text:
\section{Introduction}
\label{sec:introduction}
Contrastive Learning (CL) emerges to be a promising paradigm for learning data representations without labeled data \citep{InfoNCE,infomax}. Recently, it has achieved impressive results and gradually closed the gap between supervised and unsupervised learning, hopefully leading to a new era that resolves the hunger for labeled data in the deep learning field \citep{moco,simclrv2,wang2021residual}. However, despite its intriguing empirical success, a theoretical understanding of how contrastive learning actually works in practice is still under-explored.

The general methodology of contrastive learning is quite simple, that is to maximize the similarity between augmented views of the same image (\textit{a.k.a.~} positive samples), and minimize the similarity between that of two random images (\textit{a.k.a.~} negative samples). Intuitively, it is \emph{an instance discrimination task} (differing each image from others) instead of \emph{a classification task} (clustering images from the same class together and differing with other classes). Nevertheless, as shown in Figure \ref{fig:clusted-features}, CL representations are also class-separated. Therefore, understanding how the pretraining task (CL) and the downstream task (classification) interact plays a central role in both theoretical understandings and practical designings of contrastive methods.

Previously, \citet{arora} and \citet{jason} have tried to establish guarantees on the classification performance for self-supervised representations. However, their analysis relies heavily on the assumption that the two positive samples, as augmented views of the same image, are (nearly) conditionally independent on the class $y$. However, this is hardly practical as the augmented views are still strongly input-dependent (see Figure \ref{fig:positive-samples}). In fact, if the conditional independence is satisfied, the unsupervised task will become as informative as the supervised task, making this discussion almost unnecessary. This motivates us to find more practical and weaker assumptions to understand how contrastive learning actually works (even without conditional independence). To achieve this, we need to re-examine the contrastive learning process. Previously, \citet{wang} show that CL objective involves two goals: alignment (for positive samples) and uniformity (for negative samples). Nevertheless, we show that there exist bad cases where the features could still have poor performance even with perfect alignment and uniformity. Thus, contradictory to the common belief of contrastive learning as learning invariance, we note that invariance alone is inadequate to learning useful representations for downstream tasks.

In this paper, we provide a novel understanding of contrastive learning that requires only practical and minimal assumptions, while also guarantee class-separated representations. Our core insight hinges on the observation that contrastive learning usually adopts much more aggressive data augmentations than that in supervised learning \citep{moco,simclr}. As shown in Figure \ref{fig:positive-samples}, we notice that aggressive random cropping of two images can generate views that are very much alike that we could even hardly tell them apart, \textit{e.g.,~} the wheels of two different cars. In other words, there will be support overlap between different intra-class images through aggressively augmented views of them, a phenomenon we call \textit{augmentation overlap}. Thus, the alignment of positive samples will also cluster all the intra-class samples together, and lead to class-separated representations. From our perspective, the role of data augmentation is to create a certain degree of ``chaos'' between intra-class samples, and the role of contrastive loss is to ``climb the ladder of chaos'', \textit{i.e.,~} the process that we gradually cluster intra-class samples by aligning positive samples.

Following this intuition, we develop a new theory for understanding the effectiveness of contrastive learning from the perspective of augmentation overlap. Specifically, we derive the upper and lower bounds for its downstream performance and show how the two bounds will asymptotically converge with our assumptions on augmentation overlap. Driven by this analysis, we further discuss how varying augmentation will affect the performance of contrastive learning from both synthetic and real-world datasets, and show that the results align well with our theory. In summary,
\begin{itemize}
    \item We characterize the failure of the previous analysis of contrastive learning, and develop a new understanding through the augmentation overlap effect. Compared to existing theories on contrastive learning, ours can provide guidance to the practical designing of contrastive methods and evaluation metrics.
    \item We establish general guarantees (both upper and lower bounds) for the downstream performance without assumptions on conditional independence. And we further show how the two bounds could asymptotically converge under our less restrictive assumptions.
    \item We provide a quantitative discussion on the effect of augmentation strength, which verifies our theory from both theoretical and empirical aspects. Motivated by our theory, we further propose a new unsupervised evaluation metric for contrastive learning named ARC and show that it aligns well with downstream performance on real-world datasets.
\end{itemize}

\begin{figure}
    \centering
    \subfigure[Contrastive learning learns clustered features.]{
    \label{fig:clusted-features}
    \includegraphics[width=0.6\textwidth]{figures/tsne_change.pdf}}
    \hfill
    \subfigure[Intra-class samples are more alike via augmented views.]{
    \label{fig:positive-samples}
    \includegraphics[width=0.3\textwidth]{figures/positive.pdf}}

    \vspace{-0.1in}
    \caption{(a) t-SNE visualization of representations before and after contrastive learning. Each point denotes a sample and its color denotes its class. (b) Applying aggressive data augmentations \citep{simclr} to four images from ImageNet (two are cars and two are pens). The 1st column shows the raw (center-cropped) images and the 2-5th colums show the augmented ones.}
    \label{fig:my_label}
    \vspace{-0.1in}
\end{figure}

\section{Related Work}
\label{sec:related-work}

\textbf{Contrastive Learning in Practice.} Contrastive self-supervised learning originates from a mutual information perspective of representation learning \citep{InfoNCE,infomax}, and soon becomes a general learning paradigm that contrasts between positive and negative pairs \citep{moco,simclr}. It is rapidly closing the performance gap between unsupervised and supervised learning on large-scale dataset like ImageNet \citep{mocov3}, and outperforms supervised learning when combined with a few (\textit{e.g.,~} 10\%) labels \citep{simclrv2}.
Several recent works show that similar performance could be achieved without negative samples by adopting certain training techniques \citep{BYOL,simsiam}.

\textbf{Understanding Contrastive Learning Objectives.} Both the original InfoNCE loss \citep{InfoNCE} and its InfoMax variants \citep{infomax,bounds} are designed as variational estimates of the mutual information between inputs and representations, but these estimators are shown to have poor bias-variance trade-offs \citep{song2019understanding}. Instead, \citet{wang} simply understand contrastive learning through the two terms in the InfoNCE loss: alignment of positive samples and uniformity of negative samples. However, as we show later, this perspective is also insufficient to explain the effectiveness of contrastive learning, and we should take the interplay between augmentation and alignment into consideration.

