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<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="utf-8"/>
<title>Parameter-free Mirror Descent</title>
<meta name="author" content="Andrew Jacobsen, Ashok Cutkosky"/>
<style type="text/css">
.underline { text-decoration: underline; }
</style>
<link rel="stylesheet" href="file:////home/andrew/reveal.js/dist/reveal.css"/>
<link rel="stylesheet" href="file:////home/andrew/reveal.js/dist/theme/night.css" id="theme"/>
<link rel="stylesheet" href="./presentation.css"/>
<script type="text/javascript" src="https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.5/MathJax.js?config=TeX-AMS-MML_HTMLorMML"></script>
</head>
<body>
<div class="reveal">
<div class="slides">
<section id="sec-title-slide">
<h1 class="title">Parameter-free Mirror Descent</h1><h2 class="author">Andrew Jacobsen, Ashok Cutkosky</h2><p class="date">Created: 2022-06-17 Fri 11:10</p>
</section>
<p>
\(
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</p>
\(
\newcommand{\orange}[1]{\Color{orange}{#1}}
\newcommand{\skyblue}[1]{\Color{skyblue}{#1}}
\newcommand{\salmon}[1]{\Color{salmon}{#1}}
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\newcommand{\wt}{w_t}
\)
<section>
<section id="slide-orgc93b7fd">
<h2 id="orgc93b7fd">Online Learning</h2>
<ul>
<li class="fragment appear">“Learning from a stream of data”</li>
<li class="fragment appear">Learning problem formalized as a game:
<ul>
<li class="fragment appear">on each round \(t\in[1,T]\)
<ul>
<li class="fragment appear">learner chooses a \(\color{salmon}{w_t\in W}\)</li>
<li class="fragment appear">adversary reveals a convex <i>loss function</i> \(\color{skyblue}{\ell_t(\cdot)}\)</li>
<li class="fragment appear">learner incurs a loss of \(\color{skyblue}{\ell_t(}\color{salmon}{w_t}\color{skyblue}{)}\)</li>
</ul></li>
</ul></li>
<li class="fragment appear"><p>
<b>(Static) Regret</b>: compare \(\color{salmon}{\text{your total loss}}\) to that of \(\color{skyblue}{\text{any fixed decision }\cmp}\)
</p>
<div>
\begin{align*}
R_T(\cmp) = \color{salmon}{\sum_{t=1}^T \ell_t(w_t)} - \color{skyblue}{\sum_{t=1}^T\ell_t(\cmp)}
\end{align*}
</div>
<ul>
<li class="fragment appear"><i>e.g.</i> in a bounded domain, usually care about \(\cmp=\argmin_x\sumtT\ell_t(x)\)</li>
</ul></li>
</ul>
</section>
</section>
<section>
<section id="slide-org6521315">
<h2 id="org6521315">Parameter-free Learning</h2>
<ul>
<li class="fragment appear">Gradient Descent: \(\wtpp = \wt-\eta\skyblue{\grad\ell_t(w_t)}\)
<ul>
<li class="fragment appear"><p>
Setting step-size \(\eta=\frac{\orange{\norm{u}}}{\sqrt{\skyblue{\sumtT\norm{\grad\ell_t(w_t)}^2}}}\) guarantees optimal regret:
</p>
<div>
\begin{align*}
R_T(\orange{\cmp})\le\orange{\norm{\cmp}}\sqrt{\skyblue{\sumtT\norm{\grad\ell_t(w_t)}^2}}
\end{align*}
</div></li>
