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6.824 2006 Lecture 2: I/O Concurrency
Recall timeline
[draw this time-line]
Time-lines for CPU, disk, network
How can we use the system's resources more efficiently?
What we want is *I/O concurrency*
Ability to overlap I/O wait with other useful work.
In web server case, I/O wait mostly for net transfer to ... | https://ocw.mit.edu/courses/6-824-distributed-computer-systems-engineering-spring-2006/218dc61c38c02d26cb8dcb9193268a4c_lec2_concurrency.pdf |
than I/O concurrency: 2x, not
100x
In general, very hard to program to get good scaling.
Usually easier to buy two separate computers, which we *will* talk
about.
Multiple process problems
Cost of starting a new process (fork()) may be high.
Cite as: Robert Morris, course materials for 6.824 Distributed Compu... | https://ocw.mit.edu/courses/6-824-distributed-computer-systems-engineering-spring-2006/218dc61c38c02d26cb8dcb9193268a4c_lec2_concurrency.pdf |
threads are usually expensive, just like processes
Kernel has to help create each thread
Kernel has to help with each context switch?
So it knows which thread took a fault...
lock/unlock must go through kernel, but bad for them to be slow
Many O/S do not provide kernel-supported threads, not portable
User-le... | https://ocw.mit.edu/courses/6-824-distributed-computer-systems-engineering-spring-2006/218dc61c38c02d26cb8dcb9193268a4c_lec2_concurrency.pdf |
maybe: disk read()
Why are non-blocking system calls hard in general?
Typical system call implementation, inside the kernel:
[sys_read.c]
Can we just return to user program instead of wait_for_disk?
No: how will kernel know where to continue?
ie. should it run userspace code or continue in the kernel syscall... | https://ocw.mit.edu/courses/6-824-distributed-computer-systems-engineering-spring-2006/218dc61c38c02d26cb8dcb9193268a4c_lec2_concurrency.pdf |
8.701
0. Introduction
0.4 Literature
Introduction to Nuclear
and Particle Physics
Markus Klute - MIT
1
Recommended Books
© Cambridge University Press. All rights reserved. This
content is excluded from our Creative Commons license.
For more information, see https://ocw.mit.edu/fairuse.
Introduction to High Energy P... | https://ocw.mit.edu/courses/8-701-introduction-to-nuclear-and-particle-physics-fall-2020/21ca5d10c04a4587015d09877caf84a9_MIT8_701f20_lec0.4.pdf |
Welcome
back
to 8.033!
Image Courtesy of Wikipedia.
Summary of last lecture:
• Space/time unification
• More 4-vectors: U, K
• Doppler effect, aberration
• Proper time, rest length,
timelike, spacelike, null
The Three Types of 4-Vectors:
SPACELIKE
NULL
TIMELIKE
Future
Past
DXth Dx > 0
|Dx| > cDt
DXth Dx = 0
|Dx| ... | https://ocw.mit.edu/courses/8-033-relativity-fall-2006/21e5641958305fcb6e704fe8b59b404d_lecture7_kinem4.pdf |
dilaton examples:
GRB’s
SUPERNOVAE
CLOCKS ON PLANES
GPS
GPS uses a constellation of 24 “NAVSTAR” satellites that
are 11,000 miles above the earth's surface.
How GPS receivers calculate your location:
The positioning process:
1.
Satellite 1 transmits a signal that
contains data on its location in
space and the exac... | https://ocw.mit.edu/courses/8-033-relativity-fall-2006/21e5641958305fcb6e704fe8b59b404d_lecture7_kinem4.pdf |
8.04: Quantum Mechanics
Massachusetts Institute of Technology
Professor Allan Adams
2013 February 7
Lecture 2
Experimental Facts of Life
Assigned Reading:
E&R 16,7, 21,2,3,4,5, 3all NOT 4all!!!
1all, 23,5,6 NOT 2-4!!!
Li.
12,3,4 NOT 1-5!!!
Ga.
3
Sh.
We all know atoms are made of:
• electrons
– Cathode... | https://ocw.mit.edu/courses/8-04-quantum-physics-i-spring-2013/220bfdfab6ca1826045b587d03dc9624_MIT8_04S13_Lec02.pdf |
≈ (3646 angstrom) · 1 −
−1
4
n2
for n ∈ {3, 4, 5, . . .} .
(0.1)
Rydberg and Ritz then found that
λ−1 = R · (n1 − n2 ) for ni ∈ Z, n2 > n1
−2
−2
(0.2)
where R is the Rydberg constant dependent on the particular element but independent of
the emission series. Where did that come from? 1
Experimental result #3... | https://ocw.mit.edu/courses/8-04-quantum-physics-i-spring-2013/220bfdfab6ca1826045b587d03dc9624_MIT8_04S13_Lec02.pdf |
3
Figure 2: Photoelectric effect experimental schematic
Figure 3: Photoelectric dependence of I on V : expectation (top) versus reality (bottom)
Einstein’s interpretation of this is that light comes in packets of definite energy E = hν, the
intensity is proportional to the number of such packets, and the kinetic ene... | https://ocw.mit.edu/courses/8-04-quantum-physics-i-spring-2013/220bfdfab6ca1826045b587d03dc9624_MIT8_04S13_Lec02.pdf |
. But the intensity shows an interference pattern. This implies that amplitudes,
rather than intensities, add.
Let us try to investigate the fringe widths in the interference pattern further. Let us suppose
that light from the two slits start in phase. If they coincide at a single point on the screen,
they remain i... | https://ocw.mit.edu/courses/8-04-quantum-physics-i-spring-2013/220bfdfab6ca1826045b587d03dc9624_MIT8_04S13_Lec02.pdf |
2πfy
λD
)).
fy = Dλn
as expected. Note that maxima correspond to constructive interference, while minima cor
respond to destructive interference. This comes from the fact that amplitudes add, and the
intensity is the square of the amplitude.
The point is that in 8.03, you did the double-slit experiment and saw t... | https://ocw.mit.edu/courses/8-04-quantum-physics-i-spring-2013/220bfdfab6ca1826045b587d03dc9624_MIT8_04S13_Lec02.pdf |
that electrons interfere like waves even with themselves! If they were really
particles, they would have followed only one of two paths: the path from the top slit to the
end, or the path from the bottom slit to the end. We could use a wall to check which one
is happening. Yet this produces the exact same conundrum ... | https://ocw.mit.edu/courses/8-04-quantum-physics-i-spring-2013/220bfdfab6ca1826045b587d03dc9624_MIT8_04S13_Lec02.pdf |
found the phenomenon of Bragg scattering. The path length difference between one
layer of the crystal and the next is
for a square crystal of length f. Constructive interference occurs when
Δf = 2f sin(θ)
This means
Δf = λn.
