30 KiB
Towards Gradient Inversion
Note, this chapter is very preliminary - to be finalized
In the following we will question some fundamental aspects of the formulations so far, namely the update step computed via gradients. To re-cap, the approaches explained in the previous chapters either dealt with purely supervised training, integrated the physical model as a physical loss term or included it via differentiable physics (DP) operators embedded into the training graph. For supervised training with data from physical simulations standard procedure apply. The latter two methods are more relevant in the context of this book. They share similarities, but in the loss term case, the physics evaluations are only required at training time. For DP approaches, the solver itself is usually also employed at inference time, which enables an end-to-end training of NNs and numerical solvers. All three approaches employ first-order derivatives to drive optimizations and learning processes, and the latter two also using them for the physics terms. This is a natural choice from a deep learning perspective, but we haven’t questioned at all whether this is actually the best choice.
Not too surprising after this introduction: A central insight of the
following chapter will be that regular gradients are often a
sub-optimal choice for learning problems involving physical
quantities. It turns out that both supervised and DP gradients have
their pros and cons, and leave room for custom methods that are aware of
the physics operators. In particular, we’ll show in the following how
scaling problems of DP gradients affect NN training. Then, we’ll also
illustrate how multi-modal problems (as hinted at in
{doc}intro-teaser) negatively influence NNs. Finally, we’ll
explain several alternatives to prevent these problems. It turns out
that a key property that is missing in regular gradients is a proper
inversion of the Jacobian matrix.
```{admonition} A preview of this chapter :class: tip
Below, we’ll proceed in the following steps: - Show how scaling issues and multi-modality can negatively affect NN training. - Spoiler: What was missing in our training runs with GD or Adam so far is a proper inversion of the Jacobian matrix. - We’ll explain two alternatives to prevent these problems: an analytical full-, and a numerical half-inversion.
XXX notes, open issues XXX
- GD - is "diff. phys." , rename? add supervised before?
- comparison notebook: add legends to plot
- double check func-comp w QNewton, "later" derivatives of backprop means what?
## Traditional optimization methods
As before, let $L(x)$ be a scalar loss function, subject to minimization. The goal is to compute a step in terms of the input parameters $x$ , denoted by $\Delta x$. The different versions of $\Delta x$ will be denoted by a subscript.
All NNs of the previous chapters were trained with gradient descent (GD) via backpropagation. GD with backprop was also employed for the PDE solver (_simulator_) $\mathcal P$.
% , with an evaluation chain $L(\mathcal P(x))$.
When we simplify the setting, and leave out the NN for a moment, this gives the minimization problem
$\text{arg min}_{x} L(x)$ with $L(x) = \frac 1 2 \| \mathcal P(x) - y^* \|_2^2$.
As a central quantity, we have the composite gradient
$(\partial L / \partial x)^T$ of the loss function $L$:
$$
\Big( \frac{\partial L}{\partial x} \Big)^T =
\Big( \frac{\partial \mathcal P}{\partial x} \Big)^T
\Big( \frac{\partial L}{\partial \mathcal P} \Big)^T
$$ (loss-deriv)
We've shown that using $\partial L/\partial x$ works, but
in the field of classical optimization, other algorithms are more widely used than GD: popular are so-called Quasi-Newton methods, which use fundamentally different updates.
Hence, in the following we'll revisit GD along with Quasi-Newton methods and Inverse Jacobians as a third alternative. We'll focus on the pros and cons of the different methods on a theoretical level. Among others, it's interesting to discuss why classical optimization algorithms aren't widely used for NN training despite having some obvious advantages.
Note that we exclusively consider multivariate functions, and hence all symbols represent vector-valued expressions unless noted otherwise.
%techniques such as Newton's method or BFGS variants are commonly used to optimize numerical processes since they can offer better convergence speed and stability. These methods likewise employ gradient information, but substantially differ from GD in the way they compute the update step, typically via higher order derivatives.
