diff --git a/egs/librispeech/ASR/pruned_transducer_stateless7/optim.py b/egs/librispeech/ASR/pruned_transducer_stateless7/optim.py
index a0d9eacc9..65cfeedc3 100644
--- a/egs/librispeech/ASR/pruned_transducer_stateless7/optim.py
+++ b/egs/librispeech/ASR/pruned_transducer_stateless7/optim.py
@@ -144,7 +144,6 @@ param_rms_smooth1: Smoothing proportion for parameter matrix, if assumed rank of
             size_lr_scale=0.1,
             min_lr_factor=(0.05, 0.01, 0.01),
             max_lr_factor=(10.0, 40.0, 10.0),
-            #param_pow=(0.99999, 0.99999, 0.99999),
             param_pow=(1.0, 1.0, 1.0),
             param_rms_smooth0=0.4,
             param_rms_smooth1=0.2,
@@ -154,7 +153,10 @@ param_rms_smooth1: Smoothing proportion for parameter matrix, if assumed rank of
             scalar_max=2.0,
             size_update_period=4,
             lr_update_period=(200, 1000),
-            grad_cov_period=3,
+            # grad_cov_period does not affect speed very much.  If __name__ ==
+            # "__main__", keeping this coprime to 20 which is the number of
+            # sample pairs used in the test code
+            grad_cov_period=(3 if __name__ == "__main__" else 5),
             max_block_size=1024,
     ):
 
@@ -464,7 +466,6 @@ param_rms_smooth1: Smoothing proportion for parameter matrix, if assumed rank of
 
         scale_step.masked_fill_(is_too_small, size_lr * size_update_period)
         scale_step.masked_fill_(is_too_large, -size_lr * size_update_period)
-
         delta = state["delta"]
         # the factor of (1-beta1) relates to momentum.
         delta.add_(p * scale_step, alpha=(1-beta1))
@@ -891,12 +892,13 @@ param_rms_smooth1: Smoothing proportion for parameter matrix, if assumed rank of
                 # A check.
                 # Z_prime is supposed to satisfy: Z_prime G_prime Z_prime = P_prime  (eqn:2)
                 P_prime_check = torch.matmul(Z_prime * G_prime.unsqueeze(-2), Z_prime)
-                diff_ratio = ((P_prime - P_prime_check) ** 2).sum().sqrt() // (P_prime ** 2).sum().sqrt()
-                if diff_ratio > 0.001:
-                    logging.warn(f"Z_prime does not satisfy its definition, diff_ratio = {diff_ratio}, size={size}")
+                diff_ratio_l2 = ((P_prime - P_prime_check) ** 2).sum().sqrt() // (P_prime ** 2).sum().sqrt()
+                diff_ratio_l1 = (P_prime - P_prime_check).abs().sum() // P_prime.abs().sum()
+                if diff_ratio_l2 > 0.00001 or diff_ratio_l1 > 0.03:
+                    logging.warn(f"Z_prime does not satisfy its definition, diff_ratio_l{1,2} = {diff_ratio_l1.item(),diff_ratio_l2.item()}, size={size}")
             Z = torch.matmul(U_g, torch.matmul(Z_prime, U_g.transpose(2, 3)))
             # OK, Z is the SPD transform that maps G to P, as in Z G Z = P.
-            # We just need the basis that diagonalizes this.
+            # We just need the basis U_z that diagonalizes this.
             U_z, S, _ = _svd(Z)
             if True:
                 skip = 10 if S.shape[-1] > 40 else 1
@@ -1268,7 +1270,6 @@ param_rms_smooth1: Smoothing proportion for parameter matrix, if assumed rank of
         # slower update at the start will help stability anyway.
         bias_correction2 = 1 - beta2 ** (state["step"] + 1)
         denom = (exp_avg_sq / bias_correction2).sqrt() + eps
-        alpha = -lr * (1-beta1) / denom
 
         delta = state["delta"]
         delta.add_(grad / denom, alpha=-lr*(1-beta1))