<li class="fragment appear">Impossible to match without prior knowledge of \(\orange{\cmp}\)</li>
<li class="fragment appear"><p>
An algorithm is \(\plum{\textit{``parameter-free''}}\) if can match this bound \(\plum{\textit
{up to log terms}}\)
</p>
<div>
\begin{align*}
R_T(\orange{\cmp})\le O\brac{\orange{\norm{\cmp}}\sqrt{\skyblue{\sumtT\norm{\grad\ell_t(w_t)}^2}\plum{\log(\norm{u}T+1)}}}
\end{align*}
</div>
<ul>
<li class="fragment appear"><i>ex.</i> coin-betting, FTRL, potential-based methods
<ul>
<li class="fragment appear"><i>“approximate”</i> the best-in-hindsight decision</li>
</ul></li>
</ul></li>
</ul></li>
</ul>
</section>
</section>
<section>
<section id="slide-orgc5f8a58">
<h2 id="orgc5f8a58">\(A \Color{orange}{\textrm{ Fixed Comparator}}\) Isn’t Always Appropriate</h2>
<ul>
<li class="fragment appear"><p>
<b>Illustrative Example:</b> Predict next observation \(\Color{royalblue}{y_t\sim\mathcal{N}(\mu_t,\sigma^2)}\)
</p>
<img src="windowed_online_rw_baseline1.gif", width=1200, height=700,align=right></li>
<li class="fragment appear">\(\ell_t(x)=\half(y_t-x)^2\)</li>
</ul>
</section>
</section>
<section>
<section id="slide-org8b1a8f6">
<h2 id="org8b1a8f6">\(A \Color{orange}{\textrm{ Fixed Comparator}}\) Isn’t Always Appropriate</h2>
<ul>
<li><p>
<b>Illustrative Example:</b> Predict next observation \(\Color{royalblue}{y_t\sim\mathcal{N}(\mu_t,\sigma^2)}\)
</p>
<img src="windowed_online_rw_baseline2.gif", width=1200, height=700,align=right></li>
<li>\(\ell_t(x)=\half(y_t-x)^2\)</li>
</ul>
</section>
</section>
<section>
<section id="slide-orge967861">
<h2 id="orge967861">Dynamic Regret</h2>
<ul>
<li><p>
<b>Dynamic Regret</b>: compare your total loss to the total loss of any other \(\color{orange}{\text{sequence}}\)
</p>
<div>
\begin{align*}
R_T(\Color{orange}{\cmp_1,\ldots,\cmp_T}) = \sum_{t=1}^T\ell_t(w_t)-\sum_{t=1}^T\ell_t(\color{orange}{\cmp_t})
\end{align*}
</div></li>
</ul>
<img src="blank.svg", width=800, height=466,align=right>
<img src="blank.svg", width=800, height=466,align=right>
<img src="blank.svg", width=800, height=500,align=right>
</section>
</section>
<section>
<section id="slide-org763f391">
<h2 id="org763f391">Dynamic Regret</h2>
<ul>
<li><p>
<b>Dynamic Regret</b>: compare your total loss to the total loss of any other \(\color{orange}{\text{sequence}}\)
</p>
<div>
\begin{align*}
R_T(\Color{orange}{\cmp_1,\ldots,\cmp_T}) = \sum_{t=1}^T\ell_t(w_t)-\sum_{t=1}^T\ell_t(\color{orange}{\cmp_t})
\end{align*}
</div></li>
</ul>
<img src="windowed_online_rw_baseline4.gif", width=800, height=466,align=right>
<img src="blank.svg", width=800, height=466,align=right>
<img src="blank.svg", width=800, height=500,align=right>
</section>
</section>
<section>
<section id="slide-org49fb1bc">
<h2 id="org49fb1bc">Dynamic Regret</h2>
<ul>
<li><p>
<b>Dynamic Regret</b>: compare your total loss to the total loss of any other \(\color{orange}{\text{sequence}}\)