\textbf{Understanding Downstream Generalization.} \citet{arora} propose the first theoretical guarantees by bridging the contrastive and classification objectives.
\citet{jason} further link the reconstruction-based objective to the downstream objective. However, both \citet{arora} and \citet{jason} rely on the unrealistic assumption that the positive samples are (nearly) conditionally independent. \citet{huang2021towards} establish bounds by assuming a very small intra-class support diameter, which is also not practical. Besides, some also explore the information-theoretical perspectives for analyzing contrastive learning {\citep{InfoMin,tsai2020self,tosh2020contrastive,tosh21a}}, though their mutual information assumptions are hard to verify.
{Recently, similar to our analysis, \citet{haochen2021provable} also study the augmentation graph and establish guarantees in terms of graph connectivity. Our work differs to theirs mainly in three aspects: 1) our analysis is applicable for the widely adopted InfoNCE and CE losses, while theirs is developed for their own spectral loss; 2) ours starts from the alignment and uniformity perspective while theirs starts from the matrix decomposition perspective; 3) our theory is empirically verified and inspires a useful evaluation metric for data augmentation, while their analysis focusing on minimizing the decomposition error is farther from the practical designing of positive and negative samples. In a nutshell, compared to previous discussions, our theory has a closer connection to the actual contrastive learning process, and we verify the feasibility of each assumption with empirical evidence. }

\section{Limitations of Previous Understandings}

We begin by introducing the basic notations and common practice of contrastive learning in the image classification task. In general, it has two stages, unsupervised pretraining, and supervised finetuning. In the first stage, with $N$ unlabeled samples $\mathcal{D}_u=\{x_i\}_{i=1}^N$, we pretrain an encoder mapping from the $d$-dimensonal input space to a unit hypersphere $f\in{\mathcal{F}}:{\mathbb{R}}^d\to{\mathbb{S}}^{m-1}$ in the $m$-dimensional space. In the second stage, we evaluate the learned representations $z$ with the labeled data $\mathcal{D}_l=\{(x_i,y_i)\}$ where labels $y_i\in\{1,\dots,K\}$. Specifically, we fix the encoder and learn a linear classification head $g:{\mathcal{R}}^m\to{\mathcal{R}}^K$ on top from $\tilde\mathcal{D}_l=\{(z,y)|z=f(x)\in{\mathcal{R}}^m\}$.

\textbf{Contrastive Pretraining}.
Taking a training example $x\in\mathcal{D}_u$, we draw its positive sample $x^+=t(x)$ by applying a random data augmentation $t\sim{\mathcal{T}}$, and draw $M$ randomly augmented samples $\{x_i^-\}_{i=1}^M$ from $\mathcal{D}_u$ as its negative samples. Then, we can learn the encoder $f$ with the widely used InfoNCE loss
\citep{InfoNCE}
\begin{equation}\small
\begin{aligned}
{\mathcal{L}}_{\rm NCE}(f)
=&\mathbb{E}_{p(x,x^+)}\mathbb{E}_{\{p(x_i^-)\}}\left[-\log\frac{\exp(f(x)^\top f(x^+))}{\sum_{i=1}^M\exp(f(x)^\top f(x^-_i))}\right].\\
\end{aligned}
\label{eqn:infonce-frac}
\end{equation}
Let $p(x)$ be the data distribution, $p(x,x^+)$ be the joint distribution of positive pairs, and we simply assume  $p(x,x^+)=p(x^+,x)$ and $p(x)=\int p(x,x^+)d x^+,\forall\ x\in{\mathbb{R}}^d$ following \citet{wang}.

\noindent\textbf{Linear Evaluation}. To evaluate the learned representations by contrastive learning, we usually adopt the
Cross Entropy (CE) loss \citep{simclr} for a labeled pair $(x,y)\in{\mathcal{D}}_l$
\begin{equation}\small
{\mathcal{L}}_{\rm CE}(f,g)=\mathbb{E}_{p(x,y)}\left[
-\log\frac{\exp\left(f(x)^\top w_y\right)}{\sum_{i=1}^K\exp\left(f(x)^\top w_i\right)}\right],
\label{eq:linear-evaluation}
\end{equation}
with a linear classifier $g(z)=Wz$ where $W=[w_1,w_2,\dots,w_K]$.

\subsection{Existing Theoretical Assumptions and Their Limitations}

As discussed above, there are some previous understandings on how contrastive learning yields good performance, and they mainly differ by their theoretical assumptions.

\begin{wrapfigure}{r}{0.35\textwidth}
 \centering
\includegraphics[width=0.35\textwidth]{figures/wang.pdf}
\caption{Contrastive learning may learn class inseparable features even with perfect aligned postive samples and uniform negative samples. Colors denote classes.
}
\label{fig:wang-couterexample}
\vspace{-0.2in}
\end{wrapfigure}

First, \cite{wang} interpret the first and second terms of the InfoNCE loss (Eq.~\ref{eqn:infonce-frac}) as they are aiming at the following two properties: 1) {alignment} (the nominator): positive samples $x,x^+$ has similar features, \textit{i.e.,~} $f(x)\approx f(x^+)$; 2) {uniformity} (the denominator): features are roughly uniformly distributed in the unit hypersphere ${\mathbb{S}}^{m-1}$. In particular, they show that InfoNCE can be minimized with 1) perfect alignment  and 2) perfect uniformity. However, as we illustrate in Figure \ref{fig:wang-couterexample}, the features could still have very poor downstream performance in the finite sample scenario. This issue can be described rigorously by the following proposition.

\begin{prop}[Class-uniform Features Also Minimize the InfoNCE Loss]
For $N$ training examples of $K$ classes, consider the case when features $\{f(x_i)\}_{i=1}^N$ are randomly distributed in ${\mathbb{S}}^{m-1}$ with maximal uniformity  {(\textit{i.e.,~}, minimizing the 2nd term of Eq.~\ref{eqn:infonce-frac})} while also satisfying $\forall x_i,x_i^+\sim p(x,x^+), f(x_i)=f(x^+_i)$. Because we have {these two properties}, the InfoNCE loss achieves its minimum. However, the downstream classification accuracy is at most $1/K+\varepsilon$ and $\varepsilon$ is nearly zero when $N$ is large enough.
\label{prop:wang-couterexample}
\end{prop}
\vspace{-0.05in}
Proofs can be found in Appendix \ref{sec:ommited-proofs}. This proposition indicates that
the instance discrimination task (alignment + uniformity) alone cannot guarantee {the learning of} class-discriminative features as desired in the downsteam classification.
Instead, \citet{arora} and \cite{jason} both establish the relationship between pretraining and classification objectives and provide guarantees for the downstream performance. In fact, the two works both assume the conditional independence of the two positive samples, \textit{i.e.,~} $p(x,x^+|y)=p(x|y)p(x^+|y)$. However, this assumption is too strong as it is hardly practical. As shown in Figure \ref{fig:positive-samples}, augmented views from the same class are not actually independent as views from the same sample are more alike than that from other samples.

\section{New Augmentation Overlap Theory for Contrastive Learning}
\label{sec:overlap-theory}
The analysis above motivates us to find a \emph{minimal} and \emph{practical} assumption: 1) it is enough to guarantee good performance on downstream tasks; 2) it is less restrictive than the i.i.d. assumptions as in \citet{arora} and \cite{jason}.

