Adds C++ and nn quantization utilities (#230)

* Add C++ de-/quantize ops
* Add quantize functions to the docs and tests
* Add a QuantizedLinear module

docs/src/python/nn/layers.rst

@@ -26,3 +26,4 @@ Layers
    RoPE
    MultiHeadAttention
    Sequential
+   QuantizedLinear

docs/src/python/ops.rst

@@ -34,6 +34,7 @@ Operations
    conv2d
    cos
    cosh
+   dequantize
    divide
    equal
    erf
@@ -73,6 +74,8 @@ Operations
    partition
    pad
    prod
+   quantize
+   quantized_matmul
    reciprocal
    reshape
    round

mlx/ops.cpp

@@ -2649,4 +2649,111 @@ array quantized_matmul(
   return out;
 }
 
+std::tuple<array, array, array> quantize(
+    const array& w,
+    int groups /* = 128 */,
+    int width /* = 4 */,
+    StreamOrDevice s /* = {} */) {
+  if (w.ndim() != 2) {
+    throw std::invalid_argument("[quantize] Only matrices supported for now");
+  }
+
+  if ((w.shape(0) % 32) != 0) {
+    throw std::invalid_argument(
+        "[quantize] All dimensions should be divisible by 32 for now");
+  }
+
+  if ((w.shape(-1) % groups) != 0) {
+    std::ostringstream msg;
+    msg << "[quantize] The last dimension of the matrix needs to be divisible by "
+        << "the quantization group size " << groups
+        << ". However the provided matrix"
+        << " has shape " << w.shape();
+    throw std::invalid_argument(msg.str());
+  }
+
+  // Compute some constants used for the quantization
+  int n_bins = (1 << width) - 1; // 2**width - 1
+  int el_per_int = 32 / width;
+  array shifts = power(array(2, uint32), arange(0, 32, width, uint32, s), s);
+  shifts = reshape(shifts, {1, 1, -1}, s);
+
+  // Compute scales and biases
+  array packed_w = reshape(w, {w.shape(0), w.shape(1) / groups, groups}, s);
+  array w_max = max(packed_w, /* axis= */ -1, /* keepdims= */ true, s);
+  array w_min = min(packed_w, /* axis= */ -1, /* keepdims= */ true, s);
+  array delta = divide(subtract(w_max, w_min, s), array(n_bins, w.dtype()), s);
+  array scales = squeeze(delta, -1, s);
+  array biases = squeeze(w_min, -1, s);
+
+  // Quantize and pack w
+  packed_w =
+      astype(round(divide(subtract(packed_w, w_min, s), delta, s), s), uint32);
+  packed_w = reshape(packed_w, {w.shape(0), -1, el_per_int}, s);
+  packed_w = sum(
+      multiply(packed_w, shifts, s), /* axis= */ 2, /* keepdims= */ false, s);
+
+  return std::make_tuple(packed_w, scales, biases);
+}
+
+array dequantize(
+    const array& w,
+    const array& scales,
+    const array& biases,
+    int groups /* = 128 */,
+    int width /* = 4 */,
+    StreamOrDevice s /* = {} */) {
+  if (w.ndim() != 2 || scales.ndim() != 2 || biases.ndim() != 2) {
+    throw std::invalid_argument("[dequantize] Only matrices supported for now");
+  }
+
+  if ((w.shape(0) % 32) != 0) {
+    throw std::invalid_argument(
+        "[dequantize] All dimensions should be divisible by 32 for now");
+  }
+
+  if (w.shape(0) != scales.shape(0) || w.shape(0) != biases.shape(0)) {
+    throw std::invalid_argument(
+        "[dequantize] Shape of scales and biases does not match the matrix");
+  }
+
+  if (w.dtype() != uint32) {
+    throw std::invalid_argument(
+        "[dequantize] The matrix should be given as a uint32");
+  }
+
+  // Compute some constants for the dequantization
+  int el_per_int = 32 / width;
+
+  if (w.shape(1) * el_per_int != scales.shape(1) * groups) {
+    std::ostringstream msg;
+    msg << "[dequantize] Shape of scales and biases does not match the matrix "
+        << "given the quantization parameters. Provided matrix of shape "
+        << w.shape() << " and scales/biases of shape " << scales.shape()
+        << " with groups=" << groups << " and width=" << width << ".";
+    throw std::invalid_argument(msg.str());
+  }
+
+  // Extract the pieces from the passed quantized matrix
+  std::vector<array> parts;
+  for (int start = 0; start < 32; start += width) {
+    // TODO: Implement bitwise operators for integral types
+    int shift_left = 32 - (start + width);
+    int shift_right = shift_left + start;
+    array p = multiply(w, array(1 << shift_left, uint32), s);
+    p = floor_divide(p, array(1 << shift_right, uint32), s);
+    p = expand_dims(p, -1, s);
+    parts.push_back(p);
+  }
+  array w_full = concatenate(parts, -1, s);
+
+  // Dequantize
+  w_full = reshape(w_full, {w.shape(0), -1, groups}, s);
+  w_full = multiply(w_full, expand_dims(scales, -1, s), s);
+  w_full = add(w_full, expand_dims(biases, -1, s), s);
+  w_full = reshape(w_full, {w.shape(0), -1}, s);
+
+  return w_full;
+}
+
 } // namespace mlx::core

