CRPS-Learning

# CRPS-Learning
## Jonathan Berrisch, Florian Ziel
### University of Duisburg-Essen
### 2021-11-17

---

class:middle

# Outline

- [Motivation](#motivation)
- [The Framework of Prediction under Expert Advice](#pred_under_exp_advice)
- [The Continious Ranked Probability Score](#crps)
- [Optimality of (Pointwise) CRPS-Learning](#crps_optim)
- [A Simple Probabilistic Example](#simple_example)
- [The Proposed CRPS-Learning Algorithm](#proposed_algorithm)
- [Simulation Results](#simulation)
- [Possible Extensions](#extensions)
- [Application Study](#application)
- [Wrap-Up](#conclusion)
- [References](#references)

---

# Motivation

The Idea:
- Combine multiple forecasts instead of choosing one

- Combination weights may vary over **time**, over the **distribution** or **both**

2 Popular options for combining distributions:
- Combining across quantiles (this paper)
  - Horizontal aggregation, vincentization
- Combining across probabilities
  - Vertical aggregation

]

![](data:image/png;base64,#index_files/figure-html/unnamed-chunk-1-1.svg)

]

.panel[.panel-name[Distribution]
![](data:image/png;base64,#index_files/figure-html/unnamed-chunk-2-1.svg)

]]
]

---

# The Framework of Prediction under Expert Advice

### The sequential framework

Each day, `$t = 1, 2, ... T$`
- The **forecaster** receives predictions `$\widehat{X}_{t,k}$` from `$K$` **experts**
- The **forecaster** assings weights `$w_{t,k}$` to each **expert**
- The **forecaster** calculates her prediction:
`\begin{equation}
    \widetilde{X}_{t} = \sum_{k=1}^K w_{t,k} \widehat{X}_{t,k}.
    \label{eq_forecast_def}
\end{equation}`
- The realization for `$t$` is observed
]

- The experts can be institutions, persons, or models
- The forecasts can be point-forecasts (i.e., mean or median) or full predictive distributions
- We do not need any assumptions concerning the underlying data
- <a id='cite-cesa2006prediction'></a><a href='#bib-cesa2006prediction'>Cesa-Bianchi and Lugosi (2006)</a>

]

---

Weights are updated sequentially according to the past performance of the `$K$` experts.

`\begin{equation}
    R_{t,k}  = \widetilde{L}_{t} - \widehat{L}_{t,k} =  \sum_{i = 1}^t \ell(\widetilde{X}_{i},Y_i) - \ell(\widehat{X}_{i,k},Y_i)
    \label{eq_regret}
\end{equation}`

The cumulative regret:
- Indicates the predictive accuracy of the expert `$k$` until time `$t$`.
- Measures how much the forecaster *regrets* not having followed the expert's advice

Popular loss functions for point forecasting <a id='cite-gneiting2011making'></a><a href='#bib-gneiting2011making'>Gneiting (2011a)</a>:
.pull-left[
- `$\ell_2$`-loss `$\ell_2(x, y) = | x -y|^2$`
  - optimal for mean prediction 
]
.pull-right[
- `$\ell_1$`-loss `$\ell_1(x, y) = | x -y|$` 
  - optimal for median predictions 
]

---

# Popular Algorithms and the Risk

### Popular Aggregation Algorithms

#### The naive combination

`\begin{equation}
    w_{t,k}^{\text{Naive}} = \frac{1}{K}
\end{equation}`

#### The exponentially weighted average forecaster (EWA)

`\begin{align}
    w_{t,k}^{\text{EWA}} & = \frac{e^{\eta R_{t,k}} }{\sum_{k = 1}^K e^{\eta R_{t,k}}}
    =
    \frac{e^{-\eta \ell(\widehat{X}_{t,k},Y_t)} w^{\text{EWA}}_{t-1,k} }{\sum_{k = 1}^K e^{-\eta \ell(\widehat{X}_{t,k},Y_t)} w^{\text{EWA}}_{t-1,k} }
    \label{eq_ewa_general}
\end{align}`

]

### Optimality

In stochastic settings, the cumulative Risk should be analyzed <a id='cite-wintenberger2017optimal'></a><a href='#bib-wintenberger2017optimal'>Wintenberger (2017)</a>:
`\begin{align}
    &\underbrace{\widetilde{\mathcal{R}}_t = \sum_{i=1}^t \mathbb{E}[\ell(\widetilde{X}_{i},Y_i)|\mathcal{F}_{i-1}]}_{\text{Cumulative Risk of Forecaster}} \\
    &\underbrace{\widehat{\mathcal{R}}_{t,k} = \sum_{i=1}^t \mathbb{E}[\ell(\widehat{X}_{i,k},Y_i)|\mathcal{F}_{i-1}]}_{\text{Cumulative Risk of Experts}}
    \label{eq_def_cumrisk}
\end{align}`

]

---

# Optimal Convergence

### The selection problem
`\begin{equation}
    \frac{1}{t}\left(\widetilde{\mathcal{R}}_t - \widehat{\mathcal{R}}_{t,\min} \right) \stackrel{t\to \infty}{\rightarrow} a \quad \text{with} \quad a \leq 0.
    \label{eq_opt_select}
\end{equation}`
The forecaster is asymptotically not worse than the best expert `$\widehat{\mathcal{R}}_{t,\min}$`.

### The convex aggregation problem

`\begin{equation}
    \frac{1}{t}\left(\widetilde{\mathcal{R}}_t - \widehat{\mathcal{R}}_{t,\pi} \right) \stackrel{t\to \infty}{\rightarrow} b \quad \text{with} \quad b \leq 0 .
    \label{eq_opt_conv}
\end{equation}`
The forecaster is asymptotically not worse than the best convex combination `$\widehat{X}_{t,\pi}$` in hindsight (**oracle**).
]

Optimal rates with respect to selection \eqref{eq_opt_select} and convex aggregation \eqref{eq_opt_conv}  <a href='#bib-wintenberger2017optimal'>Wintenberger (2017)</a>:

