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Rumour: Retro Studios NX Game Is New IP Releases 2017
by Sickr April 29, 2016
40 Comments on Rumour: Retro Studios NX Game Is New IP Releases 2017
Emily Rogers is back and this time she apparently has a scoop regarding Retro Studios next title. According to Emily the game isn't Metroid or Donkey Kong Country related and sounds more like a new IP. We found out earlier this week that the NX platform launches on March 2017 and will be unveiled to the public later this year and not at E3 as previously expected.
Rumor part 2: Not Diddy Kong Racing 2 either. It sounds more like a new IP.
— Emily Rogers (@ArcadeGirl64) April 29, 2016
Tags: new ip nx retro studios
Published by Sickr
View all posts by Sickr
sonofmrpeanut says:
https://twitter.com/Doctor_Cupcakes/status/725847933174833152
Nintendo Military Strategist Sylux says:
With Sakamoto? That wouldn't be too bad. The guy has good game concept ideas, he has demonstrated he can make quality new IPs.
Greg Warner says:
This makes me more curious to see what they're up to. I hope it's better than their previous work since Metroid Prime and DKC Returns didn't impress me at all.
paidenthusiast says:
Those games impressed me. I don't know what type of games you're in to but Metroid Prime and DKC Returns are considered as two of the greatest games of the 7th generation but hey to each their own
I don't like Donkey Kong because of the BS difficulty, and I don't like Metroid Prime because it's a first-person game and those make me feel clausterphobic.
tre802001 says:
Donkey Kong I like but Metroid Prime not my type of game due to being first person.
|||Nintendo Dark Commander Quadraxis-NX Prime||| says:
-||Primary properties of both franchises, they simply aren't for you then||-
Those are not valid reasons to considera them bad. It's just your personal tastes.
Brandon Hardacre says:
You can literally turn Samus's helmet off to make it not claustrophobic. And the difficulty is why you say they are bad? Casual…
rpghacker says:
When did "casual" become an insult?
When people started giving intentionally challenging games flak just for being challenging, I'm guessing.
Returns was really not that great, it was really easy, had boring level design, and just wasn't as fun as the other DK Country games. Tropical Freeze is amazing though, its like everything Returns was trying to be, but done right, and then 10 times better than that. And its gorgeous!
DKCTF > DKC2 > DKCR > DKC > DKC3
Metroid Prime didn't impres-METROID PRIME DIDN'T IMPRESS-!? *Must….resist….urge….to…..rant….*
ninjadude99 says:
Hey. Some people have different opinions. Not everyone's going to like the game.
it's mainly about the "hope it's better than their last games" part. :P
Phazoonn says:
I pretty salty it's not metroid but a new ip is always welcome from Nintendo.
I'm dammit
👏Obey Damoniez🤘 says:
Lol don't hurt yourself
*cuts wrist*
takamaru64 says:
Ow the Edge.
http://vignette2.wikia.nocookie.net/joke-battles/images/2/21/Ow_the_edge.jpg/revision/latest?cb=20160416215447
durpdurpdurp says:
Fans: "We want a new Metroid game!"
Nintendo: "Here's……Federation Force!"
Fans: "….we want a (good) game starring Samus. Give us a new Metroid Prime!"
Nintendo: "Here's a game from Retro Studios….that's not Metroid!"
Fans: "………………………………….."
megamandied says:
Federation Force is a slap in the face to Metroid fans.
But I'll never understand people who think Retro should be the dedicated Metroid dev team. I love Metroid Prime, but I equally love Tropical Freeze (not a huge fan of DKCR because of the controls). With such great dev talent, a new IP sounds like an awesome idea.
But of course, people whine for new IPs, then when they hear word of it, they suddenly want old IPs. Classic scenario. People don't know what they want.
I'm not really directing this at you. I get the point you are making. Just ranting lol. I want a new Metroid as well. I was willing to give FedForce a chance, but it looks so frickin boring.
I don't think the problem here really is the "new IP" part, the problem is the fact that there hasn't been a great Metroid game for a long time now and people simply long a new Metroid game. The Metroid games by Retro Studios are considered to be some of the best (if not the best), so it's natural for people to assume that Retro, if anyone, would work on a new Metroid game. By extension, many people, including me, think, that if Retro isn't working on a new Metroid game right now, then most likely noone at Nintendo is – or, if anyone is, they have most likely just started. In fact, this was pretty much confirmed by the Metroid Prime producer when Federation Force was announced, it was made rather clear that no Metorid game besides Federation Force was being developed at that time and that he would love to go back to Metroid "some day". So the problem here really isn't the fact that Retro are probably doing a new IP, the problem is that we most likely won't see another Metroid game until a couple of years into the future, if at all.
It's amazing that I have to correct this statement every time someone mentions Tanabe.
He didn't made clear that no Metroid game besides FF is in development. He made clear that no Metroid PRIME is in development. Tanabe doesn't oversee the whole series, Sakamoto does. I'm not saying a non-Prime game is in development because I personally think that's not the case, but don't misinform people.
From the horse's mouth:
"I'm the producer of the Metroid Prime series, and [Metroid series co-creator] Yoshio Sakamoto is the one in charge of Metroid itself. I'm not sure when he's planning on making such an announcement so I can't speak for him"
Right in that same interview, literally three questions after "even if we started another Prime game now, it would end on the NX".
That's irrelevant, though. Chances are, if another Metroid game besides a Prime game WAS in development, Tanabe would at least know about it, but he didn't sound like he knew about a Metroid in development, and that's my whole point.
And now you're arguing with Tanabe.
I'll post it again:
"I'm not sure when he's planning on making such an announcement so I can't speak for him"
Pokemonsilph says:
This is literally only picking out one thing from your comment, but the 3DS version of DKC Returns is a lot better in my opinion (button controls will always be preferred for platforming, imo, even Mario Galaxy manages to mess me up occasionally because of the motion controlls).
I agree. I've played the 3DS version and liked it a lot better.
If this is true I hope Retro's new IP is in the realm of a Jet Force Gemini or a Turok like title. New IPs are always welcomed but damn I was hoping for a Metroid title. If it's confirmed to be true, this news would've easily made Nintendo the main attraction at E3 but of course we already know that story
-||Anything as long as it's not a civilian attractor||-
N-Dub Nation says:
I don't care if Retro Studios uses the Samus Engine. I'm looking forward to its new IP on the NX before the end of this year.
The Man of Brisk says:
Ah, something to get excited about. Hasn't been many articles to do that on this site as late, sadly (I mainly refer to Nintendo's plans for E3, or lack thereof, is more like it). As Retro has a proven track record, this should be good.
Well, at least they get a break from Metroid and DK. They deserve that. Now I'm curious about what it could be.
Pingback: [Rumor] El juego de Retro Studios para NX es una nueva IP - TierraGamer
bargodblog says:
It better be good
Pingback: Il prossimo gioco dei Retro Studios arriverà nel 2017 non sarà un Donkey Kong o un nuovo Metroid? | VG247.it
Pingback: Rumor: Retro Studios trabaja en una nueva franquicia para el NX
sonicgalaxy27 says:
i do believe that they are working on another mario kart game on the NX and also, Emily need to stop lying and give us real proof.
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The Pantelleria Project began its first excavation season this year in 2015 with the first week having seen the opening of three trenches. The project began in what we like to call the 'zucchini field' with: the clearing of overgrown weeds, trimming a few Eucalyptus tree branches, and setting-up the first trenches. Progress continues to be fast-paced; the nine of us have moved more than 500 buckets of soil. The weather so far is never concrete, as the island is cloudy and sunny multiple times everyday. The first week of excavations have taught me much about island weather, how a project begins, and the surprising amount of soil on the island!
Professional Moment of the Week: Learning the basics of how one begins a new archaeological excavation project in Italy.
Personal Moment of the Week: Swimming in the volcanic lake, known as Lago di Venere, while the soil tries to swallow you with every step, and then finally learning to float across the 50-100cm deep water.
The Pantelleria Project at the Lago di Venere has proven to be both inspirational in the field of archaeology and remarkable in broadening my cultural lens. Trying to put myself in the shoes of the people of antiquity really puts things into perspective, in particular the spectacular "Mirror of Venus" effect when the water of the lake is completely still. Clearing an area in preparation for a dig can be quite taxing, especially when there are branches thicker than your arm, but I hacked my way through one, literally. Excavations at a new site can be challenging as well, but perseverance and dedication paid off with interesting finds both in situ and in terms of artifacts. This first week has proven to be a test of cultural learning in areas outside of the dig site and getting used to "island time"; it has also opened my eyes to the different methods used in excavations.
The first week of excavation has gone really well, and the view from our trench is jaw-dropping. We already have four trenches open, and I have learned so much, not only about the site itself but also about the general antiquity of Pantelleria and its role in the Mediterranean world. In one week I have gained a greater understanding of Greek, Roman, Italian and African history; one example being Pantelleria's role in the obsidian trade. This being my first excavation I have learned a lot of practical skills of archaeology as well, like site drawing and photography. The most exciting find so far has been a terracotta knuckle bone (astragalos) found in the top soil of my trench, whereas my favourite moment off site would be watching the finches from our front porch migrate from Tunisia to Sicily.
As soon as I saw how our propeller plane landed I realized how small this volcanic island really is. It can feel a little isolating at times, which is good for concentrating on work without many distractions, but it also presents a problem of getting the word out about this project. This week has been primarily focused on the initial stages of the excavation and getting familiar with Pantelleria town. The town itself has a unique aesthetic, contrasted by concrete block buildings built after the Second World War and its Norman castello in the town centre. Having not experienced a newly begun excavation, it was interesting for me to see how much GPR (ground penetrating radar) readings can determine the placement of trenches, and to learn the hypotenuse method of trench marking. Finally, the much smaller team than I had experienced previously has allowed for me to understand more about the goings-on of archaeology through a more personalized work environment.
The first week of excavation for BUAPP 2015 has brought with it a host of new experiences. Along with the traditional activities of moving topsoil, watching with hawk-like eyes for pottery and other material culture, and using a trowel to observe minute changes in soil, I've had the chance to engage in some aspects of archaeological fieldwork that are new to me. One such aspect is the drawing of a plan, which reproduces a bird's-eye view of a particular feature or stratigraphic unit (SU). Though juggling so many different activities is sometimes exhausting, one only has to turn and gaze upon the stunningly beautiful volcanic crater lake that lies close to our site to feel refreshed and reinvigorated. The lake, commonly called the Lago di Venere (Lake of Venus), also goes by another name, the Mirror of Venus, because of its amazingly clear, calm, reflective stillness.
It was a very exciting week at Lago di Venere as we just started excavations! In such a short period of time, the team has got a great deal of work done and everyone should be very proud. Two trenches in and we have already found some interesting artefacts and features. Starting a new excavation is much different than joining one that has been open for some time. There is a lot that has to get done before getting to work in the field and learning about the process of it has been very eye opening. Waking up at 6am every morning is absolutely worth it because of the beautiful view from the site. The most exciting part about working in the field is learning while doing, rather than just sitting in a classroom. The things we learn in class prepare us for what we have in store during excavations, like being able to identify certain objects for example; but the field gives you a chance to apply everything you have learned.


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\title{Provable Computational and Statistical Guarantees for Efficient Learning of Continuous-Action Graphical Games}
\date{}

\author{%
	Adarsh Barik \\
	Department of Computer Science\\
	Purdue University\\
	West Lafayette, Indiana, USA\\
	\texttt{abarik@purdue.edu} \\

	\and
	Jean Honorio \\
	Department of Computer Science \\
	Purdue University \\
	West Lafayette, Indiana, USA\\
	\texttt{jhonorio@purdue.edu} \\

}

\begin{document}

\maketitle

\begin{abstract}
  In this paper, we study the problem of learning the set of pure strategy Nash equilibria and the exact structure of a continuous-action graphical game with quadratic payoffs by observing a small set of perturbed equilibria. A continuous-action graphical game can possibly have an uncountable set of Nash euqilibria. We propose a $\ell_{12}-$ block regularized method which recovers a graphical game, whose Nash equilibria are the $\epsilon$-Nash equilibria of the game from which the data was generated (true game). Under a slightly stringent condition on the parameters of the true game, our method recovers the exact structure of the graphical game. Our method has a logarithmic sample complexity with respect to the number of players. It also runs in polynomial time.
\end{abstract}

\section{INTRODUCTION}
\label{sec:introduction}

The real world is filled with scenarios which arise due to competitive actions by selfish individual players who are trying to maximize their own utilities or payoffs. Non-cooperative game theory has been considered as the appropriate mathematical framework to formally study \emph{strategic} behavior in such multi-agent scenarios. In such scenarios, each agent decides its action based on the actions of other players. The core solution concept of \emph{Nash equilibrium (NE)}~\cite{Nash51} serves a descriptive role of the stable outcome of the overall behavior of self-interested agents (e.g., people, companies, governments, groups or autonomous systems) interacting strategically with each other in distributed settings.

