1. Yi Wu (CMU)
Joint work with
ParikshitGopalan (MSR SVC)
Ryan ODonnell (CMU)
David Zuckerman (UT Austin)
Pseudorandom Generators for Halfspaces
TexPoint fonts used in EMF.
Read the TexPoint manual before you delete this box.: AAAAA

2. Outline
Introduction
Pseudorandom Generators
Halfspaces
Pseudorandom Generators for Halfspaces
Our Result
Proof
Conclusion
2

3. Deterministic Algorithm
Program
Input
Output
The algorithm deterministically outputs the correct result.
3

4. Randomized Algorithm
Program
Input
Output
Random Bits.
The algorithm outputs the correct result with high probability.
4

5. Primality testing
ST-connectivity
Order statistics
Searching
Polynomial and matrix identity verification
Interactive proof systems
Faster algorithms for linear programming
Rounding linear program solutions to integer
Minimum spanning trees
shortest paths minimum cuts
Counting and enumeration
Matrix permanent
Counting combinatorial structures
Primality testing
ST-connectivity
Order statistics
Searching
Polynomial and matrix identity verification
Interactive proof systems
Faster algorithms for linear programming
Rounding linear program solutions to integer
Minimum spanning trees
shortest paths minimum cuts
Counting and enumeration
Matrix permanent
Counting combinatorial structures
Primality testing
ST-connectivity
Order statistics
Searching
Polynomial and matrix identity verification
Interactive proof systems
Faster algorithms for linear programming
Rounding linear program solutions to integer
Minimum spanning trees
shortest paths minimum cuts
Counting and enumeration
Matrix permanent
Counting combinatorial structures
Primality testing
ST-connectivity
Order statistics
Searching
Polynomial and matrix identity verification
Interactive proof systems
Faster algorithms for linear programming
Rounding linear program solutions to integer
Minimum spanning trees
shortest paths minimum cuts
Counting and enumeration
Matrix permanent
Counting combinatorial structures
Randomized Algorithms
5

6. Is Randomness Necessary?
Open Problem:
Can we simulate every randomized polynomial time algorithm by a deterministic polynomial time algorithm (the BPP P cojecture)?
Derandomization of randomized algorithms.
Primality testing [AKS]
ST-connectivity [Reingold]
Quadratic residues [?]
6

7. How to generate randomness?
Question: How togenerate randomness for every randomized algorithm?
Simpler Question: How to generate pseudorandomness for some class of programs?
7

8. Pseudorandom Generator (PRG)
Both program Answer Yes/No with almost the same probability
Yes /No
Yes/ No
Input
Program
Input
Program
n pseudorandom bit
PRG
Quality of the PRG: number of seed
n random bit
Seed
k {-1,1} of the form
h(x) = sgn(w1x1++wnxn- )
where w1,, wn, R.

• Well-studied in complexity theory
• 13. Widely used in Machine Learning: Perceptron, Winnow, boosting, Support Vector Machines, Lasso, Liner Regression.

12

14. Product Distribution
For halfspace h(x), x is sampled from some product distribution; i.e., each xi is independently sampled from distribution Di .
For example, each Dican be
Uniform distribution on {-1,1}
Uniform distribution on [-1,1]
Gaussian Distribution
13

15. Index
Introduction
Pseudorandom Generators
Halfspaces
Pseudorandom Generators for Halfspaces
Main Result
Proof
Conclusion
14

16. PRG for halfspaces
Both program Answer Yes/No with almost the same probability
Yes/No
Yes/No
h(x) = sign(w1x1++wnxn-)
h(x) = sign(w1x1++wnxn-)
Pseudorandom Variable
x1, x2 xn
PRG
x1, x2 xnfrom some product distribution
k