`\begin{align}
    \frac{1}{t}\left(\widetilde{\mathcal{R}}_t - \widehat{\mathcal{R}}_{t,\min} \right) & =
    \mathcal{O}\left(\frac{\log(K)}{t}\right)\label{eq_optp_select}
\end{align}`
`\begin{align}
    \frac{1}{t}\left(\widetilde{\mathcal{R}}_t - \widehat{\mathcal{R}}_{t,\pi} \right)  & =
    \mathcal{O}\left(\sqrt{\frac{\log(K)}{t}}\right)
    \label{eq_optp_conv}
\end{align}`

Algorithms can statisfy both \eqref{eq_optp_select} and \eqref{eq_optp_conv} depending on:

- The loss function
- Regularity conditions on `$Y_t$` and `$\widehat{X}_{t,k}$`
- The weighting scheme

]

---

name:crps

## Optimality

EWA satisfies optimal selection convergence \eqref{eq_optp_select} in a deterministic setting if:
- Loss `$\ell$` is exp-concave
- Learning-rate `$\eta$` is chosen correctly

Those results can be converted to stochastic iid settings <a id='cite-kakade2008generalization'></a><a href='#bib-kakade2008generalization'>Kakade and Tewari (2008)</a> <a id='cite-gaillard2014second'></a><a href='#bib-gaillard2014second'>Gaillard, Stoltz, and Van Erven (2014)</a>.

Optimal convex aggregation convergence \eqref{eq_optp_conv} can be satisfied by applying the kernel-trick:
`\begin{align}
\ell^{\nabla}(x,y) = \ell'(\widetilde{X},y) x
\end{align}`
`$\ell'$` is the subgradient of `$\ell$` at forecast combination `$\widetilde{X}$`.

]

## Probabilistic Setting

**An appropriate choice:**

`\begin{align*}
    \text{CRPS}(F, y) & = \int_{\mathbb{R}} {(F(x) - \mathbb{1}\{ x > y \})}^2 dx
    \label{eq_crps}
\end{align*}`

It's strictly proper <a id='cite-gneiting2007strictly'></a><a href='#bib-gneiting2007strictly'>Gneiting and Raftery (2007)</a>.

Using the CRPS, we can calculate time-adaptive weights `$w_{t,k}$`. However, what if the experts' performance varies in parts of the distribution?

`\begin{align*}
    \text{CRPS}(F, y) = 2 \int_0^{1}  \text{QL}_p(F^{-1}(p), y) \, d p.
    \label{eq_crps_qs}
\end{align*}`

... to combine quantiles of the probabilistic forecasts individually using the quantile-loss QL.

]

---

# CRPS-Learning Optimality

The same algorithm satisfies that there exist a `$C>0$` such that for `$x>0$` it holds that
`\begin{equation}
    P\left( \frac{1}{t}\left(\widetilde{\mathcal{R}}_t - \widehat{\mathcal{R}}_{t,\min} \right) \leq
    C\left(\frac{\log(K)+\log(\log(Gt))+ x}{\alpha t}\right)^{\frac{1}{2-\beta}} \right) \geq
    1-2e^{-x}
    \label{eq_boa_opt_select}
\end{equation}`

if `$Y_t$` is bounded, the considered loss `$\ell$` is convex `$G$`-Lipschitz and weak exp-concave in its first coordinate.

---

# A Probabilistic Example

Simple Example:

`\begin{align}
    Y_t               & \sim \mathcal{N}(0,\,1)                     \\
    \widehat{X}_{t,1} & \sim \widehat{F}_{1}  = \mathcal{N}(-1,\,1) \\
    \widehat{X}_{t,2} & \sim \widehat{F}_{2}  = \mathcal{N}(3,\,4)
    \label{eq:dgp_sim1}
\end{align}`

- True weights vary over `$p$`
- Figures show the ECDF and calculated weights using `$T=25$` realizations
- Pointwise solution creates rough estimates
- Pointwise is better than constant
- Smooth solutions may be better than pointwise

]

]

]]
]

---
  
# The B-Smooth Procedure

Represent weights as linear combinations of bounded basis functions:

`\begin{align}
    w_{t,k} = \sum_{l=1}^L \beta_{t,k,l} \varphi_l = \boldsymbol \beta_{t,k}' \boldsymbol \varphi
\end{align}`

A popular choice are are B-Splines as local basis functions

`$\boldsymbol \beta_{t,k}$` is calculated using a reduced regret matrix:

`$\underbrace{\boldsymbol r_{t}}_{\text{LxK}} = \frac{L}{P} \underbrace{\boldsymbol B'}_{\text{LxP}} \underbrace{\left({\boldsymbol{QL}}_{\mathcal{P}}^{\nabla}(\widetilde{\boldsymbol X}_{t},Y_t)- {\boldsymbol{QL}}_{\mathcal{P}}^{\nabla}(\widehat{\boldsymbol X}_{t},Y_t)\right)}_{\text{PxK}}$`

If `$L = P$` it holds that `$\boldsymbol \varphi = \boldsymbol{I}$` 
For `$L = 1$` we receive constant weights

]

We receive the constant solution for high values of `$\lambda$` when setting `$d=1$`

]

---
    
# The P-Smooth Procedure

Penalized cubic B-Splines for smoothing weights:

Let `$\varphi=(\varphi_1,\ldots, \varphi_L)$` be bounded basis functions on `$(0,1)$` Then we approximate `$w_{t,k}$` by

`\begin{align}
w_{t,k}^{\text{smooth}} = \sum_{l=1}^L \beta_l \varphi_l = \beta'\varphi
\end{align}`

with parameter vector `$\beta$`. The latter is estimated penalized `$L_2$`-smoothing which minimizes

`\begin{equation}
    \| w_{t,k} - \beta' \varphi  \|^2_2 + \lambda \| \mathcal{D}^{d}  (\beta' \varphi)  \|^2_2
    \label{eq_function_smooth}
\end{equation}`

with differential operator `$\mathcal{D}$`

Computation is easy, since we have an analytical solution

]

We receive the constant solution for high values of `$\lambda$` when setting `$d=1$`

]

---

name:proposed_algorithm

# The Proposed CRPS-Learning Algorithm

**Initialization:**

Array of expert predicitons: `$\widehat{X}_{t,p,p}$`

Vector of Prediction targets: `$Y_t$`

Starting Weights: `$\boldsymbol w_0=(w_{0,1},\ldots, w_{0,K})$`

Penalization parameter: `$\lambda\geq 0$`

B-spline and penalty matrices `$\boldsymbol B$` and `$\boldsymbol D$` on `$\mathcal{P}= (p_1,\ldots,p_M)$`

