The document discusses concept learning as search and describes two algorithms:
1) Find-S searches the hypothesis space in a general-to-specific ordering to find the most specific hypothesis consistent with the training data.
2) Candidate-Elimination maintains both general and specific boundaries of the version space and iteratively updates them based on positive and negative examples to find all consistent hypotheses.
CMSC 56 | Lecture 12: Recursive Definition & Algorithms, and Program Correctnessallyn joy calcaben
Recursive Definition & Algorithms, and Program Correctness
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 23, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
Linear Discriminant Analysis (LDA) Under f-Divergence MeasuresAnmol Dwivedi
For more details, please have a look at:
1. https://www.mdpi.com/1099-4300/24/2/188
2. https://ieeexplore.ieee.org/document/9518004
Abstract:
In statistical inference, the information-theoretic performance limits can often be expressed in terms of a notion of divergence between the underlying statistical models (e.g., in binary hypothesis testing, the total error probability is equal to the total variation between the models). As the data dimension grows, computing the statistics involved in decision-making and the attendant performance limits (divergence measures) face complexity and stability challenges. Dimensionality reduction addresses these challenges at the expense of compromising the performance (divergence reduces due to the data processing inequality for divergence). This paper considers linear dimensionality reduction such that the divergence between the models is \emph{maximally} preserved. Specifically, the paper focuses on the Gaussian models and characterizes an optimal projection of the data onto a lower-dimensional subspace with respect to four $f$-divergence measures (Kullback-Leibler, $\chi^2$, Hellinger, and total variation). There are two key observations. First, projections are not necessarily along the dominant modes of the covariance matrix of the data, and even in some situations, they can be along the least dominant modes. Secondly, under specific regimes, the optimal design of subspace projection is identical under all the $f$-divergence measures considered, rendering a degree of universality to the design independent of the inference problem of interest.
My thesis integrates perspectives from text comprehension and multimedia learning theories. Results provide evidence for a linear contiguity effect and a text cohesion effect as new multimedia design principles. Publications are forthcoming.
Tutorial on Belief Propagation in Bayesian NetworksAnmol Dwivedi
The goal of this mini-project is to implement belief propagation algorithms for posterior probability inference and most probable explanation (MPE) inference for the Bayesian Network with binary values in which the Conditional Probability Table for each random-variable/node is given.
CMSC 56 | Lecture 12: Recursive Definition & Algorithms, and Program Correctnessallyn joy calcaben
Recursive Definition & Algorithms, and Program Correctness
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 23, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
Linear Discriminant Analysis (LDA) Under f-Divergence MeasuresAnmol Dwivedi
For more details, please have a look at:
1. https://www.mdpi.com/1099-4300/24/2/188
2. https://ieeexplore.ieee.org/document/9518004
Abstract:
In statistical inference, the information-theoretic performance limits can often be expressed in terms of a notion of divergence between the underlying statistical models (e.g., in binary hypothesis testing, the total error probability is equal to the total variation between the models). As the data dimension grows, computing the statistics involved in decision-making and the attendant performance limits (divergence measures) face complexity and stability challenges. Dimensionality reduction addresses these challenges at the expense of compromising the performance (divergence reduces due to the data processing inequality for divergence). This paper considers linear dimensionality reduction such that the divergence between the models is \emph{maximally} preserved. Specifically, the paper focuses on the Gaussian models and characterizes an optimal projection of the data onto a lower-dimensional subspace with respect to four $f$-divergence measures (Kullback-Leibler, $\chi^2$, Hellinger, and total variation). There are two key observations. First, projections are not necessarily along the dominant modes of the covariance matrix of the data, and even in some situations, they can be along the least dominant modes. Secondly, under specific regimes, the optimal design of subspace projection is identical under all the $f$-divergence measures considered, rendering a degree of universality to the design independent of the inference problem of interest.
My thesis integrates perspectives from text comprehension and multimedia learning theories. Results provide evidence for a linear contiguity effect and a text cohesion effect as new multimedia design principles. Publications are forthcoming.
Tutorial on Belief Propagation in Bayesian NetworksAnmol Dwivedi
The goal of this mini-project is to implement belief propagation algorithms for posterior probability inference and most probable explanation (MPE) inference for the Bayesian Network with binary values in which the Conditional Probability Table for each random-variable/node is given.
