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<!DOCTYPE html>
<html>
<head>
<title>L2F</title>
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<body>
<textarea id="source">
class: center, middle
name:opening
## Lifelong Learning Forests
.center[
Joshua T. Vogelstein
<br>
[[email protected]](mailto:jovo at jhu dot edu)
| <http://brainx.io/L2F>
<br><br><br>
<img src="images/logo_jhu.png" STYLE="HEIGHT:100px;"/>
]
---
## Goals of this Talk
- Motivation
- Background
- Intuition
- Applications
- Everyone in room understands
Please ask questions!
---
class: center, middle
# Motivation
---
<video width="320" height="550" controls>
<source src="images/lion_knocking.mp4" type="video/mp4">
</video>
---
### A motivating example: psychopathy
- .blue[truth]: this guy is a psychopath
- .blue[sample]: psychopath index is high
- .blue[action]: kill him, keep him in jail, or release him
- .blue[loss]: cost of jailing
- .blue[risk]: false positives are quite bad!
Goal: release people if they are not "too" dangerous
--
But then....
- .blue[truth]: cured of psychopathy
- .blue[sample]: new index, now says low
- .blue[action]: can start treatment
- .blue[loss]: cost of treatment
- .blue[risk]: treating people that don't get better is not so bad
Goal: Can the judge (.blue[learner]) updates her ruling?
---
class: center, middle
# Background
---
### Statistical Decision Theory
| object | space | definition |
|:--- |:--- |
| $\xi \sim P_\xi$ | $\Xi$ | .blue[true] latent state |
| $z \sim P_{z \vert \xi} = h(\xi)$ | $ \mathcal{Z}$ | noisily .blue[sampled] state
| $a$ | $\mathcal{A}$ | possible .blue[actions]
| $g: \mathcal{Z} \to \mathcal{A}$ | $\mathcal{G}$ | decision rule
| $\ell(g(z), a) \to \mathbb{R}_+$ | $\mathcal{L}$ | .blue[loss] function
| $f_n: \mathcal{Z}^n \times \mathcal{L} \to \mathcal{G}$ | $\mathcal{F}$ | .blue[learner]
| $\mathbb{R}_P [ \ell (\hat{g}(z; f_n), a)]$ | $\mathcal{R}$ | .blue[risk] functional
- Let $\varepsilon_n = \mathbb{R}[f_n] - \mathbb{R}^{*}$ be the .blue[residual]
- Goal: choose $f_n$ such that for many $P \in \mathcal{P}$,
$\varepsilon_n \rightarrow 0$
---
## Lifelong Learning Theory
- At $t=t'$, any of the following may change: $P\_{\xi}, h, \mathcal{Z}, \mathcal{A}, \ell, \mathbb{R}$
- We know when $\mathcal{Z}, \mathcal{A}, \ell, \mathbb{R}$ changes
- But not necessarily if $P=P\_{z,\xi}$ changes
- Goal: choose $f_n$ such that for many $P \in \mathcal{P}$
<!-- 1. $\varepsilon^{t<t^*}_n \rightarrow 0$ for many $P \in \mathcal{P}$
2. -->
| constraint | definition |
|:--- |:--- |
| $\varepsilon_n(t) \rightarrow 0$ | consistency |
| $\varepsilon_n(t>t') = \varepsilon_n(t)$ | no catestrophic forgetting
| $\varepsilon_{n+m}(t>t') \rightarrow 0$| continual learning
| $\\#(f_{n+m}) - \\#(f_n) \in \mathcal{O}(\log m)$ | space efficient
What class of $f_n$ might have these properties?
---
## Random Forests Might!
- Excellent empirical performance
- Caruana et al. 2006 (ICML): "With excellent performance on all eight metrics, calibrated boosted trees were the best learning algorithm overall. .blue[Random forests] are close second."
- Caruana et al. 2008 (ICML): "the method that performs consistently well across all dimensions is .blue[random forests]."
- Delgado et al. 2014 (JMLR): "The classifiers most likely to be the bests are the .blue[random forest]."
--
- Strong theoretical performance
- .blue[consistent]: $\varepsilon_n(RF_n) \rightarrow 0$ for any $P \in \mathcal{P}$
- .blue[space efficient]: $\\#(RF_n) \in \mathcal{O}(T n )$
---
### Brief Introduction to Random Forests
- Let $z_i=(x_i,y_i)$, where $ x_i \in \mathbb{R}^p$, $y_i \in \{0,1\}$
- Given $z_i$ for $i \in [n]$,
- For T trees:
- Subsample $n' < n$ samples
- At each node $j$, select feature $d_j$ and threshold $\tau_j$ to split
- For each child of $j$, repeat until criteria is met
- Tree is encoded by $( d_j,\tau_j )_j$
All the magic is in choosing the $d_j$'s and $\tau_j$'s
---
### A Simple Example: XOR
<img src="images/sample_XOR.png" style="height: 200px;"/>
--
<img src="images/Fig1_posteriors.svg" style="height: 200px;"/>
---
class: center, middle
# Intuition
---
### Scenario 1: Semisupervised
- Given $(x_i,y_i)$ for $i \in [n]$, $\quad x_i \in \mathbb{R}^p$, $y_i \in \{0,1\}$
