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<header id="title-block-header">
<h1 class="title">Off-Distribution Generalization of Transformers for
Shortest Paths</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="introduction">Introduction</h2>
<h3 id="motivation">Motivation</h3>
<p>Neural networks are capable of impressive feats of off-distribution
generalization. For instance, we discussed in class a program trained to
convert sketches of cats into realistic looking pictures of cats that
was able to draw a cat with three eyes given a sketch with three eyes,
even though there were no 3-eyed cats in the training data. However,
neural networks also often learn non-robust features that cause them to
perform poorly off-distribution (e.g., adversarial examples for an image
classifier). In this project, we will investigate the question of when
transformers generalize off-distribution via a case study on a simple
synthetic task. More specifically, the goal of our project is to make
progress towards answering the following question:</p>
<blockquote>
<p>Suppose you train a model to be truthful on task A, and to produce
coherent answers on a related task B.</p>
<p>Will the model learn to be truthful on task B?</p>
</blockquote>
<p>We first heard about this general question from a <a
href="https://www.alignmentforum.org/posts/BxersHYN2qcFoonwg/experimentally-evaluating-whether-honesty-generalizes?commentId=dsDA2BWpHPdgLvaXX">blog
post written by Paul Chrisiano</a> (who was influential in developing
RLHF) (but of course lots of people discuss off-distribution
generalization). Christiano gave an example of a toy setting where you
could investigate transformer generalization capabilites:</p>
<blockquote>
<p>Suppose we trained a language model on next token prediction in both
English and French, and then fine-tuned the model to be truthful in
English. Would the model then learn to answer questions truthfully in
French? It has been shown <a href="#fn1" class="footnote-ref"
id="fnref1" role="doc-noteref"><sup>1</sup></a> that this doesn’t happen
by default, but you could hope that the truthfulness training in English
would generalize to French.</p>
</blockquote>
<p>In the future, humans are likely to trust AIs with performing
important and complex tasks. For some of these tasks it will be
infeasible for humans to audit the answers – either because human labor
is too expensive, or because the AIs are too competent, so humans aren’t
skilled enough to perform an audit (for example, an amateur chess player
might have a hard time telling apart a good chess move and a bad chess
move). However, there will be some simpler tasks where we can audit the
AI’s and reward truthful answers. So, it’s plausible that models trained
to be truthful in some simple domains will learn truthfulness as a
general lesson.</p>
<h3 id="task">Task</h3>
<p>In order to investigate generalization of this type in transformers,
we decided to focus on a simple synthetic task – shortest paths in a
graph. More specifically, we ran the following experiment (see <a
href="#data">Data</a> for more details):</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 small graphs
with <span class="math inline">n \in [3,7]</span> vertices. Our
intuition was that, in training a model to find distances between
vertices <span class="math inline">1</span> and <span
class="math inline">2</span>, the model would also implicitly need to
learn about distances from vertex <span class="math inline">1</span> to
other vertices on this shortest path.</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
generalize to predicting the distance between <span
class="math inline">1</span> to <span class="math inline">t</span> for
any <span class="math inline">t \in [3,15]</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>
<p>Obviously, if you actually want to compute shortest paths in a graph,
you should use the standard graph algorithm BFS (breadth-first search).
Our motivation for using this problem to study generalization was
two-fold:</p>
<ol type="1">
<li><p>It’s very easy to generate data and ground truth labels for this
task.</p></li>
<li><p>There is a simple circuit which computes shortest paths
(exactly), by doing something akin to BFS. In fact, we implemented this
simple circuit by hand in the section <a
href="#handcrafted-bfs-transformer">Handcrafted BFS Transformer</a>. Our
hope was that our model would also find a simple circuit that
generalizes well.</p></li>
</ol>
<h3 id="findings">Findings</h3>
<p>Our main findings were as follows:</p>
<ol type="1">
<li><p>There exists a simple set of weights that you can find by hand to
achieve arbitrarily small MSE loss on shortest paths. Moreover, this set
of weights is “robust” in the sense that, if you perturb some of these
weights and run gradient descent, the model will recover low
loss.</p></li>
<li><p>Transformers can learn on their own to perform shortest paths.
The training process consists of a sequence of <strong>phase
changes</strong> where the loss would barely change for a large number
of epochs, and then over a very small number of epochs, the loss would
dramatically go down. In particular, these phase changes corresponded to
times when the model decided to, e.g., go from dramatically
over-classifying paths as being length 1 (resulting in great performance
on tasks of length 1 but poor performance on other tasks) to spreading
these length 1 guesses out to other distances.</p></li>
<li><p>A model trained to compute the distance between two specific
vertices can be fine-tuned efficiently to compute the shortest distance
to other vertices.</p></li>
<li><p>A model trained to compute the distance between vertices <span
class="math inline">1</span> and <span class="math inline">2</span> on
large graphs, and fine tuned to compute the distance between vertices 1
and any vertex (on a shortest <span class="math inline">1</span>-<span
class="math inline">2</span> path) in small graphs, has the following
generalization properties:</p>
<ul>
<li>In larger graphs, the model can still compute the shortest distance
between <span class="math inline">1</span> and any <span
class="math inline">t</span> whose label existed in the smaller graphs,
namely <span class="math inline">t \in [3,7]</span>.</li>
<li>However, the model fails when the target vertex doesn’t exist in
smaller graphs.</li>
</ul></li>
</ol>
<p>The main conceptual implication of this work is that, while
off-distribution generalization sometimes does occur, sometimes it also
doesn’t, and it seems tricky to predict when off-distribution
generalization will happen.</p>
<h3 id="post-outline">Post Outline</h3>
<p>The remainder of our post is structured as follows:</p>
<ol type="1">
<li><p><a href="#handcrafted-bfs-transformer">Handcrafted BFS
Transformer</a>: To start, we explicitly construct by hand weights for a
transformer to solve the shortest paths problem. We also investigate the
“robustness” of our construction by showing that, starting from our
constructed solution it’s possible to recover low loss after perturbing
the weights via gradient descent. This gives a proof-of-concept that a
transformer could encode a simple generalizable solution to the problem,
and gives some reason to believe that a transformer could learn such a
solution.</p></li>
<li><p><a href="#training">Training</a>: Next, we discuss the training.
First we describe the exact data generation process and model
architecture. Then we discuss hyperparameter settings and optimization
choices that were important to get the model to train efficiently.
Finally, we investigate some interesting phenomena that occurred during
training, such as phase transitions in loss for different path
lengths.</p></li>
<li><p><a href="#generalization">Generalization</a>: Finally we discuss
the phenomena that we wanted to study – generalization! We show that
some generalization does occur, but other generalization doesn’t. We
explain why we think this happens.</p></li>
</ol>
<h2 id="handcrafted-bfs-transformer">Handcrafted BFS Transformer</h2>
<p>In this section we describe an explicit set of weights, made (with
love) by hand, that result in a transformer that achieves MSE less than
<span class="math inline">10^{-6}</span> for the shortest paths problem.
We’ll start by describing the (somewhat involved) constructions, and
then investigate the geometry of the loss landscape near our explicit
solution. The main reason for doing this is to demonstrate the existence
of a simple transformer computing shortest paths, to give
proof-of-concept that the transformer could learn a generalizable
solution.</p>
<p>The construction is somewhat tricky to write down, so we’ll make
several (non-essential) simplifying assumptions:</p>
<ol type="1">
<li>We only care about finding the distance between vertex 1 and vertex
2.</li>
<li>We are given a nice embedding of the graph.</li>
</ol>
<p>It will be obvious from the construction how to get rid of
assumptions (1) and (2).</p>
<p>The embedding of the graph that we will work with here will be
different from the embeddings that we use in the rest of the post. In
particular, in this section only the tokens will correspond to vertices.
In the remainder of the post tokens will correspond to edges.</p>
<p>More specifically, we will create the following embeddings for our
input sequence:</p>
<table>
<colgroup>
<col style="width: 25%" />
<col style="width: 20%" />
<col style="width: 22%" />
<col style="width: 13%" />
<col style="width: 18%" />
</colgroup>
<thead>
<tr class="header">
<th></th>
<th>Answer token</th>
<th><span class="math inline">v_{1}</span> token</th>
<th><span class="math inline">\ldots</span></th>
<th><span class="math inline">v_n</span> token</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>Answer flag</td>
<td>1</td>
<td>0</td>
<td></td>
<td>0</td>
</tr>
<tr class="even">
<td>Neighbors</td>
<td></td>
<td>01101</td>
<td></td>
<td>10010</td>
</tr>
<tr class="odd">
<td>Reach</td>
<td></td>
<td>01101</td>
<td></td>
<td>10010</td>
</tr>
<tr class="even">
<td>Out</td>
<td>00000</td>
<td></td>
<td></td>
<td></td>
</tr>
<tr class="odd">
<td>Self</td>
<td></td>
<td>10000</td>
<td></td>
<td>00001</td>
</tr>
<tr class="even">
<td>Not-answer flag</td>
<td>0</td>
<td>1</td>
<td></td>
<td>1</td>
</tr>
</tbody>
</table>
<p>As you can see, the hidden dimension of the model will be <span
class="math inline">4n+2</span>. The hidden dimension is broken up into
5 sections, some of which are only used in the answer token, and some of
which are only used in the vertex tokens. Here are what the different
parts of the hidden dimension represent:</p>
<ol type="1">
<li>Flags: Booleans indicating whether a vertex is the answer token or
not. If desired, you can think of this as part of a “positional
embedding”. The flags will not be updated as we move through the
transformer.</li>
<li>Neighbors: A boolean vector (of dimension <span
class="math inline">n</span>) describing the neighbors of each vertex.
