Local Loss Optimization in the Infinite Width: Stable Parameterization of Predictive Coding Networks and Target Propagation

## Abstract

Abstract Local learning, which trains a network through layer-wise local targets and losses, has been studied as an alternative to backpropagation (BP) in neural computation. However, its algorithms often become more complex or require additional hyperparameters because of the locality, making it challenging to identify desirable settings in which the algorithm progresses in a stable manner.
To provide theoretical and quantitative insights, we introduce the maximal update parameterization ( μ 𝜇 \mu italic_μ P) in the infinite-width limit for two representative designs of local targets: predictive coding (PC) and target propagation (TP). We verified that μ 𝜇 \mu italic_μ P enables hyperparameter transfer across models of different widths.
Furthermore, our analysis revealed unique and intriguing properties of μ 𝜇 \mu italic_μ P that are not present in conventional BP. By analyzing deep linear networks, we found that PC’s gradients interpolate between first-order and Gauss-Newton-like gradients, depending on the parameterization.
We demonstrate that, in specific standard settings, PC in the infinite-width limit behaves more similarly to the first-order gradient.
For TP, even with the standard scaling of the last layer, which differs from classical μ 𝜇 \mu italic_μ P, its local loss optimization favors the feature learning regime over the kernel regime.

## 1 Introduction

Deep learning has achieved remarkable performance by building upon the backpropagation (BP) algorithm and developing architectures specialized for it [Rumelhart et al., , LeCun et al., , ]. BP, however, is not always a suitable method for more general objectives, such as biologically plausible computation [Lillicrap et al., , Bredenberg et al., ] or efficient distributed computation [Amid et al., ]. A representative alternative is local loss optimization, a type of credit assignment problem, in which loss functions are defined layer-wise, and targets are set locally. The basic formulation involves performing regression on target signals at each layer to reduce the global error across the entire network: Predictive Coding networks, usually referred to as PC, generate their targets through the internal dynamics of inference [Whittington and Bogacz, , Song et al., , Salvatori et al., ], while Target Propagation (TP)
generates them using feedback networks [Bengio, , Lee et al., , Ernoult et al., ].

In many cases, the use of local losses requires additional hyperparameters (HPs) and their careful tuning, making the algorithm configuration significantly more complicated compared to that of BP.
For example, PC requires not only the usual HPs, such as learning rate and initialization of weight parameters, but also those for the inference phase, such as the initialization of the state and the number of inference sequences. These HPs are primary considerations and have been reported as critical for ensuring stable training behavior [Pinchetti et al., , Alonso et al., , Rosenbaum, ].
A few analyses have succeeded in providing theoretical intuition for such local learning algorithms by introducing specific conditions or additional corrections that bridge them to classical optimization formulations [Song et al., , Alonso et al., , Meulemans et al., ]. However, such conditions are not always met in practice and may not be commonly shared across the entire family of methods.
To develop local learning that is more easily manageable across a broader range of settings, it is promising to establish a theoretical foundation that enables the analysis of natural learning dynamics under fewer constraints.

For standard BP, deep learning theory offers insights into the universal properties of learning [Bahri et al., , Bartlett et al., ]. A key research focus in this area is understanding learning in the infinite-width limit, including studies on neural tangent kernel (NTK) and feature learning regimes [Jacot et al., , Chizat et al., , Mei et al., , Bordelon and Pehlevan, ]. In particular, Yang and Hu [] provided a unified perspective on the parameterizations that realize these learning regimes and proposed maximal update parameterization ($\mu$P) as a unique scaling of HPs, such as random initialization and learning rates, that achieves feature learning in the infinite-width limit. Building on this developing theoretical foundation, we expect to gain universal insight into local learning, which has not yet been systematically analyzed.

In this work, we derive the $\mu$P for PC and TP and investigate hyperparameter transfer (the so-called $\mu$Transfer) across different widths. Our contributions are summarized as follows:

- •
While it is known that PC inference trivially reduces to gradient computation of BP under the fixed prediction assumption (FPA), a technical and heuristic condition, there is generally no guarantee that PC will reduce to BP, making it highly non-trivial to identify its $\mu$P. We first consider PC with a single sequential inference and reveal the $\mu$P even without FPA (Theorem 11). We also empirically verify the $\mu$Transfer of learning rates, showing that the optimal learning rate does not depend on the order of width.
- •
Second, for a more general context involving multiple inference sequences, we consider the convergence of the inference phase. We find that, for deep linear networks, we can explicitly obtain the local targets and losses at the fixed point of the inference, which depend on inference step sizes (Theorem 4.2). Interestingly, it takes a similar form to the Gauss-Newton (GN) gradient, but it can reduce to the conventional first-order gradient descent (GD) depending on the parameterization and step sizes. We find that the eventual gradient is closer to GD for sufficiently wide neural networks under standard experimental settings with $\mu$P. We also confirm that a larger inference step size, identified through this analysis, enhances $\mu$Transfer of HPs.
- •
Finally, we derive $\mu$P for both TP and its variant difference target propagation (DTP) assuming linear feedback networks (Theorem 19). We reveal a distinct property that differs from BP and PC; the feedback network of (D)TP changes the preferable scale of the last layer compared to the usual $\mu$P and causes the absence of the kernel regime. In this sense, (D)TP favors feature learning more strongly than other learning methods.

Thus, this study provides a solid and qualitative foundation for the further development of local learning schemes in large-scale neural networks in the future.

## 2 Related Work

Local learning: Most research on local learning stems from the exploration of biologically plausible learning [Lillicrap et al., ], with PC and TP following this line.
As deep learning has evolved, local learning has also begun to focus on large-scale networks, and some models have achieved performances close to those trained with BP [Ernoult et al., , Ren et al., ].
Several algorithms are inherently structured to resemble the BP chain [Akrout et al., ] or to estimate first-order gradients [Scellier and Bengio, ]. In contrast, PC relies on an inference phase, and TP uses a feedback network, both of which are quite different from BP and seems to be fundamental designs for using local targets.
However, their optimization properties are still not well understood.
Alonso et al. [] proposed a modified PC as a proximal point algorithm (implicit SGD), though it requires additional corrections and adaptive rescaling [Alonso et al., ].
Innocenti et al. [] proposed an inference phase computed by a GN method, but it requires a quadratic approximation of the local loss around a special initialization.
As discussed in the next section, bridging to such classical optimization requires strong conditions that may deviate significantly from the original purpose and algorithm [Rosenbaum, , Meulemans et al., ].

Infinite width and $\mu$P: While the NTK regime guarantees the existence of learning dynamics in the infinite-width limit and its global convergence, it reduces to just a kernel method [Jacot et al., , Lee et al., ]. To realize feature learning in the infinite-width limit, Yang and Hu [] proposed $\mu$P, which is a non-trivial scaling of HPs with respect to the width. From a theoretical perspective, this serves as a parameterization that enables the dynamics of feature learning, such as those described by the mean-field regime [Mei et al., ] or the dynamical mean-field theory [Bordelon and Pehlevan, ]. For more applications,
$\mu$P or its extension has been validated across various architectures [Yang et al., , Vyas et al., , Everett et al., ].
It covers not only the naive first-order gradient but also entry-wise adaptive optimizers such as Adam [Yang and Littwin, ] and second-order optimization methods like K-FAC [Ishikawa and Karakida, ]. There has been little previous work on the infinite-width analysis of local learning. Bordelon and Pehlevan [] formulated (direct) feedback alignment and (supervised) Hebbian learning using dynamical mean-field theory, which are rather close to BP.

## 3 Preliminaries

In this section, we provide an overview of local learning and $\mu$P in an $L$-layer fully connected neural network:

$$ $h_{l}=\phi\left(u_{l}\right),\quad u_{l}=W_{l}h_{l-1}\quad(l=1,\dots,L),$ (1) $$

where $W_{l}\in\mathbb{R}^{M_{l}\times M_{l-1}}$ are weight matrices, $h_{l},u_{l}\in\mathbb{R}^{M_{l}\times N}$ are activations and $N$ is the number of samples, independent of the order of width $M$. We set the width of the hidden layers to $M_{l}=M$ for $(l=1,\dots,L-1)$ for simplicity.
To keep the notation concise, for non-linear networks, we set $M_{L}=1$ ; however, we can easily generalize to $M_{L}=\Theta(1)$.
The activation function $\phi(\cdot)$ is usually assumed to be differentiable and polynomially bounded for some theoretical reasons within the $\mu$P framework [Yang and Hu, ].

### 3.1 Overview of Local Learning

#### 3.1.1 Predictive Coding

Predictive Coding updates both the states and weights to minimize the following free-energy function [Whittington and Bogacz, , Song et al., , Salvatori et al., ]:

$$ $\mathcal{F}(v,W)=\gamma_{L}\mathcal{L}(y,W_{L}\phi(v_{L-1}))+\sum_{l=1}^{L-1} \gamma_{l}\frac{1}{2}\left\|v_{l}-W_{l}\phi(v_{l-1})\right\|^{2}.$ (2) $$

To distinguish the internal state from the forward signal propagation $u_{l}$, we denote this state as $v_{l}$.
Although this algorithm was originally derived from the variational Bayes formulation, it has been extended beyond the scope of the original framework, aiming instead to develop inference computations that work more effectively in practice.
PC is composed of two phases: an inference phase, in which the per-layer states $v_{l}$ are updated and a learning phase, in which weights $W_{l}$ are updated.
Its update rule for the inference phase is given by

$$ $\displaystyle v_{l,s+1}$ $\displaystyle=v_{l,s}-\frac{\partial\mathcal{F}}{\partial v_{l}}=v_{l,s}- \gamma_{l}e_{l,s}+\gamma_{l+1}\phi^{\prime}\left(v_{l,s}\right)\circ W_{l+1}^{ \top}e_{l+1,s},$ (3) $$

where we define $e_{l,s}:=v_{l,s}-W_{l}\phi(v_{l-1,s})$ and $\circ$ denotes the Hadamard product.
The update rule for the learning phase is given by

$$ $\displaystyle W_{l,t+1}$ $\displaystyle=W_{l,t}-\eta_{l}\frac{\partial\mathcal{F}}{\partial W_{l}}=W_{l, t}+\eta_{l}\gamma_{l}e_{l,s}\phi(v_{l-1,s})^{\top}.$ (4) $$

Note that the inference time index $s$ and the parameter update index $t$ are distinct with $s$ resetting to 0 at each $t$. We usually omit the step size $\gamma_{l}$ in Eq. (4) in implementation.
Generally, weights are updated after multiple inference steps, while incremental version of PC (iPC), which updates the weights after just a single inference, has also been proposed [Salvatori et al., ].
The internal state can be updated simultaneously across all layers or computed sequentially in a specified order. In the first part of the next section, we focus on the Sequential Inference (SI) method, where $e_{l,s}$ is computed sequentially by propagating from the output layer to the input layer. For more details on this difference, see Algorithm 1 in the Appendix.

Empirically, to improve the trainability, PC often relies on assumptions that are either rational or, at times, unrealistic. One such reasonable assumption is the initialization method for $v_{l,0}$, which is used to improve the convergence [Song et al., , Alonso et al., , Rosenbaum, ]:

Technique (i): Forward initialization (F-ini).
At each training step $t$, $v_{l,0}$ is initialized such that $v_{l,0}=u_{l,t}$ , which ensures $e_{l,0}=0$.

Generally, the gradient computation of PC does not match that of BP. However, under F-ini and SI, it reduces to BP by adopting the following rather technical assumption [Millidge et al., , Rosenbaum, ]:

Technique (ii): Fixed prediction assumption (FPA).
Replace $\phi\left(v_{l-1,s}\right)$ with $\phi\left(v_{l-1,0}\right)$ during the inference phase.

Under FPA, the inference is given by
${{e}}_{l,s+1}=(1-\gamma_{l}){{e}}_{l,s}+\gamma_{l+1}\phi^{\prime}\left(v_{l,0}
\right)\circ W_{l+1}^{\top}{{e}}_{l+1,s}.$
By substituting F-ini, one can easily verify that this sequential inference computes $\nabla_{u_{l}}\mathcal{L}$.

#### 3.1.2 Target Propagation

In target propagation (TP), $\hat{h}_{L}=h_{L}-\hat{\eta}\nabla_{h_{L}}\mathcal{L}$ is propagated through the feedback network, which generates local targets $\hat{h}_{l}$ as follows:

$$ $\hat{h}_{l}=g_{l}(\hat{h}_{l+1}),\quad g_{l}(x)=\psi(Q_{l}x)\quad(l=1,...,L-1),$ (5) $$

where $Q_{l}\in\mathbb{R}^{M_{l-1}\times M_{l}}$ are weight matrices. We also analyze the Difference Target Propagation (DTP), a variant of TP, whose definition is provided in the appendix.
The feedback network is trained to minimize the following reconstruction loss:

$$ $\mathcal{L}_{\textbf{rec}}(Q_{l})=\left\|g_{l}\left(f_{l}(h_{l-1})\right)-h_{l -1}\right\|^{2},$ (6) $$

where $f_{l}(x)=\phi(W_{l}x)$.
TP updates the weights $W_{l}$ to minimize the following local loss $\|e_{l}\|^{2}:=\|\hat{h}_{l}-h_{l}\|^{2}$.
The gradient of this local loss provides the update rule for the learning phase as
$W_{l,t+1}=W_{l,t}-\eta_{l}\phi^{\prime}(W_{l,t}{h}_{l-1})\circ e_{l}{h}_{l-1}^
{\top}.$
For a so-called invertible network, TP computes the Gauss-Newton Target (GNT), i.e., $e_{l}^{\text{GNT}}=(\delta_{l}\delta_{l}^{\top}+\rho I)^{-1}\delta_{l}{e}_{L}$ where $\delta_{l}=\nabla_{u_{l}}u_{L}$ is the BP signal [Meulemans et al., ]
(^1^11The final gradient $d\mathcal{F}/dW_{l}$ is equivalent to the special case of K-FAC [Martens and Grosse, ] where the preconditioners are applied only to the backward signals.).
Note that the assumption of the invertible network is restrictive
because the invertible network requires invertible activation functions, regular weight matrices, and the training tp converge to the solution of $g_{l}(\hat{{h}}_{l+1})=f_{l+1}^{-1}(\hat{{h}}_{l+1})=W_{l+1}^{-1}\phi^{-1}(
\hat{{h}}_{l+1})$. For general networks,
(D)TP does not necessarily lead to the GNT.

Remark on a connection between PC and TP.
Some previous studies have argued that PC yields GNT-like solutions, and thus can be connected to TP [Alonso et al., , Millidge et al., ].
These works attempt to gain an intuitive insight from the fixed point equation for each layer:

$$ $\displaystyle h_{l}^{*}$ $\displaystyle=\left(W_{l+1}^{\top}W_{l+1}+\gamma_{l+1}/\gamma_{l}I\right)^{-1} \left(W_{l+1}^{\top}h_{l+1}^{*}+\gamma_{l+1}/\gamma_{l}W_{l}h_{l-1}^{*}\right),$ (7) $$

where $h_{l}^{*}$ means $\phi(v_{l}^{*})$.
For $\gamma_{l+1}/\gamma_{l}\ll 1$, we approximate $h_{l}^{*}\approx W_{l+1}^{\dagger}h_{l+1}^{*}$. If we multiply this approximation across layers, the naive expectation is that $h_{l}^{*}\approx\prod_{i=l+1}^{L}W_{i}^{\dagger}e_{L}^{*}$, which corresponds to the GNT for linear networks.
Thus, we can intuitively see that the PC may be linked to the GNT, although its exact connection requires careful limit operations across layers. Additionally, taking the limits $\gamma_{l+1}/\gamma_{l}\ll 1$ for all layers means the exponential decay of $\gamma_{l}$ with depth, rasing concerns regarding its practical relevance.
For $\gamma_{l}=1$, Innocenti et al. [] has recently derived an explicit formulation of the free energy at the fixed point using an unfolding calculation of a hierarchical Gaussian model. This fomulation shows that the obtained gradient differs from that of the exact GNT, supporting the idea that the connection to GNT would be weak.

