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\begin{document}

% paper title
\title{Lecture Notes for September 23th}
\date{}

\author{
    Dmitry Berenson\\
}



\maketitle

\subsection*{Projected Gradient Descent Regret Bounds}

Distance between optimal weight vector $w^*$ and the weight vector at time $t$ ($w_t$).

\begin{equation}
	D(w_t, w^*)	= (w_t - w^*)^T(w_t - w^*)
\end{equation}

\noindent then the distance after the update for the next time step is:


\begin{equation}
	D(w_{t+1},w^*) - D(w_t,w^*) = \left|\left|w_t - \alpha \nabla l_t - w^*\right|\right|^2 - \left|\left|w_t - w^*\right|\right|^2
\end{equation}

\noindent Now we substitute $z_t = w_t - w^*$ and get

\begin{equation}
	D(w_{t+1},w^*) - D(w_t,w^*) = z_t^2 - 2\alpha \nabla l_t^T z_t + \alpha^2 \nabla l_t^2 - z_t^2
\end{equation}

\noindent canceling and substituting in for $z_t$ gives us:


\begin{equation}
	D(w_{t+1},w^*) - D(w_t,w^*) = - 2\alpha \nabla l_t^T(w_t - w^*) + \alpha^2 \nabla l_t^2
\end{equation}

\noindent Now we can sum over time to get the distance up until time $t$.

\begin{equation}
	\sum_t \left(D(w_{t+1},w^*) - D(w_t,w^*)\right) = - 2\alpha \sum_t \left(\nabla l_t^T(w_t - w^*)\right) + \sum_t \left(\alpha^2 \nabla l_t^2\right)
\end{equation}

\begin{equation}
	\sum_t \left(D(w_{t+1},w^*) - D(w_t,w^*)\right) = D(w_{n},w^*) - D(w_0,w^*) 
\end{equation}

\noindent The regret is bounded by:

\begin{equation}
	R(w^*) \leq \sum_t \left(\nabla l_t^T(w_t - w^*)\right) = -\frac{1}{2\alpha} \left[D(w_{n},w^*) - D(w_0,w^*) - \sum_t \alpha^2 \nabla l_t^2\right]
\end{equation}

\noindent Next, we need to assume that the loss at any time step is bounded by a constant $G$:

\begin{equation}
	 \nabla l_t^2 \leq G
\end{equation}

\noindent The bound on the reward then becomes:

\begin{equation}
	R(w^*) \leq \frac{\alpha}{2} G^2 n + \frac{1}{2\alpha}D(w_{0},w^*) - \frac{1}{2\alpha}D(w_n,w^*) 
\end{equation}

\noindent Since we want an upper bound and the last term can only decrease this bound, we can eliminate it.

\begin{equation}
	R(w^*) \leq \frac{\alpha}{2} G^2 n + \frac{1}{2\alpha}D(w_{0},w^*)
\end{equation}

\noindent Since we are dealing with a bounded convex set of $w$ vectors, let us bound $D(w_{0},w^*)$ by the size of the set $F^2$. Now we need to pick $\alpha$ to minimize regret. To do this, we take the derivative of the above equation w.r.t $\alpha$ and set it equal to 0. Some algebra gives us

\begin{equation}
	\alpha = \frac{F}{G\sqrt{n}}
\end{equation}

\noindent and the regret is thus bounded by\footnote{Drew says the right bound is actually $R(w^*) \leq 2FG\sqrt{n}$}

\begin{equation}
	R(w^*) \leq FG\sqrt{n}
\end{equation}

\noindent note that the regret grows sub-linearly with time which means that as $t \rightarrow \infty$:

\begin{equation}
	\frac{R_t}{t} \longrightarrow 0
\end{equation}

\noindent Thus this algorithm is a ``no-regret'' algorithm.

\subsection*{Portfolio Optimization}

\noindent We want to do nearly as well as the best Constantly Rebalancing Portfolio. The weight vector (percent invested in a given stock) is subject to the constraints:

\begin{align}
w_i \geq 0\\
\sum_i w_i = 1
\end{align}

\noindent The loss is:

\begin{equation}
	l_t(w^*) = \log{\sum_i w_i^* r_i^t}
\end{equation}

\noindent We make the no-junk-bond assumption which is:

\begin{equation}
	r_{\mathrm{max}} \geq r_i \geq r_{\mathrm{min}} > 0
\end{equation}

\noindent the regret is bounded by:

\begin{equation}
R(\mathrm{Investment}) \leq \sqrt{2}\sqrt{n}\left(\frac{r_{\mathrm{max}}}{r_{\mathrm{min}}}\right)^2\sqrt{T}
\end{equation}
\end{document}


