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min_norm_solvers.py
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min_norm_solvers.py
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"""
FROM: https://github.com/isl-org/MultiObjectiveOptimization
"""
import numpy as np
import torch
from tqdm import tqdm
class MinNormSolver:
MAX_ITER = 1000
STOP_CRIT = 1e-5
def _min_norm_element_from2(v1v1, v1v2, v2v2):
"""
Analytical solution for min_{c} |cx_1 + (1-c)x_2|_2^2
d is the distance (objective) optimzed
v1v1 = <x1,x1>
v1v2 = <x1,x2>
v2v2 = <x2,x2>
"""
if v1v2 >= v1v1:
# Case: Fig 1, third column
gamma = 0.999
cost = v1v1
return gamma, cost
if v1v2 >= v2v2:
# Case: Fig 1, first column
gamma = 0.001
cost = v2v2
return gamma, cost
# Case: Fig 1, second column
gamma = -1.0 * ( (v1v2 - v2v2) / (v1v1+v2v2 - 2*v1v2) )
cost = v2v2 + gamma*(v1v2 - v2v2)
return gamma, cost
def _min_norm_2d(vecs, dps):
"""
Find the minimum norm solution as combination of two points
This is correct only in 2D
ie. min_c |\sum c_i x_i|_2^2 st. \sum c_i = 1 , 1 >= c_1 >= 0 for all i, c_i + c_j = 1.0 for some i, j
"""
dmin = 1e8
for i in range(len(vecs)):
for j in range(i+1,len(vecs)):
if (i,j) not in dps:
dps[(i, j)] = 0.0
for k in range(len(vecs[i])):
dps[(i,j)] += torch.mul(vecs[i][k], vecs[j][k]).sum().data.cpu()
dps[(j, i)] = dps[(i, j)]
if (i,i) not in dps:
dps[(i, i)] = 0.0
for k in range(len(vecs[i])):
dps[(i,i)] += torch.mul(vecs[i][k], vecs[i][k]).sum().data.cpu()
if (j,j) not in dps:
dps[(j, j)] = 0.0
for k in range(len(vecs[i])):
dps[(j, j)] += torch.mul(vecs[j][k], vecs[j][k]).sum().data.cpu()
c,d = MinNormSolver._min_norm_element_from2(dps[(i,i)], dps[(i,j)], dps[(j,j)])
if d < dmin:
dmin = d
sol = [(i,j),c,d]
return sol, dps
def _min_norm_2d_accelerated(vecs, dps):
dmin = 1e8
# V2
vec = torch.stack(vecs)
M = torch.matmul(vec, vec.T)
# print("-------debug---------", len(vecs))
for i in range(len(vecs)):
if (i,i) not in dps:
dps[(i,i)] = M[i,i].item()
for j in range(i+1,len(vecs)):
if (j,j) not in dps:
dps[(j,j)] = M[j,j].item()
if (i,j) not in dps:
dps[(i,j)] = dps[(j,i)] = M[i,j].item()
c,d = MinNormSolver._min_norm_element_from2(dps[(i,i)], dps[(i,j)], dps[(j,j)])
# print("------debug--------- i-j {}-{}, d {}, dmin {}".format(i, j, d, dmin))
if d < dmin:
dmin = d
sol = [(i,j),c,d]
# V1
# for i in range(len(vecs)):
# if (i,i) not in dps:
# dps[(i,i)] = torch.dot(vecs[i], vecs[i])
# for j in range(i+1,len(vecs)):
# if (j,j) not in dps:
# dps[(j,j)] = torch.dot(vecs[j], vecs[j])
# if (i,j) not in dps:
# dps[(i, j)] = torch.dot(vecs[i], vecs[j])
# dps[(j, i)] = dps[(i, j)]
# c,d = MinNormSolver._min_norm_element_from2(dps[(i,i)], dps[(i,j)], dps[(j,j)])
# if d < dmin:
# dmin = d
# sol = [(i,j),c,d]
return sol, dps
def _projection2simplex(y):
"""
Given y, it solves argmin_z |y-z|_2 st \sum z = 1 , 1 >= z_i >= 0 for all i
"""
m = len(y)
sorted_y = np.flip(np.sort(y), axis=0)
tmpsum = 0.0
tmax_f = (np.sum(y) - 1.0)/m
for i in range(m-1):
tmpsum+= sorted_y[i]
tmax = (tmpsum - 1)/ (i+1.0)
if tmax > sorted_y[i+1]:
tmax_f = tmax
break
return np.maximum(y - tmax_f, np.zeros(y.shape))
def _next_point(cur_val, grad, n):
proj_grad = grad - ( np.sum(grad) / n )
tm1 = -1.0*cur_val[proj_grad<0]/proj_grad[proj_grad<0]
tm2 = (1.0 - cur_val[proj_grad>0])/(proj_grad[proj_grad>0])
skippers = np.sum(tm1<1e-7) + np.sum(tm2<1e-7)
t = 1
if len(tm1[tm1>1e-7]) > 0:
t = np.min(tm1[tm1>1e-7])
if len(tm2[tm2>1e-7]) > 0:
t = min(t, np.min(tm2[tm2>1e-7]))
next_point = proj_grad*t + cur_val
next_point = MinNormSolver._projection2simplex(next_point)
return next_point
def find_min_norm_element(vecs):
"""
Given a list of vectors (vecs), this method finds the minimum norm element in the convex hull
as min |u|_2 st. u = \sum c_i vecs[i] and \sum c_i = 1.
