The document discusses extending the barrier method to optimization problems with generalized inequalities. It introduces logarithmic barrier functions for generalized inequalities and defines the central path. Points on the central path give dual feasible solutions. The barrier method can be applied to find an epsilon-optimal solution in O(log(1/epsilon)) iterations, each requiring O(sqrt(n)) Newton iterations under self-concordance assumptions. Complexity is analyzed using self-concordant functions and properties of generalized logarithms.
EM 알고리즘을 jensen's inequality부터 천천히 잘 설명되어있다
이것을 보면, LDA의 Variational method로 학습하는 방식이 어느정도 이해가 갈 것이다.
옛날 Andrew Ng 선생님의 강의노트에서 발췌한 건데 5년전에 본 것을
아직도 찾아가면서 참고하면서 해야 된다는 게 그 강의가 얼마나 명강의였는지 새삼 느끼게 된다.
EM 알고리즘을 jensen's inequality부터 천천히 잘 설명되어있다
이것을 보면, LDA의 Variational method로 학습하는 방식이 어느정도 이해가 갈 것이다.
옛날 Andrew Ng 선생님의 강의노트에서 발췌한 건데 5년전에 본 것을
아직도 찾아가면서 참고하면서 해야 된다는 게 그 강의가 얼마나 명강의였는지 새삼 느끼게 된다.
Fractal Dimension of Space-time Diagrams and the Runtime Complexity of Small ...Hector Zenil
Complexity measures are designed to capture complex behaviour and to quantify how complex that particular behaviour is. If a certain phenomenon is genuinely complex this means that it does not all of a sudden becomes simple by just translating the phenomenon to a different setting or framework with a different complexity value. It is in this sense that we expect different complexity measures from possibly entirely different fields to be related to each other. This work presents our work on a beautiful connection between the fractal dimension of space-time diagrams of Turing machines and their time complexity. Presented at Machines, Computations and Universality (MCU) 2013, Zurich, Switzerland.
Fractal dimension versus Computational ComplexityHector Zenil
We investigate connections and tradeoffs between two important complexity measures: fractal dimension and computational (time) complexity. We report exciting results applied to space-time diagrams of small Turing machines with precise mathematical relations and formal conjectures connecting these measures. The preprint of the paper is available at: http://arxiv.org/abs/1309.1779
We approach the screening problem - i.e. detecting which inputs of a computer model significantly impact the output - from a formal Bayesian model selection point of view. That is, we place a Gaussian process prior on the computer model and consider the $2^p$ models that result from assuming that each of the subsets of the $p$ inputs affect the response. The goal is to obtain the posterior probabilities of each of these models. In this talk, we focus on the specification of objective priors on the model-specific parameters and on convenient ways to compute the associated marginal likelihoods. These two problems that normally are seen as unrelated, have challenging connections since the priors proposed in the literature are specifically designed to have posterior modes in the boundary of the parameter space, hence precluding the application of approximate integration techniques based on e.g. Laplace approximations. We explore several ways of circumventing this difficulty, comparing different methodologies with synthetic examples taken from the literature.
Authors: Gonzalo Garcia-Donato (Universidad de Castilla-La Mancha) and Rui Paulo (Universidade de Lisboa)
Z Transform And Inverse Z Transform - Signal And SystemsMr. RahüL YøGi
The z-transform is the most general concept for the transformation of discrete-time series.
The Laplace transform is the more general concept for the transformation of continuous time processes.
For example, the Laplace transform allows you to transform a differential equation, and its corresponding initial and boundary value problems, into a space in which the equation can be solved by ordinary algebra.
The switching of spaces to transform calculus problems into algebraic operations on transforms is called operational calculus. The Laplace and z transforms are the most important methods for this purpose.
