The document discusses modeling stochastic gradient descent (SGD) using stochastic differential equations (SDEs). It outlines SGD, random walks, Wiener processes, and SDEs. It then covers continuous-time SGD and controlled SGD, modeling SGD as an SDE. It provides an example of modeling quadratic loss functions with SGD as an SDE. Finally, it discusses the effects of learning rate and batch size on generalization when modeling SGD as an SDE.
Control of Discrete-Time Piecewise Affine Probabilistic Systems using Reachab...Leo Asselborn
This presentation proposes an algorithmic approach to
synthesize stabilizing control laws for discrete-time piecewise
affine probabilistic (PWAP) systems based on computations of
probabilistic reachable sets. The considered class of systems
contains probabilistic components (with Gaussian distribution)
modeling additive disturbances and state initialization. The
probabilistic reachable state sets contain all states that are
reachable with a given confidence level under the effect of
time-variant control laws. The control synthesis uses principles
of the ellipsoidal calculus, and it considers that the system
parametrization depends on the partition of the state space. The
proposed algorithm uses LMI-constrained semi-definite programming
(SDP) problems to compute stabilizing controllers,
while polytopic input constraints and transitions between regions
of the state space are considered. The formulation of
the SDP is adopted from a previous work in [1] for switched
systems, in which the switching of the continuous dynamics
is triggered by a discrete input variable. Here, as opposed
to [1], the switching occurs autonomously and an algorithmic
procedure is suggested to synthesis a stabilizing controller. An
example for illustration is included.
Robust Control of Uncertain Switched Linear Systems based on Stochastic Reach...Leo Asselborn
This presentation proposes an approach to algorithmically synthesize control strategies for
set-to-set transitions of uncertain discrete-time switched linear systems based on a combination
of tree search and reachable set computations in a stochastic setting. For given Gaussian
distributions of the initial states and disturbances, state sets wich are reachable to a chosen
confidence level under the effect of time-variant hybrid control laws are computed by using
principles of the ellipsoidal calculus. The proposed algorithm iterates over sequences of the
discrete states and LMI-constrained semi-definite programming (SDP) problems to compute
stabilizing controllers, while polytopic input constraints are considered. An example for illustration is included.
Probabilistic Control of Switched Linear Systems with Chance ConstraintsLeo Asselborn
An approach to algorithmically synthesize control
strategies for set-to-set transitions of uncertain discrete-time
switched linear systems based on a combination of tree search
and reachable set computations in a stochastic setting is
proposed in this presentation. The initial state and disturbances
are assumed to be Gaussian distributed, and a time-variant
hybrid control law stabilizes the system towards a goal set.
The algorithmic solution computes sequences of discrete states
via tree search and the continuous controls are obtained
from solving embedded semi-definite programs (SDP). These
program taking polytopic input constraints as well as timevarying
probabilistic state constraints into account. An example
for demonstrating the principles of the solution procedure with
focus on handling the chance constraints is included.
Subgradient Methods for Huge-Scale Optimization Problems - Юрий Нестеров, Cat...Yandex
We consider a new class of huge-scale problems, the problems with sparse subgradients. The most important functions of this type are piecewise linear. For optimization problems with uniform sparsity of corresponding linear operators, we suggest a very efficient implementation of subgradient iterations, the total cost of which depends logarithmically in the dimension. This technique is based on a recursive update of the results of matrix/vector products and the values of symmetric functions. It works well, for example, for matrices with few nonzero diagonals and for max-type functions.
We show that the updating technique can be efficiently coupled with the simplest subgradient methods. Similar results can be obtained for a new non-smooth random variant of a coordinate descent scheme. We also present promising results of preliminary computational experiments.
Probabilistic Control of Uncertain Linear Systems Using Stochastic ReachabilityLeo Asselborn
This presentation proposes an approach to algorithmically synthesize control strategies for
set-to-set transitions of discrete-time uncertain systems based on reachable set computations in
a stochastic setting. For given Gaussian distributions of the initial states and disturbances, state
sets wich are reachable to a chosen confidence level under the effect of time-variant control laws
are computed by using principles of the ellipsoidal calculus. The proposed algorithm iterates over
LMI-constrained semi-definite programming problems to compute probabilistically stabilizing
controllers, while ellipsoidal input constraints are considered. An example for illustration is included.
