From a6d88a18a4bfd46ab0dd487730b72dcbc3b8b5b8 Mon Sep 17 00:00:00 2001 From: Yuren Hao Date: Wed, 15 Jul 2026 10:32:26 -0500 Subject: assets/: organize into decks|figs|hw|refs|misc; scripts + gitignore repointed Co-Authored-By: Claude Fable 5 Claude-Session: https://claude.ai/code/session_014FAPDWQ49M5Ye3NpTndTpn --- assets/ept_method_intro.tex | 606 -------------------------------------------- 1 file changed, 606 deletions(-) delete mode 100644 assets/ept_method_intro.tex (limited to 'assets/ept_method_intro.tex') diff --git a/assets/ept_method_intro.tex b/assets/ept_method_intro.tex deleted file mode 100644 index 31d458d..0000000 --- a/assets/ept_method_intro.tex +++ /dev/null @@ -1,606 +0,0 @@ -\documentclass[11pt]{article} - -\usepackage[margin=1in]{geometry} -\usepackage{amsmath,amssymb} -\usepackage{bm} -\usepackage[round]{natbib} -\usepackage{enumitem} -\usepackage{booktabs} -\usepackage[colorlinks=true,linkcolor=blue,citecolor=blue,urlcolor=blue]{hyperref} - -% --- light-weight notation --------------------------------------------------- -\newcommand{\R}{\mathbb{R}} -\newcommand{\C}{\mathbb{C}} -\renewcommand{\Re}{\operatorname{Re}} -\newcommand{\xin}{x_{\mathrm{in}}} -\newcommand{\zstar}{z^{\ast}} -\newcommand{\zbar}{\bar{z}} -\newcommand{\Fnc}{F_{\mathrm{nc}}} -\newcommand{\Jnc}{J_{\mathrm{nc}}} -\newcommand{\half}{\tfrac12} -\newcommand{\grad}{\nabla} -\newcommand{\dd}{\,\mathrm{d}} -\newcommand{\inner}[2]{\langle #1,\, #2\rangle} -\DeclareMathOperator{\Attn}{Attn} -\DeclareMathOperator{\FFN}{FFN} -\DeclareMathOperator{\softmax}{softmax} -\DeclareMathOperator{\LSE}{LSE} -\DeclareMathOperator{\LN}{LN} -\DeclareMathOperator{\jvp}{jvp} -\DeclareMathOperator{\vjp}{vjp} -\DeclareMathOperator*{\argmin}{arg\,min} - -\title{\bf Training a Transformer Language Model with Equilibrium Propagation:\\ -from energy-based EP to non-conservative, holomorphic, tracking-AEP} -\author{Method introduction (internal)} -\date{2026-06-21} - -\begin{document} -\maketitle - -\begin{abstract} -We train a transformer-class language model in which \emph{both} attention and the -feed-forward network learn \emph{without backpropagation through the computation}, -using Equilibrium Propagation (EP). This note is written for a reader who knows -\emph{classic} energy-based EP \citep{scellier2017} --- the two-phase free/nudged -relaxation of a conservative, symmetric-Jacobian system --- but has not met the -non-conservative / asymmetric / holomorphic extensions. We first recall why classic -EP \emph{requires} a conservative system, then show that softmax self-attention -breaks that requirement (independent $Q,K,V$ give an asymmetric Jacobian). We then -introduce, from first principles, the pieces that repair this: the -\emph{asymmetric / adjoint} EP correction $J\!\to\!J^{\!\top}$ -\citep{scurria2026}; the \emph{holomorphic} EP estimator \citep{laborieux2022}; -the \emph{Convergent Energy Transformer} (CET) route \citep{hoier2026} that -sidesteps the problem by making attention conservative; and finally \emph{our} -recipe: a damped non-conservative equilibrium-transformer block, trained with -\emph{tracking-AEP} (re-linearizing the correction at the moving common-mode -midpoint) plus a residual-driven stabilization stack. We report what is solidly -validated --- component gradients match backprop at cosine $0.99$--$1.0$, and EP -trains the block stably and competitively with a backprop transformer at equal -parameters on a character-level LM --- and clearly mark the larger-scale work -(the $C{=}512$ ``residual-defense'' line) as \emph{ongoing}. -\end{abstract} - -\tableofcontents - -%============================================================================== -\section{Recap: classic energy-based EP and why it needs a conservative system} -\label{sec:classic} - -\paragraph{Setup.} -Classic EP \citep{scellier2017} trains a dynamical system whose state -$z\in\R^{d}$ relaxes, under a fixed input/clamp, to the minimum of a scalar -\emph{energy} $E(z,\theta)$. Two ideas make it a learning rule. - -\paragraph{Two phases.