Empirical¶
Compute empirical NNGP and NTK; approximate functions via Taylor series.
All functions in this module are applicable to any JAX functions of proper
signatures (not only those from nt.stax
).
NNGP and NTK are computed using empirical_nngp_fn
, empirical_ntk_fn
, or
empirical_kernel_fn
(for both). The kernels have a very specific output shape
convention that may be unexpected. Further, NTK has multiple implementations
that may perform differently depending on the task. Please read individual
functions’ docstrings.
Example
>>> from jax import random
>>> import neural_tangents as nt
>>> from neural_tangents import stax
>>>
>>> key1, key2, key3 = random.split(random.PRNGKey(1), 3)
>>> x_train = random.normal(key1, (20, 32, 32, 3))
>>> y_train = random.uniform(key1, (20, 10))
>>> x_test = random.normal(key2, (5, 32, 32, 3))
>>>
>>> # A narrow CNN.
>>> init_fn, f, _ = stax.serial(
>>> stax.Conv(32, (3, 3)),
>>> stax.Relu(),
>>> stax.Conv(32, (3, 3)),
>>> stax.Relu(),
>>> stax.Conv(32, (3, 3)),
>>> stax.Flatten(),
>>> stax.Dense(10)
>>> )
>>>
>>> _, params = init_fn(key3, x_train.shape)
>>>
>>> # Default setting: reducing over logits; pass `vmap_axes=0` because the
>>> # network is iid along the batch axis, no BatchNorm. Use default
>>> # `implementation=1` since the network has few trainable parameters.
>>> kernel_fn = nt.empirical_kernel_fn(f, trace_axes=(-1,),
>>> vmap_axes=0, implementation=1)
>>>
>>> # (5, 20) np.ndarray test-train NNGP/NTK
>>> nngp_test_train = kernel_fn(x_test, x_train, 'nngp', params)
>>> ntk_test_train = kernel_fn(x_test, x_train, 'ntk', params)
>>>
>>> # Full kernel: not reducing over logits.
>>> kernel_fn = nt.empirical_kernel_fn(f, trace_axes=(), vmap_axes=0)
>>>
>>> # (5, 20, 10, 10) np.ndarray test-train NNGP/NTK namedtuple.
>>> k_test_train = kernel_fn(x_test, x_train, params)
>>>
>>> # A wide FCN with lots of parameters
>>> init_fn, f, _ = stax.serial(
>>> stax.Flatten(),
>>> stax.Dense(1024),
>>> stax.Relu(),
>>> stax.Dense(1024),
>>> stax.Relu(),
>>> stax.Dense(10)
>>> )
>>>
>>> _, params = init_fn(key3, x_train.shape)
>>>
>>> # Use implicit differentiation in NTK: `implementation=2` to reduce
>>> # memory cost, since the network has many trainable parameters.
>>> ntk_fn = nt.empirical_ntk_fn(f, vmap_axes=0, implementation=2)
>>>
>>> # (5, 5) np.ndarray test-test NTK
>>> ntk_test_train = ntk_fn(x_test, None, params)
>>>
>>> # Compute only output variances:
>>> nngp_fn = nt.empirical_nngp_fn(f, diagonal_axes=(0,))
>>>
>>> # (20,) np.ndarray train-train diagonal NNGP
>>> nngp_train_train_diag = nngp_fn(x_train, None, params)
- neural_tangents.utils.empirical.empirical_kernel_fn(f, trace_axes=(- 1,), diagonal_axes=(), vmap_axes=None, implementation=1)[source]¶
Returns a function that computes single draws from NNGP and NT kernels.
WARNING: resulting kernel shape is nearly
zip(f(x1).shape, f(x2).shape)
subject totrace_axes
anddiagonal_axes
parameters, which make certain assumptions about the outputsf(x)
that may only be true in the infinite width / infinite number of samples limit, or may not apply to your architecture. For most precise results in the context of linearized training dynamics of a specific finite-width network, set bothtrace_axes=()
anddiagonal_axes=()
to obtain the kernel exactly of shapezip(f(x1).shape, f(x2).shape)
.For networks with multiple (i.e. lists, tuples, PyTrees) outputs, in principal the empirical kernels will have terms measuring the covariance between the outputs. Here, we ignore these cross-terms and consider each output separately. Please raise an issue if this feature is important to you.