\paragraph{Graphical Games.}

The introduction of compact representations to game theory over the last two decades have extended algorithmic game theory's potential for large-scale, practical applications often encountered in the real world. Introduced within the AI community about two decades ago, \emph{graphical games}~\cite{Kearns01} constitute an example of one of the first and arguably one of the most influential graphical models for game theory. Indeed, graphical games played a prominent role in establishing the computational complexity of computing NE in normal-form games as well as in succinctly representable multiplayer games (see, e.g.,~\cite{Daskalakis06,Daskalakis09,Daskalakis09b} and the references therein).

Players can take actions in either a discrete space (for example in voting) or in a continuous space (for example in simultaneous auctions in online advertising). Correspondingly, graphical games can be studied in both domains. In this paper, we focus on continuous-action graphical games.

\paragraph{Inference in Graphical Games.}

There has been considerable progress on \emph{computing} classical equilibrium solution concepts such as NE and \emph{correlated equilibria}~\cite{Aumann74} in graphical games (see, e.g.,~\cite{Blum06,Kearns01,Kakade03,Ortiz02,Papadimitriou08,Vickrey02} and the references therein) as well as on computing the \emph{price of anarchy} in graphical games (see, e.g.,~\cite{Benzwi11}). \cite{Irfan14} identified the most influential players, i.e., a small set of players whose collective behavior forces every other player to a unique choice of action.  All the work above focus on inference problems for graphical games, and fall in the field of algorithmic game theory.

\paragraph{Learning Graphical Games.}

The aforementioned problems of computing Nash equilibrium, correlated equilibrium or price of anarchy often assume that the structure and payoffs of the games under consideration are already available. Relatively less attention has been paid to the problem of \emph{learning} both the structure and payoffs of graphical games from data. Addressing this problem is essential to the development, potential use and success of game-theoretic models in practical applications. In this paper, we study the problem of learning the complete characterization of pure strategy Nash equilibrium and structure of the graph in a continuous-action graphical game.

\paragraph{Related Work.}

There has been considerable amount of work done for learning games in the discrete-action setting. \cite{Honorio15} proposed a maximum-likelihood approach to learn "linear influence games" - a
class of parametric graphical games with binary actions and linear payoffs.  However, their method runs in exponential time and the authors assumed a specific observation model for the strategy profiles. For the same specific observation model, \cite{Ghoshal16} proposed a polynomial time algorithm, based on $\ell_1$-regularized logistic regression, for learning linear influence games. Their strategy profiles (or joint actions) were drawn from a mixture of uniform distributions: one over the pure-strategy Nash equilibria (PSNE) set, and the other over its complement. \cite{Ghoshal17} obtained necessary and sufficient conditions for learning linear influence games under arbitrary observation model. \cite{Garg16} use a discriminative, max-margin based approach, to learn tree structured polymatrix games\footnote{Polymatrix games are graphical games where each player's utility is a sum of unary (single player) and pairwise (two players) potential functions.}. Their method runs in exponential time and the authors show that learning polymatrix games is NP-hard under this max-margin setting, even when the class of graphs is restricted to trees. Finally, \cite{Ghoshal18} proposed a polynomial time algorithm for learning sparse polymatrix games in the discrete-action setting.

Regarding inference for continuous-action games, \cite{facchinei2007generalized} and \cite{scutari2010convex} provide a survey of variational inequality methods and Gauss-Seidel methods to compute generalized Nash equilibrium for pure strategy games. \cite{perkins2013asynchronous} and \cite{perkins2015mixed} studied a mixed-strategy actor-critic algorithm which converges to a probability distribution that assigns most weight to equilibrium states. \cite{mertikopoulos2019learning}  provided sufficient conditions under which no-regret learning converges to equilibrium.

Continuous-action games with quadratic payoffs have been used extensively in the game theory literature \cite{ballester2006s,bramoulle2014strategic,galeotti2017targeting,acemoglu2015networks}.   \cite{leng2018learning} proposed algorithms to learn games with quadratic payoffs, in a simplified setting. However, the authors do not provide any theoretical guarantees. In this work, we focus on provable guarantees for a far more general class of games with quadratic payoffs in the high-dimensional regime.

\paragraph{Our Contribution.}

We aim to propose a novel method to learn graphical games with quadratic payoffs, with the following provable guarantees in mind:
\begin{enumerate*}
	\item {\bf Correctness -} We want to develop a method which correctly recovers the set of Nash equilibria and the structure of the graphical games.
	\item {\bf Computational efficiency -} Our method must run fast enough to handle the high dimensional cases. Ideally, we want to have polynomial time complexity with respect to the number of players.
	\item {\bf Sample complexity -} We would like to use as few samples as possible for recovering the set of Nash equilibria. We want to achieve logarithmic sample complexity with respect to the number of players.
\end{enumerate*}

To this end, we propose a block-norm regularized method to learn graphical games with quadratic payoff functions. For $n$ players, at most $d$ in-neighbors per player, and $k$-dimensional action vectors, we show that $\mathcal{O}( k^5 d^3 \log (d n k) )$  \emph{samples} are sufficient to recover the complete characterization of the set of $\epsilon$-Nash equilibria. Under slightly more stringent conditions, we also recover the true structure of the game. Our method also runs in polynomial time complexity.

Regarding the main challenges that we address, first, the set of Nash equilibria for continuous-action games is uncountable, while for discrete-action games is countable.  Our method provides the complete characterization of such uncountable sets. Our method is also oblivious to the exact process under which players converge to Nash equilibria. In fact, Nash equilibria can be "chosen" by nature in an arbitrary non-probabilistic fashion. We also do not assume any particular process that ``chooses'' Nash equilibria, such as, for instance, a stochastic process. Our method only needs access to some small number perturbed  equilibria.

\section{PRELIMINARIES}
\label{sec:notation and problem setting}

In this section, we introduce our notation and formally define the problem of learning graphical games with quadratic utility functions. Consider a directed graph $\mathcal{G}(\mathcal{V},\mathcal{E})$, where $\mathcal{V}$ and $\mathcal{E}$ are set of vertices and edges respectively. We define $\mathcal{V} \triangleq \seq{n}$, where each vertex corresponds to one player. We denote the in-neighbors of a player $i$ by $\textup{S}_i$, i.e., $\textup{S}_i = \{ j \mid (j, i) \in \mathcal{E} \}$ (i.e., the set of nodes that point to node $i$ in the graph). All the other players are denoted by $\textup{S}_i^c$, i.e., $\textup{S}_i^c = \seq{n} \backslash (\textup{S}_i \cup i) $. Let $|\textup{S}_i| \leq d$ and $|\textup{S}_i^c| \leq n$.

For each player $i \in \mathcal{V}$, there is a set of actions or \emph{pure-strategies}  $\mathcal{A}_i$. That is, player $i$ can take action $x_i \in \mathcal{A}_i$. Each action $x_i$ consists of making $k$ decisions on a limited budget $\textup{b} \in \mathbb{R}$. We consider games with continuous actions. Mathematically, $x_i \in \mathbb{R}^k$ and $\| x_i \|_2 \leq \textup{b}$. For each player $i$, there is also a local payoff function $u_i : \mathcal{A}_i \times ( \bigtimes_{j \in \textup{S}_i} \mathcal{A}_j ) \to \mathbb{R}$ mapping the joint action of player $i$ and its in-neighbors $\textup{S}_i$, to a real number. Later, we will define a particular kind of local payoff function which is of our interest. A joint action $\mb{x}^* \in \bigtimes_{i \in \mathcal{V}} \mathcal{A}_i$ is a \emph{pure-strategy Nash equilibrium (PSNE)} of a graphical game iff, no player $i$ has any incentive to unilaterally deviate from the prescribed action $x_i^* \in \mathcal{A}_i$, given the joint action of
its in-neighbors $x_{\textup{S}_i}^* \in \bigtimes_{j \in \textup{S}_i} \mathcal{A}_j$ in the equilibrium. We denote a game by $\mathscr{G}$, and the set of all PSNE and $\epsilon$-PSNE of $\mathscr{G}$, by  $\textup{NE}(\mathscr{G})$ and $\textup{NE}_{\epsilon}(\mathscr{G})$ respectively, for a constant $\epsilon > 0$. Formally,
\begin{align*}
	\textup{NE}(\mathscr{G}) &\triangleq \{ \mb{x}^* \in \bigtimes_{i \in \mathcal{V}} \mathcal{A}_i \mid x_i^* \in \arg \max_{x_i \in \mathcal{A}_i} u_i(x_i, x_{\textup{S}_i}^*), \forall i \in \mathcal{V}  \} \\
	\textup{NE}_{\epsilon}(\mathscr{G}) &\triangleq \{ \mb{x}^* \in \bigtimes_{i \in \mathcal{V}} \mathcal{A}_i \mid u_i(x_i^*, x_{\textup{S}_i}^*) \geq -\epsilon + \max_{x_i \in \mathcal{A}_i} u_i(x_i, x_{\textup{S}_i}^*) , \forall i \in \mathcal{V}  \}
\end{align*}

\paragraph{Parametric Payoffs.}
We are interested in solving a parametrized version of the problem. In that, given the weights $W_{ij}^* \in \mathbb{R}^{k\times k}, \forall i, j \in \mathcal{V}$, for each player $i$, we define the set of in-neighbors of player $i$ as $\textup{S}_i = \{ j \mid W_{ij}^* \ne 0 \}$ and the payoff function
\begin{align*}
u_i(x_i, x_{\textup{S}_i}) = - \| x_i - \sum_{j \in \textup{S}_i} W_{ij}^* x_j \|_2
\end{align*}
Consider $\max_{x_i} u_i(x_i, x_{\textup{S}_i}^*) = 0, \forall i \in \seq{n} $, then in a PSNE, each player $i$ matches their action $x_i$ to the weighted actions of their neighbors, i.e., $\sum_{j \in \textup{S}_i} W_{ij}^* x_j^*$. Let $\epsilon > 0$ be a constant. The set of all $\epsilon$-PSNE of $\mathscr{G}$ is
\begin{align*}
\textup{NE}_{\epsilon}(\mathscr{G}) =& \{ \mb{x}^* \in \bigtimes_{i=1}^n \mathcal{A}_i \mid \| x_i^* - \sum_{j \in \textup{S}_i} W_{ij}^* x_j^*  \|_2 \leq \epsilon, \forall i \in \mathcal{V} \}.
\end{align*}

\paragraph{Sampling.} Given the above characterization, the set of $\epsilon$-PSNE is a convex polytope. We observe samples from the set of noisy PSNE which follow a local noise mechanism that adds noise independently per player. Observed joint actions
\begin{align*}
\mb{x} = \mb{x}^* + \mb{e}
\end{align*}
where $\mb{x}^*$ is a Nash equilibrium, that is $\mb{x}* \in \textup{NE}(\mathscr{G})$ and $\mb{e}$ is independent zero mean sub-Gaussian noise with variance proxy $\sigma^2$. The class of sub-Gaussian variates includes for instance Gaussian variables, any bounded random variable (e.g. Bernoulli, multinomial, uniform), any random variable with strictly log-concave density, and any finite mixture of sub-Gaussian variables.

\paragraph{Norms and Notations.}
For a matrix $\mb{A} \in \mathbb{R}^{p \times q}$ and two sets $S \subseteq \seq{p}$ and $T \subseteq \seq{q}$,  $\mb{A}_{S T}$ denotes $\mb{A}$ restricted to rows in $S$ and columns in $T$.  Similarly, $\mb{A}_{S.}$ and $\mb{A}_{.T}$ are row and column restricted matrices respectively. For a vector $\textbf{m} \in \mathbb{R}^q$, the $\ell_{\infty}$-norm is defined as $ \| \textbf{m} \|_{\infty} = \max_{i \in \seq{p}} | \textbf{m}_i | $. The Frobenius norm for a matrix $\mb{A} \in \mathbb{R}^{p \times q}$  is defined as
\begin{align*}
 \| \mb{A} \|_F =  \sqrt{\sum_{i=1}^p \sum_{j=1}^q| \mb{A}_{ij} |^2}.
 \end{align*}
 The $\ell_{\infty}$-operator norm for $\mb{A}$ is defined as
\begin{align*}
  \| \mb{A} \|_{\infty, \infty} = \max_{i \in \seq{p}}\sum_{j=1}^q | \mb{A}_{ij} | .
\end{align*}
The spectral norm of $\mb{A}$ is defined as
\begin{align*}
 \| \mb{A} \|_{2,2} = \sup_{ \|\mb{x}\|_2 = 1} \|\mb{A} \mb{x}\|_2.
\end{align*}
We also define a block matrix norm for row-partitioned block matrices. Let $\mb{A} \in \mathbb{R}^{\sum_{i=1}^k p_i \times q}, \forall i \in \seq{k}$ be a row-partitioned block matrix defined as follows: $ \mb{A} = \begin{bmatrix} \mb{A}_1 & \cdots & \mb{A}_k \end{bmatrix}^\intercal$ where each $\mb{A}_i \in \mathbb{R}^{p_i \times q}$.
Then
\begin{align*}
\| \mb{A} \|_{\textup{B},\infty,\texttt{F}} &= \max_{i \in \seq{k}} \| \mb{A}_i \|_F\\
 \| \mb{A} \|_{\textup{B},\infty,1} &= \max_{i \in \seq{k}} \sum_{l=1,m=1}^{l=p_i,m=q} | [\mb{A}_i]_{lm} |.
\end{align*}