} -\begin{itemize}[leftmargin=1.4em,itemsep=2pt] - \item \emph{Free phase.} Run the gradient dynamics $\dot z=-\grad_z E(z,\theta)$ - to the free equilibrium $\zstar=\argmin_z E(z,\theta)$, - in practice an Euler relaxation to a fixed point. - \item \emph{Nudged phase.} Add the task loss to the energy with a small strength - $\beta$, $E_\beta = E + \beta\,\ell(z)$, and relax to the nudged - equilibrium $z_\beta$. -\end{itemize} - -\paragraph{The contrastive gradient.} -EP's central identity is that the loss gradient w.r.t.\ any parameter is the -\emph{contrastive difference of $\partial E/\partial\theta$ across the two phases}: -\begin{equation} - \frac{\partial \mathcal{L}}{\partial \theta} - \;\approx\; - \frac{1}{\beta}\!\left[ - \frac{\partial E}{\partial\theta}(z_\beta,\theta) - -\frac{\partial E}{\partial\theta}(\zstar,\theta) - \right] - \qquad(\text{one-sided, bias }O(\beta)). - \label{eq:ep-onesided} -\end{equation} -Centered / symmetric nudging \citep{laborieux2021} uses $\pm\beta$ and averages, -reducing the estimator bias to $O(\beta^2)$: -\begin{equation} - \frac{\partial \mathcal{L}}{\partial \theta} - \;\approx\; - \frac{1}{2\beta}\!\left[ - \frac{\partial E}{\partial\theta}(z_{+\beta}) - -\frac{\partial E}{\partial\theta}(z_{-\beta}) - \right]. - \label{eq:ep-centered} -\end{equation} -The update is \emph{local}: each parameter reads only the two equilibria of the -terms it touches; there is no backward pass and no weight transport. As -$\beta\!\to\!0$ with a converged free phase, the EP estimate equals the -implicit/equilibrium gradient, and (in an RNN with static input) it equals the -step-wise BPTT gradient \citep{ernoult2019}. - -\paragraph{Why this needs a conservative / symmetric-Jacobian system.} -Equations \eqref{eq:ep-onesided}--\eqref{eq:ep-centered} are only valid because -the dynamics are the \emph{gradient} of a scalar energy. Write the force as -$F(z) = -\grad_z E(z)$ and its Jacobian as $J=\partial F/\partial z$. If $F$ -descends an energy, then $J = -\,\partial^2 E/\partial z^2$ is a Hessian and is -therefore \emph{symmetric}, $J=J^{\!\top}$. This symmetry is exactly what makes the -nudged perturbation a faithful surrogate for the loss \emph{adjoint}: linearizing -the nudged relaxation around $\zstar$ produces a response governed by -$(I-J)^{-1}$, and because $J=J^{\!\top}$ this self-adjoint operator is the same one -the true gradient (which involves $(I-J^{\!\top})^{-1}$) requires. We therefore -record the four implicit premises of classic EP --- the transformer will break all -four, and each fix below targets exactly one of them: -\begin{description}[leftmargin=2.6em,itemsep=2pt] - \item[(A) Conservative / symmetric.] A scalar energy $E$ exists, so $J=J^{\!\top}$. - \item[(B) Free phase converged.] The readout sits at the true fixed point; - residual $\approx 0$. - \item[(C) Small-$\beta$ linear response, clean nudge.] $\beta\!\to\!0$ is a mere - perturbation, and no non-analytic ``clamp'' contaminates the estimate. - \item[(D) The fixed point stays stable throughout training.] After every weight - update the free phase still relaxes to a stable fixed point. -\end{description} - -%============================================================================== -\section{The gap: softmax attention is non-conservative} -\label{sec:gap} - -A pre-LN transformer block computes, for a state $z$, -\begin{equation} - \Attn(z) = \softmax\!\Big(\tfrac{Q(z)K(z)^{\!\top}}{\sqrt{d}},\ \text{causal}\Big)V(z)\,W_O, - \qquad - Q=zW_Q,\ K=zW_K,\ V=zW_V, - \label{eq:attn} -\end{equation} -with \emph{independent} projections $W_Q,W_K,W_V$. The query--key coupling -$i\!\to\!j$ is governed by $W_QW_K^{\!\top}$, while $j\!\to\!i$ is governed by -$W_KW_Q^{\!\top}$; these differ, and $V$ is a third independent map. Consequently -the attention Jacobian is \emph{asymmetric}, $J_{\Attn}\neq J_{\Attn}^{\!\top}$, and -\emph{no scalar energy has this gradient}. An untied $4\times$ FFN -($W_2\,\mathrm{GELU}(W_1\cdot)$ with $W_2\neq W_1^{\!