mlx/ops.h

@@ -1041,4 +1041,20 @@ array quantized_matmul(
     int width = 4,
     StreamOrDevice s = {});
 
+/** Quantize a matrix along its last axis */
+std::tuple<array, array, array> quantize(
+    const array& w,
+    int groups = 128,
+    int width = 4,
+    StreamOrDevice s = {});
+
+/** Dequantize a matrix produced by quantize() */
+array dequantize(
+    const array& w,
+    const array& scales,
+    const array& biases,
+    int groups = 128,
+    int width = 4,
+    StreamOrDevice s = {});
+
 } // namespace mlx::core

python/mlx/nn/layers/__init__.py

@@ -38,6 +38,7 @@ from mlx.nn.layers.embedding import Embedding
 from mlx.nn.layers.linear import Linear
 from mlx.nn.layers.normalization import GroupNorm, LayerNorm, RMSNorm
 from mlx.nn.layers.positional_encoding import RoPE, SinusoidalPositionalEncoding
+from mlx.nn.layers.quantized import QuantizedLinear
 from mlx.nn.layers.transformer import (
     MultiHeadAttention,
     TransformerEncoder,

python/mlx/nn/layers/base.py

@@ -258,6 +258,44 @@ class Module(dict):
         filter_fn = filter_fn or Module.valid_parameter_filter
         self.update(self.filter_and_map(filter_fn, map_fn))
 
+    def update_modules(self, modules: dict):
+        """Replace the child modules of this :class:`Module` instance with the
+        provided ones in the dict of dicts and lists.
+
+        It is the equivalent of :meth:`Module.update` but for modules instead
+        of parameters and allows us to flexibly edit complex architectures by
+        programmatically swapping layers.
+
+        The passed in parameters dictionary need not be a full dictionary
+        similar to :meth:`parameters`. Only the provided locations will be
+        updated.
+
+        Args:
+            modules (dict): A complete or partial dictionary of the modules
+                submodules.
+        """
+
+        def apply(dst, modules):
+            if isinstance(modules, dict):
+                for k in modules:
+                    if k in dst:
+                        current_value = dst[k]
+                        new_value = modules[k]
+                        if self.is_module(current_value) and self.is_module(new_value):
+                            dst[k] = new_value
+                        elif isinstance(current_value, (dict, list)):
+                            apply(current_value, new_value)
+            elif isinstance(modules, list):
+                for i in range(len(dst)):
+                    current_value = dst[i]
+                    new_value = modules[i]
+                    if self.is_module(current_value) and self.is_module(new_value):
+                        dst[i] = new_value
+                    elif isinstance(current_value, (dict, list)):
+                        apply(current_value, new_value)
+
+        apply(self, modules)
+
     def apply_to_modules(self, apply_fn: Callable[[str, "mlx.nn.Module"], Any]):
         """Apply a function to all the modules in this instance (including this
         instance).

