prompt: We are currently solving the following issue within our repository. Here is the issue text:
  --- BEGIN ISSUE ---
  split_axis.backward fails on incomplete gradients
  When there is a None in the grad_outputs, split_axis fails to backprop the incomplete gradients.
  
  
  --- END ISSUE ---
  
  Below are some code segments, each from a relevant file. One or more of these files may contain bugs.
  --- BEGIN FILES ---
  Path: `chainer/functions/split_axis.py`
  Content:
  ```
  1 import collections
  2 
  3 import numpy
  4 
  5 from chainer import cuda
  6 from chainer import function
  7 from chainer.utils import type_check
  8 
  9 
  10 _args = 'float* y, float* x, int cdimy, int cdimx, int rdim, int coffset'
  11 _preamble = '''
  12 #define COPY(statement) \
  13     int l   = i / (rdim * cdimy);  \
  14     int c   = i / rdim % cdimy + coffset;  \
  15     int r   = i % rdim;  \
  16     int idx = r + rdim * (c + cdimx * l);  \
  17     statement;
  18 '''
  19 
  20 
  21 class SplitAxis(function.Function):
  22 
  23     """Function that splits multiple arrays towards the specified axis."""
  24 
  25     def __init__(self, indices_or_sections, axis):
  26         if not isinstance(indices_or_sections, (int, collections.Iterable)):
  27             raise TypeError('indices_or_sections must be integer or 1-D array')
  28         self.indices_or_sections = indices_or_sections
  29         self.axis = axis
  30 
  31     def check_type_forward(self, in_types):
  32         type_check.expect(in_types.size() == 1)
  33         type_check.expect(in_types[0].ndim >= self.axis)
  34 
  35         if isinstance(self.indices_or_sections, collections.Iterable):
  36             max_index = type_check.Variable(
  37                 self.indices_or_sections[-1], 'max_index')
  38             type_check.expect(in_types[0].shape[self.axis] > max_index)
  39         else:
  40             sections = type_check.Variable(
  41                 self.indices_or_sections, 'sections')
  42             type_check.expect(in_types[0].shape[self.axis] % sections == 0)
  43 
  44     def forward_cpu(self, x):
  45         if isinstance(self.indices_or_sections, collections.Iterable):
  46             cdimx = x[0].shape[self.axis]
  47             ind = list(self.indices_or_sections)
  48             ind.append(cdimx)
  49             prev_i = 0
  50             for i in ind:
  51                 cdimy = max(0, min(i, cdimx) - prev_i)
  52                 if cdimy == 0:
  53                     raise ValueError('Not support if shape contains 0')
  54                 prev_i = i
  55         return tuple(numpy.split(x[0], self.indices_or_sections, self.axis))
  56 
  57     def forward_gpu(self, x):
  58         xshape = x[0].shape
  59         self.cdimx = xshape[self.axis]
  60         self.rdim = numpy.prod(xshape[self.axis + 1:], dtype=int)
  61 
  62         if isinstance(self.indices_or_sections, collections.Iterable):
  63             ind = list(self.indices_or_sections)
  64             ind.append(self.cdimx)
  65         else:
  66             sec = self.indices_or_sections
  67             if self.cdimx % sec:
  68                 raise ValueError(
  69                     'array split does not result in an equal division')
  70             ind = numpy.arange(1, sec + 1) * (self.cdimx // sec)
  71         ys = []
  72         kernel = cuda.elementwise(
  73             _args, 'COPY(y[i] = x[idx])', 'split_fwd', preamble=_preamble)
  74         prev_i = 0
  75         for i in ind:
  76             cdimy = max(0, min(i, self.cdimx) - prev_i)
  77             s = list(xshape)
  78             s[self.axis] = cdimy
  79             y = cuda.empty(tuple(s), dtype=x[0].dtype)
  80             if cdimy == 0:
  81                 raise ValueError('Not support if shape contains 0')
  82             kernel(y, x[0], cdimy, self.cdimx, self.rdim, prev_i)
  83             prev_i = i
  84             ys.append(y)
  85         return tuple(ys)
  86 
  87     def backward_cpu(self, x, gys):
  88         return numpy.concatenate(gys, axis=self.axis),
  89 
  90     def backward_gpu(self, x, gys):
  91         gx = cuda.empty_like(x[0])
  92         coffset = 0
  93         kernel = cuda.elementwise(
  94             _args, 'COPY(x[idx] = y[i])', 'split_bwd', preamble=_preamble)
  95         for gy in gys:
  96             cdimy = gy.shape[self.axis]
  97             if cdimy != 0:
  98                 kernel(gy, gx, cdimy, self.cdimx, self.rdim, coffset)
  99             coffset += cdimy
  100         return gx,
  101 
  102 
  103 def split_axis(x, indices_or_sections, axis):
  104     """Splits given variables along an axis.
  105 
  106     Args:
  107         x (tuple of Variables): Variables to be split.
  108         indices_or_sections (int or 1-D array): If this argument is an integer,
  109             N, the array will be divided into N equal arrays along axis.
  110             If it is a 1-D array of sorted integers, it
  111             indicates the positions where the array is split.
  112         axis (int): Axis that the input array is split along.
  113 
  114     Returns:
  115         ``tuple`` or ``Variable``: Tuple of :class:`~chainer.Variable` objects
  116              if the number of outputs is more than 1 or
  117              :class:`~chainer.Variable` otherwise.
  118 
  119     .. note::
  120         This function raises ``ValueError`` if at least
  121         one of the outputs is splitted to zero-size
  122         (i.e. `axis`-th value of its shape is zero).
  123 
  124     """
  125     return SplitAxis(indices_or_sections, axis)(x)
  126
  ```
  --- END FILES ---
  