\subsection{Gap Between Contrastive Learning and Downstream Classification}

We start with an assumption on the label consistency between positive samples, that is, any pair of positive samples $(x,x^+)$ should belong to the same class.
\begin{assumption}[Label Consistency]
 $\forall\ x,x^+ \sim p(x,x^+)$, {we assume the labels are deterministic (one-hot) and consistent: $p(y|x)=p(y|x^+)$.}
 \label{assumption:label-consistency}
\end{assumption}
\vspace{-0.05in}
This is a natural and minimal assumption that is likely to hold in practice. As shown in Figure \ref{fig:positive-samples}, the widely adopted augmentations in contrastive learning \citep{simclr} like images cropping, color distortion, and horizontal flipping will hardly alter the belonging image classes.

With this minimal assumption, we can characterize the generalization gap between unsupervised and supervised learning risks. We first introduce the {mean CE loss}, $L^\mu_{\rm CE}(f)=\mathbb{E}_{p(x,y)}\left[-\log\frac{\exp\left(f(x)^\top \mu_y\right)}{\sum_{i=1}^K\exp\left(f(x)^\top \mu_i\right)}\right]$, where we use the classwise \emph{mean representation} $\mu_k=\mathbb{E}_{p(x|y=k)}[f(x)]$  as the weight $w_k$ of the classifier $g$.
It is easy to see that the mean CE loss upper bounds the CE loss, \textit{i.e.,~} $L^\mu_{\rm CE}(f)\geq\min_g {\mathcal{L}}_{\rm CE}(f,g)$ and \cite{arora} showed that the mean classifier could achieve comparable performance to learned weights. Then, we have the following upper and lower bounds on the downstream risk (measured by mean CE loss).

\begin{theorem}[Guarantees for General Encoders] If Assumption \ref{assumption:label-consistency} holds, then, for any $f \in \mathcal{F}$, its downstream classification risk ${\mathcal{L}}_{\rm CE}^\mu(f)$ can be bounded by the contrastive learning risk ${\mathcal{L}}_{\rm NCE}(f)$
\begin{equation}
\begin{aligned}
&{\mathcal{L}}_{\rm NCE}(f)- \sqrt{\var(f(x)\mid y)} -\frac{1}{2}{\var(f(x)\mid y)}-{\mathcal{O}}\left(M^{-1/2}\right)\\
\leq\ &{\mathcal{L}}_{\rm CE}^{\mu}({f})+\log(M/K)\leq
{\mathcal{L}}_{\rm NCE}(f) + \sqrt{\var(f(x)|y)} + {\mathcal{O}}\left(M^{-1/2}\right),
\end{aligned}
\end{equation}
where $\log(M/K)$ is a constant\footnote{$\log(M/K)$ could be absorbed in the loss functions by replacing $\operatorname{sum}$ with $\operatorname{mean}$ in InfoNCE and CE.}, $\mathrm{Var}(f(x)|y)=\mathbb{E}_{p(y)}\left[\mathbb{E}_{p(x|y)}\Vert f(x)-\mathbb{E}_{p(x|y)}f(x)\Vert^2\right]$ denotes the conditional (intra-class) feature variance, and ${\mathcal{O}}\left(M^{-1/2}\right)$ denotes the order of the approximation error by using $M$ negative samples.

\label{thm:general-generalization-gap}
\end{theorem}

Notably, our generalization bounds above improve over previous ones in the following aspects:
\begin{enumerate}[1)]
    \item we do not require the conditional independence assumption as in \citet{arora};
    \item we directly analyze the widely adopted InfoNCE loss (for contrastive learning) and CE loss (for supervised finetuning), while \citet{arora} are restricted to hinge and logistic objectives that have worse performance in practice \citep{simclr};
    \item the class collision error terms introduced in \citet{arora} (due to the existence of same-class samples in the negative samples) now disappear in our bounds by adopting the InfoNCE loss, which also {helps understand} why InfoNCE performs better in practice; and
    \item the bounds in \citet{arora} will become looser with more negative samples, which is contradictory to the common practice \citep{simclr}. While in our bounds, a larger $M$ indeed has a lower approximation error and helps close the generalization gap.
\end{enumerate}

{In fact, several recent works have also been devoted to resolve the last ``large-$M$'' problem \citep{ash2021investigating,merad2020contrastive}. Nevertheless, their analysis also requires the conditional independence assumption as in \citet{arora}, while we show this problem can be resolved even without conditional independence. \citet{nozawa2021understanding} also establish bounds for the InfoNCE loss, but their bounds have incompressible class collision terms while ours do not.}

Nevertheless, an important message of the theorem above is that Assumption \ref{assumption:label-consistency} alone is still \textbf{insufficient} to guarantee good downstream performance. As there are intra-class variance terms in the upper and lower bounds, when they are large enough, contrastive learning might still have inferior performance as shown in Proposition \ref{prop:wang-couterexample}.
Although the variance terms can be easily eliminated with the canonical conditional independence assumption, discussions in Section \ref{sec:introduction} have already demonstrated its impracticality.
In the next part, we will present a new understanding of how contrastive learning could control this variance term in practice.

\begin{figure}
    \centering

    \subfigure[Contrastive learning with an augmentation graph satisfying intra-class connectivity.]{
    \includegraphics[width=.42\textwidth]{figures/aug_graph.pdf}
    \label{fig:augmentation-graph}
    }
    \hfill
    \subfigure[Augmentation  graph  under  increasing  augmentation  strengthes (left to right).]{
    \includegraphics[width=.51\textwidth]{figures/aug_strength.pdf}
    \label{fig:aug-strength}
    }

    \caption{Illustrative examples of augmentation graphs, where each dot denotes a sample $x\in{\mathcal{D}}_u$ and its color denotes its class. The lighter disks denote the support of the positive samples $p(x^+|x)$. We draw a solid edge for each ${\mathcal{T}}$-connected pair.

    }

\end{figure}

\subsection{Closing the Gap with Intra-class Connectivity}
\label{sec:intra-class-connectivity}
The theorem above motivates us to study how contrastive learning could effectively control its intra-class variance and learn class-separated features.
Here, we propose a new understanding of this clustering ability through a dissection of the augmented views. In particular, we notice that although samples are different from each other, applying aggressive augmentations like that in SimCLR \citep{simclr} can largely make them more alike. For example, in Figure \ref{fig:positive-samples}, two different cars become very similar when they are both cropped to the wheels. Then, with contrastive learning, the two cars will have closer representations as they share a common view of the wheels. In other words, two different intra-class samples could be aligned together if they have overlapped augmented views. If all intra-class samples could be bridged by data augmentations, we can successfully cluster the whole class together. Below, we formalize the intuition above with the language of graphs.

