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<h1 class="title is-1 publication-title">Learning-to-Optimize via Deep Unfolded Flows</h1>
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<a href="https://asaravanos.github.io/">Augustinos D. Saravanos</a><sup>1</sup>,</span>
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<a href="https://oswinso.xyz/">Oswin So</a><sup>1</sup>,</span>
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<a href="https://sabbirahmad26.github.io/">H. M. Sabbir Ahmad</a><sup>2</sup>,</span>
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<a href="http://chuchu.mit.edu/">Chuchu Fan</a><sup>1</sup></span>
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<span class="author-block"><sup>1</sup>Massachusetts Institute of Technology     <sup>2</sup>Boston University</span>
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<p>
We introduce <b>FlowOptimizer</b>, a deep unfolded, flow-based framework for learned iterative optimization. Motivated by the expressiveness of flow models, we represent each optimization iteration via a velocity field that operates on a population of candidate solutions, i.e., a set of parallel iterates, conditioned on contextual information including their objective values and gradients, as well as population-level statistics. The velocity field is initially trained in a simulation-free manner by matching displacements from source populations to improved target ones obtained through sampling the objective. Subsequently, we unfold this velocity field as the internal iteration of an optimization sequence, and fine-tune it in an end-to-end manner by directly optimizing objective values over a targeted class of problems. Notably, FlowOptimizer is a self-supervised framework whose training relies solely on objective evaluations without requiring knowledge of solutions. We evaluate our approach on a series of tasks from standard non-convex optimization benchmarks to real-world problems from supply chain, robotics and power grid applications. FlowOptimizer consistently outperforms well-established sampling-based/gradient-based traditional optimization and learning-to-optimize methods by orders of magnitude in terms of solution quality. We further highlight its ability to be trained on low-dimensional problems and successfully generalize to substantially higher-dimensional (×10) ones.
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<!-- Section 1: Conceptual Overview (paper Fig. 1)
<h2 class="title is-3">Bridging optimization and generative modeling</h2>
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<img src="figs/overview.jpg"
class="center-image"
style="padding: 0px 50px 0px 0px; max-width: 50%;"
alt="Conceptual overview of FlowOptimizer"/>
<p>
<b>FlowOptimizer</b> learns to optimize by treating each iteration as a learned <b>transport of a population</b> of candidate solutions, driven by a flow-based velocity field.
<br><br>
From an <b>optimization</b> perspective, it bridges <b>gradient-based</b> methods (which exploit local derivative information) and <b>sampling-based</b> methods (which explore robustly but scale poorly), combining their complementary strengths in a single learnable framework.
<br><br>
From a <b>machine learning</b> perspective, it lies at the intersection of <b>learning-to-optimize</b> and <b>generative modeling</b> — leveraging the expressiveness of flow models to capture population-level optimization dynamics.
</p>
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<!-- Section 2: Core Iteration (paper Fig. 2, Eqs. 2-3) -->
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<h2 class="title is-3">One iteration = a learned population update</h2>
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<img src="figs/core_iteration.jpg"
class="center-image"
style="padding: 0px 50px 0px 0px; max-width: 50%;"
alt="FlowOptimizer core iteration"/>
<p>
At each iteration \(k\), FlowOptimizer transforms a <b>population</b> of candidate solutions \(\mathcal{X}^k=\{x^k_j\}_{j=1}^P\) into an improved one, conditioned on contextual information \(c^k\) (objective values, gradients, and population-level statistics):
<br><br>
<center>\(\mathcal{X}^{k+1} = \mathcal{T}(\mathcal{X}^k, c^k)\)</center>
<br>
We parameterize this transformation as the flow induced by a learnable, context-conditioned <b>velocity field</b> \(v_\theta\) acting jointly on the whole population:
<br><br>
<center>\(\dot{z}_t = v_\theta(z_t, t, c^k), \qquad z_{t=0}=\mathcal{X}^k\)</center>
<br>
The update is obtained by integrating the ODE and setting \(\mathcal{X}^{k+1}=z_{t=1}\). A <b>self-attention</b> architecture makes \(v_\theta\) permutation-equivariant over the population, capturing inter-member interactions. An optional base sampler can be applied first to stabilize exploration.
</p>
</div>
</div> -->
<!-- Section 3: Deep Unfolding (paper Fig. 3, Eqs. 12-13) -->
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<h2 class="title is-3">Unfolding the flow into a deep optimizer</h2>
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<img src="figs/unfolded_architecture.jpg"
class="center-image"
width="90%"
alt="deep unfolded FlowOptimizer architecture"/>
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<p>
We unroll \(K\) FlowOptimizer iterations as sequential layers of a deep network, yielding the full <b>deep unfolded</b> architecture trained end-to-end. Training proceeds in two phases:
<br><br>
<b>1. Flow-matching pre-training.</b> A single velocity field is trained in a <b>simulation-free</b> manner to match displacements from source populations to improved target populations obtained by sampling the objective — no ground-truth solutions required.