python/mlx/nn/layers/quantized.py

@@ -0,0 +1,124 @@
+# Copyright © 2023 Apple Inc.
+
+import math
+
+import mlx.core as mx
+from mlx.nn.layers.base import Module
+from mlx.nn.layers.linear import Linear
+from mlx.utils import tree_flatten, tree_map
+
+
+class QuantizedLinear(Module):
+    """Applies an affine transformation to the input using a quantized weight matrix.
+
+    It is the quantized equivalent of :class:`mlx.nn.Linear`. For now its
+    parameters are frozen and will not be included in any gradient computation
+    but this will probably change in the future.
+
+    QuantizedLinear also provides two useful classmethods to convert linear
+    layers to QuantizedLinear layers.
+
+    - :meth:`from_linear` returns a QuantizedLinear layer that applies the same
+      linear transformation up to the quantization error.
+    - :meth:`quantize_module` swaps all the linear layers of the passed module
+      with QuantizedLinear ones.
+
+    Args:
+        input_dims (int): The dimensionality of the input features
+        output_dims (int): The dimensionality of the output features
+        bias (bool): If set to ``False`` then the layer will not use a bias.
+            (default: True).
+        groups (int): The group size to use for the quantized weight. See
+            :func:`~mlx.core.quantize`. (default: 128)
+        width (int): The bit width to use for the quantized weight. See
+            :func:`~mlx.core.quantize`. (default: 4)
+    """
+
+    def __init__(
+        self,
+        input_dims: int,
+        output_dims: int,
+        bias: bool = True,
+        groups: int = 64,
+        width: int = 4,
+    ):
+        super().__init__()
+
+        # Quantization config
+        self.groups = groups
+        self.width = width
+
+        # Initialize the quantized weight
+        scale = math.sqrt(1 / input_dims)
+        weight = mx.random.uniform(
+            low=-scale,
+            high=scale,
+            shape=(output_dims, input_dims),
+        )
+        self.weight, self.scales, self.biases = mx.quantize(weight, groups, width)
+
+        # And bias if needed
+        if bias:
+            self.bias = mx.zeros((output_dims,))
+
+        # Freeze this model's parameters
+        self.freeze()
+
+    def unfreeze(self, *args, **kwargs):
+        """Wrap unfreeze so that we unfreeze any layers we might contain but
+        our parameters will remain frozen."""
+        super().unfreeze(*args, **kwargs)
+        self.freeze(recurse=False)
+
+    def _extra_repr(self):
+        out_dims, in_dims = self.weight.shape
+        in_dims *= 32 // self.width
+        return (
+            f"input_dims={in_dims}, output_dims={out_dims}, bias={'bias' in self},"
+            f"groups={self.groups}, width={self.width}"
+        )
+
+    def __call__(self, x):
+        x = mx.quantized_matmul(
+            x,
+            self.weight.T,
+            scales=self.scales,
+            biases=self.biases,
+            groups=self.groups,
+            width=self.width,
+        )
+        if "bias" in self:
+            x = x + self.bias
+        return x
+
+    @classmethod
+    def from_linear(cls, linear_layer: Module, groups: int = 64, width: int = 4):
+        """Create a QuantizedLinear layer from the parameters of a provided
+        linear layer."""
+        output_dims, input_dims = linear_layer.weight.shape
+        ql = cls(input_dims, output_dims, False, groups, width)
+        ql.weight, ql.scales, ql.biases = mx.quantize(
+            linear_layer.weight, groups, width
+        )
+        if "bias" in linear_layer:
+            ql.bias = linear_layer.bias
+
+        return ql
+
+    @classmethod
+    def quantize_module(
+        cls,
+        model: Module,
+        groups: int = 64,
+        width: int = 4,
+        linear_class_predicate=lambda m: isinstance(m, Linear),
+    ):
+        def _quantize_if_linear(m):
+            if linear_class_predicate(m):
+                return cls.from_linear(m, groups, width)
+            else:
+                return m
+
+        leaves = model.leaf_modules()
+        leaves = tree_map(_quantize_if_linear, leaves, is_leaf=Module.is_module)
+        model.update_modules(leaves)

python/mlx/utils.py

@@ -1,7 +1,7 @@
 # Copyright © 2023 Apple Inc.
 
 
-def tree_map(fn, tree, *rest):
+def tree_map(fn, tree, *rest, is_leaf=None):
     """Applies ``fn`` to the leaves of the python tree ``tree`` and
     returns a new collection with the results.
 
@@ -10,6 +10,9 @@ def tree_map(fn, tree, *rest):
     ``fn``. In that respect, :meth:`tree_map` is closer to :func:`itertools.starmap`
     than to :func:`map`.
 
