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На практике пробуем KAN – принципиально новую архитектуру нейросетей

На днях ученые из MIT показали альтернативу многослойному перцептрону (MLP). MLP с самого момента изобретения глубокого обучения лежит в основе всех нейросетей, какими мы их знаем сегодня. На его идее в том числе построены большие языковые модели и системы компьютерного зрения.

Однако теперь все может измениться. В KAN (Kolmogorov-Arnold Networks) исследователи реализовали перемещение функций активации с нейронов на ребра нейросети, и такой подход показал блестящие результаты.

0df03599e984071a838e5b2957c662b8.png

Идею KAN ученые подчерпнули из теоремы Колмогорова-Арнольда, именно в их честь и названа архитектура. Вообще говоря, исследование очень математичное, в статье 50 страниц с формулами, повсюду термины из мат.анализа, высшей алгебры, функана и прочего.

В общем, если хотите разобраться с тем, как эта сенсация работает, и при этом не сойти с ума, на нашем сайте мы, команда канала Data Secrets, написали для вас длинный и интересный разбор. Там мы на пальцах объяснили всю математику, рассказали про строение сети, привели примеры и ответили на вопрос "а почему до этого раньше никто не додумался".

Прочитайте, не пожалеете: https://datasecrets.ru/articles/9.

А эта статья - для тех, кто хочет поиграть с новой архитектурой на практике. Мы рассмотрим несколько примеров кода на Python и понаблюдаем, как KAN справляется с привычными нам задачами машинного обучения. Поехали!

Установка

Чтобы участники сообщества могли сразу же потрогать все своими руками, добрые исследователи вместе со статьей представили библиотеку pykan, благодаря которой можно запускать KAN из коробки. Именно с ней мы сегодня и будем работать.

Итак, начнем с установки. Библиотеку можно поставить привычно через pip (pip install pykan) или с помощью клонирования репозитория: 

git clone https://github.com/KindXiaoming/pykan.git
cd pykan
pip install -e .
# pip install -r requirements.txt # install requirements

Далее импортируем библиотеку с помощью from kan import * и наконец-то переходим к написанию кода!

Регрессия

Ну куда же без задачи регрессии? Ведь именно с нее началось машинное обучение в 50-х годах прошлого века... Ладно, краткие исторические справки оставим на потом.

Давайте загадаем KAN такую загадку: возьмем функцию от двух переменных f(x,y) = exp(sin(pi*x)+y^2) и попросим KAN по входам и выходам функции найти ее формулу. Это так называемая символьная регрессия. Надо сказать, что задача хоть и кажется тривиальной, но обычно математически трудна для нейросетей. 

from kan import *
# формируем KAN: 2D входы, 1D выходы, 5 скрытых нейронов, 
# кубические сплайны и сетка на 5 точках.
model = KAN(width=[2,5,1], grid=5, k=3, seed=0)

# сгенерируем датасет
f = lambda x: torch.exp(torch.sin(torch.pi*x[:,[0]]) + x[:,[1]]**2)
dataset = create_dataset(f, n_var=2)
dataset['train_input'].shape, dataset['train_label'].shape
#(torch.Size([1000, 2]), torch.Size([1000, 1]))

Сплайны в KAN – это как раз те самые обучаемые функции на ребрах. В математике сплайн – это такая гладкая кривая, кусочно-полиномиальная функция, которая на разных отрезках задается различными полиномами. Каждый сплайн аппроксимируется с помощью заданного количества точек (сетки). Чем больше точек - тем точнее аппроксимация.

Обучающую и тестовую выборки получили, значит можно обучать. Тут ничего нового – привычный метод train:

# train the model
model.train(dataset, opt="LBFGS", steps=20, lamb=0.01, lamb_entropy=10.);

Можно визуализировать KAN, который у нас получился:

1865c982f4bb2f4b3459932d1c296e9f.png

Давай посмотрим на эту картинку внимательнее. Наверху мы видим сплайн, похожий на экспоненту, а затем слева и справа наблюдаем соотвественно синус и параболу. Ничего не напоминает?

Все верно, если сложить все вместе, то получится формула которую мы загадывали: f(x,y) = exp(sin(pi*x)+y^2). Благодаря тому, что в KAN обучаются не параметры (числа), а функции, он почти идеально справляется с задачей регрессии на сложных функциях и, как показали исследователи, гораздо эффективнее генерализирует данные. В частности, в этой задаче мы получаем метрику r2 равной 0.99.

В статье исследователи также показали, как KAN помогает решать дифференциальные уравнения и (пере)открывает законы физики и математики.

