diff --git a/PyTorch/LanguageModeling/Transformer-XL/README.md b/PyTorch/LanguageModeling/Transformer-XL/README.md
index 1ab0a50d7..0ebf3a63a 100644
--- a/PyTorch/LanguageModeling/Transformer-XL/README.md
+++ b/PyTorch/LanguageModeling/Transformer-XL/README.md
@@ -1113,7 +1113,11 @@ perplexity on the test dataset.
 
 ## Performance
 
-The performance measurements in this document were conducted at the time of publication and may not reflect the performance achieved from NVIDIA’s latest software release. For the most up-to-date performance measurements, go to [NVIDIA Data Center Deep Learning Product Performance](https://developer.nvidia.com/deep-learning-performance-training-inference).
+The performance measurements in this document were conducted at the time of
+publication and may not reflect the performance achieved from NVIDIA’s latest
+software release. For the most up-to-date performance measurements, go to
+[NVIDIA Data Center Deep Learning Product
+Performance](https://developer.nvidia.com/deep-learning-performance-training-inference).
 
 ### Benchmarking
 
diff --git a/PyTorch/LanguageModeling/Transformer-XL/pytorch/inference/mem_transformer_jit.py b/PyTorch/LanguageModeling/Transformer-XL/pytorch/inference/mem_transformer_jit.py
index 4de68c6ff..aeaddfe74 100644
--- a/PyTorch/LanguageModeling/Transformer-XL/pytorch/inference/mem_transformer_jit.py
+++ b/PyTorch/LanguageModeling/Transformer-XL/pytorch/inference/mem_transformer_jit.py
@@ -122,7 +122,7 @@ def forward(self, h, attn_mask=None, mems=None):
         head_v = head_v.view(c.size(0), c.size(1), self.n_head, self.d_head)
 
         # [bsz x n_head x qlen x klen]
-        attn_score = torch.einsum('ibnd,jbnd->bnij', (head_q, head_k))
+        attn_score = torch.einsum('ibnd,jbnd->bnij', head_q, head_k)
         attn_score.mul_(self.scale)
         if attn_mask is not None:
             if attn_mask.dim() == 2:
@@ -135,7 +135,7 @@ def forward(self, h, attn_mask=None, mems=None):
         attn_prob = self.dropatt(attn_prob)
 
         # [bsz x n_head x qlen x klen] * [klen x bsz x n_head x d_head] -> [qlen x bsz x n_head x d_head]
-        attn_vec = torch.einsum('bnij,jbnd->ibnd', (attn_prob, head_v))
+        attn_vec = torch.einsum('bnij,jbnd->ibnd', attn_prob, head_v)
         attn_vec = attn_vec.contiguous().view(
             attn_vec.size(0), attn_vec.size(1), self.n_head * self.d_head)
 
@@ -262,13 +262,13 @@ def forward(self, w, r, r_w_bias, r_r_bias, attn_mask,
 
         # compute attention score
         rw_head_q = w_head_q + r_w_bias                                # qlen x bsz x n_head x d_head
-        # AC = torch.einsum('ibnd,jbnd->bnij', (rw_head_q, w_head_k))    # bsz x n_head x qlen x klen
+        # AC = torch.einsum('ibnd,jbnd->bnij', rw_head_q, w_head_k)    # bsz x n_head x qlen x klen
         rw_head_q = rw_head_q.view(qlen, bsz * self.n_head, self.d_head).permute(1, 0, 2)
         w_head_k = w_head_k.reshape(klen, bsz * self.n_head, self.d_head).permute(1, 2, 0)
         AC = torch.bmm(rw_head_q, w_head_k).view(bsz, self.n_head, qlen, klen)
 