We present recent result on the numerical analysis of Quasi Monte-Carlo quadrature methods, applied to forward and inverse uncertainty quantification for elliptic and parabolic PDEs. Particular attention will be placed on Higher
-Order QMC, the stable and efficient generation of
interlaced polynomial lattice rules, and the numerical analysis of multilevel QMC Finite Element discretizations with applications to computational uncertainty quantification.
Control of Discrete-Time Piecewise Affine Probabilistic Systems using Reachab...Leo Asselborn
This presentation proposes an algorithmic approach to
synthesize stabilizing control laws for discrete-time piecewise
affine probabilistic (PWAP) systems based on computations of
probabilistic reachable sets. The considered class of systems
contains probabilistic components (with Gaussian distribution)
modeling additive disturbances and state initialization. The
probabilistic reachable state sets contain all states that are
reachable with a given confidence level under the effect of
time-variant control laws. The control synthesis uses principles
of the ellipsoidal calculus, and it considers that the system
parametrization depends on the partition of the state space. The
proposed algorithm uses LMI-constrained semi-definite programming
(SDP) problems to compute stabilizing controllers,
while polytopic input constraints and transitions between regions
of the state space are considered. The formulation of
the SDP is adopted from a previous work in [1] for switched
systems, in which the switching of the continuous dynamics
is triggered by a discrete input variable. Here, as opposed
to [1], the switching occurs autonomously and an algorithmic
procedure is suggested to synthesis a stabilizing controller. An
example for illustration is included.
Robust Control of Uncertain Switched Linear Systems based on Stochastic Reach...Leo Asselborn
This presentation proposes an approach to algorithmically synthesize control strategies for
set-to-set transitions of uncertain discrete-time switched linear systems based on a combination
of tree search and reachable set computations in a stochastic setting. For given Gaussian
distributions of the initial states and disturbances, state sets wich are reachable to a chosen
confidence level under the effect of time-variant hybrid control laws are computed by using
principles of the ellipsoidal calculus. The proposed algorithm iterates over sequences of the
discrete states and LMI-constrained semi-definite programming (SDP) problems to compute
stabilizing controllers, while polytopic input constraints are considered. An example for illustration is included.
Probabilistic Control of Switched Linear Systems with Chance ConstraintsLeo Asselborn
An approach to algorithmically synthesize control
strategies for set-to-set transitions of uncertain discrete-time
switched linear systems based on a combination of tree search
and reachable set computations in a stochastic setting is
proposed in this presentation. The initial state and disturbances
are assumed to be Gaussian distributed, and a time-variant
hybrid control law stabilizes the system towards a goal set.
The algorithmic solution computes sequences of discrete states
via tree search and the continuous controls are obtained
from solving embedded semi-definite programs (SDP). These
program taking polytopic input constraints as well as timevarying
probabilistic state constraints into account. An example
for demonstrating the principles of the solution procedure with
focus on handling the chance constraints is included.
Subgradient Methods for Huge-Scale Optimization Problems - Юрий Нестеров, Cat...Yandex
We consider a new class of huge-scale problems, the problems with sparse subgradients. The most important functions of this type are piecewise linear. For optimization problems with uniform sparsity of corresponding linear operators, we suggest a very efficient implementation of subgradient iterations, the total cost of which depends logarithmically in the dimension. This technique is based on a recursive update of the results of matrix/vector products and the values of symmetric functions. It works well, for example, for matrices with few nonzero diagonals and for max-type functions.
We show that the updating technique can be efficiently coupled with the simplest subgradient methods. Similar results can be obtained for a new non-smooth random variant of a coordinate descent scheme. We also present promising results of preliminary computational experiments.