Hat matrix: `$$\boldsymbol{\mathcal{H}} = \boldsymbol B(\boldsymbol B'\boldsymbol B+ \lambda (\alpha \boldsymbol D_1'\boldsymbol D_1 + (1-\alpha) \boldsymbol D_2'\boldsymbol D_2))^{-1} \boldsymbol B'$$`

Cumulative Regret: `$R_{0,k} = 0$`

Range parameter: `$E_{0,k}=0$`

Starting pseudo-weights: `$\boldsymbol \beta_0 = \boldsymbol B^{\text{pinv}}\boldsymbol w_0(\boldsymbol{\mathcal{P}})$`

]]

**Core**:

for( t in 1:T ) {

&nbsp;&nbsp;&nbsp;&nbsp; `$\widetilde{\boldsymbol X}_{t} = \text{Sort}\left( \boldsymbol w_{t-1}'(\boldsymbol P) \widehat{\boldsymbol X}_{t} \right)$` .grey[\# Prediction]

&nbsp;&nbsp;&nbsp;&nbsp; `$\boldsymbol r_{t} = \frac{L}{M} \boldsymbol B' \left({\boldsymbol{QL}}_{\boldsymbol{\mathcal P}}^{\nabla}(\widetilde{\boldsymbol X}_{t},Y_t)- {\boldsymbol{QL}}_{\boldsymbol{\mathcal P}}^{\nabla}(\widehat{\boldsymbol X}_{t},Y_t)\right)$`

&nbsp;&nbsp;&nbsp;&nbsp; `$\boldsymbol E_{t}    = \max(\boldsymbol E_{t-1}, \boldsymbol r_{t}^+ + \boldsymbol r_{t}^-)$`

&nbsp;&nbsp;&nbsp;&nbsp; `$\boldsymbol V_{t}     = \boldsymbol V_{t-1} + \boldsymbol r_{t}^{ \odot 2}$`

&nbsp;&nbsp;&nbsp;&nbsp; `$\boldsymbol \eta_{t}  =\min\left( \left(-\log(\boldsymbol \beta_{0}) \odot \boldsymbol V_{t}^{\odot -1} \right)^{\odot\frac{1}{2}} , \frac{1}{2}\boldsymbol E_{t}^{\odot-1}\right)$`

&nbsp;&nbsp;&nbsp;&nbsp; `$\boldsymbol R_{t}     = \boldsymbol R_{t-1}+  \boldsymbol r_{t} \odot \left( \boldsymbol 1 - \boldsymbol \eta_{t} \odot \boldsymbol r_{t} \right)/2 + \boldsymbol E_{t} \odot \mathbb{1}\{-2\boldsymbol \eta_{t}\odot \boldsymbol r_{t} > 1\}$`

&nbsp;&nbsp;&nbsp;&nbsp; `$\boldsymbol \beta_{t}     = K \boldsymbol \beta_{0} \odot \boldsymbol {SoftMax}\left( -  \boldsymbol \eta_{t} \odot \boldsymbol R_{t} + \log( \boldsymbol \eta_{t}) \right)$`

&nbsp;&nbsp;&nbsp;&nbsp; `$\boldsymbol w_{t}(\boldsymbol P) = \underbrace{\boldsymbol B(\boldsymbol B'\boldsymbol B+ \lambda (\alpha \boldsymbol D_1'\boldsymbol D_1 + (1-\alpha) \boldsymbol D_2'\boldsymbol D_2))^{-1} \boldsymbol B'}_{\boldsymbol{\mathcal{H}}} \boldsymbol B \boldsymbol \beta_{t}$`

} .grey[\#t]

]
]

---

# Simulation Study

Data Generating Process of the [simple probabilistic example](#simple_example)

- Constant solution `$\lambda \rightarrow \infty$`
- Pointwise Solution of the proposed BOAG
- Smoothed Solution of the proposed BOAG
  - Weights are smoothed during learning
  - Smooth weights are used to calculate Regret, adjust weights, etc.

]

Deviation from best attainable `$\boldsymbol{QL}_\boldsymbol{\mathcal{P}}$` (1000 runs).

![](data:image/png;base64,#pre_vs_post.gif)

]

CRPS Values for different `$\lambda$` (1000 runs)

![](data:image/png;base64,#pre_vs_post_lambda.gif)

]

CRPS Values for different number of knots (1000 runs)

![](data:image/png;base64,#pre_vs_post_kstep.gif)

]

]
---

# Simulation Study

The same simulation carried out for different algorithms (1000 runs):

---

# Simulation Study

**New DGP:**

`\begin{align}
    Y_t               & \sim \mathcal{N}\left(\frac{\sin(0.005 \pi t )}{2},\,1\right) \\
    \widehat{X}_{t,1} & \sim      \widehat{F}_{1}  = \mathcal{N}(-1,\,1)              \\
    \widehat{X}_{t,2} & \sim       \widehat{F}_{2}  = \mathcal{N}(3,\,4) \label{eq_dgp_sim2}
\end{align}`

]

**Weights of expert 2**

]

---

name:extensions

# Possible Extensions

**Forgetting**

- Only taking part of the old cumulative regret into account
- Exponential forgetting of past regret

`\begin{align*}
    R_{t,k} & =  R_{t-1,k}(1-\xi) + \ell(\widetilde{F}_{t},Y_i) - \ell(\widehat{F}_{t,k},Y_i) \label{eq_regret_forget}
\end{align*}`

**Fixed Shares** <a id='cite-herbster1998tracking'></a><a href='#bib-herbster1998tracking'>Herbster and Warmuth (1998)</a>

- Adding fixed shares to the weights
  - Shrinkage towards a constant solution

`\begin{align*}
    \widetilde{w}_{t,k} = \rho \frac{1}{K} + (1-\rho) w_{t,k}
    \label{fixed_share_simple}.
\end{align*}`

]

**Non-Equidistant Knots**
- Non-equidistant spline-basis could be used
- Potentially improves the tail-behavior
- Destroys shrinkage towards constant

]