λ−1 =
n
.
2f sin(θ)
Davisson and Germer also observed that
√
n
2f sin(θ)
≈
2mqeV0 ... | https://ocw.mit.edu/courses/8-04-quantum-physics-i-spring-2013/220bfdfab6ca1826045b587d03dc9624_MIT8_04S13_Lec02.pdf |
3.032 Mechanical Behavior of Materials
Fall 2007
Buckling: Long, thin beam under end-loaded compression
P [N]
δ
max [m]
r
θ = dδ/dx
Lecture 3 (09.10.07)
3.032 Mechanical Behavior of Materials
Fall 2007
Beam bending: End-loaded cantilever
Images removed due to copyright restrictions.
Please see: Silva, Emilio C. C. M.,... | https://ocw.mit.edu/courses/3-032-mechanical-behavior-of-materials-fall-2007/222911da292c25b77b09b47da533315c_lec3.pdf |
3.46 PHOTONIC MATERIALS AND DEVICES
Lecture 5: Waveguide Design—Optical Fiber and Planar Waveguides
Lecture
Fiber Optics
Optical fiber ≡ core + cladding
guided if n2 > n1
power loss to cladding if < n1
n
⎛
1 ⎟⎟⎟⎟⎟
θ = sin−1⎜⎜⎜
⎜n2⎝
n2
⎠
c
⎞
JK K
each mode travels with β, v , U (x,y), P, k
single mode (small c... | https://ocw.mit.edu/courses/3-46-photonic-materials-and-devices-spring-2006/2229d0701ea240ed993e2a499432519b_3_46l5_waveguide.pdf |
⎢
⎜⎜
⎝
⎢
⎣
1
⎞2 ⎤ 2
n
⎥
1 ⎟⎟⎟⎟
⎥
n2
⎠ ⎥
⎦
2
1
= (n2 − n1 )
2 2
θ = sin−1(NA)
a
1
NA = (n2 − n2 )2 ≈ n2(2Δ)
1
2
1
2
3.46 Photonic Materials and Devices
Prof. Lionel C. Kimerling
Lecture 5: Waveguide Design—Optical Fiber and Planar Waveguides
Page 1 of 5
Lecture
θ = acceptance ∠ for fiber
a
≡ exit angle fo... | https://ocw.mit.edu/courses/3-46-photonic-materials-and-devices-spring-2006/2229d0701ea240ed993e2a499432519b_3_46l5_waveguide.pdf |
n1 ,
Condition for guiding
n k < β < n 0
1 0
2
kT = rate of change of u(r) in core
γ = rate of U(r) in cladding
kT
2 = n2 k0
2
2 − β2
2
2
2 2
γ = β −n k1 0
rate of decay high ⇒ low penetration
2
kT + γ
2
2
(n2 −n )k = (NA)2
2
0
2
1
2
k0
↑ γ ↓⇒ penetration into cladding
,
k
T
kT > NA
⋅
0 ⇒ γ imaginary, wave... | https://ocw.mit.edu/courses/3-46-photonic-materials-and-devices-spring-2006/2229d0701ea240ed993e2a499432519b_3_46l5_waveguide.pdf |
2 fiber
n2 = 1.452, Δ = 0.01, NA = 0.205
λ = 0.85 μm (GaAs)
a (core) = 25 μm
⇒ V = 37.9, M = 585
0
remove cladding ⇒ n1 = 1, NA = 1
⇒ V = 184.8,
M > 13,800
3.46 Photonic Materials and Devices
Prof. Lionel C. Kimerling
Lecture 5: Waveguide Design—Optical Fiber and Planar Waveguides
Page 3 of 5
Lecture
... | https://ocw.mit.edu/courses/3-46-photonic-materials-and-devices-spring-2006/2229d0701ea240ed993e2a499432519b_3_46l5_waveguide.pdf |
center
O
→shortest travel, slowest velocity
⇒ power low profile
n1
n2
n
( )
2
n r
n=
2
2
⎡
⎢
− ⎜⎜
1 2
⎢
⎢⎣
p
⎛ ⎞
r
⎜ ⎟⎟⎟⎟
⎝ ⎠
a
⎤
⎥Δ⎥
⎥⎦
n = n2 @ r = 0
= n1 @ r = a
r ≤ a
p = 1
p = 2
p → ∞
n2(r) linear
n2(r) quadratic
n2(r) step function
Δ =
2
2
n2 − n1
22n2
3.46 Photonic Materials and Devi... | https://ocw.mit.edu/courses/3-46-photonic-materials-and-devices-spring-2006/2229d0701ea240ed993e2a499432519b_3_46l5_waveguide.pdf |
Maslab Software
Engineering
January 5th, 2005
Yuran Lu
Agenda
Getting Started
On the Server
Using the Documentation
Design Sequence
Tools
The Maslab API
Design Principles
Threading in Java
On the Server
Put these lines in your .environment:
add 6.186
add -f java_v1.5.0
setenv JAVA_HOME /... | https://ocw.mit.edu/courses/6-186-mobile-autonomous-systems-laboratory-january-iap-2005/222e989bfe21b02a813a2c10e4a646d7_software.pdf |
Modular Design
Provides abstraction
Gives up fine-control abilities, but makes code much
more manageable
The Design Process
Top-down vs. Bottom-up
Write out specifications for each module
Write code for modules
Test each module separately as it is being written
Test overall system for function... | https://ocw.mit.edu/courses/6-186-mobile-autonomous-systems-laboratory-january-iap-2005/222e989bfe21b02a813a2c10e4a646d7_software.pdf |
Thread, Runnable, wait(), notify(), sleep(),
yield()
Must take care to avoid deadlock
Synchronization in Threading
Allows blocks of code to be mutually
exclusive
Writing to the same object from two
threads at the same time will cause your
program to break | https://ocw.mit.edu/courses/6-186-mobile-autonomous-systems-laboratory-january-iap-2005/222e989bfe21b02a813a2c10e4a646d7_software.pdf |
Matrices
We have already defined what we mean by a matrix. In this section,
we introduce algebraic operations into the set of matrices.
Definition. If A and B are two matrices of the same size, say
k by n, we define A t B to be the k by n matrix obtained by adding
the corresponding entries of A and B, m d we defin... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/224273817cd62ecaecef64bf6c607878_MIT18_024s11_ChB1notes.pdf |
the j-th
column of B .