         rr_head_q = w_head_q + r_r_bias
-        # BD = torch.einsum('ibnd,jnd->bnij', (rr_head_q, r_head_k))     # bsz x n_head x qlen x klen
+        # BD = torch.einsum('ibnd,jnd->bnij', rr_head_q, r_head_k)     # bsz x n_head x qlen x klen
         rr_head_q = rr_head_q.permute(2, 1, 0, 3).reshape(self.n_head, bsz * qlen, self.d_head)
         r_head_k = r_head_k.permute(1, 2, 0).view(self.n_head, self.d_head, klen)
         BD = torch.bmm(rr_head_q, r_head_k).view(self.n_head, bsz, qlen, klen).permute(1, 0, 2, 3)
@@ -290,7 +290,7 @@ def forward(self, w, r, r_w_bias, r_r_bias, attn_mask,
         attn_prob = self.dropatt(attn_prob)
 
         # compute attention vector
-        # attn_vec = torch.einsum('bnij,jbnd->ibnd', (attn_prob, w_head_v))
+        # attn_vec = torch.einsum('bnij,jbnd->ibnd', attn_prob, w_head_v)
         attn_prob = attn_prob.view(bsz * self.n_head, qlen, klen)
         w_head_v = w_head_v.permute(1, 2, 0, 3).reshape(bsz * self.n_head, klen, self.d_head)
         attn_vec = torch.bmm(attn_prob, w_head_v).permute(1, 0, 2).view(qlen, bsz, self.n_head, self.d_head)
@@ -358,11 +358,11 @@ def forward(self, w, r_emb, r_w_bias, r_bias, attn_mask=None, mems=None):
         r_bias = r_bias.t()
 
         # compute attention score
-        rw_head_q = w_head_q + r_w_bias[None]                        # qlen x bsz x n_head x d_head
+        rw_head_q = w_head_q + r_w_bias[None]                      # qlen x bsz x n_head x d_head
 
-        AC = torch.einsum('ibnd,jbnd->bnij', (rw_head_q, w_head_k))  # bsz x n_head x qlen x klen
-        B_ = torch.einsum('ibnd,jnd->bnij', (w_head_q, r_emb))       # bsz x n_head x qlen x klen
-        D_ = r_bias[None, :, None, :]                                # 1   x n_head x    1 x klen
+        AC = torch.einsum('ibnd,jbnd->bnij', rw_head_q, w_head_k)  # bsz x n_head x qlen x klen
+        B_ = torch.einsum('ibnd,jnd->bnij', w_head_q, r_emb)       # bsz x n_head x qlen x klen
+        D_ = r_bias[None, :, None, :]                              # 1   x n_head x    1 x klen
         BD = self._rel_shift(B_ + D_)
 
         # [bsz x qlen x klen x n_head]
@@ -381,7 +381,7 @@ def forward(self, w, r_emb, r_w_bias, r_bias, attn_mask=None, mems=None):
         attn_prob = self.dropatt(attn_prob)
 
         # compute attention vector
-        attn_vec = torch.einsum('bnij,jbnd->ibnd', (attn_prob, w_head_v))
+        attn_vec = torch.einsum('bnij,jbnd->ibnd', attn_prob, w_head_v)
 
         # [qlen x bsz x n_head x d_head]
         attn_vec = attn_vec.contiguous().view(
diff --git a/PyTorch/LanguageModeling/Transformer-XL/pytorch/inference/proj_adaptive_softmax_jit.py b/PyTorch/LanguageModeling/Transformer-XL/pytorch/inference/proj_adaptive_softmax_jit.py
index e50b2f0ba..5eb63482e 100644
--- a/PyTorch/LanguageModeling/Transformer-XL/pytorch/inference/proj_adaptive_softmax_jit.py
+++ b/PyTorch/LanguageModeling/Transformer-XL/pytorch/inference/proj_adaptive_softmax_jit.py
@@ -146,7 +146,7 @@ def _compute_logit(self, hidden, weight, bias, proj: Optional[torch.Tensor]):
         if proj is None:
             logit = F.linear(hidden, weight, bias=bias)
         else:
-            logit = torch.einsum('bd,de,ev->bv', (hidden, proj, weight.t()))
+            logit = torch.einsum('bd,de,ev->bv', hidden, proj, weight.t())
             if bias is not None:
                 logit = logit + bias
         return logit
diff --git a/PyTorch/LanguageModeling/Transformer-XL/pytorch/mem_transformer.py b/PyTorch/LanguageModeling/Transformer-XL/pytorch/mem_transformer.py
index 5b8145ca6..39a25977c 100644
--- a/PyTorch/LanguageModeling/Transformer-XL/pytorch/mem_transformer.py
+++ b/PyTorch/LanguageModeling/Transformer-XL/pytorch/mem_transformer.py
@@ -125,7 +125,7 @@ def forward(self, h, attn_mask=None, mems=None):
         head_v = head_v.view(c.size(0), c.size(1), self.n_head, self.d_head)
 