\textbf{Notations.} A graph ${\mathcal{G}}$ is represented by a tuple ${\mathcal{G}}=({\mathcal{V}},{\mathcal{E}})$ where ${\mathcal{V}}=(v_1,v_2,\dots,v_N)$ is a set of vertices and ${\mathcal{E}}\subseteq{\mathcal{V}}\times{\mathcal{V}}$ is a set of edges. A path is a sequence of edges that joins a sequence of vertices, \textit{e.g.,~} $v_{i_1}- v_{i_2}-\cdots- v_{i_k}$. We say that two vertices $v$ and $u$ are connected if ${\mathcal{G}}$ contains a path from $v$ to $u$.  A graph is said to be connected if every pair of vertices in the graph is connected. Two graphs are said to be disjoint if any pair of inter-graph vertices are not connected.

To begin with, we define the concept of ${\mathcal{T}}$-connectivity of sample pairs, which describes whether two samples could be connected via the augmentation overlap of their augmented views.

\begin{definition}[${\mathcal{T}}$-connectivity] Given a collection of augmentations ${\mathcal{T}}=\{t\mid t:{\mathbb{R}}^d\to{\mathbb{R}}^d\}$, we say that two different images $x_i,x_j\in{\mathbb{R}}^d$ are \emph{${\mathcal{T}}$-connected} if they have overlapped views:  $\supp(p(x_i^+|x_i))\bigcap\supp(p(x_j^+|x_j))\neq\varnothing$, or equivalently,  $\exists\ t_i,t_j\in{\mathcal{T}}$ such that  $t_i(x_i)=t_j(x_j)$.
\end{definition}
Then, we can define an augmentation graph of all training samples in terms of their ${\mathcal{T}}$-connectivity.
\begin{definition}[Augmentation Graph]
Given a set of $N$ samples ${\mathcal{D}}=\{x_i\}_{i=1}^N$ and an augmentation set ${\mathcal{T}}=\{t\mid t:{\mathbb{R}}^d\to{\mathbb{R}}^d\}$, we can define an augmentation graph ${\mathcal{G}}({\mathcal{D}},{\mathcal{T}})=({\mathcal{V}},{\mathcal{E}})$ as
\begin{itemize}
    \item we take the $N$ natural samples as the vertices of the graph, \textit{i.e.,~} ${\mathcal{V}}=\{x_i\}_{i=1}^N$;
    \item there exists an edge $e_{ij}$ between two vertices $x_i$ and $x_j$ if they are ${\mathcal{T}}$-connected.
\end{itemize}
\end{definition}

Based on these concepts, we introduce the following assumption that  with a proper choice of data augmentations, all intra-class samples could form a connected graph, as depicted in Figure \ref{fig:augmentation-graph}.
\begin{assumption}[Intra-class Connectivity]
Given a training set ${\mathcal{D}}_u$, there exists an appropriate augmentation set ${\mathcal{T}}$ such that the  augmentation graph ${\mathcal{G}}({\mathcal{D}}_u,{\mathcal{T}})$ is class-wise connected, \textit{i.e.,~} $\forall\ k\in\{1,\dots,K\}$, the subgraph ${\mathcal{G}}_k$ (graph ${\mathcal{G}}$ restricted to vertices in class $k$) is connected.
\label{assumption:intra-class connectivity}
\end{assumption}
Comparing to \cite{arora} and \cite{jason} that require (nearly) conditional independence $p(x,x^+|y)=p(x|y)p(x^+|y)$, ours only requires the connectivity of intra-class samples as in Figure \ref{fig:positive-samples}, and does not need them to be conditionally independent.

To make this analysis technically simpler, we make another assumption that we can align positive samples perfectly by minimizing the InfoNCE loss. In practice, the alignment loss can typically be minimized up to a small error $\varepsilon$, and we have appended a more involved discussion of this weak alignment scenario in Appendix \ref{sec:appendix-weak-alignment}. For now, we focus on the simplified perfect alignment scenario.
\begin{assumption}[Perfect Alignment]
At the minimizer $f^\star$ of the InfoNCE loss, we can achieve perfect alignment, \textit{i.e.,~}
$\forall\ x,x^+\sim p(x,x^+), f^\star(x)=f^\star(x^+)$.
\label{assumption:perfect-alignment}
\end{assumption}
\begin{prop}
Under Assumptions \ref{assumption:intra-class connectivity} \& \ref{assumption:perfect-alignment}, by minimizing the InfoNCE loss we can conclude that the conditional variance terms vanish {at the minimizer $f^\star$}, \textit{i.e.,~}
\begin{equation}
 \var(f^\star(x)\mid y)=0.
\end{equation}
\label{prop:variance-vanish}
\end{prop}
\vspace{-0.1in}
Intuitively, for samples in each class $k$, if the corresponding subgraph ${\mathcal{G}}_k$ is connected, there exists a path connecting every intra-class pairs $(x_i,x_j)$, as shown in Figure \ref{fig:augmentation-graph}. Consequently, aligning the positive pairs will also align all samples on the path, and eventually align $x_i$ and $x_j$. In this way, all intra-class samples can be clustered together and the intra-class variance shrinks to zero (under Assumption \ref{assumption:perfect-alignment}). Besides, because proper data augmentation will not cause inter-class augmentation overlap (Assumption \ref{assumption:label-consistency}), inter-class samples can be well separated with the uniformity term. As a result, we can attain alignment of intra-class samples while maximizing the uniformity of inter-class samples. According to Theorem \ref{thm:general-generalization-gap}, we will have an asymptotically closed generalization gap (with more negative samples $M\to\infty$) for the encoder that minimizes the contrastive loss.
\begin{theorem}[Guarantees for the Optimal Encoder]
If Assumption \ref{assumption:label-consistency}, \ref{assumption:intra-class connectivity} \& \ref{assumption:perfect-alignment} hold and $f$ is $L$-smooth, then, for the minimizer $f^\star=\argmin{\mathcal{L}}_{\rm NCE}(f)$, its classification risk can be upper and lower bounded by its contrastive risk as
\begin{equation}
\begin{aligned}
&{\mathcal{L}}_{\rm NCE}(f^\star)-{\mathcal{O}}\left(M^{-1/2}\right)\leq&{\mathcal{L}}_{\rm CE}^{\mu}({f^\star})+\log(M/K)\leq {\mathcal{L}}_{\rm NCE}(f^\star)+ {\mathcal{O}}\left(M^{-1/2}\right).
\end{aligned}
\end{equation}
\label{thm:closed-generalization-gap}
\end{theorem}
We note that different to previous bounds that hold for any $f\in{\mathcal{F}}$ as in Theorem \ref{thm:general-generalization-gap}, our results here only stand for the minimizer of the contrastive loss $f^\star$. This indicates that the InfoNCE loss alone cannot simply guarantee good downstream performance, and the learning dynamics matters for the contrastive learning to learn useful features.