Probabilistic Control of Uncertain Linear Systems Using Stochastic ReachabilityLeo Asselborn
This presentation proposes an approach to algorithmically synthesize control strategies for
set-to-set transitions of discrete-time uncertain systems based on reachable set computations in
a stochastic setting. For given Gaussian distributions of the initial states and disturbances, state
sets wich are reachable to a chosen confidence level under the effect of time-variant control laws
are computed by using principles of the ellipsoidal calculus. The proposed algorithm iterates over
LMI-constrained semi-definite programming problems to compute probabilistically stabilizing
controllers, while ellipsoidal input constraints are considered. An example for illustration is included.
We present recent result on the numerical analysis of Quasi Monte-Carlo quadrature methods, applied to forward and inverse uncertainty quantification for elliptic and parabolic PDEs. Particular attention will be placed on Higher
-Order QMC, the stable and efficient generation of
interlaced polynomial lattice rules, and the numerical analysis of multilevel QMC Finite Element discretizations with applications to computational uncertainty quantification.
In real-world scenarios, decision making can be a very challenging task even for modern computers. Generalized reinforcement learning (GRL) was developed to facilitate complex decision making in highly dynamical systems through flexible policy generalization mechanisms using kernel-based methods. GRL combines the use of sampling, kernel functions, stochastic process, non-parametric regression and functional clustering.
Representation of signals & Operation on signals
(Time Reversal, Time Shifting , Time Scaling, Amplitude scaling, Signal addition, Signal Multiplication)
Introduction to Neural Networks and Deep Learning from ScratchAhmed BESBES
If you're willing to understand how neural networks work behind the scene and debug the back-propagation algorithm step by step by yourself, this presentation should be a good starting point.
We'll cover elements on:
- the popularity of neural networks and their applications
- the artificial neuron and the analogy with the biological one
- the perceptron
- the architecture of multi-layer perceptrons
- loss functions
- activation functions
- the gradient descent algorithm
At the end, there will be an implementation FROM SCRATCH of a fully functioning neural net.
code: https://github.com/ahmedbesbes/Neural-Network-from-scratch
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/
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Empowering the Data Analytics Ecosystem: A Laser Focus on Value
The data analytics ecosystem thrives when every component functions at its peak, unlocking the true potential of data. Here's a laser focus on key areas for an empowered ecosystem:
1. Democratize Access, Not Data:
Granular Access Controls: Provide users with self-service tools tailored to their specific needs, preventing data overload and misuse.
Data Catalogs: Implement robust data catalogs for easy discovery and understanding of available data sources.
2. Foster Collaboration with Clear Roles:
Data Mesh Architecture: Break down data silos by creating a distributed data ownership model with clear ownership and responsibilities.
Collaborative Workspaces: Utilize interactive platforms where data scientists, analysts, and domain experts can work seamlessly together.
3. Leverage Advanced Analytics Strategically:
AI-powered Automation: Automate repetitive tasks like data cleaning and feature engineering, freeing up data talent for higher-level analysis.
Right-Tool Selection: Strategically choose the most effective advanced analytics techniques (e.g., AI, ML) based on specific business problems.
4. Prioritize Data Quality with Automation:
Automated Data Validation: Implement automated data quality checks to identify and rectify errors at the source, minimizing downstream issues.
Data Lineage Tracking: Track the flow of data throughout the ecosystem, ensuring transparency and facilitating root cause analysis for errors.
5. Cultivate a Data-Driven Mindset:
Metrics-Driven Performance Management: Align KPIs and performance metrics with data-driven insights to ensure actionable decision making.
Data Storytelling Workshops: Equip stakeholders with the skills to translate complex data findings into compelling narratives that drive action.
Benefits of a Precise Ecosystem:
Sharpened Focus: Precise access and clear roles ensure everyone works with the most relevant data, maximizing efficiency.
Actionable Insights: Strategic analytics and automated quality checks lead to more reliable and actionable data insights.
Continuous Improvement: Data-driven performance management fosters a culture of learning and continuous improvement.
Sustainable Growth: Empowered by data, organizations can make informed decisions to drive sustainable growth and innovation.
By focusing on these precise actions, organizations can create an empowered data analytics ecosystem that delivers real value by driving data-driven decisions and maximizing the return on their data investment.