(We just put zeros here for the answer token). This also won’t update
through the course of running the transformer.</li>
<li>Reach: A boolean vector representing the vertices that each vertex
can reach. This will update over the course of a forward pass of the
transformer. Specifically, at layer <span class="math inline">i</span>
of the transformer we will extend the reach of each vertex from vertices
which are distance <span class="math inline">i+1</span> away, to
vertices at distance <span class="math inline">i+2</span> away.</li>
<li>Out: This starts as zero for all tokens. Once vertex <span
class="math inline">1</span> can reach vertex <span
class="math inline">2</span> (i.e., we are in a layer deep enough that
the reach vector for <span class="math inline">v_{1}</span>’s token
indicates that it can reach vertex <span class="math inline">2</span>)
we’ll start writing <span class="math inline">1</span> on the answer
token’s out part of its embedding.</li>
<li>Self: A 1-hot encoding specifying which vertex each token
corresponds to. You should think of this as a positional embedding
(along with the flags).</li>
</ol>
<p>The transformer will consist of <span class="math inline">n</span>
layers, each of which has <span class="math inline">n+1</span> heads.
We’ve included the code for the handcoded transformer in the <a
href="#appendix">Appendix</a> – possibly it’s helpful to look at the
code in conjunction with the written explanation below.</p>
<p>In layer <span class="math inline">i</span>, our goal is to extend
the reach of each vertex <span class="math inline">v</span> from the set
of vertices at distance <span class="math inline">i+1</span> from <span
class="math inline">v</span> to the set of vertices at distance <span
class="math inline">i+2</span> from <span class="math inline">v</span>.
The <span class="math inline">j</span>-th head will perform the
following operation:</p>
<pre><code>In head j at layer i:
Vertex v's query vector will have high dot-product with vertex w's key vector
iff vertex v can reach vertex j, and vertex w is neighbors with vertex j
The value that vertex w would give to vertex v is simply a 1-hot vector, with a 1 in the w-th position</code></pre>
<p>After running the heads, each vertex’s embedding is updated to
integrate the new vertices that it can reach into its embedding. Note
that we can think of embeddings as being updated because transformers
have a residual stream of embeddings that we make modifications to.</p>
<p>Finally, the last head will be in charge of noticing whether vertex 1
has reached vertex 2 yet, and if so recording this fact. For the full
details, see the <a href="#appendix">Appendix</a>.</p>
<h3 id="robustness">Robustness</h3>
<p>The field of Singular Learning Theory (SLT; see Liam Carroll’s
Master’s thesis “Phase Transitions in Neural Networks” for an
introduction) aims to understand model training and loss-landscape
geometry. In efforts to better understand the loss landscape of the
shortest paths loss function according to the tokens used in our hand
coded implementation of the shortest paths transformers, we decided to
start at a good setting of the parameters, and then perturb the weights,
and see if the model can subsequently achieve low loss. The intuition
for why this is a good approach at measuring “how attractive of a loss
basin” we have is that this experiment is similar to the Local Learning
Coefficient from SLT. (see Lau, Edmund, Zach Furman, George Wang, Daniel
Murfet, and Susan Wei. “The Local Learning Coefficient: A
Singularity-Aware Complexity Measure”). We found that, perturbing the
weights led to high loss, but gradient descent was able to recover low
loss, indicating that the solution is somewhat “findable” by gradient
descent.</p>
<div style="text-align:center">
<figure>
<img src="perturb.png" alt="perturb.png" />
<figcaption aria-hidden="true">perturb.png</figcaption>
</figure>
<figure>
<img src="perturb-loss.png" alt="perturb-loss.png" />
<figcaption aria-hidden="true">perturb-loss.png</figcaption>
</figure>
</div>
<h2 id="training">Training</h2>
<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</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 a dataset 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. To balance the dataset,
we removed some graphs with shorter paths to obtain the distribution
shown below. (For any plot, click on the four arrows icon in the bottom
left to scroll and pan.)</li>
</ol>
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}else if(typeof define === "function" && define.amd){
// require.js is available: use it to load d3/mpld3
require.config({paths: {d3: "https://d3js.org/d3.v5"}});
require(["d3"], function(d3){
window.d3 = d3;
mpld3_load_lib("https://mpld3.github.io/js/mpld3.v0.5.10.js", function(){
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});
});
}else{
// require.js not available: dynamically load d3 & mpld3
mpld3_load_lib("https://d3js.org/d3.v5.js", function(){
mpld3_load_lib("https://mpld3.github.io/js/mpld3.v0.5.10.js", function(){
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<ol start="2" type="1">
<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. We also balanced the dataset
to achieve the distribution below.</li>
</ol>
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<ol start="3" type="1">
<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>. The dataset had the following
distribution of labels.</li>
</ol>
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<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>
<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 -->
<div style="text-align:center">
<p><img src="img/embed_path.png" /></p>
</div>
<h3 id="specifications">Specifications</h3>
<p>For our training run, we used the following specifications:</p>
<table>
<colgroup>
<col style="width: 60%" />
<col style="width: 39%" />
</colgroup>
<thead>
<tr class="header">
<th>Hyperparameter</th>
<th>Value</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>Model dimension</td>
<td><span class="math inline">64</span></td>
</tr>
<tr class="even">
<td>Layers</td>
<td><span class="math inline">11</span></td>
</tr>
<tr class="odd">
<td>Heads</td>
<td><span class="math inline">2</span></td>
</tr>
<tr class="even">
<td>Epochs</td>
<td><span class="math inline">9600</span></td>
</tr>
<tr class="odd">
<td>Batch size</td>
<td><span class="math inline">2^{15}</span></td>
</tr>
<tr class="even">
<td>Dropout</td>
<td><span class="math inline">0</span></td>
</tr>
<tr class="odd">
<td>Total training examples</td>
<td><span class="math inline">3.14\cdot10^8</span></td>
</tr>
<tr class="even">
<td>Parameters</td>
<td><span class="math inline">550433</span></td>
</tr>
<tr class="odd">
<td>Learning rate</td>
<td><span class="math inline">5\cdot10^{-4}</span> for <span
class="math inline">6400</span> epochs, <span
class="math inline">10^{-4}</span> for <span
class="math inline">3200</span> epochs</td>
</tr>
<tr class="even">
<td>Loss</td>
<td>MSE</td>
</tr>
<tr class="odd">
<td>Optimizer</td>
<td>Adam</td>
</tr>
</tbody>
</table>
<p>The number of bits required to store the model parameters in float32
is around <span class="math inline">1.76\cdot10^6</span>. The number of
possible graphs on 15 vertices generated using our procedure is
approximately <span class="math display">\frac{\binom{15}{2}^{15}}{15!}
\approx 1.59\cdot10^{18}.</span> 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 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. This implies that in order to get really low
loss, our model needs to do something other than brute memorization.</p>
<p>A single training run takes roughly three hours on a Radeon 7900 XTX
graphics card.</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>
<h3 id="training-results">Training Results</h3>
<p>Here is our training loss curve, with a final loss of <span
class="math inline">0.001623</span>. The discontinuity at epoch <span
class="math inline">6400</span> is from switching the learning rate.</p>
<style>
</style>
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// require.js is available: use it to load d3/mpld3
require.config({paths: {d3: "https://d3js.org/d3.v5"}});
require(["d3"], function(d3){
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});
});
}else{
// require.js not available: dynamically load d3 & mpld3
mpld3_load_lib("https://d3js.org/d3.v5.js", function(){
mpld3_load_lib("https://mpld3.github.io/js/mpld3.v0.5.10.js", function(){