\subsection{\textcolor{black}{Rethinking the Role of Data Augmentations}}

Our analysis above suggests a new understanding of the role of data augmentations in contrastive learning. Conventionally, the success of contrastive learning is usually attributed to learning {invariance} \textit{w.r.t.~} various data augmentations by matching positive examples. However, as shown in Proposition \ref{prop:wang-couterexample}, matching positive pairs alone is theoretically inadequate to learn useful features. Indeed, assuming that an ideal encoder that possesses invariance \emph{a priori} does exist, like invariance to translation (CNNs), rotation \citep{rotation}, and scaling \citep{scale}, do we obtain class-discriminative features simply by random initialization? Still NO, since these low-level properties are independent of high-level class information that we want to learn. Thus,
the reason why contrastive learning works {cannot simply be attributed to the invariance learning principle}.

We instead believe that the role of data augmentation is to create a certain degree of ``chaos'' between different intra-class samples (Figure \ref{fig:positive-samples}) such that they become more alike (or formally, ${\mathcal{T}}$-connected). In this way, the chaos serves as a ``ladder'' for bridging intra-class samples together when labels are absent, and the mission of the contrastive loss is to ``climb this ladder'', that is, aligning intra-class samples by aligning the overlapped positive samples, as shown in Figure \ref{fig:augmentation-graph}. Therefore, from our perspective, instance discrimination by contrastive learning is actually \textbf{a surrogate} for the classification task, and the surrogate {can} complete its misson when the ladder of chaos is complete (or formally, when intra-class connectivity holds).

\section{\textcolor{black}{Quantifying} the Influence of Augmentation Strength}

We have shown that with appropriate augmentations, we can derive guarantees on downstream performance. However, in practice, as illustrated in Figure \ref{fig:aug-strength}, there could be cases where augmentations are either too weak (intra-class features cannot be clustered together as in Figure \ref{fig:wang-couterexample}) or too strong (inter-class features will also collapse to the same point) and lead to sub-optimal results.
In this section, we further provide a quantitative analysis of how different strength of data augmentation will affect the final performance, both theoretically and empirically.

\subsection{Characterization on Random Augmentation Graph}
\label{sec:random-augmentation}

\begin{figure}
    \centering
    \includegraphics[width=\textwidth]{figures/tsne_random_new.pdf}
    \caption{t-SNE visualization of features learned with different augmentation strength $r$ on the random augmentation graph experiment. Each dot denotes a sample and its color denotes its class.}
    \label{fig:random-tsne}
\end{figure}

In practice, there are various data augmentation types that are hard to be described precisely. For the ease of analysis, we consider a simple case where for each class $k$, there are  $N$ samples uniformly distributed around the cluster center $c_k$ on a hypersphere ${\mathbb{S}}^d$. We then augment each sample $x_i$ with random samples in a hyper-disk of radius $r$ on the hypersphere.

In Appendix \ref{sec:theoretical-random-graph}, we provide theoretical analysis on how different augmentation strength (measured by $r$) will affect the connectivity of the augmentation as a function of the number of samples $N$, the position of the cluster centers $c_k$ and input dimensions $d$. {In particular, the minimal $r$ for the graph to be connected decreases as $N$ increases, so large-scale datasets can bring better connectivity. Meanwhile, the required $r$ also increases as $d$ increases, so we need more samples or stronger augmentations for large-size inputs.
}
Here, we show our simulation results by applying contrastive learning to the problem above. From Figure \ref{fig:random-tsne}, we can see that when $r=0$ (no augmentation), the features are mixed together and hardly (linearly) separable, which corresponds to the under-overlap case in Figure \ref{fig:aug-strength}. As we increase $r$ from $0$ to $0.1$, the features become more and more discriminative. And when $r$ is too large {($r=1.5$)}, the inter-class features become mixed and inseparable again (over-overlap). {In Appendix \ref{sec:visualization-augmentation-graph}, we  provide visualization results of the augmentation graphs, which also align well with our analysis.}
Overall, our theoretical and {empirical} discussions verify our theory that intra-class augmentation overlap with a proper amount of data augmentation is crucial for contrastive learning to work well.

\begin{figure}[t]
    \centering
    \subfigure[ACR {\bm{s}} aug-strength r.]{
    \label{fig:ACR-augment-strength}
    \includegraphics[width=0.3\textwidth]{gaussian_graph/figures/simclr_ACR.pdf}}
    \subfigure[ACR while training (r=0.01).]{
    \label{fig:ACR-r-0-01}
    \includegraphics[width=0.3\textwidth]{gaussian_graph/figures/training_process_ch.pdf}}
    \subfigure[ACR while training (r=0.92).]{
    \label{fig:ACR-r-0-92}

    \includegraphics[width=0.3\textwidth]{gaussian_graph/figures/training_process_ch1.pdf}}
    \caption{(a) Average Confusion Rate (ACR) and downstream accuracy {\bm{s}} different augmentation strength (before training). (b,c): ACR and downstream accuracy while training. }

    \label{fig:ACR}
\end{figure}

\subsection{New Surrogate Metrics for Augmentation Overlap}
\label{sec:surrogate-metrics}

From our theory and analysis above, we see that the augmentation overlap between intra-class samples indeed matters from contrastive learning to generalize better. Inspired by this, we propose the Confusion Ratio metric as a measure of the degree of augmentation overlap. Specifically, for an unlabeled dataset ${\mathcal{D}}_u$ with $N$ samples, we randomly augment each raw sample $x_i\in{\mathcal{D}}_u$ for $C$ times, and get an augmented set $\widetilde{{\mathcal{D}}}_u=\{x_{ij},i\in[N],j\in[C]\}$. Then, for each $x_{ip}\in\widetilde{\mathcal{D}}_u$ that is an augmented view of $x_i\in{\mathcal{D}}_u$, denoting its $k$-nearest neighbors in $\widetilde{\mathcal{D}}_u$ in the feature space of $f$ as ${\mathcal{N}}_k(x_{ip},f)$ and other augmented views from the same image as ${\mathcal{C}}(x_{ip})=\{x_{ij},j\neq p\}$, we can define its Confusion Ratio (CR) as the ratio of augmented views from \textbf{different} raw samples in its $k$-nearest neighbors,
\begin{equation}
\operatorname{CR}(x_{ij},f)=\frac{\#[{\mathcal{N}}_k(x_{ip},f)\setminus {\mathcal{C}}(x_{ip})]}{\# {\mathcal{N}}_k(x_{ip},f)}\in[0,1].
\end{equation}
We also define its average as Average Confusion Ratio (ACR):
\begin{equation}
\operatorname{ACR}(f)=\mathbb{E}_{x_{ij}\sim\widetilde{\mathcal{D}}_u}\operatorname{CR}(x_{ij,f}).
\end{equation}
When augmentation overlap happens, the nearest neighbors could be augmented views from a different sample, leading to a higher ACR. Thus, ACR measures the degree of augmentation overlap, and a higher ACR indicates a higher degree of augmentation overlap. Here we take $k=1$ by default.