### 3.2 μ 𝜇 \mu italic_μ P and Learning Regimes

The abc-parameterization scales the parameters by width, as follows [Yang and Hu, ]:

$$ ${W}_{l}={w}_{l}/M^{a_{l}},\quad{w}_{l}\sim\mathcal{N}(0,{\sigma^{\prime 2}}/M^ {2b_{l}}),\quad\eta_{l}=\eta_{l}^{\prime}/M^{c_{l}}.$ (8) $$

$\mu$P and its conditions:
Consider the temporal change of $u_{l}$ by the parameter update:

$$ $\Delta u_{l,t}:=u_{l,t}-u_{l,0}=\Theta\left(1/M^{r_{l}}\right),$ (9) $$

where $\Theta(\cdot)$ denotes the order with respect to the width and $x=\Theta(M^{a})$ means $\sqrt{\|x\|^{2}/M}=\Theta(M^{a})$ for $x\in\mathbb{R}^{M}$.
Yang and Hu [] characterized the condition where $r_{l}=0$ for all layers as the feature learning regime and the condition where $r_{l<L}>0$ and $r_{L}$ = 0 as the kernel regime.
The parametrization that induces the feature learning regime is called $\mu$P, while the NTK parametrization corresponds to the kernel regime with $r_{l<L}=1/2$. The training dynamics are referred to as stable when they neither vanish nor explode as the network width increases; the parameterization that achieves this is called a stable parameterization.
Yang and Hu [] demonstrated that the concrete evaluation of feature learning with $r_{l}=0$ reduces to the following two conditions:

###### Condition 3.1 ( $W_{l}$ updated maximally) .

$\Delta W_{l,t}h_{l-1,t}=\Theta(1)$
where $\Delta W_{l,t}:=W_{l,t}-W_{l,0}$.

###### Condition 3.2 ( $W_{L}$ initialized maximally) .

$W_{L,0}\Delta u_{L-1,t}=\Theta(1).$

Note that the original derivation of the parametrization that satisfies the above conditions is based on the first (infinitesimal) one-step update of the parameters [Yang and Hu, , Ishikawa and Karakida, ].
Our work also follows the same approach.

###### Condition 3.1 ( W l subscript 𝑊 𝑙 W_{l} italic_W start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT updated maximally) .

###### Condition 3.2 ( W L subscript 𝑊 𝐿 W_{L} italic_W start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT initialized maximally) .

#### 3.2.1 μ 𝜇 \mu italic_μ P for Gauss-Newton Target

The following work has recently derived the $\mu$P scaling, including both first-order and second-order optimizations.

###### Proposition 3.3 ( Ishikawa and Karakida [ 2024 ] ) .

Consider the first one-step update by the GNT:
$W_{l,1}=W_{l,0}-\eta_{l}\phi^{\prime}(W_{l,t}{h}_{l-1})\circ(\delta_{l}\delta_
{l}^{\top}+\rho I)^{-e_{B}}\delta_{l}\text{diag}({e}_{L})h_{l-1}^{\top}$. In the infinite-width limit,
this update admits the $\mu$P for feature learning at

$$ $\left\{\,\begin{aligned} &\theta_{1}=e_{B}-1,\quad\theta_{1<l<L}=e_{B},\quad \theta_{L}=1\\ &b_{1}=0,\quad b_{1<l<L}=1/2,\quad b_{L}=1,\end{aligned}\right.$ (10) $$

where $\theta_{l}:=2a_{l}+c_{l}$. We obtain $\mu$P of SGD for $e_{B}=0$, and that of GNT for $e_{B}=1$.

**Table 1: Parameterization ($b_{l},c_{l}$). PC with $\bar{\gamma}_{L}=0$ reduces to SGD’s $\mu$P, while one with $\bar{\gamma}_{L}=-1$ reduces to GNT’s. TP has the distinctive property of $b_{L}=1/2$.**
| Layer | SP | SGD | GNT | PC | TP |
| --- | --- | --- | --- | --- | --- |
| Input | $(0,0)$ | $(0,-1)$ | $(0,0)$ | $(0,-\bar{\gamma_{L}}-1)$ | $(0,0)$ |
| Hidden | $(1/2,0)$ | $(1/2,0)$ | $(1/2,1)$ | $(1/2,-\bar{\gamma_{L}})$ | $(1/2,1)$ |
| Output | $(1/2,0)$ | $(1,1)$ | $(1,1)$ | $(1,1)$ | $(1/2,1)$ |

More precisely, we can also allow $b_{L}\geq 1$ for the feature learning regime. However such initialization eventually reduces to the case of $b_{L}=1$ in the next parameter update. Thus, we effectively summarize it as $b_{L}=1$.
The scaling of $b_{L}=1$ implies that a smaller initialization is required compared to the standard parametrization (SP), which is PyTorch’s default, for sufficiently wide neural networks.
It can also be immediately verified that we can set $a_{l}=0$ due to shift invariance without loss of generality.
In Table 1, we summarize the $\mu$P from previous work and our results obtained in the following sections.

###### Proposition 3.3 ( Ishikawa and Karakida [ 2024 ] ) .

## 4 Feature Learning of predictive coding

### 4.1 μ 𝜇 \mu italic_μ P of PC with single-shot sequential inference

Figure: Figure 1: $\mu$P enables the transfer of learning rates across widths. (Left) PC reduces to SGD, when F-ini, FPA and SI are applied. In fact, using the $\mu$P of SGD, learning rates are successfully transferred across different widths. (Right) Even without FPA, our $\mu$P of PC also allows $\mu$Transfer across widths. In this case, inference is performed only once and the difference in test accuracy between $\bar{\gamma}_{L}=0$ and $\bar{\gamma}_{L}=1$ is small. Both figures show results for training a 3-layer MLP on FashionMNIST.
Refer to caption: x1.png

As noted in the preliminary section, PC involves such techniques as F-ini, SI, and FPA, which must be clearly distinguished when deriving the $\mu$P.
It is well-established that when F-ini, SI, and FPA are all assumed, PC reduces to the gradient computation of BP, and the $\mu$P should match the one from standard BP. Figure 1 shows that when F-ini, FPA, and SI are applied, the $\mu$P of BP can be directly transferred to PC.

However, this may not hold for general PC and BP as there is no guarantee of their equivalence. To explore this, we first remove FPA. Although initialization (F-ini) and sampling (SI) are inherently arbitrary, the justification for FPA is unclear from both machine learning and biological perspectives. In PC without FPA, we find the $\mu$P as follows:

###### Theorem 4.1 ( $\mu$ P for PC (informal)) .

Let the inference step sizes be $\gamma_{l<L}=\Theta(1)$ and $\gamma_{L}=\gamma^{\prime}/M^{\bar{\gamma}_{L}}$ with a positive constant $\gamma^{\prime}$.
Consider the first one-step update of the learning parameters after a first single-shot SI with F-initialization.
Then, PC admits the $\mu$P for feature learning at

$$ $\left\{\,\begin{aligned} &\theta_{1}=-\bar{\gamma}_{L}-1,\quad\theta_{1<l<L}=- \bar{\gamma}_{L}\geq 0,\quad\theta_{l=L}=1,\\ &b_{1}=0,\quad b_{l<L}=1/2,\quad b_{L}=1.\end{aligned}\right.$ (11) $$

Rough sketch of the derivation.
Section A.2 of the Appendix presents a detailed and comprehensive derivation.
It is based on the perturbation approach, which applies to general networks with nonlinear activation functions.
This method is inspired by the previous work that derived the $\mu$P by evaluating Conditions 3.1 and 3.2 using the perturbations, such as $\partial_{\eta^{\prime}}(\Delta W_{l,1}h_{l-1,1})\big{|}_{\eta^{\prime}=0}=
\Theta(1)$. This allows for a systematic and transparent derivation.
In PC, we extend the perturbation argument to the inference step size and require

$$ $\partial_{\gamma^{\prime}}\partial_{\eta^{\prime}}(\Delta W_{l,1}h_{l-1,1}) \big{|}_{\eta^{\prime}=\gamma^{\prime}=0}=\Theta(1),$ (12) $$

which is an example of Condition 3.1 for the hidden layer.
Under the assumption of F-ini ($e_{l,0}=0$), by putting $\delta_{l}=\nabla_{u_{l}}\mathcal{L}$, we obtain $u_{l,1}-u_{l,0}=-\prod_{i=l+1}^{L}\gamma_{i}\delta_{l}$, and

$$ $\displaystyle e_{l,1}$ $\displaystyle=(u_{l,0}-\prod\nolimits_{i=l+1}^{L}\gamma_{i}\delta_{l})-\phi(W_ {l,0}(u_{l-1,0}-\prod\nolimits_{i=l}^{L}\gamma_{i}\delta_{l-1})).$ (13) $$

For the hidden layers, this leads to

$$ $\displaystyle\partial_{\gamma^{\prime}}\partial_{\eta^{\prime}}(\Delta W_{l,1} h_{l-1,1})\big{|}_{\eta^{\prime}=\gamma^{\prime}=0}$ $\displaystyle=\frac{1}{M^{\theta_{l}+\bar{\gamma}_{L}}}\left(-\delta_{l}+\phi^ {\prime}(W_{l}u_{l-1,0})\circ W_{l}\delta_{l-1}\right)h_{l-1,0}^{\top}h_{l-1,0}.$ (14) $$

We can similarly evaluate the other layers. Eventually, the $\mu$P’s conditions become

$$ $\displaystyle\theta_{1}+\bar{\gamma}_{L}+(a_{L}+b_{L})=0,\quad\theta_{l}+\bar{ \gamma}_{L}-1+(a_{L}+b_{L})=0,$ (15) $\displaystyle\theta_{L}-1=0,\quad a_{L}+b_{L}-1=0.$ (16) $$

The last condition comes from Condition 3.2. We can derive the NTK parameterization of PC in the same way.

As Figure 1 demonstrates, the obtained $\mu$P supports $\mu$Transfer in PC without FPA. This means that the optimal hyperparameters tuned for smaller-width models can be effectively re-used in larger-width models.
Additionally, consistent with previous work, we observed the empirical rule of “wider is better” in $\mu$P [Yang et al., ], where test accuracy improves as the network width increases.

Thus far, when considering the parameter gradients, it appears that $\bar{\gamma}_{L}$ as a free parameter can be absorbed into the learning rate, allowing the feature learning dynamics to remain stable. However, as the following analysis shows, $\gamma_{L}$ modifies the preconditioning of the computed gradients, which may influence $\mu$Transfer of both $\gamma_{L}$ itself and the learning rate. The analysis also demonstrates the validity of setting $\gamma_{l<L}=\Theta(1)$.

###### Theorem 4.1 ( μ 𝜇 \mu italic_μ P for PC (informal)) .

### 4.2 Analysis with Linear Network

Figure: Figure 2: (Left) Comparison of gradients with the analytical solution of a linear network. (a) $\bar{\gamma}_{L}=0$ yields gradients closer to BP gradient (which means GD in this experiment) compared to $\bar{\gamma}_{L}=-1$. (b) As $M_{L}$ approaches 1, PC’s gradient converges to BP’s. (c) As $M_{l}$ increases, the PC gradient approaches BP’s. (Right) In a nonlinear MLP, PC’s gradient also approaches BP’s when $\bar{\gamma}_{L}=0$. This experiment was conducted using MLP with Tanh on the FashionMNIST dataset.
Refer to caption: x3.png

In the previous section, we derived the $\mu$P under the assumption that inference is performed only once using F-ini and SI.
However, in practice, the inference phase typically involves multiple update steps.
To address this, we found that it is possible to explicitly derive the following general solutions (fixed points) of the inference phase for linear networks.
See Section A.3 for the derivation.

###### Theorem 4.2 .

Suppose an L-layered linear network, where we put $e_{l}^{*}=v_{l}^{*}-W_{l}v_{l-1}^{*}$, with $*$ denoting the fixed point of the inference process(3). The following holds:

$$ $\displaystyle e_{l}^{*}$ $\displaystyle=\frac{\gamma_{L}}{\gamma_{l}}W_{L:l+1}^{\top}(I+C_{\gamma}(W))^{ -1}(W_{L:1}x-y),\quad C_{\gamma}(W):=\sum_{i=2}^{L}\frac{\gamma_{L}}{\gamma_{i -1}}W_{L:i}W_{L:i}^{\top}$ (17) $\displaystyle v_{l}^{*}$ $\displaystyle=W_{l:1}x+\left(\frac{\gamma_{L}}{\gamma_{l}}W_{L:l+1}^{\top}+ \sum_{i=2}^{l-1}\frac{\gamma_{L}}{\gamma_{i-1}}W_{l:i}W_{L:i}^{\top}\right)(I+ C_{\gamma}(W))^{-1}(y-f).$ (18) $$

and $e_{L}^{*}=y-W_{L}v_{L-1}^{*}=(I+C_{\gamma}(W))^{-1}(W_{L:1}x-y)$
where $W_{L:i}=W_{L}W_{L-1}...W_{i}$.

From this general solution, we can also confirm the following property of the infinite width.

###### Corollary 4.3 .

Suppose the setting of Theorem 4.2, $\gamma_{l<L}=\Theta(1)$ and $a_{L}+b_{L}=1$. In the infinite-width limit, the PC’s gradient reduces to GD for $\gamma_{L}=\Theta(1)$. For $\gamma_{L}=\Theta(M)$, the preconditioner part $C_{\gamma}$ remains of order 1.

The exact solutions $e_{l}^{*}$ provide much clearer insight into the gradient computation compared to Eq. (7), which was previously argued but not explicitly solved. First, it becomes evident that PC does not generally coincide with GNT. Consequently, PC is also generally different from TP.
In fact, PC coincides with GNT only for the input layer in a shallow network (i.e., $L=2$), where the update vector for PC corresponds to a GNT update with a damping term. Although PC does not entirely coincide with GNT, it is noteworthy that the scaling of $c_{l}$ in $\mu$P for $\bar{\gamma_{L}}=-1$ matches that of GNT. In contrast, for $\bar{\gamma}_{L}=0$, the PC’s gradient aligns with the (S)GD. Because the preconditioner part scales as $C_{\gamma}=O(1/M)$ in the infinite-width limit, we observe that $e_{l}^{*}=\delta_{l}$, which reduces to the GD. Naturally, $\mu$P matches that of GD in this case.