\top}$) is non-conservative for -the same reason. Premise~(A) fails. - -Empirically this is not a cosmetic issue: with an asymmetric $J$ the nudged phase -relaxes under $J$ but the correct loss adjoint needs $J^{\!\top}$, so the raw EP -contrast is \emph{biased}. Measured against the true backprop gradient, uncorrected -EP gives an attention-parameter cosine of only $\approx 0.25$ (essentially the -wrong direction), even though the loss-adjacent output projection looks fine. (This -is the same pathology that limits feedback alignment, which only trains the layer -right before the loss and leaves $Q/K/V$ at cosine $\approx 0.25$ and the upstream -FFN at $\approx -0.01$.) - -There are two ways out, and we will use the second: -\begin{enumerate}[leftmargin=1.6em,itemsep=2pt] - \item \textbf{Energy route} (make attention conservative): fold attention into a - scalar energy with a \emph{tied} value, so $F=-\grad E$ and classic EP is - exactly valid. This is the CET route (\S\ref{sec:cet-energy}); it costs the - $Q\!\neq\!K$ asymmetry and the free value that make attention expressive. - \item \textbf{Force route} (keep real attention, repair the \emph{estimator}): - leave \eqref{eq:attn} as a non-conservative \emph{force} and add a - correction that turns $J$ into $J^{\!\top}$ in the nudged phase. This is the - AEP route (\S\ref{sec:aep}), and it is what our block uses. -\end{enumerate} - -%============================================================================== -\section{AEP, holomorphic EP, and the force-form readout} -\label{sec:aep} - -\subsection{Force-form (vector-field) EP} -\label{sec:vf} -The first step is to drop the energy and write the dynamics directly as a force -$F(z)$, relaxing $\dot z=F(z)$ to a fixed point $\zstar$. The parameter gradient is -then read off a \emph{vector-field} (VF) contrast \citep{scurria2026}: -\begin{equation} - \frac{\partial\mathcal{L}}{\partial\theta} - \;\approx\; - \frac{\partial}{\partial\theta}\,\big\langle a,\ F(\zstar;\theta)\big\rangle, - \qquad - a \;=\; \frac{z_{-\beta}-z_{+\beta}}{2\beta}\ \approx\ -\frac{\dd \zstar}{\dd\beta}, - \label{eq:vf} -\end{equation} -where $a$ is the centered contrast (the ``adjoint state'') read from the two nudged -equilibria, and the right-hand side is \emph{one} autograd call evaluated at the -fixed point only --- per-term local bookkeeping, \emph{not} backprop through the -relaxation steps. Every term of the block (attention, FFN, LayerNorm affines, and -the embeddings, which enter through the input clamp $-(z-\xin)$) is a term of the -same $F$, so \eqref{eq:vf} trains them jointly with no per-module schedule. - -\paragraph{Attribution / honest caveat.} -The force-form VF readout \eqref{eq:vf} is \emph{not ours}: it is the baseline of -\citet{scurria2026}. Crucially it \emph{collapses on its own} for a non-conservative -system (their CIFAR-10 VF reaches chance, $10\%$; MNIST $64\%$ vs.\ $92.7\%$), -exactly mirroring our measured cosine $\approx 0.25$ for uncorrected attention. VF -is therefore the ``starting point that fails''; what rescues it is the next step. - -\subsection{The AEP correction: \texorpdfstring{$J\!\to\!J^{\!\top}$}{J to J transpose}} -\label{sec:aep-corr} -For a non-conservative $F$, the nudged relaxation linearized at $\zstar$ runs under -$J=\partial F/\partial z$, but the true adjoint requires $J^{\!\top}$. \emph{Asymmetric -EP} (AsymEP) \citep{scurria2026} repairs this by adding to the nudged force a term -that subtracts twice the antisymmetric part of the Jacobian. With -$v=z-\zstar$ and $\Jnc$ the Jacobian of the \emph{non-conservative} part $\Fnc$, -\begin{equation} - \mathrm{corr}(z) \;=\; \Jnc\,v - \Jnc^{\!\top} v - \;=\; (\Jnc-\Jnc^{\!\top})\,v - \;=\; 2\,A_J\,v, - \qquad - A_J \equiv \tfrac12\big(\Jnc-\Jnc^{\!\top}\big), - \label{eq:aep} -\end{equation} -which is \emph{mathematically identical} to their $-2A_J(\zstar)(z-\zstar)$. The -nudged force becomes $f \;=\; F(z) \mp \beta\,\grad_z\ell(z) - \mathrm{corr}(z)$, -so the attention part of the nudged linearization is replaced as -\begin{equation} - J\,v \;-\; (J-J^{\!\top})\,v \;=\; J^{\!\top} v , -\end{equation} -i.e.\ \emph{$J$ is turned into $J^{\!