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Fast Algorithms
for Analyzing Massive Data
Alexander Gray
Georgia Institute of Technology
www.fast-lab.org
The FASTlab
Fundamental Algorithmic and Statistical Tools Laboratory
www.fast-lab.org
1.
Alexander Gray: Assoc Prof, Applied Math + CS; PhD CS
2.
Arkadas Ozakin: Research Scientist, Math + Physics; PhD Physics
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
Dongryeol Lee: PhD student, CS + Math
Ryan Riegel: PhD student, CS + Math
Sooraj Bhat: PhD student, CS
Nishant Mehta: PhD student, CS
Parikshit Ram: PhD student, CS + Math
William March: PhD student, Math + CS
Hua Ouyang: PhD student, CS
Ravi Sastry: PhD student, CS
Long Tran: PhD student, CS
Ryan Curtin: PhD student, EE
Ailar Javadi: PhD student, EE
Anita Zakrzewska: PhD student, CS
+ 5-10 MS students and undergraduates
7 tasks of
machine learning / data mining
1. Querying: spherical range-search O(N), orthogonal range-search
2.
3.
4.
5.
6.
7.
O(N), nearest-neighbor O(N), all-nearest-neighbors O(N2)
Density estimation: mixture of Gaussians, kernel density
estimation O(N2), kernel conditional density estimation O(N3)
Classification: decision tree, nearest-neighbor classifier O(N2),
kernel discriminant analysis O(N2), support vector machine O(N3) ,
Lp SVM
Regression: linear regression, LASSO, kernel regression O(N2),
Gaussian process regression O(N3)
Dimension reduction: PCA, non-negative matrix factorization,
kernel PCA O(N3), maximum variance unfolding O(N3); Gaussian
graphical models, discrete graphical models
Clustering: k-means, mean-shift O(N2), hierarchical (FoF)
clustering O(N3)
Testing and matching: MST O(N3), bipartite cross-matching
O(N3), n-point correlation 2-sample testing O(Nn), kernel embedding
7 tasks of
machine learning / data mining
1. Querying: spherical range-search O(N), orthogonal range-search
2.
3.
4.
5.
6.
7.
O(N), nearest-neighbor O(N), all-nearest-neighbors O(N2)
Density estimation: mixture of Gaussians, kernel density
estimation O(N2), kernel conditional density estimation O(N3)
Classification: decision tree, nearest-neighbor classifier O(N2),
kernel discriminant analysis O(N2), support vector machine O(N3), Lp
SVM
Regression: linear regression, LASSO, kernel regression O(N2),
Gaussian process regression O(N3)
Dimension reduction: PCA, non-negative matrix factorization,
kernel PCA O(N3), maximum variance unfolding O(N3); Gaussian
graphical models, discrete graphical models
Clustering: k-means, mean-shift O(N2), hierarchical (FoF)
clustering O(N3)
Testing and matching: MST O(N3), bipartite cross-matching
O(N3), n-point correlation 2-sample testing O(Nn), kernel embedding
7 tasks of
machine learning / data mining
1. Querying: spherical range-search O(N), orthogonal range-search O(N), nearest2.
3.
4.
5.
6.
7.
neighbor O(N), all-nearest-neighbors O(N2)
Density estimation: mixture of Gaussians, kernel density estimation O(N2), kernel
conditional density estimation O(N3), submanifold density estimation [Ozakin
& Gray, NIPS 2010], O(N3), convex adaptive kernel estimation [Sastry &
Gray, AISTATS 2011] O(N4)
Classification: decision tree, nearest-neighbor classifier O(N2), kernel discriminant
analysis O(N2), support vector machine O(N3) , Lp SVM, non-negative SVM
[Guan et al, 2011]
Regression: linear regression, LASSO, kernel regression O(N2), Gaussian process
regression O(N3)
Dimension reduction: PCA, non-negative matrix factorization, kernel PCA O(N3),
maximum variance unfolding O(N3); Gaussian graphical models, discrete graphical
models, rank-preserving maps [Ouyang and Gray, ICML 2008] O(N3);
isometric separation maps [Vasiiloglou, Gray, and Anderson MLSP 2009]
O(N3); isometric NMF [Vasiiloglou, Gray, and Anderson MLSP 2009] O(N3);
functional ICA [Mehta and Gray, 2009], density preserving maps [Ozakin
and Gray, in prep] O(N3)
Clustering: k-means, mean-shift O(N2), hierarchical (FoF) clustering O(N3)
Testing and matching: MST O(N3), bipartite cross-matching O(N3), n-point
correlation 2-sample testing O(Nn), kernel embedding
7 tasks of
machine learning / data mining
1. Querying: spherical range-search O(N), orthogonal range-search
2.