Second, the order of $e_{l}^{*}$ in the analytical solution for the linear network matches the order of $e_{l,1}$ as derived in Theorem 11. Therefore, this theorem implies that in linear networks, the $\mu$P of PC would remains unchanged regardless of the presence of F-ini or the number of inference iterations.
Moreover, as proved in Section A.3.2, the orders of $e_{l}^{*}$ and $e_{l,1}$ align only when $\gamma_{l<L}=\Theta(1)$. In practical settings with multiple inferences, it is desirable for the $\mu$P to be consistent both when performing a single inference and after the inference has fully converged. Therefore, setting $\gamma_{l<L}=\Theta(1)$ is reasonable.

Additionally, we found that the dimension of the last layer plays a key role in determining the similarity between PC and BP. According to the solution for linear networks, when $M_{L}=1$, the PC’s gradient aligns with GD. Figure 2 shows numerical results confirming that for $M_{L}=1$, the gradient direction always corresponds to GD, and for $M_{l}\gg M_{L}=\Theta(1)$, the gradient approaches GD as well. We observed that both GN and BP get much closer to each other for sufficiently large widths. In other words, even when we realize GNT by setting $\bar{\gamma}_{L}=-1$, it has a quite close direction. A detailed view of the cosine similarity at the large width is shown in Figure 2(c). This result seems reasonable because
in the context of second-order optimization, it has also been reported that GNT tends to collapse into an identity matrix owing to damping [Benzing, ].
In summary, while PC’s gradient switches between GD and GNT depending on the parameterization, it is important to highlight that GNT behaves similar to GD in the infinite-width limit.

As a minor extension, we can also analyze the nudge-type loss of PC defined by Eq. (S.86)  [Alonso et al., , Millidge et al., , Pinchetti et al., ]. In this case, the damping term $I$ in Eq. (17), is replaced by $(1+\gamma/\beta)I$. Thus, the dependence on the parameterization remains essentially the same as that of the naive PC. Further discussion on nudge-type PC can be found in Appendix A.3.3.

###### Theorem 4.2 .

###### Corollary 4.3 .

### 4.3 Stability of inference phase

Figure: Figure 3: (Left) $\hat{\gamma_{L}}=-1$ steadily reduces the local loss as width increases. We observed the inference loss in a randomly initialized linear network for various $\bar{\gamma}_{L}$. For $\bar{\gamma}_{L}=-1$, the inference loss consistently decreases with increasing width. (Right) The "wider is better" trend holds for $\mu$P with $\hat{\gamma_{L}}=-1$. With F-ini, this trend holds for $\mu$P regardless of the $\bar{\gamma}_{L}$ value. However, without F-ini, the benefits of $\bar{\gamma}_{L}=-1$ become particularly prominent.
Refer to caption: x5.png

Figure: Figure 4: (Left) $\mu$P can transfer the learning rate across widths (without F-ini). We trained a 3-layer CNN on FashionMNIST with 100 inference iterations. Without F-ini, the stability of the inference becomes more crucial. As a result, unlike the single-shot SI with F-ini shown in Figure 1, the stability provided by $\bar{\gamma}_{L}=-1$ becomes critical. Note that additional experiments with different settings, including other architectures and loss functions, are presented in Section B.1.2 of the Appendix. (Right) $\Delta h$ remains consistent across widths during training. We confirm that the condition $\Delta h=\Theta(1)$ required by $\mu$P holds throughout the training.
Refer to caption: x7.png

Figure: Figure 5: $\mu$P with $\bar{\gamma}_{L}=-1$ performs consistently well, regardless of $\gamma_{l}$. When $\gamma_{l}$ is small ($\gamma_{l}=0.01$), $\mu$P with $\bar{\gamma}_{L}=0$ performs poorly, while $\mu$P with $\bar{\gamma}_{L}=-1$ shows significantly better performance. This difference is likely due to slower inference convergence in $\mu$P with $\bar{\gamma}_{L}=0$. For larger values of $\gamma_{l}$ ($\gamma_{l}=1$), both $\mu$P configurations exhibit high accuracy. However, for $\mu$P with $\bar{\gamma}_{L}=0$, $\gamma_{L}$ does not transfer effectively across widths, whereas $\mu$P with $\bar{\gamma}_{L}=-1$ demonstrates the successful transfer of $\gamma_{L}$ across widths.
Refer to caption: x9.png

To ensure feature learning in SGD, the $\mu$P framework requires stable activations, i.e., $\Delta u_{l<L}=\Theta(1)$.
It seems natural to apply this requirement to the inference phase of PC. That is, let us suppose $u_{l<L,s}$ varies by $\Theta(1)$ during the inference.
Note that in Eq.(3) at $L-1$, the feedforward signal from the lower layer is $\gamma_{L-1}e_{L-1,s}=\Theta(1)$, and the error feedback from the last layer is $\gamma_{L}\phi^{\prime}\left(u_{L-1,s}\right)\circ W_{L}^{\top}e_{L,s}=\Theta
\left(1/M^{\bar{\gamma}_{L}+b_{L}-1}\right)$.
Both terms should be of order $\Theta(1)$ for the inference to successfully merge both feedforward and feedback signals.
When $b_{L}=1$, this condition requires $\gamma_{L}=\Theta(M)$, and we can expect the local loss in the last layer $e_{L}$ to decrease most prominently during the inference.
Additionally for $\gamma_{l<L}=\Theta(1)$, the inference remains stable for layers $l<L-1$.
Empirical results in Figure 3 (left) comfirm that when $\bar{\gamma}_{L}=-1$, the inference loss decreases consistently as the width increase, verifying that the “wider is better” hypothesis holds even in inference. This facilitates the hyperparameter transfer of $\gamma_{L}$ for the inference dynamics.

We also observe the benefits of using $\bar{\gamma}_{L}=-1$ for the parameter updates.
Without F-ini, the convergence of inference usually deteriorates for SP, making inference stability specially critical in this scenario.
As shown in Figure 3 (right), the “wider is better” trend holds with F-ini regardless of $\bar{\gamma}_{L}$. However, without F-ini, this trend holds only when $\bar{\gamma}_{L}=-1$.
Figure 4 demonstrates that the $\mu$Transfer of the learning rate holds for $\bar{\gamma}_{L}=-1$. Additionally, Figure 5 indicates that $\bar{\gamma}_{L}=-1$ is also preferable from the perspective of $\mu$Transfer of $\gamma_{L}$.

## 5 Feature Learning of target propagation

### 5.1 μ 𝜇 \mu italic_μ P of TP

Figure: Figure 6: In Target Prop, using $\mu$P ensures that $\Delta h_{l}$ remains consistent across widths. This figure shows the RMS norm of $\Delta h_{l}$ during training. For SP, $\Delta h_{l}$ in the input layer diminishes as the width increases, while $\Delta h_{l}$ in the output layer diverges with increasing width. Consequently, the training dynamics become unstable. In contrast, with $\mu$P, $\Delta h_{l}$ remains consistent across different widths in both the input and output layers.
Refer to caption: x10.png

As overviewed in Section 3.1.2, TP reduces to GNT in the highly restrictive case of invertible networks. However, TP is not equivalent to GNT or BP in general cases [Meulemans et al., , Ernoult et al., ]. While TP involves two networks trained using different manners, and one may feel it challenging to obtain a stable parameterization for learning, we demonstrate that, under the assumption that the feedback network uses a linear activation function $\psi$, we can systematically derive $\mu$P for both TP and DTP.

###### Theorem 5.1 ( $\mu$ P for TP and DTP (informal)) .

Consider a linear feedback network. The forward network is allowed to have nonlinear activation functions. After the first training phase of $Q_{l}$, take the first one-step update of $W$. Then,
we obtain the $\mu$P as follows:

$$ $\left\{\,\begin{aligned} &c_{1}=0,\quad c_{1<l<L}=1,\quad c_{L}=1,\\ &b_{1}=0,\quad b_{1<l<L}=1/2,\quad b_{L}=1/2.\end{aligned}\right.$ (19) $$

The derivation is presented in Section A.4.1.
Note that the linear feedback network has trained weights in a pseudo-inverse form, that is, $Q_{l}^{*}=h_{l-1}(h_{l}^{\top}h_{l}+\mu I)^{-1}h_{l}^{\top}$.
Stable parametrization can also be discussed for the training of the feedback network. For further details, see Section A.5.

Figure 6 tracks $\Delta h_{l}$ during training. In $\mu$P, $\Delta h_{l}$ remains consistent across different widths, whereas in SP, $\Delta h_{l}$ either diverges or diminishes as the width changes. As demonstrated in Figure 7,
using the $\mu$P for TP results in the $\mu$Transfer appropriately across widths.

Figure: Figure 7: (Left) $\mu$P can transfer the learning rate across widths in TP. We trained a 3-layer MLP on FashionMNIST. (Right) TP does not have kernel regime. We measured $\omega_{L}=\log_{M}(\|W_{L,0}\Delta h_{L-1,T}\|_{\text{RMS}}/{\|W_{L,0}\|_{ \text{RMS}}\|\Delta h_{L-1,T}\|_{\text{RMS}}})$ across different parametrizations following Everett et al. []. In the infinite-width limit, $\omega_{L}$ converges to $\alpha$. Therefore, in TP, where $\omega_{L}$ remains fixed at 1/2 even as the width increases, the kernel regime disappears.
Refer to caption: x11.png

###### Theorem 5.1 ( μ 𝜇 \mu italic_μ P for TP and DTP (informal)) .

### 5.2 Disappearance of the Kernel Regime

It is notable that $\mu$P in the previous work of the gradient methods requires $b_{L}=1$; in TP, $\mu$P requires $b_{L}=1/2$. For the usual gradient methods, a stable parametrization with $b_{L}=1/2$ leads to the kernel regime. This raises the question: does a kernel regime exist in TP? Interestingly, in TP, the kernel regime disappears (see Corollary A.8 for the details).

Rough sketch of derivation.
Condition 3.2 must hold to achieve stable learning in the hidden layers. Note that this condition is required in both the feature learning and kernel regimes. By expressing $\Delta h_{L-1}=\Theta(1/M^{r})$ , we obtain

$$ $\displaystyle a_{L}+b_{L}+r-\alpha=0.$ (20) $$

It is known that when the inner product $W_{L}\Delta h_{L-1}$ follows the Law of Large Numbers (LLN), $\alpha=1$, and when it follows the Central Limit Theorem (CLT), $\alpha=1/2$ [Everett et al., ].
Additionally, to prevent the output of the last layer from exploding, it is necessary that $h_{L}=O(1)$, that is, $a_{L}+b_{L}\geq 1/2$. Consequently, $r\leq\alpha-1/2$.
In BP, the dependence between $W_{L}$ and $\Delta h_{L-1}$ results in $\alpha=1$ by the LLN. We have $r\leq 1/2$, allowing for the kernel regime.
In contrast, in TP, updating the feedforward network weights does not induce a dependence between $W_{L}$ and $\Delta h_{L-1}$, leading to $\alpha=1/2$. This is because the gradient propagated from the last layer is computed based on the feedback weight $Q^{*}_{L}=h_{L-1}\left(h_{L}^{\top}h_{L}+\mu I\right)^{-1}h_{L}^{\top}$. This corresponds to a regression of the last layer by using the downstream layer as the basis. Consequently, $r\leq 0$ and the kernel regime cannot be achieved in TP.

Note that $\alpha=1/2$ in TP is also empirically confirmed in Figure 7.
TP seems to be the first example in the infinite-width limit where $b_{L}=1/2$ induces feature learning.

## 6 Conclusion

In this work, we revealed $\mu$P for local loss optimization that can effectively scale toward the infinite width in a stable manner, supported by our analysis of linear networks. Our study covers two of the most fundamental settings: the local targets computed during the inference phase (i.e., PC) and the feedback network (i.e., TP). Although neither method generally reduces to BP or GNT, making gradient computation non-trivial, we identified the $\mu$P and highlighted its intriguing properties, such as the gradient switching depending on the parameterization and the disappearance of the kernel regime. Additionally, we empirically confirmed that the derived $\mu$P facilitates hyperparameter transfer across widths.

##### Limitation and future direction.

The derivation of $\mu$P assumes a one-step gradient and linear networks, although this prerequisite is not unique to our work [Yang, , Yang et al., ].
Ensuring the existence of feature learning dynamics for more general steps in the infinite width limit would require the development of a tensor program. However, handling the dependencies between variables that differ from standard BP, such as those arising from the inference phase and feedback pass, is non-trivial and presents an interesting direction for future research.
Additionally, it would also be valuable to explore the learning dynamics of local learning and its convergence properties by extending the infinite width theory or further analyzing of linear networks.
We believe that understanding the universal behavior of large-scale limits will provide a foundation for the development of more effective algorithms.

## Appendix A Derivation of Stable Parameterization

### A.1 Conditions

We follow the derivation based on the infinitesimal one-step update from random initialization [Yang and Hu, , Ishikawa and Karakida, ], which involves taking the limit of a sufficiently small coefficient of the learning rate $\eta^{\prime}$. This formulation clarifies the proof and enables the systematic derivation of $\mu$P across various problems.
In the infinitesimal formulation, Conditions 3.1 and 3.2 are expressed as follows:

###### Condition A.1 ( $W_{l}$ updated maximally) .

$$ $\partial_{\eta^{\prime}}\Delta W_{l}h_{l-1,1}\big{|}_{\eta^{\prime}=0}=\Theta(1)$ (S.1) $$

where $\Delta W_{l}:=W_{l,1}-W_{l,0}$.

###### Condition A.2 ( $W_{L}$ initialized maximally) .

$$ $\partial_{\eta^{\prime}}W_{L,0}\Delta u_{L-1,1}\big{|}_{\eta^{\prime}=0}= \Theta(1).$ (S.2) $$

In addition, the stability of learning is defined as follows (see Definition H.4 in Yang and Hu [] for more detail):

###### Definition A.3 (Stability of learning) .

$$ $\Delta h_{l,t}=O(1).$ (S.3) $$

As described in the previous work [Yang and Hu, , Ishikawa and Karakida, ], Condition 3.1 (or A.1) is required from the expansion of Eq. (9) by the parameter update, yielding

$$ $\partial_{\eta^{\prime}}\Delta W_{l}h_{l-1,1}\big{|}_{\eta^{\prime}=0}=\Theta( 1/M^{r_{l}}),$ (S.4) $$

for abc-parameterization. The stability requires

$$ $r_{l}\geq 0$ (S.5) $$

and, in particular, feature learning is characterized by $r_{l}=0$. Condition 3.2 (or S.2) is required to eliminate a uninteresting case in which the hidden layer provides no contribution to the network output. Both NTK and feature learning regimes are characterized by this condition.

As is common in the $\mu$P theory, we also assume that the firing activities are of order 1 at random initialization:

###### Assumption A.4 .

$u_{l,0},h_{l,0}=\Theta(1)$   ($l<L$),  $f_{0}=u_{L,0}=O(1)$.

As shown in Theorem H.6 of Yang and Hu [], this assumption immediately leads to

$$ $\displaystyle a_{1}+b_{1}=0,\quad a_{1<l<L}+b_{1<l<L}=1/2,\quad a_{L}+b_{L} \geq 1/2.$ (S.6) $$

###### Condition A.1 ( W l subscript 𝑊 𝑙 W_{l} italic_W start_POSTSUBSCRIPT italic_l end_POSTSUBSCRIPT updated maximally) .

###### Condition A.2 ( W L subscript 𝑊 𝐿 W_{L} italic_W start_POSTSUBSCRIPT italic_L end_POSTSUBSCRIPT initialized maximally) .

