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<header id="title-block-header">
<h1 class="title">Discovering Graph Algorithms Using Transformers</h1>
</header>
<!-- Guidelines: https://www.dropbox.com/scl/fi/bet8enscln8ue36kd8t17/final_project_guidelines.pdf?rlkey=knd19cnumk51ho1y9crno56ib&e=2&dl=0 -->
<div style="text-align:center">
<p>Anthony Wang, Alek Westover, Kevin Zhao</p>
<p>{xy,alekw,kevinmz}@mit.edu</p>
</div>
<h2 id="motivation">Motivation</h2>
<p>Transformers–the architecture that powers LLMs–can do incredible
feats: trained on hundreds of gigabytes of raw text, they can learn to
hold natural conversations, reason about the physical world, and write
code. Skeptics argue that LLMs are simply memorizing their datasets
without gaining any deeper understanding. For instance, GPT’s o1 model,
achieving 90th percentile on Codeforces, struggles with simple but
bizarre algorithms problems such as “find the subarray of a 2D array
with the minimum average”. In this project, we hope to explore
<strong>when off-distribution generalization happens in a
transformer</strong>. Paul Christiano proposed an experiment <a
href="https://www.alignmentforum.org/posts/BxersHYN2qcFoonwg/experimentally-evaluating-whether-honesty-generalizes?commentId=dsDA2BWpHPdgLvaXX">here</a>
about shortest paths in a graph to investigate this, so we decided to
become the first to implement his experiment and put transformers’
generalization abilities to the test.</p>
<p>LLMs are notorious for making up complete nonsense, so we also hope
that our project can shed light on when truthfulness generalizes. It’s
generally desirable for LLMs to output true statements. One current
approach for ensuring this is to have a human in the loop rewarding the
model for true outputs (e.g. RLHF). However, humans can be poor judges
of truthfulness and have many cognitive biases and superficial
heuristics. A further challenge is that as LLMs become more capable,
there might not even exist experts that are good judges of whether the
models outputs, such as difficult mathematical proofs, are truthful. For
instance, most Task Rabbit workers would probably be hard pressed to
evaluate whether a difficult mathematical proof produced by an LLM is
true. The entire mathematical community has been known on occasion to <a
href="https://en.wikipedia.org/wiki/Grunwald%E2%80%93Wang_theorem">believe
false statements for many years</a>.</p>
<p>One possible solution is to reward an LLM for truthful behavior on
simple inputs, and then hope that the LLM generalizes its truthful
behavior for more complex inputs where humans cannot provide helpful
labels. Deep learning models can be remarkably good at off-distribution
generalization–for instance, a model trained to transform hand drawn
cats into images of cats might be able to handle a “cat” with three eyes
in an intuitive way. We might hope that generalizing truthfully is
simple, thus promoted by “Occam’s Razor”.</p>
<h2 id="related-work">Related Work</h2>
<p>COMMENT FROM ALEK – please remove all mentions of graph neural
networks – that is BS: there is no actual reason why you’d ever use a
Neural network to solve shortest paths, the point of choosing a
synthetic task is because there is a <strong>simple ground
truth</strong> which makes it easy to evaluate whether or not our model
is performing correctly. We’d also hoped that the simplicity of the task
would make it more feasible to do with a limited compute budget, but
apparently this task was too hard for our architecture.</p>
<p>There has been some research into the algorithmic optimization of
GNNs and how they may solve real-world issues; however, none of the
related work targets using generic machine learning methods to solve
graph problems.</p>
<ul>
<li><p>Cappart et al. has researched more into the Combinatorial
Optimization of GNNs and developed algorithms for related tasks, thus
facilitating machine learning <span class="citation"
data-cites="DBLP:journals/corr/abs-2102-09544">(<a
href="#ref-DBLP:journals/corr/abs-2102-09544"
role="doc-biblioref">Cappart et al. 2021</a>)</span>. Their results are
mostly algorithmic so we develop further by trading a bit of accuracy
for much faster computation in such tasks.</p></li>
<li><p>Tutsoy uses a graph-theory-based approach to model the
epidemiological characteristics of infectious diseases, such as COVID-19
<span class="citation" data-cites="10.1109/TPAMI.2023.3256421">(<a
href="#ref-10.1109/TPAMI.2023.3256421" role="doc-biblioref">Tutsoy
2023</a>)</span>. We understand from his paper how GNN optimization may
also be useful in researching novel diseases.</p></li>
</ul>
<h2 id="task">Task</h2>
<p>Our synthetic task is simple: compute the distance between various
vertices in an input graph. To test off-distribution generalization, our