---

# Application to NN Power Price Forecasts

Apply BOA to Normal and JSU Power price forecasts:

```r
mod <- online_mv(
  y = Y,
  experts = experts,
  tau = 1:99 / 100
)
```

This yields:

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<table class="gt_table">
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  </thead>
  <thead class="gt_col_headings">
    <tr>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">norm</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">jsu</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">comb</th>
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  </thead>
  <tbody class="gt_table_body">
    <tr><td class="gt_row gt_right">1.4449</td>
<td class="gt_row gt_right">1.4594</td>
<td class="gt_row gt_right">1.4438</td></tr>
  </tbody>
  
  
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</div>

Figure shows most recent JSU weights

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]

---

# Application to NN Power Price Forecasts

Combination consistently outperforms

```r
mod <- online_mv(
  y = Y,
  experts = experts,
  tau = 1:99 / 100,
  smooth_pr = list(
    penalized = list(
      lambda = c(-Inf, 2^(-2:15), 2^20)
    ) # lambda = 4096 was selected
  ),
  smooth_mv = list(
    penalized = list(
      lambda = c(-Inf, 2^(-2:15), 2^20)
    ) # lambda = 8 was selected
  ),
  forget_regret = c(0, 2^seq(-11, -4, 1))
  # .15625 was selected
)
```

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</style>
<table class="gt_table">
  <thead class="gt_header">
    <tr>
      <th colspan="8" class="gt_heading gt_title gt_font_normal gt_bottom_border" style>CRPS Values</th>
    </tr>
    
