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<!DOCTYPE html>
<html>
<head>
<title>IID++</title>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8" />
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<textarea id="source">
#### Surprise! IID++<br>Out of Distribution & <br>Prospective Learning
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, Pratik Chaudhari, <br>Carey E. Priebe, Timothy Verstynen, Konrad Kording, <br>The Future Learning Collective
<!-- | Joshua T. Vogelstein <br> -->
<!-- [Microsoft Research](https://www.microsoft.com/en-us/research/): Weiwei Yang | Jonathan Larson | Bryan Tower | Chris White -->
<img src="images/neurodata_blue.png" width="20%" style="vertical-align: top; " >
<!-- <img src="images/jhu.png" width="8%" style="vertical-align: top"> -->
---
#### Outline
- Classical Learning
- Out of Distribution (OoD) Learning
- Prospective Learning
---
#### What is learning?
<img src="images/glivenko-cantelli-33.png" width="640">
<img src="images/GV2.png" width="640">
---
#### A more modern definition
- Assume $Z\_i \sim^{iid} F, \quad i \in [n]$
- Let $L((Z_i)^n) \rightarrow h_n$
- Let $R$ denote risk, eg, expected loss
- Let $R^*$ be Bayes optimal risk
- $L$ learns iff $\, \exists N$ s.t. $\forall n > N$
<!-- & $\delta, \epsilon > 0$ -->
$$\mathbb{P}[ R(h\_n) - R^* < \epsilon ] \geq 1 - \delta$$
- GC thm: for any $F$, $\exists L$ s.t. error ↘ as $n$ ↗
- implication: more data is better!
---
#### So learning is solved?
- No.
---
#### What's the problem
<img src="images/Hand2006.png" width="640">
Future (validation) data is sampled from a different distribution
---
#### Real world failures
<img src="images/google_flu_trends.jpg" width="640">
Can't model real world phenomema very well
---
#### Natural Intelligence failures
<img src="images/ScienceVogelstein23.png" width="640">
Can't replicate the intelligence of even the simplest organisms
---
## The Value of Out of Distribution Data
.footnote[.small[De Silva, et al., The Value of OoD Data. ICML, 2023]]
---
#### What is OoD Learning?
- Assume source/OoD $Z\_i \sim^{iid} F, \quad i \in [m]$
- Assume target $Z\_i \sim^{iid} F', \quad m < i \leq n+m$
- Let $L((Z\_i)^{n+m}) \rightarrow h\_{n+m}$
- Let $R$ denote risk, eg, expected loss wrt $F'$
- Let $R^*$ be Bayes optimal risk wrt $F'$
- $L$ OoD learns from $F$ about $F'$ if
$$\mathbb{P}[ R(h\_{n+m}) - R^* < \epsilon ] \geq 1 - \delta$$
- Question: does more data from $F$ help?
---
#### Let's take a poll
- Let $E=\mathbb{E}[ R(h\_{n+m}) - R^*]$ be generalization error on $F'$
- Expectation wrt $n$ samples from $F$ and $m$ samples from $F'$
Who thinks as $m$ increases, for fixed $n$, E
--
1. decreases
--
2. increases
--
3. depends on the relationship between $F$ and $F'$
---
#### Let's see: the simplest example
- $F$ is two gaussians
- $F'$ is two gaussians shifted by $\Delta$
<!-- <img src="https://github.com/neurodata/ood-tl/blob/main/reports/figures/gausstask_fig.png?raw=true" width="700"> -->
<img src="images/ood-1-summary-plot-a.png" width="300">
<!--  -->
---
#### E is non-monotonic in $m$!
<!-- <img src="https://github.com/neurodata/ood-tl/blob/main/reports/figures/gaussian_task_analytical_plot.svg?raw=true" width="640"> -->
<img src="images/ood-1-summary-plot-b.png" height="300">
- For a fixed $\Delta$, error can be non-monotonic wrt OoD sample size $m$
- implications: more pre-training can hurt!
---
#### A Bias/Variance Explanation
<img src="images/2-bias-var-breakdown-plot.png" width="500">
---
#### Synthetic image data examples
<img src="images/9-simdata-plot.png" width="640">
---
#### Trying to break it
<img src="images/11-effect-of-hyp-aug-pt.png" width="640">
---
#### Summary so far
- OoD learning is different in kind from in-distribution learning
- More OoD data does not necessarily help or hurt, it depends on sample sizes and distributions.