{Here, to measure the augmentation strength in real-world datasets, following the common practice \citep{simclr}, we adopt the $\rm RandomResizedCrop$ operator with scale range $[a,b]$ for data augmentation, and we define its strength of augmentation as $r = (1-b) + (1-a)$ (a comparison with other kinds of augmentations, \textit{e.g.,~} color jittering, can be found in Appendix \ref{sec:evaluation-augmentation}).}
As shown in Figure \ref{fig:ACR-augment-strength}, ACR (augmentation overlap) indeed increases with the strength of data augmentations, and only a moderate ACR achieves the best accuracy, which is consistent with our theory discussed above. Besides, we also plot the change of ACR along the training process in Figure \ref{fig:ACR-r-0-01} \& \ref{fig:ACR-r-0-92}. We can notice that for weak augmentations, the initial ACR is low, and it rapidly decreases to zero and seldom changes while training, which leads to poor test accuracy. Instead, with proper augmentations, the initial ACR is higher, and it gradually decreases to zero and obtains good accuracy. This is also consistent with our theory that we need a certain amount of augmentation overlap for contrastive learning to work well. At the beginning, this will lead to a higher ACR, but as training continues,
better alignment (lower ACR) will help bring up the test accuracy.

\textbf{Average Relative Confusion (ARC).} In the discussion above, we notice that ACR itself does not indicate the test accuracy, but the relative change of ACR before and after training can be used as such an indicator. A large change of ACR means a large change of augmentation overlap, which indicates that the contrastive loss can actually cluster intra-class samples together through overlapped views. Based on this observation, we propose Average Relative Confusion (ARC) as
\begin{equation}
\operatorname{ARC}=\frac{1-\operatorname{ACR}(f_{\rm final})}{1-\operatorname{ACR}(f_{\rm init})},
\end{equation}
a ratio calculated with the initial ACR of the initialized model $f_{\rm init}$ and the final ACR of the pretrained model $f_{\rm final}$.
A higher ARC indicates that the contrastive learning process faces a hard task (augmentation overlap) at the beginning (high initial ACR), while successfully clustering intra-class samples with good alignment of positive samples at the end (lo final ACR).
Therefore, a higher ARC score should correspond to higher downstream accuracy.

\begin{figure}[t]
    \centering
    \subfigure[CIFAR-10 (SimCLR)]{

    \includegraphics[width=0.235\textwidth]{gaussian_graph/simclr_mixup.png}}
    \subfigure[CIFAR-10 (BYOL)]{

    \includegraphics[width=0.235\textwidth]{gaussian_graph/BYOL_mixup.png}}
    \subfigure[CIFAR-100 (SimCLR)]{

    \includegraphics[width=0.235\textwidth]{gaussian_graph/simclr_mixup_cifari100.png}}
    \subfigure[STL-10 (SimCLR)]{

    \includegraphics[width=0.235\textwidth]{gaussian_graph/simclr_mixup_stl10.png}}

    \caption{Average Relative Confusion (ARC) and downstream accuracy {\bm{s}} different augmentation strength on different datasets (CIFAR-10, CIFAR-100, and STL-10) with different contrastive learning methods: SimCLR \citep{simclr} and BYOL \citep{BYOL}.}
    \label{fig:ARC}
    \vspace{-0.1in}
\end{figure}

\begin{figure}[t]
    \centering
    \subfigure[$k=10$ ]{

    \includegraphics[width=0.3\textwidth]{gaussian_graph/simclr_mixup_k=10.png}}
    \subfigure[$k=20$ ]{

    \includegraphics[width=0.3\textwidth]{gaussian_graph/simclr_mixup_k=20.png}}
    \subfigure[$k=100$]{

    \includegraphics[width=0.3\textwidth]{gaussian_graph/simclr_mixup_k=100.png}}

    \caption{Average Relative Confusion (ARC) and downstream accuracy {\bm{s}} different augmentation strength on CIFAR-10 (SimCLR) with different number of nearest neighbors $k$.}
    \label{fig:ARC-k}
    \vspace{-0.15in}
\end{figure}

As shown in Figure \ref{fig:ARC} \& \ref{fig:ARC-k}, as augmentations become stronger, ARC scores indeed align well with the change of downstream accuracy across 1) different datasets, 2) different contrastive methods, and 3) different choices of $k$. This justifies our understanding of contrastive learning through augmentation overlap.
Meanwhile, as the calculation of ARC only involves unsupervised data, it could serve as a good surrogate metric for evaluating contrastive learning without using labeled data. Compared to previous evaluation methods like linear classification (Eq.~\ref{eq:linear-evaluation}), our ARC metric is more preferable as 1) it is theoretically motivated; 2) it does not need labeled data; 3) it does not need to learn additional modules like linear classifiers or rotation tasks \citep{selfaugment}. More experimental details can be found in Appendix \ref{sec:additional-experiments}.

\section{Conclusion}
In this paper, we have proposed a new understanding of contrastive learning through a revisiting of the role of data augmentations. In particular, we notice the aggressive data augmentation applied in contrastive learning can significantly increase the augmentation overlap between intra-class samples, and as a result, by aligning positive samples, we can also cluster inter-class samples together. Based on this insight, we develop a new augmentation overlap theory that could guarantee good downstream performance without relying on conditional independence and obtain asymptotically closed gaps. With this perspective, we also characterize how different augmentation strength affects downstream performance with both random graphs and real-world datasets. Last but not least, we also develop a new surrogate metric for evaluating contrastive learning without labels and show that it aligns well with downstream performance. Overall, we believe that we pave a new way for understanding contrastive learning with insights on the designing of contrastive methods and evaluation metrics.

\section*{Acknowledgement}
Yisen Wang is partially supported by the National Natural Science Foundation of China under Grant 62006153, Project 2020BD006 supported by PKU-Baidu Fund, and Huawei Technologies Inc.
Jiansheng Yang is supported by the National Science Foundation of China under Grant No. 11961141007.
Zhouchen Lin is supported by the NSF China (No. 61731018), NSFC Tianyuan Fund for Mathematics (No. 12026606), Project 2020BD006 supported by PKU-Baidu Fund, and Qualcomm.

