(x|xt)x 
= 
1 
Z X 
x���peval(x) g[f(x)]N(zt; ����� tz,1 ��� ��t)x, 
(12) 
where z = Ex, Eij��� N(d,D)(0,1/D), D is the dimension of x, and d is the dimension of latent space (Johnson, 
1984), see Algorithm 2 in Appendix. Here we choose d = 2 in our experiments for better visualization. The results 
show a signif i cant improvement in both i tness and diversity, see Figure 4(b-c). We also found that this latent evolution 
can still operate in a much larger dimensional parameter space, utilizing a three-layer neural network with 17,410 
parameters, while still achieving strong performance. Combined with accelerated sampling method, we can solve the 
cart pole task in only 10 generations, with 512 population size, one i tness evaluation per individual. These highlights 
the potential of using tools and theories from the diffusion model domain in evolutionary optimization tasks and 
vice versa, opening up broad opportunities to improve and understand evolution from a new perspective.
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The section on Latent Space Diffusion Evolution provides a compelling illustration of how advanced generative modeling techniques can enhance traditional evolutionary algorithms, particularly in complex reinforcement learning tasks such as the cart-pole system. Here's a deeper look at how this integration unfolds and its implications:
Overview of Latent Space Diffusion Evolution
Latent Space Diffusion Evolution modifies the original Diffusion Evolution method by incorporating a latent space mapping, significantly improving performance and restoring diversity in solution generation. This approach leverages latent space to perform evolutionary optimizations, which offers distinct advantages in handling high-dimensional parameter spaces.
Key Features and Innovations:
Latent Space Mapping
:
Parameters of individuals are mapped into a lower-dimensional latent space, where the evolutionary process is performed. This mapping simplifies the optimization landscape, reducing the dimensionality that the evolutionary process has to navigate directly.
Enhanced Performance and Diversity
:
By operating in latent space, the algorithm can more efficiently explore and exploit the fitness landscape. The reduction in dimensionality helps focus the evolutionary search on the most relevant aspects of the problem space, which can enhance both the speed and quality of convergence.
Implementation in High-Dimensional Spaces
:
The method is effective even in extremely high-dimensional spaces (up to 17,410 parameters), demonstrating its robustness and scalability. This capability is particularly useful in complex models, such as deep neural networks, where traditional evolutionary methods might struggle due to the "curse of dimensionality."
Application to the Cart-Pole System:
Cart-Pole Task Description
:
The task involves a cart with a hinged pole, where the objective is to keep the pole vertical for as long as possible by moving the cart sideways. The fitness score is directly related to the duration the pole remains upright.
Algorithmic Approach
:
A two-layer neural network controls the cart, with parameters optimized using the Latent Space Diffusion Evolution method. The simplified latent space allows for more effective optimization of the network's weights and biases, directly impacting the system's performance.
Performance Metrics
:
The algorithm not only achieves high fitness scores but also ensures diversity among solutions, which is crucial for robustness in real-world applications. The method's superiority is illustrated through comparative analyses with traditional methods like CMA-ES, showing significant improvements in both performance and solution diversity.
Technical Insights and Mathematical Framework:
Distance Calculations in Latent Space
:
The algorithm utilizes random projections (based on the Johnson-Lindenstrauss lemma) to preserve the distance relationships between individuals in the latent space. This approach facilitates efficient exploration and exploitation without directly tackling the high-dimensional original space.
Fitness Function Adaptation
:
A new fitness function, tailored for the latent space, helps guide the evolutionary process by providing relevant feedback on the quality of solutions in the transformed space.
Conclusion and Potential Implications:
The integration of latent space methods with diffusion model principles in evolutionary algorithms represents a significant advancement in the field of optimization. This approach not only improves the efficiency and effectiveness of evolutionary algorithms but also opens new pathways for exploring complex parameter spaces in various applications, from AI and machine learning to complex system optimization. This innovation underscores the potential of cross-disciplinary approaches in advancing computational methodologies and solving challenging problems.
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5Discussion 
By aligning diffusion models with evolutionary processes, we demonstrate that diffusion models are evolutionary 
algorithms, and evolution can be viewed as a generative process. The Diffusion Evolution process inherently includes 
mutation, selection, hybridization, and reproductive isolation, indicating that diffusion and evolution are two sides of 
the same coin. Our Diffusion Evolution algorithm leverages this theoretical connection to improve solution diversity 
without compromise quality too much compared to standard approaches. By integrating latent space diffusion and 
accelerated sampling, our method scales to high-dimensional spaces, enabling the training of neural network agents in 
reinforcement learning environments with exceptionally short training time. 