Классификация

Тут все еще интереснее. Но все по порядку. Снова сгенерируем игрушечный датасет (в сообществе его прозвали "две луны"): 

from kan import KAN
import matplotlib.pyplot as plt
from sklearn.datasets import make_moons
import torch
import numpy as np

dataset = {}
train_input, train_label = make_moons(n_samples=1000, shuffle=True, noise=0.1, random_state=None)
test_input, test_label = make_moons(n_samples=1000, shuffle=True, noise=0.1, random_state=None)

dataset['train_input'] = torch.from_numpy(train_input)
dataset['test_input'] = torch.from_numpy(test_input)
dataset['train_label'] = torch.from_numpy(train_label[:,None])
dataset['test_label'] = torch.from_numpy(test_label[:,None])

X = dataset['train_input']
y = dataset['train_label']
plt.scatter(X[:,0], X[:,1], c=y[:,0])
5155a97cacd4873802177f8bef70a4de.png

Для начала давайте немного развлечемся и решим задачу, как будто это регрессия: будем предсказывать некоторое число, округлять его и сравнивать с реальной меткой класса. 

model = KAN(width=[2,1], grid=3, k=3)

def train_acc():
    return torch.mean((torch.round(model(dataset['train_input'])[:,0]) == dataset['train_label'][:,0]).float())

def test_acc():
    return torch.mean((torch.round(model(dataset['test_input'])[:,0]) == dataset['test_label'][:,0]).float())

results = model.train(dataset, opt="LBFGS", steps=20, metrics=(train_acc, test_acc));
results['train_acc'][-1], results['test_acc'][-1]
# (1.0, 1.0)

По последней строке видно: KAN справился идеально. Если заглянуть глубже, то мы увидим, что (опять же с помощью обучения функций) сетка вывела для себя "формулу ответа", которая и помогает ей безупречно справится с задачей:

fb7a1cef13d9d41efcb22454a59434b3.png

А теперь попробуем по-взрослому, с кросс-энтропией, логитами и argmax. Вот код, в котором мы немного подправляем размерности в датасете и обучаем KAN:

dataset = {}
train_input, train_label = make_moons(n_samples=1000, shuffle=True, noise=0.1, random_state=None)
test_input, test_label = make_moons(n_samples=1000, shuffle=True, noise=0.1, random_state=None)

dataset['train_input'] = torch.from_numpy(train_input)
dataset['test_input'] = torch.from_numpy(test_input)
dataset['train_label'] = torch.from_numpy(train_label)
dataset['test_label'] = torch.from_numpy(test_label)

X = dataset['train_input']
y = dataset['train_label']

model = KAN(width=[2,2], grid=3, k=3)

def train_acc():
    return torch.mean((torch.argmax(model(dataset['train_input']), dim=1) == dataset['train_label']).float())

def test_acc():
    return torch.mean((torch.argmax(model(dataset['test_input']), dim=1) == dataset['test_label']).float())

results = model.train(dataset, opt="LBFGS", steps=20, metrics=(train_acc, test_acc), loss_fn=torch.nn.CrossEntropyLoss());

Точность в этом случае немного ниже, но все еще достаточно хороша: 0.9660. Кстати, вот так можно посмотреть на формулы KAN (для каждого класса формула своя):

lib = ['x','x^2','x^3','x^4','exp','log','sqrt','tanh','sin','abs']
model.auto_symbolic(lib=lib)
formula1, formula2 = model.symbolic_formula()[0]

В данном случае они получаются такими:

4643d50b1be61e78fe438abd8b45a8bc.png464af9df1ed304432f80ba308222fc08.png

Заключение

В статье мы рассмотрели, как запустить KAN для привычных задач регрессии и классификации и немного заглянули "под капот" архитектуры. Если хотите больше примеров – загляните в документацию или в репозитория проекта, там лежат очень красивые и понятные ноутбуки, в которых можно найти туториалы по библиотеке и кейсы использования KAN.

Авторы KAN доказали, что ему требуется во много раз меньше нейронов, чтобы достичь точности MLP. Также KAN гораздо лучше генерализует данные и лучше справляется с аппроксимацией сложных математических функций (мы увидели это на примерах), у него, можно сказать, "технический склад ума".

Однако у архитектуры есть бутылочное горлышко: KAN учится медленнее MLP примерно в 10 раз. Возможно, это станет серьезным камнем преткновения, а возможно инженеры быстро научатся оптимизировать эффективность таких сетей.

Больше новостей из мира машинного обучения можно найти в нашем телеграм-канале. Подписывайтесь, чтобы быть в курсе: @data_secrets.