         # [bsz x n_head x qlen x klen]
-        attn_score = torch.einsum('ibnd,jbnd->bnij', (head_q, head_k))
+        attn_score = torch.einsum('ibnd,jbnd->bnij', head_q, head_k)
         attn_score.mul_(self.scale)
         if attn_mask is not None:
             if attn_mask.dim() == 2:
@@ -138,7 +138,7 @@ def forward(self, h, attn_mask=None, mems=None):
         attn_prob = self.dropatt(attn_prob)
 
         # [bsz x n_head x qlen x klen] * [klen x bsz x n_head x d_head] -> [qlen x bsz x n_head x d_head]
-        attn_vec = torch.einsum('bnij,jbnd->ibnd', (attn_prob, head_v))
+        attn_vec = torch.einsum('bnij,jbnd->ibnd', attn_prob, head_v)
         attn_vec = attn_vec.contiguous().view(
             attn_vec.size(0), attn_vec.size(1), self.n_head * self.d_head)
 
@@ -264,10 +264,10 @@ def forward(self, w, r, r_w_bias, r_r_bias, attn_mask=None, mems=None):
 
         # compute attention score
         rw_head_q = w_head_q + r_w_bias                                # qlen x bsz x n_head x d_head
-        AC = torch.einsum('ibnd,jbnd->bnij', (rw_head_q, w_head_k))    # bsz x n_head x qlen x klen
+        AC = torch.einsum('ibnd,jbnd->bnij', rw_head_q, w_head_k)      # bsz x n_head x qlen x klen
 
         rr_head_q = w_head_q + r_r_bias
-        BD = torch.einsum('ibnd,jnd->bnij', (rr_head_q, r_head_k))     # bsz x n_head x qlen x klen
+        BD = torch.einsum('ibnd,jnd->bnij', rr_head_q, r_head_k)       # bsz x n_head x qlen x klen
         BD = self._rel_shift(BD)
 
         # [bsz x n_head x qlen x klen]
@@ -285,7 +285,7 @@ def forward(self, w, r, r_w_bias, r_r_bias, attn_mask=None, mems=None):
         attn_prob = self.dropatt(attn_prob)
 
         # compute attention vector
-        attn_vec = torch.einsum('bnij,jbnd->ibnd', (attn_prob, w_head_v))
+        attn_vec = torch.einsum('bnij,jbnd->ibnd', attn_prob, w_head_v)
 
         # [qlen x bsz x n_head x d_head]
         attn_vec = attn_vec.contiguous().view(
@@ -350,11 +350,11 @@ def forward(self, w, r_emb, r_w_bias, r_bias, attn_mask=None, mems=None):
         r_bias = r_bias.t()
 
         # compute attention score
-        rw_head_q = w_head_q + r_w_bias[None]                        # qlen x bsz x n_head x d_head
+        rw_head_q = w_head_q + r_w_bias[None]                      # qlen x bsz x n_head x d_head
 
