-
Notifications
You must be signed in to change notification settings - Fork 8
/
NSF-Simons-MoDL_24.html
573 lines (318 loc) · 13.2 KB
/
NSF-Simons-MoDL_24.html
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
<!DOCTYPE html>
<html>
<head>
<title>PL</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8" />
<link rel="stylesheet" href="fonts/quadon/quadon.css">
<link rel="stylesheet" href="fonts/gentona/gentona.css">
<!-- <link rel="stylesheet" href="slides_style_v2.css"> -->
<link rel="stylesheet" href="slides_style_i.css">
<script type="text/javascript" src="assets/plotly/plotly-latest.min.js"></script>
</head>
<body>
<textarea id="source">
#### Probably approximately correct in the future:<br> "Prospective Learning"
<br><br>
Joshua T. Vogelstein <br>
<!-- , [JHU](https://www.jhu.edu/) <br> -->
<!-- Co-PI: Vova Braverman, [JHU](https://www.jhu.edu/) <br> -->
Ashwin de Silva, Rahul Ramesh, Rubing Yang, Pratik Chaudhari
<!-- | Joshua T. Vogelstein <br> -->
<!-- [Microsoft Research](https://www.microsoft.com/en-us/research/): Weiwei Yang | Jonathan Larson | Bryan Tower | Chris White -->
.ye[In memory of Sheldon Caplis]
<img src="images/neurodata_blue.png" width="20%" style="vertical-align: top; " >
<!-- <img src="images/jhu.png" width="8%" style="vertical-align: top"> -->
---
#### Outline
- Motivation
- Formalization
- Theorization
- Experimentation
- Deliberation
---
# .center[Motivation]
---
#### What the &*#*@ is learning?
--
- Learning is an evolved property
--
- It enabled organisms to make better decisions (on average, in their niche), about how to act *in the future* based on the past
--
- This works because the future was (at least) partially predictable
--
- Biology evolved many different learning algorithms for different contexts
- behavioral learning
- associational learning
- reinforcement learning
- sensorimotor learning
- imitation learning
---
#### How do we *model* learning?
- What we call "learning" in AI is a formal model of a natural phenomenon
- And, there are many complimentary formal definitions, e.g.,
- PAC learning
- Online Learning
- Reinforcement learning
- But, it is not actually "learning" as observed in the world, it is a model
- George Box: "all models are wrong, some are useful"
---
#### Probably Almost Correct Learning
--
- Nearly 100 years old model
--
- Work horse of modern AI revolution (useful)
--
- Yet, the assumptions (IID) are wrong (and dumb)
---
#### Can we do any better?
- We start from a different assumption
- Data: random process, not random variable
- Goal: dynamic objective, not fixed objective
- This more complicated model reduces bias and adds variance
- Let's see some examples
- In each, $Z$ is a Bernoulli process
---
#### The classic
$Z_t$ is IID
<img src="images/pl_case1_seq.png" width="640">
---
#### The switcher
$Z_t$ is independent, but not identical
<img src="images/pl_case2_seq.png" width="640">
---
#### The data dependent switcher
$Z_t$ neither independent nor identically distributed
<img src="images/pl_case3_seq.png" width="640">
---
#### The decision dependent switcher
$Z_t$ depends on past data and decisions
<img src="images/pl_case4_seq.png" width="640">
---
#### Focus
The remainder of this talk will focus on the case where:
- $Z\_t \sim F\_t$,
- where $Z\_t \perp Z\_{t'}$ and
- $F\_t \neq F\_{t'}$ for some $t \neq t'$
---
# .center[Formalization]
---
#### Data model
- $z_t = (x_t, y_t) \in \mathcal{X} \times \mathcal{Y}$
- $z = (z\_t)\_{t \in \mathbb{N}}$ is a realization of a stochastic process $Z = (Z\_t)\_{t \in \mathbb{N}}$
- let $z\_{\leq t}$ denote the past and $z\_{>t}$ denote the future
---
#### Hypothesis class
- hypothesis sequence $h=(h\_t)\_{t \in \mathbb{N}}$
- $h\_t \in \mathcal{Y}^{\mathcal{X}} \subset \mathcal{H}_t$
- $\mathcal{H} := \mathcal{H}\_1 \subseteq \mathcal{H}\_2 \subseteq \mathcal{H}\_3 \cdots$
- $h \in \mathcal{H} \subset (\mathcal{Y}^{\mathcal{X}})^{\mathbb{N}}$
---
#### Learner
- Map from history to hypothesis sequence:
$$L: z_{\leq t} \mapsto h$$
---
#### Prospective loss, Risk, and expected Risk
- Prospective loss:
$$
\bar \ell\(h, Z) = \limsup\_{\tau \to \infty} \frac{1}{\tau} \sum\_{s=1}^{\tau} \ell (s, h\_{s} (X\_{s}), Y\_{s})
$$
where $\ell: \mathbb{N} \times \mathcal{Y} \times \mathcal{Y} \mapsto [0,1]$ is a bounded, monotonically decaying (in time) loss function.
