code: def gradient_descent_mse_ensemble(
      kernel_fn: KernelFn,
      x_train: jnp.ndarray,
      y_train: jnp.ndarray,
      learning_rate: float = 1.0,
      diag_reg: float = 0.0,
      diag_reg_absolute_scale: bool = False,
      trace_axes: Axes = (-1,),
      **kernel_fn_train_train_kwargs,
  ):
      r"""Predicts the gaussian embedding induced by gradient descent on MSE loss.
  
      This is equivalent to an infinite ensemble of infinite-width networks after
      marginalizing out the initialization, if `kernel_fn` is the kernel function of
      the infinite-width network. Note that `kernel_fn` can in principle also be an
      empirical / Monte Carlo finite-width kernel function, but in this case the
      returned output will not have a simple interpretation (unless these functions
      are used to approximate the infinite-width kernel).
  
      Note that first invocation of the returned `predict_fn` will be slow and
      allocate a lot of memory for its whole lifetime, as the kernel computation,
      and either eigendecomposition (`t` is a scalar or an array) or Cholesky
      factorization (`t=None`) of `kernel_fn(x_train, None, get)` is performed and
      cached for future invocations (or both, if the function is called on both
      finite and infinite (`t=None`) times).
  
      Args:
        kernel_fn:
          A kernel function that computes NNGP and/or NTK. Must have a signature
          `kernel_fn(x1, x2, get, **kernel_fn_kwargs)` and return a
          :class:`~neural_tangents.Kernel` object or a `namedtuple` with `nngp`
          and/or `ntk` attributes. Therefore, it can be an `AnalyticKernelFn`, but
          also a `MonteCarloKernelFn`, or an `EmpiricalKernelFn` (but only
          `nt.empirical_kernel_fn` and not `nt.empirical_ntk_fn` or
          `nt.empirical_nngp_fn`, since the latter two do not accept a `get`
          argument). Note that for meaningful outputs, the kernel function must
          represent or at least approximate the infinite-width kernel.
  
        x_train:
          training inputs.
  
        y_train:
          training targets.
  
        learning_rate:
          learning rate, step size.
  
        diag_reg:
          a scalar representing the strength of the diagonal regularization for
          `kernel_fn(x_train, None, get)`, i.e. computing
          `kernel_fn(x_train, None, get) + diag_reg * I` during Cholesky
          factorization or eigendecomposition.
  
        diag_reg_absolute_scale:
          `True` for `diag_reg` to represent regularization in absolute units,
          `False` to be
          `diag_reg * jnp.mean(jnp.trace(kernel_fn(x_train, None, get)))`.
  
        trace_axes:
          `f(x_train)` axes such that `kernel_fn(x_train, None, get)`,
          `kernel_fn(x_test, x_train, get)`[, and `kernel_fn(x_test, None, get)`]
          lack these pairs of dimensions and are to be interpreted as
          :math:`\Theta \otimes I`, i.e. block-diagonal along `trace_axes`. These
          can can be specified either to save space and compute, or to even improve
          approximation accuracy of the infinite-width or infinite-samples limit,
          since in these limits the covariance along channel / feature / logit
          axes indeed converges to a constant-diagonal matrix. However, if you
          target linearized dynamics of a specific finite-width network,
          `trace_axes=()` will yield most accurate result.
  
        **kernel_fn_train_train_kwargs:
          optional keyword arguments passed to `kernel_fn`. For train-train kernel,
          these are passed to `kernel_fn` without changes. For test-test kernel,
          they are passed to `kernel_fn`, unless overwritten by a similar
          `**kernel_fn_test_test_kwargs` arguments passed to the `predict_fn`
          function call. Finally, for test-train kernel, values that are tuples of
          arrays (destined for calls of the finite-width network on training and
          testing data) will be tuples of values combined from
          `**kernel_fn_train_train_kwargs` and `**kernel_fn_test_test_kwargs`, and
          all other values must match.
  