###### Definition A.3 (Stability of learning) .

###### Assumption A.4 .

### A.2 Predictive Coding with single-shot SI

###### Theorem A.5 (Stable parameterization for PC) .

Set inference step sizes $\gamma_{l<L}=\Theta(1)$ and $\gamma_{L}=\gamma^{\prime}/M^{\bar{\gamma}_{L}}$ with a positive constant $\gamma^{\prime}$.
Suppose F-initialization and single-shot sequential inference, and consider one-step update of parameters after the inference. For infinitesimal step sizes $\gamma_{L}^{\prime}$ and $\eta^{\prime}$, PC admits the $\mu$P for feature learning at

$$ $\left\{\,\begin{aligned} &c_{1}=-\bar{\gamma}_{L}-1,\quad c_{1<l<L}=-\bar{ \gamma}_{L},\quad c_{l=L}=1,\\ &b_{1}=0,\quad b_{l>L}=1/2,\quad b_{l}=1.\end{aligned}\right.$ (S.7) $$

Additionally, it admits the NTK parametrization at

$$ $\left\{\,\begin{aligned} &c_{1}=-\bar{\gamma}_{L},\quad c_{1<l<L}=1-\bar{ \gamma}_{L},\quad c_{L}=1\\ &b_{1}=0,\quad b_{l<L}=1/2.\end{aligned}\right.$ (S.8) $$

Proof.
Assuming F-ini, considering the single-shot SI for $v_{l}$, we have

$$ $\displaystyle v_{l,1}$ $\displaystyle=v_{l,0}+\gamma_{l+1}\phi^{\prime}\left(v_{l,0}\right)\circ W_{l+ 1}^{\top}e_{l+1,1}$ (S.9) $\displaystyle=v_{l,0}+\gamma_{l+1}\phi^{\prime}\left(v_{l,0}\right)\circ W_{l+ 1}^{\top}(v_{l+1,1}-W_{l+1}{h}_{l,0})$ (S.10) $\displaystyle=v_{l,0}+\gamma_{l+1}\phi^{\prime}\left(v_{l,0}\right)\circ W_{l+ 1}^{\top}(v_{l+1,1}-v_{l+1,0}),$ (S.11) $$

for $l<L$ where $e_{L,{0}}=y-W_{L}v_{L-1,0}=:\delta_{L}$. To keep the notation concise, we set $M_{L}=1$ in this proof. A generalization for $M_{L}=\Theta(1)$ is possible.
Next, we define

$$ $\delta_{l<L}:=\partial f/\partial u_{l}.$ (S.12) $$

Note that a batch gradient can be used with $N$ training samples where $N=O(1)$. One can regard $v_{l}$ as an $M\times N$ matrix in the derivation.

Using

$$ $v_{l,1}-v_{l,0}=-\prod_{i=l+1}^{L}\gamma_{i}\delta_{l},$ (S.13) $$

we have

$$ $\displaystyle e_{l,1}$ $\displaystyle:=v_{l,1}-\phi(W_{l}v_{l-1,1})$ (S.14) $\displaystyle=(u_{l,0}-\prod_{i=l+1}^{L}\gamma_{i}\delta_{l})-\phi(W_{l}(u_{l- 1,0}-\prod_{i=l}^{L}\gamma_{i}\delta_{l-1}))$ (S.15) $$

for $l=1,...,L-1$. Recall that $v_{l,0}=u_{l,0}$ for F-ini.
For $l=L$,

$$ $\displaystyle e_{L,1}$ $\displaystyle:=y-W_{L}h_{L-1,1}$ (S.16) $\displaystyle=y-W_{L}\phi(u_{L-1,0}-\gamma_{L}\delta_{L-1}\text{diag}(\delta_{ L}))$ (S.17) $$

where $\text{diag}(x)$ denotes a diagonal matrix whose diagonal entries are given by $x$.
The above equation comes from

$$ $\displaystyle v_{L-1,1}$ $\displaystyle=u_{L-1,0}-\frac{\gamma_{L}}{2}\nabla_{v_{L-1}}\|y-W_{L}h_{L-1}\| ^{2}$ (S.18) $\displaystyle=u_{L-1,0}-{\gamma_{L}}\phi^{\prime}_{L-1}\circ W_{L}^{\top} \delta_{L}=u_{L-1,0}-{\gamma_{L}}\delta_{L-1}\text{diag}(\delta_{L}).$ (S.19) $$

The first one-step update of the weight is expressed as

$$ $\Delta W_{l,1}=\frac{\eta^{\prime}}{M^{2a_{l}+c_{l}}}e_{l,1}h_{l-1,1}^{\top},$ (S.20) $$

In PC, in addition to the usual learning rate $\eta$, there also exists $\gamma$.
Therefore, in addition to the infinitesimal update of the learning rate $\eta$ for the weight update, we also consider the infinitesimal inference step size $\gamma_{L}$. By applying the perturbation of $\gamma_{L}$ to Conditions A.1 and S.2, we derive

$$ $\displaystyle\Delta U_{l}+\partial_{\gamma_{L}^{\prime}}\Delta U_{l}\big{|}_{ \gamma_{L}^{\prime}=0}\gamma_{L}^{\prime}=\Theta(1),$ (S.21) $\displaystyle\Delta V_{L}+\partial_{\gamma_{L}^{\prime}}\Delta V_{L}\big{|}_{ \gamma_{L}^{\prime}=0}\gamma_{L}^{\prime}=\Theta(1)$ (S.22) $$

where we define

$$ $\Delta U_{l}:=\partial_{\eta^{\prime}}\Delta W_{l,1}h_{l-1,s=1}\big{|}_{\eta^{ \prime}=0},$ (S.23) $$

$$ $\Delta V_{L}:=\partial_{\eta^{\prime}}W_{L,0}\Delta h_{l-1,s=1}\big{|}_{\eta^{ \prime}=0}.$ (S.24) $$

It is noteworthy that we retain the zero-th order terms, namely, $\Delta U_{l}$ and $\Delta V_{l}$ in the conditions.
This is because, even without the inference phase, parameter updates can progress while the internal states remain at their initialization. Therefore, even if the maximalization of the order is less than $\Theta(1)$ in the first-order perturbation terms, stable learning can still occur. Since $\mu$P aims to maximize the order of updates as much as possible, we require the first-order terms of Eqs. (S.21,S.22) to be $\Theta(1)$ whenever possible.

We introduce the following kernel matrix:

$$ $K^{A}_{l}:=h_{l}^{\top}h_{l}/M.$ (S.25) $$

For the random initialization $W_{l}$, from Eq. (S.6), we asymptotically obtain

$$ $K^{A}_{l}=\Theta(1),\quad K^{A}_{L}=\Theta(1/M^{2(a_{L}+b_{L})})$ (S.26) $$

in the infinite-width limit [Yang, ].

On Condition A.1.

(i) Case of $1<l<L$.

$$ $\displaystyle\partial_{\gamma_{L}^{\prime}}\Delta U_{l}\big{|}_{\gamma^{\prime }_{L}=0}$ $\displaystyle=\partial\left(\frac{1}{M^{\theta_{l}}}e_{l,s=1}h_{l-1,s=1}^{\top }h_{l-1,s=1}\right)\big{|}_{\gamma_{L}^{\prime}=0}$ (S.27) $\displaystyle\quad=\frac{1}{M^{\theta_{l}}}\partial(e_{l,s=1})h_{l-1,s=1}^{ \top}h_{l-1,s=1}\big{|}_{\gamma_{L}^{\prime}=0}+\frac{1}{M^{\theta_{l}}}e_{l,s =1}\partial(h_{l-1,s=1}^{\top}h_{l-1,s=1})\big{|}_{\gamma_{L}^{\prime}=0}$ $\displaystyle\quad=\frac{1}{M^{\theta_{l}}}\partial(e_{l,s=1})\big{|}_{\gamma_ {L}^{\prime}=0}h_{l-1,s=0}^{\top}h_{l-1,s=0}$ (S.28) $\displaystyle\quad=\frac{1}{M^{\theta_{l}+\bar{\gamma}_{L}-1}}\left(-\delta_{l }+\phi^{\prime}(W_{l}u_{l-1,0})\circ W_{l}\delta_{l-1}\right)K^{A}_{l-1}$ (S.29) $$

where we used $e_{l<L,s=1}\big{|}_{\gamma_{L}^{\prime}=0}=0$ and $h_{l,s=1}\big{|}_{\gamma_{L}^{\prime}=0}=h_{l,s=0}$. Since $\delta_{l<L}=\Theta(W_{L})=\Theta(1/M^{a_{L}+b_{L}})$ and $\Delta U_{l}\sim M^{\theta_{l}+a_{L}+b_{L}-1}$, we have

$$ $\Delta U_{l}+\partial_{\gamma_{L}^{\prime}}\Delta U_{l}\big{|}_{\gamma_{L}^{ \prime}=0}\gamma_{L}^{\prime}\sim 1/M^{\text{min}\{\theta_{l}+a_{L}+b_{L}-1, \theta_{l}+\bar{\gamma}_{L}+a_{L}+b_{L}-1\}}.$ (S.30) $$

The similarity symbol (“$\sim$”) denotes that the left-hand side is of the same order as the right-hand side. Note that
if the first-order term becomes negligible, the contribution of the inference phase disappears in the parameter update. To maximize the order of the first-order term, we require

$$ $\bar{\gamma}_{L}\leq 0$ (S.31) $$

and obtain

$$ $r_{l}=\theta_{l}+\bar{\gamma}_{L}+a_{L}+b_{L}-1.$ (S.32) $$

(ii) Case of $l=1$.

$$ $\displaystyle\partial_{\gamma_{L}^{\prime}}\Delta U_{l}\big{|}_{\gamma^{\prime }_{L}=0}$ $\displaystyle=-\frac{1}{M^{\theta_{l}+\bar{\gamma}_{L}}}\delta_{l}K^{A}_{0}.$ (S.33) $\displaystyle\sim 1/M^{\theta_{1}+\bar{\gamma}_{L}+a_{L}+b_{L}}$ (S.34) $$

Here, we used $e_{l<L,s=1}\big{|}_{\gamma_{L}^{\prime}=0}=0$ and $h_{l,s=1}\big{|}_{\gamma_{L}^{\prime}=0}=h_{l,s=0}$.
Similar to the case of $1<l<L$, Condition (S.21) leads to $\bar{\gamma}\leq 0$ and

$$ $r_{l}=\theta_{1}+\bar{\gamma}_{L}+a_{L}+b_{L}.$ (S.35) $$

(iii) Case of $l=L$.

$$ $\displaystyle\partial_{\gamma_{L}^{\prime}}\Delta U_{L}\big{|}_{\gamma^{\prime }_{L}=0}$ (S.36) $\displaystyle\quad=\frac{1}{M^{\theta_{L}}}\partial(e_{L,s=1})h_{L-1,s=1}^{ \top}h_{L-1,s=1}\big{|}_{\gamma_{L}^{\prime}=0}+\frac{1}{M^{\theta_{L}}}e_{L,s =1}\partial(h_{L-1,s=1}^{\top}h_{L-1,s=1})\big{|}_{\gamma_{L}^{\prime}=0}$ (S.37) $\displaystyle\quad=\frac{1}{M^{\theta_{L}}}\partial(e_{L,s=1})\big{|}_{\gamma_ {L}^{\prime}=0}h_{L-1,s=0}^{\top}h_{L-1,s=0}+\frac{1}{M^{\theta_{L}}}e_{L,s=1} \partial(h_{L-1,s=1}^{\top}h_{L-1,s=1})\big{|}_{\gamma_{L}^{\prime}=0}$ (S.38) $\displaystyle\quad=\frac{1}{M^{\theta_{L}+\bar{\gamma}_{L}-1}}\phi^{\prime}(W_ {L}u_{L-1,0})\circ W_{L}\delta_{L-1}K^{A}_{L-1}+\frac{1}{M^{\theta_{L}}}\delta _{L}\partial(h_{L-1,s=1}^{\top}h_{L-1,s=1})\big{|}_{\gamma_{L}^{\prime}=0}$ (S.39) $\displaystyle\quad=\frac{1}{M^{\theta_{L}+\bar{\gamma}_{L}-1}}\phi^{\prime}(W_ {L}u_{L-1,0})\circ W_{L}\delta_{L-1}K^{A}_{L-1}+\frac{2}{M^{\theta_{L}}}\delta _{L}h_{L-1,s=1}^{\top}\partial(h_{L-1,s=1})\big{|}_{\gamma_{L}^{\prime}=0}$ (S.40) $$

Note that from Eq. (S.17), we have

$$ $e_{L,s=1}\big{|}_{\gamma_{L}^{\prime}=0}=-(W_{L}(\phi^{\prime}_{L-1}\circ \delta_{L-1}))\circ\delta_{L}/M^{\bar{\gamma}_{L}}.$ (S.41) $$

Since $e_{L,s=1}\big{|}_{\gamma_{L}^{\prime}=0}=\delta_{L}\neq 0$,

$$ $W_{L}(\phi^{\prime}_{L-1}\circ\delta_{L-1}))\circ\delta_{L}=\Theta(1/M^{2(a_{L }+b_{L})-1}).$ (S.42) $$

For the second term in Eq. (S.40), we have

$$ $\displaystyle h_{L-1,s=1}^{\top}\partial(h_{L-1,s=1})\big{|}_{\gamma_{L}^{ \prime}=0}$ $\displaystyle=h_{L-1,s=0}^{\top}\partial\phi(W_{L-1}v_{L-2,1})\big{|}_{\gamma_ {L}^{\prime}=0}$ (S.43) $\displaystyle=h_{L-1}^{\top}(\phi^{\prime}_{L-1}\circ W_{L-1}\partial v_{L-2,1 })\big{|}_{\gamma_{L}^{\prime}=0}$ (S.44) $\displaystyle=-\frac{\gamma_{L-1}}{M^{\bar{\gamma}_{L}}}h_{L-1}^{\top}(\phi^{ \prime}_{L-1}\circ W_{L-1}\delta_{L-2})$ (S.45) $$

where we used $v_{L-2,1}-v_{L-2,0}=-\gamma_{L-1}\gamma_{L}\delta_{L-2}$ from Eq. (S.13).
Let us recall that a variable without an index indicates the initial state at $s=0$.
The variable $\delta_{L-1}$ includes $W_{L}$ whereas $h_{L-1}$ is independent of it. Therefore, by applying the CLT with respect to $W_{L}$, we have

$$ $h_{L-1,s=1}^{\top}\partial(h_{L-1,s=1})\big{|}_{\gamma_{L}^{\prime}=0}\sim 1/M ^{\bar{\gamma}_{L}+a_{L}+b_{L}-1/2}.$ (S.46) $$

Then,

$$ $\partial_{\gamma_{L}^{\prime}}\Delta U_{L}\big{|}_{\gamma^{\prime}=0}\sim 1/M^ {\min\{\theta_{L}+\bar{\gamma}_{L}+2(a_{L}+b_{L})-2,\theta_{L}+\bar{\gamma}_{L }+a_{L}+b_{L}-1/2\}}.$ (S.47) $$

In contrast, we have

$$ $\Delta U_{L}\sim 1/M^{\theta_{L}-1}.$ (S.48) $$

Comparing the zero-th and first order terms (S.47,S.48), we obtain

$$ $\min\{\theta_{L}-1,\theta_{L}+\bar{\gamma}_{L}+2(a_{L}+b_{L})-2,\theta_{L}+ \bar{\gamma}_{L}+a_{L}+b_{L}-1/2\}=0$ (S.49) $$

Because $a_{L}+b_{L}-1/2\geq 0$ from Eq. (S.6), we obtain

$$ $\theta_{L}-1=0.$ (S.50) $$

On Condition A.2.

$$ $\displaystyle\partial_{\gamma_{L}^{\prime}}\Delta V_{L}\big{|}_{\gamma_{L}^{ \prime}=0}$ $\displaystyle=W_{L,0}(\phi^{\prime}(u_{L-1})\circ\partial_{\gamma_{L}^{\prime} }\partial_{\eta^{\prime}}(\Delta W_{L-1,1}h_{L-2,1})\big{|}_{\eta^{\prime}=0, \gamma_{L}^{\prime}=0})$ $\displaystyle\quad=e_{M}(\delta_{L-1}\circ\frac{1}{M^{\theta_{L-1}}}\partial_{ \gamma^{\prime}}(\Delta W_{L-1}h_{L-2})\big{|}_{\gamma_{L}^{\prime}=0})$ (S.51) $\displaystyle\quad=\Theta(1/M^{a_{L}+b_{L}+r_{L-1}-1})$ (S.52) $$

where $e_{M}$ denotes an $M$-dimensional vector with all entries equal to 1.
Note that the product with $e_{M}$ means the summation over $M$.