- Parameters
f (
Callable
[[Any
,Union
[List
[ndarray
],Tuple
[ndarray
, …],ndarray
]],Union
[List
[ndarray
],Tuple
[ndarray
, …],ndarray
]]) – the function whose NTK we are computing.f
should have the signaturef(params, inputs, **kwargs)
and should return annp.ndarray
outputs.trace_axes (
Union
[int
,Sequence
[int
]]) – output axes to trace the output kernel over, i.e. compute only the trace of the covariance along the respective pair of axes (one pair for each axis intrace_axes
). This allows to save space and compute if you are only interested in the respective trace, but also improve approximation accuracy if you know that covariance along these pairs of axes converges to aconstant * identity matrix
in the limit of interest (e.g. infinite width or infiniten_samples
). A common use case is the channel / feature / logit axis, since activation slices along such axis are i.i.d. and the respective covariance along the respective pair of axes indeed converges to a constant-diagonal matrix in the infinite width or infiniten_samples
limit. Also related to “contracting dimensions” in XLA terms. (https://www.tensorflow.org/xla/operation_semantics#dotgeneral)diagonal_axes (
Union
[int
,Sequence
[int
]]) – output axes to diagonalize the output kernel over, i.e. compute only the diagonal of the covariance along the respective pair of axes (one pair for each axis indiagonal_axes
). This allows to save space and compute, if off-diagonal values along these axes are not needed, but also improve approximation accuracy if their limiting value is known theoretically, e.g. if they vanish in the limit of interest (e.g. infinite width or infiniten_samples
). If you further know that on-diagonal values converge to the same constant in your limit of interest, you should specify these axes intrace_axes
instead, to save even more compute and gain even more accuracy. A common use case is computing the variance (instead of covariance) along certain axes. Also related to “batch dimensions” in XLA terms. (https://www.tensorflow.org/xla/operation_semantics#dotgeneral)vmap_axes (
Optional
[Tuple
[Union
[List
[int
],Tuple
[int
, …],int
,None
],Union
[List
[int
],Tuple
[int
, …],int
,None
],Dict
[str
,Union
[List
[int
],Tuple
[int
, …],int
,None
]]]]) –applicable only to NTK.
A triple of
(in_axes, out_axes, kwargs_axes)
passed tovmap
to evaluate the empirical NTK in parallel ove these axes. Precisely, providing this argument implies thatf(params, x, **kwargs)
equals to a concatenation alongout_axes
off
applied to slices ofx
and**kwargs
alongin_axes
andkwargs_axes
. In other words, it certifies thatf
can be evaluated as avmap
without_axes=out_axes
overx
(alongin_axes
) and those arguments in**kwargs
that are present inkwargs_axes.keys()
(alongkwargs_axes.values()
).For example if
_, f, _ = nt.stax.Aggregate()
,f
is called viaf(params, x, pattern=pattern)
. By default, inputsx
, patternspattern
, and outputs off
are all batched along the leading0
dimension, and each outputf(params, x, pattern=pattern)[i]
only depends on the inputsx[i]
andpattern[i]
. In this case, we can passvmap_axes=(0, 0, dict(pattern=0)
to specify along which dimensions inputs, outputs, and keyword arguments are batched respectively.This allows us to evaluate Jacobians much more efficiently. If
vmap_axes
is not a triple, it is interpreted asin_axes = out_axes = vmap_axes, kwargs_axes = {}
. For example a very common use case isvmap_axes=0
for a neural network with leading (0
) batch dimension, both for inputs and outputs, and no interactions between different elements of the batch (e.g. no BatchNorm, and, in the case ofnt.stax
, also no Dropout). However, if there is interaction between batch elements or no concept of a batch axis at all,vmap_axes
must be set toNone
, to avoid wrong (and potentially silent) results.implementation (
int
) –applicable only to NTK.