%```{figure} resources/placeholder.png
%---
%height: 220px
%name: pg-training
%---
%TODO, visual overview of PG training
%```
## Gradient descent
The optimization updates $\Delta x_{\text{GD}}$ of GD scale with the derivative of the objective w.r.t. the inputs,
$$
\Delta x_{\text{GD}} = -\eta \cdot \frac{\partial L}{\partial x}
$$ (GD-update)
where $\eta$ is the scalar learning rate.
The Jacobian $\frac{\partial L}{\partial x}$ describes how the loss reacts to small changes of the input.
Surprisingly, this very widely used update has a number of undesirable properties that we'll highlight in the following. Note that we've naturally applied this update in supervised settings such as {doc}`supervised-airfoils`, but we've also used it in the differentiable physics approaches. E.g., in {doc}`diffphys-code-sol` we've computed the derivative of the fluid solver. In the latter case, we've still only updated the NN parameters, but the fluid solver Jacobian was part of {eq}`GD-update`, as shown in {eq}`loss-deriv`.
**Units** 📏
A first indicator that something is amiss with GD is that it inherently misrepresents dimensions.
Assume two parameters $x_1$ and $x_2$ have different physical units.
Then the GD parameter updates scale with the inverse of these units because the parameters appear in the denominator for the GD update above ($\cdots / \partial x$).
The learning rate $\eta$ could compensate for this discrepancy but since $x_1$ and $x_2$ have different units, there exists no single $\eta$ to produce the correct units for both parameters.
One could argue that units aren't very important for the parameters of NNs, but nonetheless it's unnerving from a physics perspective that they're wrong, and it hints at some more fundamental problems.
**Function sensitivity** 🔍
GD has also inherent problems when functions are not _normalized_.
This can be illustrated with a very simple example:
consider the function $L(x) = c \cdot x$.
Then the parameter updates of GD scale with $c$, i.e. $\Delta x_{\text{GD}} = \eta \cdot c$, and
$L(x+\Delta x_{\text{GD}})$ will even have terms on the order of $c^2$.
If $L$ is normalized via $c=1$, everything's fine. But in practice, we'll often
have $c \ll 1$, or even worse $c \gg 1$, and then our optimization will be in trouble.
More specifically, if we look at how the loss changes, the expansion around $x$ for
the update step of GD gives:
$L(x+\Delta x_{\text{GD}}) = L(x) + \Delta x_{\text{GD}} \frac{\partial L}{\partial x} + \cdots $.
This first-order step causes a change in the loss of
$\big( L(x) - L(x+\Delta x_{\text{GD}}) \big) = -\eta \cdot (\frac{\partial L}{\partial x})^2 + \mathcal O(\Delta x^2)$. Hence the loss changes by the squared derivative, which leads to the $c^2$ factor mentioned above. Even worse, in practice we'd like to have a normalized quantity here. For a scaling of the gradients by $c$, we'd like our optimizer to compute a quantity like $1/c^2$, in order to get a reliable update from the gradient.
This demonstrates that
for sensitive functions, i.e. functions where _small changes_ in $x$ cause _large_ changes in $L$, GD counter-intuitively produces large $\Delta x_{\text{GD}}$. This causes even larger steps in $L$, and leads to exploding gradients.
For insensitive functions where _large changes_ in the input don't change the output $L$ much, GD produces _small_ updates, which can lead to the optimization coming to a halt. That's the classic _vanishing gradients_ problem.
Such sensitivity problems can occur easily in complex functions such as deep neural networks where the layers are typically not fully normalized.
Normalization in combination with correct setting of the learning rate $\eta$ can be used to counteract this behavior in NNs to some extent, but these tools are not available when optimizing physics simulations.
Applying normalization to a simulation anywhere but after the last solver step would destroy the state of the simulation.