  Please first localize the bug based on the issue statement, and then generate a patch according to the `git diff` format fenced by three backticks.
  
  Here is an example:
  
  ```diff
  diff --git a/examples/server_async.py b/examples/server_async.py
  --- a/examples/server_async.py
  +++ b/examples/server_async.py
  @@ -313,4 +313,4 @@
   
   
   if __name__ == "__main__":
  -    asyncio.run(run_async_server("."), debug=True)
  +    asyncio.run(run_async_server(), debug=True)
  diff --git a/examples/server_sync.py b/examples/server_sync.py
  --- a/examples/server_sync.py
  +++ b/examples/server_sync.py
  @@ -313,5 +313,5 @@
   
   
   if __name__ == "__main__":
  -    server = run_sync_server(".")
  +    server = run_sync_server()
       server.shutdown()
  
  ```
  
golden_diff: diff --git a/chainer/functions/split_axis.py b/chainer/functions/split_axis.py
  --- a/chainer/functions/split_axis.py
  +++ b/chainer/functions/split_axis.py
  @@ -1,6 +1,7 @@
   import collections
   
   import numpy
  +import six
   
   from chainer import cuda
   from chainer import function
  @@ -85,14 +86,25 @@
           return tuple(ys)
   
       def backward_cpu(self, x, gys):
  -        return numpy.concatenate(gys, axis=self.axis),
  +        if any(gy is None for gy in gys):
  +            gx = numpy.zeros_like(x[0])
  +            gxs = numpy.split(gx, self.indices_or_sections, self.axis)
  +            for gxi, gy in six.moves.zip(gxs, gys):
  +                if gy is None:
  +                    continue
  +                gxi[:] = gy
  +            return gx,
  +        else:
  +            return numpy.concatenate(gys, axis=self.axis),
   
       def backward_gpu(self, x, gys):
  -        gx = cuda.empty_like(x[0])
  +        gx = cuda.zeros_like(x[0])
           coffset = 0
           kernel = cuda.elementwise(
               _args, 'COPY(x[idx] = y[i])', 'split_bwd', preamble=_preamble)
           for gy in gys:
  +            if gy is None:
  +                continue
               cdimy = gy.shape[self.axis]
               if cdimy != 0:
                   kernel(gy, gx, cdimy, self.cdimx, self.rdim, coffset)
  
