[SYSTEMDS-3821] Add GELU Activation Function (Approximation)

This patch introduces the Gaussian Error Linear Unit (GELU) activation
function to SystemDS as a built-in operation. The implementation uses the
widely adopted approximate formulation (https://arxiv.org/abs/1606.08415).

This patch is a part of a series of commits to support popular Transformer
architectures in SystemDS. The GELU activation the most commonly used
activation functions in models like BERT and GPT.

Closes #2177

scripts/nn/layers/gelu.dml

@@ -0,0 +1,70 @@
+#-------------------------------------------------------------
+#
+# Licensed to the Apache Software Foundation (ASF) under one
+# or more contributor license agreements.  See the NOTICE file
+# distributed with this work for additional information
+# regarding copyright ownership.  The ASF licenses this file
+# to you under the Apache License, Version 2.0 (the
+# "License"); you may not use this file except in compliance
+# with the License.  You may obtain a copy of the License at
+#
+#   http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing,
+# software distributed under the License is distributed on an
+# "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
+# KIND, either express or implied.  See the License for the
+# specific language governing permissions and limitations
+# under the License.
+#
+#-------------------------------------------------------------
+
+/*
+ * Gaussian Error Linear Unit (GELU) nonlinearity layer.
+ */
+
+source("nn/layers/tanh.dml") as tanh
+
+forward = function(matrix[double] X)
+    return (matrix[double] out) {
+  /*
+   * Computes the forward pass for a GELU nonlinearity layer, via 
+   * its tanh approximation.
+   *
+   * Performs an element-wise evaluation of 
+   *    `GELU(x) = x * CDF(x)`.
+   * where CDF is the cumulative distribution function of the 
+   * standard normal distribution:
+   *    `CDF(x) = 0.5 * (1 + erf(x/sqrt(2)))`
+   * This implementation uses the tanh approximation:
+   *    `CDF(x) =~ 0.5 * (1 + tanh(sqrt(2/pi) * (x + 0.044715x^3)))`
+   *
+   * Inputs:
+   *  - X: Inputs, of shape (any, any).
+   *
+   * Outputs:
+   *  - out: Outputs, of same shape as `X`.
+   */
+  cdf = 0.5 * (1 + tanh(sqrt(2 / pi) * (X + 0.044715 * X^3)))
+  out = cdf * X
+}
+
+backward = function(matrix[double] dout, matrix[double] X)
+    return (matrix[double] dX) {
+  /*
+   * Computes the backward pass for a GELU nonlinearity layer, via 
+   * its tanh approximation.
+   *
+   * Inputs:
+   *  - dout: Gradient wrt `out` from upstream, of same shape as `X`.
+   *  - X: Previous input data matrix, of shape (any, any).
+   *
+   * Outputs:
+   *  - dX: Gradient wrt `X`, of same shape as `X`.
+   */
+   a = sqrt(2 / pi)
+   b = 0.044715
+   T = tanh(a * (X + b * X^3))
+   dT = 1 - T^2
+   dX = dout * (0.5 * (1 + T) + 0.5 * X * dT * a * (1 + 3 * b * X^2))
+}

src/test/java/org/apache/sysds/test/applications/nn/NNComponentTest.java

@@ -124,6 +124,11 @@ public void resnet() {
 		run("resnet_bottleneck.dml");
 	}
 
+	@Test
+	public void gelu() {
+		run("gelu.dml");
+	}
+
 	@Override
 	protected void run(String name) {
 		super.run("component/" + name);

src/test/scripts/applications/nn/component/gelu.dml

@@ -0,0 +1,66 @@
+#-------------------------------------------------------------
+#
+# Licensed to the Apache Software Foundation (ASF) under one
+# or more contributor license agreements.  See the NOTICE file
+# distributed with this work for additional information
+# regarding copyright ownership.  The ASF licenses this file
+# to you under the Apache License, Version 2.0 (the
+# "License"); you may not use this file except in compliance
+# with the License.  You may obtain a copy of the License at
+#
+#   http://www.apache.org/licenses/LICENSE-2.0
+#
+# Unless required by applicable law or agreed to in writing,
+# software distributed under the License is distributed on an
+# "AS IS" BASIS, WITHOUT WARRANTIES OR CONDITIONS OF ANY
+# KIND, either express or implied.  See the License for the
+# specific language governing permissions and limitations
+# under the License.
+#
+#-------------------------------------------------------------
+
+source("nn/layers/gelu.dml") as gelu
+source("src/test/scripts/applications/nn/util.dml") as test_util
+
+gelu_test1 = function() {
+    print("Testing GELU, test 1")
+
+    X = matrix("1.  -0.5
+                0.   2.", rows=2, cols=2)
+    dout = matrix("1 1
+                1 1", rows=2, cols=2)
+    out_expected = matrix("0.841192  -0.154286
+                        0.         1.9545977", rows=2, cols=2)
+    gradient_expected = matrix("1.0829641  0.13263011
+                                0.5        1.0860993", rows=2, cols=2)
+
+    out = gelu::forward(X)
+
+    test_util::check_all_close(out, out_expected, 0.00001)
+
+    gradient = gelu::backward(dout, X)
+    test_util::check_all_close(gradient, gradient_expected, 0.00001)
+}
+
+gelu_test2 = function() {
+    print("Testing GELU, test 2")
+
+    X = matrix("0.5 -1.5
+                1.  -2.", rows=2, cols=2)
+    dout = matrix("1 1
+                1 1", rows=2, cols=2)
+    out_expected = matrix("0.345714   -0.10042843
+                        0.841192   -0.04540229", rows=2, cols=2)
+    gradient_expected = matrix("0.8673699  -0.1277108
+                                1.0829641  -0.08609922", rows=2, cols=2)
+
+    out = gelu::forward(X)
+
+    test_util::check_all_close(out, out_expected, 0.00001)
+
+    gradient = gelu::backward(dout, X)
+    test_util::check_all_close(gradient, gradient_expected, 0.00001)
+}
+
+gelu_test1()
+gelu_test2()