\top}$}, restoring the correct adjoint and hence the -exact gradient for $Q\!\neq\!K$ attention. Two structural facts make this cheap and -local: -\begin{itemize}[leftmargin=1.4em,itemsep=2pt] - \item \emph{The symmetric (conservative) parts cancel.} The damping $-c\,z$ has - Jacobian $-cI$ (symmetric), the FFN-as-Hopfield-energy and the input clamp - are symmetric, so they contribute $0$ to $A_J$. Thus a \emph{single} - correction on the attention term repairs the \emph{whole} block; FFN/clamp - ride along in the conservative part and are already exact under VF. - \item \emph{It is matrix-free.} We never build $\Jnc$. Each nudged step uses one - Jacobian-vector product and one vector-Jacobian product, - $\Jnc v=\jvp(\Fnc,\zstar,v)$ and $\Jnc^{\!\top} v=\vjp(\Fnc,\zstar,v)$. -\end{itemize} - -\paragraph{Attribution.} -The correction \eqref{eq:aep} is \citet{scurria2026}'s, \emph{not} ours. Their scope -is feedforward / Hopfield nets on static MNIST/CIFAR with an \emph{explicitly -constructed} Jacobian, no attention, no sequence model, and no stability controller. -\emph{Ours on this line} is: (i) the matrix-free $\jvp/\vjp$ form (their explicit -Jacobian is infeasible at transformer state dimension $B\!\cdot\!T\!\cdot\!C$); -(ii) the application to data-dependent \emph{softmax attention}; (iii) the -combination with holomorphic estimation (\S\ref{sec:holo}); (iv) the common-mode -\emph{tracking} variant (\S\ref{sec:tracking}); and (v) the transformer-LM -application together with the stability stack (\S\ref{sec:stab}). - -\paragraph{Validity window.} -The correction is linearized \emph{at $\zstar$}, so the nudged trajectory must stay -inside the linear-response window. At $\varepsilon{=}0.1$ a nudge horizon -$T_2\!\approx\!20$ is comfortably inside; $T_2\gtrsim 60$ can leave it (\S\ref{sec:stab}). - -\subsection{Holomorphic EP: variance-reduced, higher-order estimates} -\label{sec:holo} -The $\pm\beta$ contrast trades bias against noise: small $\beta$ shrinks the -$O(\beta^2)$ bias but amplifies the $1/\beta$ noise on $(z_{-\beta}-z_{+\beta})/2\beta$. -Holomorphic EP \citep{laborieux2022} removes this trade-off by replacing the two -real points with $N$ points on a \emph{complex circle}, -$\beta_k = r\,e^{2\pi i k/N}$, relaxing the \emph{holomorphically extended} dynamics -and reading the contrast off a discrete Cauchy integral: -\begin{equation} - a \;=\; -\,\Re\!\left[\frac{1}{Nr}\sum_{k=0}^{N-1} e^{-i\phi_k}\,(z_k-\zstar)\right], - \qquad \phi_k=\tfrac{2\pi k}{N}, - \label{eq:holo} -\end{equation} -whose bias is $O(r^{N})$ instead of $O(r^{2})$ --- so $r$ may be $5$--$10\times$ -larger at equal bias, cutting the $1/\beta$ noise by the same factor. The -holomorphic extension is built by hand (complex LayerNorm with non-conjugate -variance, softmax as a ratio of exponentials, the $\tanh$-form GELU which is an -entire function); the AEP correction \eqref{eq:aep} is \emph{real-linear in $v$}, so -it preserves holomorphy and is applied to the real and imaginary parts separately. -No clamps appear inside the holomorphic nudge --- clamps are non-analytic and would -destroy the $O(r^N)$ bias order. This addresses premise~(C). \citep{laborieux2022} -is the source; we add only the combination with the AEP correction and with softmax -attention. - -%============================================================================== -\section{The equilibrium-transformer block (and the CET alternative)} -\label{sec:block} - -\subsection{Our damped, non-conservative block (\texttt{thick})} -\label{sec:thick} -The state is $z\in\R^{B\times T\times C}$, one vector per token position. Inference -is a relaxation to a fixed point under a \emph{single force} $F$, -$z\leftarrow z+\varepsilon F(z)$ for $T_1$ steps ($\varepsilon{=}0.1$, $T_1{\approx}150$), -after which logits $=\zstar W_h$. The force is a pre-LN transformer block written as -a force rather than a layer stack: -\begin{equation} - F(z) = - \underbrace{-(z-\xin)}_{\text{input clamp}} - +\underbrace{\Attn(\LN_1(z))}_{\text{causal MHSA},\ W_Q,W_K,W_V,W_O} - +\underbrace{W_2\mathrm{GELU}(W_1\LN_2(z)+b_1)+b_2}_{\text{untied }4\times\text{ FFN}} - -\underbrace{c\,z}_{\text{damping}}. - \label{eq:thick} -\end{equation} -Here $\xin=\mathrm{tok}[\mathrm{idx}]+\mathrm{pos}$ is the (trained) input -embedding, clamped as a boundary condition through the $-(z-\xin)$ term; this is the -same fixed-point map a Deep Equilibrium model \citep{bai2019} uses. The block is -strongly non-conservative ($Q\!