experiment has three steps.</p>
<ol type="1">
<li><strong>Pre-train</strong> a transformer to predict the distance
between vertices <span class="math inline">1</span> and <span
class="math inline">2</span> in graphs with <span class="math inline">n
\in [3,15]</span> vertices.</li>
</ol>
<div style="text-align:center">
<p><img src="img/train.svg" /></p>
</div>
<ol start="2" type="1">
<li><strong>Fine-tune</strong> a transformer to predict the distances
between vertex <span class="math inline">1</span> to <span
class="math inline">t</span> for any <span class="math inline">t</span>
on the shortest path from <span class="math inline">1</span> to <span
class="math inline">2</span>, but only do fine-tuning on graphs with
<span class="math inline">n \in [3,7]</span> vertices.</li>
</ol>
<div style="text-align:center">
<p><img src="img/finetune.svg" /></p>
</div>
<ol start="3" type="1">
<li><strong>Test</strong> whether the transformer can accurately predict
the distances between <span class="math inline">1</span> to <span
class="math inline">t</span> for any <span class="math inline">t \leq
7</span> on the shortest path from <span class="math inline">1</span> to
<span class="math inline">2</span> for graphs with <span
class="math inline">n \in [3,15]</span> vertices.</li>
</ol>
<div style="text-align:center">
<p><img src="img/test.svg" /></p>
</div>
<h3 id="algorithm-for-shortest-paths">Algorithm for Shortest Paths</h3>
<p>The standard algorithm to find the shortest path in a graph between
vertices <span class="math inline">u</span> and <span
class="math inline">v</span> is <strong>breadth-first search
(BFS)</strong>, taught in every intro algorithms class. Initially, BFS
starts at <span class="math inline">u</span>, and at each phase,
explores a farther layer of vertices from <span
class="math inline">u</span>. During a phase, BFS goes through every
vertex in the current layer and adds any of their unvisited neighbors to
the next layer. The algorithm terminates once we reach <span
class="math inline">v</span> or if the next layer is empty. For a graph
with <span class="math inline">V</span> vertices and <span
class="math inline">E</span> edges, the runtime of BFS is <span
class="math inline">O(V + E)</span>. BFS gives us an easy and fast way
to find the ground truth answer for any graph, so that we can verify the
accuracy of our machine learning approach.</p>
<p>We hope that our model can learn BFS or some other simple,
generalizable algorithm for shortest paths, because the model can’t just
pull some magic number out of a hat but intuitively needs to count or
list the vertices on the shortest path from <span
class="math inline">1</span> to <span class="math inline">2</span>. In
fact, we will show how to hand-craft a set of weights to implement BFS
in a transformer, so it’s indeed theoretically possible for a
transformer to achieve 100% accuracy.</p>
<h3 id="data">Data</h3>
<p>We’ll represent an <span class="math inline">n</span> vertex, <span
class="math inline">m</span> edge unweighted, undirected graph as
sequence of the endpoints of the <span class="math inline">m</span>
edges, so <span
class="math inline">[a_1,b_1,a_2,b_2,\ldots,a_m,b_m]</span> represents a
graph with the edges <span class="math inline">\{(a_i,b_i)\}</span>
where <span class="math inline">a_i < b_i</span> for <span
class="math inline">1 \leq i \leq m</span>. All sequences are padded to
the same length using the padding token <span
class="math inline">0</span>.</p>
<p>The full input to our model additionally includes the target vertex
<span class="math inline">t</span> after the padding tokens. The label
to an input is the length of the shortest path from <span
class="math inline">1</span> to <span class="math inline">t</span>. If
no such path exists, we define the length to be <span
class="math inline">n+1</span> which represents infinity. For example,
the input <span class="math inline">[1, 3, 3, 4, 2, 4, 2, 3, 0, 0, 0, 0,
3]</span> has the label <span class="math inline">2</span>.</p>
<div style="text-align:center">
<p><img src="img/finetune.svg" /></p>
</div>
<p>We have three datasets for each step.</p>
<ol type="1">
<li><strong>Pre-train data</strong>: For each <span
class="math inline">n \in [3,15]</span>, we generated an equal number of
graphs on <span class="math inline">n</span> vertices, with <span
class="math inline">t = 2</span>. Each graph was created by choosing
<span class="math inline">n</span> random edges.</li>
<li><strong>Fine-tune data</strong>: For each <span
class="math inline">n \in [3,7]</span>, we generated an equal number of
graphs on <span class="math inline">n</span> vertices each with a random
<span class="math inline">t</span> on the shortest path from <span
class="math inline">1</span> to <span class="math inline">2</span>.