Finally, from Conditions 1 and 2, the $\mu$P is given by

$$ $\displaystyle\theta_{1}+\bar{\gamma}_{L}+a_{L}+b_{L}=0\quad(l=1),$ (S.53) $\displaystyle\theta_{l}+\bar{\gamma}_{L}+a_{L}+b_{L}-1=0\quad(1<l<L),$ (S.54) $\displaystyle\theta_{L}-1=0\quad(l=L)$ (S.55) $\displaystyle a_{L}+b_{L}-1=0,$ (S.56) $$

and $\bar{\gamma}_{L}\leq 0$. That is,

$$ $\left\{\,\begin{aligned} &c_{1}=-\bar{\gamma}_{L}-1,\quad c_{1<l<L}=-\bar{ \gamma}_{L}\geq 0,\quad c_{l=L}=1,\\ &b_{1}=0,\quad b_{l>L}=1/2,\quad b_{l}=1.\end{aligned}\right.$ (S.57) $$

We can also derive the NTK parameterization, which is a commonly used term for the kernel regime for $r_{l<L}=1/2$ [Yang and Hu, ]:

$$ $\displaystyle\theta_{1}+\bar{\gamma}_{L}+a_{L}+b_{L}=1/2\quad(l=1),$ (S.58) $\displaystyle\theta_{l}+\bar{\gamma}_{L}+a_{L}+b_{L}-1=1/2\quad(1<l<L),$ (S.59) $\displaystyle\theta_{L}-1=0,\quad(l=L)$ (S.60) $\displaystyle a_{L}+b_{L}-1/2=0.$ (S.61) $$

∎

It is noteworthy that the gradient computed by Eq. (S.15) differs from $\delta_{l}$ in standard GD, implying that the NTK matrix also deviates from $\nabla_{\theta}f^{\top}\nabla_{\theta}f$. Even in this case, the NTK regime can emerge with a certain modified kernel composed of $e_{l}$ and $h_{l}$. A similar situation arises in the NTK regime of second-order optimization [Karakida and Osawa, ]. Altough the preconditioner modifies the NTK matrix, the linearization of the model still holds, allowing the emargence of the kernel regime.

###### Theorem A.5 (Stable parameterization for PC) .

### A.3 Fixed points of PC in Linear Networks

#### A.3.1 Proof for Theorem 4.2

In this section, we analyze the fixed point of the inference phase using a linear network:

$$ $\displaystyle f(x)=W_{l}{W}_{L-1}...{W}_{1}x.$ (S.62) $$

Even for linear networks, the properties of the fixed points have rarely been analyzed. An exception is a recent study by Innocenti et al. []. They explicitly derived the free energy at a fixed point to analyze the parameter loss landscape of a naive PC. However, their analysis uses an unfolding calculation of a hierarchical Gaussian model to directly derive the free energy. Although this is an elegant derivation, it is not a method for explicitly obtaining the fixed points themself. Additionally, since their proof is based on $\gamma=1$, we need another method to determine the dependence on the inference size.
Here, we provide a derivation of the states at the fixed point that can be used more generally for various inference sizes and adding a nudge term (in Section A.3.3).

${\it Proof.}$
We consider the inference of naive PC:

$$ $\displaystyle F(v_{1},...,v_{L})=\frac{\gamma_{L}}{2}\|y-W_{L}v_{L-1}\|^{2}+ \sum_{l=1}^{L-1}\frac{\gamma_{l}}{2}\|v_{l}-W_{l}v_{l-1}\|^{2}.$ (S.63) $$

Taking $\frac{\partial F}{\partial v_{l}}=0$, we obtain the following fixed-point equations:

$$ $\displaystyle-\gamma_{L}W_{L}^{\top}(y-W_{L}v_{L-1})+\gamma_{L-1}(v_{L-1}-W_{L -1}v_{L-2})$ $\displaystyle=0,\quad(l=L)$ (S.64) $\displaystyle-\gamma_{l}W_{l}^{\top}(v_{l}-W_{l}v_{l-1})+\gamma_{l-1}(v_{l-1}- W_{l-1}v_{l-2})$ $\displaystyle=0,\quad(1<l<L)$ (S.65) $\displaystyle-\gamma_{2}W_{2}^{\top}(v_{2}-W_{2}v_{1})+\gamma_{1}(v_{1}-W_{1}x)$ $\displaystyle=0\quad(l=1).$ (S.66) $$

These equations are summarized in the following matrix form:

$$ $\displaystyle\begin{bmatrix}I&O&\dots&\dots&O\\ -W_{L}^{\top}&I&O&\dots&O\\ O&-W_{L-1}^{\top}&I&\dots&O\\ \vdots&\ddots&\ddots&&\vdots\\ O&\dots&O&-W_{2}^{\top}&I\end{bmatrix}\begin{bmatrix}\gamma_{L-1}e_{L-1}^{*}\\ \gamma_{L-2}e_{L-2}^{*}\\ \vdots\\ \gamma_{2}e_{2}^{*}\\ \gamma_{1}e_{1}^{*}\end{bmatrix}=\begin{bmatrix}\gamma_{L}W_{L}^{\top}e_{L}^{* }\\ O\\ O\\ \vdots\\ O\end{bmatrix}$ (S.67) $$

where $e_{l}^{*}:=v_{l}^{*}-W_{l}v_{l-1}^{*}$ and $e_{L}^{*}:=y-W_{L}v_{L-1}^{*}$.

Here, we use the following lemma:

###### Lemma A.6 .

Define

$$ $A_{L}:=\begin{bmatrix}I&O&\dots&\dots&O\\ -W_{L}^{\top}&I&O&\dots&O\\ O&-W_{L-1}^{\top}&I&\dots&O\\ \vdots&\ddots&\ddots&&\vdots\\ O&\dots&O&-W_{2}^{\top}&I\end{bmatrix}.$ (S.68) $$

Its inverse matrix is given by

$$ $\displaystyle A_{L}^{-1}=\begin{bmatrix}I&O&\dots&\dots&O\\ W_{L}^{\top}&I&O&\dots&O\\ W_{L-1:L}^{\top}&W_{L-1}^{\top}&I&\dots&O\\ \vdots&\ddots&\ddots&&\vdots\\ W_{2:L}^{\top}&\dots&W_{2:3}^{\top}&W_{2}^{\top}&I\end{bmatrix}.$ (S.69) $$

Proof. One can easily derive this inverse matrix. A simple derivation is achived by induction. We can express

$$ $A_{L}=\begin{bmatrix}I&O\\ K&A_{L-1}\end{bmatrix}$ (S.70) $$

where $K^{\top}=[W_{L},O,...,O]$. Suppose that the inverse of $A_{L-1}$ is given by Eq. (S.69). Then,

$$ $A_{L}^{-1}=\begin{bmatrix}I&O\\ KA_{L-1}^{-1}&A_{L-1}^{-1}\end{bmatrix}.$ (S.71) $$

Since $KA_{L-1}^{-1}=[W_{L},W_{L-1:L},...,W_{2:L}]^{\top}$, the inversion of $A_{L}$ is also given by Eq. (S.69).

∎

By using Lemma A.6, we can transform Eq. (S.67) as follows:

$$ $\displaystyle\begin{bmatrix}e_{L-1}^{*}\\ e_{L-2}^{*}\\ \vdots\\ e_{2}^{*}\\ e_{1}^{*}\end{bmatrix}=\gamma_{L}\begin{bmatrix}\frac{1}{\gamma_{L-1}}W_{L}^{ \top}e_{L}^{*}\\ \frac{1}{\gamma_{L-2}}W_{L-1:L}^{\top}e_{L}^{*}\\ \vdots\\ \\ \frac{1}{\gamma_{2}}W_{2:L}^{\top}e_{L}^{*}\end{bmatrix}.$ (S.72) $$

Altough this equation can not be solved explicitly for $v_{l-1}^{*}$,
we can, nonetheless, solve it by summing over $e_{l}^{*}$ as follows:

$$ $\displaystyle v_{L-1}^{*}-W_{L-1:1}x$ (S.73) $\displaystyle\qquad=xe_{L-1}^{*}+W_{L-1}e_{L-2}^{*}+\dots+W_{L-1:2}e_{1}^{*}$ $\displaystyle\qquad=\gamma_{L}\left(\frac{1}{\gamma_{L-1}}I+\frac{1}{\gamma_{L -2}}W_{L-1}W_{L-1}^{\top}+\dots+\frac{1}{\gamma_{1}}W_{L-1:2}W_{L-1:2}^{\top} \right)W_{L}^{\top}e_{L}^{*}.$ (S.74) $$

This leads to

$$ $\displaystyle v_{L-1}^{*}=\left(I+\frac{\gamma_{L}}{\gamma_{L-1}}W_{L}^{\top}W _{L}+\frac{\gamma_{L}}{\gamma_{L-2}}W_{L-1}W_{L-1}^{\top}W_{L}^{\top}W_{L}+ \dots+\frac{\gamma_{L}}{\gamma_{1}}W_{L-1:2}W_{L-1:2}^{\top}W_{L}^{\top}W_{L} \right)^{-1}$ $\displaystyle\cdot\left(W_{L-1:1}x+\gamma_{L}\left(\frac{1}{\gamma_{L-1}}I+ \frac{1}{\gamma_{L-2}}W_{L-1}W_{L-1}^{\top}+\dots+\frac{1}{\gamma_{1}}W_{L-1:2 }W_{L-1:2}^{\top}\right)W_{L}^{\top}y\right).$ (S.75) $$

Set an $M_{L}\times M_{L}$ matrix

$$ $C_{\gamma}(W):=\sum_{i=2}^{L}\frac{\gamma_{L}}{\gamma_{i-1}}W_{L:i}W_{L:i}^{ \top}.$ (S.76) $$

Then,

$$ $\displaystyle e_{L}^{*}$ $\displaystyle=y-W_{L}v_{L-1}^{*}$ (S.77) $\displaystyle=y-\left(I+C_{\gamma}(W)\right)^{-1}\left(W_{L:1}x+\left(I+C_{ \gamma}(W)\right)y\right)$ $\displaystyle=\left(I+C_{\gamma}(W)\right)^{-1}(y-f).$ (S.78) $$

Thus, at the fixed point, the local loss is explicitly obtained as

$$ $e_{l}^{*}=\frac{\gamma_{L}}{\gamma_{l}}W_{L:l+1}^{\top}(I+C_{\gamma}(W))^{-1}( y-f).$ (S.79) $$

We can also obtain $v_{l}^{*}$. From $e^{*}_{1}$, we have

$$ $v_{1}^{*}=W_{1}x+\frac{\gamma_{L}}{\gamma_{1}}W_{L:2}^{\top}(I+C_{\gamma}(W))^ {-1}(y-f).$ (S.80) $$

By induction, we have

$$ $\displaystyle v_{l}^{*}$ $\displaystyle=e_{l}^{*}+W_{l}v_{l-1}^{*}$ (S.81) $\displaystyle=W_{l:1}x+\left(\frac{\gamma_{L}}{\gamma_{l}}W_{L:l+1}^{\top}+ \sum_{i=2}^{l-1}\frac{\gamma_{L}}{\gamma_{i-1}}W_{l:i}W_{L:i}^{\top}\right)(I+ C_{\gamma}(W))^{-1}(y-f).$ (S.82) $$

∎

###### Lemma A.6 .

#### A.3.2 Balance condition determining γ l < L subscript 𝛾 𝑙 𝐿 \gamma_{l<L} italic_γ start_POSTSUBSCRIPT italic_l < italic_L end_POSTSUBSCRIPT

Here, we consider the order of $e_{l,1}$ with respect to $\gamma_{l<L}$. For a linear network with one-shot SI, we obtain

$$ $\displaystyle e_{l,1}$ $\displaystyle=-\prod_{i=l+1}^{L}\gamma_{i}(\delta_{l}-\gamma_{l}W_{l}\delta_{l -1})\sim 1/M^{\min(0,\bar{\gamma_{l}})+\sum_{i=l+1}^{L}\bar{\gamma}_{i}}$ (S.83) $$

In contrast, recall that the order of $e_{l}^{*}$ at the fixed point is

$$ $\displaystyle e_{l}^{*}$ $\displaystyle\sim 1/M^{-\bar{\gamma}_{l}+\min(0,\bar{\gamma}_{l})}.$ (S.84) $$

Therefore, to satisfy $e_{l,1}\sim e_{l}^{*}$, the following is necessary:

$$ $\sum_{i=l+1}^{L}\bar{\gamma}_{i}=-\bar{\gamma}_{l}$ (S.85) $$

for all $l<L$. This is equivalent to $\bar{\gamma}_{l}=0$ for all $l<L$.
Thus, $e_{l,1}\sim e_{l}^{*}$ holds if and only if $\bar{\gamma}_{l<L}=0$.

#### A.3.3 Nudged Predictive Coding

Predictive Coding with a nudge term updates the state $v_{l}$ and weights $W_{l}$ to minimize the following free energy function [Alonso et al., , Millidge et al., , Pinchetti et al., ]:

$$ $\mathcal{F}=\beta\mathcal{L}(y,v_{L})+\sum_{l=1}^{L}\gamma_{l}\frac{1}{2}\left \|v_{l}-W_{l}\phi\left(v_{l-1}\right)\right\|^{2}.$ (S.86) $$

Here, $\beta$ is a nudge coefficient parameter that covers some variants of the PC algorithms in the previous work.
We can extend Theorem 4.2 to the nudged PC.

###### Theorem A.7 .

Suppose an L-layered linear network and put $e_{l}^{*}=v_{l}^{*}-W_{l}v_{l-1}^{*}$, where $*$ denotes the fixed point of the inference given by Eq. (S.86). The following holds:

$$ $\displaystyle e_{l}^{*}=v_{l}^{*}$ $\displaystyle-W_{l}v_{l-1}^{*}=\frac{\gamma_{L}}{\gamma_{l}}W_{L:l+1}^{\top} \left(I+\frac{\gamma_{L}}{\beta}I+C_{\gamma}(W)\right)^{-1}(W_{L:1}x-y)$ (S.87) $\displaystyle e_{L}^{*}=v_{L}^{*}$ $\displaystyle-W_{L}v_{L-1}^{*}=\left(I+\frac{\gamma_{L}}{\beta}I+C_{\gamma}(W) \right)^{-1}(W_{L:1}x-y)$ (S.88) $$

where $W_{L:i}=W_{L}W_{L-1}...W_{i}$.