Adjusting the learning rate is also difficult in practice, e.g. when simulation parameters at different time steps are optimized simultaneously or when the magnitude of the simulation output varies w.r.t. the initial state.
**Convergence near optimum** 💎
Finally, the loss landscape of any differentiable function necessarily becomes flat close to an optimum,
as the gradient approaches zero upon convergence.
Therefore $\Delta x_{\text{GD}} \rightarrow 0$ as the optimum is approached, resulting in slow convergence.
This is an important point, and we will revisit it below. It's also somewhat surprising at first, but it can actually
stabilize the training. On the other hand, it makes the learning process difficult to control.
## Quasi-Newton methods
Quasi-Newton methods, such as BFGS and its variants, employ the gradient $\frac{\partial L}{\partial x}$ and the inverse of the Hessian $\frac{\partial^2 L}{\partial x^2}$ for the update
$$
\Delta x_{\text{QN}} = -\eta \cdot \left( \frac{\partial^2 L}{\partial x^2} \right)^{-1} \frac{\partial L}{\partial x} ,
$$ (quasi-newton-update)
where $\eta$, the scalar step size, takes the place of GD's learning rate. As a further improvement, it is typically determined via a line search in many Quasi-Newton methods (we'll leave out this step for now).
This construction solves some of the problems of gradient descent from above, but has other drawbacks.
**Units and Sensitivity** 📏
Quasi-Newton methods definitely provide a much better handling of physical units than GD.
The quasi-Newton update from equation {eq}`quasi-newton-update`
produces the correct units for all parameters to be optimized.
As a consequence, $\eta$ can stay dimensionless.
If we now consider how the loss changes via
$L(x+\Delta x_{\text{QN}}) = L(x) + -\eta \cdot \left( \frac{\partial^2 L}{\partial x^2} \right)^{-1} \frac{\partial L}{\partial x} \frac{\partial L}{\partial x} + \cdots $ , the second term correctly cancels out the $x$ quantities, and leaves us with a scalar update in terms of $L$. Thinking back to the example with a scaling factor $c$ from the GD section, the inverse Hessian in Newton's methods successfully gives us a factor of $1/c^2$ to couteract the undesirable scaling of our updates.
**Convergence near optimum** 💎
Quasi-Newton methods also exhibit much faster convergence when the loss landscape is relatively flat.
Instead of slowing down, they take larger steps, even when $\eta$ is fixed.
This is thanks to the eigenvalues of the inverse Hessian, which scale inversely with the eigenvalues of the Hessian, and hence increase with the flatness of the loss landscape.
**Consistency in function compositions**
So far, quasi-Newton methods address both shortcomings of GD.
However, similar to GD, the update of an intermediate space still depends on all functions before that.
This behavior stems from the fact that the Hessian of a composite function carries non-linear terms of the gradient.
Consider a function composition $L(y(x))$, with $L$ as above, and an additional function $y(x)$.
Then the Hessian $\frac{d^2L}{dx^2} = \frac{\partial^2L}{\partial y^2} \left( \frac{\partial y}{\partial x} \right)^2 + \frac{\partial L}{\partial y} \cdot \frac{\partial^2 y}{\partial x^2}$ depends on the square of the inner gradient $\frac{\partial y}{\partial x}$.
This means that the Hessian is influenced by the _later_ derivatives of a backprop pass,
and as a consequence, the update of any latent space is unknown during the computation of the gradients.
% chain of function evaluations: Hessian of an outer function is influenced by inner ones; inversion corrects and yields quantity similar to IG, but nonetheless influenced by "later" derivatives
**Dependence on Hessian** 🎩
In addition, a fundamental disadvantage of quasi-Newton methods that becomes apparent from the discussion above is their dependence on the Hessian. It plays a crucial role for all the improvements discussed so far.
The first obvious drawback is the _computational cost_.
While evaluating the exact Hessian only adds one extra pass to every optimization step, this pass involves higher-dimensional tensors than the computation of the gradient.
