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\def\can{{\mathrm{can}}}
\def\coker{\operatorname{coker}\nolimits}
\def\End{\operatorname{End}\nolimits}
\def\eps{\varepsilon}
\def\gr{\operatorname{gr}\nolimits}
\def\Hom{\operatorname{Hom}\nolimits}
\def\id{\operatorname{id}\nolimits}
\def\idun{\mathbf{1}}
\def\iso{\buildrel \sim\over\to}
\def\opp{{\operatorname{opp}\nolimits}}
\def\rev{\operatorname{rev}}
\def\CC{{\mathcal{C}}}
\def\CD{{\mathcal{D}}}
\def\CE{{\mathcal{E}}}
\def\CL{{\mathcal{L}}}
\def\GS{{\mathfrak{S}}}

\newtheorem*{problem}{Problem}

\title{Second Batch Community Experiment --- Week \#1 Problems}
\date{\today}
\author{First Proof}
\begin{document}

\maketitle

\section{Problem C1}

We consider two online learning frameworks. In regret minimization (RM), the learner has a nonempty compact convex action set and the losses are convex, Lipschitz, and differentiable. In gradient equilibrium (GEQ), the learner’s action set is the Euclidean space and the gradients (or vector fields more generally) are bounded and restorative. Prove or disprove: a RM problem can always be solved using a GEQ learner, with no asymptotic loss in the learner’s error rate, and vice versa. Formulate reductions using standard assumptions in the online learning literature and state any additional assumptions that are made.

\section{Problem C2}

Let $\CC$ be a category. We will denote by $c$ the identity map $\id_c$ of
an object $c$ of $\CC$.

\smallskip
Let $k$ be a field. By category, we mean a category enriched in $k$-vector spaces.

\smallskip
Given $\CL$ a category, we denote by $\CL^\wedge$ the category of functors
from $\CL^\opp$ to the category of $k$-vector spaces.
The fully faithful functor $\CL\to\CL^\vee,\ l\mapsto \Hom(-,l)$
identifies $\CL$ with a full subcategory of $\CL^\wedge$.

\smallskip
Let $\CC$ be a monoidal category. The unit object of $\CC$ will be denoted by
$\idun$. We write $c_1c_2$ for the tensor
product $c_1\otimes c_2$ of two objects of $\CC$. Similarly, we write
$f_1f_2$ for the tensor product of two maps $f_1\otimes f_2$.

\smallskip
A $\CC$-module is a strict monoidal functor from $\CC$ to the
category of endofunctors of a category. A $(\CC,\CC)$-bimodule is a
$(\CC\otimes\CC^{\rev})$-module, where $\CC^{\rev}$ is the monoidal category with same underlying
category as $\CC$ but with $c\otimes^{\rev}c'=c'\otimes c$.

\medskip
We consider the symmetric monoidal category $\mathcal{F}$ of vector spaces $V$ with exhaustive filtrations $0=V^{\le -1}\subset V^{\le 0}\subset\cdots \subset V$.

Let $\mathcal{C}$ (resp. $\mathcal{D}$) be a monoidal (resp. symmetric monoidal) $\mathcal{F}$-enriched category and $\Phi:\gr\mathcal{D}\xrightarrow{\sim}\gr\mathcal{C}$ be an equivalence of monoidal categories enriched in graded vector spaces.

\medskip
Assume there is a full subcategory $\CD_1$ of $\CD^{\le 0}$ with the following properties:
\begin{itemize}
	\item[(i)] $\CD_1$ generates the monoidal category $\CD^{\le 0}$
	\item[(ii)] $\Hom_\CD(d_1\cdots d_n,d'_1\cdots d'_{n'})=0$ for all
		$d_1,\ldots,d_n,d'_1,\ldots,d'_{n'}\in\CD_1$ and all $n\neq n'$
	\item[(iii)] for all $d_1,\ldots,d_n,d'_1,\ldots,d_n$ and $r\ge 0$, there is an isomorphism
		\begin{align*}
			\bigoplus_{\substack{w\in\GS_n\\ \ell(w)=r}}
		\Hom^{\le 0}_{\CD}(d_1\cdots d_n,d'_{w(1)}\cdots d'_{w(n)})
			&\iso \bigl(\Hom_{\CD}^{\le r}/\Hom_{\CD}^{\le r-1}\bigr)
		(d_1\cdots d_n,d'_1\cdots d'_n) \\
			(f_w)_w&\mapsto \sum_w \sigma_{w^{-1}}\circ f_w
		\end{align*}
		where $\sigma_{w^{-1}}$ is the symmetry isomorphism corresponding to $w^{-1}$
	\item[(iv)] Consider $d_1,d_2,d_3,d'_1,d'_2,d'_3\in\CD_1$,
		$\gamma\in\Hom^{\le 0}_{\CD}(d_1d_2d_3,d_1d_3d_2)$ and $\gamma'\in
		\Hom^{\le 0}_{\CD}(d'_1d'_2d'_3,d'_1d'_3d'_2)$. If
		$u_1u_3u_2\circ \gamma=\gamma'\circ u_1u_2u_3$ for all
		$u_i\in\Hom_\CD(d_i,d'_i)$, $i\in\{1,2,3\}$, then $\gamma=\gamma'=0$.
\end{itemize}



\smallskip
Let $\delta \CC$ be the free
$(\CC,\CC)$-bimodule generated by one
object $\ast$ and
by maps $\alpha_c:\ast c\to c\ast$ for $c\in \CC$ modulo the
relations
$$\alpha_\idun=\id_\ast$$
$$\alpha_{cc'}=c\alpha_{c'}\circ \alpha_c c':\ast cc'\to cc'\ast$$
$$(f\ast)\circ\alpha_c=\alpha_{c'}\circ (\ast f):*c\to c'*
\text{ for }f\in\Hom_\CC(c,c').$$

\begin{problem}
Show that there is a functor $\Delta$ from $\CC$ to the category of
functors of $(\CC,\CC)$-bimodules from $\delta\CC$ to $(\delta\CC)^\wedge$
	such that $\Delta(c_0)(\ast)$ is a quotient of $c_0\ast$ and
	$\Delta(c_0)(\alpha_c)\neq 0$ if $cc_0{\not\simeq}0$.
\end{problem}

\end{document}