Schematically,
This definition seems rather strange, but it is in fact e,xtremely
useful. Motivation will come later! One important justification for this
definition is the fact that this product operation satisfies some of the familar
"laws of algebra" :
Theorem '1. Matrix-multiplication ... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/224273817cd62ecaecef64bf6c607878_MIT18_024s11_ChB1notes.pdf |
= I a b
n
s=l is sj
n
+ I
a c
s=l is sj'
The other distributivity formula and the homogeneity formula are proved similarly.
We leave them as exercises.
Now let us verify associativity.
-
If A is k by n and B is n by p, then
.
A
B is k by p. The product (AeB) C is thus defined
provided C has size p by q. The... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/224273817cd62ecaecef64bf6c607878_MIT18_024s11_ChB1notes.pdf |
.. + 0 + TiQ l j + O+ . . a + 0= Q5.
1s
c =
ij
= 1. We conclude that
Ar, entirely similar proof shows that B * I = B if B has rn columns.
m
Remark.
I f A
B
i s d e f i n e d , t h e n B
A need n o t b e
d e f i n e d . And e v e n i f it i s d e f i n e d , t h e two p r c d u c t s need n o t
b e e q u a l ... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/224273817cd62ecaecef64bf6c607878_MIT18_024s11_ChB1notes.pdf |
is called
k and j = l , - - = , n ,
.
a system of k linear equations 9 n unknowns:
A solution of this system is a vector X = (xl,...,x ) that satisfies each
equation. The solution -set of the system consists of all such vectors; it is
a subset of Vn .
n
We wish to determine whether this systemhas a solution, and ... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/224273817cd62ecaecef64bf6c607878_MIT18_024s11_ChB1notes.pdf |
system
-
2 x + y + z = l
This system has a solution; in fact, it has mora than one solution. In
solving this sytem, we can ignore the third equation, since it is the sum of the
first two. Then we can assign a value to y arbitrarily, say y = t, and solve
-
* -
the first two equations for x and z. We obtain the resul... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/224273817cd62ecaecef64bf6c607878_MIT18_024s11_ChB1notes.pdf |
.setof the system A'X = C.
Proof. Exchanging rows i and j of both matrices has the effect of
simply exchanging equations i and j of the system. Replacing row i by itself
plus c times row j has the effect of replacing the ith equation by itself
plus c times the jth equation. And multiplying rcw i by a non-zero scala... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/224273817cd62ecaecef64bf6c607878_MIT18_024s11_ChB1notes.pdf |
3. Let A be a matrix of size k by n. Let r be the rank
-
of A. Then the solution space of the system of equations A 0 X = 2 is
a subspace of Vn of dimension n - r.
Proof. The preceding theorem tells us that we can apply elementary
operations to both the matrices A and 0 without changing the solution set.
-
Applying ... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/224273817cd62ecaecef64bf6c607878_MIT18_024s11_ChB1notes.pdf |
merely of the single
term 9! )
-
k!t us pause to consider an example.
EJ-ample3 . Let A be the 4 by 5 matrix given on p.A20. The
equation A'X = 0 represents a system of 4 equations in 5 unknowns. Nc~w A
-
-
reduces by row operations to the reduced echelon matrix
Here the pivots appear in columns 1,2 and 4; thus J i... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/224273817cd62ecaecef64bf6c607878_MIT18_024s11_ChB1notes.pdf |
coefficents %
of these vectors, then X = 0 if and only if each % (for k in K)
-
equals 0. This is easy. Consider the first expression for X tkat we wrote down,
where each component of X is a linear combination of the unknowns rc-
The kth component of X is simply 5 . It follows that the equation X = 2
implies in part... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/224273817cd62ecaecef64bf6c607878_MIT18_024s11_ChB1notes.pdf |
X is a column matrix such that A'X = C, then A.(X - ? ) = 2 , and
ccnversely. The solution space of the system A'X = 0 is a subspace of
of dimension m = n - r; let All...,A be a basis for it. Then X is a solution
m
of the system A'X = C if and only if X - P is a linear combination of the
'n
-
vectors Air that is, ... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/224273817cd62ecaecef64bf6c607878_MIT18_024s11_ChB1notes.pdf |
of C 1 in row k. If c' is not zero, there are no
values of xl,...,x satisfying this equation, so the system has no solution.
k
k
n
#
Let us choose C * to be a k by 1 mstrix whose last entry is non-zero.
Then apply the same elementary operations as before, in reverse order, to
both B and C*. These operations tra... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/224273817cd62ecaecef64bf6c607878_MIT18_024s11_ChB1notes.pdf |
,
2
for j in J, appears in only one equation of the system, we can solve for each
x in terms of the numbers c; a d the unknowns
j
values arbitrarily to the \ and thus obtain a particular solution of the
. We can now assign
system. The theorem follows.
The procedure just described actually does much more than wa... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/224273817cd62ecaecef64bf6c607878_MIT18_024s11_ChB1notes.pdf |
n, show that
therearevalues of C such that the system A'X = C has no solution.
/---
2, Ccnsider the matrix A of p. A23. (a) Find the general solution
of the system AbX = 2. (b) Does the system A - X = C have a solution for
arbitrary C?
3 . Repeat Exercise 2 for the matrices C, D l and E of p. A23.
4. &t B be the ... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/224273817cd62ecaecef64bf6c607878_MIT18_024s11_ChB1notes.pdf |
basis Al,...,A ,B1,...,B for all of Vn . Show the vectors
r
A.B1 , ..., A.3r span R; this follows from the fact that A.Ai = 0 for
all i. Show these vectors are independent.]
m
-
(c) Cclnclude that if r € k, there are vectors C in Vk such
that the system A - X = C has no solution; while if r = kt this system
has a... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/224273817cd62ecaecef64bf6c607878_MIT18_024s11_ChB1notes.pdf |
Vn is as the solution set of a
system of linear equations
where the rows of A are independent. I f A has size k by n, then
the plane in question has dimension n - k. The equation is called a
caretesian form for the equation of a plane. (If the rows of A were not
independent, then the solutionsetwould be either emp... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/224273817cd62ecaecef64bf6c607878_MIT18_024s11_ChB1notes.pdf |
its orthogonal complement has dimension n - k. Fwthermore, W is the
orthogonal complement of W ; that is, (W ) = W.
I
L I
Proof. Ttiat wL has dimension n - k is an imnediate consequence of
Theorem 3 j for W is the row space of a k by n matrix A with independent rows
Ai , whence w L is the solution space of the sys... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/224273817cd62ecaecef64bf6c607878_MIT18_024s11_ChB1notes.pdf |
using
the Gass-Jordan algorithm; and then one writes down the equation B.X = B O P . - .
*
m
k
We now turn to the special case of V3, whose model is the familiar
3-dimensional space in which we live. In this space, we have only lines
(1-planes) and planes (2-planes) to deal with. A P ~we can use either the
parame... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/224273817cd62ecaecef64bf6c607878_MIT18_024s11_ChB1notes.pdf |
+ a2(x2 - p2) + a3(x3 - P3 ) = 0 .