</p>
<div>
\begin{align*}
R_T(\Color{orange}{\cmp_1,\ldots,\cmp_T}) = \sum_{t=1}^T\ell_t(w_t)-\sum_{t=1}^T\ell_t(\color{orange}{\cmp_t})
\end{align*}
</div></li>
</ul>
<img src="windowed_online_rw_baseline4.gif", width=800, height=466,align=right>
<img src="windowed_online_rw_baseline3.gif", width=800, height=466,align=right>
<ul>
<li class="fragment appear"><p>
<i>complexity</i> of the sequence quantified by \(\Color{orange}{\textbf{Path Length}}\)
</p>
<div>
\begin{align*}
\Color{orange}{P_T} = \sum_{t=2}^T\norm{\color{orange}{\cmp_t-\cmp_\tmm}}
\end{align*}
</div></li>
</ul>
</section>
</section>
<section>
<section id="slide-orgd59bf58">
<h2 id="orgd59bf58">Dynamic Regret (Continued)</h2>
<ul>
<li class="fragment appear"><p>
In <b>Bounded domains</b> with \(\max_{x\in\cX}\norm{x}= D\), Gradient descent with a fixed step-size \(\eta\) guarantees
</p>
<p>
\(
\qquad\qquad R_T^{\eta}(\orange{\cmp_1,\ldots,\cmp_T})\le O\brac{\frac{\Color{orange}{D^2+DP_T}}{2\eta}+\frac{\eta}{2}\Color{skyblue}{\sumtT\norm{\grad\ell_t(w_t)}^2}}
\)
</p></li>
<li class="fragment appear"><p>
Choosing \(\eta^*=\sqrt{\frac{\orange{D^2+DP_T}}{\skyblue{\sumtT\norm{\grad\ell_t(w_t)}^2}}}\) guarantees (optimal) dynamic regret:
</p>
<p>
\(
\qquad\qquad R_T^{\eta^*}(\orange{\cmp_1,\ldots,\cmp_T})\le O\brac{\sqrt{\Color{orange}{(D^2+DP_T)}\skyblue{\sumtT\norm{\grad\ell_t(w_t)}^2}}}
\)
</p></li>
<li class="fragment appear">Run (projected) gradient descent with many different step-sizes in parallel,
combine them using an experts algorithm (Zhang <i>et al.</i>, 2018)
<ul>
<li>using \(\Log{T}\) experts, one can guarantee the optimal
bound up to log terms</li>
</ul></li>
</ul>
</section>
</section>
<section>
<section id="slide-orgc61b755">
<h2 id="orgc61b755">Trickier: <i>Unbounded</i> domains</h2>
<ul>
<li class="fragment appear">in an <span class="underline">unbounded domain</span> gradient descent <span class="underline">cannot</span> guarantee \(R_T\le \tilde O(\sqrt{\skyblue{T}\orange{P_T}})\)
<ul>
<li><b>(Theorem 3)</b> In fact, we show that “FTRL-like” algorithms cannot guarantee \(\tilde O\left(\sqrt{\skyblue{T}\orange{P_T}}\right)\) dynamic regret</li>
</ul></li>
<li class="fragment appear">If you are very clever, it is possible to fix this issue considering a larger pool of \(O(T\Log{T})\) experts (and a smarter meta-algorithm)</li>
</ul>
<ul>
<li class="fragment appear"><i>i.e.</i> run <i>projected</i> gradient descent with many different step-sizes <i>and</i> “every reasonable domain size”</li>
<li class="fragment appear">Luo <i>et al.</i> 2022: Corralling a Larger Band of Bandits: A Case Study on \(\Color{gold}{\text{Switching Regret}}\) for Linear Bandits
<ul>
<li>\(\color{gold}{\text{Also here at COLT!}}\)</li>
</ul></li>
</ul>
</section>
</section>
<section>