  </thead>
  <thead class="gt_col_headings">
    <tr>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">norm</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">jsu</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">n-</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">n+</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">j-</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">j+</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">comb</th>
      <th class="gt_col_heading gt_columns_bottom_border gt_left" rowspan="1" colspan="1">h</th>
    </tr>
  </thead>
  <tbody class="gt_table_body">
    <tr><td class="gt_row gt_right" style="background-color: #7CC665; color: #000000;">1.4449</td>
<td class="gt_row gt_right" style="background-color: #81C866; color: #000000;">1.4594</td>
<td class="gt_row gt_right" style="background-color: #FCA85E; color: #000000;">2.4058</td>
<td class="gt_row gt_right" style="background-color: #FFEB9D; color: #000000;">2.1155</td>
<td class="gt_row gt_right" style="background-color: #FA9153; color: #000000;">2.4808</td>
<td class="gt_row gt_right" style="background-color: #FFF0A6; color: #000000;">2.0809</td>
<td class="gt_row gt_right" style="background-color: #73C364; color: #000000;">1.4174</td>
<td class="gt_row gt_left">all</td></tr>
    <tr><td class="gt_row gt_right" style="background-color: #026C39; color: #FFFFFF;">0.9902</td>
<td class="gt_row gt_right" style="background-color: #036F3B; color: #FFFFFF;">1.0036</td>
<td class="gt_row gt_right" style="background-color: #FEE08B; color: #000000;">2.1853</td>
<td class="gt_row gt_right" style="background-color: #FDFEBD; color: #000000;">1.9754</td>
<td class="gt_row gt_right" style="background-color: #FECC7A; color: #000000;">2.2674</td>
<td class="gt_row gt_right" style="background-color: #FAFDB8; color: #000000;">1.9561</td>
<td class="gt_row gt_right" style="background-color: #006837; color: #FFFFFF;">0.9738</td>
<td class="gt_row gt_left">1</td></tr>
    <tr><td class="gt_row gt_right" style="background-color: #036F3B; color: #FFFFFF;">1.0044</td>
<td class="gt_row gt_right" style="background-color: #05723C; color: #FFFFFF;">1.0150</td>
<td class="gt_row gt_right" style="background-color: #FEE28E; color: #000000;">2.1728</td>
<td class="gt_row gt_right" style="background-color: #FFFBB8; color: #000000;">2.0126</td>
<td class="gt_row gt_right" style="background-color: #FECF7C; color: #000000;">2.2553</td>
<td class="gt_row gt_right" style="background-color: #FFFDBC; color: #000000;">1.9944</td>
<td class="gt_row gt_right" style="background-color: #016B39; color: #FFFFFF;">0.9867</td>
<td class="gt_row gt_left">2</td></tr>
    <tr><td class="gt_row gt_right" style="background-color: #0C7E43; color: #FFFFFF;">1.0691</td>
<td class="gt_row gt_right" style="background-color: #0D8044; color: #FFFFFF;">1.0777</td>
<td class="gt_row gt_right" style="background-color: #FEE391; color: #000000;">2.1637</td>
<td class="gt_row gt_right" style="background-color: #FFEDA1; color: #000000;">2.1003</td>
<td class="gt_row gt_right" style="background-color: #FED17E; color: #000000;">2.2466</td>
<td class="gt_row gt_right" style="background-color: #FFF0A6; color: #000000;">2.0823</td>
<td class="gt_row gt_right" style="background-color: #0A7A41; color: #FFFFFF;">1.0532</td>
<td class="gt_row gt_left">3</td></tr>
    <tr><td class="gt_row gt_right" style="background-color: #097A40; color: #FFFFFF;">1.0508</td>
<td class="gt_row gt_right" style="background-color: #0A7B41; color: #FFFFFF;">1.0567</td>
<td class="gt_row gt_right" style="background-color: #FFF0A6; color: #000000;">2.0804</td>
<td class="gt_row gt_right" style="background-color: #FFEA9C; color: #000000;">2.1182</td>
<td class="gt_row gt_right" style="background-color: #FEE491; color: #000000;">2.1604</td>
<td class="gt_row gt_right" style="background-color: #FFEDA1; color: #000000;">2.0981</td>
<td class="gt_row gt_right" style="background-color: #07763E; color: #FFFFFF;">1.0354</td>
<td class="gt_row gt_left">4</td></tr>
    <tr><td class="gt_row gt_right" style="background-color: #108446; color: #FFFFFF;">1.0952</td>
<td class="gt_row gt_right" style="background-color: #118747; color: #FFFFFF;">1.1072</td>
<td class="gt_row gt_right" style="background-color: #FEE492; color: #000000;">2.1584</td>
<td class="gt_row gt_right" style="background-color: #FFF1A7; color: #000000;">2.0755</td>
<td class="gt_row gt_right" style="background-color: #FED380; color: #000000;">2.2386</td>
<td class="gt_row gt_right" style="background-color: #FFF5AE; color: #000000;">2.0508</td>
<td class="gt_row gt_right" style="background-color: #0F8345; color: #FFFFFF;">1.0879</td>
<td class="gt_row gt_left">5</td></tr>
    <tr><td class="gt_row gt_right" style="background-color: #4DAE5C; color: #000000;">1.2991</td>
<td class="gt_row gt_right" style="background-color: #51B05D; color: #000000;">1.3096</td>
<td class="gt_row gt_right" style="background-color: #FEDA86; color: #000000;">2.2087</td>
<td class="gt_row gt_right" style="background-color: #FFF1A8; color: #000000;">2.0722</td>
<td class="gt_row gt_right" style="background-color: #FEC876; color: #000000;">2.2852</td>
<td class="gt_row gt_right" style="background-color: #FFF8B2; color: #000000;">2.0321</td>
<td class="gt_row gt_right" style="background-color: #4CAE5B; color: #000000;">1.2941</td>
<td class="gt_row gt_left">6</td></tr>
    <tr><td class="gt_row gt_right" style="background-color: #A6D96A; color: #000000;">1.5783</td>
<td class="gt_row gt_right" style="background-color: #A7D96B; color: #000000;">1.5834</td>
<td class="gt_row gt_right" style="background-color: #FEBB6B; color: #000000;">2.3376</td>
<td class="gt_row gt_right" style="background-color: #FED682; color: #000000;">2.2261</td>
<td class="gt_row gt_right" style="background-color: #FCA85E; color: #000000;">2.4062</td>
<td class="gt_row gt_right" style="background-color: #FEE08B; color: #000000;">2.1852</td>
<td class="gt_row gt_right" style="background-color: #A3D86A; color: #000000;">1.5692</td>
<td class="gt_row gt_left">7</td></tr>
    <tr><td class="gt_row gt_right" style="background-color: #C2E57C; color: #000000;">1.6908</td>
<td class="gt_row gt_right" style="background-color: #C3E67D; color: #000000;">1.6944</td>
<td class="gt_row gt_right" style="background-color: #FDAC60; color: #000000;">2.3958</td>
<td class="gt_row gt_right" style="background-color: #FEC070; color: #000000;">2.3145</td>
<td class="gt_row gt_right" style="background-color: #FA9756; color: #000000;">2.4615</td>
<td class="gt_row gt_right" style="background-color: #FECA78; color: #000000;">2.2762</td>
<td class="gt_row gt_right" style="background-color: #BFE47A; color: #000000;">1.6790</td>
<td class="gt_row gt_left">8</td></tr>
    <tr><td class="gt_row gt_right" style="background-color: #AFDD6F; color: #000000;">1.6132</td>
<td class="gt_row gt_right" style="background-color: #B2DE72; color: #000000;">1.6268</td>
<td class="gt_row gt_right" style="background-color: #FA9655; color: #000000;">2.4659</td>
<td class="gt_row gt_right" style="background-color: #FEE28E; color: #000000;">2.1745</td>
<td class="gt_row gt_right" style="background-color: #F77F4A; color: #000000;">2.5379</td>
<td class="gt_row gt_right" style="background-color: #FFE898; color: #000000;">2.1364</td>
<td class="gt_row gt_right" style="background-color: #A4D86A; color: #000000;">1.5743</td>
<td class="gt_row gt_left">9</td></tr>
    <tr><td class="gt_row gt_right" style="background-color: #A4D86A; color: #000000;">1.5718</td>
<td class="gt_row gt_right" style="background-color: #A7DA6B; color: #000000;">1.5855</td>
<td class="gt_row gt_right" style="background-color: #FB9B57; color: #000000;">2.4508</td>
<td class="gt_row gt_right" style="background-color: #FFE594; color: #000000;">2.1499</td>
<td class="gt_row gt_right" style="background-color: #F8844D; color: #000000;">2.5223</td>
<td class="gt_row gt_right" style="background-color: #FFEC9F; color: #000000;">2.1089</td>
<td class="gt_row gt_right" style="background-color: #99D369; color: #000000;">1.5370</td>
<td class="gt_row gt_left">10</td></tr>
    <tr><td class="gt_row gt_right" style="background-color: #9DD569; color: #000000;">1.5481</td>
<td class="gt_row gt_right" style="background-color: #A1D769; color: #000000;">1.5628</td>
<td class="gt_row gt_right" style="background-color: #FCA05A; color: #000000;">2.4340</td>
<td class="gt_row gt_right" style="background-color: #FEE492; color: #000000;">2.1602</td>
<td class="gt_row gt_right" style="background-color: #F9894F; color: #000000;">2.5063</td>
<td class="gt_row gt_right" style="background-color: #FFEA9C; color: #000000;">2.1184</td>
<td class="gt_row gt_right" style="background-color: #97D268; color: #000000;">1.5292</td>
<td class="gt_row gt_left">11</td></tr>
