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∥P ) ⇐
(Repeated pro
jection property)
∥ ) =
⎧
⎪⎪
)
∥
D Q∗ P min ∈E D Q P
⎨
⎪
( ∥Q∗)
⎩⎪D
Q
(
(Q∗∗∥Q∗) = minQ
D
∈E2
Q
(
1
136
Indeed, by first tilting from P to Q∗ we find
P [ ˆP ∈ E1 ∩ E2] ≤
≤
2−nD(Q∗
Q∗ P
2 nD
(
−
∥P )Q∗[P
ˆ
ˆQ P
[
∥ ) ∗
∈ E
1
E ]
∩ 2
∈ E
]
2
(12.21)
(12.22)
and
from here proceed by tilting from Q∗ to ... | https://ocw.mit.edu/courses/6-441-information-theory-spring-2016/70ca499b4657031ee80dedff6f4a5a22_MIT6_441S16_chapter_12.pdf |
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18.034, Honors Differential Equations
Prof. Jason Starr
Lecture 8
2/20/04
1. Did the autonomous ODE problem
y' =
4 y -y
2
1
0
-1
+
-
-
+
2. Quickly talked about the “piecewise linear envelope” interpretation of existence/ uniqueness
thm. Talked about difficulty w/ carrying out Picard iterates algorithm. Sequel... | https://ocw.mit.edu/courses/18-034-honors-differential-equations-spring-2004/70f7a61cdf850d6a98b52e063708b7a5_lec8.pdf |
2.997 DecisionMaking in LargeScale Systems
MIT, Spring 2004
February 25
Handout #10
Lecture Note 7
1 RealTime Value Iteration
Recall the realtime value iteration (RTVI) algorithm
choose xk+1 = f (xk , uk, wk)
choose ut in some fashion
update Jk+1(xk) = (T Jk)(xk),
Jk+1(x) = (T Jk)(x), ∀ x = xk
We thus hav... | https://ocw.mit.edu/courses/2-997-decision-making-in-large-scale-systems-spring-2004/712e86c6571618f73c085f8da5a897cd_lec_7_v1.pdf |
state and action a chosen. Therefore we circumvent the need
to perform expectations
y Pa(x, y)J(y) associated with greedy policies.
�
We define the operator
(HQ)(x, a) = ga(x) + α
�
y
Pa(x, y) min Q(y, a�)
a�
(3)
It is easy to show that the operator H has the same properties as operator T defined in previous le... | https://ocw.mit.edu/courses/2-997-decision-making-in-large-scale-systems-spring-2004/712e86c6571618f73c085f8da5a897cd_lec_7_v1.pdf |
)
a�
However, note that such an algorithm should not be expected to converge; in particular, Qt(xt+1, a�) is a
y Put (x, y) mina� Qt(y, a�). We consider a smallstep version of this scheme, where the
noisy estimate of
noise is attenuated:
�
�
Qt+1(xt, at) = (1 − γt)Qt(xt, at) + γt gat (xt) + α min Qt(xt+1, a�) ... | https://ocw.mit.edu/courses/2-997-decision-making-in-large-scale-systems-spring-2004/712e86c6571618f73c085f8da5a897cd_lec_7_v1.pdf |
i + 1
(Hrt + wi+1).
Therefore, rt+1 = rt
obtaining the final form
(k) . Finally, we may consider replacing samples Hrt + wi with samples Hr(i−1) + wi,
t
rt+1 = (1 − γt)rt + γt(Hrt + wt).
A simple application of these ideas involves estimating the expected value of a random variable by drawing
i.i.d. samples.
Examp... | https://ocw.mit.edu/courses/2-997-decision-making-in-large-scale-systems-spring-2004/712e86c6571618f73c085f8da5a897cd_lec_7_v1.pdf |
noisy
term γtw decreases with time, motivating the condition (7).
We will consider two approaches to analyzing the stochastic approximation algorithm
rt+1 = (1 − γt)rt + γt(Hrt + wt)
= rt + γt(Hrt + wt − rt)
= rt + γtS(rt, wt)
(8)
where we define S(rt, wt) = Hrt + wt − rt. The two approaches are
1. Lyapunov func... | https://ocw.mit.edu/courses/2-997-decision-making-in-large-scale-systems-spring-2004/712e86c6571618f73c085f8da5a897cd_lec_7_v1.pdf |
(1 − γt)�rt − r∗
2 − (1 − α)γt�rt − r∗
rt − r∗
�
�
2 + γt�F rt − r∗
2 + αγt�rt − r∗
�
2
�
≤
=
2.
�
�
2
Therefore, �rt − r∗
�
2 is nonincreasing and bounded below by zero. Thus, �rt − r∗
� →
t
0
2 −→ � ≥ 0. Then
0 ≤ �
rt+1 − r∗
� ≤ �
2
rt − r∗
2 − (1 − α)γt�rt − r∗
�
rt − r∗
�
2 − (1 − α)γt�
rt−1 − r∗
2 − (1 − α)(... | https://ocw.mit.edu/courses/2-997-decision-making-in-large-scale-systems-spring-2004/712e86c6571618f73c085f8da5a897cd_lec_7_v1.pdf |
Definition 1 We call function V a Lyapunov function if V satisfies
(a) V (·) ≥ 0
(b) (�r V )T S(r) ≤ 0
(c) V (r) = 0 ⇔ S(r) = 0
3.1.2 Stochastic Case
The argument used for convergence in the stochastic case parallels the argument used in the deterministic
case. Let Ft denote all information that is available at sta... | https://ocw.mit.edu/courses/2-997-decision-making-in-large-scale-systems-spring-2004/712e86c6571618f73c085f8da5a897cd_lec_7_v1.pdf |
we may have V (rt+1) > V (rt) occasionally. Therefore we need an stochastic counterpart to the
result that “every nonincreasing sequence bounded below converges.” The stochastic counterpart of interest
to our analysis is given below.
Theorem 1 (Supermartingale Convergence Theorem) Suppose that Xt, Yt and Zt are nonn... | https://ocw.mit.edu/courses/2-997-decision-making-in-large-scale-systems-spring-2004/712e86c6571618f73c085f8da5a897cd_lec_7_v1.pdf |
MIT 6.972 Algebraic techniques and semidefinite optimization
February 7, 2006
Lecturer: Pablo A. Parrilo
Scribe: Pablo A. Parrilo
Lecture 1
1
Introduction: what is this course about?
In this course we aim to understand the properties, both mathematical and computational, of sets
defined by polynomial equations an... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/7190173dff88ad51d942add6026d0d4f_lecture_01.pdf |
our stops along the way will include:
• Linear optimization, second order cones, semidefinite programming
• Algebra: groups, fields, rings
• Univariate polynomials
•
Resultants and discriminants
• Hyperbolic polynomials
• Sum of squares
• Ideals, varieties, Groebner bases, Nullstellensatz
• Quantifier elimination... | https://ocw.mit.edu/courses/6-972-algebraic-techniques-and-semidefinite-optimization-spring-2006/7190173dff88ad51d942add6026d0d4f_lecture_01.pdf |
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