<br><br>
<b>2. End-to-end fine-tuning.</b> The unfolded layers are jointly optimized by directly minimizing the objective values of the population iterates. The per-iteration loss balances the <i>best</i> and <i>mean</i> objective in the population,
<br><br>
<center>\(\ell_k(\mathcal{X}^k) = \alpha\,\mathrm{Best}(f(\mathcal{X}^k)) + (1-\alpha)\,\mathrm{Mean}(f(\mathcal{X}^k))\)</center>
<br>
and the total training loss aggregates these across all unfolded iterations with weights \(w_k\):
<br><br>
<center>\(\mathcal{L}_{\mathrm{FT}} = \sum_{k=1}^{K} w_k\,\ell_k(\mathcal{X}^k)\)</center>
<br>
This makes FlowOptimizer a fully <b>self-supervised</b> optimizer that learns purely from objective evaluations.
</p> -->
<!-- Results 1: Standard Benchmarks (paper Fig. 4) -->
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<h2 class="title is-3">Results: standard optimization benchmarks</h2>
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class="center-image"
width="90%"
alt="standard optimization benchmark results"/>
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<b>Highly non-convex benchmarks (Ackley, Rastrigin, Levy, \(n=30\)).</b> FlowOptimizer reaches significantly lower objective error in shorter wall-clock time, with reduced variance across runs, while gradient-based and L2O baselines stagnate in poor local minima and sampling-based methods converge slowly.
</p>
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<!-- Results 2: Real-World Problems (paper Table 1) -->
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<h2 class="title is-3">Results: real-world optimization problems</h2>
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<p>
Across robotics, power grid, and supply chain problems, FlowOptimizer consistently achieves the lowest error, with the performance gap <b>widening as the problem dimension grows</b>. The table reports the average ratio \(\mathrm{Best}(f(\mathcal{X}^k))/f(\mu^0)\) (lower is better); the best result per row is in <b>bold</b>.
</p>
<table class="table is-bordered is-striped is-narrow is-hoverable is-fullwidth has-text-centered"> --> -->
<!-- <thead>
<tr>
<th>Problem Class</th>
<th>Dim. \(n\)</th>
<th>RS</th>
<th>CEM</th>
<th>CMA-ES</th>
<th>MS-GD</th>
<th>MS-NAG</th>
<th>L2O-GD</th>
<th>FlowOpt (ours)</th>
</tr>
</thead>
<tbody>
<tr>
<td rowspan="2">Robotic Arm</td>
<td>10</td><td>0.18</td><td>0.09</td><td>0.008</td><td>0.014</td><td>0.021</td><td>0.012</td><td><b>0.003</b></td>
</tr>
<tr>
<td>30</td><td>0.31</td><td>0.18</td><td>0.04</td><td>0.18</td><td>0.17</td><td>0.12</td><td><b>0.007</b></td>
</tr>
<tr>
<td rowspan="2">Power Grid</td>
<td>20</td><td>0.53</td><td>0.14</td><td>0.03</td><td>0.05</td><td>0.03</td><td>0.03</td><td><b>5e-4</b></td>
</tr>
<tr>
<td>50</td><td>0.83</td><td>0.31</td><td>0.11</td><td>0.23</td><td>0.17</td><td>0.08</td><td><b>1.2e-3</b></td>
</tr>
<tr>
<td rowspan="2">Supply Chain</td>
<td>40</td><td>0.47</td><td>0.09</td><td>0.05</td><td>0.07</td><td>0.03</td><td>0.013</td><td><b>0.007</b></td>
</tr>
<tr>
<td>80</td><td>0.68</td><td>0.14</td><td>0.06</td><td>0.09</td><td>0.03</td><td>0.027</td><td><b>0.013</b></td>
</tr>
</tbody>
</table>
</div>
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<!-- Results 3: Generalization (paper Fig. 5) -->
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<h2 class="title is-3">Results: generalization to higher dimensions</h2>
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<p align="center">
<b>Trained on 20D, deployed on 200D (\(\times 10\)).</b> The dimension-agnostic variant of FlowOptimizer retains strong performance despite the tenfold increase in dimensionality, while gradient-based, sampling-based, and L2O baselines degrade severely under the same wall-clock budget.
</p>
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<h2 class="title is-3">Related Work</h2>
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<p>
This work is built on our previous work <a href="https://mit-realm.github.io/gcbfplus/">GCBF+</a> and <a href="https://mit-realm.github.io/gcbf-website/">GCBFv0</a>, which eliminates the requirement of a performant nominal policy and the knowledge of dynamics.
<br>
<br>
For the multi-agent constrained optimal control problem, we can also directly solve it without a CBF (better sampling efficiency but less robustness) using Distributed Epigraph Form MARL (<a href="https://mit-realm.github.io/def-marl/">Def-MARL</a>).
<br>
<br>
For a survey of the field of learning safe control for multi-robot systems, see <a rel="survey" href="https://arxiv.org/pdf/2311.13714.pdf">this paper</a>.
</p>
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</section> --> -->
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<pre><code>Citation coming soon.</code></pre>
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