Novel Power Grid Reduction Method based on L1 ... - Semantic Scholar
Novel Power Grid Reduction Method based on L1 Regularization Ye Wang, Meng Li, Xinyang Yi, Zhao Song, Michael Orshansky, Constantine Caramanis University of Texas at Austin [email protected]
June 3, 2015
1 / 12
Power Grid Reduction: Motivation • Large-scale power grids
• Simulation time determined by size of power grid: number of
ports and elements (R,L,C) • Speed up simulation/analysis by reducing grid size I
Focus only on steady-state analysis (DC), hope to extend to other types of analysis (Transient, AC)
2 / 12
Power Grid Reduction: Formulation • Port relation (Ohm’s law) by LG
LG v = i • Admittance matrix (Graph Laplacian)
P k,k6=i ωik , if i = j, LG (i, j) = −ωij , if i 6= j and {i, j} ∈ E , 0, otherwise. • Goal: want a sparse approximation LG 0 I With far fewer non-zeros I Preserve similar port relation
3 / 12
State-of-the-Art Methods • Krylov subspace based methods [FL04,LS06] I Project the original system onto a low-rank Krylov subspace for efficiency • Time-Constant Equilibration Reduction (TICER) [She99] I Eliminate low-degree nodes by connecting their neighbors • Algebraic multigrid methods [SAN03] I Reduce the number of nodes and edges simultaneously • Sampling based spectral sparsification approach [ZFZ14] I In time O(m log n/2 ), find an -power approximation G 0 of O(n log n/2 ) edges satisfies: (1 − )v T LG v ≤ v T LG 0 v ≤ (1 + )v T LG v , ∀v ∈ Rn . • They all try to build sparsifiers preserving LG v = i for all
i ∈ Rn ... Is that necessary? 4 / 12
Our Key Observation • In practice, currents delivered from ports do not vary
unboundedly I
Peak values of currents can be estimated from system-level description or transistor-level simulation
• The actual space is a small subset of the entire space i2
i2_max i1_max
i1
• How to utilize the range information not explored
before? I
For more sparsity and accuracy of reduced power grids 5 / 12
Our Main Contribution • Propose an efficient method that allows using range
information for better sparsification • Leverage recent advances of `1 regularization to drive sparsity • We call it graph Sparsification by `1 regularization on
Laplacian (SparseLL)
6 / 12
First Attempt for Sparsification • Objective function: averaged error in the given range
Z
[k(LG − LG 0 )v k22 ]dv
min LG 0
I
ΩV
Allow to incorporate the range information
• Constraints: sparsity specified by `0 -norm (number of
non-zeros) kLG 0 k0 ≤ m0 • Non-linear and non-convex in both objective and constraints,
hard to solve...
7 / 12
Reformulation as Stochastic Optimization • Integral discretization by deterministic mesh requires
exponential number of samples Z N 1 X 2 k(LG − LG 0 )vi k22 [k(LG − LG 0 )v k2 ]dv ≈ N ΩV i=1
• Randomized discretization leverages fast convergence from
stochastic gradient descent (SGD)
Mesh Discretization
I I
SGD
Sample vi ∼ ΩV , calculate gradient, update psolution Converge to optimal solution with rate O( 1/t) for t iterations 8 / 12
`1 Regularization for Sparsity • `0 constraints are combinatorial and non-convex: result in an
NP-hard problem • `1 norm is the tightest while being convex relaxation of `0
norm 1
1
1
-1
1
-1
-1
-1
L0 norm
L1 norm
• Sparsity encouraged by spiky `1 norm
1
1
1
-1 -1
||x||2=1
1
-1 -1
non-sparse
||x||1=1
sparse 9 / 12
Complete SparseLL Formulation • Objective: regularized empirical risk function
min LG 0
I
N 1 X k(LG − LG 0 )vi k22 + λkL0G k1 N i=1
Parameter λ controls the degree of sparsity
• Constraints:
LG 0 (i, j) ≤ 0, with i 6= j, LG 0 = LT G0, n X LG 0 (i, j) = 0, ∀i ∈ {1, 2, . . . , n}. j=1
10 / 12
Experimental Results • Optimizing a sample 17-node power grid 1
10
Average Current Error
Spectral SparseLL
0
10
−1
10
−2
10
8
10
12
14
16
18
20
22
24
26
Number of Edges in Sparsified Graphs
• A smaller error while significantly reducing # of edges rand1 rand2 rand3 ibmpg1
Nodes 100 500 1000 5388
Edges 4000 100000 400000 27000
Spectral [ZFZ14] Error Edges 5.40% 1031 4.44% 8120 4.80% 15114 3.80% 6703
Error 0.18% 0.07% 0.03% 0.01%
SparseLL Edges Err Reduction 996 30X 8021 60X 14213 160X 6570 380X 11 / 12
Summary • Identified and specified a realistic range for currents • Formulated power grid reduction as a convex optimization
problem I I