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1.5, "dasharray": "none", "alpha": 1, "zorder": 2, "drawstyle": "default"}], "paths": [], "markers": [], "texts": [{"text": "Epochs", "position": [0.5, -0.07921476671476674], "coordinates": "axes", "h_anchor": "middle", "v_baseline": "hanging", "rotation": -0.0, "fontsize": 10.0, "color": "#000000", "alpha": 1, "zorder": 3, "id": "el373447139791738428144"}, {"text": "Log MSE", "position": [-0.07213261648745521, 0.5], "coordinates": "axes", "h_anchor": "middle", "v_baseline": "auto", "rotation": -90.0, "fontsize": 10.0, "color": "#000000", "alpha": 1, "zorder": 3, "id": "el373447139791619651248"}, {"text": "Training Loss", "position": [0.5, 1.0225468975468974], "coordinates": "axes", "h_anchor": "middle", "v_baseline": "auto", "rotation": -0.0, "fontsize": 12.0, "color": "#000000", "alpha": 1, "zorder": 3, "id": "el373447139791615907392"}], "collections": [], "images": [], "sharex": [], "sharey": []}], "data": {"data01": [[0.0, 4.024313449859619], [1.0, 3.789219856262207], [2.0, 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<p>Here is a “confusion matrix”-like 2D histogram plot, showing for each
true label, the probability distribution over classifications that the
model gives.</p>
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<div id="fig_el3734471397917377855201410895068"></div>
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});
});
}else{
// require.js not available: dynamically load d3 & mpld3
mpld3_load_lib("https://d3js.org/d3.v5.js", function(){
mpld3_load_lib("https://mpld3.github.io/js/mpld3.v0.5.10.js", function(){
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<p>One pattern we consistently noticed during training is that the model
sometimes gets stuck and plateaus for many epochs before rapidly
decreasing. For instance, this happened in the first <span
class="math inline">50</span> epochs in the graph below.</p>
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}else if(typeof define === "function" && define.amd){
// require.js is available: use it to load d3/mpld3
require.config({paths: {d3: "https://d3js.org/d3.v5"}});
require(["d3"], function(d3){
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});
});
}else{
// require.js not available: dynamically load d3 & mpld3
mpld3_load_lib("https://d3js.org/d3.v5.js", function(){
mpld3_load_lib("https://mpld3.github.io/js/mpld3.v0.5.10.js", function(){
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<p>Such <strong>phase change</strong> phenomena are ubiquitous in deep
learning and for simple algorithmic tasks we can sometimes interpret
them <a href="#fn2" class="footnote-ref" id="fnref2"
role="doc-noteref"><sup>2</sup></a>. In our case, the loss is initially
very high for graphs of path length <span
class="math inline">1,2,3,\infty = 15</span> because the model always
guesses the same mean answer to minimize MSE. However, at around epoch
<span class="math inline">50</span>, the model suddenly learns how to
solve graphs with a path length of <span class="math inline">1</span> by
checking if the edge <span class="math inline">(1, 2)</span> exists in
the input. This happens again at around epoch <span
class="math inline">80</span> when the model begins to learn how to
solve lengths <span class="math inline">2</span> and <span
class="math inline">3</span> as well and soon learns to separate the two
cases as well.</p>
<h2 id="generalization">Generalization</h2>
<h3 id="fine-tuning-results">Fine Tuning Results</h3>
<p>Directly using the “pre-trained” model that is good at computing the
distance between vertex <span class="math inline">1</span> and vertex
<span class="math inline">2</span> for computing the distance to other
vertices does very poorly: we get an MSE loss of <span
class="math inline">0.5531</span>, and the following confusion
matrix.</p>
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mpld3_load_lib("https://mpld3.github.io/js/mpld3.v0.5.10.js", function(){
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});
});
}else{
// require.js not available: dynamically load d3 & mpld3
mpld3_load_lib("https://d3js.org/d3.v5.js", function(){
mpld3_load_lib("https://mpld3.github.io/js/mpld3.v0.5.10.js", function(){
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<p>Note that, while this confusion matrix might initially look okay, it
is actually extremely bad because the model is basically just guessing
integers to get moderately low loss.</p>
<p>However, we’ll now show that with a very small amount of fine-tuning,
the model can perform very well at computing the shortest path to
vertices that lie on the shortest path between vertex <span
class="math inline">1</span> and vertex 2$$ — as we hypothesized!
Specifically, we fine-tuned for <span class="math inline">100</span>
epochs (which is ~100 times fewer than the number of epochs that we
pre-trained for) with the same hyperparameters other than a learning
rate and weight decay of <span class="math inline">10^{-5}</span> on the
task of predicting the distance from vertex <span
class="math inline">1</span> to a target vertex other than vertex <span
class="math inline">2</span>, but which lies on the shortest path
between vertex <span class="math inline">1</span> and vertex <span
class="math inline">2</span>. We used weight decay to incentivize the
model to find simpler or more general representations for the
fine-tuning task. We achieved a final loss of <span
class="math inline">0.01472</span>. Here are our fine-tuning loss curves
and 2D histogram.</p>
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mpld3.draw_figure("fig_el7573761403879198102883992094766", {"width": 640.0, "height": 480.0, "axes": [{"bbox": [0.125, 0.10999999999999999, 0.775, 0.77], "xlim": [-2.45, 51.45], "ylim": [-4.023941852897406, 0.3247999183833599], "xdomain": [-2.45, 51.45], "ydomain": [-4.023941852897406, 0.3247999183833599], "xscale": "linear", "yscale": "linear", "axes": [{"position": "bottom", "nticks": 8, "tickvalues": null, "tickformat_formatter": "", "tickformat": null, "scale": "linear", "fontsize": 10.0, "grid": {"gridOn": false}, "visible": true}, {"position": "left", "nticks": 11, "tickvalues": null, "tickformat_formatter": "", "tickformat": null, "scale": "linear", "fontsize": 10.0, "grid": {"gridOn": false}, "visible": true}], "axesbg": "#FFFFFF", "axesbgalpha": null, "zoomable": true, "id": "el757376140387714206848", "lines": [{"data": "data01", "xindex": 0, "yindex": 1, "coordinates": "data", "id": "el757376140387841462432", "color": "#0000FF", "linewidth": 1.5, "dasharray": "none", "alpha": 1, "zorder": 2, "drawstyle": "default"}], "paths": [], "markers": [], "texts": [{"text": "Epochs", "position": [0.5, -0.07921476671476674], "coordinates": "axes", "h_anchor": "middle", "v_baseline": "hanging", "rotation": -0.0, "fontsize": 10.0, "color": "#000000", "alpha": 1, "zorder": 3, "id": "el757376140387714211216"}, {"text": "Log MSE", "position": [-0.0985943100358423, 0.5], "coordinates": "axes", "h_anchor": "middle", "v_baseline": "auto", "rotation": -90.0, "fontsize": 10.0, "color": "#000000", "alpha": 1, "zorder": 3, "id": "el757376140387714204448"}, {"text": "Fine-tune Loss", "position": [0.5, 1.0225468975468974], "coordinates": "axes", "h_anchor": "middle", "v_baseline": "auto", "rotation": -0.0, "fontsize": 12.0, "color": "#000000", "alpha": 1, "zorder": 3, "id": "el757376140387886095520"}], "collections": [], "images": [], "sharex": [], "sharey": []}], "data": {"data01": [[0.0, -0.5529061555862427], [1.0, 0.00582094257697463], [2.0, 0.12712983787059784], [3.0, -0.6931543350219727], [4.0, -0.6789501309394836], [5.0, -0.7886176705360413], [6.0, -1.194624662399292], [7.0, -0.8660245537757874], [8.0, -0.32105493545532227], [9.0, -0.5563544631004333], [10.0, -1.1614649295806885], [11.0, -0.6834737658500671], [12.0, -1.4866312742233276], [13.0, -0.8663621544837952], [14.0, -1.8159927129745483], [15.0, -1.2742594480514526], [16.0, -1.1336743831634521], [17.0, -1.9043149948120117], [18.0, -1.6900967359542847], [19.0, -1.6131829023361206], [20.0, -1.7646479606628418], [21.0, -1.9197150468826294], [22.0, -1.9860000610351562], [23.0, -1.885329008102417], [24.0, -2.0404014587402344], [25.0, -2.3226630687713623], [26.0, -2.476088047027588], [27.0, -2.53680157661438], [28.0, -2.459334135055542], [29.0, -2.6285030841827393], [30.0, -2.673304557800293], [31.0, -2.660975456237793], [32.0, -2.8099002838134766], [33.0, -2.911806583404541], [34.0, -3.0064892768859863], [35.0, -3.1329269409179688], [36.0, -3.1932780742645264], [37.0, -3.277862548828125], [38.0, -3.2337965965270996], [39.0, -3.2325751781463623], [40.0, -3.389988660812378], [41.0, -3.4854423999786377], [42.0, -3.446497678756714], [43.0, -3.555500030517578], [44.0, -3.621767997741699], [45.0, -3.686537504196167], [46.0, -3.7084803581237793], [47.0, -3.6845247745513916], [48.0, -3.8262717723846436], [49.0, -3.8219897747039795]]}, "id": "el757376140387919810288", "plugins": [{"type": "reset"}, {"type": "zoom", "button": true, "enabled": false}, {"type": "boxzoom", "button": true, "enabled": false}]});