${\it Proof.}$
Put

$$ $\displaystyle F(v_{1},...,v_{L})=\beta\|y-v_{L}\|^{2}+\sum_{l=1}^{L}\frac{ \gamma_{l}}{2}\|v_{l}-W_{l}v_{l-1}\|^{2}.$ (S.89) $$

Taking $\frac{\partial F}{\partial v_{l}}=0$, we have

$$ $\displaystyle\beta(v_{L}-y)+\gamma_{L}(v_{L}-W_{L}v_{L-1})$ $\displaystyle=0$ (S.90) $\displaystyle-\gamma_{l}W_{l}^{\top}(v_{l}-W_{l}v_{l-1})+\gamma_{l-1}(v_{l-1}- W_{l-1}v_{l-2})$ $\displaystyle=0\quad(1<l\leq L)$ (S.91) $\displaystyle-\gamma_{2}W_{2}^{\top}(v_{2}-W_{2}v_{1})+\gamma_{1}(v_{1}-W_{1}x)$ $\displaystyle=0\quad(l=1).$ (S.92) $$

Putting $\chi=v_{L}-y$, the system of equations can be written in a matrix form as follows:

$$ $\displaystyle\begin{bmatrix}I&O&\dots&\dots&O\\ -W_{L}^{\top}&I&O&\dots&O\\ O&-W_{L-1}^{\top}&I&\dots&O\\ \vdots&\ddots&\ddots&&\vdots\\ O&\dots&O&-W_{2}^{\top}&I\end{bmatrix}\begin{bmatrix}\gamma_{L}e_{L}^{*}\\ \gamma_{L-1}e_{L-1}^{*}\\ \vdots\\ \gamma_{2}e_{2}^{*}\\ \gamma_{1}e_{1}^{*}\end{bmatrix}=\begin{bmatrix}-\beta\chi^{*}\\ O\\ O\\ \vdots\\ O\end{bmatrix}.$ (S.93) $$

From Lemma A.6, the above equation is transformed into

$$ $\displaystyle\begin{bmatrix}e_{L}^{*}\\ e_{L-1^{*}}\\ \vdots\\ e_{2}^{*}\\ e_{1}^{*}\end{bmatrix}=-\beta\begin{bmatrix}\frac{1}{\gamma_{L}}\chi^{*}\\ \frac{1}{\gamma_{L-1}}W_{L}^{\top}\chi^{*}\\ \vdots\\ \\ \frac{1}{\gamma_{1}}W_{2:L}^{\top}\chi^{*}\end{bmatrix}.$ (S.94) $$

Take the following summation:

$$ $\displaystyle e_{L}^{*}+W_{L}e_{L-1}^{*}+W_{L-1}e_{L-2}^{*}+\dots+W_{L-1:2}e_{ 1}^{*}$ $\displaystyle=v_{L}^{*}-W_{L:1}x$ $\displaystyle=-\frac{\beta}{\gamma_{L}}\left(I+C_{\gamma}(W)\right)\chi^{*}.$ (S.95) $$

Thus, we can explicitly obtain $v_{L}$ as

$$ $\displaystyle v_{L}^{*}=\left(\frac{\beta}{\gamma_{L}}I+\frac{\beta}{\gamma_{L }}C_{\gamma}(W)\right)^{-1}\left(W_{L:1}x+\frac{\beta}{\gamma_{L}}\left(I+C_{ \gamma}(W)\right)y\right).$ (S.96) $$

Thus, $\chi$ can be written as follows:

$$ $\displaystyle\chi^{*}$ $\displaystyle=y-v_{L}^{*}$ (S.97) $\displaystyle=y-\left(\frac{\beta}{\gamma_{L}}I+\frac{\beta}{\gamma_{L}}C_{ \gamma}(W)\right)^{-1}\left(W_{L:1}x+\frac{\beta}{\gamma_{L}}\left(I+C_{\gamma }(W)\right)y\right)$ (S.98) $\displaystyle=\left(I+\frac{\beta}{\gamma_{L}}I+\frac{\beta}{\gamma_{L}}C_{ \gamma}(W)\right)^{-1}(y-f).$ (S.99) $$

From the above, we conclude that

$$ $\displaystyle e_{l}^{*}$ $\displaystyle=\frac{\beta}{\gamma_{l}}W_{L:l+1}^{\top}\left(I+\frac{\beta}{ \gamma_{L}}I+\frac{\beta}{\gamma_{L}}C_{\gamma}(W)\right)^{-1}(y-f)$ (S.100) $\displaystyle=\frac{\gamma_{L}}{\gamma_{l}}W_{L:l+1}^{\top}\left(I+\frac{ \gamma_{L}}{\beta}I+C_{\gamma}(W)\right)^{-1}(y-f)$ (S.101) $$

at the fixed point.

∎

###### Theorem A.7 .

### A.4 Target Propagation

#### A.4.1 Derivation of Theorem 19 .

Assume that the feedback network is linear: $g_{l}(x)=Q_{l}x$. Here, we consider a reconstruction loss with L2 regularization:

$$ $\displaystyle l(Q_{l,s})=\|Q_{l,s}{\phi(W_{l-1}h_{l-1})}-h_{l-1}\|^{2}+\mu_{l} \|Q_{l,s}\|^{2}$ (S.102) $$

with $\mu_{l}\geq 0$.
Note that while some work adds noise to $h_{l-1}$, it does not affect the order; therefore, we will ignore it in this derivation.
As described below, by taking the ridge-less limit of $\mu$, we can evaluate the parameterization of the original TP and Difference Target Propagation (DTP) in a clear and unified manner.
Considering the fixed point for $Q_{L}$, since $\frac{\partial l(Q_{l})}{\partial Q_{l}}=0$ holds, we have

$$ $Q_{l}^{*}=h_{l-1}(h_{l}^{\top}h_{l}+\mu_{l}I)^{-1}h_{l}^{\top}$ (S.103) $$

where $h_{l}=\phi(W_{l}h_{l-1})$. The feedback network is given by the network with Eq. (S.103). As a side note, this weight is essentially the same as the pseudo-inverse weight, which is known as an extension of the Hebbian weight [Kanter and Sompolinsky, ].

Local targets of DTP.
DTP is an improved methods of TP, where $\hat{h}_{L}$ is propagated as follows:

$$ $\hat{h}_{i}=g_{i}^{\text{diff }}\left(\hat{h}_{i+1},h_{i+1},h_{i}\right)=g_{i} \left(\hat{h}_{i+1}\right)-\left(g_{i}\left(h_{i+1}\right)-h_{i}\right).$ (S.104) $$

For the last layer, the error is given by

$$ $\hat{h}_{L}=h_{L}+\beta(y-h_{L})$ (S.105) $$

For a linear feedback network, we have

$$ $\displaystyle\hat{h}_{l}$ $\displaystyle=g_{l+1}\left(\hat{h}_{l+1}\right)-\left(g_{l}\left(h_{l+1}\right )-h_{l}\right)$ (S.106) $\displaystyle=h_{l}+Q_{l+1}(\hat{h}_{l+1}-h_{l+1})$ (S.107) $\displaystyle=h_{l}+Q_{l+1}((h_{l+1}+Q_{l+2}(\hat{h}_{l+2}-h_{l+2}))-h_{l+1})$ (S.108) $\displaystyle=h_{l}+Q_{l+1}Q_{l+2}(\hat{h}_{l+2}-h_{l+2})$ (S.109) $\displaystyle=h_{l}-\beta\prod_{i=l+1}^{L}Q_{l}\delta_{L}.$ (S.110) $$

Therefore, at the equilibrium point for $Q_{l}$, for $l\leq L-1$, we have

$$ $\displaystyle\hat{h}_{l}-h_{l}$ $\displaystyle=-\beta\prod_{i=l+1}^{L}Q_{l}^{*}\delta_{L}$ (S.111) $\displaystyle=-\beta h_{l}\prod_{i=l+1}^{L-1}(h_{i}^{\top}h_{i}+\mu_{l}I)^{-1} h_{i}^{\top}h_{i}(h_{L}^{\top}h_{L}+\mu_{l}I)^{-1}h_{L}^{\top}\delta_{L}$ (S.112) $\displaystyle=-\beta h_{l}\prod_{i=l+1}^{L-1}(K^{A}_{i}+\mu_{i}^{\prime}I)^{-1 }K^{A}_{i}(K^{A}_{L}+\mu_{L}^{\prime}I)^{-1}h_{L}^{\top}\delta_{L}$ (S.113) $$

where $\delta_{L}=\partial\mathcal{L}/\partial f$. To avoid an uninteresting change of order, we introduce $\mu_{l}=\mu_{l}^{\prime}/M^{\bar{\mu}_{l}}$ and require that it have the same order as $K_{l}^{A}$.
This is essentially equivalent to the valid condition argued in Ishikawa and Karakida [], which requires the damping term to have the same order as the preconditioner in the second-order optimization.
We note that we can take the ridge-less limit $\mu_{l}^{\prime}\rightarrow 0+$ because $K^{A}_{l}$ (S.26)
is typically set to be regular at random initialization in the neural tangent kernel literature [Jacot et al., , Yang, ]. For instance, this holds true for normalized input samples with $\|x\|=1$.

Local targets of original TP.
The signal propagation in the feedback network is

$$ $\displaystyle\hat{h}_{l}$ $\displaystyle=Q_{l}^{*}\cdots Q_{L}^{*}\hat{h}_{L}$ (S.114) $\displaystyle=h_{l}\prod_{i=l+1}^{L-1}(h_{i}^{\top}h_{i}+\mu_{i}I)^{-1}h_{i}^{ \top}h_{i}(h_{L}^{\top}h_{L}+\mu_{L}I)^{-1}h_{L}^{\top}(h_{L}-\beta\delta_{L})$ (S.115) $\displaystyle\rightarrow h_{l}-\beta h_{l}\prod_{i=l+1}^{L-1}(K^{A}_{i})^{-1}K _{i}^{A}(K_{L}^{A})^{-1}h_{L}^{\top}\delta_{L}\quad(\mu_{l}^{\prime} \rightarrow 0+).$ (S.116) $$

Thus, the target is reduced to essentially the same as that in DTP (S.113) and we can treat both in the same manner.

On Condition A.1.

$$ $\displaystyle\partial_{\eta^{\prime}}\Delta W_{L-1}h_{L-2,1}\big{|}_{\eta^{ \prime}=0}$ $\displaystyle=\frac{1}{M^{\theta_{L-1}}}(\hat{h}_{L-1}-h_{L-1})h_{L-1}^{\top}h _{L-1}$ (S.117) $\displaystyle=-\frac{1}{M^{\theta_{L-1}-1}}\beta h_{L-1}(h_{L}^{\top}h_{L}+\mu _{L}I)^{-1}h_{L}^{\top}\delta_{L}K^{A}_{L-1}$ (S.118) $\displaystyle=-\frac{1}{M^{\theta_{L-1}-1}}\beta h_{L-1}(K^{A}+\mu_{L}^{\prime }I)^{-1}(K^{A}-M^{-1}h_{L}^{\top}y)K^{A}_{L-1}.$ (S.119) $$

Similarly, when $\hat{h}_{l}-h_{l}=-\beta\prod_{i=l+1}^{L}Q_{i}\delta_{L}$, we have

$$ $\displaystyle\partial_{\eta^{\prime}}\Delta W_{l}h_{l-1,1}\big{|}_{\eta^{ \prime}=0}$ $\displaystyle=\frac{1}{M^{\theta_{l}}}(\hat{h}_{l}-h_{l})h_{l-1}^{\top}h_{l-1}$ (S.120) $\displaystyle=-\frac{1}{M^{\theta_{l}-1}}\beta h_{l}\prod_{i=l+1}^{L}(h_{i}^{ \top}h_{i}+\mu_{i}I)^{-1}h_{i}^{\top}h_{i}(h_{L}^{\top}h_{L}+\mu_{L}I)^{-1}h_{ L}^{\top}\delta_{L}K^{A}_{l-1}$ (S.121) $\displaystyle=-\frac{1}{M^{\theta_{l}-1}}\beta h_{l}\prod_{i=l+1}^{L}(K^{A}_{i }+\mu_{i}^{\prime}I)^{-1}K^{A}_{i}(K^{A}_{L}+\mu_{L}^{\prime}I)^{-1}(K^{A}_{L} -M^{-1}h_{L}^{\top}y)K^{A}_{l-1}$ (S.122) $$

for $l=1,...,L-2$.
On the right-hand side, $h_{L}\sim 1/M^{a_{L}+b_{L}-1/2}$, and from Eq. (S.26), we have

$$ $\displaystyle K^{A}_{L}-M^{-1}h_{L}^{\top}y$ $\displaystyle\sim\max\{1/M^{2(a_{L}+b_{L})},1/M^{a_{L}+b_{L}+1/2}\}$ (S.123) $\displaystyle=1/M^{a_{L}+b_{L}+1/2}.$ (S.124) $$

In the last line, we used $a_{L}+b_{L}\geq 1/2$.
Here, from Assumption A.4, which states that $a_{L}+b_{L}\geq 1/2$, we obtain

$$ $r_{l}=\left\{\,\begin{aligned} &\theta_{1}-a_{L}-b_{L}+1/2\quad(l=1),\\ &\theta_{l}-1-a_{L}-b_{L}+1/2\quad(1<l<L),\\ &\theta_{L}-1\quad(l=L).\end{aligned}\right.$ (S.125) $$

On Condition A.2.

$$ $\displaystyle\partial_{\eta^{\prime}}(W_{L,0}\Delta h_{L-1,1})\big{|}_{\eta^{ \prime}=0}$ $\displaystyle=\frac{1}{M^{\theta_{L-1}}}W_{L,0}(\hat{h}_{L-1}-h_{L-1})h_{L-1}^ {\top}$ (S.126) $\displaystyle=-\frac{1}{M^{\theta_{L-1}-1}}\beta h_{L}(h_{L}^{\top}h_{L}+\mu_{ L}^{\prime}I)^{-1}h_{L}^{\top}\delta_{L}K^{A}_{L-1}$ (S.127) $\displaystyle=-\frac{1}{M^{\theta_{L-1}-1}}\beta h_{L}(K^{A}_{L}+\mu_{L}^{ \prime}I)^{-1}(K^{A}_{L}-M^{-1}h_{L}^{\top}y)K^{A}_{L-1}.$ (S.128) $$

Thus, its order is

$$ $\displaystyle\partial_{\eta^{\prime}}(W_{L,0}\Delta h_{L-1,1})\big{|}_{\eta^{ \prime}=0}$ $\displaystyle\sim 1/M^{\theta_{L-1}-1+(a_{L}+b_{L}-1/2)-2(a_{L}+b_{L})+(a_{L}+ b_{L}+1/2)}$ (S.129) $\displaystyle=1/M^{\theta_{L-1}-1=r_{L-1}+(a_{L}+b_{L})-1/2}.$ (S.130) $$

Finally, from Conditions A.1 and A.2, the $\mu$P is given by

$$ $\displaystyle\theta_{1}-a_{L}-b_{L}+1/2=0\quad(l=1),$ (S.131) $\displaystyle\theta_{l}-a_{L}-b_{L}-1/2=0\quad(1<l<L),$ (S.132) $\displaystyle\theta_{L}-1=0\quad(l=L),$ (S.133) $\displaystyle a_{L}+b_{L}-1/2=0.$ (S.134) $$

∎

#### A.4.2 Disappearance of Kernel regime

###### Corollary A.8 .

Stable learning satisfying Condition A.2 leads to $r_{L-1}=0$ for TP and DTP.