    <tr><td class="gt_row gt_right" style="background-color: #A0D669; color: #000000;">1.5604</td>
<td class="gt_row gt_right" style="background-color: #A5D96A; color: #000000;">1.5777</td>
<td class="gt_row gt_right" style="background-color: #F9894F; color: #000000;">2.5063</td>
<td class="gt_row gt_right" style="background-color: #FEE28E; color: #000000;">2.1749</td>
<td class="gt_row gt_right" style="background-color: #F57044; color: #000000;">2.5809</td>
<td class="gt_row gt_right" style="background-color: #FFE898; color: #000000;">2.1338</td>
<td class="gt_row gt_right" style="background-color: #9CD469; color: #000000;">1.5462</td>
<td class="gt_row gt_left">12</td></tr>
    <tr><td class="gt_row gt_right" style="background-color: #B4DF73; color: #000000;">1.6328</td>
<td class="gt_row gt_right" style="background-color: #B9E176; color: #000000;">1.6538</td>
<td class="gt_row gt_right" style="background-color: #F77E4A; color: #000000;">2.5389</td>
<td class="gt_row gt_right" style="background-color: #FECC7A; color: #000000;">2.2658</td>
<td class="gt_row gt_right" style="background-color: #F16740; color: #000000;">2.6119</td>
<td class="gt_row gt_right" style="background-color: #FED783; color: #000000;">2.2239</td>
<td class="gt_row gt_right" style="background-color: #B0DD70; color: #000000;">1.6195</td>
<td class="gt_row gt_left">13</td></tr>
    <tr><td class="gt_row gt_right" style="background-color: #BEE379; color: #000000;">1.6748</td>
<td class="gt_row gt_right" style="background-color: #C3E57C; color: #000000;">1.6930</td>
<td class="gt_row gt_right" style="background-color: #F7804B; color: #000000;">2.5327</td>
<td class="gt_row gt_right" style="background-color: #FEBA6B; color: #000000;">2.3406</td>
<td class="gt_row gt_right" style="background-color: #F26941; color: #000000;">2.6038</td>
<td class="gt_row gt_right" style="background-color: #FEC372; color: #000000;">2.3028</td>
<td class="gt_row gt_right" style="background-color: #B9E176; color: #000000;">1.6546</td>
<td class="gt_row gt_left">14</td></tr>
    <tr><td class="gt_row gt_right" style="background-color: #AADB6D; color: #000000;">1.5961</td>
<td class="gt_row gt_right" style="background-color: #AFDD70; color: #000000;">1.6149</td>
<td class="gt_row gt_right" style="background-color: #F8854D; color: #000000;">2.5196</td>
<td class="gt_row gt_right" style="background-color: #FEDB87; color: #000000;">2.2063</td>
<td class="gt_row gt_right" style="background-color: #F46C43; color: #000000;">2.5916</td>
<td class="gt_row gt_right" style="background-color: #FEE28F; color: #000000;">2.1705</td>
<td class="gt_row gt_right" style="background-color: #A0D669; color: #000000;">1.5609</td>
<td class="gt_row gt_left">15</td></tr>
    <tr><td class="gt_row gt_right" style="background-color: #B8E176; color: #000000;">1.6523</td>
<td class="gt_row gt_right" style="background-color: #BEE379; color: #000000;">1.6739</td>
<td class="gt_row gt_right" style="background-color: #EE613D; color: #000000;">2.6334</td>
<td class="gt_row gt_right" style="background-color: #FFE695; color: #000000;">2.1475</td>
<td class="gt_row gt_right" style="background-color: #E44D33; color: #FFFFFF;">2.7051</td>
<td class="gt_row gt_right" style="background-color: #FFEB9D; color: #000000;">2.1147</td>
<td class="gt_row gt_right" style="background-color: #ABDB6D; color: #000000;">1.6006</td>
<td class="gt_row gt_left">16</td></tr>
    <tr><td class="gt_row gt_right" style="background-color: #A2D76A; color: #000000;">1.5672</td>
<td class="gt_row gt_right" style="background-color: #A8DA6B; color: #000000;">1.5888</td>
<td class="gt_row gt_right" style="background-color: #F57446; color: #000000;">2.5707</td>
<td class="gt_row gt_right" style="background-color: #FFF7B2; color: #000000;">2.0356</td>
<td class="gt_row gt_right" style="background-color: #EC5E3B; color: #000000;">2.6457</td>
<td class="gt_row gt_right" style="background-color: #FFFCBA; color: #000000;">2.0011</td>
<td class="gt_row gt_right" style="background-color: #92D068; color: #000000;">1.5140</td>
<td class="gt_row gt_left">17</td></tr>
    <tr><td class="gt_row gt_right" style="background-color: #BFE47A; color: #000000;">1.6772</td>
<td class="gt_row gt_right" style="background-color: #C2E57C; color: #000000;">1.6915</td>
<td class="gt_row gt_right" style="background-color: #F16740; color: #000000;">2.6129</td>
<td class="gt_row gt_right" style="background-color: #FFE899; color: #000000;">2.1303</td>
<td class="gt_row gt_right" style="background-color: #E65135; color: #FFFFFF;">2.6899</td>
<td class="gt_row gt_right" style="background-color: #FFEFA4; color: #000000;">2.0897</td>
<td class="gt_row gt_right" style="background-color: #B2DE72; color: #000000;">1.6285</td>
<td class="gt_row gt_left">18</td></tr>
    <tr><td class="gt_row gt_right" style="background-color: #FFE99B; color: #000000;">2.1245</td>
<td class="gt_row gt_right" style="background-color: #FFE898; color: #000000;">2.1341</td>
<td class="gt_row gt_right" style="background-color: #B51427; color: #FFFFFF;">2.9295</td>
<td class="gt_row gt_right" style="background-color: #F8864E; color: #000000;">2.5147</td>
<td class="gt_row gt_right" style="background-color: #A50026; color: #FFFFFF;">2.9936</td>
<td class="gt_row gt_right" style="background-color: #FA9253; color: #000000;">2.4767</td>
<td class="gt_row gt_right" style="background-color: #FFF0A6; color: #000000;">2.0790</td>
<td class="gt_row gt_left">19</td></tr>
    <tr><td class="gt_row gt_right" style="background-color: #DAF08D; color: #000000;">1.7891</td>
<td class="gt_row gt_right" style="background-color: #DCF08F; color: #000000;">1.7960</td>
<td class="gt_row gt_right" style="background-color: #F57546; color: #000000;">2.5656</td>
<td class="gt_row gt_right" style="background-color: #FEC675; color: #000000;">2.2921</td>
<td class="gt_row gt_right" style="background-color: #EE623D; color: #000000;">2.6319</td>
<td class="gt_row gt_right" style="background-color: #FED17E; color: #000000;">2.2481</td>
<td class="gt_row gt_right" style="background-color: #D4ED88; color: #000000;">1.7635</td>
<td class="gt_row gt_left">20</td></tr>
    <tr><td class="gt_row gt_right" style="background-color: #5EB861; color: #000000;">1.3525</td>
<td class="gt_row gt_right" style="background-color: #62BA62; color: #000000;">1.3641</td>
<td class="gt_row gt_right" style="background-color: #FED07E; color: #000000;">2.2497</td>
<td class="gt_row gt_right" style="background-color: #FBFDBA; color: #000000;">1.9638</td>
<td class="gt_row gt_right" style="background-color: #FEBD6E; color: #000000;">2.3264</td>
<td class="gt_row gt_right" style="background-color: #F3FAAE; color: #000000;">1.9188</td>
<td class="gt_row gt_right" style="background-color: #58B45F; color: #000000;">1.3312</td>
<td class="gt_row gt_left">21</td></tr>
    <tr><td class="gt_row gt_right" style="background-color: #209A51; color: #FFFFFF;">1.1864</td>
<td class="gt_row gt_right" style="background-color: #2A9D53; color: #FFFFFF;">1.2058</td>
<td class="gt_row gt_right" style="background-color: #FEC271; color: #000000;">2.3075</td>
<td class="gt_row gt_right" style="background-color: #DAEF8D; color: #000000;">1.7877</td>
<td class="gt_row gt_right" style="background-color: #FDAD61; color: #000000;">2.3902</td>
<td class="gt_row gt_right" style="background-color: #D2EC86; color: #000000;">1.7524</td>
<td class="gt_row gt_right" style="background-color: #16904C; color: #FFFFFF;">1.1449</td>
<td class="gt_row gt_left">22</td></tr>
    <tr><td class="gt_row gt_right" style="background-color: #118747; color: #FFFFFF;">1.1070</td>
<td class="gt_row gt_right" style="background-color: #158D4B; color: #FFFFFF;">1.1325</td>
<td class="gt_row gt_right" style="background-color: #FEC877; color: #000000;">2.2820</td>
<td class="gt_row gt_right" style="background-color: #D6EE89; color: #000000;">1.7693</td>
<td class="gt_row gt_right" style="background-color: #FDB466; color: #000000;">2.3648</td>
<td class="gt_row gt_right" style="background-color: #CDEA83; color: #000000;">1.7358</td>
<td class="gt_row gt_right" style="background-color: #0D8043; color: #FFFFFF;">1.0754</td>
<td class="gt_row gt_left">23</td></tr>
    <tr><td class="gt_row gt_right" style="background-color: #3BA556; color: #FFFFFF;">1.2453</td>
<td class="gt_row gt_right" style="background-color: #46AA59; color: #000000;">1.2767</td>
<td class="gt_row gt_right" style="background-color: #FB9F59; color: #000000;">2.4381</td>
<td class="gt_row gt_right" style="background-color: #D5ED88; color: #000000;">1.7649</td>
<td class="gt_row gt_right" style="background-color: #F8864E; color: #000000;">2.5149</td>
<td class="gt_row gt_right" style="background-color: #CDEA83; color: #000000;">1.7334</td>
<td class="gt_row gt_right" style="background-color: #1D9950; color: #FFFFFF;">1.1797</td>
<td class="gt_row gt_left">24</td></tr>
  </tbody>
  