We call this the equation of the plane throuqh
--
p = (pll P2' p3) with
normal vector N = (a a2, a3 1.
WE have thus proved the first half of the following theorem:
Theorem% If M is a 2-plane in V3, then M has a cartesian
equation of the form
a x + a x + a x - b ,
1 1 2 2 3 3 -
w... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/224273817cd62ecaecef64bf6c607878_MIT18_024s11_ChB1notes.pdf |
normal vectors are independent.
Proof. Take a cartesian equation for each plane; collectively,
they form a system A - X = C of three equations in three unknowns.
The rows of A are the normal vectors. The solution space of the system
(which consists of the points common to all three planes) consists of a
a single p... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/224273817cd62ecaecef64bf6c607878_MIT18_024s11_ChB1notes.pdf |
if N-P # b. Thus the intersection of L and M is either all of L, or
it is empty.
On the other hand, if L is not parallel to M, then N - A # 0.
In this case the equation can be solved uniquely for t. Thus the intersection
of L and M consists of a single point. a
r Ex-ample 5. Ccnsider the plane M = M(P;A,B) in V3 ,... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/224273817cd62ecaecef64bf6c607878_MIT18_024s11_ChB1notes.pdf |
a parametric equation for the line of intersection of the
planes of Exercise 3.
6. Write a cartesian equation for the plane through P = (-1,0,2)
and Q = (3,1,5) that is parallel to the line through R = (1,1,1)with
direction vector A = (1,3,4).
7. Write cartesian equations for the plane M(P;A,B) in V4,
where P = (... | https://ocw.mit.edu/courses/18-024-multivariable-calculus-with-theory-spring-2011/224273817cd62ecaecef64bf6c607878_MIT18_024s11_ChB1notes.pdf |
§ 1. Information measures: entropy and divergence
Review: Random variables
• Two methods to describe a random variable (R.V.) X:
1. a function X Ω
∶
→
X
2. a distribution
PX
on
from the probability space Ω,
some measurable space (X ,
F
)
.
( F
)
, P to a target space
X
.
• Convention: capital letter – RV (e.g. X); smal... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/2243edffb30f57181ed97dcb77691580_MIT6_441S16_chapter_1.pdf |
., the entropy of H(PX
∣Y =y) averaged over PY .
Note:
8
1
(
PX Y X Y
∣ )
∣
]
,
• Q: Why such definition, why log, why entropy?
Name comes from thermodynamics. Definition is justified by theorems in this course (e.g.
operationally by compression), but also by a number of experiments. For example, we can
measure time it t... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/2243edffb30f57181ed97dcb77691580_MIT6_441S16_chapter_1.pdf |
�� (p)i
H(X) =
p ⋅ pi log
∞
∑
i=0
= log
1
p
+ p ⋅ log
ppi(i log
1
p
+ log
)
1
p
1
p ⋅ pi
1
p
⋅
=
∞
∑
i=0
1 − p
p2
=
h(p)
p
Example (Infinite entropy): Can H(X) = +∞? Yes, P[X = k] =
c
k ln2 k , k = 2, 3, ⋯
9
011/2 Review: Convexity
• Convex
∈ [
all α
set: A subset S of some vector space is convex if x, y S
0, 1
1 α.)
≜ ... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/2243edffb30f57181ed97dcb77691580_MIT6_441S16_chapter_1.pdf |
f EX Ef X
⇒ f (EX
)
conv
=
unless X is a constant (X E
ex
(
Ef (X)
f (EX)
)
)
<
X
Ef (X
a.s.)
Famous puzzle: A man says, ”I am the average height and average weight of the
population. Thus, I am an average man.” However, he is still considered to be a little
overweight. Why?
Answer: The weight is roughly proportional t... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/2243edffb30f57181ed97dcb77691580_MIT6_441S16_chapter_1.pdf |
X1, . . . , Xn mutually independent
(1.1)
(1.2)
Proof. 1. Expectation of non-negative function
2. Jensen’s inequality
3. H only depends on the values of PX , not locations:
H(
) = H(
)
4. Later (Lecture 2)
1
= E[ log
5. E log
PXY (X,Y )
))
(
6. Intuition: X
, f X contains the same
is 1-1. Thus by 3 and 5:
PX (X)⋅PY ∣X
... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/2243edffb30f57181ed97dcb77691580_MIT6_441S16_chapter_1.pdf |
[
/
1 2, P X b
w for su
ceed by asking “X b?”. If not, ask “X c?”, after which we will kno
/ ×
+
/
+ / ×
/
1 2 1 4 2 1 8 3 1 8 3
of
minimal
)
= ] =
=
questions
1 4, and P X c
is
that the
=
average
+
)
1
probab
The
)
X.
(
] = /
bits
num
ber
∈ {
H
×
+
=
=
(
(
/
[
[
1.1.1 Entropy: axiomatic characterization
One might wond... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/2243edffb30f57181ed97dcb77691580_MIT6_441S16_chapter_1.pdf |
H(X, Y ) = H(X)+H(
∑
)
=
.
pi and m
Equivalently, Hmn r11, . . . , rmn Hm p1, . . . , pm
∑
i=1 rij
= qj.
) ≤
(
(
) +
Y ) if X ⊥⊥ Y . Equivalently, H p1q1, . . . , pmqn Hm(p1, . . . , pm
mn(
) ≤
)+
)
Hn q1, . . . , qn .
(
then Hm(p1, . . . , p
[CT06, p. 53] and the reference therein.
i=1 pi log
m) = ∑m
1
pi
is the only ... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/2243edffb30f57181ed97dcb77691580_MIT6_441S16_chapter_1.pdf |
body (that is, every such process needs to be helped by expending some amount of
external work).
Notice that there is something annoying about the second law as compared to the first law. In
the first law there is a quantity that is conserved, and this is somehow logically easy to accept. The
second law seems a bit harde... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/2243edffb30f57181ed97dcb77691580_MIT6_441S16_chapter_1.pdf |
1
pj
,
where k is the Boltzmann constant, we assume that each particle can only be in one of (cid:96) molecular
states (e.g. spin up/down, or if we quantize the phase volume into (cid:96) subcubes) and pj is the fraction
of particles in j-th molecular state.