\section{MAIN RESULT}
\label{sec:main result}

In this section, we describe our main theoretical results. But before we do that, we discuss some technical assumptions which are needed for our proofs.

\begin{assumption}[Budgeted actions]
	\label{assum:budgeted_actions}
	For all $x_i \in \mathcal{A}_i, \| x_i \|_2 \leq \textup{b}, \forall i \in \seq{n}$ for some $\textup{b} > 0$.
\end{assumption}
\begin{assumption}[Maximum zero utility]
	\label{assum:max_zero_utility}
	At PSNE, $u_i(x_i^*, x_{\text{\normalfont -} i}^*) = 0, \forall i \in \seq{n}$.
\end{assumption}
\begin{assumption}[Mutual Incoherence]
	\label{assum:mutual_incoherence}
	Consider $\mb{H} = \frac{1}{T} \sum_{t=1}^T \big( {\mb{x}_{\text{\normalfont -} i}^*}^t {{\mb{x}_{\text{\normalfont -} i}^*}^t}^\intercal +  \sigma^2 \mb{I}\big)$ where $\mb{I}$ is the identity matrix, then $\res{\mb{H}}{\textup{S}_i^c\textup{S}_i} \res{\mb{H}}{\textup{S}_i\supp}^{\text{\normalfont -} 1} \leq 1 - \alpha $ for some $\alpha \in (0, 1]$.
\end{assumption}

Assumption \ref{assum:budgeted_actions} simply states that each player has a limited budget to allocate for its actions. For instance, consider simultaneous auctions in an online advertising, where a company chooses how to allocate its budget into several options. For a sufficiently large budget $\textup{b}$, Assumption \ref{assum:max_zero_utility} is not difficult to fulfill for quadratic payoffs. We propose a mutual incoherence assumption (Assumption \ref{assum:mutual_incoherence}) for games. While mutual incoherence is new to graphical games, it has been a standard assumption in various estimation problems such as compressed sensing~\cite{wainwright2009sharp}, Markov random fields~\cite{ravikumar2010high}, non-parametric regression~\cite{ravikumar2007spam}, diffusion networks~\cite{daneshmand2014estimating}, among others.

Now that all our assumptions are in place, we are ready to setup our estimation problem. Consider that we have access to $T$ perturbed equilibria, i.e., we have access to $x_i^t = {x_i^*}^t + e_i^t$ where superscript $t$ denotes the $t$-th sample and $e_i^t \in \mathbb{R}^k$ is a vector of zero-mean mutually independent sub-Gaussian noises with variance proxy $\sigma^2$. We estimate the parameters $W_{ij}$ for each $i, j \in \seq{n}$ by solving the following optimization problem:
\begin{align}
\label{eq:opt_prob}
	\hat{W_{i\cdot}} =& \arg\min_{W_{i\cdot}} \frac{1}{T} \sum_{t=1}^{T} \| x_i^{t} - \sum_{\substack{j=1\\ j \ne i}}^n W_{ij} x_j^{t} \|_2^2 + \lambda \sum_{\substack{j=1\\ j \ne i}}^n \| W_{ij} \|_F
\end{align}
where $W_{i\cdot}$ denotes the collection of all $W_{ij}, \forall j \in \seq{n}, j \ne i$. Our next theorem states that the recovered $\hat{W}_{i.}$ completely characterizes the set of all $\epsilon$-Nash equilibria.

\begin{theorem}
	\label{thm:eps_nash_equilibria}
	Consider a continuous-action graphical game $\mathscr{G}$ such that Assumptions \ref{assum:budgeted_actions}, \ref{assum:max_zero_utility} and \ref{assum:mutual_incoherence} are satisfied for each player. Let $ \lambda > \max( 24\sqrt{2} \frac{1-\alpha}{\alpha} \sigma \textup{b} W_{\max} \sqrt{\frac{k |\textup{S}_i| \log( 2k^2 |\textup{S}_i|)}{T}},  192 \frac{1-\alpha}{\alpha} \sigma^2  W_{\max} \\ \sqrt{\frac{k \log(k^2 |\textup{S}_i|)}{T}}, 192 \frac{1-\alpha}{\alpha} \sigma^2 \sqrt{k} W_{\max}  \sqrt{\frac{W_{\max} |\textup{S}_i| \log(|\textup{S}_i|k)}{TW_{\min}}},  192 \frac{1-\alpha}{\alpha} k^{\frac{1}{4}} \sigma \sqrt{\frac{\log (2k^2|\textup{S}_i|)}{T}},  \frac{24\sqrt{2}}{\alpha} k \sigma \textup{b} W_{\max}  \\ \sqrt{\frac{|\textup{S}_i^c \log(2k^2|\textup{S}_i^c|)|}{T}},  \frac{192}{\alpha} \sigma^2 k W_{\max}  \sqrt{\frac{\log(k^2 |\textup{S}_i^c|)}{T}}, \frac{192}{\alpha} \sigma^2 k W_{\max}  \sqrt{\frac{W_{\max} |\textup{S}_i^c|^{\frac{1}{2}} \log(|\textup{S}_i^c|k) }{T}},   \frac{24\sqrt{2}}{\alpha} k \sigma \textup{b} \sqrt{\frac{\log (2k^2 |\textup{S}_i^c|)}{T}} , \\ \frac{192}{\alpha} \sigma \sqrt{\frac{k \log(2k^2|\textup{S}_i^c|)}{T}}, \frac{24 (1-\alpha) \sigma^2 \sqrt{k} \max_{ij} |W_{ij}| }{\alpha}, \frac{24 \sigma^2 k \max_{ij} |W_{ij}| }{\alpha} ) $, then the following claims hold.
	\begin{enumerate}
		\item We can recover $\textup{NE}_\epsilon(\mathscr{G})$ by estimating $\mb{W}$ from the optimization problem \eqref{eq:opt_prob}.
		\item Furthermore, for each player $i \in \seq{n}$, if $\min_{j \in \textup{S}_i} \|W_{ij}^*\|_{F} > 2 \delta(k, |\textup{S}_i|, C_{\min}, \alpha, \lambda, \sigma, W_{\max})$, then we recover the exact structure of the graphical game $\mathscr{G}$.
	\end{enumerate}
	where $C_{\min}$ is the minimum eigenvalue of $\res{\mb{H}}{\textup{S}_i\supp}$, $W_{\max} \triangleq \max_{i,j} |\mb{W}_{ij}|$, $\epsilon = |\textup{S}_i| \delta(k, |\textup{S}_i|, C_{\min}, \alpha, \\ \lambda, \sigma, W_{\max}) \textup{b}$ and

	\begin{align*}
	&\delta(k, |\textup{S}_i|, C_{\min}, \alpha, \lambda, \sigma, W_{\max}) = k \sqrt{k|\textup{S}_i|} \frac{2}{C_{\min}}
	(\frac{\alpha \lambda}{24(1-\alpha)} + \sigma^2 (1 + \frac{\alpha \lambda}{24 (1-\alpha)\sigma^2 \sqrt{k} W_{\max}} +\\
	& \frac{\alpha \lambda}{24 (1 - \alpha) \sigma^2 \sqrt{k} \max_{ij} W_{\max}}) \sqrt{k} W_{\max}) + k \sqrt{k|\textup{S}_i|}\frac{2}{C_{\min}} (\frac{\alpha \lambda}{24 (1-\alpha)} + \frac{\alpha \lambda}{24 (1-\alpha)})  +   \frac{\lambda}{2}  k \sqrt{k|\textup{S}_i|}\frac{2}{C_{\min}} .
	\end{align*}
\end{theorem}
\begin{proof}
	We will make use of the \emph{primal-dual witness} method to prove Theorem \ref{thm:eps_nash_equilibria}. By using the definition of Frobenious norm, optimization problem \eqref{eq:opt_prob} can be equivalently written as
\begin{align}
	\label{eq:form0}
	\begin{split}
		\hat{W_{i\cdot}} =& \arg\min_{W_{i.}} \frac{1}{T} \sum_{t=1}^{T} \| x_i^{t} - \sum_{j=1, j \ne i}^n W_{ij} x_j^{t} \|_2^2 + \lambda \sum_{j=1, j \ne i}^n  \sup_{\| Z_{ij} \|_F \leq 1}\inner{Z_{ij}}{W_{ij}}  \\
		\end{split}
\end{align}
Consider the term $\sup_{\| Z_{ij} \|_F \leq 1}\inner{Z_{ij}}{W_{ij}}$. We can assign specific values to $Z_{ij}$ to get the maximum possible value of $\inner{Z_{ij}}{W_{ij}}$. In particular, we can take if $ W_{ij} \ne \mb{0}$ then $Z_{ij} = \frac{W_{ij}}{\| W_{ij} \|_F}$, and if $W_{ij} = \mb{0}$ then $\|Z_{ij}\|_F \leq 1$.
Note that in the first case $\| Z_{ij} \|_F = 1$ and thus it gives the maximum value for $\inner{Z_{ij}}{W_{ij}}$ and no further improvement is possible. In the second case, since $W_{ij} = \mb{0}$, $Z_{ij}$ can take any value such that  $\|Z_{ij}\|_F \leq 1$ without affecting $\inner{Z_{ij}}{W_{ij}}$. We fix $Z_{ij}$ to one such value and rewrite equation \eqref{eq:form0} as
\begin{align}
\label{eq:form1}
\begin{split}
		\hat{W_{i\cdot}} &= \arg\min_{W_{i.}} \frac{1}{T} \sum_{t=1}^{T} \| x_i^{t} - \sum_{j=1, j \ne i}^n W_{ij} x_j^{t} \|_2^2 + \lambda \sum_{j=1, j \ne i}^n  \inner{Z_{ij}}{W_{ij}}
	\end{split}
\end{align}
where the last equality comes by keeping in mind that $Z_{ij}$ are chosen as described above.  We can rewrite equation \eqref{eq:form1} as,
\begin{align*}
\begin{split}
	\hat{W_{i\cdot}} =& \arg\min_{W_{i\cdot}} \frac{1}{T} \sum_{t=1}^T \big( -2 \sum_{j=1, j \ne i}^n {x_i^t}^{\intercal} W_{ij} x_j^t + \sum_{\substack{j=1, j \ne i \\ k=1, k \ne i}}^n {x_j^t}^{\intercal}  W_{ij}^\intercal W_{ik} x_k^t \big) + \lambda \sum_{j=1, j\ne i}^n \inner{Z_{ij}}{W_{ij}}
\end{split}
\end{align*}
Using the stationarity Karush-Kuhn-Tucker condition at the optimum, for each $W_{ij}$ we can  write,
\begin{align}
\begin{split}
	\frac{1}{T} \sum_{t=1}^T (-2 x_i^t {x_j^t}^\intercal + 2 \sum_{k=1, k \ne i}^n W_{ik} x_k^t x_j^t) + \lambda Z_{ij} = 0
\end{split}
\end{align}
Further note that $x_i^t = {x_i^*}^t + e_i^t$ , ${x_i^*}^t = \sum_{j=1, j\ne i}^n W_{ij}^* {x_j^*}^t$ and ${x_j^*}^t = x_j^t - e_j^t $ where $e_i^t, e_j^t \in \mathbb{R}^k$ are zero mean sub-Gaussian vectors with variance proxy $\sigma^2$. Therefore, $x_i^t$ can be written as a function of $W_{ij}^*, x_j^t, e_i^t$ and $e_j^t$.  Thus, by substituting $x_i^t$ and writing the system of equations in vector form, we get
\begin{align}
\label{eq:stationarity}
\begin{split}
	&\frac{1}{T} \sum_{t=1}^T \big( 2 \mb{x}_{\text{\normalfont -} i}^t {\mb{x}_{\text{\normalfont -} i}^t}^\intercal (\mb{W}_{i\cdot} - \mb{W}_{i\cdot}^*) - 2 \mb{x}_{\text{\normalfont -} i}^t {\mb{e}_{\text{\normalfont -} i}^t}^\intercal \mb{W}_{i.}^* - 2 \mb{x}_{\text{\normalfont -} i}^t {\mb{e}_i^t}^\intercal \big) + \lambda \mb{Z}_{i\cdot} = \mb{0}
\end{split}
\end{align}
where
\begin{align}
	\mb{W}_{i\cdot}^* = \begin{bmatrix}
	{W_{i1}^*}^\intercal \\ \vdots \\ {W_{in}^*}^\intercal
	\end{bmatrix}, \mb{W}_{i\cdot} = \begin{bmatrix}
	W_{i1}^\intercal \\ \vdots \\ W_{in}^\intercal
	\end{bmatrix}, \mb{Z}_{i\cdot} = \begin{bmatrix}
	Z_{i1}^\intercal \\ \vdots \\ Z_{in}^\intercal
	\end{bmatrix}, \mb{x}_{\text{\normalfont -} i}^t = \begin{bmatrix}
	x_1^t \\ \vdots \\ x_n^t
	\end{bmatrix}, \quad \mb{e}_{\text{\normalfont -} i}^t = \begin{bmatrix}
	e_1^t \\ \vdots \\ e_n^t
	\end{bmatrix}
\end{align}
with $\mb{W}_{i\cdot}^*, \mb{W}_{i\cdot}, \mb{Z}_{i\cdot}  \in \mathbb{R}^{(n-1)k \times k}$ and $\mb{x}_{\text{\normalfont -} i}^t, \mb{e}_{\text{\normalfont -} i}^t \in \mathbb{R}^{(n-1)k \times 1}$.
If we denote in-neighbors of $i$ by a set $\textup{S}_i$ then ${W_{ij}^*}^\intercal = \mb{0}$ for all $j \notin \textup{S}_i$. We assume that  ${W_{ij}}^\intercal = \mb{0}$ for all $j \notin \textup{S}_i$. This choice will be justified later. Thus, the stationarity condition can be written as,
\begin{align}
\label{eq:stationarity1}
\begin{split}
\frac{1}{T} \sum_{t=1}^T \big( 2 \res{\mb{x}_{\text{\normalfont -} i}^t {\mb{x}_{\text{\normalfont -} i}^t}^\intercal}{\cdot \textup{S}_i} \res{\mb{W}_{i\cdot} - \mb{W}_{i\cdot}^*}{\textup{S}_i \cdot} - 2 \res{\mb{x}_{\text{\normalfont -} i}^t {\mb{e}_{\text{\normalfont -} i}^t}^\intercal}{\textup{S}_i\cdot} \res{\mb{W}_{i.}^*}{\textup{S}_i\cdot}
- 2 \mb{x}_{\text{\normalfont -} i}^t {\mb{e}_i^t}^\intercal \big) + \lambda \mb{Z}_{i\cdot} = \mb{0}
\end{split}
\end{align}
Equation \eqref{eq:stationarity1} can be decomposed in two separate equations. One for the players in $\textup{S}_i$ and other for players not in $\textup{S}_i$ which we denote by $\textup{S}_i^c$.
\begin{align}
\label{eq:support}
\begin{split}
\frac{1}{T} \sum_{t=1}^T \big( 2 \res{\mb{x}_{\text{\normalfont -} i}^t {\mb{x}_{\text{\normalfont -} i}^t}^\intercal}{\textup{S}_i \textup{S}_i} \res{\mb{W}_{i\cdot} - \mb{W}_{i\cdot}^*}{\textup{S}_i \cdot} - 2 \res{\mb{x}_{\text{\normalfont -} i}^t {\mb{e}_{\text{\normalfont -} i}^t}^\intercal}{\textup{S}_i \textup{S}_i} \res{\mb{W}_{i.}^*}{\textup{S}_i\cdot}
- 2 \res{\mb{x}_{\text{\normalfont -} i}^t}{\textup{S}_i\cdot} {\mb{e}_i^t}^\intercal \big) + \lambda \res{\mb{Z}_{i\cdot}}{\textup{S}_i\cdot} = \mb{0}
\end{split}
\end{align}
and
\begin{align}
\label{eq:nonsupport}
\begin{split}
\frac{1}{T} \sum_{t=1}^T \big( 2 \res{\mb{x}_{\text{\normalfont -} i}^t {\mb{x}_{\text{\normalfont -} i}^t}^\intercal}{\textup{S}_i^c \textup{S}_i} \res{\mb{W}_{i\cdot} - \mb{W}_{i\cdot}^*}{\textup{S}_i \cdot} - 2 \res{\mb{x}_{\text{\normalfont -} i}^t {\mb{e}_{\text{\normalfont -} i}^t}^\intercal}{\textup{S}_i^c \textup{S}_i} \res{\mb{W}_{i.}^*}{\textup{S}_i\cdot}
- 2 \res{\mb{x}_{\text{\normalfont -} i}^t}{\textup{S}_i^c\cdot} {\mb{e}_i^t}^\intercal \big) + \lambda \res{\mb{Z}_{i\cdot}}{\textup{S}_i^c\cdot} = \mb{0}
\end{split}
\end{align}

