A PPT about Maths Basic Fact.
Sebuah presentasi tentang fakta dasar bilangan. Disajikan dalam Mini Conference In house Training di Sekolah Ciputra, 14 Februari 2015.
2012 Honda Odyssey Brochure in Tampa Florida DealerWesleyChapelHonda
We have a great selection of new 2012 Honda Odyssey. Our business is built on referrals, and we are always trying to ’Build Relationships That Last’. Wesley Chapel Honda, home of the ’Nationwide Lifetime Warranty’. Wesley Chapel Honda, serving the greater Tampa Bay area.
A PPT about Maths Basic Fact.
Sebuah presentasi tentang fakta dasar bilangan. Disajikan dalam Mini Conference In house Training di Sekolah Ciputra, 14 Februari 2015.
2012 Honda Odyssey Brochure in Tampa Florida DealerWesleyChapelHonda
We have a great selection of new 2012 Honda Odyssey. Our business is built on referrals, and we are always trying to ’Build Relationships That Last’. Wesley Chapel Honda, home of the ’Nationwide Lifetime Warranty’. Wesley Chapel Honda, serving the greater Tampa Bay area.
2024 State of Marketing Report – by HubspotMarius Sescu
https://www.hubspot.com/state-of-marketing
· Scaling relationships and proving ROI
· Social media is the place for search, sales, and service
· Authentic influencer partnerships fuel brand growth
· The strongest connections happen via call, click, chat, and camera.
· Time saved with AI leads to more creative work
· Seeking: A single source of truth
· TLDR; Get on social, try AI, and align your systems.
· More human marketing, powered by robots
ChatGPT is a revolutionary addition to the world since its introduction in 2022. A big shift in the sector of information gathering and processing happened because of this chatbot. What is the story of ChatGPT? How is the bot responding to prompts and generating contents? Swipe through these slides prepared by Expeed Software, a web development company regarding the development and technical intricacies of ChatGPT!
Product Design Trends in 2024 | Teenage EngineeringsPixeldarts
The realm of product design is a constantly changing environment where technology and style intersect. Every year introduces fresh challenges and exciting trends that mold the future of this captivating art form. In this piece, we delve into the significant trends set to influence the look and functionality of product design in the year 2024.
How Race, Age and Gender Shape Attitudes Towards Mental HealthThinkNow
Mental health has been in the news quite a bit lately. Dozens of U.S. states are currently suing Meta for contributing to the youth mental health crisis by inserting addictive features into their products, while the U.S. Surgeon General is touring the nation to bring awareness to the growing epidemic of loneliness and isolation. The country has endured periods of low national morale, such as in the 1970s when high inflation and the energy crisis worsened public sentiment following the Vietnam War. The current mood, however, feels different. Gallup recently reported that national mental health is at an all-time low, with few bright spots to lift spirits.
To better understand how Americans are feeling and their attitudes towards mental health in general, ThinkNow conducted a nationally representative quantitative survey of 1,500 respondents and found some interesting differences among ethnic, age and gender groups.
Technology
For example, 52% agree that technology and social media have a negative impact on mental health, but when broken out by race, 61% of Whites felt technology had a negative effect, and only 48% of Hispanics thought it did.
While technology has helped us keep in touch with friends and family in faraway places, it appears to have degraded our ability to connect in person. Staying connected online is a double-edged sword since the same news feed that brings us pictures of the grandkids and fluffy kittens also feeds us news about the wars in Israel and Ukraine, the dysfunction in Washington, the latest mass shooting and the climate crisis.
Hispanics may have a built-in defense against the isolation technology breeds, owing to their large, multigenerational households, strong social support systems, and tendency to use social media to stay connected with relatives abroad.
Age and Gender
When asked how individuals rate their mental health, men rate it higher than women by 11 percentage points, and Baby Boomers rank it highest at 83%, saying it’s good or excellent vs. 57% of Gen Z saying the same.