1.1.3* Entropy: submodularity
Recall that [
of S. A set funct... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/2243edffb30f57181ed97dcb77691580_MIT6_441S16_chapter_1.pdf |
also monotone:
⊂
T1 T2
(cid:212)⇒
) ≤
H XT1 H XT2
(
(
)
.
n, let us denote by Γn the
So fixing
submodular set-functions
[ ]
all non-empty subsets of [n], Γn is a closed
via an obvious enumeration of
on n . Note that
n 1. Similarly, let us denote by Γn the set of all set-functions corresponding to
2
−
∗
convex cone in R
... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/2243edffb30f57181ed97dcb77691580_MIT6_441S16_chapter_1.pdf |
). Let X n be discrete n-dimensional RV and denote Hk(X n
1
n
)
(
k
¯
) =
¯
H
H(XT ) – the average entropy of a k-subset of coordinates. Then k
k is decreasing in k:
n]
)
k
∑
⊂(
T
[
¯Hn ≤ ⋯ ≤
1
n
1 ¯
Hk
k
⋯ ≤
¯
H1 .
¯
Furthermore, the sequence Hk is increasing and concave
in the sense of decreasing slope:
Proof. Denote... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/2243edffb30f57181ed97dcb77691580_MIT6_441S16_chapter_1.pdf |
11
).
14
Note: Han’s inequality holds for any submodular set-function.
Example: Another submodular set-function is
Han’s
inequality for this one reads
S
↦ I(XS; XSc) .
0 =
1
n
In ≤ ⋯ ≤
1
k
Ik⋯ ≤ I1 ,
where Ik = 1
n
)
(
k
X n.
∑S∶∣S∣=k I(XS; XSc) – gauges the amount of k-subset coupling in the random vector
1.2 Diverge... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/2243edffb30f57181ed97dcb77691580_MIT6_441S16_chapter_1.pdf |
⋅
log
0
0
=
0
(2)
∃a ∶ Q(a) = 0, P (a) > 0 ⇒ D(P ∥Q) = ∞
15
• A = R
k, P and Q have densities fP and fQ
D(P
Q) =
∥
⎧
⎪⎪
⎨
⎪⎪
⎩
Rk log P (xk
f
)
∫
Q xk)
f (
+∞
P (xk)dx
f
k
, Leb{fP > 0, fQ = 0
} =
0
, otherwise
•
A — measurable space:
D(P ∥Q
) =
⎧
⎪⎪
⎨
⎪
⎪
⎩
dP
dQ log dP
dQ = EP log dP
dQ
Q
E
+∞
, P
≪ Q
, otherwise
(A... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/2243edffb30f57181ed97dcb77691580_MIT6_441S16_chapter_1.pdf |
nite values) D(P ∥Q) can be ∞
are
consistent since D P Q supΠ D PΠ QΠ where Π is a finite partition of the underlying
space
also when P ≪ Q, but the two case of D P ∥
(
(proof: later)
( ∥ )
= +∞
)
,
A
Q
=
∥
)
(
s
P Q) ≠ D(Q∥P )
• (Asymmetry)
=
/
1 2, Q H
be absolutely sure; Upon observing HHT, know for sure it is P .
D ... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/2243edffb30f57181ed97dcb77691580_MIT6_441S16_chapter_1.pdf |
allis etc. Inequalities between
various f -divergences such as (1.14) was once an active field of research. It was made largely
irrelevant by a work of Harremo¨es and Vajda [HV11] giving a simple method for obtaining
best possible inequalities between any two f -divergences.
Theorem 1.4 (H v.s. D). If distribution P is ... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/2243edffb30f57181ed97dcb77691580_MIT6_441S16_chapter_1.pdf |
2
σ2
0
+
σ2
1
2
σ
0
]
− 1 log
e
Example (Vector Gaussian):
A =
k
C
D(Nc(m1, Σ1
)∥Nc(m0, Σ0)) = log det
1 + (
m1 − m0)H Σ−1
0 (m1 − m0
Σ0
tr Σ 1
−
0 Σ
+ (
− log det Σ
− )
I log e
1
(1.17)
(1.18)
)
log e
17
d (p ||q )p1qd (p ||q )q1p−log q−log q− (assume det Σ0 ≠ 0).
Note: The definition of D(P ∥Q
measures), in which case... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/2243edffb30f57181ed97dcb77691580_MIT6_441S16_chapter_1.pdf |
negative, take values of
(
)
±∞
or even be
undefined2.
Nevertheless, differential entropy shares
many properties with the usual entropy:
Theorem 1.5 (Properties of differential entropy). Assume that all differential entropies appearing
below exists and are finite (in particular all RVs have pdfs and conditional pdfs). Then ... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/2243edffb30f57181ed97dcb77691580_MIT6_441S16_chapter_1.pdf |
and hence measurable.
n
−(−1) n.
c
en2
Leb{AS} ≤ Leb{KS}
18
Proof. Let X n be uniformly distributed on K. Then h(X n) =
a1
an where
× ⋯
×
Then, we have by 1. in Theorem 1.5
log a
= (
i h X
∣
i X
i
1
− )
.
log Leb{K}
. Let A be rectangle
On the other hand, by the chain rule
h(XS) ≤ log Leb{KS
}
n
h(XS) = ∑
i 1
=
∑
∈S
i... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/2243edffb30f57181ed97dcb77691580_MIT6_441S16_chapter_1.pdf |
MIT OpenCourseWare
https://ocw.mit.edu
6.441 Information Theory
Spring 2016
For information about citing these materials or our Terms of Use, visit: https://ocw.mit.edu/terms. | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/2243edffb30f57181ed97dcb77691580_MIT6_441S16_chapter_1.pdf |
15.093J Optimization Methods
Lecture 4: The Simplex Method II
Slide 1
Slide 2
Slide 3
Slide 4
1 Outline
• Revised Simplex method
• The full tableau implementation
• Finding an initial BFS
• The complete algorithm
• The column geometry
• Computational efficiency
2 Revised Simplex
Initial data: A, b, c
1. St... | https://ocw.mit.edu/courses/15-093j-optimization-methods-fall-2009/227bd50822ccfd66c653bccf0a3e11fe_MIT15_093J_F09_lec04.pdf |
1
−1
0
1
1 0 0
1 −1 0 0 −1
1
1 1 0
1
1 0 1
1 0 −1 0
1 0
0 0
1 0
−1 1
0 0 −1 1
2.2 Practical issues
• Numerical Stability
B−1 needs to be computed from scratch once in a while, as errors accu
mulate
• Sparsity
B−1 is represented in terms of sparse triangular matrices
3 Full tableau implementa... | https://ocw.mit.edu/courses/15-093j-optimization-methods-fall-2009/227bd50822ccfd66c653bccf0a3e11fe_MIT15_093J_F09_lec04.pdf |
1.6
0.4 −0.6
0
1
0 −0.6
0.4
0
0.4 −0.6
0.4
0.4
136
4
4
4
x3 =
x1 =
x2 =
0
0
1
0
0
0
0
1
x 3
.