+    The keyword argument ``is_leaf`` decides what constitutes a leaf from
+    ``tree`` similar to :func:`tree_flatten`.
+
     .. code-block:: python
 
         import mlx.nn as nn
@@ -26,21 +29,28 @@ def tree_map(fn, tree, *rest):
         fn (Callable): The function that processes the leaves of the tree
         tree (Any): The main python tree that will be iterated upon
         rest (Tuple[Any]): Extra trees to be iterated together with tree
+        is_leaf (Optional[Callable]): An optional callable that returns True if
+            the passed object is considered a leaf or False otherwise.
 
     Returns:
         A python tree with the new values returned by ``fn``.
     """
-    if isinstance(tree, list):
+    if is_leaf is not None and is_leaf(tree):
+        return fn(tree, *rest)
+    elif isinstance(tree, list):
         return [
-            tree_map(fn, child, *(r[i] for r in rest)) for i, child in enumerate(tree)
+            tree_map(fn, child, *(r[i] for r in rest), is_leaf=is_leaf)
+            for i, child in enumerate(tree)
         ]
     elif isinstance(tree, tuple):
         return tuple(
-            tree_map(fn, child, *(r[i] for r in rest)) for i, child in enumerate(tree)
+            tree_map(fn, child, *(r[i] for r in rest), is_leaf=is_leaf)
+            for i, child in enumerate(tree)
         )
     elif isinstance(tree, dict):
         return {
-            k: tree_map(fn, child, *(r[k] for r in rest)) for k, child in tree.items()
+            k: tree_map(fn, child, *(r[k] for r in rest), is_leaf=is_leaf)
+            for k, child in tree.items()
         }
     else:
         return fn(tree, *rest)

python/src/ops.cpp

@@ -3035,4 +3035,101 @@ void init_ops(py::module_& m) {
         Returns:
           result (array): The result of the multiplication of ``x`` with ``w``.
       )pbdoc");
+  m.def(
+      "quantize",
+      &quantize,
+      "w"_a,
+      py::pos_only(),
+      "groups"_a = 128,
+      "width"_a = 4,
+      py::kw_only(),
+      "stream"_a = none,
+      R"pbdoc(
+        quantize(w: array, /, groups: int = 128, width: int = 4, *, stream: Union[None, Stream, Device] = None) -> Tuple[array, array, array]
+
+        Quantize the matrix ``w`` using ``width`` bits per element.
+
+        Note, every ``groups`` elements in a row of ``w`` are quantized
+        together. Hence, number of columns of ``w`` should be divisible by
+        ``groups``. In particular, the rows of ``w`` are divided into groups of
+        size ``groups`` which are quantized together.
+
+        .. warning::
+
+          ``quantize`` currently only supports 2D inputs with dimensions which are multiples of 32
+
+        Formally, for a group of :math:`g` consecutive elements :math:`w_1` to
+        :math:`w_g` in a row of ``w`` we compute the quantized representation
+        of each element :math:`\hat{w_i}` as follows
+
+        .. math::
+
+          \begin{aligned}
+            \alpha &= \max_i w_i \\
+            \beta &= \min_i w_i \\
+            s &= \frac{\alpha - \beta}{2^b - 1} \\
+            \hat{w_i} &= \textrm{round}\left( \frac{w_i - \beta}{s}\right).
+          \end{aligned}
+
+        After the above computation, :math:`\hat{w_i}` fits in :math:`b` bits
+        and is packed in an unsigned 32-bit integer from the lower to upper
+        bits. For instance, for 4-bit quantization we fit 8 elements in an
+        unsigned 32 bit integer where the 1st element occupies the 4 least
+        significant bits, the 2nd bits 4-7 etc.
+
+        In order to be able to dequantize the elements of ``w`` we also need to
+        save :math:`s` and :math:`\beta` which are the returned ``scales`` and
+        ``biases`` respectively.
+
+        Args:
+          w (array): Matrix to be quantized
+          groups (int, optional): The size of the group in ``w`` that shares a
+            scale and bias. (default: 128)
+          width (int, optional): The bitwidth of the elements in ``w``.
+            (default: 4)
+
+        Returns:
+          (tuple): A tuple containing
+
+            - w_q (array): The quantized version of ``w``
+            - scales (array): The scale to multiply each element with, namely :math:`s`
+            - biases (array): The biases to add to each element, namely :math:`\beta`
+      )pbdoc");
+  m.def(
+      "dequantize",
+      &dequantize,
+      "w"_a,
+      py::pos_only(),
+      "scales"_a,
+      "biases"_a,
+      "groups"_a = 128,
+      "width"_a = 4,
+      py::kw_only(),
+      "stream"_a = none,
+      R"pbdoc(
+        dequantize(w: array, /, scales: array, biases: array, groups: int = 128, width: int = 4, *, stream: Union[None, Stream, Device] = None) -> array
+
+        Dequantize the matrix ``w`` using the provided ``scales`` and
+        ``biases`` and the ``groups`` and ``width`` configuration.
+
+        Formally, given the notation in :func:`quantize`, we compute
+        :math:`w_i` from :math:`\hat{w_i}` and corresponding :math:`s` and
+        :math:`\beta` as follows
+
+        .. math::
+
+          w_i = s \hat{w_i} - \beta
+
+        Args:
+          w (array): Matrix to be quantized
+          scales (array): The scales to use per ``groups`` elements of ``w``
+          biases (array): The biases to use per ``groups`` elements of ``w``
+          groups (int, optional): The size of the group in ``w`` that shares a
+            scale and bias. (default: 128)
+          width (int, optional): The bitwidth of the elements in ``w``.
+            (default: 4)
+
+        Returns:
+          result (array): The dequantized version of w
+      )pbdoc");
 }

