neural_tangents.empirical_ntk_fn
- neural_tangents.empirical_ntk_fn(f, trace_axes=(-1,), diagonal_axes=(), vmap_axes=None, implementation=NtkImplementation.JACOBIAN_CONTRACTION, _j_rules=True, _s_rules=True, _fwd=None)[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 (
ApplyFn
) – the function whose NTK we are computing. It should have the signaturef(params, x, **kwargs)
whereparams
is aPyTree
,x
is aPyTree
, andf
should also return aPyTree
.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 (
Union
[Any
,None
,tuple
[Optional
[Any
],Optional
[Any
],dict
[str
,Optional
[Any
]]]]) –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 (
Union
[NtkImplementation
,int
]) – AnNtkImplementation
value (or anint
0
,1
,2
, or3
). See theNtkImplementation
docstring for details._j_rules (
bool
) – Internal debugging parameter, applicable only whenimplementation
isSTRUCTURED_DERIVATIVES
(3
) orAUTO
(0
). Set toTrue
to allow custom Jacobian rules for intermediary primitivedy/dw
computations for MJJMPs (matrix-Jacobian-Jacobian-matrix products). Set toFalse
to use JVPs or VJPs, via JAX’sjax.jacfwd
orjax.jacrev
. Custom Jacobian rules (True
) are expected to be not worse, and sometimes better than automated alternatives, but in case of a suboptimal implementation setting it toFalse
could improve performance._s_rules (
bool
) – Internal debugging parameter, applicable only whenimplementation
isSTRUCTURED_DERIVATIVES
(3
) orAUTO
(0
). Set toTrue
to allow efficient MJJMp rules for structureddy/dw
primitive Jacobians. In practice should be set toTrue
, and setting it toFalse
can lead to dramatic deterioration of performance._fwd (
Optional
[bool
]) – Internal debugging parameter, applicable only whenimplementation
isSTRUCTURED_DERIVATIVES
(3
) orAUTO
(0
). Set toTrue
to allowjax.jvp
in intermediary primitive Jacobiandy/dw
computations,False
to always usejax.vjp
.None
to decide automatically based on input/output sizes. Applicable when_j_rules=False
, or when a primitive does not have a Jacobian rule. Should be set toNone
for best performance.
- Return type:
- Returns:
A function
ntk_fn
that computes the empirical ntk.