Functions Representations
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 13, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Functions Representations
CMSC 56 | Discrete Mathematical Structure for Computer Science
October 13, 2018
Instructor: Allyn Joy D. Calcaben
College of Arts & Sciences
University of the Philippines Visayas
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
Covers supervised learning and discriminative algorithms. Includes: Linear Regression, The LMS Algorithm, Probabalistic interpretations, Classification, Logistic Regression, Underfitting and Overfitting.
Reinforcement Learning: Hidden Theory and New Super-Fast AlgorithmsSean Meyn
A tutorial, and very new algorithms -- more details on arXiv and at NIPS 2017 https://arxiv.org/abs/1707.03770
Part of the Data Science Summer School at École Polytechnique: http://www.ds3-datascience-polytechnique.fr/program/
---------
2018 Updates:
See Zap slides from ISMP 2018 for new inverse-free optimal algorithms
Simons tutorial, March 2018 [one month before most discoveries announced at ISMP]
Part I (Basics, with focus on variance of algorithms)
https://www.youtube.com/watch?v=dhEF5pfYmvc
Part II (Zap Q-learning)
https://www.youtube.com/watch?v=Y3w8f1xIb6s
Big 2017 survey on variance in SA:
Fastest convergence for Q-learning
https://arxiv.org/abs/1707.03770
You will find the infinite-variance Q result there.
Our NIPS 2017 paper is distilled from this.
QMC algorithms usually rely on a choice of “N” evenly distributed integration nodes in $[0,1)^d$. A common means to assess such an equidistributional property for a point set or sequence is the so-called discrepancy function, which compares the actual number of points to the expected number of points (assuming uniform distribution on $[0,1)^{d}$) that lie within an arbitrary axis parallel rectangle anchored at the origin. The dependence of the integration error using QMC rules on various norms of the discrepancy function is made precise within the well-known Koksma--Hlawka inequality and its variations. In many cases, such as $L^{p}$ spaces, $1<p<\infty$, the best growth rate in terms of the number of points “N” as well as corresponding explicit constructions are known. In the classical setting $p=\infty$ sharp results are absent for $d\geq3$ already and appear to be intriguingly hard to obtain. This talk shall serve as a survey on discrepancy theory with a special emphasis on the $L^{\infty}$ setting. Furthermore, it highlights the evolution of recent techniques and presents the latest results.
Embracing GenAI - A Strategic ImperativePeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Francesca Gottschalk - How can education support child empowerment.pptxEduSkills OECD
Francesca Gottschalk from the OECD’s Centre for Educational Research and Innovation presents at the Ask an Expert Webinar: How can education support child empowerment?
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Normal Labour/ Stages of Labour/ Mechanism of LabourWasim Ak
Normal labor is also termed spontaneous labor, defined as the natural physiological process through which the fetus, placenta, and membranes are expelled from the uterus through the birth canal at term (37 to 42 weeks
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
Safalta Digital marketing institute in Noida, provide complete applications that encompass a huge range of virtual advertising and marketing additives, which includes search engine optimization, virtual communication advertising, pay-per-click on marketing, content material advertising, internet analytics, and greater. These university courses are designed for students who possess a comprehensive understanding of virtual marketing strategies and attributes.Safalta Digital Marketing Institute in Noida is a first choice for young individuals or students who are looking to start their careers in the field of digital advertising. The institute gives specialized courses designed and certification.