Gen Z spends the most amount of time on social media, so the notion that social media negatively affects mental health appears to be correlated. Unfortunately, Gen Z is also the generation that’s least comfortable discussing mental health concerns with healthcare professionals. Only 40% of them state they’re comfortable discussing their issues with a professional compared to 60% of Millennials and 65% of Boomers.
Race Affects Attitudes
As seen in previous research conducted by ThinkNow, Asian Americans lag other groups when it comes to awareness of mental health issues. Twenty-four percent of Asian Americans believe that having a mental health issue is a sign of weakness compared to the 16% average for all groups. Asians are also considerably less likely to be aware of mental health services in their communities (42% vs. 55%) and most likely to seek out information on social media (51% vs. 35%).
AI Trends in Creative Operations 2024 by Artwork Flow.pdfmarketingartwork
This article is all about what AI trends will emerge in the field of creative operations in 2024. All the marketers and brand builders should be aware of these trends for their further use and save themselves some time!
A report by thenetworkone and Kurio.
The contributing experts and agencies are (in an alphabetical order): Sylwia Rytel, Social Media Supervisor, 180heartbeats + JUNG v MATT (PL), Sharlene Jenner, Vice President - Director of Engagement Strategy, Abelson Taylor (USA), Alex Casanovas, Digital Director, Atrevia (ES), Dora Beilin, Senior Social Strategist, Barrett Hoffher (USA), Min Seo, Campaign Director, Brand New Agency (KR), Deshé M. Gully, Associate Strategist, Day One Agency (USA), Francesca Trevisan, Strategist, Different (IT), Trevor Crossman, CX and Digital Transformation Director; Olivia Hussey, Strategic Planner; Simi Srinarula, Social Media Manager, The Hallway (AUS), James Hebbert, Managing Director, Hylink (CN / UK), Mundy Álvarez, Planning Director; Pedro Rojas, Social Media Manager; Pancho González, CCO, Inbrax (CH), Oana Oprea, Head of Digital Planning, Jam Session Agency (RO), Amy Bottrill, Social Account Director, Launch (UK), Gaby Arriaga, Founder, Leonardo1452 (MX), Shantesh S Row, Creative Director, Liwa (UAE), Rajesh Mehta, Chief Strategy Officer; Dhruv Gaur, Digital Planning Lead; Leonie Mergulhao, Account Supervisor - Social Media & PR, Medulla (IN), Aurelija Plioplytė, Head of Digital & Social, Not Perfect (LI), Daiana Khaidargaliyeva, Account Manager, Osaka Labs (UK / USA), Stefanie Söhnchen, Vice President Digital, PIABO Communications (DE), Elisabeth Winiartati, Managing Consultant, Head of Global Integrated Communications; Lydia Aprina, Account Manager, Integrated Marketing and Communications; Nita Prabowo, Account Manager, Integrated Marketing and Communications; Okhi, Web Developer, PNTR Group (ID), Kei Obusan, Insights Director; Daffi Ranandi, Insights Manager, Radarr (SG), Gautam Reghunath, Co-founder & CEO, Talented (IN), Donagh Humphreys, Head of Social and Digital Innovation, THINKHOUSE (IRE), Sarah Yim, Strategy Director, Zulu Alpha Kilo (CA).
Trends In Paid Search: Navigating The Digital Landscape In 2024Search Engine Journal
The search marketing landscape is evolving rapidly with new technologies, and professionals, like you, rely on innovative paid search strategies to meet changing demands.
It’s important that you’re ready to implement new strategies in 2024.
Check this out and learn the top trends in paid search advertising that are expected to gain traction, so you can drive higher ROI more efficiently in 2024.
You’ll learn:
- The latest trends in AI and automation, and what this means for an evolving paid search ecosystem.
- New developments in privacy and data regulation.
- Emerging ad formats that are expected to make an impact next year.
Watch Sreekant Lanka from iQuanti and Irina Klein from OneMain Financial as they dive into the future of paid search and explore the trends, strategies, and technologies that will shape the search marketing landscape.