As $\frac{\partial^2 L}{\partial x^2}$ grows with the square of the parameter count, both its evaluation and its inversion become very expensive for large systems.
Many algorithms therefore avoid computing the exact Hessian and instead approximate it by accumulating the gradient over multiple update steps.
The memory requirements also grow quadratically.
The quasi-Newton update above additionally requires the _inverse_ Hessian matrix. Thus, a Hessian that is close to being non-invertible typically causes numerical stability problems, while inherently non-invertible Hessians require a fallback to a first order GD update.
Another related limitation of quasi-Newton methods is that the objective function needs to be _twice-differentiable_.
While this may not seem like a big restriction, note that many common neural network architectures use ReLU activation functions of which the second-order derivative is zero.
%
Related to this is the problem that higher-order derivatives tend to change more quickly when traversing the parameter space, making them more prone to high-frequency noise in the loss landscape.
```{note}
_Quasi-Newton Methods_
are still a very active research topic, and hence many extensions have been proposed that can alleviate some of these problems in certain settings. E.g., the memory requirement problem can be sidestepped by storing only lower-dimensional vectors that can be used to approximate the Hessian. However, these difficulties illustrate the problems that often arise when applying methods like BFGS.% %, and how to they can be used to improve the training of neural networks.
Inverse Gradients
As a first step towards fixing the aforementioned issues, we’ll consider what we’ll call inverse gradients (IGs). Unfortunately, they come with their own set of problems, which is why they only represent an intermediate step (we’ll revisit them in a more practical form later on).
Instead of L (which is scalar), let’s consider optimization problems for a generic, potentially non-scalar function y(x). This will typically be the physical simulator \mathcal P later on, but to keep things general and readable, we’ll call it y for now. This setup implies an inverse problem: for y = \mathcal P(x) we want to find an x given a target y^*. We define the update
\Delta x_{\text{IG}} = \frac{\partial x}{\partial y} \cdot \Delta y. (IG-def)
to be the IG update. Here, the Jacobian \frac{\partial x}{\partial y}, which is similar to the inverse of the GD update above, encodes how the inputs must change in order to obtain a small change \Delta y in the output. % The crucial step is the inversion, which of course requires the Jacobian matrix to be invertible. This is a problem somewhat similar to the inversion of the Hessian, and we’ll revisit this issue below. However, if we can invert the Jacobian, this has some very nice properties.
Note that instead of using a learning rate, here the step size is determined by the desired increase or decrease of the value of the output, \Delta y. Thus, we need to choose a \Delta y instead of an \eta. This \Delta y will show up frequently in the following equations, and make them look quite different to the ones above at first sight. Effectively, \Delta y plays the same role as the learning rate, i.e., it controls the step size of the optimization.
% Positive Aspects Units 📏
IGs scale with the inverse derivative. Hence the updates are automatically of the same units as the parameters without requiring an arbitrary learning rate: \frac{\partial x}{\partial y} times \Delta y has the units of x.
Function sensitivity 🔍
They also don’t have problems with normalization as the parameter updates from the example L(x) = c \cdot x above now scale with c^{-1}. Sensitive functions thus receive small updates while insensitive functions get large updates.
Convergence near optimum and function compositions 💎
IGs show the opposite behavior of GD close to an optimum: they produce updates that still progress the optimization. This leads to fast convergence, as we will demonstrate in more detail below.
% Consistency in function compositions
Additionally, IGs are consistent in function composition. The change in x is \Delta x_{\text{IG}} = \Delta L \cdot \frac{\partial x}{\partial y} \frac{\partial y}{\partial L} and the approximate change in y is \Delta y = \Delta L \cdot \frac{\partial y}{\partial x} \frac{\partial x}{\partial y} \frac{\partial y}{\partial L} = \Delta L \frac{\partial y}{\partial L}. % In the example in table~, the change \Delta y is the same no matter what space is used as optimization target. The change in intermediate spaces is independent of their respective dependencies, at least up to first order. Consequently, the change to these spaces can be estimated during backprop, before all gradients have been computed.