for beginners, providing thorough training in areas such as SEO, digital communication marketing, and PPC training in Noida. After finishing the program, students receive the certifications recognised by top different universitie, setting a strong foundation for a successful career in digital marketing.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
2. Concept Learning as Search
• We assume that the concept lies in the
hypothesis space. So we search for a
hypothesis belonging to this hypothesis
space that best fits the training examples,
such that the output given by the hypothesis
is same as the true output of concept
• Hence the search has achieved the
learning of the actual concept using the
given training set
3. Concept Learning as Search
• In short:
Assume , search for an that best fits D, such
that xi D, h(xi) = c(xi)
Where c is the concept we are trying to determine (the
output of the training set)
H is the hypothesis space
D is the training set
h is the hypothesis
xi is the ith instance of Instance space
Hc Hh
4. Ordering of Hypothesis Space
• General to Specific Ordering of Hypothesis
Space
• Most General Hypothesis:
– hg< ?, ? >
• Most Specific Hypothesis:
– hs< Ø , Ø >
5. Ordering of Hypothesis Space
SK = < T, BP >, T = { H, N, L } and BP = { H, N, L }
< ?, ? >
< H, ? > < N, ? > < L, ? > < ?, H > < ?, N > < ?, L >
< H, H >< H, N >< H, L > < N, H >< N, N >< N, L > < L, H >< L, N >< L, L >
< Ø , Ø >
6. Find-S Algorithm
• FIND-S finds the most specific hypothesis
possible within the version space given a
set of training data
• Uses the general-to-specific ordering for
searching through the hypotheses space
7. Find-S Algorithm
Initialize hypothesis h to the most specific hypothesis in H
(the hypothesis space)
For each positive training instance x (i.e. output is 1)
For each attribute constraint ai in h
If the constraint ai is satisfied by x
Then do nothing
Else
Replace ai in h by the next more
general constraint that is satisfied by x
Output hypothesis h
8. Find-S Algorithm
To illustrate this algorithm, let us assume that the learner is given the sequence of
following training examples from the SICK domain:
D T BP SK
x1 H H 1
x2 L L 0
x3 N H 1
The first step of FIND-S is to initialize hypothesis h to the most specific hypothesis in
H:
h = < Ø , Ø >
9. Find-S Algorithm
D T BP SK
x1 H H 1
First training example is positive:
But h = < Ø , Ø > fails over this first instance
Because h(x1) = 0, since Ø gives us 0 for any attribute
value
Since h = < Ø , Ø > is so specific that it doesn’t give even one single instance
as positive, so we change it to next more general hypothesis that fits this
particular first instance x1 of the training data set D to
h = < H , H >
10. Find-S Algorithm
< ?, ? >
< H, ? > < N, ? > < L, ? > < ?, H > < ?, N > < ?, L >
< H, H >< H, N >< H, L > < N, H >< N, N >< N, L > < L, H >< L, N >< L, L >
< Ø , Ø >
SK = < T, BP >, T = { H, N, L } and BP = { H, N, L }
11. Find-S Algorithm
D T BP SK
x1 H H 1
x2 L L 0
Upon encountering the second example; in this case a negative example, the algorithm makes no
change to h. In fact, the FIND-S algorithm simply ignores every negative example
So the hypothesis still remains: h = < H , H >
12. Find-S Algorithm
D T BP SK
x1 H H 1
x2 L L 0
x3 N H 1
Final Hypothesis:
h = < ?, H >
What does this hypothesis state?
This hypothesis will term all the future patients which have BP = H as SICK for all the
different values of T
13. Find-S Algorithm
< ?, ? >
< H, ? > < N, ? > < L, ? > < ?, H > < ?, N > < ?, L >
< H, H >< H, N >< H, L > < N, H >< N, N >< N, L > < L, H >< L, N >< L, L >
< Ø , Ø >
D T BP SK
x1 H H 1
x2 L L 0
x3 N H 1
14. Candidate-Elimination Algorithm
• Although FIND-S does find a consistent
hypothesis
• In general, however, there may be more
hypotheses consistent with D; of which
FIND-S only finds one
• Candidate-Elimination finds all the
hypotheses in the Version Space
15. Version Space (VS)
• Version space is a set of all the
hypotheses that are consistent with all the
training examples
• By consistent we mean
h(xi) = c(xi) , for all instances belonging to
training set D
16. Version Space
Let us take the following training set D:
D T BP SK
x1 H H 1
x2 L L 0
x3 N N 0
Another representation of this set D:
BP
H - - 1
N - 0 -
L 0 - -
L N H T
17. Version Space
Is there a hypothesis that can generate this D:
BP
H - - 1
N - 0 -
L 0 - -
L N H T
One of the consistent hypotheses can be h1 = < H, H >
BP
H 0 0 1
N 0 0 0
L 0 0 0
L N H T
18. Version Space
There are other hypotheses consistent with D, such as h2 = < H, ? >
There’s another hypothesis, h3 = < ?, H >
BP
H 1 1 1
N 0 0 0
L 0 0 0
L N H T
BP
H 0 0 1
N 0 0 1
L 0 0 1
L N H T
19. Version Space
• Version space is denoted as
VS H,D = {h1, h2, h3}
• This translates as: Version space is a