If you’re looking to assess your paid search strategy and design an industry-aligned plan for 2024, then this webinar is for you.
5 Public speaking tips from TED - Visualized summarySpeakerHub
From their humble beginnings in 1984, TED has grown into the world’s most powerful amplifier for speakers and thought-leaders to share their ideas. They have over 2,400 filmed talks (not including the 30,000+ TEDx videos) freely available online, and have hosted over 17,500 events around the world.
With over one billion views in a year, it’s no wonder that so many speakers are looking to TED for ideas on how to share their message more effectively.
The article “5 Public-Speaking Tips TED Gives Its Speakers”, by Carmine Gallo for Forbes, gives speakers five practical ways to connect with their audience, and effectively share their ideas on stage.
Whether you are gearing up to get on a TED stage yourself, or just want to master the skills that so many of their speakers possess, these tips and quotes from Chris Anderson, the TED Talks Curator, will encourage you to make the most impactful impression on your audience.
See the full article and more summaries like this on SpeakerHub here: https://speakerhub.com/blog/5-presentation-tips-ted-gives-its-speakers
See the original article on Forbes here:
http://www.forbes.com/forbes/welcome/?toURL=http://www.forbes.com/sites/carminegallo/2016/05/06/5-public-speaking-tips-ted-gives-its-speakers/&refURL=&referrer=#5c07a8221d9b
ChatGPT and the Future of Work - Clark Boyd Clark Boyd
Everyone is in agreement that ChatGPT (and other generative AI tools) will shape the future of work. Yet there is little consensus on exactly how, when, and to what extent this technology will change our world.
Businesses that extract maximum value from ChatGPT will use it as a collaborative tool for everything from brainstorming to technical maintenance.
For individuals, now is the time to pinpoint the skills the future professional will need to thrive in the AI age.
Check out this presentation to understand what ChatGPT is, how it will shape the future of work, and how you can prepare to take advantage.
A brief introduction to DataScience with explaining of the concepts, algorithms, machine learning, supervised and unsupervised learning, clustering, statistics, data preprocessing, real-world applications etc.
It's part of a Data Science Corner Campaign where I will be discussing the fundamentals of DataScience, AIML, Statistics etc.
Time Management & Productivity - Best PracticesVit Horky
Here's my presentation on by proven best practices how to manage your work time effectively and how to improve your productivity. It includes practical tips and how to use tools such as Slack, Google Apps, Hubspot, Google Calendar, Gmail and others.
The six step guide to practical project managementMindGenius
The six step guide to practical project management
If you think managing projects is too difficult, think again.
We’ve stripped back project management processes to the
basics – to make it quicker and easier, without sacrificing
the vital ingredients for success.
“If you’re looking for some real-world guidance, then The Six Step Guide to Practical Project Management will help.”
Dr Andrew Makar, Tactical Project Management
2024 State of Marketing Report – by HubspotMarius Sescu
https://www.hubspot.com/state-of-marketing
· Scaling relationships and proving ROI
· Social media is the place for search, sales, and service
· Authentic influencer partnerships fuel brand growth
· The strongest connections happen via call, click, chat, and camera.
· Time saved with AI leads to more creative work
· Seeking: A single source of truth
· TLDR; Get on social, try AI, and align your systems.
· More human marketing, powered by robots
ChatGPT is a revolutionary addition to the world since its introduction in 2022. A big shift in the sector of information gathering and processing happened because of this chatbot. What is the story of ChatGPT? How is the bot responding to prompts and generating contents? Swipe through these slides prepared by Expeed Software, a web development company regarding the development and technical intricacies of ChatGPT!
Product Design Trends in 2024 | Teenage EngineeringsPixeldarts
The realm of product design is a constantly changing environment where technology and style intersect. Every year introduces fresh challenges and exciting trends that mold the future of this captivating art form. In this piece, we delve into the significant trends set to influence the look and functionality of product design in the year 2024.
