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<meta name="title" content="SPHERE: Mitigating the Loss of Spectral Plasticity in Mixture-of-Experts for Deep Reinforcement Learning" />
<meta name="keywords" content="SPHERE, ICML 2026, continual reinforcement learning, mixture-of-experts, spectral plasticity, HumanoidBench, MetaWorld" />
<meta name="author" content="Lirui Luo, Guoxi Zhang, Hongming Xu, Cong Fang, Qing Li" />
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<title>SPHERE: Mitigating the Loss of Spectral Plasticity in Mixture-of-Experts for Deep Reinforcement Learning</title>
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<meta name="citation_title" content="SPHERE: Mitigating the Loss of Spectral Plasticity in Mixture-of-Experts for Deep Reinforcement Learning" />
<meta name="citation_author" content="Luo, Lirui" />
<meta name="citation_author" content="Zhang, Guoxi" />
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<meta name="citation_author" content="Fang, Cong" />
<meta name="citation_author" content="Li, Qing" />
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<span class="nav-brand">SPHERE</span>
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<div class="d-flex align-items-center gap-1 flex-wrap">
<a class="nav-pill" href="#tldr">TL;DR</a>
<a class="nav-pill" href="#interactive">Mechanism</a>
<a class="nav-pill" href="#figures">Results</a>
<a class="nav-pill" href="#bibtex">BibTeX</a>
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<header class="py-4">
<div class="hero-card">
<div class="hero-grid">
<div class="hero-title">
<h1 class="display-title">
SPHERE: Mitigating the Loss of Spectral Plasticity in Mixture-of-Experts for Deep Reinforcement Learning
</h1>
<div class="hero-authors">
<div class="authors">
<span><a href="https://liruiluo.github.io/" target="_blank" rel="noopener noreferrer">Lirui Luo</a></span>
<span>·</span>
<span><a href="https://guoxizhang.com/" target="_blank" rel="noopener noreferrer">Guoxi Zhang</a></span>
<span>·</span>
<span><a href="https://sbx126.github.io/" target="_blank" rel="noopener noreferrer">Hongming Xu</a></span>
<span>·</span>
<span><a href="https://congfang-ml.github.io/" target="_blank" rel="noopener noreferrer">Cong Fang</a></span>
<span>·</span>
<span><a href="https://liqing.io" target="_blank" rel="noopener noreferrer">Qing Li</a></span>
</div>
<div class="affiliations">
<span>Peking University</span>
<span>·</span>
<span>BIGAI</span>
</div>
</div>
<div class="venue-line" aria-label="Accepted paper at ICML 2026">
<span>Accepted to <strong>ICML 2026</strong></span>
</div>
<div class="hero-links">
<a class="resource-chip" href="https://arxiv.org/abs/2605.04712" target="_blank" rel="noopener noreferrer" aria-label="SPHERE paper on arXiv">
<span class="resource-icon" aria-hidden="true">arXiv</span>
<span>Paper</span>
</a>
<a class="resource-chip primary" href="https://github.com/sphere-rl/sphere" target="_blank" rel="noopener noreferrer" aria-label="SPHERE GitHub code repository">
<span class="resource-icon" aria-hidden="true"></></span>
<span>Code</span>
</a>
<a class="resource-chip" href="https://docs.google.com/presentation/d/1Ji2C37MLRT6mKIrKlSkV_K9kmAVCUhNvGgtvdEIdaXw/edit?usp=sharing" target="_blank" rel="noopener noreferrer" aria-label="SPHERE Google Slides presentation">
<span class="resource-icon" aria-hidden="true">PPT</span>
<span>Slides</span>
</a>
<a class="resource-chip" href="static/sphere-icml2026-poster.pdf" target="_blank" rel="noopener noreferrer" aria-label="SPHERE ICML 2026 poster PDF">
<span class="resource-icon" aria-hidden="true">POST</span>
<span>Poster</span>
</a>
</div>
</div>
</div>
</div>
</header>
<main id="content" class="pb-5">
<section class="section-card tldr-card" id="tldr">
<div class="section-heading-row">
<div>
<h2 class="section-title">TL;DR</h2>
<p class="section-text mb-0">
SPHERE studies why Mixture-of-Experts (MoE) policies lose their ability to adapt in continual RL. It shows that learning updates collapse into too few directions, then keeps expert features diverse so later tasks remain learnable.