<section id="slide-org8abae28">
<h2 id="org8abae28">Our Approach</h2>
<ul>
<li class="fragment appear"><p>
We instead approach the problem from a <span class="underline">mirror descent-based</span> perspective
</p>
<div>
\begin{align*}
w_\tpp=\argmin_w \inner{\grad\ell_t(w_t),w}+\Color{plum}{\phi_t(w)}+D_{\psi_t}(w|wt)
\end{align*}
</div>
<ul>
<li class="fragment appear">Key: additional \(\color{plum}{\textit{composite objective }\phi_t(w)}\), carefully chosen to
ensure stability</li>
</ul></li>
<li class="fragment appear"><p>
<b>(Proposition 1)</b> We design an algorithm which guarantees for any \(\eta\)
</p>
<div>
\begin{align*}
R_T^\eta(\orange{\cmp_1,\ldots,\cmp_T})\le\tilde{O}\brac{\frac{\Color{orange}{\max_t\norm{\cmp_t}+P_T}}{\eta}+\eta\skyblue{\sumtT\norm{\grad\ell_t(w_t)}^2}\orange{\norm{\cmp_t}}}
\end{align*}
</div></li>
<li class="fragment appear"><p>
<b>(Theorem 4)</b> “combining” many such algorithms for a set reasonable of \(\eta\)’s guarantees
</p>
<div>
\begin{align*}
R_T(\orange{\cmp_1,\ldots,\cmp_T})&\le \tilde O\brac{\sqrt{\brac{\orange{\max_t\norm{\cmp_t}+P_T}}\skyblue{\sumtT\norm{\grad\ell_t(w_t)}^2}\orange{\norm{\cmp_t}}}}
\end{align*}
</div>
<ul>
<li class="fragment appear">prior works (bounded domain):
\(
R_T(\orange{\cmp_1,\ldots,\cmp_T})\le \tilde O\brac{\sqrt{\brac{\orange{D^2+DP_T}}\skyblue{\sumtT\norm{\grad\ell_t(w_t)}^2}}}
\)
<ul>
<li>unexpected improvement: improved adaptivity to the individual \(\orange{\norm{\cmp_t}}\)!</li>
</ul></li>
</ul></li>
</ul>
</section>
</section>
<section>
<section id="slide-org5fd56a3">
<h2 id="org5fd56a3">Our Approach: Proof Sketch</h2>
<ul>
<li>\(
R_T^\eta(\orange{\cmp_1,\ldots,\cmp_T})\le\tilde{O}\brac{\frac{\Color{orange}{\max_t\norm{\cmp_t}+P_T}}{\eta}+\eta\skyblue{\sumtT\norm{\grad\ell_t(w_t)}^2}\orange{\norm{\cmp_t}}}\)
<ul>
<li class="fragment appear">Notice! \(R_T^\eta(\orange{0,\ldots,0})\le O(1)\) (constant)!
<ul>
<li class="fragment appear">Run algorithm for many \(\eta\in\cS\) in parallel and play \(\plum{w_t=\sum_{\eta}\wt^\eta}\).</li>
<li class="fragment appear"><p>
Then for any \(\widetilde\eta\in\cS\):
</p>
<div>
\begin{align*}
R_T(\orange{\cmp_1,\ldots\cmp_T})
&\overset{\text{(convexity)}}{\le}
\sumtT\inner{\grad\ell_t(w_t), \plum{w_t}-\orange{\cmp_t}}
\le
\sumtT\inner{\grad\ell_t(w_t), \plum{\sum_\eta w_t^\eta}-\orange{\cmp_t}}\\
\end{align*}
</div></li>
</ul></li>
</ul></li>
</ul>
<img src="blank.svg", width=1600, height=500,align=left>
</section>
</section>
<section>
<section id="slide-org2393acd">
<h2 id="org2393acd">Our Approach: Proof Sketch</h2>
<ul>
<li>\(
R_T^\eta(\orange{\cmp_1,\ldots,\cmp_T})\le\tilde{O}\brac{\frac{\Color{orange}{\max_t\norm{\cmp_t}+P_T}}{\eta}+\eta\skyblue{\sumtT\norm{\grad\ell_t(w_t)}^2}\orange{\norm{\cmp_t}}}\)
<ul>
<li>Notice! \(R_T^\eta(\orange{0,\ldots,0})\le O(1)\) (constant)!