3.
4.
5.
6.
7.
O(N), nearest-neighbor O(N), all-nearest-neighbors O(N2)
Density estimation: mixture of Gaussians, kernel density
estimation O(N2), kernel conditional density estimation O(N3)
Classification: decision tree, nearest-neighbor classifier O(N2),
kernel discriminant analysis O(N2), support vector machine O(N3) ,
Lp SVM
Computational
Regression: linear regression, kernel regression O(N2),
Problem!
Gaussian process regression
O(N3), LASSO
Dimension reduction: PCA, non-negative matrix factorization,
kernel PCA O(N3), maximum variance unfolding O(N3), Gaussian
graphical models, discrete graphical models
Clustering: k-means, mean-shift O(N2), hierarchical (FoF)
clustering O(N3)
Testing and matching: MST O(N3), bipartite cross-matching
O(N3), n-point correlation 2-sample testing O(Nn), kernel embedding
The “7 Giants” of Data
(computational problem types)
[Gray, Indyk, Mahoney, Szalay, in National Acad of Sci
Report on Analysis of Massive Data, in prep]
1. Basic statistics: means, covariances, etc.
2. Generalized N-body problems: distances,
geometry
3.
4.
5.
6.
7.
Graph-theoretic problems: discrete graphs
Linear-algebraic problems: matrix operations
Optimizations: unconstrained, convex
Integrations: general dimension
Alignment problems: dynamic prog, matching
7 general strategies
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6.
7.
Divide and conquer / indexing (trees)
Function transforms (series)
Sampling (Monte Carlo, active learning)
Locality (caching)
Streaming (online)
Parallelism (clusters, GPUs)
Problem transformation
(reformulations)
1. Divide and conquer
• Fastest approach for:
– nearest neighbor, range search (exact) ~O(logN) [Bentley
1970], all-nearest-neighbors (exact) O(N) [Gray & Moore, NIPS
2000], [Ram, Lee, March, Gray, NIPS 2010], anytime nearest neighbor
(exact) [Ram & Gray, SDM 2012], max inner product [Ram & Gray,
under review]
– mixture of Gaussians [Moore, NIPS 1999], k-means [Pelleg and
Moore, KDD 1999], mean-shift clustering O(N) [Lee & Gray, AISTATS
2009], hierarchical clustering (single linkage, friends-of-friends)
O(NlogN) [March & Gray, KDD 2010]
– nearest neighbor classification [Liu, Moore, Gray, NIPS 2004],
kernel discriminant analysis O(N) [Riegel & Gray, SDM 2008]
– n-point correlation functions ~O(Nlogn) [Gray & Moore, NIPS 2000],
[Moore et al. Mining the Sky 2000], multi-matcher jackknifed npcf [March
& Gray, under review]
3-point correlation
(biggest previous:
20K)
n=2: O(N)
n=3: O(Nlog3)
VIRGO
simulation data,
N = 75,000,000
naïve: 5x109 sec.
(~150 years)
multi-tree: 55 sec.
(exact)
n=4: O(N2)
3-point correlation
Naive - O(Nn)
(estimated)
2 point cor.
100 matchers
3 point cor.
243 matchers
Single
Multi-bandwidth
bandwidth
[March & Gray
[Gray & Moore
in prep 2010]
2000, Moore et
new
al. 2000]
2.0 x 107 s 352.8 s
56,000
1.1 x 1011
s
891.6 s
1.23 x 108
4.96 s
71.1
13.58 s
65.6
2.3 x 1014 14530 s
503.6 s
4 point
cor.