-        AC = torch.einsum('ibnd,jbnd->bnij', (rw_head_q, w_head_k))  # bsz x n_head x qlen x klen
-        B_ = torch.einsum('ibnd,jnd->bnij', (w_head_q, r_emb))       # bsz x n_head x qlen x klen
-        D_ = r_bias[None, :, None, :]                                # 1   x n_head x    1 x klen
+        AC = torch.einsum('ibnd,jbnd->bnij', rw_head_q, w_head_k)  # bsz x n_head x qlen x klen
+        B_ = torch.einsum('ibnd,jnd->bnij', w_head_q, r_emb)       # bsz x n_head x qlen x klen
+        D_ = r_bias[None, :, None, :]                              # 1   x n_head x    1 x klen
         BD = self._rel_shift(B_ + D_)
 
         # [bsz x qlen x klen x n_head]
@@ -372,7 +372,7 @@ def forward(self, w, r_emb, r_w_bias, r_bias, attn_mask=None, mems=None):
         attn_prob = self.dropatt(attn_prob)
 
         # compute attention vector
-        attn_vec = torch.einsum('bnij,jbnd->ibnd', (attn_prob, w_head_v))
+        attn_vec = torch.einsum('bnij,jbnd->ibnd', attn_prob, w_head_v)
 
         # [qlen x bsz x n_head x d_head]
         attn_vec = attn_vec.contiguous().view(
diff --git a/PyTorch/LanguageModeling/Transformer-XL/pytorch/run_enwik8_base.sh b/PyTorch/LanguageModeling/Transformer-XL/pytorch/run_enwik8_base.sh
index db542a835..b6f3a9d6b 100755
--- a/PyTorch/LanguageModeling/Transformer-XL/pytorch/run_enwik8_base.sh
+++ b/PyTorch/LanguageModeling/Transformer-XL/pytorch/run_enwik8_base.sh
@@ -1,5 +1,7 @@
 #!/bin/bash
 
+export OMP_NUM_THREADS=1
+
 if [[ $1 == 'train' ]]; then
     echo 'Run training...'
     python train.py \
diff --git a/PyTorch/LanguageModeling/Transformer-XL/pytorch/run_enwik8_large.sh b/PyTorch/LanguageModeling/Transformer-XL/pytorch/run_enwik8_large.sh
index 5db67bf76..539e3fae0 100755
--- a/PyTorch/LanguageModeling/Transformer-XL/pytorch/run_enwik8_large.sh
+++ b/PyTorch/LanguageModeling/Transformer-XL/pytorch/run_enwik8_large.sh
@@ -1,5 +1,7 @@
 #!/bin/bash
 
+export OMP_NUM_THREADS=1
+
 if [[ $1 == 'train' ]]; then
     echo 'Run training...'
     python train.py \
diff --git a/PyTorch/LanguageModeling/Transformer-XL/pytorch/run_lm1b_base.sh b/PyTorch/LanguageModeling/Transformer-XL/pytorch/run_lm1b_base.sh
index e4aebeff1..b0a988d56 100755
--- a/PyTorch/LanguageModeling/Transformer-XL/pytorch/run_lm1b_base.sh
+++ b/PyTorch/LanguageModeling/Transformer-XL/pytorch/run_lm1b_base.sh
@@ -1,5 +1,7 @@
 #!/bin/bash
 
+export OMP_NUM_THREADS=1
+
 if [[ $1 == 'train' ]]; then
     echo 'Run training...'
     python train.py \
diff --git a/PyTorch/LanguageModeling/Transformer-XL/pytorch/run_lm1b_large.sh b/PyTorch/LanguageModeling/Transformer-XL/pytorch/run_lm1b_large.sh
index f8b330ae6..7ac4db2c7 100755
--- a/PyTorch/LanguageModeling/Transformer-XL/pytorch/run_lm1b_large.sh
+++ b/PyTorch/LanguageModeling/Transformer-XL/pytorch/run_lm1b_large.sh
@@ -1,5 +1,7 @@
 #!/bin/bash
 
+export OMP_NUM_THREADS=1
+
 if [[ $1 == 'train' ]]; then
     echo 'Run training...'
     python train.py \
diff --git a/PyTorch/LanguageModeling/Transformer-XL/pytorch/run_multinode_wt103_large.sh b/PyTorch/LanguageModeling/Transformer-XL/pytorch/run_multinode_wt103_large.sh
index 35a531353..782d1a95e 100644
--- a/PyTorch/LanguageModeling/Transformer-XL/pytorch/run_multinode_wt103_large.sh
+++ b/PyTorch/LanguageModeling/Transformer-XL/pytorch/run_multinode_wt103_large.sh
@@ -14,6 +14,8 @@
 # See the License for the specific language governing permissions and
 # limitations under the License.
 