Note that even Newton’s method with its inverse Hessian didn’t fully get this right. The key here is that if the Jacobian is invertible, we’ll directly get the correctly scaled direction at a given layer, without helper quantities such as the inverse Hessian.
% Limitations Dependence on the inverse Jacobian 🎩
So far so good. The above properties are clearly advantageous, but unfortunately IGs require the inverse of the Jacobian, \frac{\partial x}{\partial y}. It is only well-defined for square Jacobians, meaning for functions y with the same inputs and output dimensions. In optimization, however, the input is typically high-dimensional while the output is a scalar objective function. And, somewhat similar to the Hessians of quasi-Newton methods, even when the \frac{\partial y}{\partial x} is square, it may not be invertible.
Thus, we now consider the fact that inverse gradients are linearizations of inverse functions and show that using inverse functions provides additional advantages while retaining the same benefits.
% — split — ?
Inverse simulators
So far we’ve discussed the problems of existing methods, and a common theme among the methods that do better, Newton and IGs, is that the regular gradient is not sufficient. We somehow need to address it’s problems with some form of inversion. Before going into details of NN training and numerical methods to perform this inversion, we will consider one additional “special” case that will further illustrate the need for inversion: if we can make use of an inverse simulator, this likewise addresses many of the inherent issues of GD. It actually represents the ideal setting for computing update steps for the physics simulation part.
Let y = \mathcal P(x) be a forward simulation, and \mathcal P(y)^{-1}=x denote its inverse. In contrast to the inversion of Jacobian or Hessian matrices from before, \mathcal P(^{-1} denotes a full inverse of all functions of \mathcal P. Employing the inverse solver in the minimization problem above yields
\text{arg min}_{x} \frac 1 2 \| \mathcal P^{-1}(y^*) \|_2^2 , (pg-inverse-problem)
which, somewhat surprisingly, is not a minimization problem anymore if we consider single cases with one x,y^* pair. We basically just need to solve the inverse problem by evaluating \mathcal P^{-1}(y^*) to obtain x. As we plan to bring back NNs and more complex scenarios soon, let’s assume that we are still dealing with a collection of y^* targets, and non-obvious solutions x. One example could be that we’re looking for an x that yields multiple y^* targets with minimal distortions in terms of L^2.
Now, instead of evaluating \mathcal P^{-1} once to obtain the solution, we can iteratively update a current approximation of the solution x_0 with an update that we’ll call \Delta x_{\text{PG}} when employing the inverse physical simulator.
It also turns out to be a good idea to employ a local inverse that is conditioned on an initial guess for the solution x. We’ll denote this local inverse with \mathcal P^{-1}(y^*; x). As there are potentially very different x-space locations that result in very similar y^*, we’d like to find the one closest to the current guess. This is important to obtain well behaved solutions in multi-modal settings, where we’d like to avoid the solution manifold to consist of a set of very scattered points.
Equipped with these changes, we can formulate an optimization problem where a current state of the optimization x_0, with y_0 = \mathcal P(x_0), is updated with
\frac{\Delta x_{\text{PG}} }{\Delta y} \equiv \big( \mathcal P^{-1} (y_0 + \Delta y; x_0) - x_0 \big) . (PG-def)
Here the step in y-space, \Delta y, is either the full distance y^*-y_0 or a part of it, in line with the
learning rate from above, the the y-step used for IGs. When applying the update
\Delta x_{\text{PG}} = \mathcal P^{-1}(y_0 +
\Delta y; x_0) - x_0 it will produce \mathcal P(x_0 + \Delta x) = y_0 + \Delta y
exactly, despite \mathcal P being a
potentially highly nonlinear function. When rewriting this update in the
typical gradient format, \frac{\Delta
x_{\text{PG}}}{\Delta y} replaces the gradient from the IG update
above {eq}IG-def, and gives \Delta x.