</p>
</div>
<span class="pill-hint">Phenomenon → diagnosis → regularization</span>
</div>
<div class="tldr-grid">
<div class="tldr-item">
<span class="tldr-k">Problem</span>
<span class="tldr-v">A policy trained across many tasks can stop adapting to later tasks.</span>
</div>
<div class="tldr-item">
<span class="tldr-k">Method</span>
<span class="tldr-v">A regularizer that keeps expert features diverse instead of collapsed.</span>
</div>
<div class="tldr-item">
<span class="tldr-k">Result</span>
<span class="tldr-v">Better continual-control performance and healthier feature geometry across tasks.</span>
</div>
</div>
</section>
<section class="section-card interactive-card" id="interactive">
<div class="section-heading-row mechanism-heading">
<div>
<h2 class="section-title">Mechanism: Why Policies Stop Learning in Continual RL</h2>
<p class="section-text mb-0">
The mechanism story starts with the observed learning slowdown, connects it to collapsed update directions, and then shows how SPHERE keeps those directions more diverse.
</p>
</div>
<span class="pill-hint">Phenomenon → spectral collapse → SPHERE</span>
</div>
<div class="figure-card lead-figure mb-3">
<div class="figure-lead">
<div class="figure-kv">
<div class="kv">
<span class="kv-k">Phenomenon</span>
<span class="kv-v">In continual RL, policies keep receiving new experience but can stop changing effectively.</span>
</div>
<div class="kv">
<span class="kv-k">Diagnosis</span>
<span class="kv-v">The update geometry loses rank, meaning learning concentrates into too few functional directions.</span>
</div>
<div class="kv">
<span class="kv-k">SPHERE</span>
<span class="kv-v">SPHERE keeps the weighted expert-feature Gram more isotropic, which helps preserve diverse update directions.</span>
</div>
</div>
</div>
<picture>
<source srcset="figures_sphere/teaser.webp" type="image/webp" />
<img class="img-fluid rounded" src="figures_sphere/teaser.png" width="1737" height="1145" alt="Overview showing continual-RL plasticity loss, spectral collapse, and SPHERE regularization" loading="lazy" decoding="async" />
</picture>
</div>
<div class="d-flex align-items-center justify-content-between flex-wrap gap-2">
<h3 class="viz-subsection-title mb-0">Update Geometry: Collapse vs. SPHERE</h3>
<div class="pill-hint">Use the slider · or press Play</div>
</div>
<p class="section-text">
The visualization makes the diagnosis concrete. A unit sphere of possible gradient directions becomes an ellipsoid after multiplication by the
empirical neural tangent kernel (eNTK) matrix <span class="math-inline-soft">K</span>. When <span class="math-inline-soft">K</span> becomes low-rank, one axis shrinks toward zero and the ellipsoid degenerates
toward a near-plane or line; SPHERE keeps the spectrum more isotropic.
</p>
<p class="section-text">
Top row (<span class="math-inline-soft">∇fL</span>) shows the input sphere of directions; bottom row
(<span class="math-inline-soft">K∇fL</span>) shows the stretching process.
</p>
<div class="viz-data-note" aria-label="Interactive visualization data source">
<div class="viz-data-note-k">Real-data interactive panel</div>
<div class="viz-data-note-v">
This interactive animation is generated from the underlying HumanoidBench experiment data used for Fig. 1. For each task, the
top-3 eigenvalues of <span class="math-inline-soft">K</span> shape the ellipsoid, and gradient-direction samples
are projected into the same 3D eigenspace so the slider shows how the measured update geometry evolves.