Again, each graph was created by choosing <span
class="math inline">n</span> random edges.</li>
<li><strong>Generalization test data</strong>: The same as the fine-tune
data, except we sample <span class="math inline">n \in [3,15]</span> and
<span class="math inline">t \leq 7</span>.</li>
</ol>
<p>We wrote some Python code to generate the data during the training
loop, but Python is excruciatingly slow and data generation wasted a lot
of training time. Our solution was to pre-generate the data before
training using a multithreaded version of our Python code.</p>
<h2
id="complicated-explicit-transformer-formula-for-shortest-paths">Complicated
explicit transformer formula for shortest paths</h2>
<p>TODO: Kevin or Alek</p>
<div class="sourceCode" id="cb1"><pre class="sourceCode py"><code class="sourceCode python"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="co"># Configuration</span></span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a>NVTXS <span class="op">=</span> <span class="dv">16</span></span>
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a>MAXDIST <span class="op">=</span> NVTXS <span class="op">+</span> <span class="dv">1</span></span>
<span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a>AVGDEG <span class="op">=</span> <span class="dv">2</span></span>
<span id="cb1-5"><a href="#cb1-5" aria-hidden="true" tabindex="-1"></a>SEQLEN <span class="op">=</span> NVTXS <span class="op">+</span> <span class="dv">1</span></span>
<span id="cb1-6"><a href="#cb1-6" aria-hidden="true" tabindex="-1"></a>HIDDENDIM <span class="op">=</span> <span class="dv">4</span> <span class="op">*</span> NVTXS <span class="op">+</span> <span class="dv">2</span></span>
<span id="cb1-7"><a href="#cb1-7" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-8"><a href="#cb1-8" aria-hidden="true" tabindex="-1"></a><span class="co"># Start indices for different sections of the input data</span></span>
<span id="cb1-9"><a href="#cb1-9" aria-hidden="true" tabindex="-1"></a>START_REACH <span class="op">=</span> NVTXS <span class="op">+</span> <span class="dv">1</span></span>
<span id="cb1-10"><a href="#cb1-10" aria-hidden="true" tabindex="-1"></a>START_OUT <span class="op">=</span> <span class="dv">2</span> <span class="op">*</span> NVTXS <span class="op">+</span> <span class="dv">1</span></span>
<span id="cb1-11"><a href="#cb1-11" aria-hidden="true" tabindex="-1"></a>START_SELF <span class="op">=</span> <span class="dv">3</span> <span class="op">*</span> NVTXS <span class="op">+</span> <span class="dv">1</span></span>
<span id="cb1-12"><a href="#cb1-12" aria-hidden="true" tabindex="-1"></a>SRC_FLAG_IDX <span class="op">=</span> START_SELF</span>
<span id="cb1-13"><a href="#cb1-13" aria-hidden="true" tabindex="-1"></a>ANS_FLAG_IDX <span class="op">=</span> <span class="dv">0</span></span>
<span id="cb1-14"><a href="#cb1-14" aria-hidden="true" tabindex="-1"></a>NOTANS_FLAG_IDX <span class="op">=</span> <span class="op">-</span><span class="dv">1</span></span>
<span id="cb1-15"><a href="#cb1-15" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-16"><a href="#cb1-16" aria-hidden="true" tabindex="-1"></a>BIG <span class="op">=</span> <span class="dv">20</span></span>
<span id="cb1-17"><a href="#cb1-17" aria-hidden="true" tabindex="-1"></a>SUPABIG <span class="op">=</span> <span class="dv">100</span></span>
<span id="cb1-18"><a href="#cb1-18" aria-hidden="true" tabindex="-1"></a>MED <span class="op">=</span> <span class="dv">10</span></span>
<span id="cb1-19"><a href="#cb1-19" aria-hidden="true" tabindex="-1"></a>CURSE <span class="op">=</span> <span class="dv">5</span></span>
<span id="cb1-20"><a href="#cb1-20" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-21"><a href="#cb1-21" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> SillyTransformer(nn.Module):</span>
<span id="cb1-22"><a href="#cb1-22" aria-hidden="true" tabindex="-1"></a> <span class="kw">def</span> <span class="fu">__init__</span>(<span class="va">self</span>, device):</span>
<span id="cb1-23"><a href="#cb1-23" aria-hidden="true" tabindex="-1"></a> <span class="bu">super</span>().<span class="fu">__init__</span>()</span>
<span id="cb1-24"><a href="#cb1-24" aria-hidden="true" tabindex="-1"></a> <span class="va">self</span>.device <span class="op">=</span> device</span>
