neural_tangents.predict.gradient_descent_mse(k_train_train, y_train, learning_rate=1.0, diag_reg=0.0, diag_reg_absolute_scale=False, trace_axes=(-1,))[source]

Predicts the outcome of function space gradient descent training on MSE.

Solves in closed form for the continuous-time version of gradient descent.

Uses the closed-form solution for gradient descent on an MSE loss in function space detailed in [,*] given a Neural Tangent or Neural Network Gaussian Process Kernel over the dataset. Given NNGP or NTK, this function will return a function that predicts the time evolution for function space points at arbitrary time[s] (training step[s]) t. Note that these time[s] (step[s]) are continuous and are interpreted in units of the learning_rate so absolute_time = learning_rate * t, and the scales of learning_rate and t are interchangeable.

Note that first invocation of the returned predict_fn will be slow and allocate a lot of memory for its whole lifetime, as either eigendecomposition (t is a scalar or an array) or Cholesky factorization (t=None) of k_train_train is performed and cached for future invocations (or both, if the function is called on both finite and infinite (t=None) times).

[*] “Neural Tangent Kernel: Convergence and Generalization in Neural Networks

[**] “Wide Neural Networks of Any Depth Evolve as Linear Models Under Gradient Descent


>>> import neural_tangents as nt
>>> #
>>> t = 1e-7
>>> kernel_fn = nt.empirical_ntk_fn(f)
>>> k_train_train = kernel_fn(x_train, None, params)
>>> k_test_train = kernel_fn(x_test, x_train, params)
>>> #
>>> predict_fn = nt.predict.gradient_descent_mse(k_train_train, y_train)
>>> #
>>> fx_train_0 = f(params, x_train)
>>> fx_test_0 = f(params, x_test)
>>> #
>>> fx_train_t, fx_test_t = predict_fn(t, fx_train_0, fx_test_0,
>>>                                    k_test_train)
  • k_train_train (ndarray) – kernel on the training data. Must have the shape of zip(y_train.shape, y_train.shape) with trace_axes absent.

  • y_train (ndarray) – targets for the training data.

  • learning_rate (float) – learning rate, step size.

  • diag_reg (float) – a scalar representing the strength of the diagonal regularization for k_train_train, i.e. computing k_train_train + diag_reg * I during Cholesky factorization or eigendecomposition.

  • diag_reg_absolute_scale (bool) – True for diag_reg to represent regularization in absolute units, False to be diag_reg * np.mean(np.trace(k_train_train)).

  • trace_axes (Union[int, Sequence[int]]) – f(x_train) axes such that k_train_train lacks these pairs of dimensions and is to be interpreted as \(\Theta \otimes I\), 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.

Return type



A function of signature predict_fn(t, fx_train_0, fx_test_0, k_test_train) that returns output train [and test] set[s] predictions at time[s] t.