With objective as average current error in that range Use `1 norm to encourage sparsity
• Solved the problem using an efficient SGD algorithm
12 / 12
June 3, 2015
1 / 12
Power Grid Reduction: Motivation • Large-scale power grids
• Simulation time determined by size of power grid: number of
ports and elements (R,L,C) • Speed up simulation/analysis by reducing grid size I
Focus only on steady-state analysis (DC), hope to extend to other types of analysis (Transient, AC)
2 / 12
Power Grid Reduction: Formulation • Port relation (Ohm’s law) by LG
LG v = i • Admittance matrix (Graph Laplacian)
P k,k6=i ωik , if i = j, LG (i, j) = −ωij , if i 6= j and {i, j} ∈ E , 0, otherwise. • Goal: want a sparse approximation LG 0 I With far fewer non-zeros I Preserve similar port relation
3 / 12
State-of-the-Art Methods • Krylov subspace based methods [FL04,LS06] I Project the original system onto a low-rank Krylov subspace for efficiency • Time-Constant Equilibration Reduction (TICER) [She99] I Eliminate low-degree nodes by connecting their neighbors • Algebraic multigrid methods [SAN03] I Reduce the number of nodes and edges simultaneously • Sampling based spectral sparsification approach [ZFZ14] I In time O(m log n/2 ), find an -power approximation G 0 of O(n log n/2 ) edges satisfies: (1 − )v T LG v ≤ v T LG 0 v ≤ (1 + )v T LG v , ∀v ∈ Rn . • They all try to build sparsifiers preserving LG v = i for all
i ∈ Rn ... Is that necessary? 4 / 12
Our Key Observation • In practice, currents delivered from ports do not vary
unboundedly I
Peak values of currents can be estimated from system-level description or transistor-level simulation
• The actual space is a small subset of the entire space i2
i2_max i1_max
i1
• How to utilize the range information not explored
before? I
For more sparsity and accuracy of reduced power grids 5 / 12
Our Main Contribution • Propose an efficient method that allows using range
information for better sparsification • Leverage recent advances of `1 regularization to drive sparsity • We call it graph Sparsification by `1 regularization on
Laplacian (SparseLL)
6 / 12
First Attempt for Sparsification • Objective function: averaged error in the given range
Z
[k(LG − LG 0 )v k22 ]dv
min LG 0
I
ΩV
Allow to incorporate the range information
• Constraints: sparsity specified by `0 -norm (number of
non-zeros) kLG 0 k0 ≤ m0 • Non-linear and non-convex in both objective and constraints,
hard to solve...
7 / 12
Reformulation as Stochastic Optimization • Integral discretization by deterministic mesh requires
exponential number of samples Z N 1 X 2 k(LG − LG 0 )vi k22 [k(LG − LG 0 )v k2 ]dv ≈ N ΩV i=1
• Randomized discretization leverages fast convergence from
stochastic gradient descent (SGD)
Mesh Discretization
I I
SGD
Sample vi ∼ ΩV , calculate gradient, update psolution Converge to optimal solution with rate O( 1/t) for t iterations 8 / 12
`1 Regularization for Sparsity • `0 constraints are combinatorial and non-convex: result in an
NP-hard problem • `1 norm is the tightest while being convex relaxation of `0
norm 1
1
1
-1
1
-1
-1
-1
L0 norm
L1 norm
• Sparsity encouraged by spiky `1 norm
1
1
1
-1 -1
||x||2=1
1
-1 -1
non-sparse
||x||1=1
sparse 9 / 12
Complete SparseLL Formulation • Objective: regularized empirical risk function
min LG 0
I
N 1 X k(LG − LG 0 )vi k22 + λkL0G k1 N i=1
Parameter λ controls the degree of sparsity
• Constraints:
LG 0 (i, j) ≤ 0, with i 6= j, LG 0 = LT G0, n X LG 0 (i, j) = 0, ∀i ∈ {1, 2, . . . , n}. j=1
10 / 12
Experimental Results • Optimizing a sample 17-node power grid 1
10
Average Current Error
Spectral SparseLL
0
10
−1
10
−2
10
8
10
12
14
16
18
20
22
24
26
Number of Edges in Sparsified Graphs
• A smaller error while significantly reducing # of edges rand1 rand2 rand3 ibmpg1
Nodes 100 500 1000 5388
Edges 4000 100000 400000 27000
Spectral [ZFZ14] Error Edges 5.40% 1031 4.44% 8120 4.80% 15114 3.80% 6703
Error 0.18% 0.07% 0.03% 0.01%
SparseLL Edges Err Reduction 996 30X 8021 60X 14213 160X 6570 380X 11 / 12
Summary • Identified and specified a realistic range for currents • Formulated power grid reduction as a convex optimization
problem I I
With objective as average current error in that range Use `1 norm to encourage sparsity
• Solved the problem using an efficient SGD algorithm
12 / 12