How Race, Age and Gender Shape Attitudes Towards Mental HealthThinkNow
Mental health has been in the news quite a bit lately. Dozens of U.S. states are currently suing Meta for contributing to the youth mental health crisis by inserting addictive features into their products, while the U.S. Surgeon General is touring the nation to bring awareness to the growing epidemic of loneliness and isolation. The country has endured periods of low national morale, such as in the 1970s when high inflation and the energy crisis worsened public sentiment following the Vietnam War. The current mood, however, feels different. Gallup recently reported that national mental health is at an all-time low, with few bright spots to lift spirits.
To better understand how Americans are feeling and their attitudes towards mental health in general, ThinkNow conducted a nationally representative quantitative survey of 1,500 respondents and found some interesting differences among ethnic, age and gender groups.
Technology
For example, 52% agree that technology and social media have a negative impact on mental health, but when broken out by race, 61% of Whites felt technology had a negative effect, and only 48% of Hispanics thought it did.
While technology has helped us keep in touch with friends and family in faraway places, it appears to have degraded our ability to connect in person. Staying connected online is a double-edged sword since the same news feed that brings us pictures of the grandkids and fluffy kittens also feeds us news about the wars in Israel and Ukraine, the dysfunction in Washington, the latest mass shooting and the climate crisis.
Hispanics may have a built-in defense against the isolation technology breeds, owing to their large, multigenerational households, strong social support systems, and tendency to use social media to stay connected with relatives abroad.
Age and Gender
When asked how individuals rate their mental health, men rate it higher than women by 11 percentage points, and Baby Boomers rank it highest at 83%, saying it’s good or excellent vs. 57% of Gen Z saying the same.
Gen Z spends the most amount of time on social media, so the notion that social media negatively affects mental health appears to be correlated. Unfortunately, Gen Z is also the generation that’s least comfortable discussing mental health concerns with healthcare professionals. Only 40% of them state they’re comfortable discussing their issues with a professional compared to 60% of Millennials and 65% of Boomers.
Race Affects Attitudes
As seen in previous research conducted by ThinkNow, Asian Americans lag other groups when it comes to awareness of mental health issues. Twenty-four percent of Asian Americans believe that having a mental health issue is a sign of weakness compared to the 16% average for all groups. Asians are also considerably less likely to be aware of mental health services in their communities (42% vs. 55%) and most likely to seek out information on social media (51% vs. 35%).
AI Trends in Creative Operations 2024 by Artwork Flow.pdfmarketingartwork
This article is all about what AI trends will emerge in the field of creative operations in 2024. All the marketers and brand builders should be aware of these trends for their further use and save themselves some time!
A report by thenetworkone and Kurio.
The contributing experts and agencies are (in an alphabetical order): Sylwia Rytel, Social Media Supervisor, 180heartbeats + JUNG v MATT (PL), Sharlene Jenner, Vice President - Director of Engagement Strategy, Abelson Taylor (USA), Alex Casanovas, Digital Director, Atrevia (ES), Dora Beilin, Senior Social Strategist, Barrett Hoffher (USA), Min Seo, Campaign Director, Brand New Agency (KR), Deshé M. Gully, Associate Strategist, Day One Agency (USA), Francesca Trevisan, Strategist, Different (IT), Trevor Crossman, CX and Digital Transformation Director; Olivia Hussey, Strategic Planner; Simi Srinarula, Social Media Manager, The Hallway (AUS), James Hebbert, Managing Director, Hylink (CN / UK), Mundy Álvarez, Planning Director; Pedro Rojas, Social Media Manager; Pancho González, CCO, Inbrax (CH), Oana Oprea, Head of Digital Planning, Jam Session Agency (RO), Amy Bottrill, Social Account Director, Launch (UK), Gaby Arriaga, Founder, Leonardo1452 (MX), Shantesh S Row, Creative Director, Liwa (UAE), Rajesh Mehta, Chief Strategy Officer; Dhruv Gaur, Digital Planning Lead; Leonie Mergulhao, Account Supervisor - Social Media & PR, Medulla (IN), Aurelija Plioplytė, Head of Digital & Social, Not Perfect (LI), Daiana Khaidargaliyeva, Account Manager, Osaka Labs (UK / USA), Stefanie Söhnchen, Vice President Digital, PIABO Communications (DE), Elisabeth Winiartati, Managing Consultant, Head of Global Integrated Communications; Lydia Aprina, Account Manager, Integrated Marketing and Communications; Nita Prabowo, Account Manager, Integrated Marketing and Communications; Okhi, Web Developer, PNTR Group (ID), Kei Obusan, Insights Director; Daffi Ranandi, Insights Manager, Radarr (SG), Gautam Reghunath, Co-founder & CEO, Talented (IN), Donagh Humphreys, Head of Social and Digital Innovation, THINKHOUSE (IRE), Sarah Yim, Strategy Director, Zulu Alpha Kilo (CA).