<span id="cb1-25"><a href="#cb1-25" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-26"><a href="#cb1-26" aria-hidden="true" tabindex="-1"></a> <span class="cf">with</span> torch.no_grad():</span>
<span id="cb1-27"><a href="#cb1-27" aria-hidden="true" tabindex="-1"></a> <span class="co"># Initialize weight parameters with specific configurations</span></span>
<span id="cb1-28"><a href="#cb1-28" aria-hidden="true" tabindex="-1"></a> <span class="va">self</span>.mostKs <span class="op">=</span> nn.ParameterList()</span>
<span id="cb1-29"><a href="#cb1-29" aria-hidden="true" tabindex="-1"></a> <span class="va">self</span>.mostQs <span class="op">=</span> nn.ParameterList()</span>
<span id="cb1-30"><a href="#cb1-30" aria-hidden="true" tabindex="-1"></a> <span class="va">self</span>.mostVs <span class="op">=</span> nn.ParameterList()</span>
<span id="cb1-31"><a href="#cb1-31" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span> head <span class="kw">in</span> <span class="bu">range</span>(<span class="dv">1</span>, NVTXS <span class="op">+</span> <span class="dv">1</span>):</span>
<span id="cb1-32"><a href="#cb1-32" aria-hidden="true" tabindex="-1"></a> Q <span class="op">=</span> nn.Parameter(torch.zeros((<span class="dv">2</span>, HIDDENDIM), device<span class="op">=</span>device))</span>
<span id="cb1-33"><a href="#cb1-33" aria-hidden="true" tabindex="-1"></a> Q[<span class="dv">0</span>, START_REACH <span class="op">-</span> <span class="dv">1</span> <span class="op">+</span> head] <span class="op">=</span> SUPABIG</span>
<span id="cb1-34"><a href="#cb1-34" aria-hidden="true" tabindex="-1"></a> Q[<span class="dv">1</span>, NOTANS_FLAG_IDX] <span class="op">=</span> <span class="dv">1</span></span>
<span id="cb1-35"><a href="#cb1-35" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-36"><a href="#cb1-36" aria-hidden="true" tabindex="-1"></a> K <span class="op">=</span> nn.Parameter(torch.zeros((<span class="dv">2</span>, HIDDENDIM), device<span class="op">=</span>device))</span>
<span id="cb1-37"><a href="#cb1-37" aria-hidden="true" tabindex="-1"></a> K[<span class="dv">0</span>, head] <span class="op">=</span> <span class="dv">1</span></span>
<span id="cb1-38"><a href="#cb1-38" aria-hidden="true" tabindex="-1"></a> K[<span class="dv">1</span>, ANS_FLAG_IDX] <span class="op">=</span> BIG</span>
<span id="cb1-39"><a href="#cb1-39" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-40"><a href="#cb1-40" aria-hidden="true" tabindex="-1"></a> V <span class="op">=</span> nn.Parameter(torch.zeros((NVTXS, HIDDENDIM), device<span class="op">=</span>device))</span>
<span id="cb1-41"><a href="#cb1-41" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(NVTXS):</span>
<span id="cb1-42"><a href="#cb1-42" aria-hidden="true" tabindex="-1"></a> V[i, START_SELF <span class="op">+</span> i] <span class="op">=</span> <span class="dv">1</span></span>
<span id="cb1-43"><a href="#cb1-43" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-44"><a href="#cb1-44" aria-hidden="true" tabindex="-1"></a> <span class="va">self</span>.mostKs.append(K)</span>
<span id="cb1-45"><a href="#cb1-45" aria-hidden="true" tabindex="-1"></a> <span class="va">self</span>.mostQs.append(Q)</span>
<span id="cb1-46"><a href="#cb1-46" aria-hidden="true" tabindex="-1"></a> <span class="va">self</span>.mostVs.append(V)</span>
<span id="cb1-47"><a href="#cb1-47" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-48"><a href="#cb1-48" aria-hidden="true" tabindex="-1"></a> <span class="va">self</span>.weirdKs <span class="op">=</span> nn.ParameterList()</span>
<span id="cb1-49"><a href="#cb1-49" aria-hidden="true" tabindex="-1"></a> <span class="va">self</span>.weirdQs <span class="op">=</span> nn.ParameterList()</span>
<span id="cb1-50"><a href="#cb1-50" aria-hidden="true" tabindex="-1"></a> <span class="va">self</span>.weirdVs <span class="op">=</span> nn.ParameterList()</span>
<span id="cb1-51"><a href="#cb1-51" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span> layer <span class="kw">in</span> <span class="bu">range</span>(NVTXS):</span>
<span id="cb1-52"><a href="#cb1-52" aria-hidden="true" tabindex="-1"></a> K <span class="op">=</span> nn.Parameter(torch.zeros((<span class="dv">3</span>, HIDDENDIM), device<span class="op">=</span>device))</span>