+export OMP_NUM_THREADS=1
+
 if [[ $1 == 'train' ]] || [[ $1 == 'all' ]]; then
     echo 'Run training...'
     python train.py \
diff --git a/PyTorch/LanguageModeling/Transformer-XL/pytorch/run_text8_base.sh b/PyTorch/LanguageModeling/Transformer-XL/pytorch/run_text8_base.sh
index 7058f77bb..6a38fb66a 100755
--- a/PyTorch/LanguageModeling/Transformer-XL/pytorch/run_text8_base.sh
+++ b/PyTorch/LanguageModeling/Transformer-XL/pytorch/run_text8_base.sh
@@ -1,5 +1,7 @@
 #!/bin/bash
 
+export OMP_NUM_THREADS=1
+
 if [[ $1 == 'train' ]]; then
     echo 'Run training...'
     python train.py \
diff --git a/PyTorch/LanguageModeling/Transformer-XL/pytorch/run_text8_large.sh b/PyTorch/LanguageModeling/Transformer-XL/pytorch/run_text8_large.sh
index cfc84df1c..8309409d4 100755
--- a/PyTorch/LanguageModeling/Transformer-XL/pytorch/run_text8_large.sh
+++ b/PyTorch/LanguageModeling/Transformer-XL/pytorch/run_text8_large.sh
@@ -1,5 +1,7 @@
 #!/bin/bash
 
+export OMP_NUM_THREADS=1
+
 if [[ $1 == 'train' ]]; then
     echo 'Run training...'
     python train.py \
diff --git a/PyTorch/LanguageModeling/Transformer-XL/pytorch/run_wt103_base.sh b/PyTorch/LanguageModeling/Transformer-XL/pytorch/run_wt103_base.sh
index 275da37d6..f68408db7 100755
--- a/PyTorch/LanguageModeling/Transformer-XL/pytorch/run_wt103_base.sh
+++ b/PyTorch/LanguageModeling/Transformer-XL/pytorch/run_wt103_base.sh
@@ -14,6 +14,8 @@
 # See the License for the specific language governing permissions and
 # limitations under the License.
 
+export OMP_NUM_THREADS=1
+
 if [[ "$1" == 'train' ]]; then
     echo 'Run training...'
     python -m torch.distributed.launch --nproc_per_node="$2" train.py \
diff --git a/PyTorch/LanguageModeling/Transformer-XL/pytorch/run_wt103_large.sh b/PyTorch/LanguageModeling/Transformer-XL/pytorch/run_wt103_large.sh
index bead4587e..c0350cc48 100755
--- a/PyTorch/LanguageModeling/Transformer-XL/pytorch/run_wt103_large.sh
+++ b/PyTorch/LanguageModeling/Transformer-XL/pytorch/run_wt103_large.sh
@@ -14,6 +14,8 @@
 # See the License for the specific language governing permissions and
 # limitations under the License.
 