  
</table>
</div>
]
]

---

# Insights Into Extended Model

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]

]

---
name: conclusion

# Wrap-Up

Potential Downsides:
- Pointwise optimization can induce quantile crossing
  - Can be solved by sorting the predictions

Upsides:
- Pointwise learning outperforms the Naive solution significantly
- Online learning is much faster than batch methods
- Smoothing further improves the predictive performance
- Asymptotically not worse than the best convex combination

]

Important:
- The choice of the learning rate is crucial
- The loss function has to meet certain criteria

The [<svg aria-hidden="true" role="img" viewBox="0 0 496 512" style="height:1em;width:0.97em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M165.9 397.4c0 2-2.3 3.6-5.2 3.6-3.3.3-5.6-1.3-5.6-3.6 0-2 2.3-3.6 5.2-3.6 3-.3 5.6 1.3 5.6 3.6zm-31.1-4.5c-.7 2 1.3 4.3 4.3 4.9 2.6 1 5.6 0 6.2-2s-1.3-4.3-4.3-5.2c-2.6-.7-5.5.3-6.2 2.3zm44.2-1.7c-2.9.7-4.9 2.6-4.6 4.9.3 2 2.9 3.3 5.9 2.6 2.9-.7 4.9-2.6 4.6-4.6-.3-1.9-3-3.2-5.9-2.9zM244.8 8C106.1 8 0 113.3 0 252c0 110.9 69.8 205.8 169.5 239.2 12.8 2.3 17.3-5.6 17.3-12.1 0-6.2-.3-40.4-.3-61.4 0 0-70 15-84.7-29.8 0 0-11.4-29.1-27.8-36.6 0 0-22.9-15.7 1.6-15.4 0 0 24.9 2 38.6 25.8 21.9 38.6 58.6 27.5 72.9 20.9 2.3-16 8.8-27.1 16-33.7-55.9-6.2-112.3-14.3-112.3-110.5 0-27.5 7.6-41.3 23.6-58.9-2.6-6.5-11.1-33.3 2.6-67.9 20.9-6.5 69 27 69 27 20-5.6 41.5-8.5 62.8-8.5s42.8 2.9 62.8 8.5c0 0 48.1-33.6 69-27 13.7 34.7 5.2 61.4 2.6 67.9 16 17.7 25.8 31.5 25.8 58.9 0 96.5-58.9 104.2-114.8 110.5 9.2 7.9 17 22.9 17 46.4 0 33.7-.3 75.4-.3 83.6 0 6.5 4.6 14.4 17.3 12.1C428.2 457.8 496 362.9 496 252 496 113.3 383.5 8 244.8 8zM97.2 352.9c-1.3 1-1 3.3.7 5.2 1.6 1.6 3.9 2.3 5.2 1 1.3-1 1-3.3-.7-5.2-1.6-1.6-3.9-2.3-5.2-1zm-10.8-8.1c-.7 1.3.3 2.9 2.3 3.9 1.6 1 3.6.7 4.3-.7.7-1.3-.3-2.9-2.3-3.9-2-.6-3.6-.3-4.3.7zm32.4 35.6c-1.6 1.3-1 4.3 1.3 6.2 2.3 2.3 5.2 2.6 6.5 1 1.3-1.3.7-4.3-1.3-6.2-2.2-2.3-5.2-2.6-6.5-1zm-11.4-14.7c-1.6 1-1.6 3.6 0 5.9 1.6 2.3 4.3 3.3 5.6 2.3 1.6-1.3 1.6-3.9 0-6.2-1.4-2.3-4-3.3-5.6-2z"/></svg> profoc](https://profoc.berrisch.biz/) R Package:

- Implements all algorithms discussed above
- Is written using RcppArmadillo <svg aria-hidden="true" role="img" viewBox="0 0 448 512" style="height:1em;width:0.88em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:#000000;overflow:visible;position:relative;"><path d="M190.5 66.9l22.2-22.2c9.4-9.4 24.6-9.4 33.9 0L441 239c9.4 9.4 9.4 24.6 0 33.9L246.6 467.3c-9.4 9.4-24.6 9.4-33.9 0l-22.2-22.2c-9.5-9.5-9.3-25 .4-34.3L311.4 296H24c-13.3 0-24-10.7-24-24v-32c0-13.3 10.7-24 24-24h287.4L190.9 101.2c-9.8-9.3-10-24.8-.4-34.3z"/></svg> its fast
- Accepts vectors for most parameters
  - The best parameter combination is chosen online
- Implements 
  - Forgetting, Fixed Share
  - Different loss functions + gradients

]
]

<a href="https://github.com/BerriJ" class="github-corner" aria-label="View source on Github"><svg width="80" height="80" viewBox="0 0 250 250" style="fill:#f2f2f2; color:#212121; position: absolute; top: 0; border: 0; right: 0;" aria-hidden="true"><path d="M0,0 L115,115 L130,115 L142,142 L250,250 L250,0 Z"></path><path d="M128.3,109.0 C113.8,99.7 119.0,89.6 119.0,89.6 C122.0,82.7 120.5,78.6 120.5,78.6 C119.2,72.0 123.4,76.3 123.4,76.3 C127.3,80.9 125.5,87.3 125.5,87.3 C122.9,97.6 130.6,101.9 134.4,103.2" fill="currentColor" style="transform-origin: 130px 106px;" class="octo-arm"></path><path d="M115.0,115.0 C114.9,115.1 118.7,116.5 119.8,115.4 L133.7,101.6 C136.9,99.2 139.9,98.4 142.2,98.6 C133.8,88.0 127.5,74.4 143.8,58.0 C148.5,53.4 154.0,51.2 159.7,51.0 C160.3,49.4 163.2,43.6 171.4,40.1 C171.4,40.1 176.1,42.5 178.8,56.2 C183.1,58.6 187.2,61.8 190.9,65.4 C194.5,69.0 197.7,73.2 200.1,77.6 C213.8,80.2 216.3,84.9 216.3,84.9 C212.7,93.1 206.9,96.0 205.4,96.6 C205.1,102.4 203.0,107.8 198.3,112.5 C181.9,128.9 168.3,122.5 157.7,114.1 C157.9,116.9 156.7,120.9 152.7,124.9 L141.0,136.5 C139.8,137.7 141.6,141.9 141.8,141.8 Z" fill="currentColor" class="octo-body"></path></svg></a><style>.github-corner:hover .octo-arm{animation:octocat-wave 560ms ease-in-out}@keyframes octocat-wave{0%,100%{transform:rotate(0)}20%,60%{transform:rotate(-25deg)}40%,80%{transform:rotate(10deg)}}@media (max-width:500px){.github-corner:hover .octo-arm{animation:none}.github-corner .octo-arm{animation:octocat-wave 560ms ease-in-out}}</style>

???

Execution Times:

T = 5000

Opera:

Ml-Poly > 157 ms
Boa     > 212 ms

Profoc:

Ml-Poly > 17
BOA     > 16

---

[<svg aria-hidden="true" role="img" viewBox="0 0 576 512" style="height:1em;width:1.12em;vertical-align:-0.125em;margin-left:auto;margin-right:auto;font-size:inherit;fill:currentColor;overflow:visible;position:relative;"><path d="M552 64H88c-13.255 0-24 10.745-24 24v8H24c-13.255 0-24 10.745-24 24v272c0 30.928 25.072 56 56 56h472c26.51 0 48-21.49 48-48V88c0-13.255-10.745-24-24-24zM56 400a8 8 0 0 1-8-8V144h16v248a8 8 0 0 1-8 8zm236-16H140c-6.627 0-12-5.373-12-12v-8c0-6.627 5.373-12 12-12h152c6.627 0 12 5.373 12 12v8c0 6.627-5.373 12-12 12zm208 0H348c-6.627 0-12-5.373-12-12v-8c0-6.627 5.373-12 12-12h152c6.627 0 12 5.373 12 12v8c0 6.627-5.373 12-12 12zm-208-96H140c-6.627 0-12-5.373-12-12v-8c0-6.627 5.373-12 12-12h152c6.627 0 12 5.373 12 12v8c0 6.627-5.373 12-12 12zm208 0H348c-6.627 0-12-5.373-12-12v-8c0-6.627 5.373-12 12-12h152c6.627 0 12 5.373 12 12v8c0 6.627-5.373 12-12 12zm0-96H140c-6.627 0-12-5.373-12-12v-40c0-6.627 5.373-12 12-12h360c6.627 0 12 5.373 12 12v40c0 6.627-5.373 12-12 12z"/></svg> CRPS-Learning](https://arxiv.org/abs/2102.00968)

---
name:references

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Cesa-Bianchi, N. and G. Lugosi (2006). _Prediction, learning, and
games_. Cambridge university press.

Gaillard, P., G. Stoltz, and T. Van Erven (2014). "A second-order bound
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Gaillard, P. and O. Wintenberger (2018). "Efficient online algorithms
for fast-rate regret bounds under sparsity". In: _Advances in Neural
Information Processing Systems_. , pp. 7026-7036.

Gneiting, T. (2011a). "Making and evaluating point forecasts". In:
_Journal of the American Statistical Association_ 106.494, pp. 746-762.

Gneiting, T. and A. E. Raftery (2007). "Strictly proper scoring rules,
prediction, and estimation". In: _Journal of the American statistical
Association_ 102.477, pp. 359-378.

Herbster, M. and M. K. Warmuth (1998). "Tracking the best expert". In:
_Machine learning_ 32.2, pp. 151-178.

Kakade, S. M. and A. Tewari (2008). "On the Generalization Ability of
Online Strongly Convex Programming Algorithms." In: _NIPS_. , pp.
801-808.

---

# References 2

Wintenberger, O. (2017). "Optimal learning with Bernstein online
aggregation". In: _Machine Learning_ 106.1, pp. 119-141.

---

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