Trends In Paid Search: Navigating The Digital Landscape In 2024Search Engine Journal
The search marketing landscape is evolving rapidly with new technologies, and professionals, like you, rely on innovative paid search strategies to meet changing demands.
It’s important that you’re ready to implement new strategies in 2024.
Check this out and learn the top trends in paid search advertising that are expected to gain traction, so you can drive higher ROI more efficiently in 2024.
You’ll learn:
- The latest trends in AI and automation, and what this means for an evolving paid search ecosystem.
- New developments in privacy and data regulation.
- Emerging ad formats that are expected to make an impact next year.
Watch Sreekant Lanka from iQuanti and Irina Klein from OneMain Financial as they dive into the future of paid search and explore the trends, strategies, and technologies that will shape the search marketing landscape.
If you’re looking to assess your paid search strategy and design an industry-aligned plan for 2024, then this webinar is for you.
5 Public speaking tips from TED - Visualized summarySpeakerHub
From their humble beginnings in 1984, TED has grown into the world’s most powerful amplifier for speakers and thought-leaders to share their ideas. They have over 2,400 filmed talks (not including the 30,000+ TEDx videos) freely available online, and have hosted over 17,500 events around the world.
With over one billion views in a year, it’s no wonder that so many speakers are looking to TED for ideas on how to share their message more effectively.
The article “5 Public-Speaking Tips TED Gives Its Speakers”, by Carmine Gallo for Forbes, gives speakers five practical ways to connect with their audience, and effectively share their ideas on stage.
Whether you are gearing up to get on a TED stage yourself, or just want to master the skills that so many of their speakers possess, these tips and quotes from Chris Anderson, the TED Talks Curator, will encourage you to make the most impactful impression on your audience.
See the full article and more summaries like this on SpeakerHub here: https://speakerhub.com/blog/5-presentation-tips-ted-gives-its-speakers
See the original article on Forbes here:
http://www.forbes.com/forbes/welcome/?toURL=http://www.forbes.com/sites/carminegallo/2016/05/06/5-public-speaking-tips-ted-gives-its-speakers/&refURL=&referrer=#5c07a8221d9b
ChatGPT and the Future of Work - Clark Boyd Clark Boyd
Everyone is in agreement that ChatGPT (and other generative AI tools) will shape the future of work. Yet there is little consensus on exactly how, when, and to what extent this technology will change our world.
Businesses that extract maximum value from ChatGPT will use it as a collaborative tool for everything from brainstorming to technical maintenance.
For individuals, now is the time to pinpoint the skills the future professional will need to thrive in the AI age.
Check out this presentation to understand what ChatGPT is, how it will shape the future of work, and how you can prepare to take advantage.
A brief introduction to DataScience with explaining of the concepts, algorithms, machine learning, supervised and unsupervised learning, clustering, statistics, data preprocessing, real-world applications etc.
It's part of a Data Science Corner Campaign where I will be discussing the fundamentals of DataScience, AIML, Statistics etc.