= (
Slide 14
Slide 15
= (
)
= (4,4,4)
.
.
.
= (
)
x 1
.
= (
)
x 2
4 Comparison of implementations
Slide 16
Full tableau
Revised simplex
Memory
Worst-case time
Best-case time
O(mn)
O(mn)... | https://ocw.mit.edu/courses/15-093j-optimization-methods-fall-2009/227bd50822ccfd66c653bccf0a3e11fe_MIT15_093J_F09_lec04.pdf |
s.t. Ax + y = b
x, y ≥ 0
3. If cost > 0 ⇒ LOP infeasible; stop.
4. If cost = 0 and no artificial variable is in the basis, then a BFS was found.
5. Else, all yi
∗ = 0, but some are still in the basis. Say we have AB(1), . . . , AB(k)
in basis k < m. There are m − k additional columns of A to form a basis.
6. Drive ... | https://ocw.mit.edu/courses/15-093j-optimization-methods-fall-2009/227bd50822ccfd66c653bccf0a3e11fe_MIT15_093J_F09_lec04.pdf |
II.
2. Compute the reduced costs of all variables for this initial basis, using the
cost coefficients of the original problem.
3. Apply the simplex method to the original problem.
6.1 Possible outcomes
1. Infeasible: Detected at Phase I.
2. A has linearly dependent rows: Detected at Phase I, eliminate redundant
rows.
... | https://ocw.mit.edu/courses/15-093j-optimization-methods-fall-2009/227bd50822ccfd66c653bccf0a3e11fe_MIT15_093J_F09_lec04.pdf |
1
ǫxi−1 ≤ xi ≤ 1 − ǫxi−1,
i = 2, . . . , n
x3
x
2
x2
(a)
x1
x1
(b)
Theorem
• The feasible set has 2n vertices
• The vertices can be ordered so that each one is adjacent to and has lower
cost than the previous one.
• There exists a pivoting rule under which the simplex method requires
2n − 1 changes of basis... | https://ocw.mit.edu/courses/15-093j-optimization-methods-fall-2009/227bd50822ccfd66c653bccf0a3e11fe_MIT15_093J_F09_lec04.pdf |
- 2.12 Lecture Notes -
H. Harry Asada
Ford Professor of Mechanical Engineering
Fall 2005
Introduction to Robotics, H. Harry Asada
1
Chapter 1
Introduction
Many definitions have been suggested for what we call a robot. The word may conjure up various
level... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/228dbf33cc6dc0f84c0d87e00b31a410_chapter1.pdf |
2
issue in manufacturing innovation for a few decades, and numerical control has played a central
role in increasing system flexibility. Contemporary industrial robots are programmable machines
that can perform different operations by simply modifying stored data, a feature that has evolved
from the application of n... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/228dbf33cc6dc0f84c0d87e00b31a410_chapter1.pdf |
is that of a
numerically controlled manipulator, where the human operator and the master manipulator in the
figure are replaced by a numerical controller.
Figure removed for copyright reasons.
See Figure 1-4 in Asada and Slotine, 1986.
Figure 1-3 White body assembly lines using spot welding robots
1.2 Creation of... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/228dbf33cc6dc0f84c0d87e00b31a410_chapter1.pdf |
configuration of the manipulator arm. Coriolis and centrifugal effects
are prominent when the manipulator arm moves at high speeds. The kinematic and dynamic
complexities create unique control problems that are not adequately handled by standard linear
control techniques, and thus make effective control system desig... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/228dbf33cc6dc0f84c0d87e00b31a410_chapter1.pdf |
or modifications of control actions are
provided when the resultant motion is not adequate, or when unexpected events occur during the
operation. The human operator is, therefore, an essential part of the control loop. When the
operator is eliminated from the control system, all the planning and control commands mus... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/228dbf33cc6dc0f84c0d87e00b31a410_chapter1.pdf |
and motion
commands.
o
Figure 1-7 Remote-center compliance hand
A detailed understanding of the underlying principles and "know-how" involved in the task must
be developed in order to use industrial robots effectively, while there is no such need for making
control strategies explicit when the assembly and grindi... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/228dbf33cc6dc0f84c0d87e00b31a410_chapter1.pdf |
order to adapt itself to diverse terrain
conditions. See Figure 1-10.
Photo removed for copyright reasons.
Figure 1-8 Automatically guided vehicle for meal delivery in hospitals
Photo removed for copyright reasons.
Figure 1-9 Honda’s P3 humanoid robot
Navigation is another critical functionality needed for mobile... | https://ocw.mit.edu/courses/2-12-introduction-to-robotics-fall-2005/228dbf33cc6dc0f84c0d87e00b31a410_chapter1.pdf |
Deep Learning/Double Descent
Gilbert Strang
MIT
October, 2019
1/32
Number of Weights
2/32
N = 40
N = 4000
10
8
6
4
2
0
-2
-4
-6
-3
-2
-1
0
x
1
2
3
3/32
Fit training data by a Learning function F
We are given training data : Inputs v, outputs w
Example Each v is an image of a number
w = 0, 1, . . . , 9
The vector v d... | https://ocw.mit.edu/courses/18-085-computational-science-and-engineering-i-summer-2020/22a453f2f41f9c34ad274b7d7da9a0aa_MIT18_085Summer20_lec_GS.pdf |
LU (Akvk−1 + bk)
Weights for layer k Ak = matrix and bk = offset vector
v0 = training data / v1, . . . , vℓ−1 hidden layers / vℓ = output
5/32
Deep Neural Networks
1 Key operation
2 Key rule
3 Key algorithm
4 Key subroutine
5 Key nonlinearity ReLU (y) = max (y, 0) = ramp function
Composition F = F3(F2(F1(x, v0)))
Chain... | https://ocw.mit.edu/courses/18-085-computational-science-and-engineering-i-summer-2020/22a453f2f41f9c34ad274b7d7da9a0aa_MIT18_085Summer20_lec_GS.pdf |
(cid:12)
(cid:12)
Classification problem : true = 1 or
(cid:17)
(cid:16)
1
−
Regression problem : true = vector
Gradient descent xk+1 = arg min
||
Stochastic descent xk+1 = arg min
xk
sk
L(xk)
−
xk
∇
sk
ℓ
∇
−
||
xk, vk
0
(cid:12)
(cid:12)
(cid:12)
(cid:12)
(cid:0)
(cid:1)(cid:12)
(cid:12)
(cid:12)
(cid:12)
7/32
Key com... | https://ocw.mit.edu/courses/18-085-computational-science-and-engineering-i-summer-2020/22a453f2f41f9c34ad274b7d7da9a0aa_MIT18_085Summer20_lec_GS.pdf |
. . . generalize to unseen test data ?