Proof. From Eq. (S.130), we have

$$ $r_{L-1}+(a_{L}+b_{L})-1/2=0.$ (S.135) $$

From Eq. (S.6), $a_{L}+b_{L}\geq 1/2$ and we have $r_{L-1}\geq 0$.
In contrast, from the stability of learning, we have $r_{L-1}\leq 0$. Thus, $r_{L-1}=0$.
∎

Note that, precisely speaking, $r_{L-1}=0$ does not necessarily imply $r_{l<L-1}=0$. However, it is often considered unnatural (or uninteresting) to examine cases in which the progress of learning depends on individual layers. Therefore, the $\mu$P typically assumes a uniform parameterization, meaning $r_{l<L}=r$ (see Theorem G.4 of Yang and Hu []).
In this sense, $r_{L-1}=0$ indicates the disappearance of the kernel regime.

One might be surprised by the fact that $b_{L}=1/2$ is allowed in the feature learning and that the kernel regime disappears.
Note that the feedback weight in the last layer (S.103) essentially differs from $W_{L}$.
The feedback weight recieves $h_{L}$ as input whereas $W_{l}$ recieves $h_{L-1}$.
This makes

$$ $Q_{L}\sim 1/M^{1/2-(a_{L}+b_{L})}$ (S.136) $$

$W_{L}\sim 1/M^{a_{L}+b_{L}}$.
The gradient is proportional to $Q_{L}$ in TP and $W_{L}$ in BP.
The feedback weight contributes more significantly to TP’s gradient when $a_{L}+b_{L}\geq 1/2$.
This eventually makes the index $r$ of the hidden layer (S.125) get quite large even for $a_{L}+b_{L}=1/2$.
We also need to be careful about the order of
condition A.2 (S.130).
Because the feedback weight (S.103) has a lower alignment exponent [Everett et al., ], this
causes the condition 2 of TP to be smaller than
that of SGD (or K-FAC), i.e., $1/M^{r_{L-1}+(a_{L}+b_{L})-1}$.
Therefore, the stable feature learning is possible even for $a_{L}+b_{L}=1/2$.

###### Corollary A.8 .

##### Remark on zero head initialization.

Related to the size of $b_{L}$, the parameter initialization with $b_{L}>1/2$
($b_{L}>1$ for SGD) reduces to the $\mu$P of $b_{L}=1/2$ ($b_{L}=1$ for SGD) because $W_{l,0}$ becomes negligible compared to $\Delta{W}_{l,0}$.
To illustrate the intuition, let us introduce the case where the weight of the last layer in a feedforward network is initialized as $W_{L}=O$, that is, $b_{L}=\infty$.

In this case, only the last layer is updated during the first step because $Q_{L}^{*}=O$ does not propagate the local error to the downstream layers. After the first-step update, the weight is given by

$$ $\displaystyle W_{L,1}=-\frac{\eta^{\prime}}{M^{\theta_{L}}}\delta_{L}h_{L-1}^{ \top}.$ (S.137) $$

and $W_{l<L,1}=W_{l<L,0}$.
Thus,

$$ $\displaystyle\partial_{\eta^{\prime}}\Delta W_{l}h_{l-1,1}\big{|}_{\eta^{ \prime}=0}$ $\displaystyle=-\frac{1}{M^{\theta_{L}}}\delta_{L}h_{L-1}^{\top}h_{L-1}$ (S.138) $\displaystyle=-\frac{1}{M^{\theta_{L}-1}}\delta_{L}K^{A}_{L-1}.$ (S.139) $$

and

$$ $\displaystyle h_{L,1}=\Theta(1/M^{\theta_{L}-1}),\quad K^{A}_{L}=\Theta(1/M^{2 \theta_{L}-1}),$ (S.140) $$

Substituting these into Eqs. (S.119,S.122), we obtain

$$ $r_{l}=\left\{\,\begin{aligned} &\theta_{1}-\theta_{L}+1\quad(l=1),\\ &\theta_{l}-\theta_{L}\quad(1<l<L),\\ &\theta_{L}-1\quad(l=L).\end{aligned}\right.$ (S.141) $$

From Eq. (S.128),

$$ $\displaystyle\partial_{\eta^{\prime}}(W_{L,0}\Delta h_{L-1,1})\big{|}_{\eta^{ \prime}=0}=\Theta(1/M^{\theta_{L-1}-1}).$ (S.142) $$

Finally, Conditions A.1 and A.2 lead to

$$ $\displaystyle\theta_{1}-(\theta_{L}-1)=0\quad(l=1),$ (S.143) $\displaystyle\theta_{l}-1-(\theta_{L}-1)=0\quad(1<l<L),$ (S.144) $\displaystyle\theta_{L}-1=0$ (S.145) $$

Thus, the $\mu$P is the same as in the case of random head initialization.

### A.5 Stable parametrization for feedback network

The feedback network minimizes the following loss function:

$$ $L(Q_{l})=\frac{1}{2M_{l-1}}\|\phi(Q_{l}h_{l})-h_{l-1}\|^{2}.$ (S.146) $$

where dividing by $M_{L-1}$ is to ensure that $L(Q_{l})=\Theta(1)$, which is the default setting in PyTorch.
We consider the parametrization in the feedback network:

$$ $Q_{l}\sim\mathcal{N}(0,{\sigma^{\prime 2}}/M^{2\bar{q}_{l}}),\quad\tau_{l}= \frac{\tau_{l}^{\prime}}{M^{\bar{\tau}_{l}}},$ (S.147) $$

where $\tau_{l}$ denotes the learning rate for the feedback network.

To ensure that the update $\Delta Q_{l}h_{l}=\Theta(1)$ holds, we have

$$ $\displaystyle\Delta Q_{l}h_{l}$ $\displaystyle=\frac{\tau^{\prime}}{M^{\bar{\tau}_{l}+1}}\left(\phi(Q_{l}h_{l}) -h_{l-1}\right)\phi^{\prime}(Q_{l}h_{l})h_{l}^{\top}h_{l}.$ (S.148) $$

Here, because $h_{l<L}^{\top}h_{l<L}=\Theta(M)$ and $h_{L}^{\top}h_{L}=\Theta(1/M^{2b_{L}-1})$, assuming $\Delta Q_{l}h_{l}=\Theta(1/M^{r_{l}})$, we obtain:

$$ $r_{l}=\left\{\,\begin{aligned} &\bar{\tau}_{l}\quad(1<l<L),\\ &\bar{\tau}_{L}+2b_{L}\quad(l=L)\end{aligned}\right.$ (S.149) $$

Therefore,

$$ $\displaystyle\bar{\tau}_{l<L}=0,\quad\bar{\tau}_{L}=-2b_{L}.$ (S.150) $$

If we optimize the feedback network for one step, we have

$$ $\displaystyle Q_{l<L,1}$ $\displaystyle=Q_{l<L,0}-\frac{\tau^{\prime}}{M^{\bar{\tau}_{l}+1}}h_{l-1}h_{l} ^{\top}\sim\frac{1}{M^{\min(1,q_{l})}},$ (S.151) $\displaystyle Q_{L,1}$ $\displaystyle=Q_{L,0}-\frac{\tau^{\prime}}{M^{\bar{\tau}_{L}+1}}h_{L-1}h_{L}^{ \top}\sim\frac{1}{M^{\min(1/2-b_{L},q_{L})}}.$ (S.152) $$

And,

$$ $\displaystyle Q_{l<L}^{*}$ $\displaystyle=h_{l-1}(h_{l}^{\top}h_{l}+\mu I)^{-1}h_{l}^{\top}\sim\frac{1}{M},$ (S.153) $\displaystyle Q_{L}^{*}$ $\displaystyle=h_{L-1}(h_{L}^{\top}h_{L}+\mu I)^{-1}h_{L}^{\top}\sim\frac{1}{M^ {1/2-b_{L}}}.$ (S.154) $$

Therefore, in this case, $Q_{l,1}=Q_{l}^{*}$ holds when

$$ $\bar{q}_{l<L}\geq 1,\quad\bar{q}_{L}\geq 1/2-b_{L}.$ (S.155) $$

## Appendix B Additional Experiments

### B.1 Predictive Coding

#### B.1.1 Linear Network

Figure: Figure S.1 : During training of the linear network, it converges to the analytical solution. We trained a 3-layer linear network using synthetic data.
[Uncaptioned image]: x13.png

Figure: Figure S.2 : As $M_{L}$ approaches 1, the update vector in PC converges to that of BP. We conducted inference training on a 3-layer linear network and measured the similarity between PC and BP. The results demonstrate that PC approaches BP as $M_{L}$ decreases and $\bar{\gamma}_{L}$ increases.
[Uncaptioned image]: x14.png

In Figure S.1, we measure the similarity of the inference vector with BP, GNT, and the analytical solution. With fewer inference iterations, the model behaves more like BP; however, as the the number of iterations increases, the model converges toward the analytical solution. Furthermore, as the middle layer width $M_{l}$ increases, the gap between GNT and BP decreases.
Figure S.2 further demonstrates that reducing the output dimension $M_{L}$ brings the model closer to BP. However, increasing $M_{L}$ moves the model away from BP, though this divergence is more gradual as $\bar{\gamma}_{L}$ approaches zero.

#### B.1.2 Additional Experiments on μ 𝜇 \mu italic_μ transfer for PC

##### Architecture

In the main text, we primarily focused on MLP and CNN. However, our $\mu$P is architecture-independent. The results for VGG5 are presented in Figure S.3.
Furthermore, the $\mu$Transfer observed in Figure 4 also holds for MLP, as demonstrated in Figure S.4.

Figure: Figure S.3 : In VGG5, the learning rate also transfers across widths. In SP, the optimal learning rate shifts based on model width, whereas in $\mu$P, it remains fixed. Additionally, we trained with two different $\gamma_{l}$ values, and under $\mu$P ($\bar{\gamma}_{L}=-1$), the learning rate consistently transfers across widths, regardless of $\gamma_{l}$. The model was trained for 40 epochs on 1024 samples from FashionMNIST.
Refer to caption: x15.png

Figure: Figure S.4 : Without F-ini, $\mu$P with $\bar{\gamma}_{L}=-1$ transfers the learning rates across width also in MLP. We trained 3-layer MLP on FashionMNIST without F-ini.
Refer to caption: x17.png

##### Loss Type

Figure: Figure S.5 : $\mu$P for PC transfers learning rates across widths even with Cross Entropy. We train 3-layer MLP for 40 epochs on 1024 samples from FashionMNIST.
Refer to caption: x19.png

In the main text, we mainlyu used MSE loss. However, this can be replaced with cross-entropy (CE) loss.
As demonstrated in Figure S.5, $\mu$P for PC also transfers the learning rate across widths when using cross-entropy loss.

##### Optimizer

Figure: Figure S.6 : $\mu$P with $\bar{\gamma}_{L}=-1$ can constantly transfer the learning rates across width We confirmed $\mu$Transfer when training PC with Adam to update parameters in both MLP and CNN. In the training of MLP without F-ini, we observe that $\mu$P with $\bar{\gamma}_{L}=-1$ consistently stabilizes training and performs well. All experiments were conducted on FashionMNIST with 1024 samples.
Refer to caption: x20.png

Figure: Figure S.7 : When training with Adam, muP with $\bar{\gamma}_{L}=-1$ transfer $\gamma_{L}$ across width. We trained a 3-layer MLP on FashionMNIST with Adam.
Refer to caption: x26.png

In this paper, we primarily focus on weight updates using SGD.
However, it is also possible to update the weights using Adam instead of SGD. In this case, the corresponding $\mu$P is as follows:

$$ $\left\{\,\begin{aligned} &b_{1}=0,\quad b_{l>1}=1/2,\quad b_{L}=1,\\ &c_{1}=0,\quad c_{l>1}=1.\end{aligned}\right.$ (S.156) $$

For Adam, the scaling of $b_{l}$ and $c_{l}$ does not depend on $\bar{\gamma}_{L}$.
Additionally, in Adam, the gradients are normalized, which means that $\mu$P remains unchanged regardless of whether the gradients are generated by BP or PC.
When considering the stability of the inference, scaling with respect to $\bar{\gamma}_{L}$ can be treated in the same manner as in the case of SGD.

$$ $\bar{\gamma}_{l<L}=0,\quad\bar{\gamma}_{L}=1.$ (S.157) $$

##### Train sample

Figure: Figure S.8 : The results of $\mu$P for PC are independent of the number of training samples. We train 3-layer MLP on FashionMNIST with full training samples. The stability of $\mu$P holds even with all training samples.
Refer to caption: x27.png

We reduced the number of training samples in most of the graphs for $\mu$Transfer.
By reducing the number of training samples, finite-width models are known to behave more similarly to infinite-width models, as has often been seen in papers examining the theoretical aspects of feature learning [Geiger et al., , Ishikawa and Karakida, ].
However, even when training on the full dataset, $\mu$P remains stable across widths, as shown in Figure S.8.

##### Inference Loss

Figure: Figure S.9 : When evaluating the loss after inference, only $\mu$P with $\bar{\gamma}_{L}=-1$ satisfies the empirical rule of “wider is better” Regardless of whether F-ini is applied, $\mu$P with $\bar{\gamma}_{L}=-1$ consistently reduces the loss during inference with stability. We traine 3-layer MLP on FashionMNIST.
Refer to caption: x30.png

When considering the stability of inference, we can observe the loss before updating the parameters after inference.
As shown in Figure S.9, when training a 3-layer MLP on FashionMNIST, only $\mu$P with $\bar{\gamma}_{L}=-1$ consistently reduces the inference loss as the model width increases.”

##### Base width and inference iterations

In $\mu$-transfer, some research set the width of the smaller model used for tuning the learning rate, as the base width, denoted by $M^{\prime}$, and adjusts the learning rate using $\eta_{l}=\eta_{l}^{\prime}/(M/M^{\prime})^{c_{l}}$.
As shown in Figure S.10, the choice of $M^{\prime}$ (a smaller $M^{\prime}$) can sometimes make $\mu$P with $\bar{\gamma}_{L}=0$ more sensitive.

As shown in Figure S.11, the shift in the optimal learning rate at $\gamma_{L}=0$ with $M^{\prime}=128$ becomes more evident as the number of inference iterations increases. This is likely because with more iterations, the dynamics of inference play a more critical role in weight updates. In summary, to achieve stable $\mu$Transfer independent of the base width and the number of inference iterations, we should use $\mu$P with $\gamma_{L}=-1$.