\neq\!K$, untied FFN), and AEP makes EP exact for it. - -\paragraph{Why the $-c\,z$ damping is the key recipe move.} -Raw attention at high gain has \emph{no} fixed point: the residual floors at -$\sim\!3\times10^{-2}$ and the relaxation never settles, so the entire EP family -(corrected or not) cannot even start (there is no $\zstar$ to nudge around). Adding -$-c\,z$ ($c\!\geq\!1$) makes the map contractive enough to \emph{create a stable -fixed point at any attention strength}, while leaving the map non-conservative -(independent $Q/K/V$ are untouched). Critically, the damping's Jacobian $-cI$ is -symmetric, so it \emph{cancels in $A_J$} \eqref{eq:aep}: it buys a fixed point -without polluting the AEP correction, which still sees only attention's -non-reciprocal part. Together, ``damping $+$ AEP'' is the minimal recipe that makes -real attention EP-trainable, taking the attention-parameter cosine from -$\approx 0.25$ (uncorrected) to $0.99$--$1.0$ even at high gain. - -\paragraph{A subtlety for LN-inside blocks.} -Because LayerNorm sits \emph{inside} \eqref{eq:thick} and its Jacobian scales like -$1/\sigma(z)$, large damping shrinks $\|\zstar\|$ and thereby \emph{inflates} the -effective Jacobian (measured: plain-relax residual $8.8\times10^{-3}$ at $c{=}0$ -vs.\ $3.4\times10^{-2}$ at $c{=}2$). So for \texttt{thick} we keep $c$ small ($c{=}1$) -and the actual stabilizer is the Jacobian-norm penalty of \S\ref{sec:stab}, not the -damping. (For a simpler ``thin'' variant whose FFN is an energy-based modern-Hopfield -memory and whose attention is a raw damped force, the damping \emph{is} required.) - -\subsection{The CET / energy route (the conservative alternative)} -\label{sec:cet-energy} -\textbf{CET} here means the \emph{Convergent Energy Transformer} of -\citet{hoier2026} --- an energy-based transformer block, trained with EP, that we -reproduced (on masked image completion) as the prior SOTA for ``EP $+$ attention''. -Its trick is to make attention \emph{conservative} so classic EP applies with -\emph{no} correction: attention is folded into a scalar energy -\begin{equation} - E_{\mathrm{att}}(z) \;=\; - -\frac{1}{\gamma}\sum_{\text{heads},\,i} - \LSE_{j}\!\big(\gamma\, q_i\!\cdot\!k_j\big) - \quad(\text{causal-masked}), - \label{eq:cet} -\end{equation} -whose force \emph{ties the value to the key} ($v\!\equiv\!k$), plus a confinement -$\tfrac12 c\|z\|^2$ (because $E_{\mathrm{att}}$ is unbounded below) and a -modern-Hopfield memory energy $E_{\mathrm{mem}}(z)=-\sum\mathrm{relu}(zW_m)^2$ -playing the role of the FFN (its force is a \emph{tied}-weight squared-ReLU MLP). On -this energy $F=-\grad E$ exactly, so classic EP is valid with symmetric Jacobian and -no AEP. In our reproduction EP matched truncated-BPTT (``EP $\approx$ TBPTE'', -gradient cosine $0.99$). The trade-off is expressivity: the tied value and -reciprocal coupling are the least expressive form of attention. Under \emph{exact} -gradients on the LM, this conservative route (and a monotone-DEQ variant -\citep{winston2020}) costs $\approx 0.15$--$0.2$ CE relative to the non-conservative -\texttt{thick} block --- which is precisely why we pay for the AEP machinery and keep -real attention. - -%============================================================================== -\section{Our recipe: tracking-AEP and the stabilization stack} -\label{sec:recipe} - -\subsection{Tracking-AEP: re-linearize at the moving common mode} -\label{sec:tracking} -The AEP correction \eqref{eq:aep} is frozen at $\zstar$. Near a good solution this -becomes the binding error: as the model sharpens, the true gradient shrinks below -the \emph{bias floor} of the frozen linearization, and the highly non-normal block -Jacobian makes that floor large (we measure $\|\Jnc v-\Jnc^{\!