1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
430
431
432
433
434
435
436
437
438
439
440
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
|
<!DOCTYPE html>
<html xmlns="http://www.w3.org/1999/xhtml" lang="" xml:lang="">
<head>
<meta charset="utf-8" />
<meta name="generator" content="pandoc" />
<meta name="viewport" content="width=device-width, initial-scale=1.0, user-scalable=yes" />
<title>6.7960 Project: Investigating Off-Distribution Generalization of Transformers</title>
<style>
html {
color: #1a1a1a;
background-color: #fdfdfd;
}
body {
margin: 0 auto;
max-width: 36em;
padding-left: 50px;
padding-right: 50px;
padding-top: 50px;
padding-bottom: 50px;
hyphens: auto;
overflow-wrap: break-word;
text-rendering: optimizeLegibility;
font-kerning: normal;
}
@media (max-width: 600px) {
body {
font-size: 0.9em;
padding: 12px;
}
h1 {
font-size: 1.8em;
}
}
@media print {
html {
background-color: white;
}
body {
background-color: transparent;
color: black;
font-size: 12pt;
}
p, h2, h3 {
orphans: 3;
widows: 3;
}
h2, h3, h4 {
page-break-after: avoid;
}
}
p {
margin: 1em 0;
}
a {
color: #1a1a1a;
}
a:visited {
color: #1a1a1a;
}
img {
max-width: 100%;
}
svg {
height: auto;
max-width: 100%;
}
h1, h2, h3, h4, h5, h6 {
margin-top: 1.4em;
}
h5, h6 {
font-size: 1em;
font-style: italic;
}
h6 {
font-weight: normal;
}
ol, ul {
padding-left: 1.7em;
margin-top: 1em;
}
li > ol, li > ul {
margin-top: 0;
}
blockquote {
margin: 1em 0 1em 1.7em;
padding-left: 1em;
border-left: 2px solid #e6e6e6;
color: #606060;
}
code {
font-family: Menlo, Monaco, Consolas, 'Lucida Console', monospace;
font-size: 85%;
margin: 0;
hyphens: manual;
}
pre {
margin: 1em 0;
overflow: auto;
}
pre code {
padding: 0;
overflow: visible;
overflow-wrap: normal;
}
.sourceCode {
background-color: transparent;
overflow: visible;
}
hr {
background-color: #1a1a1a;
border: none;
height: 1px;
margin: 1em 0;
}
table {
margin: 1em 0;
border-collapse: collapse;
width: 100%;
overflow-x: auto;
display: block;
font-variant-numeric: lining-nums tabular-nums;
}
table caption {
margin-bottom: 0.75em;
}
tbody {
margin-top: 0.5em;
border-top: 1px solid #1a1a1a;
border-bottom: 1px solid #1a1a1a;
}
th {
border-top: 1px solid #1a1a1a;
padding: 0.25em 0.5em 0.25em 0.5em;
}
td {
padding: 0.125em 0.5em 0.25em 0.5em;
}
header {
margin-bottom: 4em;
text-align: center;
}
#TOC li {
list-style: none;
}
#TOC ul {
padding-left: 1.3em;
}
#TOC > ul {
padding-left: 0;
}
#TOC a:not(:hover) {
text-decoration: none;
}
code{white-space: pre-wrap;}
span.smallcaps{font-variant: small-caps;}
div.columns{display: flex; gap: min(4vw, 1.5em);}
div.column{flex: auto; overflow-x: auto;}
div.hanging-indent{margin-left: 1.5em; text-indent: -1.5em;}
/* The extra [class] is a hack that increases specificity enough to
override a similar rule in reveal.js */
ul.task-list[class]{list-style: none;}
ul.task-list li input[type="checkbox"] {
font-size: inherit;
width: 0.8em;
margin: 0 0.8em 0.2em -1.6em;
vertical-align: middle;
}
/* CSS for syntax highlighting */
pre > code.sourceCode { white-space: pre; position: relative; }
pre > code.sourceCode > span { line-height: 1.25; }
pre > code.sourceCode > span:empty { height: 1.2em; }
.sourceCode { overflow: visible; }
code.sourceCode > span { color: inherit; text-decoration: inherit; }
div.sourceCode { margin: 1em 0; }
pre.sourceCode { margin: 0; }
@media screen {
div.sourceCode { overflow: auto; }
}
@media print {
pre > code.sourceCode { white-space: pre-wrap; }
pre > code.sourceCode > span { display: inline-block; text-indent: -5em; padding-left: 5em; }