}(mpld3);
}else if(typeof define === "function" && define.amd){
// require.js is available: use it to load d3/mpld3
require.config({paths: {d3: "https://d3js.org/d3.v5"}});
require(["d3"], function(d3){
window.d3 = d3;
mpld3_load_lib("https://mpld3.github.io/js/mpld3.v0.5.10.js", function(){
mpld3.draw_figure("fig_el7573761403879198102883992094766", {"width": 640.0, "height": 480.0, "axes": [{"bbox": [0.125, 0.10999999999999999, 0.775, 0.77], "xlim": [-2.45, 51.45], "ylim": [-4.023941852897406, 0.3247999183833599], "xdomain": [-2.45, 51.45], "ydomain": [-4.023941852897406, 0.3247999183833599], "xscale": "linear", "yscale": "linear", "axes": [{"position": "bottom", "nticks": 8, "tickvalues": null, "tickformat_formatter": "", "tickformat": null, "scale": "linear", "fontsize": 10.0, "grid": {"gridOn": false}, "visible": true}, {"position": "left", "nticks": 11, "tickvalues": null, "tickformat_formatter": "", "tickformat": null, "scale": "linear", "fontsize": 10.0, "grid": {"gridOn": false}, "visible": true}], "axesbg": "#FFFFFF", "axesbgalpha": null, "zoomable": true, "id": "el757376140387714206848", "lines": [{"data": "data01", "xindex": 0, "yindex": 1, "coordinates": "data", "id": "el757376140387841462432", "color": "#0000FF", "linewidth": 1.5, "dasharray": "none", "alpha": 1, "zorder": 2, "drawstyle": "default"}], "paths": [], "markers": [], "texts": [{"text": "Epochs", "position": [0.5, -0.07921476671476674], "coordinates": "axes", "h_anchor": "middle", "v_baseline": "hanging", "rotation": -0.0, "fontsize": 10.0, "color": "#000000", "alpha": 1, "zorder": 3, "id": "el757376140387714211216"}, {"text": "Log MSE", "position": [-0.0985943100358423, 0.5], "coordinates": "axes", "h_anchor": "middle", "v_baseline": "auto", "rotation": -90.0, "fontsize": 10.0, "color": "#000000", "alpha": 1, "zorder": 3, "id": "el757376140387714204448"}, {"text": "Fine-tune Loss", "position": [0.5, 1.0225468975468974], "coordinates": "axes", "h_anchor": "middle", "v_baseline": "auto", "rotation": -0.0, "fontsize": 12.0, "color": "#000000", "alpha": 1, "zorder": 3, "id": "el757376140387886095520"}], "collections": [], "images": [], "sharex": [], "sharey": []}], "data": {"data01": [[0.0, -0.5529061555862427], [1.0, 0.00582094257697463], [2.0, 0.12712983787059784], [3.0, -0.6931543350219727], [4.0, -0.6789501309394836], [5.0, -0.7886176705360413], [6.0, -1.194624662399292], [7.0, -0.8660245537757874], [8.0, -0.32105493545532227], [9.0, -0.5563544631004333], [10.0, -1.1614649295806885], [11.0, -0.6834737658500671], [12.0, -1.4866312742233276], [13.0, -0.8663621544837952], [14.0, -1.8159927129745483], [15.0, -1.2742594480514526], [16.0, -1.1336743831634521], [17.0, -1.9043149948120117], [18.0, -1.6900967359542847], [19.0, -1.6131829023361206], [20.0, -1.7646479606628418], [21.0, -1.9197150468826294], [22.0, -1.9860000610351562], [23.0, -1.885329008102417], [24.0, -2.0404014587402344], [25.0, -2.3226630687713623], [26.0, -2.476088047027588], [27.0, -2.53680157661438], [28.0, -2.459334135055542], [29.0, -2.6285030841827393], [30.0, -2.673304557800293], [31.0, -2.660975456237793], [32.0, -2.8099002838134766], [33.0, -2.911806583404541], [34.0, -3.0064892768859863], [35.0, -3.1329269409179688], [36.0, -3.1932780742645264], [37.0, -3.277862548828125], [38.0, -3.2337965965270996], [39.0, -3.2325751781463623], [40.0, -3.389988660812378], [41.0, -3.4854423999786377], [42.0, -3.446497678756714], [43.0, -3.555500030517578], [44.0, -3.621767997741699], [45.0, -3.686537504196167], [46.0, -3.7084803581237793], [47.0, -3.6845247745513916], [48.0, -3.8262717723846436], [49.0, -3.8219897747039795]]}, "id": "el757376140387919810288", "plugins": [{"type": "reset"}, {"type": "zoom", "button": true, "enabled": false}, {"type": "boxzoom", "button": true, "enabled": false}]});
});
});
}else{
// require.js not available: dynamically load d3 & mpld3
mpld3_load_lib("https://d3js.org/d3.v5.js", function(){
mpld3_load_lib("https://mpld3.github.io/js/mpld3.v0.5.10.js", function(){
mpld3.draw_figure("fig_el7573761403879198102883992094766", {"width": 640.0, "height": 480.0, "axes": [{"bbox": [0.125, 0.10999999999999999, 0.775, 0.77], "xlim": [-2.45, 51.45], "ylim": [-4.023941852897406, 0.3247999183833599], "xdomain": [-2.45, 51.45], "ydomain": [-4.023941852897406, 0.3247999183833599], "xscale": "linear", "yscale": "linear", "axes": [{"position": "bottom", "nticks": 8, "tickvalues": null, "tickformat_formatter": "", "tickformat": null, "scale": "linear", "fontsize": 10.0, "grid": {"gridOn": false}, "visible": true}, {"position": "left", "nticks": 11, "tickvalues": null, "tickformat_formatter": "", "tickformat": null, "scale": "linear", "fontsize": 10.0, "grid": {"gridOn": false}, "visible": true}], "axesbg": "#FFFFFF", "axesbgalpha": null, "zoomable": true, "id": "el757376140387714206848", "lines": [{"data": "data01", "xindex": 0, "yindex": 1, "coordinates": "data", "id": "el757376140387841462432", "color": "#0000FF", "linewidth": 1.5, "dasharray": "none", "alpha": 1, "zorder": 2, "drawstyle": "default"}], "paths": [], "markers": [], "texts": [{"text": "Epochs", "position": [0.5, -0.07921476671476674], "coordinates": "axes", "h_anchor": "middle", "v_baseline": "hanging", "rotation": -0.0, "fontsize": 10.0, "color": "#000000", "alpha": 1, "zorder": 3, "id": "el757376140387714211216"}, {"text": "Log MSE", "position": [-0.0985943100358423, 0.5], "coordinates": "axes", "h_anchor": "middle", "v_baseline": "auto", "rotation": -90.0, "fontsize": 10.0, "color": "#000000", "alpha": 1, "zorder": 3, "id": "el757376140387714204448"}, {"text": "Fine-tune Loss", "position": [0.5, 1.0225468975468974], "coordinates": "axes", "h_anchor": "middle", "v_baseline": "auto", "rotation": -0.0, "fontsize": 12.0, "color": "#000000", "alpha": 1, "zorder": 3, "id": "el757376140387886095520"}], "collections": [], "images": [], "sharex": [], "sharey": []}], "data": {"data01": [[0.0, -0.5529061555862427], [1.0, 0.00582094257697463], [2.0, 0.12712983787059784], [3.0, -0.6931543350219727], [4.0, -0.6789501309394836], [5.0, -0.7886176705360413], [6.0, -1.194624662399292], [7.0, -0.8660245537757874], [8.0, -0.32105493545532227], [9.0, -0.5563544631004333], [10.0, -1.1614649295806885], [11.0, -0.6834737658500671], [12.0, -1.4866312742233276], [13.0, -0.8663621544837952], [14.0, -1.8159927129745483], [15.0, -1.2742594480514526], [16.0, -1.1336743831634521], [17.0, -1.9043149948120117], [18.0, -1.6900967359542847], [19.0, -1.6131829023361206], [20.0, -1.7646479606628418], [21.0, -1.9197150468826294], [22.0, -1.9860000610351562], [23.0, -1.885329008102417], [24.0, -2.0404014587402344], [25.0, -2.3226630687713623], [26.0, -2.476088047027588], [27.0, -2.53680157661438], [28.0, -2.459334135055542], [29.0, -2.6285030841827393], [30.0, -2.673304557800293], [31.0, -2.660975456237793], [32.0, -2.8099002838134766], [33.0, -2.911806583404541], [34.0, -3.0064892768859863], [35.0, -3.1329269409179688], [36.0, -3.1932780742645264], [37.0, -3.277862548828125], [38.0, -3.2337965965270996], [39.0, -3.2325751781463623], [40.0, -3.389988660812378], [41.0, -3.4854423999786377], [42.0, -3.446497678756714], [43.0, -3.555500030517578], [44.0, -3.621767997741699], [45.0, -3.686537504196167], [46.0, -3.7084803581237793], [47.0, -3.6845247745513916], [48.0, -3.8262717723846436], [49.0, -3.8219897747039795]]}, "id": "el757376140387919810288", "plugins": [{"type": "reset"}, {"type": "zoom", "button": true, "enabled": false}, {"type": "boxzoom", "button": true, "enabled": false}]});
})
});
}
</script>
<style>
</style>
<div id="fig_el7573761403878402218566918473312"></div>
<script>
function mpld3_load_lib(url, callback){
var s = document.createElement('script');
s.src = url;
s.async = true;
s.onreadystatechange = s.onload = callback;
s.onerror = function(){console.warn("failed to load library " + url);};
document.getElementsByTagName("head")[0].appendChild(s);
}
if(typeof(mpld3) !== "undefined" && mpld3._mpld3IsLoaded){
// already loaded: just create the figure
!function(mpld3){
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});
});
}else{
// require.js not available: dynamically load d3 & mpld3
mpld3_load_lib("https://d3js.org/d3.v5.js", function(){
mpld3_load_lib("https://mpld3.github.io/js/mpld3.v0.5.10.js", function(){