<ul>
<li>Run algorithm for many \(\eta\in\cS\) in parallel and play \(\plum{w_t=\sum_{\eta}\wt^\eta}\).</li>
<li><p>
Then for any \(\widetilde\eta\in\cS\):
</p>
<div>
\begin{align*}
R_T(\orange{\cmp_1,\ldots\cmp_T})
&\overset{\text{(convexity)}}{\le}
\sumtT\inner{\grad\ell_t(w_t), \plum{w_t}-\orange{\cmp_t}}
\le
\sumtT\inner{\grad\ell_t(w_t), \plum{\sum_\eta w_t^\eta}-\orange{\cmp_t}}\\
&=
\underbrace{\sumtT\inner{\grad\ell_t(w_t), \plum{w_t^{\widetilde\eta}}-\orange{\cmp_t}}}_{R_T^{\widetilde\eta}(\orange{\cmp_1,\ldots\cmp_T})}+\sum_{\eta\ne\widetilde\eta}\underbrace{\sumtT\inner{\grad\ell_t(w_t),\plum{\wt^\eta}-\orange{\zeros}}}_{R_T^\eta(\orange{\zeros,\ldots,\zeros})\le O(1)}
\end{align*}
</div></li>
</ul></li>
</ul></li>
</ul>
<img src="blank.svg", width=1200, height=500,align=right>
</section>
</section>
<section>
<section id="slide-orgd5190ce">
<h2 id="orgd5190ce">Our Approach: Proof Sketch</h2>
<ul>
<li>\(
R_T^\eta(\orange{\cmp_1,\ldots,\cmp_T})\le\tilde{O}\brac{\frac{\Color{orange}{\max_t\norm{\cmp_t}+P_T}}{\eta}+\eta\skyblue{\sumtT\norm{\grad\ell_t(w_t)}^2}\orange{\norm{\cmp_t}}}\)
<ul>
<li>Notice! \(R_T^\eta(\orange{0,\ldots,0})\le O(1)\) (constant)!
<ul>
<li>Run algorithm for many \(\eta\in\cS\) in parallel and play \(\plum{w_t=\sum_{\eta}\wt^\eta}\).</li>
<li><p>
Then for any \(\widetilde\eta\in\cS\):
</p>
<div>
\begin{align*}
R_T(\orange{\cmp_1,\ldots\cmp_T})
&\overset{\text{(convexity)}}{\le}
\sumtT\inner{\grad\ell_t(w_t), \plum{w_t}-\orange{\cmp_t}}
\le
\sumtT\inner{\grad\ell_t(w_t), \plum{\sum_\eta w_t^\eta}-\orange{\cmp_t}}\\
&=
\underbrace{\sumtT\inner{\grad\ell_t(w_t), \plum{w_t^{\widetilde\eta}}-\orange{\cmp_t}}}_{R_T^{\widetilde\eta}(\orange{\cmp_1,\ldots\cmp_T})}+\sum_{\eta\ne\widetilde\eta}\underbrace{\sumtT\inner{\grad\ell_t(w_t),\plum{\wt^\eta}-\orange{\zeros}}}_{R_T^\eta(\orange{\zeros,\ldots,\zeros})\le O(1)}\\
&=
R_T^{\widetilde\eta}(\orange{\cmp_1,\ldots,\orange{\cmp_t}})+|\cS|\cdot O(1)
\end{align*}
</div></li>
</ul></li>
</ul></li>
</ul>
<img src="blank.svg", width=1200, height=500,align=right>
</section>
</section>
<section>
<section id="slide-org5ba55ec">
<h2 id="org5ba55ec">Our Approach: Proof Sketch</h2>
<ul>
<li>\(
R_T^\eta(\orange{\cmp_1,\ldots,\cmp_T})\le\tilde{O}\brac{\frac{\Color{orange}{\max_t\norm{\cmp_t}+P_T}}{\eta}+\eta\skyblue{\sumtT\norm{\grad\ell_t(w_t)}^2}\orange{\norm{\cmp_t}}}\)
<ul>
<li>Notice! \(R_T^\eta(\orange{0,\ldots,0})\le O(1)\) (constant)!