verification_info: {"golden_diff": "diff --git a/chainer/functions/split_axis.py b/chainer/functions/split_axis.py\n--- a/chainer/functions/split_axis.py\n+++ b/chainer/functions/split_axis.py\n@@ -1,6 +1,7 @@\n import collections\n \n import numpy\n+import six\n \n from chainer import cuda\n from chainer import function\n@@ -85,14 +86,25 @@\n         return tuple(ys)\n \n     def backward_cpu(self, x, gys):\n-        return numpy.concatenate(gys, axis=self.axis),\n+        if any(gy is None for gy in gys):\n+            gx = numpy.zeros_like(x[0])\n+            gxs = numpy.split(gx, self.indices_or_sections, self.axis)\n+            for gxi, gy in six.moves.zip(gxs, gys):\n+                if gy is None:\n+                    continue\n+                gxi[:] = gy\n+            return gx,\n+        else:\n+            return numpy.concatenate(gys, axis=self.axis),\n \n     def backward_gpu(self, x, gys):\n-        gx = cuda.empty_like(x[0])\n+        gx = cuda.zeros_like(x[0])\n         coffset = 0\n         kernel = cuda.elementwise(\n             _args, 'COPY(x[idx] = y[i])', 'split_bwd', preamble=_preamble)\n         for gy in gys:\n+            if gy is None:\n+                continue\n             cdimy = gy.shape[self.axis]\n             if cdimy != 0:\n                 kernel(gy, gx, cdimy, self.cdimx, self.rdim, coffset)\n", "issue": "split_axis.backward fails on incomplete gradients\nWhen there is a None in the grad_outputs, split_axis fails to backprop the incomplete gradients.\n\n", "before_files": [{"content": "import collections\n\nimport numpy\n\nfrom chainer import cuda\nfrom chainer import function\nfrom chainer.utils import type_check\n\n\n_args = 'float* y, float* x, int cdimy, int cdimx, int rdim, int coffset'\n_preamble = '''\n#define COPY(statement) \\\n    int l   = i / (rdim * cdimy);  \\\n    int c   = i / rdim % cdimy + coffset;  \\\n    int r   = i % rdim;  \\\n    int idx = r + rdim * (c + cdimx * l);  \\\n    statement;\n'''\n\n\nclass SplitAxis(function.Function):\n\n    \"\"\"Function that splits multiple arrays towards the specified axis.\"\"\"\n\n    def __init__(self, indices_or_sections, axis):\n        if not isinstance(indices_or_sections, (int, collections.Iterable)):\n            raise TypeError('indices_or_sections must be integer or 1-D array')\n        self.indices_or_sections = indices_or_sections\n        self.axis = axis\n\n    def check_type_forward(self, in_types):\n        type_check.expect(in_types.size() == 1)\n        type_check.expect(in_types[0].ndim >= self.axis)\n\n        if isinstance(self.indices_or_sections, collections.Iterable):\n            max_index = type_check.Variable(\n                self.indices_or_sections[-1], 'max_index')\n            type_check.expect(in_types[0].shape[self.axis] > max_index)\n        else:\n            sections = type_check.Variable(\n                self.indices_or_sections, 'sections')\n            type_check.expect(in_types[0].shape[self.axis] % sections == 0)\n\n    def forward_cpu(self, x):\n        if isinstance(self.indices_or_sections, collections.Iterable):\n            cdimx = x[0].shape[self.axis]\n            ind = list(self.indices_or_sections)\n            ind.append(cdimx)\n            prev_i = 0\n            for i in ind:\n                cdimy = max(0, min(i, cdimx) - prev_i)\n                if cdimy == 0:\n                    raise ValueError('Not support if shape contains 0')\n                prev_i = i\n        return tuple(numpy.split(x[0], self.indices_or_sections, self.axis))\n\n    def forward_gpu(self, x):\n        xshape = x[0].shape\n        self.cdimx = xshape[self.axis]\n        self.rdim = numpy.prod(xshape[self.axis + 1:], dtype=int)\n\n        if isinstance(self.indices_or_sections, collections.Iterable):\n            