}
pre.numberSource code
{ counter-reset: source-line 0; }
pre.numberSource code > span
{ position: relative; left: -4em; counter-increment: source-line; }
pre.numberSource code > span > a:first-child::before
{ content: counter(source-line);
position: relative; left: -1em; text-align: right; vertical-align: baseline;
border: none; display: inline-block;
-webkit-touch-callout: none; -webkit-user-select: none;
-khtml-user-select: none; -moz-user-select: none;
-ms-user-select: none; user-select: none;
padding: 0 4px; width: 4em;
color: #aaaaaa;
}
pre.numberSource { margin-left: 3em; border-left: 1px solid #aaaaaa; padding-left: 4px; }
div.sourceCode
{ }
@media screen {
pre > code.sourceCode > span > a:first-child::before { text-decoration: underline; }
}
code span.al { color: #ff0000; font-weight: bold; } /* Alert */
code span.an { color: #60a0b0; font-weight: bold; font-style: italic; } /* Annotation */
code span.at { color: #7d9029; } /* Attribute */
code span.bn { color: #40a070; } /* BaseN */
code span.bu { color: #008000; } /* BuiltIn */
code span.cf { color: #007020; font-weight: bold; } /* ControlFlow */
code span.ch { color: #4070a0; } /* Char */
code span.cn { color: #880000; } /* Constant */
code span.co { color: #60a0b0; font-style: italic; } /* Comment */
code span.cv { color: #60a0b0; font-weight: bold; font-style: italic; } /* CommentVar */
code span.do { color: #ba2121; font-style: italic; } /* Documentation */
code span.dt { color: #902000; } /* DataType */
code span.dv { color: #40a070; } /* DecVal */
code span.er { color: #ff0000; font-weight: bold; } /* Error */
code span.ex { } /* Extension */
code span.fl { color: #40a070; } /* Float */
code span.fu { color: #06287e; } /* Function */
code span.im { color: #008000; font-weight: bold; } /* Import */
code span.in { color: #60a0b0; font-weight: bold; font-style: italic; } /* Information */
code span.kw { color: #007020; font-weight: bold; } /* Keyword */
code span.op { color: #666666; } /* Operator */
code span.ot { color: #007020; } /* Other */
code span.pp { color: #bc7a00; } /* Preprocessor */
code span.sc { color: #4070a0; } /* SpecialChar */
code span.ss { color: #bb6688; } /* SpecialString */
code span.st { color: #4070a0; } /* String */
code span.va { color: #19177c; } /* Variable */
code span.vs { color: #4070a0; } /* VerbatimString */
code span.wa { color: #60a0b0; font-weight: bold; font-style: italic; } /* Warning */
/* CSS for citations */
div.csl-bib-body { }
div.csl-entry {
clear: both;
margin-bottom: 0em;
}
.hanging-indent div.csl-entry {
margin-left:2em;
text-indent:-2em;
}
div.csl-left-margin {
min-width:2em;
float:left;
}
div.csl-right-inline {
margin-left:2em;
padding-left:1em;
}
div.csl-indent {
margin-left: 2em;
} </style>
<script defer=""
src="https://cdn.jsdelivr.net/npm/katex@0.15.1/dist/katex.min.js"></script>
<script>document.addEventListener("DOMContentLoaded", function () {
var mathElements = document.getElementsByClassName("math");
var macros = [];
for (var i = 0; i < mathElements.length; i++) {
var texText = mathElements[i].firstChild;
if (mathElements[i].tagName == "SPAN") {
katex.render(texText.data, mathElements[i], {
displayMode: mathElements[i].classList.contains('display'),
throwOnError: false,
macros: macros,
fleqn: false
});
}}});
</script>
<link rel="stylesheet"
href="https://cdn.jsdelivr.net/npm/katex@0.15.1/dist/katex.min.css" />
</head>
<body>
<header id="title-block-header">
<h1 class="title">6.7960 Project: Investigating Off-Distribution