(Early stopping / Do not overfit the data)
8/32
Key Questions
1. Optimization of the weights x = Ak and bk
2. Convergence rate of descent and accelerated descent
(when xk+1 depends on xk and xk−1 : momentum added)
3. Do the weights A1, b1 . . . generalize to unseen test data ?
(Ea... | https://ocw.mit.edu/courses/18-085-computational-science-and-engineering-i-summer-2020/22a453f2f41f9c34ad274b7d7da9a0aa_MIT18_085Summer20_lec_GS.pdf |
1 hidden layer with N neurons
9/32
1. Stochastic gradient descent optimizes weights Ak, bk
2. Backpropagation in the computational graph computes
derivatives with respect to weights x = A1, b1, . . . , Aℓ, bℓ
3. The learning function F (x, v0) = . . . F3(F2(F1(x, v)))
F1(v0) = max (A1v0 + b1, 0) = ReLU
affine map
◦
F (v... | https://ocw.mit.edu/courses/18-085-computational-science-and-engineering-i-summer-2020/22a453f2f41f9c34ad274b7d7da9a0aa_MIT18_085Summer20_lec_GS.pdf |
v1 has N components / N ReLU’s
The number of flat regions in Rm bounded by the N hyperplanes
r(N, m) =
m
i=0 (cid:18)
X
N
i
=
(cid:19)
(cid:18)
N
0
N
1
+
(cid:19)
(cid:18)
+
· · ·
+
(cid:19)
(cid:18)
N
m
(cid:19)
N = 3 folds in a plane will produce 1 + 3 + 3 = 7 pieces
Recursion r(N, m) = r(N
1, m) + r(N
1, m
1)
−
−
−
1... | https://ocw.mit.edu/courses/18-085-computational-science-and-engineering-i-summer-2020/22a453f2f41f9c34ad274b7d7da9a0aa_MIT18_085Summer20_lec_GS.pdf |
= 3
←
math.mit.edu/learni
Recursion r(N, m) = r(N
1, m) + r(N
1, m
1)
−
−
−
10/32
v0 has m components / v1 has N components / N ReLU’s
The number of flat regions in Rm bounded by the N hyperplanes
r(N, m) =
m
i=0 (cid:18)
X
N
i
=
(cid:19)
(cid:18)
N
0
N
1
+
(cid:19)
(cid:18)
+
· · ·
+
(cid:19)
(cid:18)
N
m
(cid:19)
N =... | https://ocw.mit.edu/courses/18-085-computational-science-and-engineering-i-summer-2020/22a453f2f41f9c34ad274b7d7da9a0aa_MIT18_085Summer20_lec_GS.pdf |
Big problems are underdetermined [# weights > # samples]
Stochastic Gradient Descent finds weights that generalize well
11/32
F (x) = F2(F1(x)) is continuous piecewise linear
One hidden layer of neurons : deep networks have many more
Overfitting is not desirable ! Gradient descent stops early !
“Generalization” measured... | https://ocw.mit.edu/courses/18-085-computational-science-and-engineering-i-summer-2020/22a453f2f41f9c34ad274b7d7da9a0aa_MIT18_085Summer20_lec_GS.pdf |
�
N + 2 inputs and N outputs
Each shift has a diagonal of 1’s
A = x1L + x0C + x−1R
∂y
∂x1
= Lv
∂y
∂x0
= Cv
∂y
∂x−1
= Rv
12/32
Convolutional Neural Nets (CNN)
x1 x0 x−1
x0
x1
0
x1
0
0
0
0
0
0
x−1
x0
x1
0
0
x−1
x0
0
0
0
x−1
A =
N + 2 inputs and N outputs
Each shift has a diagonal of 1’s
A = x1L + x0C +... | https://ocw.mit.edu/courses/18-085-computational-science-and-engineering-i-summer-2020/22a453f2f41f9c34ad274b7d7da9a0aa_MIT18_085Summer20_lec_GS.pdf |
1 x0−1 x−1−1
x−11
x−10
vij i, j from (0, 0) to (N +1, N +1)
Input image
Output image yij i, j from (1, 1)to (N, N )
Shifts L, C, R, U, D = Left, Center, Right, Up, Down
A convolution is a combination of shift matrices = filter = Toeplitz matrix
The coefficients in the combination will be the “weights” to be learne... | https://ocw.mit.edu/courses/18-085-computational-science-and-engineering-i-summer-2020/22a453f2f41f9c34ad274b7d7da9a0aa_MIT18_085Summer20_lec_GS.pdf |
14/32
Computing the weights x = matrices Ak, bias vectors bk
Choose a loss function ℓ to measure F (x, v)
true output
−
Total loss L =
1
N
(sum of losses for all N samples)
Compute weights x to minimize the total loss L
14/32
Here are three loss functions—Cross-entropy is a favorite loss function for
neural nets
1 Sq... | https://ocw.mit.edu/courses/18-085-computational-science-and-engineering-i-summer-2020/22a453f2f41f9c34ad274b7d7da9a0aa_MIT18_085Summer20_lec_GS.pdf |
= b
(cid:18)
b
1
−
b + 1
k
(cid:19)
yk =
(cid:18)
b
1
−
1 + b
k
(cid:19)
f (xk, yk) =
(cid:18)
b
1
−
1 + b
2k
(cid:19)
f (x0, y0)
16/32
Descent formula xk+1 = xk
sk
−
F (x) Stepsize sk = Learning rate
∇
The first descent step starts out
perpendicular to the level set. As
it crosses through lower level sets,
the functio... | https://ocw.mit.edu/courses/18-085-computational-science-and-engineering-i-summer-2020/22a453f2f41f9c34ad274b7d7da9a0aa_MIT18_085Summer20_lec_GS.pdf |
−
f (xk) + βzk−1
∇
Descent with
momentum
xk+1
zk+1
= xk
f (xk+1) =
− ∇
−
szk
βzk
ck+1
Following the
eigenvector q
λ ck+1 + dk+1=
=
(cid:20)
(cid:21)
It seems a miracle that this problem has a beautiful solution.
The optimal s and β are
−
−
(cid:21)(cid:20)
s dk
β dk (cid:20)
−
1 0
λ 1
ck+1
dk+1
= ck
1
0
−
s
β
ck
dk (ci... | https://ocw.mit.edu/courses/18-085-computational-science-and-engineering-i-summer-2020/22a453f2f41f9c34ad274b7d7da9a0aa_MIT18_085Summer20_lec_GS.pdf |
Key difference : b is replaced by √b
Ordinary
descent factor
2
1
b
−
1 + b !