6
points, galaxy
s simulation
1.58 xdata
21610
matchers
28.8
1010
2. Function transforms
• Fastest approach for:
– Kernel estimation (low-ish
dimension): dual-tree fast
Gauss transforms
(multipole/Hermite
expansions) [Lee, Gray, Moore
NIPS 2005], [Lee and Gray, UAI
2006]
– KDE and GP (kernel density
estimation, Gaussian process
regression) (high-D): random
Fourier functions [Lee and Gray,
in prep]
3. Sampling
• Fastest approach for (approximate):
– PCA: cosine trees [Holmes, Gray, Isbell, NIPS 2008]
– Kernel estimation: bandwidth learning [Holmes, Gray, Isbell, NIPS
2006], [Holmes, Gray, Isbell, UAI 2007], Monte Carlo multipole method
(with SVD trees) [Lee & Gray, NIPS 2009]
– Nearest-neighbor: distance-approx: spill trees with random proj:
[Liu, Moore, Gray, Yang, NIPS 2004], rank-approximate: [Ram, Ouyang,
Gray, NIPS 2009]
e=0%(exact),0.001%,0.01%,0.1%,1%,10%
a=0.95
4
Rank-approximate NN:
• Best meaning-retaining
approximation criterion in the face
of high-dimensional distances
• More accurate than LSH
speedup over linear search
10
3
10
2
10
1
10
0
10
bio
corel
covtype
images
mnist
phy
urand
3. Sampling
• Active learning: the
sampling can depend on
previous samples
– Linear classifiers:
rigorous framework for
pool-based active learning
30
UPAL
BMAL
VW
RAL
PL
25
20
15
[Sastry and Gray, AISTATS 2012]
• Empirically allows reduction
in the number of objects that
require labeling
• Theoretical rigor:
unbiasedness
10
5
0
50
100
150
200
250
300
4. Caching
• Fastest approach for (using disk):
– Nearest-neighbor, 2-point: Disk-based treee
algorithms in Microsoft SQL Server [Riegel, Aditya,
Budavari, Gray, in prep]
• Builds kd-tree on top of built-in B-trees
• Fixed-pass algorithm to build kd-tree
No. of points
MLDB (Dual tree)
Naive
40,000
8 seconds
159 seconds
200,000
43 seconds
3480 seconds
2,000,000
297 seconds
80 hours
10,000,000
29 mins 27 sec
74 days
20,000,000
58mins 48sec
280 days
40,000,000
112m 32 sec
2 years
5. Streaming / online
• Fastest approach for (approximate, or
streaming):
– Online learning/stochastic optimization: just use the
current sample to update the gradient
• SVM (squared hinge loss): stochastic Frank-Wolfe [Ouyang and
Gray, SDM 2010]
• SVM, LASSO, et al.: noise-adaptive stochastic
approximation [Ouyang and Gray, in prep, on arxiv], accelerated
non-smooth SGD [Ouyang and Gray, under review]
– faster than SGD
– solves step size problem
– beats all existing convergence rates
6. Parallelism
• Fastest approach for (using many machines):
– KDE, GP, n-point: distributed trees [Lee and Gray, SDM 2012], 6000+
cores; [March et al, in prep for Gordon Bell Prize 2012], 100K cores?
• Each process owns the global tree and its local tree
• First log p levels built in parallel; each process determines where to send data
• Asynchronous averaging; provable convergence
– SVM, LASSO, et al.: distributed online optimization [Ouyang and Gray,
in prep, on arxiv]
• Provable theoretical speedup for the first time
P0
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P0
P0
P1
P0
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7. Transformations
between problems
• Change the problem type:
–
–
–
Linear algebra on kernel matrices  N-body inside conjugate gradient [Gray, TR 2004]
Euclidean graphs  N-body problems [March & Gray, KDD 2010]
HMM as graph  matrix factorization [Tran & Gray, in prep]
• Optimizations: reformulate the objective and constraints:
–
–
–
–
Maximum variance unfolding: SDP via Burer-Monteiro convex relaxation [Vasiloglou, Gray,
Anderson MLSP 2009]
Lq SVM, 0<q<1: DC programming [Guan & Gray, CSDA 2011]
L0 SVM: mixed integer nonlinear program via perspective cuts [Guan & Gray, under review]
Do reformulations automatically [Agarwal et al, PADL 2010], [Bhat et al, POPL 2012]
• Create new ML methods with desired computational properties:
–
–
–
Density estimation trees: nonparametric density estimation, O(NlogN) [Ram & Gray, KDD
2011]
Local linear SVMs: nonlinear classification, O(NlogN) [Sastry & Gray, under review]
Discriminative local coding: nonlinear classification O(NlogN) [Mehta & Gray, under
review]
Software
• For academic use only: MLPACK
– Open source, C++, written by students
– Data must fit in RAM: distributed in progress
• For institutions: Skytree Server
– First commercial-grade high-performance machine learning server
– Fastest, biggest ML available: up to 10,000x faster than existing
solutions (on one machine)
– V.12, April 2012-ish: distributed, streaming
– Connects to stats packages, Matlab, DBMS, Python, etc
– www.skytreecorp.com
– Colleagues: Email me to try it out: [email protected]