Fractal Dimension of Space-time Diagrams and the Runtime Complexity of Small ...Hector Zenil
Complexity measures are designed to capture complex behaviour and to quantify how complex that particular behaviour is. If a certain phenomenon is genuinely complex this means that it does not all of a sudden becomes simple by just translating the phenomenon to a different setting or framework with a different complexity value. It is in this sense that we expect different complexity measures from possibly entirely different fields to be related to each other. This work presents our work on a beautiful connection between the fractal dimension of space-time diagrams of Turing machines and their time complexity. Presented at Machines, Computations and Universality (MCU) 2013, Zurich, Switzerland.
Fractal dimension versus Computational ComplexityHector Zenil
We investigate connections and tradeoffs between two important complexity measures: fractal dimension and computational (time) complexity. We report exciting results applied to space-time diagrams of small Turing machines with precise mathematical relations and formal conjectures connecting these measures. The preprint of the paper is available at: http://arxiv.org/abs/1309.1779
We approach the screening problem - i.e. detecting which inputs of a computer model significantly impact the output - from a formal Bayesian model selection point of view. That is, we place a Gaussian process prior on the computer model and consider the $2^p$ models that result from assuming that each of the subsets of the $p$ inputs affect the response. The goal is to obtain the posterior probabilities of each of these models. In this talk, we focus on the specification of objective priors on the model-specific parameters and on convenient ways to compute the associated marginal likelihoods. These two problems that normally are seen as unrelated, have challenging connections since the priors proposed in the literature are specifically designed to have posterior modes in the boundary of the parameter space, hence precluding the application of approximate integration techniques based on e.g. Laplace approximations. We explore several ways of circumventing this difficulty, comparing different methodologies with synthetic examples taken from the literature.
Authors: Gonzalo Garcia-Donato (Universidad de Castilla-La Mancha) and Rui Paulo (Universidade de Lisboa)
Z Transform And Inverse Z Transform - Signal And SystemsMr. RahüL YøGi
The z-transform is the most general concept for the transformation of discrete-time series.
The Laplace transform is the more general concept for the transformation of continuous time processes.
For example, the Laplace transform allows you to transform a differential equation, and its corresponding initial and boundary value problems, into a space in which the equation can be solved by ordinary algebra.
The switching of spaces to transform calculus problems into algebraic operations on transforms is called operational calculus. The Laplace and z transforms are the most important methods for this purpose.
Principle of Integration - Basic Introduction - by Arun Umraossuserd6b1fd
Notes for integral calculus. Students must read function analysis before going through this book. Read Derivative Calculus before going through this book.
A polynomial interpolation algorithm is developed using the Newton's divided-difference interpolating polynomials. The definition of monotony of a function is then used to define the least degree of the polynomial to make efficient and consistent the interpolation in the discrete given function. The relation between the order of monotony of a particular function and the degree of the interpolating polynomial is justified, analyzing the relation between the derivatives of such function and the truncation error expression. In this algorithm there is not matter about the number and the arrangement of the data points, neither if the points are regularly spaced or not. The algorithm thus defined can be used to make interpolations in functions of one and several dependent variables. The algoritm automatically select the data points nearest to the point where an interpolation is desired, following the criterion of symmetry. Indirectly, the algorithm also select the number of data points, which is a unity higher than the order of the used polynomial, following the criterion of monotony. Finally, the complete algoritm is presented and subroutines in fortran code is exposed as an addendum. Notice that there is not the degree of the interpolating polynomial within the arguments of such subroutines.
Notes on intersection theory written for a seminar in Bonn in 2010.