\bibliographystyle{iclr2022_conference}

redpajama_set_name: RedPajamaArXiv


text:
Bonham retires Danny Darwin's jersey
BONHAM — Forty years ago, baseball existed in a different state than today in Bonham.
The Warriors remained in an extended postseason drought, as reaching the high school playoffs then in Texas required winning a district title.
Danny Darwin, former major league pitcher and Bonham native, welcomes the way Texas high school baseball is today with more chances to reach the postseason.
"I think all kids should experience being in the playoffs," said Darwin. "I got to experience it with two teams in the majors, with Houston vs. the Mets in 1986 and with San Francisco vs. Florida in 1997 and in the minors. Both of those teams (we faced) ended up winning it all.
"Even then, the little kid in you still comes out. I like the idea of giving kids the chance to experience something they might never again experience."
Though the high school baseball playoffs will not start for another several weeks in Texas, those attending Friday's home game in Bonham will get to have a unique experience.
Darwin's No. 3 Bonham jersey will be retired in a ceremony at 7 p.m., prior to the District 12-3A game against Anna.
"We had board members present us with the idea to look into retiring his jersey," Bonham athletic director and head football coach Cody Bounds said. "It wasn't just my and Dr. (Marvin) Beaty's idea."
Darwin will receive a framed Warriors baseball jersey and a new banner will be displayed in his honor on the fence in the outfield at the Bonham High School baseball field, according to a BISD press release.
"Any time you receive an award like this, I consider it an honor," said Darwin, a 1974 Bonham graduate. "When you have something retired, it's not just your number, it's also for the guys you played with. It's an award I share with (them). It takes a team effort to get any kind of award.
"Back then, we had a lot of good athletes and always thought we could win each time we went out. I never played on a team that won district, but the group that I played with, they were all supportive of me."
Darwin, whose professional career spanned 20 years (1978-98) and eight teams — Texas Rangers, Milwaukee Brewers, Houston Astros, Boston Red Sox, Toronto Blue Jays, Pittsburgh Pirates, Chicago White Sox and San Francisco Giants — specifically credited a few others with helping him achieve success in baseball.
"My dad was there through my whole career, little league through high school, along with Les Partridge. Those two were very instrumental in my development," he said. "They loved kids and youth athletics. Having them there to teach and help me get better was a blessing."
Darwin's route to the majors began unintentionally.
"Glenn Gary (my teammate) invited me to try out at Grayson (County College) with him," said Darwin. "He pushed me to do it. I had no desire to do so. I thought I pitched well enough.
"I was never offered a scholarship except from SMU. It was tough back then to get a baseball scholarship. Glenn talked me into trying out for the team at Grayson. I pitched my freshman year and took off my second year. I guess I got bigger and stronger (my sophomore year), and the rest, as they say, is history."
Though 40 years have passed since Darwin last pitched for Bonham, memories of him remain.
"We are so excited and honored to have Danny Darwin coming back to Bonham, where it all started," Bounds said. "Mr. Darwin was a hard-nosed, tough and fiery athlete, and that is exactly the type of athletes we are trying to produce in Bonham today.
"Any time you can have a person like Danny Darwin around our young athletes and let them see him, talk to him and just be around him is a really good thing. This ceremony is long overdue and I am personally excited to get to be a part of it."

redpajama_set_name: RedPajamaCommonCrawl


Apple 4S went on sale after the death of Steve Jobs. People buy the latest iPhone as a tribute to the former Apple boss.
Hundreds of fans queued around city blocks in Sydney and Tokyo so that they could be one of the first to get their hands on the iPhone 4S, which has some resemblance with the previous iPhone 4 but has a better camera, faster processor and well-received voice galvanized software.
Haruko Shiraishi waited along with her yorkshire terrier Miu Miu at the very end of an eight block queue in Tokyo's smart Ginza shopping district so that she could own an Iphone. She says she wanted to buy the IPhone so that she would have something to remember Steve.
Australian Tom Mosca, the first to buy the new phone in Sydney, noted that the first thing that she would ask her his new white iPhone is the location of Steve Jobs. A lot of people believed that iPhone 4S created as a tribute for Steve.
Apple CEO Tim Cook and his executive team hope that the new iPhone launched without the firm's former visionary leader at the helm, is going to protect their global market share against competitors including Samsung.
The South Korean firm, Apple's arch-rival with smartphones powered by Google's Android software, expect it would be overtaking their rival as the world's biggest smartphone vendor when it comes to the units sold in the third quarter.
The iPhone 4S – presented to the world after the death of Jobs — got hailed as a disappointment since it fell short of being a revolution in design, but flattering reviews centered around its "Siri" voice-activated software have since aided it set a record pace in initial, online sales orders.
Apple's Asian fans did not get disappointed with their new phones, ahead of sales in Germany, France, Britain and North America.
However, despite the entire craze at Apple stores, the launch marred somewhat by extensive complaints on the Internet about issues such as downloading iOS 5 – the current version of Apple's mobile software.
The vast majority of iPhone 4S buyers at the Sydney store seemed to be past Apple customers. Many of them have bought an Apple iPhone before, and only upgrades each time are a new iPhone.
To feel the presence of Steve Jobs some of Apple fans in Sydney and Tokyo put some put up some flower, candle and photo shrines to the late Apple boss built outside the stores.


text:
Empire State Building gunfire rattles Fil-Canadian couple
Posted: August 24th, 2012 ˑ Filled under: Crime ˑ 2 Comments
Police Commissioner Ray Kelly at a press conference. Photo by Elton Lugay
By Elton Lugay
Ramon and Jessica from Canada were walking down Fifth Avenue on a warm late summer morning on August 24, but ended up crouched on the sidewalk and shaking when they heard gunshots. (The couple declined to be photographed and to give their surnames.)
They were about to cross toward the Empire State Building (ESB) when they saw a commotion of people running toward them.
"We were shocked, we paused like in the middle of the street," Jessica told The FilAm. "We were pushed, Ramon fell, I had to pick him up."
Shortly after 9 a.m. a man identified as Jeffrey Johnson, reportedly shot his office manager and others outside the ESB. Both of them died and 10 others were injured.
"He killed one person, and at least nine other people were shot," said Mayor Mike Bloomberg at a press conference. "And some may have been shot accidentally by police officers who responded immediately and, while confronting the suspect and fatally shooting him."
The injured are being treated at nearby hospitals and are expected to recover.
Police Commissioner Ray Kelly said Johnson, a laid-off employee of a clothing company, attempted to flee by hiding his gun in a black bag that he carried under his arm.
"A construction worker who had followed Johnson from West 33rd Street alerted two uniformed police officers who were on post in front of the Empire State Building's Fifth Avenue entrance as part of our counterterrorism coverage. As the two officers approached Johnson, he pulled his .45-caliber semiautomatic pistol from his bag and fired on the officers who returned fire, killing him," Kelly said at a press conference.
Bloomberg maintained that New York City may be the "safest big city" in the country, but is not immune to the national problem of gun violence.
"Every day, our police officers put their lives on the line to protect us. They did so again today, responding immediately, and they were joined by a number of civilians whose bravery and assistance probably also saved lives," he said.
This is not Ramon and Jessica's first trip to New York. Their last trip was in 2006 — with their then 13-year-old daughter – but did not have the opportunity to visit the ESB. They thought maybe they could do it this time.
"This shooting is definitely terrifying. Ngayon babalik na kami sa hotel, leaving Monday pa pero medyo nakaka-trauma, ayaw na muna siguro naming lumabas," said Jessica.
The ESB is the home of The Filipino Repoter news weekly. Sports Editor L.P. Pelayo said the newspaper's employees move in and around the building every day.
"We pray for the persons injured in today's incident. Our staff is at the area each day. Thank goodness, we are all safe," he said.
The scene outside the Empire State Building
M. Matthews wrote:
A very good report Elton. This was a major breaking story today, August 24, 2012.
Elton Lugay wrote:
Thanks for the compliment Mike, I appreciate it.