<span id="cb1-53"><a href="#cb1-53" aria-hidden="true" tabindex="-1"></a> K[<span class="dv">0</span>, NOTANS_FLAG_IDX] <span class="op">=</span> <span class="op">-</span>BIG</span>
<span id="cb1-54"><a href="#cb1-54" aria-hidden="true" tabindex="-1"></a> K[<span class="dv">0</span>, SRC_FLAG_IDX] <span class="op">=</span> BIG<span class="op">+</span>SUPABIG</span>
<span id="cb1-55"><a href="#cb1-55" aria-hidden="true" tabindex="-1"></a> K[<span class="dv">1</span>, NOTANS_FLAG_IDX] <span class="op">=</span> <span class="op">-</span>SUPABIG</span>
<span id="cb1-56"><a href="#cb1-56" aria-hidden="true" tabindex="-1"></a> K[<span class="dv">1</span>, NVTXS <span class="op">+</span> <span class="dv">2</span>] <span class="op">=</span> BIG<span class="op">+</span>SUPABIG</span>
<span id="cb1-57"><a href="#cb1-57" aria-hidden="true" tabindex="-1"></a> K[<span class="dv">1</span>, ANS_FLAG_IDX] <span class="op">=</span> <span class="op">-</span>BIG<span class="op">-</span>SUPABIG</span>
<span id="cb1-58"><a href="#cb1-58" aria-hidden="true" tabindex="-1"></a> K[<span class="dv">2</span>, ANS_FLAG_IDX] <span class="op">=</span> MED</span>
<span id="cb1-59"><a href="#cb1-59" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-60"><a href="#cb1-60" aria-hidden="true" tabindex="-1"></a> Q <span class="op">=</span> nn.Parameter(torch.zeros((<span class="dv">3</span>, HIDDENDIM), device<span class="op">=</span>device))</span>
<span id="cb1-61"><a href="#cb1-61" aria-hidden="true" tabindex="-1"></a> Q[:, ANS_FLAG_IDX] <span class="op">=</span> <span class="dv">1</span></span>
<span id="cb1-62"><a href="#cb1-62" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-63"><a href="#cb1-63" aria-hidden="true" tabindex="-1"></a> V <span class="op">=</span> nn.Parameter(torch.zeros((NVTXS, HIDDENDIM), device<span class="op">=</span>device))</span>
<span id="cb1-64"><a href="#cb1-64" aria-hidden="true" tabindex="-1"></a> V[layer, SRC_FLAG_IDX] <span class="op">=</span> <span class="dv">1</span></span>
<span id="cb1-65"><a href="#cb1-65" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-66"><a href="#cb1-66" aria-hidden="true" tabindex="-1"></a> <span class="va">self</span>.weirdKs.append(K)</span>
<span id="cb1-67"><a href="#cb1-67" aria-hidden="true" tabindex="-1"></a> <span class="va">self</span>.weirdQs.append(Q)</span>
<span id="cb1-68"><a href="#cb1-68" aria-hidden="true" tabindex="-1"></a> <span class="va">self</span>.weirdVs.append(V)</span>
<span id="cb1-69"><a href="#cb1-69" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-70"><a href="#cb1-70" aria-hidden="true" tabindex="-1"></a> <span class="kw">def</span> forward(<span class="va">self</span>, src):</span>
<span id="cb1-71"><a href="#cb1-71" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span> layer <span class="kw">in</span> <span class="bu">range</span>(NVTXS):</span>
<span id="cb1-72"><a href="#cb1-72" aria-hidden="true" tabindex="-1"></a> allKs <span class="op">=</span> [<span class="va">self</span>.weirdKs[layer]] <span class="op">+</span> [x <span class="cf">for</span> x <span class="kw">in</span> <span class="va">self</span>.mostKs]</span>
<span id="cb1-73"><a href="#cb1-73" aria-hidden="true" tabindex="-1"></a> allQs <span class="op">=</span> [<span class="va">self</span>.weirdQs[layer]] <span class="op">+</span> [x <span class="cf">for</span> x <span class="kw">in</span> <span class="va">self</span>.mostQs]</span>
<span id="cb1-74"><a href="#cb1-74" aria-hidden="true" tabindex="-1"></a> allVs <span class="op">=</span> [<span class="va">self</span>.weirdVs[layer]] <span class="op">+</span> [x <span class="cf">for</span> x <span class="kw">in</span> <span class="va">self</span>.mostVs]</span>
<span id="cb1-75"><a href="#cb1-75" aria-hidden="true" tabindex="-1"></a> head_outputs <span class="op">=</span> []</span>
<span id="cb1-76"><a href="#cb1-76" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb1-77"><a href="#cb1-77" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span> (K, Q, V) <span class="kw">in</span> <span class="bu">zip</span>(allKs, allQs, allVs):</span>
<span id="cb1-78"><a href="#cb1-78" aria-hidden="true" tabindex="-1"></a> ksrc <span class="op">=</span> torch.matmul(src, K.unsqueeze(<span class="dv">0</span>).transpose(<span class="op">-</span><span class="dv">2</span>, <span class="op">-</span><span class="dv">1</span>))</span>