Following Fulton's book the following topics are covered:
- Motivation of intersection theory
- Cones and Segre Classes
- Chern Classes
- Gauss-Bonet Formula
- Segre classes under birational morphisms
- Flat pull back
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Techniques to optimize the pagerank algorithm usually fall in two categories. One is to try reducing the work per iteration, and the other is to try reducing the number of iterations. These goals are often at odds with one another. Skipping computation on vertices which have already converged has the potential to save iteration time. Skipping in-identical vertices, with the same in-links, helps reduce duplicate computations and thus could help reduce iteration time. Road networks often have chains which can be short-circuited before pagerank computation to improve performance. Final ranks of chain nodes can be easily calculated. This could reduce both the iteration time, and the number of iterations. If a graph has no dangling nodes, pagerank of each strongly connected component can be computed in topological order. This could help reduce the iteration time, no. of iterations, and also enable multi-iteration concurrency in pagerank computation. The combination of all of the above methods is the STICD algorithm. [sticd] For dynamic graphs, unchanged components whose ranks are unaffected can be skipped altogether.
Explore our comprehensive data analysis project presentation on predicting product ad campaign performance. Learn how data-driven insights can optimize your marketing strategies and enhance campaign effectiveness. Perfect for professionals and students looking to understand the power of data analysis in advertising. for more details visit: https://bostoninstituteofanalytics.org/data-science-and-artificial-intelligence/
Chatty Kathy - UNC Bootcamp Final Project Presentation - Final Version - 5.23...John Andrews
SlideShare Description for "Chatty Kathy - UNC Bootcamp Final Project Presentation"
Title: Chatty Kathy: Enhancing Physical Activity Among Older Adults
Description:
Discover how Chatty Kathy, an innovative project developed at the UNC Bootcamp, aims to tackle the challenge of low physical activity among older adults. Our AI-driven solution uses peer interaction to boost and sustain exercise levels, significantly improving health outcomes. This presentation covers our problem statement, the rationale behind Chatty Kathy, synthetic data and persona creation, model performance metrics, a visual demonstration of the project, and potential future developments. Join us for an insightful Q&A session to explore the potential of this groundbreaking project.
Project Team: Jay Requarth, Jana Avery, John Andrews, Dr. Dick Davis II, Nee Buntoum, Nam Yeongjin & Mat Nicholas
Levelwise PageRank with Loop-Based Dead End Handling Strategy : SHORT REPORT ...Subhajit Sahu
Abstract — Levelwise PageRank is an alternative method of PageRank computation which decomposes the input graph into a directed acyclic block-graph of strongly connected components, and processes them in topological order, one level at a time. This enables calculation for ranks in a distributed fashion without per-iteration communication, unlike the standard method where all vertices are processed in each iteration. It however comes with a precondition of the absence of dead ends in the input graph. Here, the native non-distributed performance of Levelwise PageRank was compared against Monolithic PageRank on a CPU as well as a GPU. To ensure a fair comparison, Monolithic PageRank was also performed on a graph where vertices were split by components. Results indicate that Levelwise PageRank is about as fast as Monolithic PageRank on the CPU, but quite a bit slower on the GPU. Slowdown on the GPU is likely caused by a large submission of small workloads, and expected to be non-issue when the computation is performed on massive graphs.
Opendatabay - Open Data Marketplace.pptxOpendatabay
Opendatabay.com unlocks the power of data for everyone. Open Data Marketplace fosters a collaborative hub for data enthusiasts to explore, share, and contribute to a vast collection of datasets.
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Leverage these privacy-preserving datasets for training and testing AI models without compromising sensitive information. Opendatabay prioritizes transparency by providing detailed metadata, provenance information, and usage guidelines for each dataset, ensuring users have a comprehensive understanding of the data they're working with. By leveraging a powerful combination of distributed ledger technology and rigorous third-party audits Opendatabay ensures the authenticity and reliability of every dataset. Security is at the core of Opendatabay. Marketplace implements stringent security measures, including encryption, access controls, and regular vulnerability assessments, to safeguard your data and protect your privacy.