\top} v\|/\|\Jnc v\|=1.37$ -at $\zstar$). The fix is to re-linearize the antisymmetric correction not at the -frozen $\zstar$ but at the \emph{instantaneous common mode} of the two nudged -trajectories, -\begin{equation} - \zbar \;=\; \half\big(z_{+}+z_{-}\big), - \qquad - \mathrm{corr}(z) \;=\; \Jnc(\zbar)\,v - \Jnc(\zbar)^{\!\top} v, - \quad v = z-\zbar, - \label{eq:track} -\end{equation} -evaluated step-by-step as $\zbar$ moves with the nudge (run the $+$ and $-$ phases in -lockstep, recompute $\jvp/\vjp$ about the running $\zbar$). This is exact transposed -differential dynamics with no compounding linearization error, and it is loose-tolerant -(it does not demand an ultra-tight free phase). At a plateau checkpoint where the -frozen estimator had collapsed (gradient cosine vs.\ BPTT $-0.045$, batch-to-batch -self-coherence $-0.27$, magnitude ratio $\sim\!4000\times$), tracking-AEP restores -cosine $0.997$, self-coherence $+0.95$, magnitude ratio $0.9$. Tracking-AEP and the -common-mode formulation \eqref{eq:track} are \emph{ours}. - -\subsection{The validity threshold and the residual as the health signal} -\label{sec:stab} -The governing empirical fact is that the EP estimator has a \emph{validity threshold} -in the free-phase relative residual -\begin{equation} - \mathrm{res} \;=\; \frac{\|z^{+}-\zstar\|}{\|\zstar\|} - \qquad(\text{one extra relaxation step}), -\end{equation} -which is the load-bearing health signal (premise~(B)). Gradient cosine vs.\ the exact -reference degrades sharply with res: $\approx 0.85$ at $\mathrm{res}\!\sim\!5\times10^{-5}$, -batch-dependent $0.2$--$0.9$ at $10^{-3}$, and noise at $10^{-2}$. BPTT has no such -threshold (it differentiates the actual finite unroll, converged or not); \emph{this -asymmetry, and nothing deeper, is the EP-specific difficulty}. Accordingly the free -phase is run adaptively: relax to $T_1{=}150$, then continue in chunks until -$\mathrm{res}\!\le\!10^{-4}$ before nudging. We emphasize there is \emph{no} structural -``EP ceiling'': an early ``EP caps at $\sim\!2.5$'' verdict was traced to two -undertrained/invalid-regime runs and retracted. - -\subsection{The stabilization stack} -Training pushes the dynamics off the contractive manifold (premise~(D)) --- and not -only for EP: even \emph{exact} BPTT on this architecture walks off the manifold on -long horizons (residual $\to 4.7\times10^{-2}$, val CE $\to 3.0$). The stack that -keeps the system valid: -\begin{itemize}[leftmargin=1.4em,itemsep=3pt] - \item \textbf{Frozen / controlled Jacobian-norm penalty (\texttt{jacreg}).} A soft - penalty $\lambda\,\|\Jnc(\zstar)\|_F^2$, estimated matrix-free by Hutchinson - (one $\jvp$ on a random probe, differentiated w.r.t.\ $\theta$). This is - \citet{bai2021}'s DEQ-stabilization penalty, \emph{not} ours. It keeps the - free phase contractive and hence the estimator inside its validity region. - A continuous controller drives it, - $\lambda \leftarrow \mathrm{clip}\big(\lambda\,(\mathrm{res}_{\mathrm{EMA}}/\mathrm{target})^{0.3}\big)$, - on an EMA-smoothed residual (the raw residual is noisy and a multiplicative - controller on it random-walks). A key hard lesson: the controller \emph{floor} - is load-bearing and must never anneal to zero --- two independent - $\lambda\!\to\!0$ runs died identically (val CE $60$--$77$, $\mathrm{res}\!\equiv\!0$), - which post-mortem is an \emph{explosion disguised as convergence by - floating-point absorption} ($\varepsilon F<\mathrm{ulp}(z)$ freezes the - relaxation), not a benign dead state. - \item \textbf{Residual, not spectral radius, as the control signal.} The block - Jacobian is highly non-normal, so transient growth is invisible to - eigenvalues (measured $\rho(J){=}0.94$ ``stable'' while the relaxation - diverged to $\mathrm{res}\,0.21$). The one-step residual \emph{is} the - transient; we control on it. - \item \textbf{Validity gate.