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<header id="title-block-header">
<h1 class="title">6.7960 Project: Investigating Off-Distribution
Generalization of Transformers</h1>
</header>
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<div style="text-align:center">
<p>Anthony Wang, Alek Westover, Kevin Zhao</p>
<p>{xy,alekw,kevinmz}@mit.edu</p>
</div>
<h2 id="goals">Goals</h2>
<p>Recently, LLMs have been developing very fast, and with that comes
the concern of aligning the models to output true and productive
statements. One common approach for ensuring this is to have a human in
the loop rewarding the model for true outputs (e.g. RLHF), but one
drawback to this problem is that humans can be poor judges of
truthfulness. As LLMs become more capable, there might not even exist
experts that are good judges of whether the model’s outputs, such as
difficult mathematical proofs, are truthful. So, we’d like to propose a
potential solution to this issue via <strong>off-distribution
generalization</strong> - applying human-like intuition to solve
problems not in the dataset. Paul Christiano <a
href="https://www.alignmentforum.org/posts/BxersHYN2qcFoonwg/experimentally-evaluating-whether-honesty-generalizes?commentId=dsDA2BWpHPdgLvaXX">proposed
an experiment</a> about shortest paths in a graph; our project is
essentially to implement Christiano’s proposed experiment. To the best
of our knowledge, although there has been research in applying machine
learning for different variations of graph searches <span
class="citation" data-cites="10.5555/3666122.3666260">(<a
href="#ref-10.5555/3666122.3666260" role="doc-biblioref">Zang et al.
2024</a>)</span>, no one has done our exact experiment yet.</p>
<p>It is generally desirable for LLMs to output true statements. A
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. We enjoy many cognitive biases and might
employ superficial heuristics when judging truthfulness. A further
challenge is that as LLMs develop further, there might not even exist
experts that can correctly judge the accuracy and truthfulness of
sophisticated outputs such as difficult mathematical proofs.</p>
<p>One approach to solving this problem is to reward an LLM for truthful
behavior on simple inputs, and then hoping that the LLM generalizes its
truthful behavior for more complex inputs where humans cannot provide
helpful labels. Deep learning models often perform remarkable feats of
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”, and
aim to investigate that with this project.</p>
<p>COMMENT FROM KEVIN – synthesize from intorduction</p>
<h3 id="task">Task</h3>
<p>We will use a synthetic task to test our hypothesis that models will
generalize truthfully off-distribution. The synthetic task is computing
the distance between various vertices in an input graph. Our experiment
will have three parts:</p>
<ol type="1">
<li>Pre-train a transformer to predict the distance between two fixed
vertices <span class="math inline">s,t</span> on graphs with <span
class="math inline">n\in [8, 32)</span> vertices.</li>
<li>Fine-tune a transformer to predict the distances between <span
class="math inline">s,t'</span> for any <span
class="math inline">t'</span> which is on the shortest path from
<span class="math inline">s</span> to <span
class="math inline">t</span>, but only do fine-tuning on graphs with
<span class="math inline">n\in [8,16)</span> vertices.</li>
<li>Test whether the transformer can accurately predict the distances
between <span class="math inline">s,t'</span> for any <span
class="math inline">t'</span> on the shortest path from <span
class="math inline">s</span> to <span class="math inline">t</span> for
graphs with <span class="math inline">n\in [16,32)</span> vertices.</li>
</ol>
<h3 id="related-work">Related Work</h3>
<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="methods">Methods</h2>
<h3 id="algorithm-for-shortest-paths">Algorithm for Shortest Paths</h3>
<p>The standard algorithm to find the shortest path in a graph between a
source numbered as <span class="math inline">u</span> and sink numbered
as <span class="math inline">v</span> is <strong>breadth-first search
(BFS)</strong>. The BFS algorithm maintains a mapping of visited
vertices to their distances with respect to <span
class="math inline">u</span>, and each run of the algorithm goes through
all the vertices newly visited in the previous run, and for each vertex,
visits any of its unvisited neighbors. The algorithm terminates once
either <span class="math inline">v</span> is visited or the set of newly
visited vertices in a single run is empty.</p>
<p>We will use this algorithm to verify the accuracy of our machine
learning approach. Given <span class="math inline">V</span> vertices and
<span class="math inline">E</span> edges, the runtime of this algorithm
is thus <span class="math inline">O(V + E)</span>; however, a machine
learning approach may do better in time through parallelism, although at
the expense of using much more memory.</p>
<h3 id="data">Data</h3>
<p>We will 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> for
<span class="math inline">1 \leq i \leq m</span>. We will pad all
sequences to be the same length using the padding token 0.</p>
<p>The full input to our model will additionally add the target vertex
after the padding tokens. The model is tasked with predicting the length
of the shortest path between vertex 1 and the target vertex <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, an input-output pair for our model could look
like <span class="math inline">[1, 3, 3, 2, 0, 0, 0, 0, 2]</span> and
<span class="math inline">2</span> respectively.</p>
<p>We have three separate datasets.</p>
<ul>
<li><strong>Pre-train data</strong>: For each <span
class="math inline">n \in [8,32)</span>, we will generate several graphs
on <span class="math inline">n</span> vertices. We generate these graphs
by inserting <span class="math inline">2n</span> random edges into the
graph. We always set the target vertex to be <span
class="math inline">2</span> here.</li>
<li><strong>Fine-tune data</strong>: For each <span