- Time to think more carefully about how much data of different kinds to use
---
#### What about time?
- $L$ OoD learns from $F$ about $F'$ if
$$\mathbb{P}[ R(h\_{n+m}) - R^* < \epsilon ] \geq 1 - \delta$$
Does not include time.
Let's revisit learning frameworks, and see what is missing
---
## Prospective Learning: Principled Extrapolation to the Future
.footnote[.small[De Silva, et al., Prospective Learning: Principled Extrapolation to the Future. CoLLAs, 2023]]
---
#### Learning Frameworks
- "Classical" PAC Learning: best case scenario, $F\_{future} = F\_{past}$
- Online Learning: worst case scenario, <br> $F\_{future}$ could be adversarial
In both cases, the "best" thing to do is whatever would have worked best in the past.
---
#### Fancy Learning Frameworks
- OoD PAC Learning: $F\_{future} \neq F\_{past}$
- Online meta-learning: $F\_{future} \sim^{iid} G$
- Continual learning: $F\_{future} \sim^{iid} G$, with bias towards previously sampled F's
In all these cases, still no ability to predict $F\_{future}$ at all.
But what about when one can predict future F's somewhat?
---
<img src="images/seligman13.png" width="640">
Learning evolved because it improves future performance in a partially predictable dynamic world
---
#### Consider...
Moving to Berkeley vs Boston
- I imagine challenges I'll face in B-town, and how I'll overcome them
- Choose the one that minimizes expected loss integrated over my life
- But I only make this choice once
---
#### Consider the following sequence
<img src="images/task-seq.png" width="640">
--
Has anyone ever run an experiment?
---
#### Online SGD & FTL Fail
<img src="images/online_seq1.png" width="640">
--
- Online Meta & Continual would do better, but never get to zero error
- Because none of these frameworks *prospect*
---
#### Formalizing prospection
- $\mathcal{H}$ is a set of feasible hypotheses
- $t \in \mathcal{T}$ indexes time
- $P :=$ { $P\_t$ }$\_{t \geq 0}$ be a sequence of distributions
- $D\_{t'} =$ { $z\_t$ }$\_{t < t'}$ be a dataset drawn from $P$ such that $z\_t \sim P\_t$,
- $\mathcal{D}$ be the set of all possible datasets
A prospective learner $L: \mathcal{D} \times \mathcal{T} \mapsto \mathcal{H}$, so <br>
$L(D\_{t'}, t) = \hat{h}\_t^{t'}$, where $\hat{h}\_t^{t'}$ is the hypothesis at time $t > t'$ trained on data up until time $t'$
---
#### Formalizing prospection
- $\ell_t : \mathcal{H} \times \mathcal{Z} \mapsto \mathbb{R}$ is a time-varying loss function.
- Risk at time $t$ be the expected loss, $ R\_t(L(D\_{t'},t)) = \mathbb{E}\_{z \sim P\_t}\left[ \ell\_t (L(D\_{t'},t) , z) \right]$
<!-- = \mathbb{E}\_{z \sim P\_t} [ \ell\_t ( \hat{h}^{t'}\_t , z) ]$. -->
Colloquially, we desire that the expected risk decreases as $t$ increases
---
#### Strong Prospective Learnability
<img src="images/strong_PL.png" width="640">
Key differences with Strong PAC Learning:
- we care about risk integrated over the future
- this requires prospecting about (1) what the future will be like, and (2) what we will be like
---
#### Weak Prospective Learnability
<img src="images/weak_PL.png" width="640">
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
---
#### Continuum Hypothesis of Learning Hypotheses
- Let $P$ is a sequence of distributions,
- Let $\mathcal{H}$ is the set of all possible relevant hypotheses.
We conjecture that there are multiple classes of learning difficulty
1. P is PAC Learnable
2. P is weakly prospectively learnable (ie, we can do better than ERM)
3. P is strongly prospectively learnable (for some $\epsilon$)
4. P is uniformly prospectively learnable (for all $\epsilon$)
5. P is not prospectively learnable
---
#### Did we re-invent the wheel?
How is reinforcement learning different from this?
- We directly build on statistical decision theory
- RL theory assumes POMDP on dynamics, not PL
- Vanilla RL focuses on a single task, not PL
- RL focuses on inference problem, learning the parameters is a necessary sub-task, PL focuses on the learning problem
- RL algorithms require many trials, we care about zero-shot learning
---
#### Zooming out
- How do we design truly intelligent systems?
- How do we solve the AI alignment issue?
---
#### 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: Princpled Extrapolation to the Future](https://arxiv.org/abs/2004.12908), CoLLAs, 2023.
---
##### 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>
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