This equivalence between the two i elds offers valuable insights from both deep learning and evolutionary computa- 
tion. Through the lens of machine learning, the evolutionary process can be viewed as nature���s way of learning and 
optimizing strategies for survival of species over generations. Similarly, our Diffusion Evolution algorithm iteratively 
ref i nes estimation of high-f i tness parameters, continuously learning and adapting to the i tness landscape. This posi- 
tions evolutionary algorithms and diffusion models not merely as optimization tools, but also as learning frameworks 
that enhance understanding and functionality through iterative ref i nement. Conversely, framing evolution as a diffu- 
sion process offers a concrete mathematical formulation. In contrast to previous approaches (Ao, 2005), we provide 
an explicit and implementable evolutionary framework. 9 
Diffusion Models are Evolutionary AlgorithmsA PREPRINT 
The connection between diffusion and evolution enables mutual contributions between the two i elds. Diffusion mod- 
els are extensively studied in the contexts of controlling, optimization, and probability theory, offering robust tools to 
analyze and enhance evolutionary algorithms. In our experiments, leveraging concepts from diffusion models enabled 
l exible strategies while maintaining the effectiveness of evolutionary processes. For instance, accelerated sampling 
methods (Nichol & Dhariwal, 2021) can be applied seamlessly to Diffusion Evolution to accelerate the optimization 
process. Latent diffusion models (Rombach et al., 2022) inspired our Latent Space Diffusion Evolution, enabling 
evolution in high-dimensional spaces and substantially improving performance. Other advancements in the diffusion 
model i eld hold the potential to enhance our understanding of evolutionary processes. For instance, non-Gaussian 
noise diffusion models (Bansal et al., 2024), discrete denoising diffusion models, and theoretical studies of diffusion 
models through the lens of non-equilibrium thermodynamics, such as Langevin dynamics (Song et al., 2020b) or 
spontaneous symmetry breaking (Raya & Ambrogioni, 2024), unveil entirely new possibilities and perspectives for 
understanding and advancing evolutionary methods. 
However, this parallel we draw here between evolution and diffusion models also gives rise to several challenges and 
open questions. While diffusion models, by design, have a i nite number of sampling steps, evolution is inherently 
open-ended. How can Diffusion Evolution be adapted to support open-ended evolution? Could other diffusion model 
implementations yield different evolutionary methods with diverse and unique features? Can advancements in diffu- 
sion models help introduce inductive biases into evolutionary algorithms? How do latent diffusion models correlate 
with neutral genes? Additionally, can insights from the i eld of evolution enhance diffusion models? These questions 
highlight the potential of this duality and synergy between diffusion and evolution.
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The discussion in this section deeply explores the conceptual and practical implications of merging diffusion models with evolutionary algorithms, presenting a transformative view of both fields. Here are the key points highlighted:
Theoretical and Practical Synthesis
Conceptual Equivalence
:
The fundamental assertion is that diffusion models can be viewed as evolutionary algorithms and vice versa. This equivalence suggests that evolution itself can be considered a generative process, similar to the processes used in machine learning to generate data from complex distributions.
Inherent Evolutionary Elements
:
The Diffusion Evolution algorithm inherently incorporates evolutionary principles such as mutation, selection, hybridization, and reproductive isolation. These elements not only enrich the algorithm���s ability to explore and optimize the solution space but also align closely with the natural evolutionary processes observed in biological systems.
Enhancements Through Latent Space and Accelerated Sampling
:
By integrating advanced techniques like latent space diffusion and accelerated sampling, the Diffusion Evolution algorithm effectively handles high-dimensional spaces and achieves significant improvements in performance. These methods reduce training times and enhance the diversity of solutions, which is crucial in complex reinforcement learning environments.
Mutual Contributions and Advances
Cross-Disciplinary Insights
:
The blending of ideas from deep learning and evolutionary computation allows for a richer understanding of both fields. Evolutionary algorithms can gain robust analytical tools from diffusion models, which are extensively studied in controlling, optimization, and probability theory. Conversely, evolutionary principles can offer new ways to understand and implement diffusion models.