      Returns:
        A function with signature `predict_fn(t, x_test, get, compute_cov)`
        returning either mean or mean and covariance of the infinite ensemble of
        infinite-width networks outputs on `x_test` at time[s] `t`, in the `get`
        regime (`"nngp"`, `"ntk"`, or `("nngp", "ntk")`).
      """
      expm1 = _make_expm1_fn(y_train.size)
      inv_expm1 = _make_inv_expm1_fn(y_train.size)
  
      trace_axes = utils.canonicalize_axis(trace_axes, y_train)
      trace_axes = tuple(-y_train.ndim + a for a in trace_axes)
      n_trace_axes = len(trace_axes)
      last_t_axes = range(-n_trace_axes, 0)
      trace_shape = tuple(y_train.shape[a] for a in trace_axes)
  
      y_train_flat = jnp.moveaxis(
          y_train, trace_axes, last_t_axes
      ).reshape((-1,) + trace_shape)
  
      k_dd_cache = {}
  
      def get_k_train_train(get: tuple[str, ...]) -> _Kernel:
          if len(get) == 1:
              get = get[0]
              if get not in k_dd_cache:
                  k_dd_cache[get] = kernel_fn(
                      x_train, None, get, **kernel_fn_train_train_kwargs
                  )
  
          elif len(get) == 2:
              if not any(g in k_dd_cache for g in get):
                  k_dd_cache.update(
                      kernel_fn(
                          x_train,
                          None,
                          get,
                          **kernel_fn_train_train_kwargs,
                      )._asdict()
                  )  # pytype: disable=attribute-error  # jax-ndarray
              else:
                  for g in get:
                      if g not in k_dd_cache:
                          k_dd_cache[g] = kernel_fn(
                              x_train,
                              None,
                              g,
                              **kernel_fn_train_train_kwargs,
                          )
  
          else:
              raise ValueError(get)
          return _Kernel(**k_dd_cache)
  
      @lru_cache(2)
      def eigenspace(get: str):
          k_dd = getattr(get_k_train_train((get,)), get)
          k_dd = _add_diagonal_regularizer(
              utils.make_2d(k_dd), diag_reg, diag_reg_absolute_scale
          )
          evals, evecs = jnp.linalg.eigh(k_dd)
          evals = jnp.expand_dims(evals, 0)
          return evals, evecs
  
      @lru_cache(4)
      def predict_inf(get: Get):
          _, get = utils.canonicalize_get(get)
          k_dd = get_k_train_train(get)
          return gp_inference(
              k_dd,
              y_train,
              diag_reg,
              diag_reg_absolute_scale,
              trace_axes,
          )
  
      def get_kernels(
          get: Get,
          x_test: Optional[jnp.ndarray],
          compute_cov: bool,
          **kernel_fn_test_test_kwargs,
      ):
          get = _get_dependency(get, compute_cov)
          k_dd = get_k_train_train(get)
          if x_test is None:
              k_td = None
              nngp_tt = compute_cov or None
          else:
              args_train, _ = utils.split_kwargs(
                  kernel_fn_train_train_kwargs, x_train
              )
              args_test, _ = utils.split_kwargs(
                  kernel_fn_test_test_kwargs, x_test
              )
  
              def is_array(x):
                  return tree_all(
                      tree_map(
                          lambda x: isinstance(
                              x, (np.ndarray, jnp.ndarray)
                          ),
                          x,
                      )
                  )
  
              kwargs_td = dict(kernel_fn_train_train_kwargs)
              kwargs_tt = dict(kernel_fn_train_train_kwargs)
  
              for k in kernel_fn_test_test_kwargs:
                  v_tt = kernel_fn_test_test_kwargs[k]
                  v_dd = kernel_fn_train_train_kwargs[k]
  
                  if is_array(v_dd) and is_array(v_tt):
                      if (
                          isinstance(v_dd, tuple)
                          and len(v_dd) == 2
                          and isinstance(v_tt, tuple)
                          and len(v_tt) == 2
                      ):
                          v_td = (args_test[k], args_train[k])
                      else:
                          v_td = v_tt
  