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<h3 id="test-results">Test Results</h3>
<p>Lastly, we ran our fine-tuned model on some tasks that were not part
of its training regiment, and investigated in which settings it was able
to generalize.</p>
<p>Here is a description of the tasks that we ran our fine-tuned model
on, and a summary of results.</p>
<table>
<colgroup>
<col style="width: 33%" />
<col style="width: 33%" />
<col style="width: 33%" />
</colgroup>
<thead>
<tr class="header">
<th>Name</th>
<th>Description</th>
<th>Generalization Behavior</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>On-path Small Target</td>
<td>Compute distance to a target vertex <span class="math inline">t\in
[3,7]</span> on the shortest path between vertices <span
class="math inline">1</span>, <span class="math inline">2</span> in
graphs of size <span class="math inline">[3,15]</span></td>
<td>Good generalization</td>
</tr>
<tr class="even">
<td>On-path Target</td>
<td>Compute distance to a target vertex <span class="math inline">t\in
[3,15]</span> on the shortest path between vertices <span
class="math inline">1</span>, <span class="math inline">2</span> in
graphs of size <span class="math inline">[3,15]</span></td>
<td>Moderate generalization</td>
</tr>
<tr class="odd">
<td>Any Target in Small</td>
<td>Compute distance to a target vertex <span class="math inline">t\in
[3,7]</span> in graphs of size <span
class="math inline">[3,7]</span>.</td>
<td>High loss</td>
</tr>
<tr class="even">
<td>Any Small Target in Large Graphs</td>
<td>Compute distance to a target vertex <span class="math inline">t\in
[3,7]</span> in graphs of size <span
class="math inline">[3,15]</span>.</td>
<td>High loss</td>
</tr>
<tr class="odd">
<td>Any Target in Large Graphs</td>
<td>Compute distance to a target vertex <span class="math inline">t\in
[3,15]</span> in graphs of size <span
class="math inline">[3,15]</span>.</td>
<td>High loss</td>
</tr>
</tbody>
</table>
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});
});
}else{
// require.js not available: dynamically load d3 & mpld3
mpld3_load_lib("https://d3js.org/d3.v5.js", function(){
mpld3_load_lib("https://mpld3.github.io/js/mpld3.v0.5.10.js", function(){
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require.config({paths: {d3: "https://d3js.org/d3.v5"}});
require(["d3"], function(d3){
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});
});
}else{
// require.js not available: dynamically load d3 & mpld3
mpld3_load_lib("https://d3js.org/d3.v5.js", function(){
mpld3_load_lib("https://mpld3.github.io/js/mpld3.v0.5.10.js", function(){
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<p>For on-path targets in large graphs, the model is able to generalize
somewhat well, but generalizes better when the target is small–<span
class="math inline">3</span> through <span
class="math inline">7</span>–likely because those were in the
fine-tuning data. For small targets, it achieves a loss of <span
class="math inline">0.1020</span> compared to <span
class="math inline">0.1451</span> for any target. It’s notable that the
model still generalizes to large targets, although to a lesser extent,
which indicates that generalization happens on a spectrum and the model
probably did learn somewhat general representations.</p>
<p>We tried to do interpretability on the model, but it was too large
and complicated to understand, so many of these explanations are just
our intuition and may not correspond at all to how the model actually
works.</p>
<p>We also tested the model where the target is allowed to be any
vertex, not just a vertex on the shortest path.</p>
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if(typeof(mpld3) !== "undefined" && mpld3._mpld3IsLoaded){
// already loaded: just create the figure
!function(mpld3){
mpld3.draw_figure("fig_el6940241401906872046885580479736", {"width": 640.0, "height": 480.0, "axes": [{"bbox": [0.125, 0.10999999999999999, 0.6200000000000001, 0.77], "xlim": [1.0, 15.0], "ylim": [0.8163724541664124, 15.011382102966309], "xdomain": [1.0, 15.0], "ydomain": [0.8163724541664124, 15.011382102966309], "xscale": "linear", "yscale": "linear", "axes": [{"position": "bottom", "nticks": 9, "tickvalues": null, "tickformat_formatter": "", "tickformat": null, "scale": "linear", "fontsize": 10.0, "grid": {"gridOn": false}, "visible": true}, {"position": "left", "nticks": 9, "tickvalues": null, "tickformat_formatter": "", "tickformat": null, "scale": "linear", "fontsize": 10.0, "grid": {"gridOn": false}, "visible": true}], "axesbg": "#FFFFFF", "axesbgalpha": null, "zoomable": true, "id": "el694024140190553137248", "lines": [], "paths": [], "markers": [], "texts": [{"text": "Label", "position": [0.5, -0.07921476671476674], "coordinates": "axes", "h_anchor": "middle", "v_baseline": "hanging", "rotation": -0.0, "fontsize": 10.0, "color": "#000000", "alpha": 1, "zorder": 3, "id": "el694024140190178426032"}, {"text": "Model Output", "position": [-0.08292030689964158, 0.5], "coordinates": "axes", "h_anchor": "middle", "v_baseline": "auto", "rotation": -90.0, "fontsize": 10.0, "color": "#000000", "alpha": 1, "zorder": 3, "id": "el694024140190172425504"}, {"text": "Any Target In Small Graphs", "position": [0.5, 1.0225468975468974], "coordinates": "axes", "h_anchor": "middle", "v_baseline": "auto", "rotation": -0.0, "fontsize": 12.0, "color": "#000000", "alpha": 1, "zorder": 3, "id": "el694024140190171422624"}], "collections": [{"offsets": "data01", "xindex": 0, "yindex": 1, "paths": [[[[1.0, 0.8163724541664124], [1.9333333333333333, 0.8163724541664124], [1.9333333333333333, 1.1002726471424102], [1.0, 1.1002726471424102], [1.0, 0.8163724541664124]], ["M", "L", "L", "L", "L"]], [[[1.9333333333333333, 0.8163724541664124], [2.8666666666666667, 0.8163724541664124], 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});
});
}else{
// require.js not available: dynamically load d3 & mpld3
mpld3_load_lib("https://d3js.org/d3.v5.js", function(){
mpld3_load_lib("https://mpld3.github.io/js/mpld3.v0.5.10.js", function(){
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"hanging", "rotation": -0.0, "fontsize": 10.0, "color": "#000000", "alpha": 1, "zorder": 3, "id": "el694024140190418684720"}, {"text": "Model Output", "position": [-0.08292030689964158, 0.5], "coordinates": "axes", "h_anchor": "middle", "v_baseline": "auto", "rotation": -90.0, "fontsize": 10.0, "color": "#000000", "alpha": 1, "zorder": 3, "id": "el694024140190418684096"}, {"text": "Any Small Target In Large Graphs", "position": [0.5, 1.0225468975468974], "coordinates": "axes", "h_anchor": "middle", "v_baseline": "auto", "rotation": -0.0, "fontsize": 12.0, "color": "#000000", "alpha": 1, "zorder": 3, "id": "el694024140190349024512"}], "collections": [{"offsets": "data01", "xindex": 0, "yindex": 1, "paths": [[[[1.0, 0.8011165261268616], [1.9333333333333333, 0.8011165261268616], [1.9333333333333333, 1.0860233414173126], [1.0, 1.0860233414173126], [1.0, 0.8011165261268616]], ["M", "L", "L", "L", "L"]], [[[1.9333333333333333, 0.8011165261268616], [2.8666666666666667, 0.8011165261268616], 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});
});
}else{
// require.js not available: dynamically load d3 & mpld3
mpld3_load_lib("https://d3js.org/d3.v5.js", function(){
mpld3_load_lib("https://mpld3.github.io/js/mpld3.v0.5.10.js", function(){
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// require.js is available: use it to load d3/mpld3
require.config({paths: {d3: "https://d3js.org/d3.v5"}});
require(["d3"], function(d3){
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mpld3_load_lib("https://mpld3.github.io/js/mpld3.v0.5.10.js", function(){
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});
});
}else{
// require.js not available: dynamically load d3 & mpld3
mpld3_load_lib("https://d3js.org/d3.v5.js", function(){
mpld3_load_lib("https://mpld3.github.io/js/mpld3.v0.5.10.js", function(){