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    "In the crypto world, this is great news I want to share. Last year, I fell victim to a scam disguised as a safe investment option. I have invested in crypto trading platforms for about 10yrs thinking I was ensuring myself a retirement income, only to find that all my assets were either frozen, I believed my assets were secure — until I discovered that my BTC funds had been frozen and withdrawals were impossible. It was a devastating moment when I realized I had been scammed, and I thought my Bitcoin was gone forever, Everything changed when a close friend recommended the Capital Crypto Recover Service. Their professionalism, expertise, and dedication enabled me to recover my lost Bitcoin funds back — more than €560.000 DEM to my BTC wallet. What once felt impossible became a reality thanks to their support. If you have lost Bitcoin through scams, hacking, failed withdrawals, or similar challenges, don’t lose hope. I strongly recommend Capital Crypto Recover Service to anyone seeking a reliable and effective solution for recovering any wallet assets. They have a proven track record of successful reputation in recovering lost password assets for their clients and can help you navigate the process of recovering your funds. Don’t let scammers get away with your hard-earned money – contact Email: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 12.10.25 01:12 harristhomas7376

    "In the crypto world, this is great news I want to share. Last year, I fell victim to a scam disguised as a safe investment option. I have invested in crypto trading platforms for about 10yrs thinking I was ensuring myself a retirement income, only to find that all my assets were either frozen, I believed my assets were secure — until I discovered that my BTC funds had been frozen and withdrawals were impossible. It was a devastating moment when I realized I had been scammed, and I thought my Bitcoin was gone forever, Everything changed when a close friend recommended the Capital Crypto Recover Service. Their professionalism, expertise, and dedication enabled me to recover my lost Bitcoin funds back — more than €560.000 DEM to my BTC wallet. What once felt impossible became a reality thanks to their support. If you have lost Bitcoin through scams, hacking, failed withdrawals, or similar challenges, don’t lose hope. I strongly recommend Capital Crypto Recover Service to anyone seeking a reliable and effective solution for recovering any wallet assets. They have a proven track record of successful reputation in recovering lost password assets for their clients and can help you navigate the process of recovering your funds. Don’t let scammers get away with your hard-earned money – contact Email: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 12.10.25 19:53 Tonerdomark

    A crook swiped my Dogecoin. It ruined my whole world. Then Mr. Sylvester showed up. He fixed it all. He pulled back €211,000 for me. Not one cent missing from my profits. His steady cool and sharp tech know-how won back my trust. I got my money smooth and sound. After endless worry, relief hit me hard. I trusted him completely. Lost cash to a scam? Hit him up now at { yt7cracker@gmail . com }. His aid turned my life around. WhatsApp at +1 512 577 7957.

  • 12.10.25 21:36 blessing

    Writing this review is a joy. Marie has provided excellent service ever since I started working with her in early 2018. I was worried I wouldn't be able to get my coins back after they were stolen by hackers. I had no idea where to begin, therefore it was a nightmare for me. However, things became easier for me after my friend sent me to [email protected] and +1 7127594675 on WhatsApp. I'm happy that she was able to retrieve my bitcoin so that I could resume trading.

  • 13.10.25 01:11 elizabethrush89

    God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 13.10.25 01:11 elizabethrush89

    God bless Capital Crypto Recover Services for the marvelous work you did in my life, I have learned the hard way that even the most sensible investors can fall victim to scams. When my USD was stolen, for anyone who has fallen victim to one of the bitcoin binary investment scams that are currently ongoing, I felt betrayal and upset. But then I was reading a post on site when I saw a testimony of Wendy Taylor online who recommended that Capital Crypto Recovery has helped her recover scammed funds within 24 hours. after reaching out to this cyber security firm that was able to help me recover my stolen digital assets and bitcoin. I’m genuinely blown away by their amazing service and professionalism. I never imagined I’d be able to get my money back until I complained to Capital Crypto Recovery Services about my difficulties and gave all of the necessary paperwork. I was astounded that it took them 12 hours to reclaim my stolen money back. Without a doubt, my USDT assets were successfully recovered from the scam platform, Thank you so much Sir, I strongly recommend Capital Crypto Recover for any of your bitcoin recovery, digital funds recovery, hacking, and cybersecurity concerns. You reach them Call/Text Number +1 (336)390-6684 His Email: [email protected] Contact Telegram: @Capitalcryptorecover Via Contact: [email protected] His website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 14.10.25 01:15 tyleradams

    Hi. Please be wise, do not make the same mistake I had made in the past, I was a victim of bitcoin scam, I saw a glamorous review showering praises and marketing an investment firm, I reached out to them on what their contracts are, and I invested $28,000, which I was promised to get my first 15% profit in weeks, when it’s time to get my profits, I got to know the company was bogus, they kept asking me to invest more and I ran out of patience then requested to have my money back, they refused to answer nor refund my funds, not until a friend of mine introduced me to the NVIDIA TECH HACKERS, so I reached out and after tabling my complaints, they were swift to action and within 36 hours I got back my funds with the due profit. I couldn’t contain the joy in me. I urge you guys to reach out to NVIDIA TECH HACKERS on their email: [email protected]