- Prospective risk at time $t$ is, for example, expected prospective loss
$$R\_t(h)
= \mathbb{E} [\bar \ell(h,Z) \mid z\_{\leq t}] = \int \bar \ell(h,Z) \mathrm{d}{\mathbb{P}\_{Z \mid z\_{\leq t}}},$$
- Expected prospective risk at time $t$ integrates out the history
$$\mathbb{E} [R\_t(h)] = \int R\_t(h) \mathrm{d}{\mathbb{P}\_{Z\_{\leq t}}}$$
---
#### Prospective Bayes risk
A hypothesis sequence that achieves the minimal possible prospective risk, given the past, as a Bayes optimal hypothesis:
$$
R\_t^* = \inf\_{h\in \sigma(Z\_{\leq t})} R\_t(h)
$$
A Bayes optimal learner selects a Bayes optimal hypothesis sequence at every time $t$.
---
#### Components Prospective Learning
- Data: $z = (z\_t)\_{t \in \mathbb{N}}$ is a realization of a stochastic process $Z = (Z\_t)\_{t \in \mathbb{N}}$
- Hypothesis sequence: $h=(h\_t)\_{t \in \mathbb{N}} \in \mathcal{H} \subset (\mathcal{Y}^{\mathcal{X}})^{\mathbb{N}}$, where $h\_t \in \mathcal{Y}^{\mathcal{X}}$
- Learner: $L: z_{\leq t} \mapsto h$
- Prospective loss:
$
\bar \ell(h, Z) = \limsup\_{\tau \to \infty} \frac{1}{\tau} \sum\_{s=1}^{\tau} \ell (s, h\_{s} (X\_{s}), Y\_{s})
$
- Prospective risk:
$R\_t(h)
= \mathbb{E} [\bar \ell(h,Z) \mid z\_{\leq t}] = \int \bar \ell(h,Z) \mathrm{d}{\mathbb{P}\_{Z \mid z\_{\leq t}}},$
- Expected prospective risk:
$\mathbb{E} [R\_t(h)] = \int R\_t(h) \mathrm{d}{\mathbb{P}\_{Z\_{\leq t}}}$
---
#### Strong Prospective Learnability
<!-- <img src="images/strong_PL2.png" width="640"> -->
A family of stochastic processes is strongly prospectively learnable, <br>
if for any stochastic process $Z$ from this family, <br>
there exists a learner that outputs a sequence of hypotheses $h$, and a finite $t'$ <br>
such that for any $t > t'$, $\epsilon > 0$ and $\delta > 0$,
<!-- A family of stochastic processes $\mathcal{Z}$ is strongly prospectively learnable, <br> -->
<!-- if there exists a learner $L$ and a finite time $t'$ where, <br> -->
<!-- for every stochastic process $Z \in \mathcal{Z}$, <br> -->
<!-- $L$ outputs a sequence of hypotheses $h$, <br> -->
<!-- such that for any $t > t'$ and $\epsilon, \delta > 0$, -->
$$
\mathbb{P} [R_t(h) - R^*_t < \epsilon] \geq 1 - \delta.
$$
Key differences with Strong PAC Learning:
- Risk is integrated over the future
- Requires prospecting about (1) what the future will be like, and (2) what we will be like
---
#### Weak Prospective Learnability
<!-- <img src="images/weak_PL2.png" width="640"> -->
<!-- A family of stochastic processes $\mathcal{Z}$ is weakly prospectively learnable, <br>
if there exists a learner $L$, a finite time $t'$, and an $\epsilon > 0$ where, <br>
for every stochastic process $Z \in \mathcal{Z}$, <br>
$L$ outputs a sequence of hypotheses $h$, <br>
such that for any $t > t'$ and $\delta > 0$, -->
A family of stochastic processes is weakly prospectively learnable, <br>
if for any stochastic process $Z$ from this family, <br>
there exists an $\epsilon > 0$, a learner that outputs a sequence of hypotheses $h$, and a finite $t'$
<br>such that for any $t>t'$ and $\delta > 0$,
$$
\mathbb{P} [R^0_t - R_t(h) > \epsilon] \geq 1-\delta.
$$
where $R^0_t$ is the risk of the learner that always outputs $h:=\mathbb{E}[Y]$.
<!-- where $L\_{ERM} : \mathcal{D} \mapsto \mathcal{H}$ be the ERM learner, so $\bar{h}\_0^{t'} = L\_{ERM}(D\_{t'})$. -->
<!-- Key additional differences with Weak PAC Learning: -->
<!-- - we compare to an ERM learner, meaning it does not include time -->
---
# .center[Theorization]
---
#### Empirical Prospective Risk Minimization
- Time agnostic EPRM (informal):
$$ z\_{\leq t} \rightarrow \hat{h} = (\hat{h}\_{\emptyset}, \hat{h}\_{\emptyset}, \hat{h}\_{\emptyset},\cdots)$$
--
- Time aware EPRM (informal):
$$ z\_{\leq t} \rightarrow \hat{h} = (\hat{h}\_1, \hat{h}\_2, \hat{h}\_3,\cdots)$$
where $\hat{h}\_t \in \mathcal{H}\_t$ for all $t$, and $\mathcal{H}\_{t'} \subseteq \mathcal{H}\_t$ for all $t' < t$
---
#### Empirical Prospective Risk Minimization
<img src="images/time-agnostic-erm.png" width="640">
---
#### Theorem 1
There exist stochastic processes for which time-agnostic ERM is not a weak prospective learner.