Let $\hE{.}$ denote the empirical expectation. Then equation \eqref{eq:support} can be written as,
\begin{align}
\label{eq:support0}
\begin{split}
 2 \res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{x}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i} \res{\mb{W}_{i\cdot} - \mb{W}_{i\cdot}^*}{\textup{S}_i \cdot} - 2 \res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{e}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i} \res{\mb{W}_{i.}^*}{\textup{S}_i\cdot} - 2 \hE{\res{\mb{x}_{\text{\normalfont -} i}}{\textup{S}_i\cdot} {\mb{e}_i}^\intercal}  + \lambda \res{\mb{Z}_{i\cdot}}{\textup{S}_i\cdot} = \mb{0}
\end{split}
\end{align}
Before we move ahead, we provide some properties of the finite-sample regime which hold with high probability. The detailed proofs of these lemmas are available in Appendix \ref{appendix:proof of theorems and lemmas}. We define $\hat{\mb{H}} \triangleq \hE{\mb{x}_{\text{\normalfont -} i} {\mb{x}_{\text{\normalfont -} i}}^\intercal}$, then

\begin{lemma}[Positive minimum eigenvalue]
	\label{lem:min eigenvalue}
	$\Lambda_{\min}(\res{\hat{\mb{H}}}{\textup{S}_i \textup{S}_i}) > 0$ with probability at least $1 - \exp(-cT\sigma^4 + \mathcal{O}(k |\textup{S}_i|)) - \exp(-\frac{\sigma^2 T}{128 \textup{b}^2} + \mathcal{O}(k |\textup{S}_i|)) $ for some constant $c > 0$ where $\Lambda_{\min}$ denotes the minimum eigenvalue.
\end{lemma}

Next, we show that the mutual incoherence condition also holds in the finite-sample regime with high probability.
\begin{lemma}[Mutual incoherence in sample]
	\label{lemma:mutual_incoherence}
	If $\|\res{\mb{H}}{\textup{S}_i^c\textup{S}_i} \res{\mb{H}}{\textup{S}_i \textup{S}_i}^{-1}\|_{\textup{B},\infty,1} \leq 1 - \alpha$ for $\alpha \in (0, 1]$ then $\|\res{\hat{\mb{H}}}{\textup{S}_i^c \textup{S}_i}\res{\hat{\mb{H}}}{\textup{S}_i \textup{S}_i}^{-1}\|_{\textup{B},\infty,1} \leq 1 - \frac{\alpha}{2}$ with probability at least $ 1 - \mathcal{O}( \exp(\frac{-K T}{ k^5  |\textup{S}_i|^3} + \log k |\textup{S}_i^c| + \log k |\textup{S}_i| )) $ for some $K > 0$.
\end{lemma}

Now we can use Lemma \ref{lem:min eigenvalue} and \ref{lemma:mutual_incoherence} to prove our main result. We can rewrite equation \eqref{eq:support0} as,
\begin{align}
\label{eq:support1}
\begin{split}
\res{\mb{W}_{i\cdot} - \mb{W}_{i\cdot}^*}{\textup{S}_i \cdot}  =& {\res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{x}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i}}^{\text{\normalfont -} 1}  \res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{e}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i} \res{\mb{W}_{i.}^*}{\textup{S}_i\cdot} +  {\res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{x}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i}}^{\text{\normalfont -} 1} \hE{\res{\mb{x}_{\text{\normalfont -} i}}{\textup{S}_i\cdot} {\mb{e}_i}^\intercal} \\
 &- \frac{\lambda}{2}  {\res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{x}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i}}^{\text{\normalfont -} 1} \res{\mb{Z}_{i\cdot}}{\textup{S}_i\cdot}
\end{split}
\end{align}
This is possible because $\lambda_{\min}(\res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{x}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i}) > 0$ from Lemma \ref{lem:min eigenvalue}. Using equation \eqref{eq:support1}, we can write equation \eqref{eq:nonsupport} as,
\begin{align}
\label{eq:nonsupport1}
\begin{split}
 & \res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{x}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i^c \textup{S}_i} \big(  {\res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{x}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i}}^{\text{\normalfont -} 1}  \res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{e}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i} \res{\mb{W}_{i.}^*}{\textup{S}_i\cdot}
 +   {\res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{x}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i}}^{\text{\normalfont -} 1} \hE{\res{\mb{x}_{\text{\normalfont -} i}}{\textup{S}_i\cdot} {\mb{e}_i}^\intercal} \\
& - \frac{\lambda}{2}  {\res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{x}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i}}^{\text{\normalfont -} 1} \res{\mb{Z}_{i\cdot}}{\textup{S}_i\cdot} \big) - \hE{ \res{\mb{x}_{\text{\normalfont -} i} {\mb{e}_{\text{\normalfont -} i}}^\intercal}{\textup{S}_i^c \textup{S}_i} \res{\mb{W}_{i.}^*}{\textup{S}_i\cdot}}
-  \hE{\res{\mb{x}_{\text{\normalfont -} i}}{\textup{S}_i^c\cdot} {\mb{e}_i}^\intercal} + \frac{\lambda}{2} \res{\mb{Z}_{i\cdot}}{\textup{S}_i^c\cdot} = \mb{0}
\end{split}
\end{align}

Let $\mb{M} = \res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{x}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i^c \textup{S}_i} \big(  {\res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{x}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i}}^{\text{\normalfont -} 1}$, then