subset of hypothesis space H, composed
of h1, h2 and h3, that is consistent with D
• In other words version space is a group of
all hypotheses consistent with D, not just
one hypothesis we saw in the previous
case
20. Candidate-Elimination Algorithm
• Candidate Elimination works with two sets:
– Set G (General hypotheses)
– Set S (Specific hypotheses)
• Starts with:
– G0 = {< ? , ? >} considers negative examples only
– S0 = {< Ø , Ø >} considers positive examples only
• Within these two boundaries is the entire
Hypothesis space
21. Candidate-Elimination Algorithm
• Intuitively:
– As each training example is observed one by
one
• The S boundary is made more and more general
• The G boundary set is made more and more specific
• This eliminates from the version space any hypotheses found
inconsistent with the new training example
– At the end, we are left with VS
22. Candidate-Elimination Algorithm
Initialize G to the set of maximally general hypotheses in H
Initialize S to the set of maximally specific hypotheses in H
For each training example d, do
If d is a positive example
Remove from G any hypothesis inconsistent with d
For each hypothesis s in S that is inconsistent with d
Remove s from S
Add to S all minimal generalization h of s, such that
h is consistent with d, and some member of G is more general than h
Remove from S any hypothesis that is more general than another one in S
If d is a negative example
Remove from S any hypothesis inconsistent with d
For each hypothesis g in G that is inconsistent with d
Remove g from G
Add to G all minimal specializations h of g, such that
h is consistent with d, and some member of S is more specific than h
Remove from G any hypothesis that is less general than another one in G
24. Candidate-Elimination Algorithm
• Candidate Elimination works with two sets:
– Set G (General hypotheses)
– Set S (Specific hypotheses)
• Starts with:
– G0 = {< ? , ? >} considers negative examples only
– S0 = {< Ø , Ø >} considers positive examples only
• Within these two boundaries is the entire
Hypothesis space
25. Candidate-Elimination Algorithm
Initialize G to the set of maximally general hypotheses in H
Initialize S to the set of maximally specific hypotheses in H
For each training example d, do
If d is a positive example
Remove from G any hypothesis inconsistent with d
For each hypothesis s in S that is inconsistent with d
Remove s from S
Add to S all minimal generalization h of s, such that
h is consistent with d, and some member of G is more general than h
Remove from S any hypothesis that is more general than another one in S
If d is a negative example
Remove from S any hypothesis inconsistent with d
For each hypothesis g in G that is inconsistent with d
Remove g from G
Add to G all minimal specializations h of g, such that
h is consistent with d, and some member of S is more specific than h
Remove from G any hypothesis that is less general than another one in G
27. Candidate-Elimination Algorithm
D T BP SK
x1 H H 1
d1 = (<H, H>, 1) [a positive example]:
G1 = {< ?, ? >}
S1 = {< H, H >}
Remove < Ø, Ø > from S0 , since it is not consistent with d1
and add the next minimally general hypothesis from H to form
S1
G1 = G0 = {< ?, ? >}, since <?, ?> is consistent with d1; both
give positive outputs
G0 = {< ?, ? >}
S0 = {< Ø, Ø >}
28. Candidate-Elimination Algorithm
D T BP SK
x2 L L 0
Second training example is: d2 = (<L, L>, 0) [negative example]
G2 = {< H, ? >, < ?, H >}
S2 = {< H, H>}
Remove < ?, ? > from G1 , since it is not consistent with d2 and add the next
minimally specialized hypothesis from H to form G2 , keeping in mind one
rule:
S2 = S1 = {< H, H>}, since <H, H> is consistent with d2: both give negative outputs for x2
“Add to G all minimal specializations h of g, such that
h is consistent with d, and some member of S is more specific than h”
Now, observe that the immediate one step specialized hypotheses of < ?, ? > are:
{< H, ? >, < N, ? >, < L, ? >, < ?, H >, < ?, N >, < ?, L >}
G1 = {< ?, ? >}
S1 = {< H, H >}
29. Candidate-Elimination Algorithm
D T BP SK
x3 N H 1
Third and final training example is: d3 = (<N, H>, 1) [A positive example]
G3 = {< ?, H >}
S3 = {< ?, H >}
In S2, < H, H > is not consistent with d3, so we remove it and add minimally
general hypotheses than < H, H >. The two choices we have are: < H, ? >
and < ?, H >. We only keep < ?, H >, since the other one is not consistent
with d3
We see that in G2, < H, ? > is not consistent with d3, so we remove it.
However < ?, H > is consistent hence it is retained: G3 = {< ?, H >}
G2 = {< H, ? >, < ?, H >}
S2 = {< H, H>}
30. Conjunctive vs Disjuncvtive
Conjuntive Rule (ANDing)
h = < T=H AND BP = ? >
BP
H 1 1 1
N 0 0 1
L 0 0 1
L N H T
BP
H 0 0 1
N 0 0 1
L 0 0 1
L N H T
Disjuntive Rule (ORing)
h = < T=H AND BP = ?
OR
T=? AND BP = H >