<ul>
<li>Run algorithm for many \(\eta\in\cS\) in parallel and play \(\plum{w_t=\sum_{\eta}\wt^\eta}\).</li>
<li><p>
Then for any \(\widetilde\eta\in\cS\):
</p>
<div>
\begin{align*}
R_T(\orange{\cmp_1,\ldots\cmp_T})
&\overset{\text{(convexity)}}{\le}
\sumtT\inner{\grad\ell_t(w_t), \plum{w_t}-\orange{\cmp_t}}
\le
\sumtT\inner{\grad\ell_t(w_t), \plum{\sum_\eta w_t^\eta}-\orange{\cmp_t}}\\
&=
\underbrace{\sumtT\inner{\grad\ell_t(w_t), \plum{w_t^{\widetilde\eta}}-\orange{\cmp_t}}}_{R_T^{\widetilde\eta}(\orange{\cmp_1,\ldots\cmp_T})}+\sum_{\eta\ne\widetilde\eta}\underbrace{\sumtT\inner{\grad\ell_t(w_t),\plum{\wt^\eta}-\orange{\zeros}}}_{R_T^\eta(\orange{\zeros,\ldots,\zeros})\le O(1)}\\
&=
R_T^{\widetilde\eta}(\orange{\cmp_1,\ldots,\orange{\cmp_t}})+|\cS|\cdot O(1)
\end{align*}
</div>
<ul>
<li>This holds for <b>any</b> \(\widetilde\eta\in \cS\)! \(\therefore\) Just need to ensure:
<ul>
<li>there is <i>some</i> near-optimal \(\widetilde\eta\in\cS\)</li>
<li>\(|\cS|\) “small enough”</li>
</ul></li>
</ul></li>
</ul></li>
</ul></li>
</ul>
<img src="blank.svg", width=1200, height=500,align=right>
</section>
</section>
<section>
<section id="slide-orgdcdf2b8">
<h3 id="orgdcdf2b8">Our Approach: Bonus Improvements!</h3>
<ul>
<li class="fragment appear">Parameter-free \(\Color{lightgreen}{\mathit{implicit}}\) updates: \(w_\tpp=\argmin_w \Color{lightgreen}{\ell_t(w)}+\phi_t(w)+D_{\psi_t}(w|w_t)\)
<ul>
<li class="fragment appear"><p>
Enables us to design an algorithm which adapts to \(\Color{lightgreen}{\textrm{gradient variability}}\) in unbounded domains:
</p>
<div>
\begin{align*}
R_T(\orange{\cmp})\le \tilde O\brac{\orange{\norm{\cmp}}\sqrt{\sumtT\Color{lightgreen}{\norm{\grad\ell_t(w_t)-\grad\ell_\tmm(w_t)}^2}}}
\end{align*}
</div></li>
</ul></li>
<li class="fragment appear">Parameter-free + Scale-free algorithm <i>without</i> restart strategies
<ul>
<li class="fragment appear">Current state-of-the-art: FreeGrad (Mhammedi 2020)</li>
</ul></li>
</ul>
</section>
</section>
<section>
<section id="slide-orgece394f">
<h2 id="orgece394f">References</h2>
<ul>
<li>Haipeng Luo, Mengxiao Zhang, Peng Zhao, Zhi-Hua Zhou. Corralling a Larger Band of Bandits: A Case Study on Switching Regret for Linear Bandits. In <i>Conference on Learning Theory (COLT)</i>, 2022.</li>
<li>Lijun Zhang, Shiyin Lu, and Zhi-Hua Zhou. Adaptive online learning in dynamic environments. In <i>Neural Information Processing Systems</i>, 2018.</li>
<li>Zakaria Mhammedi, Wouter M. Koolen Proceedings of Thirty Third Conference on Learning Theory, PMLR 125:2858-2887, 2020.</li>
</ul>
</section>
</section>
</div>
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