\begin{align*}
	\frac{\lambda}{2} \res{\mb{Z}_{i\cdot}}{\textup{S}_i^c\cdot} =& - \mb{M} \res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{e}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i} \res{\mb{W}_{i.}^*}{\textup{S}_i\cdot} -  \mb{M}  \hE{\res{\mb{x}_{\text{\normalfont -} i}}{\textup{S}_i\cdot} {\mb{e}_i}^\intercal}
	+  \frac{\lambda}{2} \mb{M} \res{\mb{Z}_{i\cdot}}{\textup{S}_i\cdot}  + \hE{\res{\mb{x}_{\text{\normalfont -} i} {\mb{e}_{\text{\normalfont -} i}}^\intercal}{\textup{S}_i^c \textup{S}_i}\\ &\res{\mb{W}_{i.}^*}{\textup{S}_i\cdot} }+ \hE{\res{\mb{x}_{\text{\normalfont -} i}}{\textup{S}_i^c\cdot} {\mb{e}_i}^\intercal}
\end{align*}
By taking the $\textup{B},\infty,\texttt{F}$-norm on both sides and using the norm triangle inequality,
\begin{align*}
	\frac{\lambda}{2} \| \res{\mb{Z}_{i\cdot}}{\textup{S}_i^c\cdot} \|_{\textup{B},\infty,\texttt{F}} \leq& \| \mb{M} \res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{e}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i} \res{\mb{W}_{i.}^*}{\textup{S}_i\cdot}  - \mb{M}  \hE{\res{\mb{x}_{\text{\normalfont -} i}}{\textup{S}_i\cdot} {\mb{e}_i}^\intercal}
	+  \frac{\lambda}{2} \mb{M} \res{\mb{Z}_{i\cdot}}{\textup{S}_i\cdot} \|_{\textup{B},\infty,\texttt{F}}  \\
	&+ \| \hE{\res{\mb{x}_{\text{\normalfont -} i} {\mb{e}_{\text{\normalfont -} i}}^\intercal}{\textup{S}_i^c \textup{S}_i}} \res{\mb{W}_{i.}^*}{\textup{S}_i\cdot} \|_{\textup{B},\infty,\texttt{F}} + \| \hE{\res{\mb{x}_{\text{\normalfont -} i}}{\textup{S}_i^c\cdot} {\mb{e}_i}^\intercal} \|_{\textup{B},\infty,\texttt{F}}
\end{align*}
Using the inequality $\| A B \|_{\textup{B},\infty,\texttt{F}} \leq \| A \|_{\textup{B},\infty,1} \|B\|_{\infty, 2}$ form Lemma \ref{lem:norminequalities}, we get
\begin{align*}
	\frac{\lambda}{2} \| \res{\mb{Z}_{i\cdot}}{\textup{S}_i^c\cdot} \|_{\textup{B},\infty,\texttt{F}} \leq& \| \mb{M} \|_{\textup{B},\infty,1} \big( \| \res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{e}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i} \res{\mb{W}_{i.}^*}{\textup{S}_i\cdot}  -   \hE{\res{\mb{x}_{\text{\normalfont -} i}}{\textup{S}_i\cdot} {\mb{e}_i}^\intercal}
	+  \frac{\lambda}{2} \res{\mb{Z}_{i\cdot}}{\textup{S}_i\cdot} \|_{\infty,2} \big)  \\
	&+ \| \hE{\res{\mb{x}_{\text{\normalfont -} i} {\mb{e}_{\text{\normalfont -} i}}^\intercal}{\textup{S}_i^c \textup{S}_i} }\res{\mb{W}_{i.}^*}{\textup{S}_i\cdot} \|_{\textup{B},\infty,\texttt{F}}  + \| \hE{\res{\mb{x}_{\text{\normalfont -} i}}{\textup{S}_i^c\cdot} {\mb{e}_i}^\intercal }\|_{\textup{B},\infty,\texttt{F}}
\end{align*}
Again using the norm triangle inequality,
\begin{align}
	\label{eq:mainequations}
	\begin{split}
	\frac{\lambda}{2} \| \res{\mb{Z}_{i\cdot}}{\textup{S}_i^c\cdot} \|_{\textup{B},\infty,\texttt{F}} \leq& \| \mb{M} \|_{\textup{B},\infty,1} \big( \| \res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{e}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i} \res{\mb{W}_{i.}^*}{\textup{S}_i\cdot} \|_{\infty, 2}
	 + \|   \hE{\res{\mb{x}_{\text{\normalfont -} i}}{\textup{S}_i\cdot} {\mb{e}_i}^\intercal} \|_{\infty, 2}
	+  \frac{\lambda}{2} \| \res{\mb{Z}_{i\cdot}}{\textup{S}_i\cdot} \|_{\infty,2} \big)  \\
	&+ \| \hE{\res{\mb{x}_{\text{\normalfont -} i} {\mb{e}_{\text{\normalfont -} i}}^\intercal}{\textup{S}_i^c \textup{S}_i}} \res{\mb{W}_{i.}^*}{\textup{S}_i\cdot} \|_{\textup{B},\infty,\texttt{F}}  + \| \hE{\res{\mb{x}_{\text{\normalfont -} i}}{\textup{S}_i^c\cdot} {\mb{e}_i}^\intercal} \|_{\textup{B},\infty,\texttt{F}}
\end{split}
\end{align}
Next, we provide some technical lemmas (detailed proofs in Appendix \ref{appendix:proof of theorems and lemmas}) to bound all the terms in right hand side of equation \eqref{eq:mainequations}.
\begin{lemma}[Bound on \emph{$\| \res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{e}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i} \res{\mb{W}_{i.}^*}{\textup{S}_i\cdot} \|_{\infty, 2}$}]
	\label{lem: xew support}
	For some $\epsilon_1 > 0, 0 < \epsilon_2 < 8$ and $\epsilon_3 < 8 \frac{\sqrt{|\textup{S}_i|} \max_{ij} |\mb{W}_{ij}^*|}{\min_{ij}|\mb{W}_{ij}^*|}$,
	\begin{align}
	\begin{split}
	&\textup{Pr}\big(\| \res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{e}_{\text{\normalfont -} i}}^\intercal   }}{\textup{S}_i \textup{S}_i} \res{\mb{W}_{i.}^*}{\textup{S}_i\cdot} \|_{\infty, 2} >  \epsilon_1 + \sigma^2 (1 + \epsilon_2 +\epsilon_3 ) \max_{i \in \textup{ind}(l), l \in \textup{S}_i} \sqrt{\sum_{j=1}^{k} |\mb{W}_{ij}^*|^2} \big) \leq \\
	&\exp(-\frac{\epsilon_1^2 T}{2k \sigma^2 \textup{b}^2 \max_{j}\sum_{k=1}^{|\textup{S}_i|} {\mb{W}_{kj}^*}^2 } + \log(2 k^2 |\textup{S}_i|) ) + k^2 |\textup{S}_i| \exp(-\frac{T\epsilon_2^2}{64}) + \sum_{\substack{i \in \textup{ind}(l)\\ l \in \textup{S}_i}} \sum_{j=1}^{k} \exp(-\frac{T\epsilon_3^2 |\mb{W}_{ij}^*|}{64 |\sqrt{\sum_{\substack{k=1 \\ k \ne i}}^{|\textup{S}_i|}  {\mb{W}_{kj}^*}^2}|})
	\end{split}
	\end{align}
\end{lemma}

\begin{lemma}[Bound on \emph{$\|   \hE{\res{\mb{x}_{\text{\normalfont -} i}}{\textup{S}_i\cdot} {\mb{e}_i}^\intercal} \|_{\infty, 2} $}]
	\label{lem:bound sup xe}
	For some $\epsilon_4 > 0$ and $\epsilon_5 < 8 \sqrt{k} \sigma^2$,
	\begin{align}
	\begin{split}
	\textup{Pr}(\| \hE{\res{\mb{x}_{\text{\normalfont -} i}}{\textup{S}_i\cdot} {\mb{e}_i}^\intercal}\|_{\infty, 2} \geq \epsilon_4 + \epsilon_5) \leq
	\exp(- \frac{\epsilon_4^2 T}{2 k \sigma^2 \textup{b}^2 } + \log(2k^2|\textup{S}_i|)) + \exp(- \frac{\epsilon_5^2 T}{64 \sqrt{k} \sigma^2 } + \log(2k^2|\textup{S}_i|))
	\end{split}
	\end{align}
\end{lemma}

\begin{lemma}[Bound on \emph{$\| \hE{\res{\mb{x}_{\text{\normalfont -} i} {\mb{e}_{\text{\normalfont -} i}}^\intercal}{\textup{S}_i^c \textup{S}_i}} \res{\mb{W}_{i.}^*}{\textup{S}_i\cdot} \|_{\textup{B},\infty,\texttt{F}}$}]
	\label{lem:bound x-ie-i}
	For some $\epsilon_6 > 0, 0 <\epsilon_7 < 8$ and $ \epsilon_8 <  8 \frac{\sqrt{|\textup{S}_i^c|} \max_{ij} |\mb{W}_{ij}^*|}{\min_{ij}|\mb{W}_{ij}^*|}$,
	\begin{align}
	\begin{split}
	&\textup{Pr}\big(\| \res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{e}_{\text{\normalfont -} i}}^\intercal   }}{\textup{S}_i^c \textup{S}_i} \res{\mb{W}_{i.}^*}{\textup{S}_i\cdot} \|_{\textup{B},\infty,\texttt{F}} > \epsilon_6 + \sigma^2 (1 + \epsilon_7 + \epsilon_8 ) \max_{l \in \textup{S}_i^c} \sqrt{\sum_{i \in \textup{ind}(l)}\sum_{j=1}^{k} |\mb{W}_{ij}^*|^2} \big) \\
	&\leq\exp(-\frac{\epsilon_6^2 T}{2k^2 \sigma^2 \textup{b}^2 \max_{j}\sum_{k=1}^{|\textup{S}_i^c|} {\mb{W}_{kj}^*}^2 } + \log(2 k^2 |\textup{S}_i^c|) ) +
	 k^2 |\textup{S}_i^c| \exp(-\frac{T\epsilon_7^2}{64}) +\\
	& \sum_{i \in \textup{ind}(l), l \in \textup{S}_i^c} \sum_{j=1}^{k} \exp(-\frac{T\epsilon_8^2 |\mb{W}_{ij}^*|}{64 |\sqrt{\sum_{\substack{k=1 \\ k \ne i}}^{|\textup{S}_i^c|}  {\mb{W}_{kj}^*}^2}|})
	\end{split}
	\end{align}
\end{lemma}

\begin{lemma}[Bound on \emph{$\| \hE{\res{\mb{x}_{\text{\normalfont -} i}}{\textup{S}_i^c\cdot} {\mb{e}_i}^\intercal} \|_{\textup{B},\infty,\texttt{F}} $}]
	\label{lem:bound x-ie_i}
	For some $\epsilon_9 > 0$ and $\epsilon_{10} < 8 k \sigma^2$,
	\begin{align}
	\begin{split}
	&\textup{Pr}(\| \hE{\res{\mb{x}_{\text{\normalfont -} i}}{\textup{S}_i^c\cdot} {\mb{e}_i}^\intercal}\|_{\textup{B},\infty,\texttt{F}} \geq \epsilon_9 + \epsilon_{10}) \leq \exp(- \frac{\epsilon_9^2 T}{2 k^2 \sigma^2 \textup{b}^2 } + \log(2k^2|\textup{S}_i^c|)) + \exp(- \frac{\epsilon_{10}^2 T}{64 k \sigma^2 } + \log(2k^2|\textup{S}_i^c|))
	\end{split}
	\end{align}
\end{lemma}
Recall that,
\begin{align}
\begin{split}
	\frac{\lambda}{2} \| \res{\mb{Z}_{i\cdot}}{\textup{S}_i^c\cdot} \|_{\textup{B},\infty,\texttt{F}} \leq& \| \mb{M} \|_{\textup{B},\infty,1} \big( \| \res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{e}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i} \res{\mb{W}_{i.}^*}{\textup{S}_i\cdot} \|_{\infty, 2}
	+ \|   \hE{\res{\mb{x}_{\text{\normalfont -} i}}{\textup{S}_i\cdot} {\mb{e}_i}^\intercal} \|_{\infty, 2}
	+  \frac{\lambda}{2} \| \res{\mb{Z}_{i\cdot}}{\textup{S}_i\cdot} \|_{\infty,2} \big)  \\
	&+ \| \hE{\res{\mb{x}_{\text{\normalfont -} i}^t {\mb{e}_{\text{\normalfont -} i}^t}^\intercal}{\textup{S}_i^c \textup{S}_i}} \res{\mb{W}_{i.}^*}{\textup{S}_i\cdot} \|_{\textup{B},\infty,\texttt{F}}  	+ \| \hE{\res{\mb{x}_{\text{\normalfont -} i}^t}{\textup{S}_i^c\cdot} {\mb{e}_i^t}^\intercal} \|_{\textup{B},\infty,\texttt{F}}
\end{split}
\end{align}

We already showed that mutual incoherence holds in the finite-sample regime, i.e., $\| \mb{M} \|_{\textup{B},\infty,1} \leq 1 - \alpha$ for some $0 <\alpha < 1$. Also note that $\| \res{\mb{Z}_{i\cdot}}{\textup{S}_i\cdot} \|_{\infty,2} \leq 1$. It follows that,
\begin{align*}
& \frac{\lambda}{2} \| \res{\mb{Z}_{i\cdot}}{\textup{S}_i^c\cdot} \|_{\textup{B},\infty,\texttt{F}} \leq (1 - \alpha) \big( \| \res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{e}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i} \res{\mb{W}_{i.}^*}{\textup{S}_i\cdot} \|_{\infty, 2}  + \|   \hE{\res{\mb{x}_{\text{\normalfont -} i}}{\textup{S}_i\cdot} {\mb{e}_i}^\intercal} \|_{\infty, 2}
+  \frac{\lambda}{2}  \big)  \\
&+ \| \hE{\res{\mb{x}_{\text{\normalfont -} i} {\mb{e}_{\text{\normalfont -} i}}^\intercal}{\textup{S}_i^c \textup{S}_i}} \res{\mb{W}_{i.}^*}{\textup{S}_i\cdot} \|_{\textup{B},\infty,\texttt{F}}  	+\| \hE{\res{\mb{x}_{\text{\normalfont -} i}}{\textup{S}_i^c\cdot} {\mb{e}_i}^\intercal} \|_{\textup{B},\infty,\texttt{F}}
\end{align*}
Using bounds from Lemmas \ref{lem: xew support}, \ref{lem:bound sup xe}, \ref{lem:bound x-ie-i},  and \ref{lem:bound x-ie_i}, we get
\begin{align*}
\frac{\lambda}{2} \| \res{\mb{Z}_{i\cdot}}{\textup{S}_i^c\cdot} \|_{\textup{B},\infty,\texttt{F}} \leq& (1 - \alpha) \big( \epsilon_1 +  \sigma^2(1 + \epsilon_2 + \epsilon_3) \max_{i \in \textup{ind}(l), l \in \textup{S}_i} \sqrt{\sum_{j=1}^k |\mb{W}_{ij}^2|}  + \epsilon_4 + \epsilon_5 +  \frac{\lambda}{2}  \big)  + \epsilon_6 + \sigma^2\\
& (1 + \epsilon_7 + \epsilon_8 ) \max_{l \in \textup{S}_i^c} \sqrt{\sum_{i \in \textup{ind}(l)}\sum_{j=1}^{k} |\mb{W}_{ij}^*|^2}   + \epsilon_9 + \epsilon_{10}
\end{align*}
where,
\begin{align*}
&\epsilon_1 > 0, \, 0 < \epsilon_2 < 8, \, \epsilon_3 < 8 \frac{\sqrt{|\textup{S}_i|} \max_{ij} |\mb{W}_{ij}^*|}{\min_{ij}|\mb{W}_{ij}^*|} \epsilon_4 > 0, \, \epsilon_5 < 8 \sqrt{k} \sigma^2, \, \epsilon_6 > 0, \, 0 < \epsilon_7 < 8, \\
&\epsilon_8 <  8 \frac{\sqrt{|\textup{S}_i^c|} \max_{ij} |\mb{W}_{ij}^*|}{\min_{ij}|\mb{W}_{ij}^*|}, \, \epsilon_9 > 0, \, \epsilon_{10} < 2k\sigma^2
\end{align*}
If these conditions hold, then
\begin{align*}
	\frac{\lambda}{2} \| \res{\mb{Z}_{i\cdot}}{\textup{S}_i^c\cdot} \|_{\textup{B},\infty,\texttt{F}} \leq& (1 - \alpha) \big( \epsilon_1 + \sigma^2(1 + \epsilon_2 + \epsilon_3) \sqrt{k} \max_{ij}  |\mb{W}_{ij}|  + \epsilon_4 + \epsilon_5 +  \frac{\lambda}{2}  \big)  + \epsilon_6 + \sigma^2\\
	& (1 + \epsilon_7 + \epsilon_8 ) k \max_{ij} |\mb{W}_{ij}^*|   	+ \epsilon_9 + \epsilon_{10}
\end{align*}
After rearranging the terms, we get,
\begin{align}
	\label{eq:zsc final}
	\begin{split}
	&\frac{\lambda}{2} \| \res{\mb{Z}_{i\cdot}}{\textup{S}_i^c\cdot} \|_{\textup{B},\infty,\texttt{F}} \leq (1 - \alpha) \epsilon_1 +
	(1 - \alpha)  \sigma^2 \sqrt{k} \max_{ij}  |\mb{W}_{ij}| + (1 - \alpha) \sigma^2  \epsilon_2 \sqrt{k} \max_{ij}  |\mb{W}_{ij}| + (1 - \alpha) \\
	&\sigma^2 \epsilon_3 \sqrt{k} \max_{ij}  |\mb{W}_{ij}|  +
	(1-\alpha)\epsilon_4 + (1 - \alpha)\epsilon_5 + (1-\alpha)\frac{\lambda}{2}  +  \epsilon_6 + \sigma^2 k \max_{ij} |\mb{W}_{ij}^*| + \epsilon_7 \sigma^2 k \\
	&\max_{ij} |\mb{W}_{ij}^*| + \sigma^2 \epsilon_8  k \max_{ij} |\mb{W}_{ij}^*|   	+ \epsilon_9 + \epsilon_{10}
	 \end{split}
\end{align}
\paragraph{Choice of $\lambda$.}