- After time $t'$, given $x_i$ for $i = n+1, \ldots n+m$
- Assume (for now), that $P_x(t < t') = P_x(t>t')$
--
### What would Tukey do?
---
### Strategy \# 1: Update $\tau_j$'s
- ".blue[Thm]" (Telgarsky-Vattani):
- The set of local optima of Hartigan’s method is a (possibly strict) subset of local optima of Lloyd’s method.
- .blue[Algorithm]:
- Use Hartigan's method to recursively update $\tau_j$'s
--
- ".blue[Conjecture 1]":
- Updating $\tau\_j$'s this way yields $\varepsilon'\_{n+m} < \varepsilon\_{n}$
---
### Strategy \# 2: Make trees deeper
- ".blue[Thm]" (DGL): For RF to be consistent:
- Let $A(x)$ denote the "cell" containing $x$
- diam$(A(x)) \rightarrow 0$ in probability
- \# of points in $A(x) \rightarrow \infty$ in probability
- .blue[Algorithm]
- Pass each new point down the tree,
- At leaf node, decide whether to continue spliting
- If so, split as per usual
- ".blue[Conjecture 2]"
- Making trees deeper this way yields $\varepsilon'\_{n+m} < \varepsilon\_{n}$
---
### Strategy \# 3: Update $d_j$'s
- Some algorithm parameters can be estimated
- Let's think of the process at each node as a .blue[random projection]
- At each node, sample $A \sim F_A$
<img src="images/RF1.png" style="width: 100%;"/>
---
#### Strategy \# 3: Update $d_j$'s
- ".blue[Thm]" (Bayes): posterior = likelihood $\times$ prior
- $F_A$ has a low-dimensional parameterization
- $d$ is the \# of non-zeros in the matrix
- Option A: estimate .blue[which $d$'s] are doing well
- Option B: estimate .blue[which dimensions's] are doing well
- .blue[Algorithm]:
- build T trees using a prior distribution over $d$
- sample different $d$'s for each node
- estimate posterior over $d$/dimensions
- build T' more trees using posterior
- ".blue[Conjecture 3]":
- Making trees stronger this way yields $\varepsilon'\_{n} < \varepsilon\_{n}$
---
#### Strategy \# 3: Make trees stronger & less correlated
- ".blue[Thm]" (Breiman): RF error is bounded by a function of tree "strength" and "correlation"
- .blue[Algorithm]
- Add more parameters to $F_A$
- e.g., relax constraint of 1 non-zero per row
<img src="images/RerF1.png" style="width: 60%;"/>
- ".blue[Conjecture 4]"
- Making trees stronger this way yields $\varepsilon'\_{n} < \varepsilon\_{n}$
---
#### $\hat{R}\_{RerF}(n) < \hat{R}\_{RF}(n)$
- ~100 benchmark datasets from Delgado et al.
- For continuous data, RerF is significantly better
<img src="images/error_histogram_benchmarks.svg" style="height: 400px;"/>
---
#### Strategy \# 4: Make trees learn dictionaries
- Putting an identity matrix in there changes nothing
<img src="images/RF_Dict1.png" style="width: 100%;"/>
---
#### Strategy \# 3: Make trees learn dictionaries
- Changing that matrix enables RF to choose a dictionary
<img src="images/RF_Dict2.png" style="width: 100%;"/>
---
#### Strategy \# 3: Make trees learn dictionaries
- Choose dictionary based on prior information
<img src="images/RF_Dict3.png" style="width: 100%;"/>
---
#### Strategy \# 3: Make trees learn dictionaries
- ".blue[Thm]" (Sulam):
- Dictionary learning pursuit achieves the global optimum of the biconvex problem
- .blue[Algorithm]
- Choose low-dimensional dictionary based on prior knowledge
- Update posterior over dictionary after training
- ".blue[Conjecture 5]":
- Making trees stronger this way yields $\varepsilon'\_{n} < \varepsilon\_{n}$
---
#### $\hat{R}\_{S-RerF}(n) < \hat{R}\_{RerF}(n)$
<img src="images/image_stripes.svg" style="width: 100%;"/>
---
### Scenario 2: "Covariate Shift" Learning
- Given $(x_i,y_i)$ for $i \in [n]$, $\quad x_i \in \mathbb{R}^p$, $y_i \in \{0,1\}$
- after time $t'$, given another $x_i$ for $i = n+1, \ldots n+m$
- Assume that we do not know whether $$P_x(t < t') \neq P_x(t>t')$$