Accelerated
descent factor
2
√b
1
1 + √b !
−
Steepest
descent
.99
1.01
2
(cid:19)
(cid:18)
= .96
Accelerated
descent
2
.9
1.1
(cid:18)
(cid:19)
= .67
Notice that λmax/λmin = 1/b = κ is the condition number of S
19/32
Key difference : b is re... | https://ocw.mit.edu/courses/18-085-computational-science-and-engineering-i-summer-2020/22a453f2f41f9c34ad274b7d7da9a0aa_MIT18_085Summer20_lec_GS.pdf |
��t 61 data points
20/32
Stochastic Gradient Descent
Stochastic gradient descent uses a “minibatch” of the training data
Every step is much faster than using all data
We don’t want to fit the training data too perfectly (overfitting)
Choosing a polynomial of degree 60 to fit 61 data points
20/32
Stochastic Gradient Desc... | https://ocw.mit.edu/courses/18-085-computational-science-and-engineering-i-summer-2020/22a453f2f41f9c34ad274b7d7da9a0aa_MIT18_085Summer20_lec_GS.pdf |
changes to large oscillations near the solution
Kaczmarz for Ax = b with random i(k) xk+1 = xk +
bi
aT
i xk
2 ai
−
ai
||
||
21/32
Stochastic Descent Using One Sample Per Step
Early steps of SGD often converge quickly toward the solution x∗
Here we pause to look at semi-convergence : Fast start by stochastic
gradient d... | https://ocw.mit.edu/courses/18-085-computational-science-and-engineering-i-summer-2020/22a453f2f41f9c34ad274b7d7da9a0aa_MIT18_085Summer20_lec_GS.pdf |
: Explicit Formulas
vL = bL + ALvL−1
or simply w = b + Av.
The output wi is not affected by bj or Ajk if j
= i
Fully connected layer
Independent weights Ajk
∂wi
∂bj
= δij
and
∂wi
∂Ajk
= δijvk
Example
∂w1
∂b1
= 1,
(cid:20)
w1
w2
∂w1
∂b2
=
(cid:21)
(cid:20)
= 0,
(cid:21)
∂w1
∂a11
b1
b2
+
a11v1 + a12v2
a21v1 + a22v2
(cid:2... | https://ocw.mit.edu/courses/18-085-computational-science-and-engineering-i-summer-2020/22a453f2f41f9c34ad274b7d7da9a0aa_MIT18_085Summer20_lec_GS.pdf |
�x : Explicit Formulas
vL = bL + ALvL−1
or simply w = b + Av.
The output wi is not affected by bj or Ajk if j
= i
Fully connected layer
Independent weights Ajk
∂wi
∂bj
= δij
and
∂wi
∂Ajk
= δijvk
Example
∂w1
∂b1
= 1,
(cid:20)
w1
w2
∂w1
∂b2
=
(cid:21)
(cid:20)
= 0,
(cid:21)
∂w1
∂a11
b1
b2
+
a11v1 + a12v2
a21v1 + a22v2
(ci... | https://ocw.mit.edu/courses/18-085-computational-science-and-engineering-i-summer-2020/22a453f2f41f9c34ad274b7d7da9a0aa_MIT18_085Summer20_lec_GS.pdf |
/∂x : Explicit Formulas
vL = bL + ALvL−1
or simply w = b + Av.
The output wi is not affected by bj or Ajk if j
= i
Fully connected layer
Independent weights Ajk
∂wi
∂bj
= δij
and
∂wi
∂Ajk
= δijvk
Example
∂w1
∂b1
= 1,
(cid:20)
w1
w2
∂w1
∂b2
=
(cid:21)
(cid:20)
= 0,
(cid:21)
∂w1
∂a11
b1
b2
+
a11v1 + a12v2
a21v1 + a22v2
(c... | https://ocw.mit.edu/courses/18-085-computational-science-and-engineering-i-summer-2020/22a453f2f41f9c34ad274b7d7da9a0aa_MIT18_085Summer20_lec_GS.pdf |
(x)))
(cid:19)(cid:18)
dF2
dF1
(F1(x))
(cid:19)(cid:18)
dF1
dx
(x)
(cid:19)
What is the multivariable chain rule ?
Which order (forward or backward along the chain) is faster ?
24/32
Backpropagation and the Chain Rule
L(x) adds up all the losses ℓ (w
true) = ℓ (F (x, v)
true)
−
−
The partial derivatives of L with resp... | https://ocw.mit.edu/courses/18-085-computational-science-and-engineering-i-summer-2020/22a453f2f41f9c34ad274b7d7da9a0aa_MIT18_085Summer20_lec_GS.pdf |
)
dF2
dF1
(F1(x))
(cid:19)(cid:18)
dF1
dx
(x)
(cid:19)
What is the multivariable chain rule ?
Which order (forward or backward along the chain) is faster ?
24/32
Backward-mode AD is faster for M1M2w
(M1M2)w needs N 3+N 2 multiplications M1(M2w) needs only N 2+N 2
Forward (((M1M2)M3) . . . ML)w needs (L
1)N 3 + N 2
−
B... | https://ocw.mit.edu/courses/18-085-computational-science-and-engineering-i-summer-2020/22a453f2f41f9c34ad274b7d7da9a0aa_MIT18_085Summer20_lec_GS.pdf |
∂wi
∂uk
=
∂wi
∂v1
∂v1
∂uk
+
· · ·
+
∂wi
∂vn
∂vn
∂uk
=
∂wi
∂v1
(cid:18)
, . . . ,
∂wi
∂vn (cid:19)
···
(cid:18)
∂v1
∂uk
, . . . ,
∂vn
∂uk (cid:19)
Multivariable chain rule : Multiply matrices !
∂w
∂u
=
∂w
∂v
(cid:18)
∂v
∂u
(cid:19)
(cid:19) (cid:18)
26/32
Hyperparameters : The Fateful Decisions
The words ... | https://ocw.mit.edu/courses/18-085-computational-science-and-engineering-i-summer-2020/22a453f2f41f9c34ad274b7d7da9a0aa_MIT18_085Summer20_lec_GS.pdf |
Regularization = Weight decay : ℓ2 or ℓ1
Small λ : increase the variance of the error (overfitting)
Large λ : increase the bias (underfitting),
b
||
−
Ax
2 is less important
||
Deep learning with many extra weights and good hyperparameters will find
solutions that generalize, without penalty
28/32
Regularization = Weight... | https://ocw.mit.edu/courses/18-085-computational-science-and-engineering-i-summer-2020/22a453f2f41f9c34ad274b7d7da9a0aa_MIT18_085Summer20_lec_GS.pdf |
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