</div>
</div>
<div class="viz-card" id="vizRoot">
<div class="viz-toolbar">
<div class="control control-grow">
<div class="control-label">Task</div>
<div class="task-controls">
<input id="taskSlider" class="form-range task-slider" type="range" min="0" max="5" value="0" step="1" aria-label="Linear task selector" />
<button id="playBtn" class="btn btn-sm btn-primary control-primary" type="button" aria-pressed="false">Play</button>
<button id="resetBtn" class="btn btn-sm btn-outline-secondary control-secondary" type="button">Reset</button>
</div>
<div class="task-readout">
<span class="task-badge" id="taskIdxBadge">Task 5</span>
<span class="task-name" id="taskName">Run</span>
</div>
</div>
<div class="control">
<div class="control-label">Mesh</div>
<div class="form-check form-switch m-0">
<input class="form-check-input" type="checkbox" role="switch" id="meshToggle" aria-label="Show mesh surface" checked />
</div>
</div>
</div>
<div class="viz-body">
<div class="viz-canvas">
<p id="vizDescription" class="sr-only">
Interactive 3D visualization comparing baseline Top-K MoE and SPHERE update geometry across HumanoidBench tasks.
</p>
<canvas id="sphereCanvas" role="img" aria-describedby="vizDescription"></canvas>
<div class="viz-overlay">
<div class="viz-label left">
<div class="tag">Baseline (Top‑K MoE)</div>
<div class="sub">collapsed spectrum → near low-rank</div>
</div>
<div class="viz-label right">
<div class="tag sphere">SPHERE</div>
<div class="sub">isotropic spectrum → diverse updates</div>
</div>
<div class="viz-stage-labels" aria-hidden="true">
<div class="stage s1"><span class="eq">∇</span><span class="txt">f</span><span class="txt">L</span></div>
<div class="stage s2"><span class="txt">K</span><span class="eq">∇</span><span class="txt">f</span><span class="txt">L</span></div>
</div>
</div>
<div class="viz-fallback" id="vizFallback" hidden>
WebGL unavailable. Please use a WebGL-capable browser to see the 3D visualization.
</div>
</div>
</div>
</div>
</section>
<section class="section-card figures-card" id="figures">
<div class="section-heading-row figures-heading">
<div>
<h2 class="section-title">Experiments & Analysis</h2>
<p class="section-text mb-0">These panels show where continual training fails, how the update geometry collapses, how SPHERE improves performance on two control benchmarks, and which design choices matter.</p>
</div>
<span class="pill-hint">Phenomenon · Diagnosis · Performance · Design · Feature Evidence</span>
</div>
<div class="figure-story">
<article class="story-step">
<div class="story-step-index">Phenomenon</div>
<div class="story-step-body">
<div class="subsection-title">Continual RL Degrades Across Architectures</div>
<div class="figure-card">
<img
class="img-fluid rounded"
src="figures_sphere/humanoidbench_archs_success_rl_vs_crl_bars_relu.png" width="2348" height="1013" loading="lazy" decoding="async"
alt="HumanoidBench success: RL vs CRL"
/>
<div class="figure-caption">
Before introducing SPHERE, the same HumanoidBench architectures succeed less when trained continually than when trained task by task.
</div>
</div>
</div>
</article>
<article class="story-step">
<div class="story-step-index">Diagnosis</div>
<div class="story-step-body">
<div class="subsection-title">Spectral Plasticity Collapses During CRL</div>
<div class="figure-card">
<img class="img-fluid rounded" src="figures_sphere/humanoidbench_entk_erank_combined.png" width="2069" height="1103" loading="lazy" decoding="async" alt="Effective rank during training" />
<div class="figure-caption">The performance drop comes with lower eNTK effective rank: baseline updates collapse, while SPHERE keeps the update geometry better conditioned.</div>
</div>
</div>
</article>
<article class="story-step">
<div class="story-step-index">Performance</div>
<div class="story-step-body">
<div class="subsection-title">SPHERE Improves Continual Training Performance</div>
<div class="performance-stack">
<div class="figure-card performance-figure">
<div class="figure-title">MetaWorld</div>
<img
class="img-fluid rounded"
src="figures_sphere/metaworld_methods_success_rl_vs_crl_bars_trim.png" width="3120" height="1011" loading="lazy" decoding="async"
alt="MetaWorld success rate across methods under RL and CRL"
/>
<div class="figure-caption">
On MetaWorld, SPHERE narrows the gap between task-by-task training and continual training, giving the strongest average success among the compared methods.