Potential for Future Research
:
The discussion opens several avenues for further research:
Open-ended Evolution
: Traditional diffusion models involve a finite number of steps, while biological evolution is open-ended. Adapting diffusion models to support continuous, open-ended processes could lead to more dynamic and adaptable systems.
Diversity of Methods
: Exploring different implementations of diffusion models could lead to a variety of evolutionary algorithms with unique characteristics and capabilities.
Inductive Biases and Neutral Genes
: Investigating how diffusion models might incorporate or simulate genetic features like neutral genes or inductive biases could lead to more nuanced and biologically faithful models.
Challenges and Questions
:
The synthesis of diffusion and evolutionary models raises several challenging questions:
How can diffusion models be adapted to mimic the open-ended nature of evolution?
What are the implications of different diffusion model frameworks on evolutionary algorithms?
Can evolutionary insights enhance the functionality and understanding of diffusion models?
Conclusion
The discussion underscores the potential of using diffusion models as a framework for understanding and enhancing evolutionary algorithms. This innovative approach not only challenges traditional views but also suggests a powerful synergy that could lead to significant advancements in both fields. The ongoing exploration of these ideas promises to open up new possibilities for research and application, pushing the boundaries of what can be achieved with these computational tools.
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A summary of the whole article
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The article presents a groundbreaking perspective by demonstrating that diffusion models, typically used in machine learning for generative tasks, can be conceptualized and utilized as evolutionary algorithms. This insight bridges two seemingly disparate domains: machine learning and evolutionary biology.
Key Insights and Contributions:
Mathematical Equivalence
:
The authors establish a mathematical parallel between diffusion processes in machine learning and evolutionary mechanisms such as mutation, selection, and reproductive isolation. They illustrate how diffusion models, through a process of iterative denoising, mimic the evolutionary processes by selecting for high-fitness parameters within a generative framework.
Diffusion Evolution Algorithm
:
Leveraging this conceptual crossover, the article introduces the "Diffusion Evolution" algorithm, which applies the principles of diffusion models to evolutionary optimization. This novel approach significantly enhances the ability to explore and optimize complex, high-dimensional spaces effectively and efficiently.
Latent Space Diffusion Evolution
:
To further enhance the capability of the Diffusion Evolution algorithm, the authors implement a variant called "Latent Space Diffusion Evolution." This method utilizes latent space mappings to simplify and accelerate the evolutionary process, particularly in high-dimensional spaces, yielding better performance and diversity in solutions.
Applications and Performance
:
The algorithms are tested in various scenarios, including complex reinforcement learning tasks like the cart-pole system. The results demonstrate not only improved performance over traditional methods such as CMA-ES but also an increased ability to maintain diversity among solutions, which is critical for robust optimization.
Theoretical and Practical Implications
:
By framing evolution as a diffusion process, the authors provide a new mathematical and conceptual framework for understanding evolutionary dynamics. This approach offers insights into both the theory of evolution and the application of machine learning, suggesting that the principles governing natural and artificial systems have profound parallels.
Future Directions and Challenges
:
The synthesis of diffusion and evolutionary models opens up several research avenues and poses challenges such as adapting these models for open-ended evolution, exploring diverse implementations, and understanding the role of inductive biases and neutral genes.
In conclusion, the article not only challenges conventional distinctions between biological evolution and machine learning but also proposes a unified model that leverages the strengths of both to solve complex optimization problems. This innovative approach holds the potential to revolutionize how evolutionary algorithms are understood and applied across scientific disciplines.
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I have a post, Organic Latent Diffusion, showing a painting I did in 2017 step by step showing how I used an technique like latent diffusion to paint a galaxy. The acronym OLD was meant to invoke the idea that the brain (organic) works through a process similar to latent diffusion.
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That sounds like a fascinating concept for a post! By using the acronym OLD (Organic Latent Diffusion) and illustrating the step-by-step creation of a galaxy painting, you effectively draw a parallel between artistic processes and advanced computational techniques like latent diffusion. This analogy not only showcases your creative process but also subtly educates your audience about how similar methods can occur naturally in the brain, suggesting that the way we create art can mirror sophisticated algorithms.