Time Management & Productivity - Best PracticesVit Horky
Here's my presentation on by proven best practices how to manage your work time effectively and how to improve your productivity. It includes practical tips and how to use tools such as Slack, Google Apps, Hubspot, Google Calendar, Gmail and others.
The six step guide to practical project managementMindGenius
The six step guide to practical project management
If you think managing projects is too difficult, think again.
We’ve stripped back project management processes to the
basics – to make it quicker and easier, without sacrificing
the vital ingredients for success.
“If you’re looking for some real-world guidance, then The Six Step Guide to Practical Project Management will help.”
Dr Andrew Makar, Tactical Project Management
Beginners Guide to TikTok for Search - Rachel Pearson - We are Tilt __ Bright...
Incremental
1. Chapter 4
Incremental Gradient, Subgradient, and
Proximal Methods for Convex Optimization:
A Survey
Dimitri P. Bertsekas
Optimization for Machine Learning 勉強会 中川研M1 山田直敬
12年5月24日木曜日 1
2. Chapter 4
Incremental Gradient, Subgradient, and
Proximal Methods for Convex Optimization:
A Survey
4.1 Introduction
4.1.1 Some Examples of Additive Cost Problems
4.1.2 Incremental Gradient Methods - Differentiable Problems
4.1.3 Incremental Subgradient Methods - Nondifferentiable Problems
4.1.4 Incremental Proximal Methods
4.2 Incremental Subgradient-Proximal Methods
4.3 Convergence for Methods with Cyclic Order
4.4 Convergence for Methods with Randomized Order
4.5 Some Applications
4.5.1 Regularized Least Squares
4.5.2 Iterated Projection Algorithms
12年5月24日木曜日 2
3. Chapter 4
Incremental Gradient, Subgradient, and
Proximal Methods for Convex Optimization:
A Survey
4.1 Introduction
4.1.1 Some Examples of Additive Cost Problems
4.1.2 Incremental Gradient Methods - Differentiable Problems
4.1.3 Incremental Subgradient Methods - Nondifferentiable Problems
4.1.4 Incremental Proximal Methods
4.2 Incremental Subgradient-Proximal Methods
4.3 Convergence for Methods with Cyclic Order TODAY’S TOPIC
4.4 Convergence for Methods with Randomized Order
4.5 Some Applications
4.5.1 Regularized Least Squares
4.5.2 Iterated Projection Algorithms
12年5月24日木曜日 3
4. Chapter 4
Incremental Gradient, Subgradient, and
Proximal Methods for Convex Optimization:
A Survey
4.1 Introduction
4.1.1 Some Examples of Additive Cost Problems
4.1.2 Incremental Gradient Methods - Differentiable Problems
4.1.3 Incremental Subgradient Methods - Nondifferentiable Problems
4.1.4 Incremental Proximal Methods
4.2 Incremental Subgradient-Proximal Methods
4.3 Convergence for Methods with Cyclic Order
4.4 Convergence for Methods with Randomized Order
4.5 Some Applications
4.5.1 Regularized Least Squares
4.5.2 Iterated Projection Algorithms
12年5月24日木曜日 4
5. Incremental Method Additive
cost Problem
ex.) loss func.