                  elif v_dd != v_tt:
                      raise ValueError(
                          f"Same keyword argument {k} of `kernel_fn` is set to"
                          f"different values {v_dd} != {v_tt} when computing "
                          f"the train-train and test-train/test-test kernels. "
                          f"If this is your intention, please submit a feature"
                          f" request at "
                          f"https://github.com/google/neural-tangents/issues"
                      )
  
                  else:
                      v_td = v_tt
  
                  kwargs_td[k] = v_td
                  kwargs_tt[k] = v_tt
  
              k_td = kernel_fn(x_test, x_train, get, **kwargs_td)
  
              if compute_cov:
                  nngp_tt = kernel_fn(x_test, None, "nngp", **kwargs_tt)
              else:
                  nngp_tt = None
          return k_dd, k_td, nngp_tt
  
      @utils.get_namedtuple("Gaussians")
      def predict_fn(
          t: Optional[ArrayOrScalar] = None,
          x_test: Optional[jnp.ndarray] = None,
          get: Optional[Get] = None,
          compute_cov: bool = False,
          **kernel_fn_test_test_kwargs,
      ) -> dict[str, Gaussian]:
          """Return output mean and covariance on the test set at time[s] `t`.
  
          Args:
            t:
              a scalar of array of scalars of any shape. `t=None` is treated as
              infinity and returns the same result as `t=jnp.inf`, but is computed
              using linear solve for test predictions instead of eigendecomposition,
              saving time and precision.
  
            x_test:
              test inputs. `None` means to return non-regularized (`diag_reg=0`)
              predictions on the train-set inputs. For regularized predictions, pass
              `x_test=x_train`.
  
            get:
              string, the mode of the Gaussian process, either "nngp" or "ntk", or a
              tuple. `get=None` is equivalent to `get=("nngp", "ntk")`.
  
            compute_cov:
              if `True` computing both `mean` and `variance` and only `mean`
              otherwise.
  
            **kernel_fn_test_test_kwargs:
              optional keyword arguments passed to `kernel_fn`. See also
              `kernel_fn_train_train_kwargs` argument of the parent function.
  
          Returns:
            `fx_test_mean_t` or `(fx_test_mean_t, fx_test_cov_t)` if
            `compute_cov == True` with potentially additional leading time dimensions.
          """
          if get is None:
              get = ("nngp", "ntk")
  
          # train-train, test-train, test-test.
          k_dd, k_td, nngp_tt = get_kernels(
              get, x_test, compute_cov, **kernel_fn_test_test_kwargs
          )
  
          # Infinite time.
          if t is None:
              return predict_inf(get)(
                  get=get, k_test_train=k_td, k_test_test=nngp_tt
              )
  
          # Finite time.
          t = jnp.array(t) * learning_rate
          t_shape = t.shape
          t = t.reshape((-1, 1))
  
          def reshape_mean(mean):
              k = _get_first(k_dd if k_td is None else k_td)
              mean = mean.reshape(t_shape + k.shape[::2] + trace_shape)
              mean = jnp.moveaxis(mean, last_t_axes, trace_axes)
              return mean
  
          def reshape_cov(cov):
              k = _get_first(k_dd if k_td is None else k_td)
              cov_shape_t = t_shape + k.shape[::2] * 2
              return utils.zip_axes(
                  cov.reshape(cov_shape_t), len(t_shape)
              )
  
          out = {}
  
          for g in get:
              evals, evecs = eigenspace(g)
  
              # Training set.
              if k_td is None:
                  mean = jnp.einsum(
                      "ji,ti,ki,k...->tj...",
                      evecs,
                      -expm1(evals, t),
                      evecs,
                      y_train_flat,
                      optimize=_optimize(),
                  )
  