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
As Europe's leading economic powerhouse and the fourth-largest hashtag#economy globally, Germany stands at the forefront of innovation and industrial might. Renowned for its precision engineering and high-tech sectors, Germany's economic structure is heavily supported by a robust service industry, accounting for approximately 68% of its GDP. This economic clout and strategic geopolitical stance position Germany as a focal point in the global cyber threat landscape.
In the face of escalating global tensions, particularly those emanating from geopolitical disputes with nations like hashtag#Russia and hashtag#China, hashtag#Germany has witnessed a significant uptick in targeted cyber operations. Our analysis indicates a marked increase in hashtag#cyberattack sophistication aimed at critical infrastructure and key industrial sectors. These attacks range from ransomware campaigns to hashtag#AdvancedPersistentThreats (hashtag#APTs), threatening national security and business integrity.
🔑 Key findings include:
🔍 Increased frequency and complexity of cyber threats.
🔍 Escalation of state-sponsored and criminally motivated cyber operations.
🔍 Active dark web exchanges of malicious tools and tactics.
Our comprehensive report delves into these challenges, using a blend of open-source and proprietary data collection techniques. By monitoring activity on critical networks and analyzing attack patterns, our team provides a detailed overview of the threats facing German entities.
This report aims to equip stakeholders across public and private sectors with the knowledge to enhance their defensive strategies, reduce exposure to cyber risks, and reinforce Germany's resilience against cyber threats.
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.
First ever open hub for data enthusiasts to collaborate and innovate. A platform to explore, share, and contribute to a vast collection of datasets. Through robust quality control and innovative technologies like blockchain verification, opendatabay ensures the authenticity and reliability of datasets, empowering users to make data-driven decisions with confidence. Leverage cutting-edge AI technologies to enhance the data exploration, analysis, and discovery experience.
From intelligent search and recommendations to automated data productisation and quotation, Opendatabay AI-driven features streamline the data workflow. Finding the data you need shouldn't be a complex. Opendatabay simplifies the data acquisition process with an intuitive interface and robust search tools. Effortlessly explore, discover, and access the data you need, allowing you to focus on extracting valuable insights. Opendatabay breaks new ground with a dedicated, AI-generated, synthetic datasets.
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.
6. Stochastic Gradient Descent (SGD)
• Convergence of SGD
• Assume that the loss function is convex
E[L(x, ¯w) L(x, w⇤
)] o(
1
p
T
) SGD
¯w
Minimum
w⇤
The distance is
guaranteed to be
smallT : step counts
¯w : w after T steps
w⇤
: w at the minimum of L
10. Random Walk
x
t = tt = t
1/2 probability1/2 probability
t = 0
0x
⇢
P(X t = x) = 1
2
P(X t = x) = 1
2
Position of the particle at t is a random variable X t such that
11. Random Walk
t = t
t = 2 t
1/2 1/2
x x0
x x0 2 x2 x
1/4 1/2 1/4
x x0 2 x2 x
1/8 3/8 1/8
t = 3 t
3/8
3 x 3 x
X t
X2 t
X3 t
13. Diffusion
t = 0
t = T
p(x, t = 0) = N(0, 0)
p(x, t = T) = N(0, DT)
Probability density function of Xt : p(x, t)
Di↵usion equation :
@p(x, t)
@t
=
D
2
@2
p(x, t)
@x2
14. Wiener process
A stochastic process W(·) is called a Wiener process if:
(1) W(0) = 0 almost surely,
(2) W(t) W(s) ⇠ N(0, t s) for all t s 0,
(3) W(t1), W(t2) W(t1), ..., W(tn) W(tn 1) are independent random variables.
