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Jonathan Berrisch
2025-06-27 09:44:14 +02:00
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@@ -26298,12 +26298,12 @@ w_{t,k}^{\text{Naive}} = \frac{1}{K}\label{eq:naive_combination}
<p>EWA satisfies optimal selection convergence <span class="math inline">\(\eqref{eq_optp_select}\)</span> in a deterministic setting if:</p>
<p><i class="fa fa-fw fa-triangle-exclamation" style="color:var(--col_amber_9);"></i> Loss <span class="math inline">\(\ell\)</span> is exp-concave</p>
<p><i class="fa fa-fw fa-triangle-exclamation" style="color:var(--col_amber_9);"></i> Learning-rate <span class="math inline">\(\eta\)</span> is chosen correctly</p>
<p>Those results can be converted to <em>any</em> stochastic setting <span class="citation" data-cites="wintenberger2017optimal">Wintenberger (<a href="#/references" role="doc-biblioref" onclick>2017</a>)</span>.</p>
<p>Optimal convex aggregation convergence <span class="math inline">\(\eqref{eq_optp_conv}\)</span> can be satisfied by applying the kernel-trick:</p>
<p><span class="math display">\[\begin{align}
\ell^{\nabla}(x,y) = \ell&#39;(\widetilde{X},y) x
\end{align}\]</span></p>
<p><span class="math inline">\(\ell&#39;\)</span> is the subgradient of <span class="math inline">\(\ell\)</span> at forecast combination <span class="math inline">\(\widetilde{X}\)</span>.</p>
<p><span class="math inline">\(\ell&#39;\)</span> is the subgradient of <span class="math inline">\(\ell\)</span> at forecast combination <span class="math inline">\(\widetilde{X}\)</span></p>
<p>Those results can be converted to <em>any</em> stochastic setting <span class="citation" data-cites="wintenberger2017optimal">Wintenberger (<a href="#/references" role="doc-biblioref" onclick>2017</a>)</span></p>
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