              # Test set.
              else:
                  neg_inv_expm1 = -inv_expm1(evals, t)
                  ktd_g = utils.make_2d(getattr(k_td, g))
                  mean = jnp.einsum(
                      "lj,ji,ti,ki,k...->tl...",
                      ktd_g,
                      evecs,
                      neg_inv_expm1,
                      evecs,
                      y_train_flat,
                      optimize=_optimize(),
                  )
  
              mean = reshape_mean(mean)
  
              if nngp_tt is not None:
                  nngp_dd = utils.make_2d(k_dd.nngp)
  
                  # Training set.
                  if k_td is None:
                      if g == "nngp":
                          cov = jnp.einsum(
                              "ji,ti,ki->tjk",
                              evecs,
                              (
                                  jnp.maximum(evals, 0.0)
                                  * jnp.exp(
                                      -2
                                      * jnp.maximum(evals, 0.0)
                                      * t
                                      / y_train.size
                                  )
                              ),
                              evecs,
                              optimize=_optimize(),
                          )
  
                      elif g == "ntk":
                          exp = jnp.einsum(
                              "mi,ti,ki->tmk",
                              evecs,
                              jnp.exp(
                                  -jnp.maximum(evals, 0.0)
                                  * t
                                  / y_train.size
                              ),
                              evecs,
                              optimize=_optimize(),
                          )
                          cov = jnp.einsum(
                              "tmk,kl,tnl->tmn",
                              exp,
                              nngp_dd,
                              exp,
                              optimize=_optimize(),
                          )
  
                      else:
                          raise ValueError(g)
  
                  # Test set.
                  else:
                      _nngp_tt = jnp.expand_dims(
                          utils.make_2d(nngp_tt), 0
                      )
  
                      if g == "nngp":
                          cov = _nngp_tt - jnp.einsum(
                              "mj,ji,ti,ki,lk->tml",
                              ktd_g,
                              evecs,
                              -inv_expm1(evals, 2 * t),
                              evecs,
                              ktd_g,
                              optimize=_optimize(),
                          )
  
                      elif g == "ntk":
                          term_1 = jnp.einsum(
                              "mi,ti,ki,lk->tml",
                              evecs,
                              neg_inv_expm1,
                              evecs,
                              ktd_g,
                              optimize=_optimize(),
                          )
                          term_2 = jnp.einsum(
                              "mj,ji,ti,ki,lk->tml",
                              ktd_g,
                              evecs,
                              neg_inv_expm1,
                              evecs,
                              utils.make_2d(k_td.nngp),
                              optimize=_optimize(),
                          )
                          term_2 += jnp.moveaxis(term_2, 1, 2)
                          cov = jnp.einsum(
                              "tji,jk,tkl->til",
                              term_1,
                              nngp_dd,
                              term_1,
                              optimize=_optimize(),
                          )
                          cov += -term_2 + _nngp_tt
  
                      else:
                          raise ValueError(g)
  
                  out[g] = Gaussian(mean, reshape_cov(cov))
  
              else:
                  out[g] = mean
  
          return out  # pytype: disable=bad-return-type  # jnp-type
  
      return predict_fn
  
imports: import collections
  from functools import lru_cache
  from typing import Any, Callable, Generator, Iterable, NamedTuple, Optional, Protocol, Union
  import jax
  from jax import grad
  from jax.experimental import ode
  import jax.numpy
  import jax.scipy
  from jax.tree_util import tree_all
  from jax.tree_util import tree_map
  import numpy
  import scipy
  from .utils import dataclasses
  from .utils import utils
  from .utils.typing import Axes
  from .utils.typing import Get
  from .utils.typing import KernelFn
label_desc: Use image node_id 4 for calling a global function with example usage: gradient_descent_mse_ensemble(kernel_fn, x_train, y_train, learning_rate, diag_reg, diag_reg_absolute_scale, trace_axes) and returns: predict_fn, _Kernel, evals,  evecs, gp_inference, k_dd,  k_td,  nngp_tt, out, predict_inf, mean, utils, tree_all