It is quite geometric, and the main idea is the fact that if d_{ij} = min |u|_2 st u = c x_i + (1-c) x_j; the solution lies in (0, d_{i,j})
Hence, we find the best 2-task solution, and then run the projected gradient descent until convergence
"""
# Solution lying at the combination of two points
dps = {}
init_sol, dps = MinNormSolver._min_norm_2d(vecs, dps)
n=len(vecs)
sol_vec = np.zeros(n)
sol_vec[init_sol[0][0]] = init_sol[1]
sol_vec[init_sol[0][1]] = 1 - init_sol[1]
if n < 3:
# This is optimal for n=2, so return the solution
return sol_vec , init_sol[2]
iter_count = 0
grad_mat = np.zeros((n,n))
for i in range(n):
for j in range(n):
grad_mat[i,j] = dps[(i, j)]
while iter_count < MinNormSolver.MAX_ITER:
grad_dir = -1.0*np.dot(grad_mat, sol_vec)
new_point = MinNormSolver._next_point(sol_vec, grad_dir, n)
# Re-compute the inner products for line search
v1v1 = 0.0
v1v2 = 0.0
v2v2 = 0.0
for i in range(n):
for j in range(n):
v1v1 += sol_vec[i]*sol_vec[j]*dps[(i,j)]
v1v2 += sol_vec[i]*new_point[j]*dps[(i,j)]
v2v2 += new_point[i]*new_point[j]*dps[(i,j)]
nc, nd = MinNormSolver._min_norm_element_from2(v1v1, v1v2, v2v2)
new_sol_vec = nc*sol_vec + (1-nc)*new_point
change = new_sol_vec - sol_vec
if np.sum(np.abs(change)) < MinNormSolver.STOP_CRIT:
return sol_vec, nd
sol_vec = new_sol_vec
def find_min_norm_element_FW(vecs):
"""
Given a list of vectors (vecs), this method finds the minimum norm element in the convex hull
as min |u|_2 st. u = \sum c_i vecs[i] and \sum c_i = 1.
It is quite geometric, and the main idea is the fact that if d_{ij} = min |u|_2 st u = c x_i + (1-c) x_j; the solution lies in (0, d_{i,j})
Hence, we find the best 2-task solution, and then run the Frank Wolfe until convergence
"""
# Solution lying at the combination of two points
dps = {}
# init_sol, dps = MinNormSolver._min_norm_2d(vecs, dps)
init_sol, dps = MinNormSolver._min_norm_2d_accelerated(vecs, dps)
n=len(vecs)
sol_vec = np.zeros(n)
sol_vec[init_sol[0][0]] = init_sol[1]
sol_vec[init_sol[0][1]] = 1 - init_sol[1]
if n < 3:
# This is optimal for n=2, so return the solution
return sol_vec , init_sol[2]
iter_count = 0
grad_mat = np.zeros((n,n))
for i in range(n):
for j in range(n):
grad_mat[i,j] = dps[(i, j)]
for _ in range(MinNormSolver.MAX_ITER):
t_iter = np.argmin(np.dot(grad_mat, sol_vec))
v1v1 = np.dot(sol_vec, np.dot(grad_mat, sol_vec))
v1v2 = np.dot(sol_vec, grad_mat[:, t_iter])
v2v2 = grad_mat[t_iter, t_iter]
nc, nd = MinNormSolver._min_norm_element_from2(v1v1, v1v2, v2v2)
new_sol_vec = nc*sol_vec
new_sol_vec[t_iter] += 1 - nc
change = new_sol_vec - sol_vec
if np.sum(np.abs(change)) < MinNormSolver.STOP_CRIT:
return sol_vec, nd
sol_vec = new_sol_vec
iter_count += 1
return sol_vec, nd
def gradient_normalizers(grads, losses=None, normalization_type='l2'):
gn = {}
if normalization_type == 'l2':
for t in grads:
gn[t] = np.sqrt(np.sum([gr.pow(2).sum().data.cpu() for gr in grads[t]]))
elif normalization_type == 'loss':
for t in grads:
gn[t] = losses[t]
elif normalization_type == 'loss+':
for t in grads:
gn[t] = losses[t] * np.sqrt(np.sum([gr.pow(2).sum().data.cpu() for gr in grads[t]]))
elif normalization_type == 'none':
for t in grads:
gn[t] = 1.0
else:
print('ERROR: Invalid Normalization Type')
return gn