1. Presenter Koki Isokawa
Mar. 17, 2021
11.6 Problems with generalized inequalities
Reading circle on Convex Optimization - Boyd & Vandenberghe
2. Overview
Introduce how the barrier method can be extended to problems with
generalized inequalities
11.6.1 Logarithmic barrier and central path
11.6.2 Barrier method
11.6.3 Examples
11.6.4 Complexity analysis via self-concordance
2
3. Preparation: problem setting
As in section 11.1, we assume that
the function are twice continuously differentiable
with
the problem is solvable
fi
A ∈ Rp×n
rank A = p
3
where
• is convex
• are -convex
• are proper cones
f0 : Rn
→ R
fi : Rn
→ Rki, i = 1,...,m Ki
Ki ⊆ Rki
(11.38)
4. Preparation: KKT conditions for the problem
We assume that the problem (11.38) is strictly feasible, so that KKT
conditions are necessary and sufficient conditions for optimality of x⋆
4
where is the derivative of at
Dfi(x⋆
) ∈ Rki×n
fi x⋆
6. Generalized logarithm for a proper cone
We first define the analog of the logarithm, , for a proper cone
We say that is a for if
is concave, closed, twice continuously differentiable, , and
for
There is a constant such that for all , and all , .
Intuitively, behaves like a logarithm along any ray in the cone
log x K ⊆ Rq
ψ : Rq
→ R generalized logarithm K
ψ dom ψ = int K
∇2
ψ(y) ≺ 0 y ∈ int K
θ > 0 y ≻K 0 s > 0 ψ(sy) = ψ(y) + θ log s
ψ K
6
* We call the constant the degree of
θ ψ
7. Properties of generalized logarithm
A generalized logarithm is only defined up to an additive constant;
if is a generalized logarithm for , then so is , where
The ordinary logarithm is a generalized logarithm for
If , then
(11.40), which implies is -increasing (see 3.6.1). (exercise 11.15)
. (directly proved by differentiating with )
ψ K ψ + a a ∈ R
R+
y ≻K 0
∇ψ(y) ≻K* 0 ψ K
yT
∇ψ(y) = θ ψ(sy) = ψ(y) + θ log s s
7
8. Logarithmic barrier functions for generalized inequalities
Let be generalized logarithms for the cones , with
degrees .
We define the for problem (11.38) as
Convexity of follows since the functions are -increasing, and the
functions are -convex (see 3.6.2)
ψ1, …, ψm K1, …, Km
θ1, …, θm
logarithmic barrier function
ϕ ψi Ki
fi Ki
8
9. The central path
Next, we define the central path for problem (11.38)
We define the central point , for , as the minimizer of ,
subject to , i.e.,
Central points are characterized by the optimality condition
x⋆
(t) t ≥ 0 tf0 + ϕ
Ax = b
9
for some , where is the derivative of at
ν ∈ Rp
Dfi(x) fi x
10. Dual points on central path
As in the scalar case, points on the central path give dual feasible points
for the problem (11.38)
For , we define (11.42), and ,
where is the optimal dual variable in (11.41)
We will show that , together with , are dual feasible for
the original problem (11.38) in the next slide
i = 1,…, m λ⋆
i (t) =
1
t
∇ψi (−fi(x⋆
(t))) ν⋆
(t) = ν/t
ν
λ⋆
1 (t), …, λ⋆
m(t) ν⋆
(t)
10
11. Dual points on central path
The Lagrangian is minimized over by from (11.41)
The dual function evaluated at is therefore equal to
The duality gap with primal feasible point and the dual feasible
point is
x x = x⋆
(t)
g (λ⋆
(t), ν⋆
(t))
x⋆
(t)
(λ⋆
(t), ν⋆
(t)) (1/t)
m
∑
i=1
θi = θ̄/t
11
for and therefore
.
yT
∇ψi(y) = θi y ≻Ki
0
λ⋆
i (y)T
fi(x⋆
(t)) = − θi/t, i = 1,…, m
This is just like a scalar case,
except that the sum of the degrees,
appears in place of , the number of inequalities.