Figure: Figure S.10 : When the base width $M^{\prime}$ is large, $\mu$P with $\bar{\gamma}_{L}=0$ tends to fail with $\mu$transfer. We train 3-layer MLP on FashionMNIST. This suggests that $\mu$P with $\gamma_{L}=-1$ should be used when setting the base width, even with F-ini. The inference is performed for 100 iterations.
Refer to caption: x32.png

Figure: Figure S.11 : $\mu$P with $\bar{\gamma}_{L}=-1$ maintains high inference stability and successfully performs $\mu$-transfer even with a large number of inference iterations. We conducted the experiment shown in Figure S.10 with varying numbers of inference iterations. Even with a larger number of inference iterations, $\mu$P with $\gamma_{L}=-1$ consistently transfers the learning rate across different widths.
Refer to caption: x36.png

##### Sequential Inference and Synchronous Inference

Figure: Figure S.12 : $\mu$P for PC also transfers learning rates across widths in synchronous inference. $\mu$P for PC can also be applied in synchronous inference. Note that when the base width is set to 1, as in SI, the learning rate does not transfer in $\mu$P with $\bar{\gamma}_{L}=0$.
Refer to caption: x39.png

Figure: Algorithm 1 PC Algorithm (Simultaneous or Sequential inference)

In the main text, we focused on Sequential Inference, where $u_{l}$ is updated layer by layer, starting from the output layer. However, Synchronous Inference, where all layers are updated simultaneously, can also be considered.
For the differences between Sequential Inference and Synchronous Inference, see Algorithm.1.
As shown in Figure S.12, since $\mu$P for PC is validated at fixed points, it is also applicable to Synchronous Inference.

##### Additional Experiments with Figure 3

Figure: Figure S.13 : $\mu$P scales better than SP and NTK. We trained a CNN by PC with F-ini and a VGG5 on the full FashionMNIST dataset. The "wider is better" principle holds for $\mu$P.
Refer to caption: x41.png

Figure S.13 presents the results of the same experiment shown in Figure 3 (Right), but with the CIFAR-10 dataset and the VGG5 model.
It is evident that even with CIFAR-10 and VGG5, $\mu$P achieves higher accuracy compared to SP and NTK.

### B.2 Target Propagation

#### B.2.1 Additional Experiments on μ 𝜇 \mu italic_μ Transfer for PC

Figure: Figure S.14 : Learning rate transfer in Feedback networks. We demonstrate that the learning rate in feedback networks transfers effectively across widths using toy data. Both the feedforward and feedback networks include a Tanh/ReLU activation function following the linear layer.
Refer to caption: x43.png

Figure: Figure S.15 : Learning rate transfer in Feedback networks (output layer). We show that the learning rate in feedback networks transfers across widths using toy data. Unlike the hidden layers, the learning rate in the output layer does not transfer under the default setting, which requires $\mu$P scaling. Both the feedforward and feedback networks include a Tanh/ReLU activation function after the linear layer.
Refer to caption: x45.png

Figure: Figure S.16 : Even when training the feedback network with DRL, $\mu$P demonstrates greater stability compared to SP. We trained a 3-layer MLP on the FashionMNIST using DRL. While SP exhibits a shift in the maximum learning rate as the model width increases, $\mu$P consistently transfers the optimal learning rates across different widths.
[Uncaptioned image]: x47.png

Figure: Figure S.17 : $\mu$P for TP remains stable regardless of the dataset or the number of training samples. We trained a 3-layer MLP on both FashionMNIST and CIFAR-10 using the full training samples. With $\mu$P, the learning rate successfully transfers across widths, ensuring that the maximum learning rate remains consistent regardless of the model width.
Refer to caption: x48.png

##### Feedback Network

As discussed in Section A.5, stable parametrization is crucial not only for feedforward but also for feedback networks.
We verified this with $\mu$P, as shown in Figures S.14 and S.15.

##### DRL

Meulemans et al. [] proposes the difference reconstruction loss (DRL) for constructing feedback networks.
In Figure S.16, we empirically confirm that our $\mu$P works effectively with DRL when training an MLP on FashionMNIST.

##### Training samples

As with PC, in the case of TP, Figure 7 uses 1024 training samples.
Similar results were observed when using the full training dataset, as shown in Figure S.17.

## Appendix C Experimental Settings

##### Architecture and dataset

We trained the following three models:

- •
MLP: We trained a 3-layer multilayer perceptron (MLP) with Tanh activation. The MLP models do not include bias.
- •
CNN: We trained 3-layer CNN with Tanh activation.
The models consist of a two-layer convolutional layers and a linear layer.
We trained with different hidden widths where the width is proportional to the input dimension of the output layer. (For example, when the width is set to 4, the input dimension of the final layer is 512.)
Max pooling is applied after the activation function.
- •
VGG5: We trained a VGG-like model consisting of 4 convolutional blocks and 3 linear blocks, based on the structure described in [Pinchetti et al., ]. When the width is set to 8, it matches the VGG5 model in Pinchetti et al. [], with the channel sizes being [128, 256, 512, 512].

##### Dataset and batch size.

We used FahionMNIST and CIFAR-10 datasets without applying any data augmentation.
The settings for batch size and training samples were as follows:

- •
PC In the experiments on $\mu$Transfer, FashionMNIST was generally trained with 1024 training samples and a batch size of 1024, except for Figure S.8. However, when training VGG5, the batch size was reduced to 64 due to memory constraints. In the experiment verifying the scaling of $\mu$P with respect to width (Figure 3), all training samples were used, with a batch size of 1024.
- •
TP In Figures 6 and 7, we trained 3-layer MLP using 1024 training samples. Note that in Figure S.17 in the Appendix, FashionMNIST and CIFAR-10 were trained using the full datasets.

##### Training recipe

Weight decay was not applied during the parameter updates for feedforward networks.
For SGD, the momentum was set to 0.9, and for AdamW, the parameters $(\beta_{1},\beta_{2})$ were set to $(0.9,0.99)$.

- •
PC The reduction mode for the loss function was set to "sum" to align the order of all terms in the free energy function.
- •
TP For feedback networks, weight decay was set to $10^{-4}$ and the learning rate for the target was set to $\hat{\eta}=0.01$. Before starting the main training, only the feedback network was trained for 5 epochs with the feedforward network fixed.

## Peer Reviews

confidence: 1
contribution: 3
initial_rating: 6
presentation: 3
questions: 1. In the preliminaries I can't find any definition for $\mathcal{L}$, I assume it is the loss function, are there any particular assumptions about it? 
2. A precise definition of the Maximal Update Parametrization in the text would be useful, especially for readers unfamiliar with the related work. 
3. In equation 5, where is $\psi$ defined? 
4. Could the authors provide a more precise definition of stable training dynamics? 
5. Is the transfer of learning rates also observed for Cross Entropy loss? 
6. How sensitive are $\mu P$  and $\mu$Transfer to different initialization schemes?
soundness: 3
strengths: - The paper is well-written. 
- The problem is conceptually well motivated.
-  The manuscript includes both rigorous proofs and empirical experiments that validate the findings. 
- The observation that the optimal learning rate does not depend on the order of width is interesting. 
- The disappearance of the kernel regime in TP is interesting.
summary: This paper investigates local learning methods for neural networks as alternatives to backpropagation. It specifically focuses on predictive coding (PC) and target propagation (TP). The authors introduce maximal update parameterization (µP) for PC and TP in the infinite-width limit to enable hyperparameter transfer across different model widths, aiming to stabilize local loss optimization. They analyze deep linear networks to show that PC’s gradient transitions between first-order and Gauss-Newton-like gradients depending on parameterization. For TP, they find that µP favors feature learning over the kernel regime, a distinction from traditional BP.
weaknesses: 1. The derivation of $\mu P$ assumes one-step gradient 
2. The derivation also assumes linear networks 
3. It is unclear how the findings extend to more general settings

---

confidence: 2
contribution: 3
initial_rating: 6
presentation: 2
questions: The discussion of the implications of the $\mu P$ parameterization for PC after corollary 4.3 and in §4.3, on the impact of the free parameter $\gamma$, is very technical. It is not clear what the broader implications of $\gamma = 0$ or $\gamma = 1$ or the fact that the the PC gradients are more similar to those of standard backprop GD or GNT. A higher-level description of what the parameter $\gamma$ represents, and what it means to be closer to 0 or 1, would help the reader grasp the importance of the results.

- It would be worth to explicitly state that, by definition, $\mu P$ depends on the algorithm used, and therefore needs to be derived to figure out what $\mu P$ means in the context of PC and TP, which is the contribution of this paper. The sentence before the list of contributions leaves the "$\mu P$ depends on the algorithm" part implicit.

- The word "inference" is heavily overloaded in ML and a clarifying note as to what it means for PC in §2 would help. This is adressed later in §3.1 but a short parenthetical might be worth including.

- "The parameterization that induces the feature learning regime is called $\mu P$"
  This phrasing implies that there is a unique parameterization that induces feature learning. It seems that the authors instead mean that "$\mu P$ induces the feature learning regime"?

- "The training dynamics are referred to as stable when they neither vanish nor explore as the network width increases"
  What quantitiy should neither vanish nor explode?

- "the parameterization that achieves this [non-vanishing and non-exploding] is called a stable parameterization"
  The phrasing as "the parameterization that achieves this" implies that this parameterization is unique. Is this what is meant or should it be "a"?

- In table 1; reminding the reader of what the parameters $b_l$, $c_l$ and $\gamma_l$ would help, e.g. $b_l: initialization of the weights, $c_l$: scaling of the step-size. Adding a legend would help remind the reader of what SP, SGD, GNT, PC and TP mean.

- Line 271-272 "the $\mu P$ of BP can be directly transferred to PC" should also mention "and leads to parameter transfer across width", as this is what the figure shows.

- Please add that $\theta_l = 2a_l + c_l$ to the statement of Thm 4.1, assuming that those parameters indeed have the same meaning as in Prop 3.3.

- An additional experiment showing that the $\mu P$ for SGD does not lead to hyperparameter transformer on PC without FPA would strenghten the claim that the $\mu P$ for PC is necessary 

- It is not clear whether the difference between GD, (S)GD and SGD is intended to meaningful or whether they are used interchangeably. If it is not, would it make sense to standardize to GD or (S)GD?

- Figure 2 should explicitly state what is being compared; The cosine similarity measures the similarity between what and what?
soundness: 3
strengths: To my knowledge, the results are new as the infinite width limit has not received as much attention for local learning as for the typical backpropagation-based methods. The objectives of the paper are clear and the paper executes on its premises. The experimental results support the claim that the derived parameterization does achieve hyperparameter transfer.
summary: The submission presents derivation of the initialization and step-size scaling required to achieve hyperparameter transfer across width using the $\mu P$ parameterization in the infinite width limit for two local learning algorithms, predictive coding and target propagation.
weaknesses: The paper deals with two very specialized subjects, local learning and infinite-width limits. As a result, it has a difficult job in making its argument clear to a general audience. The current writing makes a reasonable attempt, but there are still points that can be clarified to help guide the reader that might be only be vaguely familiar with one of the topic. I give a list of specifics in the questions below.

---

confidence: 2
contribution: 3
initial_rating: 6
presentation: 2
questions: How is $\eta'_l$ chosen in Equation (8)? Is it tuned in the experiments?

Figure 3: what is the cause of sharp loss increase in the loss of wide networks when $\bar \gamma_L < -1$? Is there some theoretical explanation? Can be it that for $\bar \gamma_L = -1$, training would also diverge for wide enough network?
soundness: 3
strengths: Authors give a thorough analysis of parametrization in local learning, motivated by previous progress of $\mu P$ in backpropagation. The question is well-motivated by the fact that model sizes for local learning are increasing, thus it is important to understand which parameterizations are robust to scaling up the models. 

Experimental results on FashionMNIST confirm theoretical findings.
summary: This work studies parameterisation for local learning algorithms: for predictive coding and target propagation. The main goal is to understand maximal update parameterization ($\mu P$), which allows transfer of hyperparameters across different widths. Authors prove optimal $\mu P$ from one step of descent or for linear networks. Experimental results are provided based on Fashion MNIST.
weaknesses: Presentation of the paper can be improved, in particular the notation section. Choice of $\mathcal{L}$ is not introduced, notation for $\gamma_l$ should be defined before Section 3.1. Also, $\bar \gamma_L$ is an important quantity, but is only defined implicitly in Theorem 4.1.

Both sections on PC and TP were a bit hard to follow (for someone outside of the field): In PC, for example, Equation (3) should also be complemented by the case $\ell = L - 1$.

In Table 1, adding references to prior work and highlighting new contributions will be helpful.

---

confidence: 2
contribution: 3
initial_rating: 5
presentation: 1
questions: Line 200: what is abc-parametrization?

Line 278: what is $M^{\overline{\gamma}_L}$?

Line 309: what is $\mu$Transfer? A reference here would be helpful.

What would the challenges be in extending the analysis to non-linear networks? 

[HRSS19 Mean-field Langevin dynamics and energy landscape of neural networks, arXiv preprint arXiv:1905.07769] studies infinite-width neural networks via the lens of mean-field Langevin dynamics. Would it be possible to study local learning through this lens, and could this potentially mitigate the requirement to develop a tensor program (line 535)?
soundness: 3
strengths: Originality: I’m not so familiar with this subfield, but in so far as I can tell, this work seems very novel as it seems to be the first work to introduce $\mu$P for local learning. The authors provide theoretical guarantees showing that it is possible to reduce the required amount of hyperparameters in local learning. 

Quality: The theoretical results are well-justified by detailed proofs in the appendix and references to prior works. The authors provide empirical justification on real-world datasets as well.

Significance: The paper is significant as it provides justification on why local learning can match performance of models trained by back-propagation.
summary: This paper considers maximal update parameterization for local learning, an alternative to backpropagation for optimizing neural networks. In particular, the authors consider predictive coding and target propagation. 

The authors develop theory to show that hyperparameters can be transferred between networks of differing widths, matching existing guarantees for backpropagation, which helps reduce the amount of computation required for training with local learning. Further, they prove that under certain assumptions, target learning reduces to gradient descent, which implies that backpropagation and local learning will converge to the same model.
weaknesses: The content is rather inaccessible as the assumed knowledge is too specific for a broad conference such as ICLR. I have background in mean-field networks, but not local learning or $\mu$P. It would be useful to add a brief appendix on these as a background (beyond what is already described in Section 3) to help the reader understand the paper.

The paper is also hard to follow due to the amount of abbreviations. It would also be useful to add backreferences for the techniques F-ini and FPA in lines 158, 162.

Some of the theorems are missing assumptions, notation, or intuition. For example, in Prop. 3.3 (line 250), what are $\delta_l$, $e_L$? In Cor. 4.3 (line 350), why do we need $a_L + b_L = 1$? 

The experimental results are on relatively small datasets, e.g. FashionMNIST and CIFAR10: it would be nice to have validation on CIFAR100 as well.

Meta-review: The paper presents theoretical advances towards better under of local learning. The specific focus is on parameterization of local learning, for predictive coding (PC) and target propagation (TP). Based on maximal update parameterization, the paper shows that hyper parameters can be transferred across different widths, similar to related results for backpropagation.  Focusing on deep linear networks, the paper shows specific results for PC and TP. The reviewers generally appreciated the advances made in the paper, as the results are mostly clearly presented. There were some concerns regarding the specific assumptions, such as linear networks, and more broadly on the quality of the exposition, as arguably a specialized audience has knowledge of these advances. The authors have added a section in the append

Decision: Accept (Poster)