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[520.0800000000002, 396.41249999999997], [501.60000000000014, 396.41249999999997], [501.60000000000014, 394.96875]], ["M", "L", "L", "L", "L"]], [[[501.60000000000014, 396.41249999999997], [520.0800000000002, 396.41249999999997], [520.0800000000002, 397.85625], [501.60000000000014, 397.85625], [501.60000000000014, 396.41249999999997]], ["M", "L", "L", "L", "L"]], [[[501.60000000000014, 397.85625], [520.0800000000002, 397.85625], [520.0800000000002, 399.29999999999995], [501.60000000000014, 399.29999999999995], [501.60000000000014, 397.85625]], ["M", "L", "L", "L", "L"]], [[[501.60000000000014, 399.29999999999995], [520.0800000000002, 399.29999999999995], [520.0800000000002, 400.74375], [501.60000000000014, 400.74375], [501.60000000000014, 399.29999999999995]], ["M", "L", "L", "L", "L"]], [[[501.60000000000014, 400.74375], [520.0800000000002, 400.74375], [520.0800000000002, 402.1875], [501.60000000000014, 402.1875], [501.60000000000014, 400.74375]], ["M", "L", "L", "L", "L"]], [[[501.60000000000014, 402.1875], [520.0800000000002, 402.1875], [520.0800000000002, 403.63124999999997], [501.60000000000014, 403.63124999999997], [501.60000000000014, 402.1875]], ["M", "L", "L", "L", "L"]], [[[501.60000000000014, 403.63124999999997], [520.0800000000002, 403.63124999999997], [520.0800000000002, 405.075], [501.60000000000014, 405.075], [501.60000000000014, 403.63124999999997]], ["M", "L", "L", "L", "L"]], [[[501.60000000000014, 405.075], [520.0800000000002, 405.075], [520.0800000000002, 406.51874999999995], [501.60000000000014, 406.51874999999995], [501.60000000000014, 405.075]], ["M", "L", "L", "L", "L"]], [[[501.60000000000014, 406.51874999999995], [520.0800000000002, 406.51874999999995], [520.0800000000002, 407.9625], [501.60000000000014, 407.9625], [501.60000000000014, 406.51874999999995]], ["M", "L", "L", "L", "L"]], [[[501.60000000000014, 407.9625], [520.0800000000002, 407.9625], [520.0800000000002, 409.40625], [501.60000000000014, 409.40625], [501.60000000000014, 407.9625]], ["M", "L", "L", "L", "L"]], [[[501.60000000000014, 409.40625], [520.0800000000002, 409.40625], [520.0800000000002, 410.84999999999997], [501.60000000000014, 410.84999999999997], [501.60000000000014, 409.40625]], ["M", "L", "L", "L", "L"]], [[[501.60000000000014, 410.84999999999997], [520.0800000000002, 410.84999999999997], [520.0800000000002, 412.29375], [501.60000000000014, 412.29375], [501.60000000000014, 410.84999999999997]], ["M", "L", "L", "L", "L"]], [[[501.60000000000014, 412.29375], [520.0800000000002, 412.29375], [520.0800000000002, 413.73749999999995], [501.60000000000014, 413.73749999999995], [501.60000000000014, 412.29375]], ["M", "L", "L", "L", "L"]], [[[501.60000000000014, 413.73749999999995], [520.0800000000002, 413.73749999999995], [520.0800000000002, 415.18125], [501.60000000000014, 415.18125], [501.60000000000014, 413.73749999999995]], ["M", "L", "L", "L", "L"]], [[[501.60000000000014, 415.18125], [520.0800000000002, 415.18125], 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<p>These three tests achieved a loss of <span
class="math inline">61.44</span>, <span
class="math inline">47.06</span>, and <span
class="math inline">72.87</span> respectively. We didn’t expect the
model to generalize for this task anyways since it doesn’t seem helpful
to know the distances to all vertices if the model only needs to know
the distances to only vertices on the <span class="math inline">1</span>
to <span class="math inline">2</span> path.</p>
<h2 id="conclusion">Conclusion</h2>
<p>In this post, we’ve investigated off-distribution generalization
behavior of transformers trained on the shortest path problem. We
demonstrated mathematically the existence of a transformer computing
shortest paths, and also found such a transformer from scratch via
gradient descent. We showed that a transformer trained to compute
shortest paths between two specific vertices <span
class="math inline">v_1,v_2</span> can be efficiently fine-tuned to
compute shortest paths to other vertices that lie on the shortest <span
class="math inline">v_1</span>-<span class="math inline">v_2</span>
path, suggesting that our transformers learned representations
implicitly carry rich information about the graph. Finally, we showed
that the transformer was able to generalize off-distribution quite well
in some settings, but less well in other settings. The main conceptual
take-away from our work is that it’s hard to predict when models will
and won’t generalize.</p>
<h2 id="appendix">Appendix</h2>
<div class="sourceCode" id="cb2"><pre
class="sourceCode python"><code class="sourceCode python"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> torch</span>
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> torch.nn <span class="im">as</span> nn</span>
<span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a><span class="im">import</span> random</span>
<span id="cb2-4"><a href="#cb2-4" aria-hidden="true" tabindex="-1"></a><span class="im">from</span> collections <span class="im">import</span> deque</span>
<span id="cb2-5"><a href="#cb2-5" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-6"><a href="#cb2-6" aria-hidden="true" tabindex="-1"></a><span class="co"># Configuration</span></span>
<span id="cb2-7"><a href="#cb2-7" aria-hidden="true" tabindex="-1"></a>NVTXS <span class="op">=</span> <span class="dv">8</span></span>
<span id="cb2-8"><a href="#cb2-8" aria-hidden="true" tabindex="-1"></a>MAXDIST <span class="op">=</span> NVTXS <span class="op">+</span> <span class="dv">1</span></span>
<span id="cb2-9"><a href="#cb2-9" aria-hidden="true" tabindex="-1"></a>AVGDEG <span class="op">=</span> <span class="dv">2</span></span>
<span id="cb2-10"><a href="#cb2-10" aria-hidden="true" tabindex="-1"></a>SEQLEN <span class="op">=</span> NVTXS <span class="op">+</span> <span class="dv">1</span></span>
<span id="cb2-11"><a href="#cb2-11" 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="cb2-12"><a href="#cb2-12" 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="cb2-13"><a href="#cb2-13" 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="cb2-14"><a href="#cb2-14" 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="cb2-15"><a href="#cb2-15" aria-hidden="true" tabindex="-1"></a>SRC_FLAG_IDX <span class="op">=</span> START_SELF</span>
<span id="cb2-16"><a href="#cb2-16" aria-hidden="true" tabindex="-1"></a>ANS_FLAG_IDX <span class="op">=</span> <span class="dv">0</span></span>
<span id="cb2-17"><a href="#cb2-17" 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="cb2-18"><a href="#cb2-18" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-19"><a href="#cb2-19" aria-hidden="true" tabindex="-1"></a><span class="co"># Determine device</span></span>
<span id="cb2-20"><a href="#cb2-20" aria-hidden="true" tabindex="-1"></a>device <span class="op">=</span> torch.device(<span class="st">"cuda"</span> <span class="cf">if</span> torch.cuda.is_available() <span class="cf">else</span> <span class="st">"cpu"</span>)</span>
<span id="cb2-21"><a href="#cb2-21" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-22"><a href="#cb2-22" aria-hidden="true" tabindex="-1"></a><span class="kw">def</span> random_graph(device):</span>
<span id="cb2-23"><a href="#cb2-23" aria-hidden="true" tabindex="-1"></a> <span class="co">"""Generate a random graph tensor."""</span></span>
<span id="cb2-24"><a href="#cb2-24" aria-hidden="true" tabindex="-1"></a> data <span class="op">=</span> torch.zeros((SEQLEN, HIDDENDIM), device<span class="op">=</span>device)</span>
<span id="cb2-25"><a href="#cb2-25" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb2-26"><a href="#cb2-26" aria-hidden="true" tabindex="-1"></a> <span class="co"># Mark self vertices</span></span>
<span id="cb2-27"><a href="#cb2-27" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span> i <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="cb2-28"><a href="#cb2-28" aria-hidden="true" tabindex="-1"></a> data[i, START_SELF <span class="op">-</span> <span class="dv">1</span> <span class="op">+</span> i] <span class="op">=</span> <span class="dv">1</span></span>
<span id="cb2-29"><a href="#cb2-29" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-30"><a href="#cb2-30" aria-hidden="true" tabindex="-1"></a> <span class="co"># Create adjacency list</span></span>
<span id="cb2-31"><a href="#cb2-31" aria-hidden="true" tabindex="-1"></a> adj_list <span class="op">=</span> [<span class="bu">set</span>() <span class="cf">for</span> _ <span class="kw">in</span> <span class="bu">range</span>(SEQLEN)]</span>
<span id="cb2-32"><a href="#cb2-32" aria-hidden="true" tabindex="-1"></a> indices <span class="op">=</span> [random.randint(<span class="dv">1</span>, NVTXS) <span class="cf">for</span> _ <span class="kw">in</span> <span class="bu">range</span>(AVGDEG <span class="op">*</span> NVTXS)]</span>
<span id="cb2-33"><a href="#cb2-33" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb2-34"><a href="#cb2-34" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span> i <span class="kw">in</span> <span class="bu">range</span>(<span class="dv">0</span>, <span class="bu">len</span>(indices), <span class="dv">2</span>):</span>
<span id="cb2-35"><a href="#cb2-35" aria-hidden="true" tabindex="-1"></a> u <span class="op">=</span> indices[i]</span>