1
or2
.1
directly instantiates Jacobians and computes their outer product.2
uses implicit differentiation to avoid instantiating whole Jacobians at once. The implicit kernel is derived by observing that: \(\Theta = J(X_1) J(X_2)^T = [J(X_1) J(X_2)^T](I)\), i.e. a linear function \([J(X_1) J(X_2)^T]\) applied to an identity matrix \(I\). This allows the computation of the NTK to be phrased as: \(a(v) = J(X_2)^T v\), which is computed by a vector-Jacobian product; \(b(v) = J(X_1) a(v)\) which is computed by a Jacobian-vector product; and \(\Theta = [b(v)] / d[v^T](I)\) which is computed via avmap
of \(b(v)\) over columns of the identity matrix \(I\).It is best to benchmark each method on your specific task. We suggest using
1
unless you get OOMs due to large number of trainable parameters, otherwise -2
.
- Return type
Callable
[[Union
[List
[ndarray
],Tuple
[ndarray
, …],ndarray
],Union
[List
[ndarray
],Tuple
[ndarray
, …],ndarray
,None
],Union
[Tuple
[str
, …],str
,None
],Any
],Union
[List
[ndarray
],Tuple
[ndarray
, …],ndarray
]]- Returns
A function to draw a single sample the NNGP and NTK empirical kernels of a given network
f
.
- neural_tangents.utils.empirical.empirical_nngp_fn(f, trace_axes=(- 1,), diagonal_axes=())[source]¶
Returns a function to draw a single sample the NNGP of a given network
f
.The Neural Network Gaussian Process (NNGP) kernel is defined as \(f(X_1) f(X_2)^T\), i.e. the outer product of the function outputs.
WARNING: resulting kernel shape is nearly
zip(f(x1).shape, f(x2).shape)
subject totrace_axes
anddiagonal_axes
parameters, which make certain assumptions about the outputsf(x)
that may only be true in the infinite width / infinite number of samples limit, or may not apply to your architecture. For most precise results in the context of linearized training dynamics of a specific finite-width network, set bothtrace_axes=()
anddiagonal_axes=()
to obtain the kernel exactly of shapezip(f(x1).shape, f(x2).shape)
.For networks with multiple (i.e. lists, tuples, PyTrees) outputs, in principal the empirical kernels will have terms measuring the covariance between the outputs. Here, we ignore these cross-terms and consider each output separately. Please raise an issue if this feature is important to you.
- Parameters
f (
Callable
[[Any
,Union
[List
[ndarray
],Tuple
[ndarray
, …],ndarray
]],Union
[List
[ndarray
],Tuple
[ndarray
, …],ndarray
]]) – the function whose NNGP we are computing.f
should have the signaturef(params, inputs[, rng])
and should return annp.ndarray
outputs.trace_axes (
Union
[int
,Sequence
[int
]]) – output axes to trace the output kernel over, i.e. compute only the trace of the covariance along the respective pair of axes (one pair for each axis intrace_axes
). This allows to save space and compute if you are only interested in the respective trace, but also improve approximation accuracy if you know that covariance along these pairs of axes converges to aconstant * identity matrix
in the limit of interest (e.g. infinite width or infiniten_samples
). A common use case is the channel / feature / logit axis, since activation slices along such axis are i.i.d. and the respective covariance along the respective pair of axes indeed converges to a constant-diagonal matrix in the infinite width or infiniten_samples
limit. Also related to “contracting dimensions” in XLA terms. (https://www.tensorflow.org/xla/operation_semantics#dotgeneral)diagonal_axes (
Union
[int
,Sequence
[int
]]) – output axes to diagonalize the output kernel over, i.e. compute only the diagonal of the covariance along the respective pair of axes (one pair for each axis indiagonal_axes
). This allows to save space and compute, if off-diagonal values along these axes are not needed, but also improve approximation accuracy if their limiting value is known theoretically, e.g. if they vanish in the limit of interest (e.g. infinite width or infiniten_samples
). If you further know that on-diagonal values converge to the same constant in your limit of interest, you should specify these axes intrace_axes
instead, to save even more compute and gain even more accuracy. A common use case is computing the variance (instead of covariance) along certain axes. Also related to “batch dimensions” in XLA terms. (https://www.tensorflow.org/xla/operation_semantics#dotgeneral)
- Return type
Callable
[[Union
[List
[ndarray
],Tuple
[ndarray
, …],ndarray
],Union
[List
[ndarray
],Tuple
[ndarray
, …],ndarray
,None
],Any
],Union
[List
[ndarray
],Tuple
[ndarray
, …],ndarray
]]- Returns
A function to draw a single sample the NNGP of a given network
f
.