<span id="cb1-79"><a href="#cb1-79" aria-hidden="true" tabindex="-1"></a> qsrc <span class="op">=</span> torch.matmul(src, Q.unsqueeze(<span class="dv">0</span>).transpose(<span class="op">-</span><span class="dv">2</span>, <span class="op">-</span><span class="dv">1</span>))</span>
<span id="cb1-80"><a href="#cb1-80" aria-hidden="true" tabindex="-1"></a> vsrc <span class="op">=</span> torch.matmul(src, V.unsqueeze(<span class="dv">0</span>).transpose(<span class="op">-</span><span class="dv">2</span>, <span class="op">-</span><span class="dv">1</span>))</span>
<span id="cb1-81"><a href="#cb1-81" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-82"><a href="#cb1-82" aria-hidden="true" tabindex="-1"></a> scores <span class="op">=</span> torch.matmul(qsrc, ksrc.transpose(<span class="op">-</span><span class="dv">2</span>, <span class="op">-</span><span class="dv">1</span>))</span>
<span id="cb1-83"><a href="#cb1-83" aria-hidden="true" tabindex="-1"></a> attention_weights <span class="op">=</span> torch.softmax(scores, dim<span class="op">=-</span><span class="dv">1</span>)</span>
<span id="cb1-84"><a href="#cb1-84" aria-hidden="true" tabindex="-1"></a> head_output <span class="op">=</span> torch.matmul(attention_weights, vsrc)</span>
<span id="cb1-85"><a href="#cb1-85" aria-hidden="true" tabindex="-1"></a> head_outputs.append(head_output)</span>
<span id="cb1-86"><a href="#cb1-86" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-87"><a href="#cb1-87" aria-hidden="true" tabindex="-1"></a> new_reaches <span class="op">=</span> <span class="bu">sum</span>(head_outputs[<span class="dv">1</span>:])</span>
<span id="cb1-88"><a href="#cb1-88" aria-hidden="true" tabindex="-1"></a> BSZ <span class="op">=</span> new_reaches.shape[<span class="dv">0</span>]</span>
<span id="cb1-89"><a href="#cb1-89" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-90"><a href="#cb1-90" aria-hidden="true" tabindex="-1"></a> nodelta_nbrs <span class="op">=</span> torch.zeros((BSZ, SEQLEN, NVTXS <span class="op">+</span> <span class="dv">1</span>), device<span class="op">=</span><span class="va">self</span>.device)</span>
<span id="cb1-91"><a href="#cb1-91" aria-hidden="true" tabindex="-1"></a> morepadlol <span class="op">=</span> torch.zeros((BSZ, SEQLEN, <span class="dv">1</span> <span class="op">+</span> NVTXS), device<span class="op">=</span><span class="va">self</span>.device)</span>
<span id="cb1-92"><a href="#cb1-92" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-93"><a href="#cb1-93" aria-hidden="true" tabindex="-1"></a> src <span class="op">=</span> src <span class="op">+</span> torch.cat((nodelta_nbrs, new_reaches, head_outputs[<span class="dv">0</span>], morepadlol), dim<span class="op">=</span><span class="dv">2</span>)</span>
<span id="cb1-94"><a href="#cb1-94" aria-hidden="true" tabindex="-1"></a> src[:, :, START_REACH:START_REACH <span class="op">+</span> NVTXS] <span class="op">=</span> <span class="dv">2</span> <span class="op">*</span> torch.sigmoid(src[:, :, START_REACH:START_REACH <span class="op">+</span> NVTXS] <span class="op">*</span> CURSE) <span class="op">-</span> <span class="dv">1</span></span>
<span id="cb1-95"><a href="#cb1-95" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-96"><a href="#cb1-96" aria-hidden="true" tabindex="-1"></a> canreach <span class="op">=</span> src[:, <span class="dv">0</span>, START_OUT:START_OUT <span class="op">+</span> NVTXS]</span>
<span id="cb1-97"><a href="#cb1-97" aria-hidden="true" tabindex="-1"></a> final_output <span class="op">=</span> <span class="dv">1</span> <span class="op">+</span> torch.<span class="bu">sum</span>(<span class="dv">1</span> <span class="op">-</span> canreach, dim<span class="op">=</span><span class="dv">1</span>)</span>
<span id="cb1-98"><a href="#cb1-98" aria-hidden="true" tabindex="-1"></a> <span class="cf">return</span> final_output</span></code></pre></div>
<p>After much deliberation, we decided the next step for us was to
customize a transformer, writing it ourselves. We observed that we
wished for the transformer to do similar work as a BFS. As a result, we
decided to work with the following transformer, for a graph with <span