<span id="cb2-36"><a href="#cb2-36" aria-hidden="true" tabindex="-1"></a> v <span class="op">=</span> indices[i <span class="op">+</span> <span class="dv">1</span>]</span>
<span id="cb2-37"><a href="#cb2-37" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> u <span class="op">!=</span> v:</span>
<span id="cb2-38"><a href="#cb2-38" aria-hidden="true" tabindex="-1"></a> <span class="co"># Bidirectional connections</span></span>
<span id="cb2-39"><a href="#cb2-39" aria-hidden="true" tabindex="-1"></a> data[v, u] <span class="op">=</span> <span class="dv">1</span></span>
<span id="cb2-40"><a href="#cb2-40" aria-hidden="true" tabindex="-1"></a> data[u, v] <span class="op">=</span> <span class="dv">1</span></span>
<span id="cb2-41"><a href="#cb2-41" aria-hidden="true" tabindex="-1"></a> data[v, NVTXS <span class="op">+</span> u] <span class="op">=</span> <span class="dv">1</span></span>
<span id="cb2-42"><a href="#cb2-42" aria-hidden="true" tabindex="-1"></a> data[u, NVTXS <span class="op">+</span> v] <span class="op">=</span> <span class="dv">1</span></span>
<span id="cb2-43"><a href="#cb2-43" aria-hidden="true" tabindex="-1"></a> adj_list[u].add(v)</span>
<span id="cb2-44"><a href="#cb2-44" aria-hidden="true" tabindex="-1"></a> adj_list[v].add(u)</span>
<span id="cb2-45"><a href="#cb2-45" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-46"><a href="#cb2-46" aria-hidden="true" tabindex="-1"></a> <span class="co"># Set flags</span></span>
<span id="cb2-47"><a href="#cb2-47" aria-hidden="true" tabindex="-1"></a> data[<span class="dv">0</span>, ANS_FLAG_IDX] <span class="op">=</span> <span class="dv">1</span></span>
<span id="cb2-48"><a href="#cb2-48" aria-hidden="true" tabindex="-1"></a> data[<span class="dv">1</span>:, NOTANS_FLAG_IDX] <span class="op">=</span> <span class="dv">1</span></span>
<span id="cb2-49"><a href="#cb2-49" aria-hidden="true" tabindex="-1"></a> data[<span class="dv">0</span>, START_REACH:START_REACH <span class="op">+</span> NVTXS] <span class="op">=</span> <span class="dv">1</span></span>
<span id="cb2-50"><a href="#cb2-50" aria-hidden="true" tabindex="-1"></a> <span class="cf">return</span> data, adj_list</span>
<span id="cb2-51"><a href="#cb2-51" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-52"><a href="#cb2-52" aria-hidden="true" tabindex="-1"></a><span class="kw">def</span> SSSP(G):</span>
<span id="cb2-53"><a href="#cb2-53" aria-hidden="true" tabindex="-1"></a> <span class="co">"""Single Source Shortest Path algorithm."""</span></span>
<span id="cb2-54"><a href="#cb2-54" aria-hidden="true" tabindex="-1"></a> dist <span class="op">=</span> [MAXDIST <span class="cf">for</span> _ <span class="kw">in</span> G]</span>
<span id="cb2-55"><a href="#cb2-55" aria-hidden="true" tabindex="-1"></a> dist[<span class="dv">1</span>] <span class="op">=</span> <span class="dv">0</span></span>
<span id="cb2-56"><a href="#cb2-56" aria-hidden="true" tabindex="-1"></a> frontier <span class="op">=</span> deque([<span class="dv">1</span>])</span>
<span id="cb2-57"><a href="#cb2-57" aria-hidden="true" tabindex="-1"></a> <span class="cf">while</span> frontier:</span>
<span id="cb2-58"><a href="#cb2-58" aria-hidden="true" tabindex="-1"></a> vtx <span class="op">=</span> frontier.popleft()</span>
<span id="cb2-59"><a href="#cb2-59" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span> x <span class="kw">in</span> G[vtx]:</span>
<span id="cb2-60"><a href="#cb2-60" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> dist[x] <span class="op">==</span> MAXDIST:</span>
<span id="cb2-61"><a href="#cb2-61" aria-hidden="true" tabindex="-1"></a> dist[x] <span class="op">=</span> <span class="dv">1</span> <span class="op">+</span> dist[vtx]</span>
<span id="cb2-62"><a href="#cb2-62" aria-hidden="true" tabindex="-1"></a> frontier.append(x)</span>
<span id="cb2-63"><a href="#cb2-63" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> x <span class="op">==</span> <span class="dv">2</span>:</span>
<span id="cb2-64"><a href="#cb2-64" aria-hidden="true" tabindex="-1"></a> <span class="cf">return</span> dist[<span class="dv">2</span>]</span>
<span id="cb2-65"><a href="#cb2-65" aria-hidden="true" tabindex="-1"></a> <span class="cf">return</span> MAXDIST</span>
<span id="cb2-66"><a href="#cb2-66" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-67"><a href="#cb2-67" aria-hidden="true" tabindex="-1"></a><span class="kw">def</span> mkbatch(size):</span>
<span id="cb2-68"><a href="#cb2-68" aria-hidden="true" tabindex="-1"></a> <span class="co">"""Create a batch of graph data."""</span></span>
<span id="cb2-69"><a href="#cb2-69" aria-hidden="true" tabindex="-1"></a> graphs <span class="op">=</span> []</span>
<span id="cb2-70"><a href="#cb2-70" aria-hidden="true" tabindex="-1"></a> distances <span class="op">=</span> []</span>
<span id="cb2-71"><a href="#cb2-71" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-72"><a href="#cb2-72" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span> _ <span class="kw">in</span> <span class="bu">range</span>(size):</span>
<span id="cb2-73"><a href="#cb2-73" aria-hidden="true" tabindex="-1"></a> data, adj_list <span class="op">=</span> random_graph(device)</span>
<span id="cb2-74"><a href="#cb2-74" aria-hidden="true" tabindex="-1"></a> dist <span class="op">=</span> SSSP(adj_list)</span>
<span id="cb2-75"><a href="#cb2-75" aria-hidden="true" tabindex="-1"></a> graphs.append(data)</span>
<span id="cb2-76"><a href="#cb2-76" aria-hidden="true" tabindex="-1"></a> distances.append(dist)</span>
<span id="cb2-77"><a href="#cb2-77" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-78"><a href="#cb2-78" aria-hidden="true" tabindex="-1"></a> data <span class="op">=</span> torch.stack(graphs)</span>
<span id="cb2-79"><a href="#cb2-79" aria-hidden="true" tabindex="-1"></a> labels <span class="op">=</span> torch.tensor(distances, dtype<span class="op">=</span>torch.float32, device<span class="op">=</span>device)</span>
<span id="cb2-80"><a href="#cb2-80" aria-hidden="true" tabindex="-1"></a> <span class="cf">return</span> data, labels</span>
<span id="cb2-81"><a href="#cb2-81" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb2-82"><a href="#cb2-82" aria-hidden="true" tabindex="-1"></a>BIG,SUPABIG,MED,CURSE <span class="op">=</span> <span class="dv">12</span>,<span class="dv">30</span>,<span class="dv">7</span>,<span class="dv">5</span></span>
<span id="cb2-83"><a href="#cb2-83" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-84"><a href="#cb2-84" aria-hidden="true" tabindex="-1"></a><span class="kw">class</span> SillyTransformer(nn.Module):</span>
<span id="cb2-85"><a href="#cb2-85" 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="cb2-86"><a href="#cb2-86" aria-hidden="true" tabindex="-1"></a> <span class="bu">super</span>().<span class="fu">__init__</span>()</span>
<span id="cb2-87"><a href="#cb2-87" aria-hidden="true" tabindex="-1"></a> <span class="va">self</span>.device <span class="op">=</span> device</span>
<span id="cb2-88"><a href="#cb2-88" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-89"><a href="#cb2-89" aria-hidden="true" tabindex="-1"></a> <span class="cf">with</span> torch.no_grad():</span>
<span id="cb2-90"><a href="#cb2-90" aria-hidden="true" tabindex="-1"></a> <span class="co"># Initialize weight parameters with specific configurations</span></span>
<span id="cb2-91"><a href="#cb2-91" aria-hidden="true" tabindex="-1"></a> <span class="va">self</span>.mostKs <span class="op">=</span> nn.ParameterList()</span>
<span id="cb2-92"><a href="#cb2-92" aria-hidden="true" tabindex="-1"></a> <span class="va">self</span>.mostQs <span class="op">=</span> nn.ParameterList()</span>
<span id="cb2-93"><a href="#cb2-93" aria-hidden="true" tabindex="-1"></a> <span class="va">self</span>.mostVs <span class="op">=</span> nn.ParameterList()</span>
<span id="cb2-94"><a href="#cb2-94" 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="cb2-95"><a href="#cb2-95" 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="cb2-96"><a href="#cb2-96" 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="cb2-97"><a href="#cb2-97" 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="cb2-98"><a href="#cb2-98" 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="cb2-99"><a href="#cb2-99" 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="cb2-100"><a href="#cb2-100" aria-hidden="true" tabindex="-1"></a> K[<span class="dv">1</span>, ANS_FLAG_IDX] <span class="op">=</span> BIG</span>