- neural_tangents.utils.empirical.empirical_ntk_fn(f, trace_axes=(- 1,), diagonal_axes=(), vmap_axes=None, implementation=1)[source]¶
Returns a function to draw a single sample the NTK of a given network
f
.The Neural Tangent Kernel is defined as \(J(X_1) J(X_2)^T\) where \(J\) is the Jacobian \(df/dparams\) of shape
full_output_shape + params.shape
.For best performance: 1) pass
x2=None
ifx1 == x2; 2) prefer square batches (i.e `x1.shape == x2.shape
); 3) make sure to setvmap_axes
correctly. 4) try differentimplementation
values.WARNING: Resulting kernel shape is nearly
zip(f(x1).shape, f(x2).shape)
subject totrace_axes
anddiagonal_axes
parameters, which make certain assumptions about the outputsf(x)
that may only be true in the infinite width / infinite number of samples limit, or may not apply to your architecture. For most precise results in the context of linearized training dynamics of a specific finite-width network, set bothtrace_axes=()
anddiagonal_axes=()
to obtain the kernel exactly of shapezip(f(x1).shape, f(x2).shape)
.For networks with multiple (i.e. lists, tuples, PyTrees) outputs, in principal the empirical kernels will have terms measuring the covariance between the outputs. Here, we ignore these cross-terms and consider each output separately. Please raise an issue if this feature is important to you.
- Parameters
f (
Callable
[[Any
,Union
[List
[ndarray
],Tuple
[ndarray
, …],ndarray
]],Union
[List
[ndarray
],Tuple
[ndarray
, …],ndarray
]]) – the function whose NTK we are computing.f
should have the signaturef(params, inputs[, rng])
and should return annp.ndarray
outputs.trace_axes (
Union
[int
,Sequence
[int
]]) – output axes to trace the output kernel over, i.e. compute only the trace of the covariance along the respective pair of axes (one pair for each axis intrace_axes
). This allows to save space and compute if you are only interested in the respective trace, but also improve approximation accuracy if you know that covariance along these pairs of axes converges to aconstant * identity matrix
in the limit of interest (e.g. infinite width or infiniten_samples
). A common use case is the channel / feature / logit axis, since activation slices along such axis are i.i.d. and the respective covariance along the respective pair of axes indeed converges to a constant-diagonal matrix in the infinite width or infiniten_samples
limit. Also related to “contracting dimensions” in XLA terms. (https://www.tensorflow.org/xla/operation_semantics#dotgeneral)diagonal_axes (
Union
[int
,Sequence
[int
]]) – output axes to diagonalize the output kernel over, i.e. compute only the diagonal of the covariance along the respective pair of axes (one pair for each axis indiagonal_axes
). This allows to save space and compute, if off-diagonal values along these axes are not needed, but also improve approximation accuracy if their limiting value is known theoretically, e.g. if they vanish in the limit of interest (e.g. infinite width or infiniten_samples
). If you further know that on-diagonal values converge to the same constant in your limit of interest, you should specify these axes intrace_axes
instead, to save even more compute and gain even more accuracy. A common use case is computing the variance (instead of covariance) along certain axes. Also related to “batch dimensions” in XLA terms. (https://www.tensorflow.org/xla/operation_semantics#dotgeneral)vmap_axes (
Optional
[Tuple
[Union
[List
[int
],Tuple
[int
, …],int
,None
],Union
[List
[int
],Tuple
[int
, …],int
,None
],Dict
[str
,Union
[List
[int
],Tuple
[int
, …],int
,None
]]]]) –A triple of
(in_axes, out_axes, kwargs_axes)
passed tovmap
to evaluate the empirical NTK in parallel ove these axes. Precisely, providing this argument implies thatf(params, x, **kwargs)
equals to a concatenation alongout_axes
off
applied to slices ofx
and**kwargs
alongin_axes
andkwargs_axes
. In other words, it certifies thatf
can be evaluated as avmap
without_axes=out_axes
overx
(alongin_axes