mがとても大きい場合、すべてのfi(x) を計算してから勾配を求めるの
は高コスト
その代わりfi を一つずつ取り、その値を用いて少しずつ更新していく
全部足して計算するよりも性能が良くなる可能性がある。
12年5月24日木曜日 5
6. 直観的な描像
= F(x)
最適性の証明にもこの考え方を用いる
f (x)
F の降下方向にノイズが乗ったも
fi の降下方向 のと解釈できる
F(x)
12年5月24日木曜日 6
7. Chapter 4
Incremental Gradient, Subgradient, and
Proximal Methods for Convex Optimization:
A Survey
4.1 Introduction
4.1.1 Some Examples of Additive Cost Problems
4.1.2 Incremental Gradient Methods - Differentiable Problems
4.1.3 Incremental Subgradient Methods - Nondifferentiable Problems
4.1.4 Incremental Proximal Methods
4.2 Incremental Subgradient-Proximal Methods
4.3 Convergence for Methods with Cyclic Order
4.4 Convergence for Methods with Randomized Order
4.5 Some Applications
4.5.1 Regularized Least Squares
4.5.2 Iterated Projection Algorithms
12年5月24日木曜日 7
8. Example 1.1: Least Squares and Inference
f_i(x) = 二乗誤差
(a_i,b_i): given. 特徴ベクトルとラベルの組
xbar : given
12年5月24日木曜日 8
9. Example 1.1: Least Squares and Inference
L1正則化
ニューラルネットワーク
最尤推定、EM
12年5月24日木曜日 9
10. Example 1.2: Dual Optimization in Separable Problems
DUAL
m
導出は同著NONLINEAR本
5.1.6節より
凸関数
12年5月24日木曜日 10
11. Example 4.3:
Minimization of an Expected Value
- Stochastic Programming
m
= ∑ π i H (x,wi )
i=1
Hはxと確率変数w の関数
ただし、wはm通りの可能な値がある(m>>1)
P(w = wi ) = π i i = 1,..., m
このときHの期待値を最小化するようなxを求める
12年5月24日木曜日 11
12. Example 4.3:
Minimization of an Expected Value
- Stochastic Programming
あるいは wi を独立なサン
プルだと思うと、
は E{H (x,w)}の近似である
これもADDITIVE COST PROBLEMの一つ
12年5月24日木曜日 12
13. Example 4.4:
Problems with Many Constraints
制約の数が大きい場合、ペナルティ
関数を用いることで無制約化する
例 十分大きなcを設定
するのがポイント
12年5月24日木曜日 13
14. Example 4.4:
Problems with Many Constraints
{ x ∈ i=1 Xi
m
closed
set
十分大きなγを設定
12年5月24日木曜日 14
15. Example 4.5:
Distributed Incremental Optimization
- Sensor Networks
m
fusion ∑ f (x )
i k
i=1
center
f1 (xk ) f1 (xk )
f2 (xk ) ... ...
... ...
fi (xk )
同期させるため 同期を取らず、
通信のオーバーヘッド大 各センサの関数値を用いてXを決める
12年5月24日木曜日 15
16. Example 4.6:
Weber Problem in Loation Theory
x
(weighted) 1- meansとも解釈できる
12年5月24日木曜日 16
17. Chapter 4
Incremental Gradient, Subgradient, and
Proximal Methods for Convex Optimization:
A Survey
4.1 Introduction
4.1.1 Some Examples of Additive Cost Problems
4.1.2 Incremental Gradient Methods - Differentiable Problems
4.1.3 Incremental Subgradient Methods - Nondifferentiable Problems
4.1.4 Incremental Proximal Methods
4.2 Incremental Subgradient-Proximal Methods
4.3 Convergence for Methods with Cyclic Order
4.4 Convergence for Methods with Randomized Order
4.5 Some Applications
4.5.1 Regularized Least Squares
4.5.2 Iterated Projection Algorithms
12年5月24日木曜日 17
18. Incremental Gradient Methods
に対して、
f_iが微分可能とする(凸性は必要なし)
歴史は長い。”back propagation” もこの形
ikの選び方は周期的に取るか、乱択によって決める
ik = (k mod m) + 1
まんべんなく∇f_ikを評価するために、反復数はm回以上取る
12年5月24日木曜日 18
32. Chapter 4
Incremental Gradient, Subgradient, and