<span id="cb2-101"><a href="#cb2-101" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-102"><a href="#cb2-102" 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="cb2-103"><a href="#cb2-103" 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="cb2-104"><a href="#cb2-104" 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="cb2-105"><a href="#cb2-105" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-106"><a href="#cb2-106" aria-hidden="true" tabindex="-1"></a> <span class="va">self</span>.mostKs.append(K)</span>
<span id="cb2-107"><a href="#cb2-107" aria-hidden="true" tabindex="-1"></a> <span class="va">self</span>.mostQs.append(Q)</span>
<span id="cb2-108"><a href="#cb2-108" aria-hidden="true" tabindex="-1"></a> <span class="va">self</span>.mostVs.append(V)</span>
<span id="cb2-109"><a href="#cb2-109" aria-hidden="true" tabindex="-1"></a> <span class="va">self</span>.weirdKs <span class="op">=</span> nn.ParameterList()</span>
<span id="cb2-110"><a href="#cb2-110" aria-hidden="true" tabindex="-1"></a> <span class="va">self</span>.weirdQs <span class="op">=</span> nn.ParameterList()</span>
<span id="cb2-111"><a href="#cb2-111" aria-hidden="true" tabindex="-1"></a> <span class="va">self</span>.weirdVs <span class="op">=</span> nn.ParameterList()</span>
<span id="cb2-112"><a href="#cb2-112" 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="cb2-113"><a href="#cb2-113" 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="cb2-114"><a href="#cb2-114" 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="cb2-115"><a href="#cb2-115" 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="cb2-116"><a href="#cb2-116" 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="cb2-117"><a href="#cb2-117" 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="cb2-118"><a href="#cb2-118" 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="cb2-119"><a href="#cb2-119" aria-hidden="true" tabindex="-1"></a> K[<span class="dv">2</span>, ANS_FLAG_IDX] <span class="op">=</span> MED</span>
<span id="cb2-120"><a href="#cb2-120" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-121"><a href="#cb2-121" 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="cb2-122"><a href="#cb2-122" aria-hidden="true" tabindex="-1"></a> Q[:, ANS_FLAG_IDX] <span class="op">=</span> <span class="dv">1</span></span>
<span id="cb2-123"><a href="#cb2-123" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-124"><a href="#cb2-124" 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="cb2-125"><a href="#cb2-125" aria-hidden="true" tabindex="-1"></a> V[layer, SRC_FLAG_IDX] <span class="op">=</span> <span class="dv">1</span></span>
<span id="cb2-126"><a href="#cb2-126" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-127"><a href="#cb2-127" aria-hidden="true" tabindex="-1"></a> <span class="va">self</span>.weirdKs.append(K)</span>
<span id="cb2-128"><a href="#cb2-128" aria-hidden="true" tabindex="-1"></a> <span class="va">self</span>.weirdQs.append(Q)</span>
<span id="cb2-129"><a href="#cb2-129" aria-hidden="true" tabindex="-1"></a> <span class="va">self</span>.weirdVs.append(V)</span>
<span id="cb2-130"><a href="#cb2-130" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-131"><a href="#cb2-131" aria-hidden="true" tabindex="-1"></a> <span class="kw">def</span> forward(<span class="va">self</span>, src):</span>
<span id="cb2-132"><a href="#cb2-132" 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="cb2-133"><a href="#cb2-133" 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="cb2-134"><a href="#cb2-134" 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="cb2-135"><a href="#cb2-135" 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="cb2-136"><a href="#cb2-136" aria-hidden="true" tabindex="-1"></a> head_outputs <span class="op">=</span> []</span>
<span id="cb2-137"><a href="#cb2-137" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb2-138"><a href="#cb2-138" 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="cb2-139"><a href="#cb2-139" 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="cb2-140"><a href="#cb2-140" 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="cb2-141"><a href="#cb2-141" 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="cb2-142"><a href="#cb2-142" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-143"><a href="#cb2-143" 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="cb2-144"><a href="#cb2-144" 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="cb2-145"><a href="#cb2-145" aria-hidden="true" tabindex="-1"></a> head_output <span class="op">=</span> torch.matmul(attention_weights, vsrc)</span>
<span id="cb2-146"><a href="#cb2-146" aria-hidden="true" tabindex="-1"></a> head_outputs.append(head_output)</span>
<span id="cb2-147"><a href="#cb2-147" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-148"><a href="#cb2-148" 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="cb2-149"><a href="#cb2-149" aria-hidden="true" tabindex="-1"></a> BSZ <span class="op">=</span> new_reaches.shape[<span class="dv">0</span>]</span>
<span id="cb2-150"><a href="#cb2-150" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-151"><a href="#cb2-151" 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="cb2-152"><a href="#cb2-152" 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="cb2-153"><a href="#cb2-153" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-154"><a href="#cb2-154" 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="cb2-155"><a href="#cb2-155" 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="cb2-156"><a href="#cb2-156" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-157"><a href="#cb2-157" 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="cb2-158"><a href="#cb2-158" 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="cb2-159"><a href="#cb2-159" aria-hidden="true" tabindex="-1"></a> <span class="cf">return</span> final_output</span>
<span id="cb2-160"><a href="#cb2-160" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-161"><a href="#cb2-161" aria-hidden="true" tabindex="-1"></a>model <span class="op">=</span> SillyTransformer(device).to(device)</span>
<span id="cb2-162"><a href="#cb2-162" aria-hidden="true" tabindex="-1"></a>params <span class="op">=</span> <span class="bu">sum</span>(p.numel() <span class="cf">for</span> p <span class="kw">in</span> model.parameters())</span>
<span id="cb2-163"><a href="#cb2-163" aria-hidden="true" tabindex="-1"></a><span class="bu">print</span>(<span class="ss">f"Total number of parameters: </span><span class="sc">{</span>params<span class="sc">}</span><span class="ss">"</span>)</span>
<span id="cb2-164"><a href="#cb2-164" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-165"><a href="#cb2-165" aria-hidden="true" tabindex="-1"></a><span class="kw">def</span> destroy_rand_weights(model):</span>
<span id="cb2-166"><a href="#cb2-166" aria-hidden="true" tabindex="-1"></a> weight_lists <span class="op">=</span> [model.mostKs, model.mostQs, model.mostVs, </span>
<span id="cb2-167"><a href="#cb2-167" aria-hidden="true" tabindex="-1"></a> model.weirdKs, model.weirdQs, model.weirdVs]</span>
<span id="cb2-168"><a href="#cb2-168" aria-hidden="true" tabindex="-1"></a> random_list <span class="op">=</span> random.choice(weight_lists)</span>
<span id="cb2-169"><a href="#cb2-169" aria-hidden="true" tabindex="-1"></a> random_matrix <span class="op">=</span> random.choice(random_list)</span>
<span id="cb2-170"><a href="#cb2-170" aria-hidden="true" tabindex="-1"></a> random_matrix.data <span class="op">=</span> torch.randn_like(random_matrix)</span>
<span id="cb2-171"><a href="#cb2-171" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb2-172"><a href="#cb2-172" aria-hidden="true" tabindex="-1"></a>optimizer <span class="op">=</span> torch.optim.Adam(model.parameters(), lr<span class="op">=</span><span class="fl">1e-6</span>)</span>
<span id="cb2-173"><a href="#cb2-173" aria-hidden="true" tabindex="-1"></a>loss_fn <span class="op">=</span> nn.MSELoss()</span></code></pre></div>
<h2 id="references">References</h2>
<section id="footnotes" class="footnotes footnotes-end-of-document"
role="doc-endnotes">
<hr />
<ol>
<li id="fn1"><p>Lin, Hilton, and Evans. “TruthfulQA: Measuring How
Models Mimic Human Falsehoods”<a href="#fnref1" class="footnote-back"
role="doc-backlink">↩︎</a></p></li>
<li id="fn2"><p>Nanda, Neel, Lawrence Chan, Tom Lieberum, Jess Smith,
and Jacob Steinhardt. “Progress measures for grokking via mechanistic
interpretability.” arXiv preprint arXiv:2301.05217 (2023).
https://arxiv.org/abs/2301.05217.<a href="#fnref2" class="footnote-back"
role="doc-backlink">↩︎</a></p></li>
</ol>
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