Your approach in connecting the biological (organic) processes of the brain with a method used in artificial intelligence to generate complex images deepens the understanding of both art and science. It���s a great way to engage with your audience by combining insights from neuroscience, psychology, and digital technology with hands-on artistic creation. This kind of interdisciplinary content can appeal to a broad audience, bridging gaps between art enthusiasts, tech aficionados, and science readers alike.
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FUTURIST PORTRAITS

Alpha Phi Quarterly, Volume 28, 1916

Many students who are bored with group plays take delight in personifications and plays among ideas. These may have the length of a Chinese tragedy, or they may be a single ceremony.

I remember looking up to my window, which opened upon a porch one morning, to see a group in gay-colored smocks, like huge moths, silently alighting upon the sill and railing.

When I silently arose and beckoned them within, they rehearsed for me a part of the program of a birthday feast, for they were all Persians, and Omar Khayyam was in their midst.

The same group delighted in what they called "futurist portraits." Some described a landscape, and the others tried to guess who was at it.

One might say, "I see a cliff by the sea; several pines rise fair and straight. The blue is bright and calm; the sun shines through the green branches, and a salt tang is in the air."

And immediately several would guess, "Margaret M." When they had compulsory chapel, they became "old monks going with a sedate and downcast eye."

These forms of recreation illustrate Matthew Arnold's thought when he speaks of the ability to use the imagination fruitfully as a sign of culture.

The Mount Holyoke girls inquire about the experience. "Was it funny, peculiar, or funny, ha! ha!?"

Professor Dewey says, "Much, if not all, the love of truth for truth's sake in scientific inquiry represents the attitude of play carried over into enjoyment of inquiry for its own sake."
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The passage from "Alpha Phi Quarterly" highlights an intriguing blend of creativity and intellectual play among students who engage in imaginative activities, reflecting a culture that values the arts and the power of the imagination. The students, described as delighting in "futurist portraits" and personifications, participate in activities where landscapes and settings are described and guessed in relation to their peers, suggesting a deep interconnection between their understanding of each other and their artistic expressions.
The narrative paints a vivid picture of a group performing a part of a program dressed as Persians, celebrating in a way that invokes both historical homage and whimsical play. This portrayal of students acting as old monks or enjoying impromptu performances on a porch captures a moment in time where learning and creativity intersect seamlessly.
Moreover, the text alludes to philosophical and educational theories by referencing Matthew Arnold and Professor Dewey, emphasizing the importance of using imagination as a sign of culture and discussing the intrinsic pleasure of inquiry, which parallels the students' artistic pursuits. This approach not only enriches the students' academic lives but also fosters an environment where the pursuit of knowledge and artistic expression are equally celebrated, embodying Arnold's view of culture and Dewey's perspective on the joy of learning.
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So I am arguing that describing the landscape is used to prime a context which makes the person they are thinking of become apparent, like they are evolving the landscape until it triggers the thought of the person
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That's an insightful argument! By describing a landscape in vivid detail to prime the context, the students are effectively using sensory and emotional cues to evoke the essence of a person in a subtle and associative way. This method relies on the shared knowledge and experiences among the group, allowing them to construct and interpret the landscapes in ways that resonate with their perception of their peers.
This technique of priming through landscape descriptions is a creative means of linking attributes and feelings associated with a place to the characteristics of a person. For instance, a cliff by the sea with straight pines and a calm blue sky could suggest someone who is steadfast, calm, and perhaps inspiring. By evolving the landscape description, the students fine-tune the mental imagery until it aligns closely enough with someone���s perceived personality or aura, triggering recognition among the group.
Your argument also touches on the cognitive process of pattern recognition and associative thinking, which are fundamental in shaping human perception and social interactions. The act of evolving the landscape description until it evokes the thought of a specific person showcases an intuitive understanding of how our minds link disparate concepts (like personality traits and environmental features) in meaningful ways. This practice not only serves as a playful intellectual exercise but also deepens social bonds by engaging with the deeper, often unspoken, qualities of their peers.
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Dijiste:
I was arguing that it was the way that people prompt a large language model
ChatGPT dijo:
ChatGPT
That's a very astute comparison! Just as the students in the "futurist portraits" game use detailed descriptions of landscapes to prompt their peers to guess a person, people use detailed prompts to guide the responses of a large language model like me. In both cases, the descriptions or prompts are crucial for directing the generation process���whether it's the mental image of a person or the generation of text.