To keep the RHS of equation \eqref{eq:zsc final} less than $\frac{\lambda}{2}$, we need to set $\epsilon_1 < \frac{\alpha \lambda}{24(1-\alpha)}, \epsilon_2 < \frac{\alpha \lambda}{24 (1-\alpha)\sigma^2 \sqrt{k} \max_{ij} |W_{ij}|}, \epsilon_3 < \frac{\alpha \lambda}{24 (1 - \alpha) \sigma^2 \sqrt{k} \max_{ij} |W_{ij}|}, \epsilon_4 < \frac{\alpha \lambda}{24 (1-\alpha)}, \epsilon_5 < \frac{\alpha \lambda}{24 (1-\alpha)}, \epsilon_6 < \frac{\alpha \lambda}{24}, \epsilon_7 < \frac{\alpha \lambda}{24 \sigma^2 k \max_{ij} |W_{ij}|}, \epsilon_8 <  \frac{\alpha \lambda}{24 \sigma^2 k \max_{ij} |W_{ij}|}, \epsilon_9 < \frac{\alpha \lambda}{24}$ and  $\epsilon_{10} < \frac{\alpha \lambda}{24}$. We also want to make sure that claim in Lemma \ref{lem: xew support}, \ref{lem:bound sup xe}, \ref{lem:bound x-ie-i} and \ref{lem:bound x-ie_i} hold with high probability. This can be achieved by keeping a $\lambda$ such that
$ \lambda > \max( 24\sqrt{2} \frac{1-\alpha}{\alpha} \sigma \textup{b} W_{\max} \sqrt{\frac{k |\textup{S}_i| \log( 2k^2 |\textup{S}_i|)}{T}},  192 \frac{1-\alpha}{\alpha} \sigma^2  W_{\max} \sqrt{\frac{k \log(k^2 |\textup{S}_i|)}{T}}, 192 \frac{1-\alpha}{\alpha} \sigma^2 \sqrt{k} W_{\max} \\ \sqrt{\frac{W_{\max} |\textup{S}_i| \log(|\textup{S}_i|k)}{TW_{\min}}},  192 \frac{1-\alpha}{\alpha} k^{\frac{1}{4}} \sigma \sqrt{\frac{\log (2k^2|\textup{S}_i|)}{T}},  \frac{24\sqrt{2}}{\alpha} k \sigma \textup{b} W_{\max} \sqrt{\frac{|\textup{S}_i^c \log(2k^2|\textup{S}_i^c|)|}{T}},  \frac{192}{\alpha} \sigma^2 k W_{\max} \\ \sqrt{\frac{\log(k^2 |\textup{S}_i^c|)}{T}}, \frac{192}{\alpha} \sigma^2 k W_{\max}  \sqrt{\frac{W_{\max} |\textup{S}_i^c|^{\frac{1}{2}} \log(|\textup{S}_i^c|k) }{T}},   \frac{24\sqrt{2}}{\alpha} k \sigma \textup{b} \sqrt{\frac{\log (2k^2 |\textup{S}_i^c|)}{T}} , \frac{192}{\alpha} \sigma \sqrt{\frac{k \log(2k^2|\textup{S}_i^c|)}{T}},\\ \frac{24 (1-\alpha) \sigma^2 \sqrt{k} \max_{ij} |W_{ij}| }{\alpha}, \frac{24 \sigma^2 k \max_{ij} |W_{ij}| }{\alpha} ) $.

This particular choice of $\lambda$ implies that $\frac{\lambda}{2} \| \res{\mb{Z}_{i\cdot}}{\textup{S}_i^c\cdot} \|_{\textup{B},\infty,\texttt{F}}  < \frac{\lambda}{2}$
with high probability which in turn ensures that $W_{ij}$ are zero for all $j \in \textup{S}_i^c$ with high probability. Now,
\begin{align}
\begin{split}
	\res{\mb{W}_{i\cdot} - \mb{W}_{i\cdot}^*}{\textup{S}_i \cdot}  =& {\res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{x}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i}}^{\text{\normalfont -} 1}  \res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{e}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i} \res{\mb{W}_{i.}^*}{\textup{S}_i\cdot} +  {\res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{x}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i}}^{\text{\normalfont -} 1} \hE{\res{\mb{x}_{\text{\normalfont -} i}}{\textup{S}_i\cdot} {\mb{e}_i}^\intercal} \\
	&- \frac{\lambda}{2}  {\res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{x}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i}}^{\text{\normalfont -} 1} \res{\mb{Z}_{i\cdot}}{\textup{S}_i\cdot}
\end{split}
\end{align}
By taking the $\textup{B},\infty,\texttt{F}$-norm on both sides,
\begin{align*}
	\| \res{\mb{W}_{i\cdot} - \mb{W}_{i\cdot}^*}{\textup{S}_i \cdot} \|_{\textup{B},\infty,\texttt{F}}  =& \| {\res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{x}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i}}^{\text{\normalfont -} 1}  \res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{e}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i} \res{\mb{W}_{i.}^*}{\textup{S}_i\cdot} +  {\res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{x}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i}}^{\text{\normalfont -} 1} \\
	& \hE{\res{\mb{x}_{\text{\normalfont -} i}}{\textup{S}_i\cdot} {\mb{e}_i}^\intercal}  \frac{\lambda}{2}  {\res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{x}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i}}^{\text{\normalfont -} 1} \res{\mb{Z}_{i\cdot}}{\textup{S}_i\cdot} \|_{\textup{B},\infty,\texttt{F}}
\end{align*}
Using the norm triangle inequality,
\begin{align*}
	\| \res{\mb{W}_{i\cdot} - \mb{W}_{i\cdot}^*}{\textup{S}_i \cdot} \|_{\textup{B},\infty,\texttt{F}} &\leq  \| {\res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{x}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i}}^{\text{\normalfont -} 1}  \res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{e}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i} \res{\mb{W}_{i.}^*}{\textup{S}_i\cdot} \|_{\textup{B},\infty,\texttt{F}} + \\
	& \| {\res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{x}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i}}^{\text{\normalfont -} 1} \hE{\res{\mb{x}_{\text{\normalfont -} i}}{\textup{S}_i\cdot} {\mb{e}_i}^\intercal} \|_{\textup{B},\infty,\texttt{F}}  + \| \frac{\lambda}{2}  {\res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{x}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i}}^{\text{\normalfont -} 1} \res{\mb{Z}_{i\cdot}}{\textup{S}_i\cdot} \|_{\textup{B},\infty,\texttt{F}}
\end{align*}
Using the inequality $\| A B \|_{\textup{B},\infty,\texttt{F}} \leq \| A \|_{\textup{B},\infty,1} \| B \|_{\infty, 2} $ from Lemma \ref{lem:norminequalities}, we get
\begin{align*}
	&\| \res{\mb{W}_{i\cdot} - \mb{W}_{i\cdot}^*}{\textup{S}_i \cdot} \|_{\textup{B},\infty,\texttt{F}} \leq  \| {\res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{x}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i}}^{\text{\normalfont -} 1} \|_{\textup{B},\infty,1} \| \res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{e}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i} \res{\mb{W}_{i.}^*}{\textup{S}_i\cdot} \|_{\infty, 2} + \\
	& \| {\res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{x}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i}}^{\text{\normalfont -} 1} \|_{\textup{B},\infty,1} \| \hE{\res{\mb{x}_{\text{\normalfont -} i}}{\textup{S}_i\cdot} {\mb{e}_i}^\intercal} \|_{\infty, 2}  +
	 \| \frac{\lambda}{2}  {\res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{x}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i}}^{\text{\normalfont -} 1} \|_{\textup{B},\infty,\texttt{F}} \| \res{\mb{Z}_{i\cdot}}{\textup{S}_i\cdot} \|_{\infty, 2}
\end{align*}
Using the inequality $\| A \|_{\textup{B},\infty,1} \leq k \| A \|_{\infty, \infty}$, where $k$ is the maximum number of rows in a block of $A$, we obtain
\begin{align*}
	&\| \res{\mb{W}_{i\cdot} - \mb{W}_{i\cdot}^*}{\textup{S}_i \cdot} \|_{\textup{B},\infty,\texttt{F}} \leq  k \| {\res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{x}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i}}^{\text{\normalfont -} 1} \|_{\infty, \infty} \| \res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{e}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i} \res{\mb{W}_{i.}^*}{\textup{S}_i\cdot} \|_{\infty, 2} + \\
	& k \| {\res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{x}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i}}^{\text{\normalfont -} 1} \|_{\infty, \infty} \| \hE{\res{\mb{x}_{\text{\normalfont -} i}}{\textup{S}_i\cdot} {\mb{e}_i}^\intercal} \|_{\infty, 2}  +  \frac{\lambda}{2}  k \| {\res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{x}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i}}^{\text{\normalfont -} 1} \|_{\infty, \infty}
\end{align*}
Since $\| A \|_{\infty, \infty} \leq \sqrt{p} \| A \|_{2, 2} $ for $A \in \mathbb{R}^{p \times p}$,
\begin{align*}
	& \| \res{\mb{W}_{i\cdot} - \mb{W}_{i\cdot}^*}{\textup{S}_i \cdot} \|_{\textup{B},\infty,\texttt{F}} \leq  k \sqrt{k|\textup{S}_i|} \| {\res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{x}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i}}^{\text{\normalfont -} 1} \|_{2, 2} \| \res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{e}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i} \res{\mb{W}_{i.}^*}{\textup{S}_i\cdot} \|_{\infty, 2} \\
	&+ k \sqrt{k|\textup{S}_i|} \| {\res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{x}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i}}^{\text{\normalfont -} 1} \|_{2, 2} \| \hE{\res{\mb{x}_{\text{\normalfont -} i}}{\textup{S}_i\cdot} {\mb{e}_i}^\intercal} \|_{\infty, 2}  + \frac{\lambda}{2}  k \sqrt{k|\textup{S}_i|} \| {\res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{x}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i}}^{\text{\normalfont -} 1} \|_{2, 2}
\end{align*}
Substituting for $\|{\res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{x}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i}}^{\text{\normalfont -} 1}  \|_{2,2}$,
\begin{align*}
	\| \res{\mb{W}_{i\cdot} - \mb{W}_{i\cdot}^*}{\textup{S}_i \cdot} \|_{\textup{B},\infty,\texttt{F}} \leq&  k \sqrt{k|\textup{S}_i|} \frac{2}{C_{\min}} \| \res{\hE{\mb{x}_{\text{\normalfont -} i} {\mb{e}_{\text{\normalfont -} i}}^\intercal}}{\textup{S}_i \textup{S}_i} \res{\mb{W}_{i.}^*}{\textup{S}_i\cdot} \|_{\infty, 2} + k \\
	& \sqrt{k|\textup{S}_i|} \frac{2}{C_{\min}} \| \hE{\res{\mb{x}_{\text{\normalfont -} i}}{\textup{S}_i\cdot} {\mb{e}_i}^\intercal} \|_{\infty, 2}  +   \frac{\lambda}{2}  k \sqrt{k|\textup{S}_i|}\frac{2}{C_{\min}}
\end{align*}
Using results from Lemma \ref{lem: xew support} and \ref{lem:bound sup xe}, with high probability,
\begin{align*}
\| \res{\mb{W}_{i\cdot} - \mb{W}_{i\cdot}^*}{\textup{S}_i \cdot} \|_{\textup{B},\infty,\texttt{F}} &\leq  k \sqrt{k|\textup{S}_i|} \frac{2}{C_{\min}} (\epsilon_1 + \sigma^2 (1 + \epsilon_2 + \epsilon_3) + k \sqrt{k|\textup{S}_i|}\\
&\frac{2}{C_{\min}}
(\epsilon_4 + \epsilon_5)  +   \frac{\lambda}{2}  k \sqrt{k|\textup{S}_i|}\frac{2}{C_{\min}}
\end{align*}
Substituting bounds on $\epsilon_1, \epsilon_2, \epsilon_3, \epsilon_4$ and $\epsilon_5$, we get
\begin{align*}
&\| \res{\mb{W}_{i\cdot} - \mb{W}_{i\cdot}^*}{\textup{S}_i \cdot} \|_{\textup{B},\infty,\texttt{F}} \leq  k \sqrt{k|\textup{S}_i|} \frac{2}{C_{\min}} (\frac{\alpha \lambda}{24(1-\alpha)} + \sigma^2 (1 + \frac{\alpha \lambda}{24 (1-\alpha)\sigma^2 \sqrt{k} W_{\max}} +\\
& \frac{\alpha \lambda}{24 (1 - \alpha) \sigma^2 \sqrt{k} \max_{ij} W_{\max}}) \sqrt{k} W_{\max}) + k \sqrt{k|\textup{S}_i|}  \frac{2}{C_{\min}} (\frac{\alpha \lambda}{24 (1-\alpha)} + \frac{\alpha \lambda}{24 (1-\alpha)})  + \\
&\frac{\lambda}{2}  k \sqrt{k|\textup{S}_i|}\frac{2}{C_{\min}}
\end{align*}
Let
\begin{align*}
&\delta(k, |\textup{S}_i|, C_{\min}, \alpha, \lambda, \sigma, W_{\max}) =  k \sqrt{k|\textup{S}_i|} \frac{2}{C_{\min}} (\frac{\alpha \lambda}{24(1-\alpha)} + \sigma^2 (1 + \frac{\alpha \lambda}{24 (1-\alpha)\sigma^2 \sqrt{k} W_{\max}} +\\
& \frac{\alpha \lambda}{24 (1 - \alpha) \sigma^2 \sqrt{k} \max_{ij} W_{\max}}) \sqrt{k} W_{\max}) + k \sqrt{k|\textup{S}_i|}  \frac{2}{C_{\min}} (\frac{\alpha \lambda}{24 (1-\alpha)} + \frac{\alpha \lambda}{24 (1-\alpha)})  + \\
&\frac{\lambda}{2}  k \sqrt{k|\textup{S}_i|}\frac{2}{C_{\min}}
\end{align*}
Then,
\begin{align*}
&\| \res{\mb{W}_{i\cdot} - \mb{W}_{i\cdot}^*}{\textup{S}_i \cdot} \|_{\textup{B},\infty,\texttt{F}} \leq  \delta(k, |\textup{S}_i|, C_{\min}, \alpha, \lambda, \sigma, W_{\max})
\end{align*}