This expression yields a first iterative method that makes use of \mathcal P^{-1}, and as such leverages all its information, such as higher-order terms.
Summary
The update obtained with a regular gradient descent method has surprising shortcomings due to scaling issues. Classical, inversion-based methods like IGs and Newton’s method remove some of these shortcomings, with the somewhat theoretical construct of the update from inverse simulators (\Delta x_{\text{PG}}) including the most higher-order terms. As such, it is interesting to consider as an “ideal” setting for improved (inverted) update steps. It get’s all of the aspect above right: units 📏, function sensitivity 🔍, compositions, and convergence near optima 💎. It provides a scale-invariant update. This comes at the cost of requiring an expression and discretization for a local inverse solver 🎩.
In contrast to the second- and first-order approximations from Newton’s method and IGs, it can potentially take highly nonlinear effects into account. Due to the potentially difficult construct of the inverse simulator, the main goal of the following sections is to illustrate how much we can gain from including all the higher-order information. Note that all three methods successfully include a rescaling of the search direction via inversion, in contrast to the previously discussed GD training. All of these methods represent different forms of differentiable physics, though.
Before moving on to including improved updates in NN training processes, we will discuss some additional theoretical aspects, and then illustrate the differences between these approaches with a practical example.
{note} The following sections will provide an in-depth look ("deep-dive"), into optimizations with inverse solvers. If you're interested in practical examples and connections to NNs, feel free to skip ahead to {doc}`physgrad-comparison` or {doc}`physgrad-nn`, respectively.
Deep Dive into Inverse simulators
We’ll now derive and discuss the \Delta x_{\text{PG}} update in more detail. Physical processes can be described as a trajectory in state space where each point represents one possible configuration of the system. A simulator typically takes one such state space vector and computes a new one at another time. The Jacobian of the simulator is, therefore, necessarily square. As long as the physical process does not destroy information, the Jacobian is non-singular. In fact, it is believed that information in our universe cannot be destroyed so any physical process could in theory be inverted as long as we have perfect knowledge of the state. Hence, it’s not unreasonable to expect that \mathcal P^{-1} can be formulated in many settings.
% We now show that these terms can help produce more stable updates than the IG alone, provided that \mathcal P_{(x_0,z_0)}^{-1} is a sufficiently good approximation of the true inverse. % Let \mathcal P^{-1}(z) be the true inverse function to \mathcal P(x), assuming that \mathcal P is fully invertible.
Fundamental theorem of calculus
To more clearly illustrate the advantages in non-linear settings, we apply the fundamental theorem of calculus to rewrite the ratio \Delta x_{\text{PG}} / \Delta y from above. This gives,
% \begin{equation}
% \begin{aligned} % \frac{\Delta z}{\Delta x} = \frac{\int_{x_0}^{x_0+\Delta x} \frac{\partial z}{\partial x} \, dx}{\Delta x} % \end{aligned}
% where we’ve integrated over a trajectory in x, and % focused on 1D for simplicity. Likewise, by integrating over z we can obtain:
\begin{aligned} \frac{\Delta x_{\text{PG}}}{\Delta y} = \frac{\int_{y_0}^{y_0+\Delta y} \frac{\partial x}{\partial y} \, dy}{\Delta y} \end{aligned}
Here the expressions inside the integral is the local gradient, and we assume it exists at all points between y_0 and y_0+\Delta y_0. The local gradients are averaged along the path connecting the state before the update with the state after the update. The whole expression is therefore equal to the average gradient of \mathcal P between the current x and the estimate for the next optimization step x_0 + \Delta x_{\text{PG}}. This effectively amounts to smoothing the objective landscape of an optimization by computing updates that can take nonlinearities of \mathcal P into account.