ind = list(self.indices_or_sections)\n            ind.append(self.cdimx)\n        else:\n            sec = self.indices_or_sections\n            if self.cdimx % sec:\n                raise ValueError(\n                    'array split does not result in an equal division')\n            ind = numpy.arange(1, sec + 1) * (self.cdimx // sec)\n        ys = []\n        kernel = cuda.elementwise(\n            _args, 'COPY(y[i] = x[idx])', 'split_fwd', preamble=_preamble)\n        prev_i = 0\n        for i in ind:\n            cdimy = max(0, min(i, self.cdimx) - prev_i)\n            s = list(xshape)\n            s[self.axis] = cdimy\n            y = cuda.empty(tuple(s), dtype=x[0].dtype)\n            if cdimy == 0:\n                raise ValueError('Not support if shape contains 0')\n            kernel(y, x[0], cdimy, self.cdimx, self.rdim, prev_i)\n            prev_i = i\n            ys.append(y)\n        return tuple(ys)\n\n    def backward_cpu(self, x, gys):\n        return numpy.concatenate(gys, axis=self.axis),\n\n    def backward_gpu(self, x, gys):\n        gx = cuda.empty_like(x[0])\n        coffset = 0\n        kernel = cuda.elementwise(\n            _args, 'COPY(x[idx] = y[i])', 'split_bwd', preamble=_preamble)\n        for gy in gys:\n            cdimy = gy.shape[self.axis]\n            if cdimy != 0:\n                kernel(gy, gx, cdimy, self.cdimx, self.rdim, coffset)\n            coffset += cdimy\n        return gx,\n\n\ndef split_axis(x, indices_or_sections, axis):\n    \"\"\"Splits given variables along an axis.\n\n    Args:\n        x (tuple of Variables): Variables to be split.\n        indices_or_sections (int or 1-D array): If this argument is an integer,\n            N, the array will be divided into N equal arrays along axis.\n            If it is a 1-D array of sorted integers, it\n            indicates the positions where the array is split.\n        axis (int): Axis that the input array is split along.\n\n    Returns:\n        ``tuple`` or ``Variable``: Tuple of :class:`~chainer.Variable` objects\n             if the number of outputs is more than 1 or\n             :class:`~chainer.Variable` otherwise.\n\n    .. note::\n        This function raises ``ValueError`` if at least\n        one of the outputs is splitted to zero-size\n        (i.e. `axis`-th value of its shape is zero).\n\n    \"\"\"\n    return SplitAxis(indices_or_sections, axis)(x)\n", "path": "chainer/functions/split_axis.py"}], "after_files": [{"content": "import collections\n\nimport numpy\nimport six\n\nfrom chainer import cuda\nfrom chainer import function\nfrom chainer.utils import type_check\n\n\n_args = 'float* y, float* x, int cdimy, int cdimx, int rdim, int coffset'\n_preamble = '''\n#define COPY(statement) \\\n    int l   = i / (rdim * cdimy);  \\\n    int c   = i / rdim % cdimy + coffset;  \\\n    int r   = i % rdim;  \\\n    int idx = r + rdim * (c + cdimx * l);  \\\n    statement;\n'''\n\n\nclass SplitAxis(function.Function):\n\n    \"\"\"Function that splits multiple arrays towards the specified axis.\"\"\"\n\n    def __init__(self, indices_or_sections, axis):\n        if not isinstance(indices_or_sections, (int, collections.Iterable)):\n            raise TypeError('indices_or_sections must be integer or 1-D array')\n        self.indices_or_sections = indices_or_sections\n        self.axis = axis\n\n    def check_type_forward(self, in_types):\n        type_check.expect(in_types.size() == 1)\n        type_check.expect(in_types[0].ndim >= self.axis)\n\n        if isinstance(self.indices_or_sections, collections.Iterable):\n            max_index = type_check.Variable(\n                self.indices_or_sections[-1], 'max_index')\n            type_check.expect(in_types[0].shape[self.axis] > max_index)\n        else:\n            sections = type_check.Variable(\n                self.indices_or_sections, 'sections')\n            type_check.expect(in_types[0].shape[self.axis] % sections == 0)\n\n    def forward_cpu(self, x):\n        if isinstance(self.indices_or_sections, collections.Iterable):\n            cdimx = x[0].shape[self.axis]\n            ind = list(self.indices_or_sections)\n            ind.append(cdimx)\n            prev_i = 0\n            for i in ind:\n                cdimy = max(0, min(i, cdimx) - prev_i)\n                if cdimy == 0:\n                    raise ValueError('Not support if shape contains 0')\n                prev_i = i\n        return tuple(numpy.split(x[0], self.indices_or_sections, self.axis))\n\n    def forward_gpu(self, x):\n        xshape = x[0].shape\n        self.cdimx = xshape[self.axis]\n        self.rdim = numpy.prod(xshape[self.axis + 1:], dtype=int)\n\n        if isinstance(self.indices_or_sections, collections.Iterable):\n            ind = list(self.indices_or_sections)\n            ind.append(self.cdimx)\n        else:\n            sec = self.indices_or_sections\n            if self.cdimx % sec:\n                raise ValueError(\n                    'array split does not result in an equal division')\n            ind = numpy.arange(1, sec + 1) * (self.cdimx // sec)\n        ys = []\n        kernel = cuda.elementwise(\n            _args, 'COPY(y[i] = x[idx])', 'split_fwd', preamble=_preamble)\n        prev_i = 0\n        for i in ind:\n            cdimy = max(0, min(i, self.cdimx) - prev_i)\n            s = list(xshape)\n            s[self.axis] = cdimy\n            y = cuda.empty(tuple(s), dtype=x[0].dtype)\n            if cdimy == 0:\n                raise ValueError('Not support if shape contains 0')\n            kernel(y, x[0], cdimy, self.cdimx, self.rdim, prev_i)\n            prev_i = i\n            ys.append(y)\n        return tuple(ys)\n\n    def backward_cpu(self, x, gys):\n        if any(gy is None for gy in gys):\n            gx = numpy.zeros_like(x[0])\n            gxs = numpy.split(gx, self.indices_or_sections, self.axis)\n            for gxi, gy in six.moves.zip(gxs, gys):\n                if gy is None:\n                    continue\n                gxi[:] = gy\n            return gx,\n        else:\n            return numpy.concatenate(gys, axis=self.axis),\n\n    def backward_gpu(self, x, gys):\n        gx = cuda.zeros_like(x[0])\n        coffset = 0\n        kernel = cuda.elementwise(\n            _args, 'COPY(x[idx] = y[i])', 'split_bwd', preamble=_preamble)\n        for gy in gys:\n            if gy is None:\n                continue\n            cdimy = gy.shape[self.axis]\n            if cdimy != 0:\n                kernel(gy, gx, cdimy, self.cdimx, self.rdim, coffset)\n            coffset += cdimy\n        return gx,\n\n\ndef split_axis(x, indices_or_sections, axis):\n    \"\"\"Splits given variables along an axis.\n\n    Args:\n        x (tuple of Variables): Variables to be split.\n        indices_or_sections (int or 1-D array): If this argument is an integer,\n            N, the array will be divided into N equal arrays along axis.\n            If it is a 1-D array of sorted integers, it\n            indicates the positions where the array is split.\n        axis (int): Axis that the input array is split along.\n\n    Returns:\n        ``tuple`` or ``Variable``: Tuple of :class:`~chainer.Variable` objects\n             if the number of outputs is more than 1 or\n             :class:`~chainer.Variable` otherwise.\n\n    .. note::\n        This function raises ``ValueError`` if at least\n        one of the outputs is splitted to zero-size\n        (i.e. `axis`-th value of its shape is zero).\n\n    \"\"\"\n    return SplitAxis(indices_or_sections, axis)(x)\n", "path": "chainer/functions/split_axis.py"}]}