</div>
</div>
<div class="figure-card performance-figure">
<div class="figure-title">HumanoidBench</div>
<img
class="img-fluid rounded"
src="figures_sphere/humanoidbench_methods_success_rl_vs_crl_bars_relu_trim.png" width="3109" height="1004" loading="lazy" decoding="async"
alt="HumanoidBench success rate across methods under RL and CRL"
/>
<div class="figure-caption">
On HumanoidBench, the same pattern holds: SPHERE improves average success over the unregularized MoE and continual-learning baselines.
</div>
</div>
</div>
</div>
</article>
<article class="story-step">
<div class="story-step-index">Design</div>
<div class="story-step-body">
<div class="subsection-title">What Matters in SPHERE?</div>
<div class="ablation-stack">
<div class="figure-card ablation-table-card">
<div class="figure-title">HumanoidBench continual-RL design ablation</div>
<table class="ablation-table" aria-label="SPHERE design ablation on HumanoidBench under continual reinforcement learning">
<thead>
<tr>
<th>Variant</th>
<th>Average success</th>
</tr>
</thead>
<tbody>
<tr>
<td>Without SPHERE</td>
<td>0.36 ± 0.08</td>
</tr>
<tr class="best-row">
<td>With SPHERE</td>
<td>0.54 ± 0.12</td>
</tr>
<tr>
<td>All hidden expert layers</td>
<td>0.42 ± 0.07</td>
</tr>
<tr>
<td>Per-expert loss sum</td>
<td>0.40 ± 0.08</td>
</tr>
<tr>
<td>Gradient-factor regularization</td>
<td>0.43 ± 0.09</td>
</tr>
</tbody>
</table>
<div class="figure-caption">
The ablation asks which SPHERE design choices matter. The default last-layer, routing-weighted expert-feature Gram is the strongest tested design; alternatives help, but none matches the full setup.
</div>
</div>
</div>
</div>
</article>
<article class="story-step">
<div class="story-step-index">Qualitative</div>
<div class="story-step-body">
<div class="subsection-title">Feature Collapse Across Tasks</div>
<div class="figure-card">
<picture>
<source srcset="figures_sphere/case.webp" type="image/webp" />
<img class="img-fluid rounded" src="figures_sphere/case.png" width="2243" height="618" alt="Qualitative analysis: collapse vs SPHERE" loading="lazy" decoding="async" />
</picture>
<div class="figure-caption">
Using the same held-out Stair states after each task, we visualize expert features. Without SPHERE, points quickly concentrate along one dominant direction; with SPHERE, multiple directions remain active across the sequence.
</div>
</div>
</div>
</article>
<article class="story-step">
<div class="story-step-index">Feature Evidence</div>
<div class="story-step-body">
<div class="subsection-title">Weighted Expert Features Track the eNTK Rank</div>
<div class="figure-card compact-figure">
<img
class="img-fluid rounded"
src="figures_sphere/reA_reK_batch_scatter_h1_run.png" width="748" height="682" loading="lazy" decoding="async"
alt="Scatter: effective rank of weighted expert feature Gram vs effective rank of eNTK"
/>
<div class="figure-caption">
The weighted expert-feature Gram follows the same trend as the eNTK effective rank, supporting it as a practical proxy for spectral plasticity.
</div>
</div>
</div>
</article>
</div>
</section>
<section class="section-card bibtex-card" id="bibtex">
<h2 class="section-title">BibTeX</h2>
<pre class="bibtex"><code>@inproceedings{luo2026sphere,
title = {SPHERE: Mitigating the Loss of Spectral Plasticity in Mixture-of-Experts for Deep Reinforcement Learning},
author = {Luo, Lirui and Zhang, Guoxi and Xu, Hongming and Fang, Cong and Li, Qing},
booktitle = {Proceedings of the 43rd International Conference on Machine Learning},
year = {2026}
}</code></pre>
</section>
</main>
</div>
</body>
</html>