---
### Strategy 4: Weight trees
- Learn $S$ trees under $P_x(t< t')$
- Learn $S'$ trees under $P_x(t>t')$
- For a new sample $x$, let $\hat{y}= \alpha RF\_{\mathcal{S}}(x) + (1-\alpha) RF\_{\mathcal{S}'}(x)$
The magic is in how to choose $\alpha$, which depends on
- difference between $n$ and $m$
- difference between $P_x(t < t')$ and $P_x(t>t')$
---
### Strategy 4: Weight trees
- ".blue[Thm]" (Shen): Our multiscale graph correlation (MGC) test has
$$\varepsilon_n(MGC) \leq \varepsilon_n(global)$$
- .blue[Algorithm]
- Hypothesis test: $H_0: P_x(t < t') = P_x(t>t')$
- Effect size estimate: $|| P_x(t < t') - P_x(t>t')||$
- Estimate $\alpha$
- ".blue[Conjecture 6]"
- $\forall \, P\_x(t < t'), P\_x(t>t')$, with high probability
$$\varepsilon\_{n}^{\mathcal{S}, \mathcal{S'}}(\hat{\alpha}) < \varepsilon\_{n}^{\mathcal{S'}}$$
---
#### $\hat{R}\_{MGC}(n) < \hat{R}\_{global}(n)$
<img src="images/FigSimVisual.svg" style="width: 100%;"/>
---
#### $\hat{R}\_{MGC}(n) < \hat{R}\_{global}(n)$
<img src="images/FigHDPower.svg" style="width: 100%;"/>
---
### Scenario 3: The Whole Megilla
- Given $(x_i,y_i)$ for $i \in [n]$, $\quad x_i \in \mathbb{R}^p$, $y_i \in \{0,1\}$
- after time $t'$, given another $x_i$ for $i = n+1, \ldots n+m$
- .blue[everything] might be changing dynamically; define the setting as any time as
$$\mathcal{S}\_t=P\_{\xi}, h, \mathcal{Z}, \mathcal{A}, \ell, \mathbb{R}$$
- what to do?
---
### Strategy 4: Forest Daemon
- ".blue[Thm]": we got nothing here :)
- .blue[Algorithm]
- A .y[Forest Demon] stores "meta-data" for each (tree, sample, setting) triple
- The demon performs "random forestry", that is, chooses weights for each tree and sample based on the setting
$$\hat{y} = \sum_k^K \alpha^{\mathcal{S}}_k T_k(x)$$
- ".blue[Conjecture 7]"
- The forest demon solves L2M!
---
### Phase 1: Demonstrate L2Forests Can Satisfy the Criteria
1. Continual learning (updating old trees)
2. Avoids catastrophic forgetting (building new trees)
3. Goal driven perception (demon is given a new setting)
4. Select plasticity (demon determines the the relationship between current and past contexts)
5. Monitoring and safety (demon keeps track of everything required)
### Phase 2: Extend to Multimodal Data
- Requires learning multimodal trees
- Requires building a demon to integrate cross-modal information
---
class: center, middle
# Applications
---
class:center, middle
<img src="images/kent1.jpg" style="height: 100%;"/>
---
### \#1: Characterizing Psychopathy with Changing Sensors
- 10 years of scanning psychopaths (multi-modality)
- Frequent scanner updates meticulously documented (.blue[$h$])
- Estimate recidivism using the first $n$ samples of $(x_i, y_i)$
- Improve accuracy including $m$ additional $x_i$'s?
<img src="images/KentF1.png" style="width: 80%;"/>
---
### \#2: Characterizing Personality in Non-Stationary Life Span Data
- Bringing it back to 🦁
- Lifespan data: 8-80 (.blue[$P$])
- Certain phenotypics are fixed (e.g., ethnicity, gender, IQ)
- Multimodal brain measurements are dynamic with age
- Learn function using $(x_i,y_i)$ of kids
- Continue improving using only $x_i$ of adults
---
### Batch Effects are a Mess in these data
<img src="images/figure_batch.svg" style="width: 100%;"/>
---
class: top, left
.pull-left[
Theory Collaborations:
- Carey E. Priebe
- Cencheng Shen
- Minh Tang
- Tyler Tomita
- James Browne
- Randal Burns
- Mincent K. Tanzynski?
- Karl Kumbier?
Data Collaborators:
- Kent Kiel (MRN)
- Bruce Rosen (MGH)
]
.pull-left[
Love Collaborators:
- 🦁
- 😋
- 👪
- 🌍
- 🌠
Questions?
- [[email protected]](mailto:[email protected])
- [neurodata.io](http://neurodata.io)
We are hiring!
]
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