for all tn > tn 1 > · · · > t2 > t1 > 0
W(t) = Xn t is a Wiener process when t = n t, n ! 1, t ! 0
15. Wiener process
• Random Walk
t = 2 t
x x0 2 x2 x
1/4 1/2 1/4
0
1/8
t = 3 t
x x
x x
0 x x0
1/8 1/8 1/8
1/41/4
X2 t
X3 t X2 t
= X t
16. Outlines
• Stochastic Gradient Descent (SGD)
• Random Walk, Diffusion and Wiener process
• Stochastic Differential Equation (SDE)
• Continuous-time SGD & Controlled SGD
• Effects of SGD on Generalization
18. ⇢ dx(t)
dt = b(x(t)) + B(x(t))dW (t)
dt , where t > 0 and W(t) is a Wiener process
x(0) = x0
Stochastic Differential Equation (SDE)
• Stochastic Differential Equation
x0
x(t)
Trajectory samples of x
Deterministic
part
Stochastic
part
23. Continuous-time SGD & Controlled SGD
• Notation Conventions:
Gradient Descent : xk+1 = xk ⌘rf(xk)
Stochastic Gradient Descent : xk+1 = xk ⌘rf k
(xk)
f : loss function
xk : weights at step k
k : index of training sample at step k (assume batch size is 1)
fi : loss function calculated by batch i, where f(x) = (1/n)⌃n
i=1fi(x)
24. Continuous-time SGD & Controlled SGD
xk+1 xk = ⌘rf k
(xk)
xk+1 xk = ⌘rf(xk) +
p
⌘Vk
Vk =
p
⌘(rf(xk) f k
(xk))
mean of Vk : 0
covariance of Vk : ⌘⌃(xk),
where ⌃(xk) = (1/n)⌃n
i=1(rf(xk) rfi(xk))(rf(xk) rfi(xk))T
Deterministic
part
Stochastic
part minimum
Deterministic
partStochastic
part
27. Continuous-time SGD & Controlled SGD
Xt ⇠ N(x0e 2(1+⌘)t
,
⌘
1 + ⌘
(1 e 4(1+⌘)t
))
t
x
E[Xt] =
⇢
x0, when t = 0
0, when t ! 1
x0
Var[Xt] =
⇢
0, when t = 0
⌘
1+⌘ , when t ! 1
E[Xt⇤ ] =
p
Var[Xt⇤ ]
Fluctuations phase Descent phase r
⌘
1 + ⌘
31. Continuous-time SGD & Controlled SGD
• Optimal control policy
u⇤
t =
⇢
1 if a 0 or t t⇤
, ( t t⇤
is descent phase)
1
1+a(t t⇤) if a > 0 and t > t⇤
, ( t > t⇤
is fluctuations phase)
t
x
Fluctuations
phase
Descent
phase t⇤
a 0 a > 0
f(x) =
1
2
a(x b)2
, assume the covariance of f0
i is ⌘⌃(x)
32. Continuous-time SGD & Controlled SGD
• General Objective Function
f(x) and fi(x) is not necessarily quadratic, and x 2 Rd
assume f(x) ⇡
1
2
dX
i=1
a(i)(x(i) b(i))2
hold locally in x, and
⌃ ⇡ diag{⌃(1), ..., ⌃(d)} where each ⌃(i) is locally constant.
(each dimension is independent)
33. Continuous-time SGD & Controlled SGD
• Controlled SGD Algorithms
At each step k, estimate ak,(i), bk,(i) for
1
2
ak,(i)(xk,(i) bk,(i))2
.
Since rf(i) ⇡ a(i)(x(i) b(i)),
we use linear regression to estimate ak,(i), bk,(i):
1
2
ak,(i)(xk,(i) bk,(i))2
xk,(i)
xk 1,(i)ak,(i) =
gxk,(i) gk,(i)xk,(i)
x2
k,(i) x2
k,(i)
, and bk,(i) = xk,(i)
gk,(i)
ak,(i)
where gk,(i) = rf k
(xk)(i), and gk+1,(i) = k,(i)gk,(i) + (1 k,(i))gk,(i)
Exponentialmoving average
45. Effects of SGD on Generalization
• Theoretical Explanation
d✓ = g(✓)dt +
r
⌘
S
R(✓)dW(t),
where R(✓)R(✓)T
= C(✓) and C(✓) is the covariance of g(✓)
dz = ⇤zdt +
r
⌘
S
p
⇤dW(t)
Change of variables:
z : New variable, where z = V T
(✓ ✓⇤
)
✓⇤
: The parameters at the minimum
V : Orthogonal matrix of the eigen decomposition H = V ⇤V T
H : The Hession of L(✓)