} When the residual exceeds a gate, the EP update is - mathematically undefined, so we apply only the homeostat (jacreg) and skip the - nudge --- a fast recovery step. At larger scale this gate is load-bearing - (off-equilibrium EP updates poison the weights). - \item \textbf{Adaptive $T_2$ by hindsight snapshot selection.} On slow-mixing - batches a long nudge phase can diverge through non-normal transient growth, - and step-size early-stopping \emph{fails} (the transient triggers it - spuriously). Instead, run to $T_{2\max}$ in lockstep, snapshot the contrast - $a_t$ every few steps, and return the \emph{most settled} snapshot (smallest - increment of $a_t$); judging by increments of the \emph{quantity of interest} - rather than step sizes makes transient growth harmless. This is ours; it - lifts probe cosine from $0.871$ to $0.932$. -\end{itemize} - -\subsection{Ongoing: the residual-defense term (\texttt{resreg}) --- under validation} -\label{sec:resreg} -At larger width ($C{=}512$) we observe a distinct, \emph{still-open} failure that we -call the below-$2.10$ wall: frozen-jacreg, tracking-AEP EP descends to best -$\approx 2.09$ and then bifurcates within $\sim\!200$ steps (residual -$5\!\times\!10^{-3}\!\to\!0.15$, gradient cosine $0.98\!\to\!0$, CE $\to\!4{+}$), -while \emph{exact} BPTT with the identical recipe sails past to $1.72$. The diagnosed -root cause is an \emph{objective mismatch}: EP optimizes the (refined) fixed point and -never defends the finite-step residual that evaluation actually uses, whereas BPTT -differentiates the finite unroll and so implicitly rewards contraction. The diverged -state is a forward bifurcation to a \emph{limit cycle}, so more relaxation steps cannot -fix it; only a residual \emph{cost} can. The proposed fix is an explicit T1-residual -penalty on the \emph{evaluated} state $z_{150}=\mathrm{relax}(\xin,T_1)$ taken before -any refinement, -\begin{equation} - R_{\mathrm{res}} \;=\; \frac{\|\varepsilon F(z_{150})\|^2}{\|z_{150}\|^2+\varepsilon}, - \qquad - \text{gradient w.r.t.\ }\theta\text{ with }z_{150}\text{ detached}, - \label{eq:resreg} -\end{equation} -scaled task-relative and added to the EP gradient (run with the validity gate off, so -the penalty is not bypassed exactly when the residual is high). \textbf{Status: this is -ongoing.} The residual-defense term \eqref{eq:resreg} held the residual pinned at -$1$--$5\times10^{-4}$ and reached best $2.0573$ (past the wall) through only step -$\sim\!1000$ before a storage cleanup deleted the run; full re-validation toward the -$\approx 1.8$ BPTT ceiling is pending. We present it as a diagnosis $+$ proposed fix, -\emph{not} a finished result. (The objective-mismatch diagnosis, the common-mode -tracking estimator, the residual-driven controller and validity gate, and this -residual-defense term are ours.) - -%============================================================================== -\section{Established results (and what is still open)} -\label{sec:results} - -\paragraph{Solidly validated.} -\begin{itemize}[leftmargin=1.4em,itemsep=3pt] - \item \textbf{EP/AEP component gradients match backprop.} On the character LM, - AEP gives causal-attention parameters cosine $0.99$, the (Hopfield) FFN - $1.00$, and the full LM block $0.99$ vs.\ the true backprop gradient - --- versus feedback alignment at $Q/K/V\approx 0.25$, FFN $\approx -0.01$. - On the CET reproduction, global cosine $0.99$ and EP $\approx$ TBPTE on - masked-image completion. - \item \textbf{EP trains the equilibrium transformer stably, without backprop.} - With the stabilization stack, end-to-end EP runs $10\text{k}+$ steps with - zero non-finite steps. - \item \textbf{It matches/beats a BP transformer at equal parameters.