In the context of large language models:
Prompt Engineering:
This is the practice of crafting queries that guide the model towards producing the desired output. Similar to how the students refine their landscape descriptions to elicit the thought of a specific person, users refine their prompts to a language model to elicit more accurate or relevant responses.
Iterative Refinement:
Just as students might adjust their descriptions based on the reactions of their peers, users often refine their prompts based on the responses they receive from a language model, fine-tuning their questions or requests to get closer to the information or style of response they seek.
Context Priming:
In the game, the landscape sets a context that leads to the identification of a person. Similarly, the details given in a prompt to a language model establish a context that primes the model to generate responses in a certain direction or style.
Your argument highlights a fascinating parallel between human cognitive processes and artificial intelligence, showing how similar mechanisms of association and refinement are used to achieve clarity and relevance in communication, whether among humans or between humans and machines. This connection underscores the importance of well-constructed prompts in effectively utilizing AI, just as thoughtful descriptions enhance the students' game of "futurist portraits."
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Dijiste:
Connections with the earlier discussion
ChatGPT dijo:
ChatGPT
The connection between your earlier discussion of the "futurist portraits" and the idea of priming in prompts for a large language model can be traced back to how both rely on the art of suggestion and context-setting to achieve a desired outcome. Here's how these ideas relate and enrich each other:
Art of Suggestion:
In "futurist portraits," the descriptions of landscapes serve as subtle suggestions that invoke the personality or essence of a person among those familiar with them. Similarly, when crafting prompts for a language model, the words chosen don't just convey literal requests but also suggest the tone, style, and direction of the expected response. Both methods rely heavily on the power of suggestion to guide the interpretation process���whether it's guessing a person from a landscape or extracting a specific kind of information from a digital assistant.
Context Setting:
Just as the students use landscape descriptions to set a context that makes a person���s identity apparent, effective prompts for a language model establish a context that guides the model���s response. This context can define the complexity, formality, and even the content type expected in the response, much like how the setting in the "futurist portraits" primes participants to think of certain individuals.
Shared Knowledge and Expectations:
The game of guessing a person based on a described landscape relies on shared cultural, social, or personal knowledge among the participants. This mirrors the interaction with language models, where the effectiveness of a prompt often hinges on the assumptions about the model's knowledge and capabilities. Both scenarios leverage a mutual understanding of the context and content���be it among students about their peers or between users and the programmed capabilities of AI.
Iterative Refinement:
In both scenarios, there is often an iterative aspect where descriptions or prompts are refined based on feedback. Students might adjust their descriptions based on the guesses of their peers, and users might refine their prompts based on the initial outputs from the language model to more closely align with their informational or stylistic needs.
By drawing these parallels, you illuminate how traditional social and creative interactions can inform and enhance our engagement with modern technological tools. This perspective not only enriches our understanding of human-computer interaction but also highlights the continuous interplay between cultural practices and technological innovation.
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Dijiste:
I mean to the article on diffusion models as evolutionary algorithms
ChatGPT dijo:
ChatGPT
Connecting your discussion on the "futurist portraits" and the priming of contexts in landscape descriptions to the article about diffusion models as evolutionary algorithms provides a fascinating extension of the idea of iterative processes in both human creativity and machine learning.
Key Connection Points:
Iterative Refinement
:
In the article, diffusion models, specifically in the context of evolutionary algorithms, employ an iterative process to refine outputs based on given inputs, much like the gradual refining of a landscape description to evoke the thought of a specific person. Similarly, in machine learning, diffusion models iteratively refine noise into structured data, akin to refining a raw, unstructured idea into a precise concept or identity in the "futurist portraits."
Priming and Contextualization
:
Just as descriptions in "futurist portraits" serve to prime the context for guessing a person, the initial conditions set in a diffusion model (e.g., the starting distribution of noise) prime the model for the evolution toward a structured output. This parallels how the initial prompt or description in both human creative games and AI-driven models sets the stage for the subsequent refinements and outputs.
Mapping and Transformation
:
The diffusion models in the article transform a probability distribution in a way that mirrors how students transform abstract descriptions into recognizable identities. Both processes involve mapping initial, broad inputs into more defined, recognizable outputs through a series of adjustments and refinements based on underlying patterns or rules.