) and those arguments in**kwargs
that are present inkwargs_axes.keys()
(alongkwargs_axes.values()
).For example if
_, f, _ = nt.stax.Aggregate()
,f
is called viaf(params, x, pattern=pattern)
. By default, inputsx
, patternspattern
, and outputs off
are all batched along the leading0
dimension, and each outputf(params, x, pattern=pattern)[i]
only depends on the inputsx[i]
andpattern[i]
. In this case, we can passvmap_axes=(0, 0, dict(pattern=0)
to specify along which dimensions inputs, outputs, and keyword arguments are batched respectively.This allows us to evaluate Jacobians much more efficiently. If
vmap_axes
is not a triple, it is interpreted asin_axes = out_axes = vmap_axes, kwargs_axes = {}
. For example a very common use case isvmap_axes=0
for a neural network with leading (0
) batch dimension, both for inputs and outputs, and no interactions between different elements of the batch (e.g. no BatchNorm, and, in the case ofnt.stax
, also no Dropout). However, if there is interaction between batch elements or no concept of a batch axis at all,vmap_axes
must be set toNone
, to avoid wrong (and potentially silent) results.implementation (
int
) –1
or2
.1
directly instantiates Jacobians and computes their outer product.2
uses implicit differentiation to avoid instantiating whole Jacobians at once. The implicit kernel is derived by observing that: \(\Theta = J(X_1) J(X_2)^T = [J(X_1) J(X_2)^T](I)\), i.e. a linear function \([J(X_1) J(X_2)^T]\) applied to an identity matrix \(I\). This allows the computation of the NTK to be phrased as: \(a(v) = J(X_2)^T v\), which is computed by a vector-Jacobian product; \(b(v) = J(X_1) a(v)\) which is computed by a Jacobian-vector product; and \(\Theta = [b(v)] / d[v^T](I)\) which is computed via avmap
of \(b(v)\) over columns of the identity matrix \(I\).It is best to benchmark each method on your specific task. We suggest using
1
unless you get OOMs due to large number of trainable parameters, otherwise -2
.
- Return type
Callable
[[Union
[List
[ndarray
],Tuple
[ndarray
, …],ndarray
],Union
[List
[ndarray
],Tuple
[ndarray
, …],ndarray
,None
],Any
],Union
[List
[ndarray
],Tuple
[ndarray
, …],ndarray
]]- Returns
A function
ntk_fn
that computes the empirical ntk.
- neural_tangents.utils.empirical.linearize(f, params)[source]¶
Returns a function
f_lin
, the first order taylor approximation tof
.Example
>>> # Compute the MSE of the first order Taylor series of a function. >>> f_lin = linearize(f, params) >>> mse = np.mean((f(new_params, x) - f_lin(new_params, x)) ** 2)
- Parameters
f (
Callable
[…,Any
]) – A function that we would like to linearize. It should have the signaturef(params, *args, **kwargs)
where params is aPyTree
andf
should return aPyTree
.params (
Any
) – Initial parameters to the function that we would like to take the Taylor series about. This can be any structure that is compatible with the JAX tree operations.
- Return type
- Returns
A function
f_lin(new_params, *args, **kwargs)
whose signature is the same as f. Heref_lin
implements the first-order taylor series off
aboutparams
.
- neural_tangents.utils.empirical.taylor_expand(f, params, degree)[source]¶
Returns a function
f_tayl
, Taylor approximation tof
of orderdegree
.Example
>>> # Compute the MSE of the third order Taylor series of a function. >>> f_tayl = taylor_expand(f, params, 3) >>> mse = np.mean((f(new_params, x) - f_tayl(new_params, x)) ** 2)
- Parameters
f (
Callable
[…,Any
]) – A function that we would like to Taylor expand. It should have the signaturef(params, *args, **kwargs)
whereparams
is aPyTree
, andf
returns aPyTree
.params (
Any
) – Initial parameters to the function that we would like to take the Taylor series about. This can be any structure that is compatible with the JAX tree operations.degree (
int
) – The degree of the Taylor expansion.
- Return type
- Returns
A function
f_tayl(new_params, *args, **kwargs)
whose signature is the same asf
. Heref_tayl
implements thedegree
-order taylor series off
aboutparams
.