+export OMP_NUM_THREADS=1
+
 if [[ $1 == 'train' ]]; then
     echo 'Run training...'
     python -m torch.distributed.launch --nproc_per_node="$2" train.py \
diff --git a/PyTorch/LanguageModeling/Transformer-XL/pytorch/train.py b/PyTorch/LanguageModeling/Transformer-XL/pytorch/train.py
index 6dc994514..b86176fae 100644
--- a/PyTorch/LanguageModeling/Transformer-XL/pytorch/train.py
+++ b/PyTorch/LanguageModeling/Transformer-XL/pytorch/train.py
@@ -285,6 +285,13 @@ def parse_args():
     if args.batch_size % args.batch_chunk != 0:
         raise RuntimeError('Batch size needs to be divisible by batch chunk')
 
+    if (
+        args.local_batch_size is not None
+        and args.local_batch_size % args.batch_chunk != 0
+    ):
+        raise RuntimeError('Local batch size needs to be divisible by '
+                           'batch chunk')
+
     if args.fp16 and args.amp == 'apex' and 'apex' not in sys.modules:
         raise RuntimeError(
             'APEX AMP unavailable, install APEX or switch to pytorch AMP'
@@ -444,8 +451,10 @@ def evaluate(eval_iter, model, args):
         for i, (data, target, seq_len, warm) in enumerate(eval_iter):
             if args.eval_max_steps > 0 and i >= args.eval_max_steps:
                 break
-            loss, mems = model(data, target, mems)
-            loss = loss.float().mean()
+            enable_autocast = args.fp16 and args.amp == 'pytorch'
+            with torch.cuda.amp.autocast(enable_autocast):
+                loss, mems = model(data, target, mems)
+                loss = loss.float().mean().type_as(loss)
             if warm:
                 # assert (mems is None) or mems.size(1) == model.mem_len
                 total_loss += seq_len * loss.item()
@@ -735,6 +744,9 @@ def main():
         args.batch_size = world_size * args.local_batch_size
         logging.info(f'--local_batch_size was set, adjusting global batch size'
                      f' to {args.batch_size} (local_batch_size * world_size)')
+        if args.batch_size % args.batch_chunk != 0:
+            raise RuntimeError('Batch size needs to be divisible by '
+                               'batch chunk')
 
     if args.profile:
         try:
diff --git a/PyTorch/LanguageModeling/Transformer-XL/pytorch/utils/log_uniform_sampler.py b/PyTorch/LanguageModeling/Transformer-XL/pytorch/utils/log_uniform_sampler.py
index 38a778072..6394b16e6 100644
--- a/PyTorch/LanguageModeling/Transformer-XL/pytorch/utils/log_uniform_sampler.py
+++ b/PyTorch/LanguageModeling/Transformer-XL/pytorch/utils/log_uniform_sampler.py
@@ -70,9 +70,9 @@ def sample_logits(embedding, bias, labels, inputs, sampler):
     hit = (labels[:, :, None] == neg_samples).detach()
 
     true_logits = torch.einsum('ijk,ijk->ij',
-        [true_w, inputs]) + true_b - true_log_probs
+                               true_w, inputs) + true_b - true_log_probs
     sample_logits = torch.einsum('lk,ijk->ijl',
-        [sample_w, inputs]) + sample_b - samp_log_probs
+                                 sample_w, inputs) + sample_b - samp_log_probs
     sample_logits.masked_fill_(hit, -1e30)
     logits = torch.cat([true_logits[:, :, None], sample_logits], -1)
 
diff --git a/PyTorch/LanguageModeling/Transformer-XL/pytorch/utils/proj_adaptive_softmax.py b/PyTorch/LanguageModeling/Transformer-XL/pytorch/utils/proj_adaptive_softmax.py
index e890a0686..f064bcbf5 100644
--- a/PyTorch/LanguageModeling/Transformer-XL/pytorch/utils/proj_adaptive_softmax.py
+++ b/PyTorch/LanguageModeling/Transformer-XL/pytorch/utils/proj_adaptive_softmax.py
@@ -112,7 +112,7 @@ def _compute_logit(self, hidden, weight, bias, proj):
         if proj is None:
             logit = F.linear(hidden, weight, bias=bias)
         else:
-            logit = torch.einsum('bd,de,ev->bv', (hidden, proj, weight.t()))
+            logit = torch.einsum('bd,de,ev->bv', hidden, proj, weight.t())
             if bias is not None:
                 logit = logit + bias
         return logit