python/tests/test_quantized.py

@@ -6,48 +6,14 @@ import mlx.core as mx
 import mlx_tests
 
 
-def select_bits(w, width, start):
-    shift_left = 32 - (start + width)
-    shift_right = shift_left + start
-    return (w * (2**shift_left)) // (2**shift_right)
-
-
-def dequantize(w, scales, biases, width):
-    w_full = mx.concatenate(
-        [select_bits(w, width, i)[..., None] for i in range(0, 32, width)], axis=-1
-    )
-    w_full = w_full.reshape(len(w), scales.shape[-1], -1)
-    w_full = scales[..., None] * w_full + biases[..., None]
-    w_full = w_full.reshape(len(w), -1)
-
-    return w_full
-
-
-def quantize(w, width, groups):
-    w = w.reshape(len(w), -1, groups)
-    w_max = w.max(-1, keepdims=True)
-    w_min = w.min(-1, keepdims=True)
-    delta = (w_max - w_min) / (2**width - 1)
-
-    w_int = mx.round((w - w_min) / delta).astype(mx.uint32)
-    scales = delta.squeeze(-1)
-    biases = w_min.squeeze(-1)
-
-    shifts = mx.array([2**i for i in range(0, 32, width)], dtype=mx.uint32)
-    w_int = w_int.reshape(len(w), -1, 32 // width)
-    w_int = w_int * shifts[None, None]
-    packed_w = w_int.sum(-1)
-
-    return packed_w, scales, biases
-
-
 class TestQuantized(mlx_tests.MLXTestCase):
     def test_quantize_dequantize(self):
         w = mx.random.normal(shape=(128, 128))
-        w_q, scales, biases = quantize(w, 4, 64)
-        w_hat = dequantize(w_q, scales, biases, 4)
-        w_hat2 = dequantize(*quantize(w_hat, 4, 64), 4)
-        self.assertLess((w_hat - w_hat2).abs().max(), 1e-6)
+        for b in [2, 4, 8]:
+            w_q, scales, biases = mx.quantize(w, 64, b)
+            w_hat = mx.dequantize(w_q, scales, biases, 64, b)
+            errors = (w - w_hat).abs().reshape(*scales.shape, -1)
+            self.assertTrue((errors <= scales[..., None] / 2).all())
 
     def test_qmm(self):
         key = mx.random.key(0)
@@ -62,14 +28,16 @@ class TestQuantized(mlx_tests.MLXTestCase):
                             ):
                                 x = mx.random.normal(shape=(M, K), key=k1)
                                 w = mx.random.normal(shape=(N, K), key=k2)
-                                w_q, scales, biases = quantize(w, width, groups)
-                                w_hat = dequantize(w_q, scales, biases, width)
+                                w_q, scales, biases = mx.quantize(w, groups, width)
+                                w_hat = mx.dequantize(
+                                    w_q, scales, biases, groups, width
+                                )
                                 y_q = mx.quantized_matmul(
                                     x, w_q.T, scales, biases, width=width, groups=groups
                                 )
                                 y_hat = x @ w_hat.T
                                 self.assertEqual(y_q.shape, y_hat.shape)
-                                self.assertLess((y_q - y_hat).abs().max(), 0.1)
+                                self.assertLess((y_q - y_hat).abs().max(), 1e-3)
 