class="math inline">n</span> vertices <span class="math inline">v_1,
v_2, \cdots, v_n</span>:</p>
<span class="math display">\begin{array}{|c|c|c|c|c|c}
\text{ANS} & v_{1} & v_{2} & \cdots & v_{n} & \\
\hline
1 & 0 & 0 & \cdots & 0 & \text{ANS}\\ \hline
\text{ANS} & \text{NBR}_{1} & \text{NBR}_{2} & \cdots &
\text{NBR}_{n} & \text{NBR}\\ \hline
\text{ANS} & \text{REACH}_{1} & \text{REACH}_{2} & \cdots
& \text{REACH}_{n} & \text{REACH}\\ \hline
\text{ANS} & \text{SELF}_{1} & \text{SELF}_{2} & \cdots
& \text{SELF}_{n} & \text{SELF}\\ \hline
V_{\text{OUT}} & NULL& NULL& NULL& NULL&
\text{OUT}\\ \hline
0 & 1 & 1 & \cdots &1 & \text{NOT}\\ \hline
\end{array}</span>
<p>Specifically, we see that <span
class="math inline">\text{NBR}_{i}</span> is a <span
class="math inline">n \times 1</span> vector detailing which of the
vertices are neighboring vertex <span class="math inline">v_i</span>, so
the <span class="math inline">j</span>th element of <span
class="math inline">v_i</span> is <span class="math inline">1</span> if
<span class="math inline">v_i</span> and <span
class="math inline">v_j</span> are neighboring vertices, and <span
class="math inline">0</span> otherwise. Additionally, <span
class="math inline">\text{SELF}_{i}</span> is just the <span
class="math inline">n \times 1</span> vector with the <span
class="math inline">i</span>th element <span
class="math inline">1</span> and all other elements <span
class="math inline">0</span> (e.g. the one-hot encoding of the vector).
Now, at every step, the <span class="math inline">\text{REACH}_k</span>
vector for all <span class="math inline">k</span> is updated based on
the previous <span class="math inline">\text{REACH}_k</span> vector and
<span class="math inline">\text{NBR}_{k}</span> (since all entries that
are <span class="math inline">1</span> in <span
class="math inline">\text{REACH}_k\text{NBR}_{k}^T</span> must be
updated in the manner such that if the <span class="math inline">(i,
j)</span>th element of <span
class="math inline">\text{REACH}_k\text{NBR}_{k}^T</span> is <span
class="math inline">1</span>, then <span
class="math inline">\text{REACH}_i</span>’s <span
class="math inline">j</span>th column is set to <span
class="math inline">1</span>. This is equivalent to adding <span
class="math inline">\text{REACH}_k</span> to each integer <span
class="math inline">i</span> where <span
class="math inline">\text{NBR}_{k}</span>’s <span
class="math inline">i</span>th entry is nonzero.</p>
<p>This iterates through all the vertices, and at the end, we may see
what run we are on to update <span
class="math inline">V_{\text{OUT}}</span>.</p>
<h2 id="perturbing-the-weights">Perturbing the Weights</h2>
<p>SLT folks like to think about geometry of loss landscape CITE So we
did an experiment where we mess with the weights.</p>
<p>Findings: XXX</p>
<h2 id="our-model">Our Model</h2>
<h3 id="architecture">Architecture</h3>
<p>We used a standard transformer architecture. To ensure that it can in
theory learn BFS, we ensured that the number of layers in our
transformer exceeds the diameter of the input graphs.</p>
<p>Since the order of the edges in the input doesn’t matter, we did not
use positional encodings. Each edge <span
class="math inline">(a,b)</span> is embedded to dimension <span
class="math inline">d</span> where the first <span
class="math inline">\frac{d}{2}</span> elements are the learned
embedding of <span class="math inline">a</span> and the last <span
class="math inline">\frac{d}{2}</span> elements are the learned
embedding of <span class="math inline">b</span>. For the target vertex
<span class="math inline">t</span>, we pair it with the special token
<span class="math inline">TARGET</span> and embed <span
class="math inline">(t,TARGET)</span> in the same way.</p>
<!-- https://cocreate.csail.mit.edu/r/sxArTEXiAgJshznmm -->
<p><img src="img/embeddings.svg" /></p>
<h3 id="training">Training</h3>
<p>To match the BFS transformer as closely as possible, we used a model
dimension of <span class="math inline">64</span>, <span
class="math inline">11</span> layers, and <span
class="math inline">2</span> heads per layer, for a total of 550433
parameters. In 32-bit float precision, that corresponds to around <span
class="math inline">1.76\cdot10^6</span> bits. The number of possible
