CRPS-Learning

# CRPS-Learning
## Jonathan Berrisch, Florian Ziel
### University of Duisburg-Essen
### 2022-02-25

---

# 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

## Probabilistic Setting

An appropriate loss:

`\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.

]

## Optimal Convergence

Convergence rates of BOA are:

]

---

# 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 ECDFs and calculated weights using `\(T=25\)` realizations
- Pointwise is better than constant
- Pointwise solution creates rough estimates

We propose 2 smoothing procedures: 
- **P-Spline smoothing** upcoming slides
- Basis smoothing <a id='cite-BERRISCH2021'></a><a href='#bib-BERRISCH2021'>Berrisch and Ziel (2021)</a> 3.2
]

]

]
]]

---
    
# 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 `\(d=1\)`

]

---

# 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)

]]]

---

# Simulation Study

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

---

# Simulation Study

**New DGP:**

`\begin{align}
    Y_t               & \sim \mathcal{N}\left(0.15 \operatorname{asinh}(\mu_t),\,1\right) \\
     \mu_t            &= 0.99 \mu_{t-1} + \varepsilon_t\\
    \varepsilon_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_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**
- Important regions can receive more knots
- Destroys shrinkage towards constant
- Profoc utilizes the beta distribution to create knots

]

---

# Application Study: Overview

Data:

- Forecasting European emission allowances (EUA)
- Daily month-ahead prices 
- Jan 13 - Dec 20 (Phase III, 2092 Obs)

Combination methods:

- Naive, BOAG, EWAG, ML-PolyG, BMA

Tuning paramter grids:

- Smoothing Penalty: `\(\Lambda= \{0\}\cup \{2^x|x\in \{-4,-3.5,\ldots,12\}\}\)`
- Learning Rates: `\(\mathcal{E}= \{2^x|x\in \{-1,-0.5,\ldots,9\}\}\)`

]
]

]

---

# Application Study: Experts

Simple exponential smoothing with additive errors (**ETS-ANN**):

`\begin{align*}
Y_{t} = l_{t-1} + \varepsilon_t \quad \text{with} \quad l_t = l_{t-1} + \alpha \varepsilon_t \quad \text{and} \quad \varepsilon_t \sim \mathcal{N}(0,\sigma^2)
\end{align*}`

Quantile regression (**QuantReg**): For each `\(p \in \mathcal{P}\)` we assume:

`\begin{align*}
F^{-1}_{Y_t}(p) = \beta_{p,0} + \beta_{p,1} Y_{t-1} + \beta_{p,2} |Y_{t-1}-Y_{t-2}|
\end{align*}`

ARIMA(1,0,1)-GARCH(1,1) with Gaussian errors (**ARMA-GARCH**):

`\begin{align*}
Y_{t} = \mu + \phi(Y_{t-1}-\mu) + \theta \varepsilon_{t-1} + \varepsilon_t \quad \text{with} \quad \varepsilon_t = \sigma_t Z, \quad \sigma_t^2 = \omega + \alpha \varepsilon_{t-1}^2 + \beta \sigma_{t-1}^2 \quad \text{and} \quad Z_t \sim \mathcal{N}(0,1)
\end{align*}`

ARIMA(0,1,0)-I-EGARCH(1,1) with Gaussian errors (**I-EGARCH**):

`\begin{align*}
Y_{t} = \mu + Y_{t-1}  + \varepsilon_t \quad \text{with} \quad \varepsilon_t = \sigma_t Z, \quad \log(\sigma_t^2) = \omega + \alpha Z_{t-1}+ \gamma (|Z_{t-1}|-\mathbb{E}|Z_{t-1}|) + \beta \log(\sigma_{t-1}^2) \quad \text{and} \quad Z_t \sim \mathcal{N}(0,1)
\end{align*}`

ARIMA(0,1,0)-GARCH(1,1) with student-t errors (**I-GARCHt**}):

`\begin{align*}
Y_{t} = \mu + Y_{t-1}  + \varepsilon_t \quad \text{with} \quad \varepsilon_t = \sigma_t Z, \quad \sigma_t^2 = \omega + \alpha \varepsilon_{t-1}^2 + \beta \sigma_{t-1}^2 \quad \text{and} \quad Z_t \sim t(0,1, \nu)
\end{align*}`

]