There also exist stochastic processes for which time-agnostic ERM is a weak prospective learner but not a strong one.
<img src="images/time-agnostic-erm-2.png" width="640">
---
#### Theorem 1
There exist stochastic processes for which time-agnostic ERM is not a weak prospective learner.
There also exist stochastic processes for which time-agnostic ERM is a weak prospective learner but not a strong one.
<img src="images/time-agnostic-erm-2.png" width="640">
Implication: We cannot build Prospective Learners using the toolkit of PAC learning
---
#### Theorem 2 (informal):
Time-Aware EPRM is a strong prospective learner, if
1. Consistency: Bayes risk can be well approximated asymptotically by an element of $\mathcal{H}$
2. Uniform Concentration: a subsequence of losses is asymptotically equal to prospective loss
---
#### Theorem 2
<img src="images/PL_thm2.png" width="640">
---
#### Examples
1. Periodic process
2. HMMs
---
# .center[Experimentation]
---
#### Reversal Learning
- A standard problem in cognitive science
- Learn something, then the opposite
---
#### Algorithms
- Time-agnostic MLP, CNN
- Fine tuning
- Time aware MLP, CNN, Auto-Regressive Transformer
- Oracle
---
#### Time-Aware NN Strongly Prospectively Reversal Learns
<img src="images/case2_revlearn_fig.png" width="1024">
---
#### Can LLMs Prospectively Learn?
<img src="images/pl_case2_seq.png" width="640">
---
#### Can LLMs Prospectively Learn?
<img src="images/LLM_fail_switcher.png" width="500">
---
#### What was the prompt?
Consider the following sequence of outcomes generated by two Bernoulli distributions, where all even outcomes are generated by a Bernoulli distribution with parameter 'p' and odd outcomes are generated from a Bernoulli distribution with parameter '1-p'.
10101010101010101010101000101010101010101010101011101010101010101100101010101010101010101011101010
The next 20 most likely sequence of outcomes are:
---
#### Why did it fail?
Generate outcomes of 10 Bernoulli trials where 0 is generated with probability 0.75 and 1 with probability 0.25
<img src="images/LLM_fail_strip.png" width="500">
---
#### the data dependent switcher
<img src="images/pl_cases3_fig.png" width="640">
---
#### the data and decision dependent switcher
<img src="images/pl_cases4_fig.png" width="640">
---
# .center[Deliberation]
---
#### Isn't this just....
- time-series modeling / forecasting?
- online learning?
- continual/lifelong learning?
- online meta-learning?
- reinforcement learning?
- use a transformer for everything?
---
#### What's next?
- Proving which kinds of stochastic processes are strongly/weakly prospectively learnable
- Developing algorithms that provably strongly/weakly prospectively learnable
- Implementing scalable algorithms
- Deploying algorithms in real-world applications
---
#### Publications
1. De Silva et al. [The Value of Out-of-Distribution Data](https://arxiv.org/abs/2109.14501), ICML, 2023.
1. De Silva et al. [Prospective Learning: Principled Extrapolation to the Future](https://arxiv.org/abs/2004.12908), CoLLAs, 2023.
1. De Silva et al. Prospective Learning: Learning for a Dynamic Future, [preprint available upon request](mailto:[email protected]).
---
##### Acknowledgements
<img src="images/neurodata2023.jpg" width="640">
.small[NSF Simons MoDL, ONR N00014-22-1-2255, and NSF CCF 2212519]
---
##### Questions?
<img src="images/dino_yummies.jpg" width="640">
</textarea>
<!-- <script src="https://gnab.github.io/remark/downloads/remark-latest.min.js"></script> -->
<!-- <script src="remark-latest.min.js"></script> -->
<script src="remark-latest.min.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.5.1/katex.min.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.5.1/contrib/auto-render.min.js"></script>
<link rel="stylesheet" href="https://cdnjs.cloudflare.com/ajax/libs/KaTeX/0.5.1/katex.min.css">
<script type="text/javascript">
var options = {};
var renderMath = function () {
renderMathInElement(document.body);
// or if you want to use $...$ for math,
renderMathInElement(document.body, {
delimiters: [ // mind the order of delimiters(!?)
{ left: "$$", right: "$$", display: true },
{ left: "$", right: "$", display: false },
{ left: "\\[", right: "\\]", display: true },
{ left: "\\(", right: "\\)", display: false },
]
});
}
// remark.macros.scale = function (percentage) {
// var url = this;
// return '<img src="' + url + '" style="width: ' + percentage + '" />';
// };
// var slideshow = remark.create({
// Set the slideshow display ratio
// Default: '4:3'
// Alternatives: '16:9', ...
// {
// ratio: '16:9',
// });
var slideshow = remark.create(options, renderMath);
</script>
</body>
</html>