     def test_qmm_shapes(self):
         key = mx.random.key(0)
@@ -77,8 +45,8 @@ class TestQuantized(mlx_tests.MLXTestCase):
         groups = 64
         width = 4
         w = mx.random.normal(shape=(32, 128), key=k2)
-        w_q, scales, biases = quantize(w, width, groups)
-        w_hat = dequantize(w_q, scales, biases, width)
+        w_q, scales, biases = mx.quantize(w, groups, width)
+        w_hat = mx.dequantize(w_q, scales, biases, groups, width)
         for s in [(3, 128), (2, 1, 7, 128)]:
             x = mx.random.normal(shape=(3, 128), key=k1)
             y_q = mx.quantized_matmul(
@@ -86,7 +54,7 @@ class TestQuantized(mlx_tests.MLXTestCase):
             )
             y_hat = x @ w_hat.T
             self.assertEqual(y_q.shape, y_hat.shape)
-            self.assertLess((y_q - y_hat).abs().max(), 0.1)
+            self.assertLess((y_q - y_hat).abs().max(), 1e-3)
 
     def test_qmv(self):
         key = mx.random.key(0)
@@ -95,17 +63,17 @@ class TestQuantized(mlx_tests.MLXTestCase):
             for width in [2, 4, 8]:
                 for M in [512, 1024]:
                     for N in [512, 1024]:
-                        # with self.subTest(shape=(M, N), groups=groups, width=width):
-                        x = mx.random.normal(shape=(1, N), key=k1)
-                        w = mx.random.normal(shape=(M, N), key=k2)
-                        w_q, scales, biases = quantize(w, width, groups)
-                        w_hat = dequantize(w_q, scales, biases, width)
-                        y_q = mx.quantized_matmul(
-                            x, w_q.T, scales, biases, width=width, groups=groups
-                        )
-                        y_hat = x @ w_hat.T
-                        self.assertEqual(y_q.shape, y_hat.shape)
-                        self.assertLess((y_q - y_hat).abs().max(), 0.1)
+                        with self.subTest(shape=(M, N), groups=groups, width=width):
+                            x = mx.random.normal(shape=(1, N), key=k1)
+                            w = mx.random.normal(shape=(M, N), key=k2)
+                            w_q, scales, biases = mx.quantize(w, groups, width)
+                            w_hat = mx.dequantize(w_q, scales, biases, groups, width)
+                            y_q = mx.quantized_matmul(
+                                x, w_q.T, scales, biases, width=width, groups=groups
+                            )
+                            y_hat = x @ w_hat.T
+                            self.assertEqual(y_q.shape, y_hat.shape)
+                            self.assertLess((y_q - y_hat).abs().max(), 1e-3)
 
 
 if __name__ == "__main__":

tests/ops_tests.cpp

@@ -2215,3 +2215,21 @@ TEST_CASE("test linspace") {
   expected = array(std::initializer_list<float>{}, {0});
   CHECK(array_equal(x, expected).item<bool>());
 }
+
+TEST_CASE("test quantize dequantize") {
+  auto x1 = ones({128, 1});
+  auto x2 = expand_dims(arange(0, 128, float32), 0);
+  auto x = x1 * x2;
+
+  for (int i = 2; i <= 8; i *= 2) {
+    int el_per_int = 32 / i;
+    auto [x_q, scales, biases] = quantize(x, 128, i);
+    CHECK_EQ(x_q.shape(), std::vector<int>{128, 128 / el_per_int});
+    CHECK_EQ(scales.shape(), std::vector<int>{128, 1});
+    CHECK_EQ(biases.shape(), std::vector<int>{128, 1});
+
+    auto x_hat = dequantize(x_q, scales, biases, 128, i);
+    auto max_diff = max(abs(x - x_hat)).item<float>();
+    CHECK(max_diff <= 127.0 / (1 << i));
+  }
+}