The equations naturally generalize to higher dimensions by replacing the integral with a path integral along any differentiable path connecting x_0 and x_0 + \Delta x_{\text{PG}} and replacing the local gradient by the local gradient in the direction of the path.
Global and local inverse simulators
Let \mathcal P be a function with a square Jacobian and y = \mathcal P(x). A global inverse function \mathcal P^{-1} is defined only for bijective \mathcal P. If the inverse exists, it can find x for any y such that y = \mathcal P(x).
Instead of using this “perfect” inverse \mathcal P^{-1} directly, we’ll in practice often use a local inverse \mathcal P_{(x_0,y_0)}^{-1}(y; x_0), which is conditioned for the point x_0, and correspondingly on y_0=\mathcal P(x_0). This local inverse is easier to obtain, as it only needs to exist near a given y_0, and not for all y. For the generic \mathcal P^{-1} to exist \mathcal P would need to be globally invertible.
By contrast, a local inverse only needs to exist and be accurate in the vicinity of (x_0, y_0). If a global inverse \mathcal P^{-1}(y) exists, the local inverse approximates it and matches it exactly as y \rightarrow y_0. More formally, \lim_{y \rightarrow y_0} \frac{\mathcal P^{-1}(y; x_0) - P^{-1}(y)}{|y - y_0|} = 0. Local inverse functions can exist, even when a global inverse does not. Non-injective functions can be inverted, for example, by choosing the closest x to x_0 such that \mathcal P(x) = y.
For differentiable functions, a local inverse is guaranteed to exist by the inverse function theorem as long as the Jacobian is non-singular. That is because the inverse Jacobian \frac{\partial x}{\partial y} itself is a local inverse function, albeit, with being first-order, not the most accurate one. Even when the Jacobian is singular (because the function is not injective, chaotic or noisy), we can usually find good local inverse functions.
Integrating a loss function
Since introducing IGs, we’ve only considered a simulator with an
output y. Now we can re-introduce the
loss function L. As before, we consider
minimization problems with a scalar objective function L(y) that depends on the result of an
invertible simulator y = \mathcal P(x).
%In {doc}physgrad In {eq}`` we’ve introduced the inverse
gradient (IG) update, which gives \Delta x =
\frac{\partial x}{\partial L} \cdot \Delta L when the loss
function is included. Here, \Delta L
denotes a step to take in terms of the loss.
By applying the chain rule and substituting the IG \frac{\partial x}{\partial L} for the update
from the inverse physics simulator from equation
{eq}PG-def, we obtain, up to first order:
\begin{aligned} \Delta x_{\text{PG}} &= \frac{\partial x}{\partial L} \cdot \Delta L \\ &= \frac{\partial x}{\partial y} \left( \frac{\partial y}{\partial L} \cdot \Delta L \right) \\ &= \frac{\partial x}{\partial y} \cdot \Delta y \\ &= \mathcal P^{-1}(y_0 + \Delta y; x_0) - x_0 + \mathcal O(\Delta y^2) \end{aligned}
These equations show that equation {eq}PG-def is equal
to the IG from the section above up to first order, but contains
nonlinear terms, i.e. $ x_{} / y = + O(y^2) $. The accuracy of the
update depends on the fidelity of the inverse function \mathcal P^{-1}. We can define an upper limit
to the error of the local inverse using the local gradient \frac{\partial x}{\partial y}. In the worst
case, we can therefore fall back to the regular gradient.
Also, we have turned the step w.r.t. L into a step in y space: \Delta
y. However, this does not prescribe a unique way to compute \Delta y since the derivative \frac{\partial y}{\partial L} as the
right-inverse of the row-vector \frac{\partial
L}{\partial y} puts almost no restrictions on \Delta y. Instead, we use a Newton step
(equation {eq}quasi-newton-update) to determine \Delta y where \eta controls the step size of the
optimization steps. We will explain this in more detail in connection
with the introduction of NNs in the next section.