m
12. 11.6.2 Barrier method
We have seen that the key properties of the central path generalized to
problems with generalized inequalities
Computing a point on the central path needs minimizing a twice differentiable
convex function with equality constraints (it can be done with Newton s method)
With the central point we can associate a dual feasible point with
duality gap . In particular, is no more than -suboptimal
So, we can apply the barrier method in 11.3 to the problem (11.38)
The desired accuracy is achieved after , plus one initial
centering step
x⋆
(t) (λ⋆
(t), ν⋆
(t))
θ̄/t x⋆
(t) θ̄/t
ϵ
[
log(θ̄/(t(0)
ϵ))
log μ ]
12
13. Phase 1 and feasibility problems
The phase 1 methods described in 11.4 are readily extended to problems
with generalized inequalities
Let be some given, -positive vectors, for .
To determine feasibility of the equalities and generalized inequalities
we solve the problem
The optimal value determines the feasibility of the equalities and
generalized inequalities, exactly as in 11.4
ei ≻Ki
0 Ki i = 1,…, m
p̄⋆
13
15. A small SOCP with barrier method
Similar to the results for linear and geometric programming in 11.3
Linear convergence
A reasonable choice of is in the range 10-100 with around 30 Newton iterations
μ
15
17. A family of SDPs
Examine the performance of the barrier method as a function of the
problem dimensions with variable and parameter
The number of Newton steps required grows very slowly with the
problem dimensions
x ∈ Rn
A ∈ Sn
17
18. 11.6.4 Complexity analysis via self-concordance
Extend the complexity analysis of the barrier method for problems with
ordinary inequalities ( 11.5), to problems with generalized inequalities
(Preview) The number of outer iterations is given by , (plus
one initial centering step)
We want to bound the number of Newton steps required in each
centering step
We use the complexity theory of Newton s method for self-concordant functions
[
log(θ̄/(t(0)
ϵ))
log μ ]
18
19. Self-concordance assumptions
We make the same assumptions as in 11.5
The function is closed and self-concordant for all
The sublevel sets of (11.38) are bounded
Exactly as in the scalar case, we have (proved in next 3 slides)
Therefore when self-concordance and bounded sublebel set conditions hold,
the number of Newton steps per centering step in no more than
Almost equal to the analysis for scalar case, with one exception: is
instead of the number of inequalities
tf0 + ϕ t ≥ t(0)
θ̄
19
20. Generalized logarithm for dual cone
We will use conjugates to prove the bound (11.48)
Let be a generalized logarithm for the proper cone , with degree
The conjugate of the function is , which is convex, and has
domain .
We define by (11.49)
is concave, and equal to a generalized logarithm with the same parameter (exercise 11.17)
It is called the
From (11.49) we obtain the inequality (11.50), which holds for any
, with equality holding if and only
ψ K θ
−ψ (−ψ)*(v) = sup
u
(vT
u + ψ(u))
−K* = {v|v ≺K* 0}
ψ̄ ψ̄(v) = − (−ψ)*(−v) = inf
u
(vT
u + ψ(u)) domψ̄ = intK* .
ψ̄ θ
dual logarithm
ψ̄(v) + ψ(u) ≤ uT
v
u ≻K 0, v ≻K* 0 ∇ψ(u) = v
20
21. Derivation of the basic bound (1/2)
We denote as , as , as , and as
From (in 11.42) and property (11.43), we conclude that
(10.50) for the pair gives
which becomes, using logarithmic homogeneity of ,
Subtracting the equality (11.51), we get
Summing over :
x⋆
(t) x x⋆
(μt) x+
λ⋆
i (t) λi ν⋆
(t) ν
tλi = ∇ψi(−fi(x))
u = − fi(x+
), v = μtλi
ψ̄i
i
21
(11.52)
22. Derivation of the basic bound (2/2)
We also have, from the definition of the dual function,
Multiplying this inequality by and adding to the inequality (11.52), we
get
We get the desired inequality (11.48) by re-arranging
μt
22