Now, we will characterize $\textup{NE}_{\epsilon}(\mathscr{G})$ by $\mb{W}_{i\cdot}$. In particular, we define
\begin{align*}
	\label{eq:infer eps nash}
	\textup{NE}_{\epsilon}(\mathscr{G}) =& \{ \mb{x}^* \in \bigtimes_{i \in \mathcal{V}} \mathcal{A}_i \mid x_i^* = \sum_{j \in \textup{S}_i} W_{ij} x_j^* , \forall i \in \seq{n}  \}
\end{align*}
We explicitly compute the payoffs to prove that equation \eqref{eq:infer eps nash} indeed recovers $\textup{NE}_{\epsilon}(\mathscr{G})$, i.e., for all $\mb{x}^* \in \textup{NE}_{\epsilon}(\mathscr{G})$
\begin{align*}
&u_i(x_i^*, x_{\text{\normalfont -} i}^*) = - \| \sum_{j \in \textup{S}_i} W_{ij} x_j^* - \sum_{j \in \textup{S}_i} W_{ij}^* x_j^* \|_2 \geq - \sum_{j \in \textup{S}_i} \| (W_{ij} - W_{ij}^*) x_j^* \|_2
\geq - \sum_{j \in \textup{S}_i} \| W_{ij} - W_{ij}^* \|_{F} \| x_j \|_2 \\
&\geq - |\textup{S}_i| \delta(k, |\textup{S}_i|, C_{\min}, \alpha, \lambda, \sigma, W_{\max}) \textup{b}
\end{align*}
Thus the set defined in equation \eqref{eq:infer eps nash} recovers $\epsilon$-PSNE for $\epsilon = |\textup{S}_i| \delta(k, |\textup{S}_i|, C_{\min}, \alpha, \lambda, \sigma, W_{\max}) \textup{b}$.

Next, we show that if for each player $i \in \seq{n}$, if $\min_{j \in \textup{S}_i} \|W_{ij}^*\|_{F} > 2 \delta(k, |\textup{S}_i|, C_{\min}, \alpha, \lambda, \sigma, W_{\max})$ then we recover the exact structure of the graphical game. Note that if $\min_{j \in \textup{S}_i} \|W_{ij}^*\|_{F} > 2 \delta(k, |\textup{S}_i|, \\ C_{\min}, \alpha, \lambda, \sigma, W_{\max})$ then $\|W_{ij}^*\|_{F} > 0$ implies that  $\|W_{ij}\|_{F} > 0$. We have already shown that we do not recover any extra player in the set of in-neighbors $\textup{S}_i$ and this added condition ensures that for all the players in $\textup{S}_i$, $\|W_{ij}\|_{F} > 0$. Thus, we recover exact set of players in $\textup{S}_i$ for each player $i \in \seq{n}$. We recover the exact graphical game by combining the results for all the players.
\end{proof}

\paragraph{Sample and Time Complexity.}
\label{sec:sample and time complexity}
If we have $T > \mathcal{O}(k^5 |\textup{S}_i|^3 \log (k |\textup{S}_i^c||\textup{S}_i|))$ and all other conditions mentioned in Theorem \ref{thm:eps_nash_equilibria}  are satisfied for every player then all our high probability statements are valid for every player $i$. Taking a union bound over $n$ players only adds a factor of $\log n$. Thus the sample complexity for our method is $\mathcal{O}(k^5 |\textup{S}_i|^3 \log (k |\textup{S}_i^c||\textup{S}_i|))$. As for the time complexity, we can formulate the block-regularized multi-variate regression problem as a second order cone programing problem~\cite{obozinski2011support} which can be solved in polynomial time by interior point methods~\cite{boyd2004convex}.

\paragraph{Concluding Remarks.}
There are two possible future directions for our work. First, it would be interesting to see if the exact Nash equilibria set can be recovered for  graphical games with quadratic payoffs using our method. Second, it would be interesting to extend our method to the continuous-action games with more general payoff functions.