  • 14.10.25 08:46 robertalfred175

    CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 14.10.25 08:46 robertalfred175

    CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 14.10.25 08:46 robertalfred175

    CRYPTO SCAM RECOVERY SUCCESSFUL – A TESTIMONIAL OF LOST PASSWORD TO YOUR DIGITAL WALLET BACK. My name is Robert Alfred, Am from Australia. I’m sharing my experience in the hope that it helps others who have been victims of crypto scams. A few months ago, I fell victim to a fraudulent crypto investment scheme linked to a broker company. I had invested heavily during a time when Bitcoin prices were rising, thinking it was a good opportunity. Unfortunately, I was scammed out of $120,000 AUD and the broker denied me access to my digital wallet and assets. It was a devastating experience that caused many sleepless nights. Crypto scams are increasingly common and often involve fake trading platforms, phishing attacks, and misleading investment opportunities. In my desperation, a friend from the crypto community recommended Capital Crypto Recovery Service, known for helping victims recover lost or stolen funds. After doing some research and reading multiple positive reviews, I reached out to Capital Crypto Recovery. I provided all the necessary information—wallet addresses, transaction history, and communication logs. Their expert team responded immediately and began investigating. Using advanced blockchain tracking techniques, they were able to trace the stolen Dogecoin, identify the scammer’s wallet, and coordinate with relevant authorities to freeze the funds before they could be moved. Incredibly, within 24 hours, Capital Crypto Recovery successfully recovered the majority of my stolen crypto assets. I was beyond relieved and truly grateful. Their professionalism, transparency, and constant communication throughout the process gave me hope during a very difficult time. If you’ve been a victim of a crypto scam, I highly recommend them with full confidence contacting: 📧 Email: [email protected] 📱 Telegram: @Capitalcryptorecover Contact: [email protected] 📞 Call/Text: +1 (336) 390-6684 🌐 Website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 15.10.25 18:07 crypto

    Cryptocurrency's digital realm presents many opportunities, but it also conceals complex frauds. It is quite painful to lose your cryptocurrency to scam. You can feel harassed and lost as a result. If you have been the victim of a cryptocurrency scam, this guide explains what to do ASAP. Following these procedures will help you avoid further issues or get your money back. Communication with Marie ([email protected] and WhatsApp: +1 7127594675) can make all the difference.

  • 15.10.25 21:52 harristhomas7376

    "In the crypto world, this is great news I want to share. Last year, I fell victim to a scam disguised as a safe investment option. I have invested in crypto trading platforms for about 10yrs thinking I was ensuring myself a retirement income, only to find that all my assets were either frozen, I believed my assets were secure — until I discovered that my BTC funds had been frozen and withdrawals were impossible. It was a devastating moment when I realized I had been scammed, and I thought my Bitcoin was gone forever, Everything changed when a close friend recommended the Capital Crypto Recover Service. Their professionalism, expertise, and dedication enabled me to recover my lost Bitcoin funds back — more than €560.000 DEM to my BTC wallet. What once felt impossible became a reality thanks to their support. If you have lost Bitcoin through scams, hacking, failed withdrawals, or similar challenges, don’t lose hope. I strongly recommend Capital Crypto Recover Service to anyone seeking a reliable and effective solution for recovering any wallet assets. They have a proven track record of successful reputation in recovering lost password assets for their clients and can help you navigate the process of recovering your funds. Don’t let scammers get away with your hard-earned money – contact Email: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1

  • 15.10.25 21:52 harristhomas7376

    "In the crypto world, this is great news I want to share. Last year, I fell victim to a scam disguised as a safe investment option. I have invested in crypto trading platforms for about 10yrs thinking I was ensuring myself a retirement income, only to find that all my assets were either frozen, I believed my assets were secure — until I discovered that my BTC funds had been frozen and withdrawals were impossible. It was a devastating moment when I realized I had been scammed, and I thought my Bitcoin was gone forever, Everything changed when a close friend recommended the Capital Crypto Recover Service. Their professionalism, expertise, and dedication enabled me to recover my lost Bitcoin funds back — more than €560.000 DEM to my BTC wallet. What once felt impossible became a reality thanks to their support. If you have lost Bitcoin through scams, hacking, failed withdrawals, or similar challenges, don’t lose hope. I strongly recommend Capital Crypto Recover Service to anyone seeking a reliable and effective solution for recovering any wallet assets. They have a proven track record of successful reputation in recovering lost password assets for their clients and can help you navigate the process of recovering your funds. Don’t let scammers get away with your hard-earned money – contact Email: [email protected] Phone CALL/Text Number: +1 (336) 390-6684 Contact: [email protected] Website: https://recovercapital.wixsite.com/capital-crypto-rec-1

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