Proximal Methods for Convex Optimization:
A Survey
4.1 Introduction
4.1.1 Some Examples of Additive Cost Problems
4.1.2 Incremental Gradient Methods - Differentiable Problems
4.1.3 Incremental Subgradient Methods - Nondifferentiable Problems
4.1.4 Incremental Proximal Methods
4.2 Incremental Subgradient-Proximal Methods
4.3 Convergence for Methods with Cyclic Order
4.4 Convergence for Methods with Randomized Order
4.5 Some Applications
4.5.1 Regularized Least Squares
4.5.2 Iterated Projection Algorithms
12年5月24日木曜日 32
33. Chapter 4
Incremental Gradient, Subgradient, and
Proximal Methods for Convex Optimization:
A Survey
4.1 Introduction
4.1.1 Some Examples of Additive Cost Problems
4.1.2 Incremental Gradient Methods - Differentiable Problems
4.1.3 Incremental Subgradient Methods - Nondifferentiable Problems
4.1.4 Incremental Proximal Methods
4.2 Incremental Subgradient-Proximal Methods
4.3 Convergence for Methods with Cyclic Order
4.4 Convergence for Methods with Randomized Order
4.5 Some Applications
4.5.1 Regularized Least Squares
4.5.2 Iterated Projection Algorithms
12年5月24日木曜日 33
34. 4.5.1 Regularized Least Square
例えばL1正則化だと、
ここで
γ 1 '
fi (x) = x 1 ,hi (x) = (ci x − di ) とおけば
2
m 2
(SUB)GRADIENT DESCENT
CLOSED FORM
12年5月24日木曜日 34
35. Chapter 4
Incremental Gradient, Subgradient, and
Proximal Methods for Convex Optimization:
A Survey
4.1 Introduction
4.1.1 Some Examples of Additive Cost Problems
4.1.2 Incremental Gradient Methods - Differentiable Problems
4.1.3 Incremental Subgradient Methods - Nondifferentiable Problems
4.1.4 Incremental Proximal Methods
4.2 Incremental Subgradient-Proximal Methods
4.3 Convergence for Methods with Cyclic Order
4.4 Convergence for Methods with Randomized Order
4.5 Some Applications
4.5.1 Regularized Least Squares
4.5.2 Iterated Projection Algorithms
12年5月24日木曜日 35
40. gradient methodが定数ステップサイズ
で収束する条件(Nonlinear本 prop.1.2.3)
{x_k}をgradient method で得られた系列とする。
i.e. x k+1 = x k + α k d k ,where∇F(x k )'d k < 0
F(x)がLipshitz連続
∃L > 0,∀x, y ∈ , ∇F(x) − ∇F(y) ≤ L x − y
n
2 k 2 2
∃c1 ,c2 ,c1 ∇F(x ) ≤ −∇F(x )'d , d
k k k
≤ c2 ∇F(x )
k
このとき gradient descentは次のαで局所最適解
c1 (2 − ε )
に収束する ε ≤α ≤
L
12年5月24日木曜日 40
41. gradient methodが減衰ステップサイズで
収束する条件(Nonlinear本 prop.1.2.4)
{x_k}をgradient method で得られた系列とする。
i.e.
x k+1
= x + α d ,where∇F(x )'d < 0
k k k k k
F(x)がLipshitz連続 ∃L > 0,∀x, y ∈ n , ∇F(x) − ∇F(y) ≤ L x − y
2 k 2 2
∃c1 ,c2 ,c1 ∇F(x ) ≤ −∇F(x )'d , d
k k k
≤ c2 ∇F(x )
k
∞
減衰ステップサイズ α → 0,
k
∑α k
=∞
k=0
このとき gradient descentは最適値F*に収束する
inf F(x) = F *
x∈X
12年5月24日木曜日 41
42. gradientに誤差が乗っている場合の
収束性について
gradientが正確に計算できず、エラー付きでしか得られない場合の
更新式は次のようになる。
g = ∇F(x ) + e
k k k
x k+1
= x −α g
k k k
≤ δ ならば、収束先の値は F + O(δ )
*
eが有界 すなわち ∀k, e k
となる。
eがstepsizeに比例する場合、すなわち ek ≤ α k q のとき、
α->0で収束先はF*に一致する
一般論おわり
12年5月24日木曜日 42