graphs on 15 vertices generated using our procedure is approximately</p>
<p><span class="math display">\frac{\binom{15}{2}^{15}}{15!} =
1.59\cdot10^{18}.</span></p>
<p>This is because there are <span
class="math inline">\binom{15}{2}</span> choices for each of the 15
edges and we don’t care about the order of the edges. This is only an
approximation because some edges might be duplicated. Each graph has an
answer between 1 and 15 which requires around 4 bits, so memorizing all
the answers requires <span class="math inline">4\cdot1.59\cdot10^{18} =
6.36\cdot10^{18}</span> bits, which is <span
class="math inline">3.61\cdot10^{12}</span> times larger than our model
size.</p>
<p>To train the model, we used MSE loss, the Adam optimizer, a learning
rate of <span class="math inline">3\cdot10^{-4}</span>, and a batch size
of <span class="math inline">2^{15}</span> for one billion randomly
generated graphs. A training run takes roughly eight hours to run on a
Radeon 7900 XTX graphics card. Our final MSE loss was <span
class="math inline">0.000555</span>.</p>
<p>TODO: use https://mpld3.github.io/index.html to make interactive
plots</p>
<p><img src="training-loss.png" /></p>
<p><img src="training-2d-histogram.png" /></p>
<p>One pattern we consistently noticed during training is that the model
often gets stuck and plateaus for many epochs before rapidly decreasing.
For instance, this happened between epochs 100 and 300 in the graph
above:</p>
<p><img src="grokking.png" /></p>
<p>“grokking” hypothesis: it’s memorizing all length 2 paths?</p>
<p>TODO: cite Neel Nanda grokking modular addition</p>
<p>TODO: CRAZY!!! training curves for 1, 2, 3 length paths</p>
<p>One pitfall we encountered during training is that we initially used
bfloat16 to save VRAM, but our huge batch size caused loss-of-precision
problems and made training very difficult. It took us two weeks to debug
this until we found that switching to float32 improved training
significantly.</p>
<h2 id="fine-tuning-results">Fine tuning results</h2>
<p>After receiving our initial results, we fine-tuned with a learning
rate of 1e-5, also with MSE and the same batch size. Our final results
are shown in the images below.</p>
<p><img src="fine-tuning-loss.png" /></p>
<p><img src="fine-tuning-2d-histogram.png" /></p>
<p><img src="test-2d-histogram.png" /></p>
<p>TODO: get new graphs</p>
<p>It’s pretty good!!!</p>
<p>Can only generalize to target vertices from 2 to 7 since 8 through 15
didn’t appear in the fine-tune data</p>
<p>but this still means it</p>
<h2 id="conclusion">Conclusion</h2>
<p>however, a machine learning approach may do better in time through
parallelism, although at the expense of using much more memory.
<strong>TODO: ALEK: this is BS. If you want a parallel algorithm for
BFS, here’s one
https://en.wikipedia.org/wiki/Parallel_single-source_shortest_path_algorithm</strong></p>
<p>just do bfs lol</p>
<p><strong>Future Work</strong> There are a couple of other things that
we could try to learn shortest paths better and maybe see more
generalization. - Chain of thought - Train model to output a path, not
just the distance. Give it partial points for outputting anything that
is a legitimate path (i.e., consists of real edges) and more points for
getting the distance correct.</p>
<h2 class="unnumbered" id="references">References</h2>
<div id="refs" class="references csl-bib-body hanging-indent"
data-entry-spacing="0" role="list">
<div id="ref-DBLP:journals/corr/abs-2102-09544" class="csl-entry"
role="listitem">
Cappart, Quentin, Didier Chételat, Elias B. Khalil, Andrea Lodi,
Christopher Morris, and Petar Velickovic. 2021. <span>“Combinatorial
Optimization and Reasoning with Graph Neural Networks.”</span>
<em>CoRR</em> abs/2102.09544. <a
href="https://arxiv.org/abs/2102.09544">https://arxiv.org/abs/2102.09544</a>.
</div>
<div id="ref-10.1109/TPAMI.2023.3256421" class="csl-entry"
role="listitem">
Tutsoy, Onder. 2023. <span>“Graph Theory Based Large-Scale Machine
Learning with Multi-Dimensional Constrained Optimization Approaches for
Exact Epidemiological Modeling of Pandemic Diseases.”</span> <em>IEEE
Trans. Pattern Anal. Mach. Intell.</em> 45 (8): 9836–45. <a
href="https://doi.org/10.1109/TPAMI.2023.3256421">https://doi.org/10.1109/TPAMI.2023.3256421</a>.
</div>
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