class="math inline">n \in [8,16)</span>, we will generate several graphs
on <span class="math inline">n</span> vertices. We generate these graphs
by inserting <span class="math inline">2n</span> random edges into the
graph. We select the target vertex to be a random vertex on the shortest
path from <span class="math inline">1</span> to <span
class="math inline">2</span>.</li>
<li><strong>Generalization testing data</strong>: The same as the
fine-tune data, except we sample <span class="math inline">n \in
[16,32)</span> instead.</li>
</ul>
<p>We wrote some Python code to generate the data during the training
loop, but Python is slow and the data generation wasted a lot of time
during training. To get around this, we pre-generated the data before
training and made our Python code multithreaded to speed it up.</p>
<h3 id="architecture">Architecture</h3>
<p>TODO: honestly not much to say here since it’s a pretty typical
arch</p>
<p>We plan to use a standard transformer architecture. We will ensure
that the number of layers in our transformer is at least the diameter of
the graph. By doing this, we ensure that there is an extremely simple
circuit — namely BFS — that the transformer could in theory learn to
perform the task. Note that if the transformer actually learns a simple
circuit to perform this task, then it seems more likely to generalize
well. This is also our intuition for why it should be possible to fine
tune on a small amount of data for finding shortest paths to other
vertices besides <span class="math inline">2</span> – it seems like the
model should be computing these other distances as intermediate values
in its computation to find the distance to vertex <span
class="math inline">2</span>.</p>
<h3 id="embeddings">Embeddings</h3>
<p>Since the order of the edges in the input does not matter, we did not
use positional encodings. Each edge <span
class="math inline">(u,v)</span> where <span class="math inline">u <
v</span> is embedded to a dimension of <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">u</span> and the last <span
class="math inline">\frac{d}{2}</span> elements are the learned
embedding of <span class="math inline">v</span>. For the target vertex
<span class="math inline">t</span>, we also 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">t</span> and the last <span
class="math inline">\frac{d}{2}</span> are a learned embedding of a
special token.</p>
<h2 id="training">Training</h2>
<p>For our model, we used a model dimension of 64, four layers, and two
heads per layer, for a total of 200545 parameters in bfloat16 which
corresponds to around 3.2e6 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>, which is <span
class="math inline">2\cdot10^{12}</span> times larger than our model
size.</p>
<p>We used MSE loss, the Adam optimizer, a learning rate of 8e-4, and a
batch size of 131072 for 8000 unique randomly generated batches. Our
final MSE loss was approximately 0.3555.</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: training curves for 1, 2, 3 length paths</p>
<h3
id="potential-mathematical-approaches-to-shortest-paths-delete-this">Potential
Mathematical Approaches to Shortest Paths? Delete this?</h3>
<p>Another way one can think of the shortest path of a graph is using a
<em>matrix</em> to record which vertices are connected. Given vertices
numbered <span class="math inline">1</span> to <span
class="math inline">V</span>, we denote the <strong>adjacency
matrix</strong> <span class="math inline">\textbf{M}</span> of
dimensions <span class="math inline">V \times V</span> as the matrix
with element <span class="math inline">\textbf{M}_{i, j} = 1</span> if
vertices <span class="math inline">i</span> and <span
class="math inline">j</span> are connected by an edge and <span
class="math inline">\textbf{M}_{i, j} = 0</span> if they are not. Now,
we note that (1) For all <span class="math inline">k</span>, <span
class="math inline">(\textbf{M}+I)^k_{i, j} = 0</span> if and only if
there exists no path from the vertex numbered <span
class="math inline">i</span> to the vertex numbered <span
class="math inline">j</span> that is distance <span
class="math inline">k</span> or less due to Markov matrix processes. As
a result, if the distance between vertices numbered <span
class="math inline">i</span> and <span class="math inline">j</span> is
<span class="math inline">d</span>, then <span
class="math inline">\text{min}\left((\textbf{M}+I)^k_{i, j}, 1\right) =
1</span> if <span class="math inline">k \ge d</span> and <span
class="math inline">\text{min}\left((\textbf{M}+I)^k_{i, j}, 1\right) =
0</span> if <span class="math inline">k < d</span>.</p>
<p>With this information, because the distance between any two vertices
is at most <span class="math inline">V-1</span> in a graph with <span
class="math inline">V</span> vertices, we note that the
<em>distance</em> matrix turns out to be simply <span
class="math display">\textbf{D} = \textbf{1}_{V \times V} \cdot V -
\Sigma_{i=0}^{V-1}\text{min}\left((\textbf{M}+I)^k_{i, j},
1\right).</span> The runtime to compute this is <span
class="math inline">O(V)</span>, although it will take more space to
compute all powers of <span class="math inline">\textbf{M}</span>.</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>Memorization? Do some math here to compute how many bits required to
memorize 1, 2, 3</p>
<h2
id="complicated-explicit-transformer-formula-for-shortest-paths">Complicated
explicit transformer formula for shortest paths</h2>
<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>
<h2 id="alek-perturbed-experiment">Alek perturbed experiment</h2>
<h2 id="conclusion">Conclusion</h2>
<p>just do bfs lol</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>
<div id="ref-10.5555/3666122.3666260" class="csl-entry" role="listitem">
Zang, Xiao, Miao Yin, Jinqi Xiao, Saman Zonouz, and Bo Yuan. 2024.
<span>“GraphMP: Graph Neural Network-Based Motion Planning with
Efficient Graph Search.”</span> In <em>Proceedings of the 37th
International Conference on Neural Information Processing Systems</em>.
NIPS ’23. Red Hook, NY, USA: Curran Associates Inc.
</div>
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