Hessians for Spearman Rank Correlation

Question

I tried to implement the Spearman's rank correlation coefficient (wiki) as a custom objective function for xgboost. I'm using the fast-soft-sort (github) package from google for the differentiable ranking and tensorflow to automatically calculate the gradients. You can find the code below:

from fast_soft_sort.tf_ops import soft_rank
import tensorflow as tf
import numpy as np

def pearson_corr(x, y):
    
    xy_t = tf.concat([x, y], axis=0)
    mean_t = tf.reduce_mean(xy_t, axis=1, keepdims=True)
    cov_t = ((xy_t-mean_t) @ tf.transpose(xy_t-mean_t))/(x.shape[1]-1)
    cov2_t = tf.linalg.diag(1/tf.sqrt(tf.linalg.diag_part(cov_t)))
    corr_matrix = cov2_t @ cov_t @ cov2_t
    corr = tf.reduce_mean(corr_matrix) * 2 - 1 # equivalent to taking element [0][1] assuming the 2x2 corr matrix is symmetric and the diagonals are 1
    
    return corr

def spearman_corr(x, y):
    
    ranks = soft_rank(x, regularization_strength=0.1)
    corr = pearson_corr(ranks, y)
    
    return corr

def get_value_grad_and_hess(x, y, f):
    
    x_var = tf.Variable(x, dtype=tf.float32)
    y_var = tf.Variable(y, dtype=tf.float32)
        
    val, grad, hess = None, None, None

    with tf.GradientTape() as t2:
    
        with tf.GradientTape() as t1:
            
            val = f(x_var, y_var)
        
        grad = t1.gradient(val, x_var)    

    hess = t2.jacobian(grad, x_var)

    return val, grad, hess

# test with random input
x = np.random.rand(1, 10) # predictions
y = np.random.rand(1, 10) # labels

print('pearson:')
val, grad, hess = get_value_grad_and_hess(x, y, pearson_corr)
print(' value:',  val)
print(' gradient:', grad)
print(' hessian:', hess)

print('spearman:')
val, grad, hess = get_value_grad_and_hess(x, y, spearman_corr)
print(' value:',  val)
print(' gradient:', grad)
print(' hessian:', hess)

Example output:

pearson:
 value: tf.Tensor(-0.3348779, shape=(), dtype=float32)
 gradient: tf.Tensor(
[[ 0.21893269  0.16921082  0.19409613 -0.00321923  0.07347419  0.29004234
  -0.07947832 -0.7088071   0.29586902 -0.4501205 ]], shape=(1, 10), dtype=float32)
 hessian: tf.Tensor(
[[[[ 0.04441248 -0.03097764  0.02028688 -0.20294864 -0.22516166
    -0.09771542 -0.06334648  0.42131865 -0.02681065  0.16094248]]

  [[-0.03097765  0.40132353  0.04399774 -0.07797898 -0.05632872
     0.04975905 -0.07172927 -0.17790946  0.06856277 -0.14871901]]

  [[ 0.02028689  0.04399772  0.44207606 -0.06522453 -0.03210837
     0.0911998  -0.07974204 -0.30411014  0.10508882 -0.22146425]]

  [[-0.20294863 -0.077979   -0.06522458  0.27985442 -0.12591925
    -0.13325104 -0.02723934  0.31153008 -0.10839472  0.14957213]]

  [[-0.22516167 -0.05632871 -0.03210838 -0.12591931  0.23029271
    -0.10794277 -0.04108595  0.30121914 -0.07069567  0.12773061]]

  [[-0.09771542  0.04975905  0.0911998  -0.13325103 -0.10794276
     0.4497667  -0.09163402 -0.12746409  0.11477053 -0.14748882]]

  [[-0.06334649 -0.07172926 -0.07974204 -0.02723937 -0.04108596
    -0.09163402  0.35762674  0.07487351 -0.09705587  0.03933275]]

  [[ 0.4213187  -0.17790946 -0.3041101   0.31153005  0.3012191
    -0.12746407  0.07487351 -0.09769349 -0.2807703  -0.12099396]]

  [[-0.02681071  0.06856281  0.1050889  -0.10839473 -0.07069571
     0.11477058 -0.0970559  -0.28077024  0.5259669  -0.23066193]]

  [[ 0.1609425  -0.14871901 -0.22146428  0.1495721   0.12773061
    -0.14748883  0.03933276 -0.12099396 -0.23066193  0.39175004]]]], shape=(1, 10, 1, 10), dtype=float32)

spearman:
 value: tf.Tensor(-0.3408205, shape=(), dtype=float32)
 gradient: tf.Tensor(
[[ 0.13679196  0.13627169  0.15643153 -0.10963751 -0.02715444  0.2698098
   0.20591483 -0.8303905   0.26787752 -0.20591483]], shape=(1, 10), dtype=float32)
 hessian: None

As you can see the code above yields both gradient and hessian for the pearson correlation function but for the Spearman correlation the hessian is None.

Does someone have an idea why the hessian is None for the Spearman correlation?

Hessians for Spearman Rank Correlation

0 Answers0