Generalization of Transformers</h1>
</header>
<!-- Guidelines: https://www.dropbox.com/scl/fi/bet8enscln8ue36kd8t17/final_project_guidelines.pdf?rlkey=knd19cnumk51ho1y9crno56ib&e=2&dl=0 -->
<div style="text-align:center">
<p>Anthony Wang, Alek Westover, Kevin Zhao</p>
<p>{xy,alekw,kevinmz}@mit.edu</p>
</div>
<h2 id="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>
<p>It looked like the fine tuning results weren’t as meaningful because
TODO: ALEK</p>
<h2 id="customizing-a-transformer">Customizing a Transformer</h2>
<p>After much deliberation, we decided the next step for us was to
customize a transformer, writing it ourselves. We observed that we
wished for the transformer to do similar work as a BFS. As a result, we
decided to work with the following transformer, for a graph with <span
class="math inline">n</span> vertices <span class="math inline">v_1,
v_2, \cdots, v_n</span>:</p>
<span class="math display">\begin{array}{|c|c|c|c|c|c}
\text{ANS} & v_{1} & v_{2} & \cdots & v_{n} & \\
\hline
1 & 0 & 0 & \cdots & 0 & \text{ANS}\\ \hline
\text{ANS} & \text{NBR}_{1} & \text{NBR}_{2} & \cdots &
\text{NBR}_{n} & \text{NBR}\\ \hline
\text{ANS} & \text{REACH}_{1} & \text{REACH}_{2} & \cdots
& \text{REACH}_{n} & \text{REACH}\\ \hline
\text{ANS} & \text{SELF}_{1} & \text{SELF}_{2} & \cdots
& \text{SELF}_{n} & \text{SELF}\\ \hline
V_{\text{OUT}} & NULL& NULL& NULL& NULL&
\text{OUT}\\ \hline
0 & 1 & 1 & \cdots &1 & \text{NOT}\\ \hline
\end{array}</span>
<p>Specifically, we see that <span
class="math inline">\text{NBR}_{i}</span> is a <span
class="math inline">n \times 1</span> vector detailing which of the
vertices are neighboring vertex <span class="math inline">v_i</span>, so
the <span class="math inline">j</span>th element of <span
class="math inline">v_i</span> is <span class="math inline">1</span> if
<span class="math inline">v_i</span> and <span
class="math inline">v_j</span> are neighboring vertices, and <span
class="math inline">0</span> otherwise. Additionally, <span
class="math inline">\text{SELF}_{i}</span> is just the <span
class="math inline">n \times 1</span> vector with the <span
class="math inline">i</span>th element <span
class="math inline">1</span> and all other elements <span
class="math inline">0</span> (e.g. the one-hot encoding of the vector).
Now, at every step, the <span class="math inline">\text{REACH}_k</span>
vector for all <span class="math inline">k</span> is updated based on
the previous <span class="math inline">\text{REACH}_k</span> vector and
<span class="math inline">\text{NBR}_{k}</span> (since all entries that
are <span class="math inline">1</span> in <span
class="math inline">\text{REACH}_k\text{NBR}_{k}^T</span> must be
updated in the manner such that if the <span class="math inline">(i,
j)</span>th element of <span
class="math inline">\text{REACH}_k\text{NBR}_{k}^T</span> is <span
class="math inline">1</span>, then <span
class="math inline">\text{REACH}_i</span>’s <span
class="math inline">j</span>th column is set to <span
class="math inline">1</span>. This is equivalent to adding <span
class="math inline">\text{REACH}_k</span> to each integer <span
class="math inline">i</span> where <span
class="math inline">\text{NBR}_{k}</span>’s <span
class="math inline">i</span>th entry is nonzero.</p>
<p>This iterates through all the vertices, and at the end, we may see
what run we are on to update <span
class="math inline">V_{\text{OUT}}</span>.</p>
<h2 id="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>
</div>
</body>
</html>
|