---

# Application Study: Results

<table class=" lightable-material" style='font-family: "Source Sans Pro", helvetica, sans-serif; margin-left: auto; margin-right: auto;'>
 <thead>
  <tr>
   <th style="text-align:left;">   </th>
   <th style="text-align:center;"> BOAG </th>
   <th style="text-align:center;"> EWAG </th>
   <th style="text-align:center;"> ML-PolyG </th>
   <th style="text-align:center;"> BMA </th>
  </tr>
 </thead>
<tbody>
  <tr>
   <td style="text-align:left;font-weight: bold;"> pointwise </td>
   <td style="text-align:center;background-color: #99EE80 !important;"> -0.161 (0.067) </td>
   <td style="text-align:center;background-color: #D6FF80 !important;"> -0.085 (0.177) </td>
   <td style="text-align:center;background-color: #AFF580 !important;"> -0.136 (0.126) </td>
   <td style="text-align:center;background-color: #FFF980 !important;"> 0.030 (0.753) </td>
  </tr>
  <tr>
   <td style="text-align:left;font-weight: bold;"> smooth </td>
   <td style="text-align:center;background-color: #85E780 !important;"> -0.185 (0.037) </td>
   <td style="text-align:center;background-color: #D1FF80 !important;"> -0.094 (0.150) </td>
   <td style="text-align:center;background-color: #99EE80 !important;"> -0.161 (0.066) </td>
   <td style="text-align:center;background-color: #FFF980 !important;"> 0.027 (0.722) </td>
  </tr>
  <tr>
   <td style="text-align:left;font-weight: bold;"> constant </td>
   <td style="text-align:center;background-color: #AEF580 !important;"> -0.137 (0.020) </td>
   <td style="text-align:center;background-color: #E0FF80 !important;"> -0.067 (0.144) </td>
   <td style="text-align:center;background-color: #B3F780 !important;"> -0.132 (0.027) </td>
   <td style="text-align:center;background-color: #FFF880 !important;"> 0.035 (0.826) </td>
  </tr>
  <tr>
   <td style="text-align:left;font-weight: bold;"> smooth* </td>
   <td style="text-align:center;background-color: #80E680 !important;"> -0.191 (0.023) </td>
   <td style="text-align:center;background-color: #9CEF80 !important;"> -0.158 (0.025) </td>
   <td style="text-align:center;background-color: #80E680 !important;"> -0.190 (0.021) </td>
   <td style="text-align:center;background-color: #FFFE80 !important;"> -0.009 (0.333) </td>
  </tr>
</tbody>
</table>

CRPS difference to **Naive** (scaled by `\(10^4\)`) of single experts and four combination methods with four options.
Additionally, we show the p-value of the DM-test, testing against **Naive**. The smallest value is bold. We also report the optimal ex-post selection by **smooth***

]

]

.panel[.panel-name[Weights]
<img src="data:image/png;base64,#index_files/figure-html/unnamed-chunk-11-1.svg" style="display: block; margin: auto;" />
]

]]

---

# Application to Multivariate Settings

Apply BOA to (24 dimensional) day-ahead power forecasts

Forecasts from probabilistic neural networks:

Pre-Print to be released soon: Grzegorz Marcjasz, Michał Narajewski, Rafał Weron and Florian Ziel

```r
# 182 obs as burn-in
mod <- online(
  y = Y, # 736 x 24
  experts = experts, # 736, 24, 99, 2
  tau = 1:99 / 100
)
```

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<table class="gt_table">
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      <th class="gt_col_heading gt_columns_bottom_border gt_right" rowspan="1" colspan="1">jsu</th>
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    <tr><td class="gt_row gt_right">1.448</td>
<td class="gt_row gt_right">1.372</td>
<td class="gt_row gt_right">1.334</td></tr>
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]

---

# Smoothing in Multivariate Setting

Smoothing parameters can be optimized online

Below parameters chosen for illustrative purposes

```r
mod <- online(
  y = Y,
  experts = experts,
  tau = 1:99 / 100,
  p_smooth_pr = list(
    lambda = c(20, 50, 100) # 50 selected
  ),
  p_smooth_mv = list(
    lambda = c(20, 50, 100) # 50 selected
  )
)
```

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<table class="gt_table">
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    </tr>
    
  </thead>
  <thead class="gt_col_headings">
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      <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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  <tbody class="gt_table_body">
    <tr><td class="gt_row gt_right">1.448</td>
<td class="gt_row gt_right">1.372</td>
<td class="gt_row gt_right">1.333</td></tr>
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]

---

# 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
  - Best parameter combination chosen online
- Implements 
  - Forgetting, Fixed Share
  - Different loss functions + gradients
  - Multivariate Inputs

]
]

<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>

---
name:references

# References 1

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<p><cite><a id='bib-cesa2006prediction'></a><a href="#cite-cesa2006prediction">Cesa-Bianchi, N. and G. Lugosi</a>
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<p><cite><a id='bib-gaillard2018efficient'></a><a href="#cite-gaillard2018efficient">Gaillard, P. and O. Wintenberger</a>
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&ldquo;Efficient online algorithms for fast-rate regret bounds under sparsity&rdquo;.
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<em>Advances in Neural Information Processing Systems</em>.
, pp. 7026&ndash;7036.</cite></p>

<p><cite><a id='bib-gneiting2011making'></a><a href="#cite-gneiting2011making">Gneiting, T.</a>
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&ldquo;Making and evaluating point forecasts&rdquo;.
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<p><cite><a id='bib-gneiting2007strictly'></a><a href="#cite-gneiting2007strictly">Gneiting, T. and A. E. Raftery</a>
(2007).
&ldquo;Strictly proper scoring rules, prediction, and estimation&rdquo;.
In: <em>Journal of the American statistical Association</em> 102.477, pp. 359&ndash;378.</cite></p>

<p><cite><a id='bib-herbster1998tracking'></a><a href="#cite-herbster1998tracking">Herbster, M. and M. K. Warmuth</a>
(1998).
&ldquo;Tracking the best expert&rdquo;.
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<p><cite><a id='bib-wintenberger2017optimal'></a><a href="#cite-wintenberger2017optimal">Wintenberger, O.</a>
(2017).
&ldquo;Optimal learning with Bernstein online aggregation&rdquo;.
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