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<h1 class="title is-1 publication-title">Training Transformer Language Models Without Backpropagation</h1>
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<span class="author-block">Yuren Hao<sup>1</sup>,</span>
<span class="author-block">Xiang Wan<sup>2</sup>,</span>
<span class="author-block">Alexi Gladstone<sup>1</sup>,</span>
<span class="author-block">Zeyi Liu<sup>1</sup>,</span>
<span class="author-block">ChengXiang Zhai<sup>1</sup></span>
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<span class="author-block"><sup>1</sup>University of Illinois Urbana-Champaign · <sup>2</sup>Stanford University<br>2026</span>
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<li><b>The largest models trained from scratch without backpropagation at any level.</b>
Every parameter update follows the Equilibrium Propagation (EP) rule: no layer, block, or
output head is trained with a backprop rule. The models are standard transformer LMs
(OLMo2-style blocks, 32k vocabulary, FineWeb-Edu); the trained network runs ordinary forward inference.</li>
<li><b>Matches backprop within a 4-5% perplexity band at 72M parameters</b>, against a backprop
twin trained on identical data, steps, and optimizer: with multi-seed controls on both sides.</li>
<li><b>First transformer language model in the EP family, at 2.15× the size of the largest prior
EP-family result</b> (VGG10 ImageNet classifier, ~63M params): models trained up to <b>135M</b>
parameters on 2.7B tokens.</li>
<li><b>Over 5× cheaper per training step than the closest EP-family work:</b> one nudged phase
with 3 relaxation sweeps per step, versus two phases with 10 iterations each (20 total) in the
prior state of the art.</li>
<li><b>Verified gradient fidelity:</b> the EP update maintains cosine ≈0.99 to the true backprop
gradient throughout training: measured, at scale. Larger non-backprop transformers in the
literature train most parameters with <i>local backprop inside blocks</i>; ours use none.</li>
<li><b>Hardware-ready by measurement, not assumption:</b> 8-bit quantization shows no EP-specific
penalty against an equally-quantized backprop twin; under injected analog faults (1% forward noise,
10% error-channel noise, device-tolerance mismatch) the EP estimator tracks the faulted network at
cosine ≈0.97: learning co-adapts to the hardware.</li>
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Analog and physical accelerators promise order-of-magnitude energy savings for training, but they
cannot run backpropagation natively: exact gradients on a physical substrate require per-step
digitization or an adjoint copy of the hardware. Equilibrium Propagation extracts gradients from the
physics itself: two relaxations and local reads: and is the only member of its family with a
gradient-equivalence guarantee that we verify directly at scale. What the field has lacked is scale and
rigor: EP results stopped at mid-size vision models, without matched controls. We train standard
transformer language models end-to-end with EP: 72M parameters within 4-5% perplexity of a
matched backprop twin, models up to 135M, with backprop twins,
multi-seed discipline, and hardware-relevant ablations (quantization, analog faults, nudge operating
windows, energy accounting) at every stage. Scaling a physical learning rule also surfaces new science:
we identified a width-scaling loss in the EP gradient invisible to per-step alignment metrics, built an
instrument that measures it in 90 minutes per candidate recipe, mapped its dose-response law, and
demonstrated an estimator-side treatment that recovers 97% of it without touching the model or the
cost budget.
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<thead><tr><th>Model</th><th>Data</th><th>EP (val CE)</th><th>BP twin</th><th>Gap</th></tr></thead>
<tbody>
<tr><td>72M transformer LM</td><td>FineWeb-Edu, 1.4B tok</td><td>3.33</td><td>3.29</td><td>+4-5% ppl</td></tr>
<tr><td>135M transformer LM</td><td>FineWeb-Edu, 2.7B tok</td><td colspan="3">trained end-to-end, zero instability
events; scaling analysis below</td></tr>
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<p class="is-size-6 has-text-grey">Twin discipline: identical architecture, tokenizer, data order,
optimizer, steps, and evaluation; multi-seed on both sides (BP n=3, band ±0.006; EP n=2).</p>
<table class="table is-fullwidth">
<thead><tr><th>Training cost, EP family</th><th>This work</th><th>Closest EP work (VGG10, ImageNet)</th></tr></thead>
<tbody>
<tr><td>Nudged phases per step</td><td><b>1</b></td><td>2</td></tr>
<tr><td>Relaxation iterations per step</td><td><b>3</b></td><td>20</td></tr>
<tr><td>Model class and scale</td><td><b>transformer LM, 72M and 135M</b></td><td>convolutional classifier, ~63M</td></tr>
</tbody>
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<h2 class="title is-3">The scaling science</h2>
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<p>Scaling a physical learning rule surfaces phenomena backprop never meets. Between widths 512 and 768
we identified a width-scaling loss in the EP gradient: localized to the top half of the network,
invisible to every per-step alignment metric, and traced to response components that finite nudge
displacement under-reaches. We built a screening instrument that measures this leak in 90 minutes per
candidate recipe, mapped its dose-response law (logarithmic across two decades of displacement
amplification), and demonstrated a pure estimator-side treatment that closes 97% of it: no change to
the model, its inference path, or the cost budget. The same instruments provide the go/no-go protocol
for each next rung of the ladder.</p>
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<h2 class="title is-3">Hardware line</h2>
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<ul>
<li>Measured energy projection for an integrated weight-stationary realization:
<b>0.21-0.63 pJ/MAC</b> (SPICE-measured analog core + datasheet periphery), against a
0.3-1 pJ/MAC digital INT8 system envelope.</li>
<li>Single-column analog prototype: SPICE-modeled, discrete multiplying-DAC parts list: kept at the
“hardware someone can actually build” level.</li>
<li>Nudge-amplitude operating windows and their evolution over training are mapped: the
dynamic-range spec an analog implementation must meet.</li>
</ul>
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<h2 class="title is-3">Roadmap</h2>
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<p>Staged scaling with matched BP controls and hardware-relevant ablations at every rung: a 150M-600M
ladder (does the gap grow or shrink with scale: measured, not assumed), then 1B-3B; each stage
gated on the previous stage’s loss, alignment, and throughput numbers. In parallel: the
algorithm→regime map across the activity-difference family (contrastive / coupled-learning arms on
the same harness), and a bounded single-column hardware feasibility study.</p>
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<pre id="bibtex-code"><code>@misc{ept2026,
title={Training Transformer Language Models Without Backpropagation},
author={Hao, Yuren and Wan, Xiang and Gladstone, Alexi and Liu, Zeyi and Zhai, ChengXiang},
year={2026},
note={Project page}
}</code></pre>
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