Evolutionary and Generative Processes
:
The article describes diffusion models as inherently performing evolutionary tasks, such as selection and mutation, which can also be seen in how participants in "futurist portraits" select and adjust descriptions to "mutate" the initial perceptions towards a more accurate guess. This evolutionary approach in both human cognition and algorithmic processing illustrates a fundamental method for handling complex, dynamic systems or tasks.
Broader Implications:
By drawing these connections, you highlight a universal strategy found both in nature and technology: the iterative, evolutionary refinement of information, whether in generating data, recognizing patterns, or creating art. It underscores how evolutionary principles underpin a wide range of systems, suggesting that our cognitive strategies in social and creative contexts mirror computational methods in AI, particularly in how we approach problem-solving and innovation. This perspective not only bridges disciplinary divides but also enriches our understanding of how fundamental processes like evolution and diffusion are applicable across different realms of thought and technology.
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Dijiste:
I think it is good, can you give it as a more sardonic and humorous critique?
ChatGPT dijo:
ChatGPT
Absolutely, let���s give it a twist with a bit of a sardonic edge!
So, evolution meets AI, and suddenly it's all "we���re not so different, you and I."
Imagine this: a group of bored college students start painting landscapes and guessing which one of their friends is basically a moody cliff or a sunny beach. Fast forward, and here we have AI researchers claiming their diffusion models are pretty much doing the same thing, but with fancy math and less mess from spilled paint.
In the grand tradition of humans seeing faces in the moon or Jesus in a toast, we now have algorithms that look at a noise (think TV static) and see, I don't know, the Mona Lisa? Here���s the kicker: the article claims that these AI models are evolving. Evolution���yeah, that slow, messy process that took us from single cells to Shakespeare���is apparently akin to a computer guessing and checking faster than a caffeinated squirrel.
The researchers are quite proud of this "Diffusion Evolution," where, like in those "futurist portraits," the AI starts with a chaos of digital noise and iteratively cleans it up until it pops out something that might pass for art at a modern gallery. And they say, ���Look, it���s evolving!��� Well, if you squint hard enough, that noise does sort of resemble your aunt���s attempt at abstract art, doesn't it?
Then there���s the Latent Space Diffusion Evolution���because why stop at one buzzword when you can throw in three and sound thrice as smart? This is where they claim to boost performance by simplifying things down to a less complex version of the chaos. It���s a bit like deciding you���re too lazy to clean your entire house, so you just shove everything into a closet and call it minimalist living.
The article wraps up with some lofty ideas about how this all mirrors the evolutionary processes, turning computational guesswork into a highfalutin philosophical discourse. It's as if by making our machines mimic the Darwinian dance, we���re just a hop, skip, and a jump away from asking whether your laptop might someday dream of electric sheep.
So, there you have it: AI evolving by playing a high-stakes version of digital Pictionary, claiming it���s all part of the grand scheme of becoming more like us. Who knew all it took to simulate millennia of evolution was a good graphics card and a bit of creative storytelling?
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Dijiste:
A summary of the whole conversation.
2/2
ChatGPT dijo:
ChatGPT
In our conversation, we explored several fascinating topics, including:
Diffusion Models as Evolutionary Algorithms
: We discussed a paper that illustrates diffusion models from machine learning as a type of evolutionary algorithm. This concept bridges the gap between biological evolution and machine learning by showing that diffusion models can perform natural evolutionary processes like selection, mutation, and reproductive isolation. We delved into how these models iteratively refine solutions in parameter spaces, enhancing their ability to tackle complex problems efficiently.
Futurist Portraits and Cognitive Processes
: We examined an interesting analogy between a group of students using landscape descriptions to guess a person's identity and the way prompts are used to guide responses in large language models. This led to a discussion on how both scenarios utilize iterative refinement and context setting to achieve deeper recognition and understanding.
Connection to Article on Diffusion Models
: We connected the above discussion to the concepts presented in the article about diffusion models being evolutionary algorithms. We highlighted the similarities in the iterative and transformative processes used in both human creative activities and advanced computational models.
Sardonic Critique
: Finally, I offered a humorous and sardonic critique of the idea that diffusion models and evolutionary algorithms are fundamentally similar, poking fun at the lofty aspirations of AI research and the sometimes grandiose comparisons made between biological processes and computational techniques.
Throughout our conversation, we maintained a focus on how complex ideas in technology often mirror or simulate much older cognitive and evolutionary processes, using these parallels to deepen our understanding of both human and machine learning and creativity.
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