} On - Shakespeare character-LM (single block, $C{=}128$), at a fully controlled - $14$k-step comparison (Table~\ref{tab:results}): EP reaches val CE - \textbf{1.676} (multi-seed $1.680\pm0.005$, $3$ seeds); the like-for-like - standard BP transformer (matched in parameter \emph{shape} to the thick - block) reaches $1.610$; EP \emph{beats} the thinner BP baseline ($1.689$). - The total gap of $0.066$ decomposes into an architecture tax $\approx 0.025$ - (BPTT on the identical block $1.635$) and an EP-rule tax $\approx 0.041\pm0.005$ - --- real, tightly reproducible, and consistent with the measured estimator - misalignment (cosine $0.85$--$0.93$). -\end{itemize} - -\begin{table}[t] - \centering - \small - \begin{tabular}{llc} - \toprule - \textbf{training rule} & \textbf{architecture / recipe} & \textbf{best val CE}\\ - \midrule - BP & standard transformer (like-for-like for \texttt{thick}) & \textbf{1.610}\\ - BPTT $+$ $\lambda$-controller $+$ param-EMA & \texttt{thick} (exact grad, same stabilizer) & 1.635\\ - \textbf{EP} & \texttt{thick}; tracking-AEP $+$ adaptive $T_1/T_2$ & \textbf{1.676}\\ - BP & standard transformer (thin-matched) & 1.689\\ - BPTT (exact grad) & \texttt{thick}, unregularized & 2.021 (destabilizes late)\\ - random & --- & 4.174\\ - \bottomrule - \end{tabular} - \caption{Fully-controlled $14$k-step comparison on Shakespeare char-LM - (random $=\ln 65$). EP matches the architecture-controlled exact-gradient - run to within $0.041$ and beats the thin-matched BP baseline. ``BPTT as - ablation'' separates the training-rule cost (EP$-$BPTT) from the - architecture cost (BPTT$-$BP).} - \label{tab:results} -\end{table} - -\paragraph{Honest framing of the controlled comparison.} -EP beats \emph{bare} BPTT, but the controlled table shows most of that win is EP's -\emph{mandatory} stabilization loop doubling as regularization: bare exact-gradient -training walks off the contractive manifold at $14$k, and the same controller that EP -cannot live without also lifts BPTT to $1.635$. The contraction controller is good for -the equilibrium architecture regardless of training rule; EP merely forced its -discovery. - -\paragraph{Ongoing / under validation.} -The $C{=}512$ work is \emph{not} a finished result. (i) The $2.40$ plateau there is -diagnosed as a late-training EP estimator bias-floor / batch-incoherence, which -tracking-AEP breaks in training ($2.40\!\to\!2.16$, still descending in a $2500$-step -warm-start test). (ii) The below-$2.10$ wall is diagnosed as the objective mismatch of -\S\ref{sec:resreg}; the residual-defense term \eqref{eq:resreg} validated res-tight and -past the wall (best $2.0573$) \emph{only through step $\sim\!1000$} before the run was -lost, and a full re-run toward the $\approx 1.8$ BPTT ceiling is pending. These should -be read as diagnoses with promising partial evidence, not as established numbers. - -%============================================================================== -\section*{Attribution summary} -\addcontentsline{toc}{section}{Attribution summary} - -\begin{description}[leftmargin=2.2em,itemsep=2pt] - \item[Theirs.] Classic energy-based EP and centered nudging - \citep{scellier2017,laborieux2021}; EP $\equiv$ BPTT in the converged, $\beta\!\to\!0$ - limit \citep{ernoult2019}; holomorphic EP \citep{laborieux2022}; the asymmetric/AEP - correction $J\!\to\!J^{\!\top}$ \emph{and} the force-form VF readout - \citep{scurria2026}; the Jacobian-norm penalty \citep{bai2021}; DEQ - \citep{bai2019} and monotone DEQ \citep{winston2020}; the Convergent Energy - Transformer / CET \citep{hoier2026}. - \item[Ours.] The transformer application of the force route and the damping recipe - (damping $+$ AEP making real attention EP-trainable at any gain); the matrix-free - $\jvp/\vjp$ form of the correction at transformer scale and its combination with - holomorphic estimation and softmax attention; \emph{tracking-AEP} (common-mode - re-linearization, Eq.~\ref{eq:track}); the residual-driven controller, the validity - gate, and adaptive-$T_2$ snapshot selection; and the (ongoing) residual-defense term - \texttt{resreg} (Eq.~\ref{eq:resreg}) with its objective-mismatch diagnosis. -\end{description} - -%============================================================================== -\begin{thebibliography}{9} -\bibitem[Bai et al., 2019]{bai2019} - S.~Bai, J.~Z.~Kolter, V.~Koltun. - \emph{Deep Equilibrium Models}. 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