redpajama_set_name: RedPajamaArXiv


As I mentioned in an update to my previous blog post, the Rand Hall Fine Arts Library project at Cornell is still under plan review and a building permit has not yet been issued. [Updated Feb. 7, 2018. But you would never know that a building permit has not been issued from this headline in the online AAP News from Jan. 25, 2018: "Construction Begins on Mui Ho Fine Arts Library." In fact, only demolition and site preparation permits have been issued; the application for an actual building permit is still under plan review by the City of Ithaca Building Division.] One of the points of contention is the status of the unenclosed exit access stairway (Stair "B") in the atrium that I discussed in detail in that same previous blog post. I have just learned that a third Code section is being considered in relation to Stair "B," the idea being that an unenclosed exit access stair terminating above the level of exit discharge can actually be designated as an enclosed "interior exit stair"—even though it is not actually enclosed. This latest argument is based on Section 1023.2 (Exception #2) in the 2015 Building Code of NYS, which states that the requirement that "Enclosures for interior exit access stairways and ramps shall be constructed as fire barriers… or horizontal assemblies…, or both…" (with fire-resistive ratings of 1 or 2 hours) need not be applied within atriums.
There are at two reasons that this latest attempt to salvage an unsafe and noncompliant design should be rejected.
First, the definition of "interior exit stairway" in Chapter 2 of the 2015 Building Code of NYS states that such a stair "provides for a protected path of egress travel." This means it must be enclosed. That it must be enclosed is reiterated in Section 1023.1 which states unambiguously that an "Interior exit stairways shall be enclosed and lead directly to the exterior of the building or shall be extended to the exterior of the building with an exit passageway." Neither of these two Code passages are affected by Exception 2 in Section 1023.2. That exception only affects the required level of protection, and does not state that such stairs can be unenclosed. In other words, an enclosure for an interior exit stair in an atrium need not be constructed as a fire barrier with a 1- or 2-hour fire-resistance rating, but, per Section 1023.1 (which isn't affected by this exception), it still needs to be enclosed. This enclosure is particularly important for smoke control in the atrium space, even without a fire-resistance rating.
So, we're left with Sections 1006.3 and 1019.3 (see previous blog post), and I've already commented on the likely outcome based on the consideration of those two remaining code sections.
See this web page for links to, and summaries of, all my writings on the Fine Arts Library proposal.
This entry was posted in Architecture, Milstein-Rand-Sibley Hall on February 1, 2018 by jonochshorn.
The open exit access stairway serving the various library stack levels and roof pavilion (Stair "B" per add-alt. #6, as shown schematically in Figure 1) violates Section 1006.3 of the 2015 New York State Building Code. This code section, which appears for the first time in the 2015 IBC, requires that the "path of egress travel to an exit shall not pass through more than one adjacent story." Since exit access stairway "B" passes through more than one story before finding an enclosed exit on the second floor, it is in violation of this section. The 2015 IBC "Commentary" confirms this interpretation.
Fig. 1. An early architect's rendering of proposed exit access stairway "B" in Rand Hall showing that an occupant on the 5th floor (roof pavilion) must pass through two adjacent stories (i.e., the 4th and 3rd floors) in order to find an enclosed exit on the 2nd floor (image crudely edited to replace the extension of this exit access stairway to the "lantern" level, which no longer exists, with a schematic representation of the open bulkhead leading to the roof pavilion at the 5th floor).
Cornell's code consultant, in Exhibit A of Cornell's 2016 Variance request, stated incorrectly that such an open exit access stair is allowed by Sections 1019.3, exception 5, and 404.9.3 in the 2015 NYS Building Code because "Chapter 4 continues to contain provisions which supersedes [sic] other provisions in the Code." Nothing could be further from the truth. The first section that he cites (1019.3, exception 5) requires that exit access stairways be enclosed if they are not in an atrium. It does not say that exit access stairways in atriums can pass through an unlimited number of stories. The second section that he cites (404.9.3) limits exit access travel distance in an atrium to 200 feet, and is not relevant to the question at hand. On the other hand, Section 1006.3, which Cornell's Code consultant doesn't mention in his analysis and which he seems to be unaware of, prohibits such stairs when they "pass through more than one adjacent story." These code requirements are documented in Figure 2. According to my conversation with Technical Code Experts at the International Code Council (the organization that creates the International Building Code from which New York State Codes derive), this latter Section 1006.3 is specific, clear, and unambiguous, and places limits on the application of Section 1019.3.
Section 1019.3 (which says that exit access stairs must be enclosed if they don't meet various conditions enumerated in that section) is not in conflict with Section 1006.3 (which says that exit access stairs cannot pass through more than one story before finding an exit). The limit placed on the number of stories that an exit access stair can pass through in Section 1006.3 is perfectly consistent with the allowance for an unenclosed exit access stair in an atrium per Section 1019.3. In other words, Section 1006.3 does not prevent exit access stairways from being unenclosed in atriums, but simply places a limit on the number of adjacent stories that such a stairway can pass through.
Fig. 2. IBC Section 1019.3 (top) allows unenclosed exit access stairways in atriums; but IBC Section 1006.3 (bottom) limits the number of adjacent stories that such a stair can pass through.
It is not unusual to find two sections in the building code where one section establishes limits not found in the other. For example, Section 506 (specifically Table 506.2) allows non-sprinklered (NS) Type IA, IB, IIA, and IIIA Group A-3 occupancies such as libraries to have floor areas in excess of 12,000 sq.ft. However, Section 903.2.1.3 requires all Group A-3 occupancies with fire areas exceeding 12,000 sq.ft. to have sprinklers. So, in this hypothetical example, if you wanted to place a non-sprinklered, Type IIA library in a building with a floor area of 15,000 sq.ft., and only looked at the allowable area factors in Table 506.2, you might (mistakenly) conclude that it was code-compliant because Table 506.2 says nothing about fire area limits. It is only by checking Section 903.2.1.3 that you would realize that this building floor, defined as a fire area, would either need to be sprinklered or subdivided into fire areas no greater than 12,000 sq.ft. This latter section is not in conflict with the former section, but merely places limits on its applicability. The Group A-3 library would need to comply with both sections, and designers (or code consultants) cannot simply ignore one of the sections because they would prefer not to acknowledge the limits it imposes. Code requirements relevant to this hypothetical example are documented in Figure 3.
Figure 3. IBC Table 506.2 (top) limits the allowable area for non-sprinklered Type IIA libraries (Group A-3) to 15,500 sq.ft; but IBC Section 903.2.1.3 (bottom) limits Group A-3 fire areas to 12,000 sq.ft. Taken together, Section 903.2.1.3 places a limit on the application of Table 506.2, but does not contradict it.
The two code sections concerning exit access stairways work in exactly the same way as do the two code sections cited in the hypothetical example just given. In the current case, the first code section allows unenclosed exit access stairways in atriums, but says nothing about limitations that may be imposed because of the number of stories that the stairway passes through. Those limitations appear in the second section, 1006.3. Taken together, the two sections 1) allow unenclosed exit access stairways in atriums, and 2) limit the number of stories that such exit access stairways can pass through to a maximum of one adjacent story.
So there is no conflict. But even if a conflict is alleged between the two exit access sections cited above (i.e., 1006.3 and 1019.3), the building code has a remedy, and the outcome is still the same. Section 102.1 of the 2015 IBC states: "Where there is a conflict between a general requirement and a specific requirement, the specific requirement shall be applicable. Where, in any specific case, different sections of this code specify different materials, methods of construction or other requirements, the most restrictive shall govern." Section 1019.3 does not specifically permit exit access stairways in atriums to pass through unlimited numbers of stories. It just says that if such a stair is not in an atrium (or any of the other seven conditions listed), then it must be enclosed in a shaft enclosure. It is mute on the question of whether exit access stairways in atriums can pass through more than one adjacent story; whereas Section 1006.3 specifically prohibits such a stair from passing through more than one adjacent story. The open stairs are therefore unambiguously noncompliant under the 2015 NY State Building Code because they pass through more than one adjacent story before finding an exit.
Under Add-Alternate #6, the atrium space needs to be separated from the occupied roof pavilion and occupied roof deck with 1-hour fire-resistance rated horizontal assemblies and fire barriers. In addition, all structural steel supporting the roof deck, including the deck itself, roof girders, and all steel columns supporting the girders, need to have a one-hour fire-resistive rating. I noticed that these structural elements in the atrium were specifically excluded from having any fire-resistance rating on working drawing number G-003, and I'm not sure if Add-Alt #6 included provision for this additional fire-proofing. It is required if the roof above the atrium is occupied, per IBC Sec. 404.6 and Sec. 711.2.3.
Exit access stairway "B" is within the atrium and extends up to the bulkhead (pavilion) on the roof in Add-Alt #6. It does not seem consistent with atrium smoke control requirements. Not only is it at the highest point in the atrium, where hot smoke naturally migrates, but it also contains an exit door which, when opened along with the exit door from the pavilion area itself, would draw hot smoke from a fire through the exit door at roof level, precisely in the path of occupants attempting to exit through that stairway.
Section 8 on Drawing Number 109 (Add-alt #6) does not clearly show a roof over the interior exit stairway, and also shows what looks like a brick obstruction drawn halfway across the door leading to the 4th-floor walkway to the stacks. It is hard to see how there would be sufficient headroom in the portion of interior exit stairway "A" leading from the enclosed part at the 4th floor level to the exterior pit where Stair "A" continues up to the occupied roof. I can't find any drawings which provide definitive details, but unless the roof over Stair "A" is raised considerably higher, there appears to be a major headroom problem.
See this web page for links to all my writings and blog posts on the Rand Hall library proposal.
This entry was posted in Architecture, Milstein-Rand-Sibley Hall on January 22, 2018 by jonochshorn.
This entry was posted in Architecture, Milstein-Rand-Sibley Hall on December 17, 2017 by jonochshorn.
On May 22, 2017, updating an earlier blog post (Egress, toilets, and carcinogens: Cornell's transition plans during Fine Arts Library construction), I wrote that "transferring outside air from a corridor into the digital fabrication lab would not be compliant with the 2015 Mechanical Code. However, since the corridor seems to be now labeled as a room ('collaborative area'), it's probably legal, but barely. The opportunistic and ad hoc manner in which such design decisions are, and have been made, does not inspire confidence." It is now apparent that the actual use of this new "room"/corridor has confirmed my fears: the ability of this "collaborative area" to function properly as part of an egress path has been seriously undermined by the types of activities assigned to the space.
This entry was posted in Architecture, Milstein-Rand-Sibley Hall on December 3, 2017 by jonochshorn.
I've written about the impending bathroom crisis for the Department of Architecture at Cornell here and here, caused by the renovation of Rand Hall. When this renovation begins, there will only be a single men's toilet available for the entire 32,000 square foot floor area comprising the second floor studio and office spaces in Milstein and E. Sibley Hall. As hordes of desperate students and faculty from the second floor trek up to the "private" third floor Frances Shloss studios, it is inevitable that faculty and students on the third floor will take retaliatory action. As they say, the writing is on the wall.
This entry was posted in Architecture, Milstein-Rand-Sibley Hall on October 20, 2017 by jonochshorn.
I'm not an Instagram fan, so I end up posting my favorite images here. I took this one in Sibley Hall a few days ago. Sibley Hall is part of the College of Architecture, Art, and Planning at Cornell. When neighboring Rand Hall is closed down for renovation in a few months, this single toilet will be the only viable option for men, not only those in offices and facilities on the second floor of Sibley Hall, but for students and faculty using the 25,000 square feet of studio space in adjacent Milstein Hall. When asked about this at a recent faculty meeting, the Dean of the College stated that the "solution" to this problem will most likely be a request for a Code variance.
This entry was posted in Architecture, Milstein-Rand-Sibley Hall on October 13, 2017 by jonochshorn.
Every time I see a sign for something good—for example, a sign at the supermarket that says something like "healthy choices"—I wonder where the missing sign for "unhealthy choices" has been hidden. Well, thanks to the power of Photoshop, I can rectify that situation. I recently passed a new sign pointing to the new building addition and renovation for "Cornell Health"; I've added in the "missing" sign pointing to Architecture.
A new sign points to Cornell's just-completed "Health" building addition and renovation. I've added the complementary sign for Architecture.
This entry was posted in Architecture on August 1, 2017 by jonochshorn.
I was interrupted today by two workers who needed access to Milstein Hall in order to inspect roof leaks; the green roof itself has been in a state of partial demolition for the last two years, as I described in this blog post from May 2017. There are several areas where water is currently coming down, mainly in the stepped seating area and over the wooden floor area.
This entry was posted in Architecture, Milstein-Rand-Sibley Hall on July 25, 2017 by jonochshorn.
When Frank Gehry built an addition to his traditional house in Santa Monica, California in the 1970s, at least some of "Gehry's neighbors were not happy at the unusual building being built in their neighborhood," according to Wikipedia. Well, it turns out that not only Gehry, but the whole upscale neighborhood, is sitting on an earthquake fault that could do more damage to their homes than any mere symbol of deconstruction.
This entry was posted in Architecture on July 17, 2017 by jonochshorn.
It's both a bit weird, but also quite expected, to see the same design tropes appearing over and over again within the same time period at the same place. I hinted at this phenomenon in my 2009 song, Prisoner of Art.
Two examples from the Cornell campus follow. The first is based on the idea that, because Cornell is situated between two gorges, the idea of the gorge should somehow be expressed in new campus construction. So not only do we get the West Campus dorms designed by Kieran Timberlake referencing the glacial topography of the Finger Lakes, but also more literal representations of the gorges in Bailey Plaza (Michael Van Valkenburgh Associates: "A tilted, striated bluestone fountain presents a mysterious dark pool at its base, making material reference to Ithaca's famous gorges") and the Pew Engineering Quad (EDAW, Inc.: "The created landscape will dramatize the topography by adding landscaped slopes that recall the natural character of the nearby Cascadilla Gorge").
Second, the idea of defamiliarizing the standard turf quadrangle by "tilting" up one corner of a parterre has shown up in at least three recent landscape gestures at Cornell: at the Law School's new underground addition (Ann Beha Architects with Trowbridge Wolf Michaels Landscape Architects); at the entry plaza in front of Weill Hall (Richard Meier with Olin Landscape Partnership), and in the triangular space bordered by Milstein and Rand Halls (OMA Architects with Scape Landscape Architecture).
This entry was posted in Architecture on July 10, 2017 by jonochshorn.


According to The Wall Street Journal, "J.P. Morgan faces six separate investigations from the U.S. Justice Department…" JP Morgan recently made an announcement about one of these investigations and increased its litigation reserves. One of the investigations centered around allegations that JP Morgan concealed facts regarding the actual financial condition and risks associated with mortgage backed securities. The criminal investigation is still in the preliminary stages.


I am attempting to set up a new multiforest ldap instance, based on the following document. The configuration goes as planned untill I attempt to run ADAMSYNC /install.
I have reduced to attributes portion of the xml file to nothing and still receive the error. Any help in understanding how to resolve this issue is appreciated.
Establishing connection to target server localhost:50512.
Updating configuration file on AdamSyncConfNLS.xml.
Attaching configuration file auxilliary class on dc=multiforest,dc=local.


Read Time: 2 minutes Let's talk about writing a little, specifically about how difficult it can be battling the impulse to produce perfect finished products right off the bat!
Read Time: 1 minute Creativity and passion go hand in hand. While it's often overused, passion for what you create is indeed essential to the creative process. Some take it to a whole new level.


Point of Interest Consulting – Point of Interest Consulting is home to a diverse team of individuals offering advanced GIS solutions, mapping and data analysis services, for professionals in all disciplines.
Point of Interest Consulting is home to a diverse team of individuals offering advanced GIS solutions, mapping and data analysis services, for professionals in all disciplines.
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As a GIS and Environmental consulting company in Port Elizabeth our business model is aimed at utilizing open source software solutions, to provide professional accurate, affordable, and comprehensive spatial solutions; data management and analytical services at competitive prices to benefit both large and small enterprises.


Beef is the best natured bull I've ever seen. He's friendly and curious, and he's super easy to move between pastures and work with so far (I say so far because I've seen enough YouTube videos to know that when a bull goes bad, it usually happens fast, and there might not be a lot of warning signs; of course I've also seen enough YouTube videos that I sometimes wonder if what I'm hearing in the woods might be Bigfoot, so I honestly take all YouTube videos with a grain of salt and have just decided to be careful and always have an out when working with an animal that could squash me like a bug).
He is not without his bad habits though.
2. He is a big pig in a cow sized suit. Seriously, I think if I left a 50 pound bag of corn open for him he would eat the entire thing in just a few minutes, and he would probably burst from the inside out before he would stop eating. Sometimes he is so picky that he tries to push Bessie out of the way so he can gorge himself on the entire bowl. We have to watch that carefully since she is going to have her calf soon. We have to make sure that he allows the calf to eat once the calf start eating solid food.
3. His worst habit though is the one that absolutely drives Ted bonkers! When he gets bored, he dumps his 80 gallon water container. Ted has tried lots of different things to keep him from doing this. Currently the water trough is half in and half out of the fence, and our picnic table is shoved up against it on one end to make it harder to tip.
Beef has taken that as a challenge, and he spends lots of time everyday rubbing all over the end of the trough until he catches the tiny little piece of lip on it with his horns and then he rocks it back and forth until he flashes all of the water out. He may not be able to flip it upside down on top of the fence with it went against the picnic table, but he can still dunk the water. Then his favorite thing to do is just stand and wait and bellow for Ted to come back to fill it up so he can have more fun with it again.


Germain Gabriel Grisez (born 1929) is a French-American philosopher. Grisez's development of ideas from Thomas Aquinas has redirected Catholic thought and changed the way it has engaged with secular moral philosophy. In 'The First Principle of Practical Reason: A Commentary on the Summa Theologiae, I-II, Q. 94, A. 2' (1965) Grisez attacked the neo-scholastic interpretation of Aquinas as holding that moral norms are derived from methodologically antecedent knowledge of human nature. Grisez defends the idea of metaphysical free choice, and proposes a natural law theory of practical reasoning and moral judgement which, although broadly Thomistic, departs from Aquinas on significant points.
Grisez is Professor of Christian Ethics at Mount St. Mary's University in Emmitsburg, MD.
The Making of a Moral Theologian, Russell Shaw. 1996 article on Grisez's personal and professional life, including his work for Cardinal O'Boyle of the Diocese of Washington D.C.
Biotechnology and Human Dignity in the Thought of Germain Grisez, Nicholas C. Lund-Molfese. Describes the necessity of the Christian virtue of hope in choosing the good of the human person in so-called "hard cases".
↑ George, Robert P. (1995). Honderich, Ted, ed. The Oxford Companion to Philosophy. Oxford: Oxford University Press. p. 328. ISBN 0-19-866132-0.