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Sapanca is 71 m above sea level and located at 40.69° N 30.27° E. Sapanca has a population of 25224. Local time in Sapanca is +03.
Light rain (total 6mm), mostly falling on Sat afternoon. Very mild (max 13°C on Mon afternoon, min 3°C on Sun night). Wind will be generally light.
Some drizzle, heaviest during Wed morning. Warm (max 22°C on Wed afternoon, min 9°C on Tue night). Wind will be generally light.
Mostly dry. Warm (max 30°C on Sat afternoon, min 14°C on Mon night). Wind will be generally light.
* NOTE: not all weather stations near Sapanca update at the same time and we only show reports from the nearest stations that are deemed current. Weather reports submitted by any ship (SYNOP) that is close to Sapanca within an acceptable time window are also included.
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God, do I love Adventure Time. My only regret is that I am not ten years old right now so that I can watch it with even more child-like glee. Remember the old ninja turtles cartoon where two of the turtles had swords that they somehow never found cause to use? Finn has swords and by God, he uses them. In one episode, he cuts a dragon in half lengthwise, from face to tail. It is such a striking example of old-school toughness that I am surprised it made it on the air, but that is Adventure Time for you.
Also: The dragons in this world are the weirdest things I have ever seen, all bulgy and moist, with ventricle-filled sacs and visible butts. I don't think I'd have it in me to stab one in the head, but then I am not exactly the heroic type.
Then again, how heroic is it to slay a dragon? It is perhaps a bit ambiguous. Jake's surprised expression indicates the fine line between a courageous act and cold-blooded murder. Regardless, we can all bask in the glow of a true man of action, as certain of his place in the world as a dog with a dead duck in his mouth. Congratulations, Finn.
SCORE: A pile of bacon pancakes.
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Model rocketry began as a Weld County Fair 4-H project in Galeton, Colo. in the late 1960s. Being a state 4-H project, Weld County was inclined to adopt it as a county project.
Model rockets are constructed from safe materials such as cardboard, plastic and wood. They are fueled by single-use motors, which are commercially manufactured and typically contain a parachute, streamer or other recovery device that allows them to land gently for later re-flight.
Currently, model rocketry has five levels. They used to have six, but Rouse said the level five rocket was the hardest for the kids to achieve because Estes Industries was not building anything. So now, they go from level one through four, skip five, then go onto six.
Levels one and two focus on learning the theories behind model rocketry, such as Newtons Law. Kids learn about trigonometry and other physics laws as well. At the end of level one, the kids are expected to be able to sit with a judge and point to what the different parts of the model rockets are. Levels three and four focus on types of engines, building the rocket, and how to launch the rocket. Level six allows 4-Hers to design and build their own rocket!
"It's fun to watch the kids grow. We have kids that were building rockets in 4-H and have come back as parents to help their kids build rockets," Rouse said.
Rouse recalls that when he started helping with model rocketry, they used to judge the rockets and immediately launch the rockets east of the fair grandstands. This was before Fly Day competitions became a Weld County Fair tradition.
Rouse encourages 4-Hers to build two rockets each year; one is their show rocket for the demonstration and exhibition interview that is put on display during fair week while the other is launched at the Fly Day competition.
Rouse said there have been many rockets that they haven't found over the years after launching them. They have lost rockets to irrigation ditches, trees, alfalfa fields and corn fields. But most of them come back down close to the Launchpad.
Model rocketry was an open class exhibition for children and adults until 2006 when it was restricted to children between the ages of 8 to 12 or 13 to 18 only. There are two classes so that the kids are competing against others whom are close to their age. But Rouse wishes it were still for adults as well and believes the hobby has value for parents whose kids are interested in model rocketry.
Rouse also explained that model rocketry is well-liked by girls, and often, they walk away winners at 4-H events.
Rouse also encourages his 4-Hers to complete one other project aside from model rocketry; whether that be showing animals or animal science.
Teaching the children safety is one of the most important parts of model rocketry. Rouse tells each model rocketeer that they must turn 365 degrees at the Launchpad and look up at the sky to make sure there are no flying aircraft present before launching their rocket. This includes the feathered kind.
"Due to all of the water we have here, there are a lot of pelicans around. I tell the kids that they need to make sure there are no pelicans flying overhead because we don't want to shoot them down," he chuckled.
As for the number of children interested in model rocketry as a 4-H project, Rouse said Weld County has around 40 to 50 kids; which is more than most counties in Colorado. And some of them go on to have careers related to their roots in 4-H model rocketry.
Although nothing special is planned related to model rocketry at this year's fair, Rouse wants to impress upon the kids that this is the 100th Weld County Fair and that the fair has come a long way since 1918. Most importantly, Rouse loves to see the children learn about model rocketry and see the joy they get from launching their rockets.
"Even though the kids are launching smaller rockets, they act as if their rockets are big," Rouse laughed!
Come see the model rocket exhibits at the 100th Weld County Fair, July 25 through 30 at Island Grove Regional Park in Greeley.
To learn more about model rocketry, visit www.weldcountyfair.com.
Rouse initiated the rocket launch program when he started his career as a Weld County 4-H Club Model Rocketry Project Leader since many kids would bring their rockets to the fair having never launched them. Cronquist said Rouse's lighthearted ability to attract and educate 4-H youngsters is truly rare.
The Weld County Fair thanks Rouse for his continued service to the youth of Weld County through guiding and mentoring them in the 4-H model rocketry program!


Richards Benefits & Financial Services main office is in Beaver Dam and is located in the BMO Harris Bank on Park Avenue. There is public parking off of East Third Street behind our office and also on the west side of the building. We also have licensed agents in Oshkosh, Watertown & West Bend.


whyitsme 7 years, 8 months ago.
I just finished my first bottle of yeast fighters (by Twin Lab), and found it to be very effective. It is very strong, so you may want to work up to the 5 pill dose. Was taking just a couple pills for more maintenance. Seems to have a good combo of herbs, fiber and even probiotic. (also has garlic, so of course after 5 pills you keep burping the garlic. But this is the only downside).I originally bought at fred meyer for about $15 for 75 pills (20 day supply at full dose), but found for under $7 on amazon for next purchase.
So if you want to try something else for a fungicide, I'd recommend it.
The topic 'Yeast Fighters pill' is closed to new replies.


The University of Arizona Museum of Art (UAMA) presents an exhibition entitled, Reality is a Good Likeness, from June 20 through December 1.
Since 2005, Patricia Carr Morgan's work has focused on constructions of reality and cultural myth. Her provocative work allows the viewer to explore reality, myth, perception and memory through the lens of her camera.
The three series highlighted in this exhibition take on some of the perceptions that fuel the American psyche.
Out of the Past uses classic film noir as backdrop, while the new images trace the history of photographic process from the beginning to the digital age. united states (title purposely in lower case) looks at iconic Western films that have shaped the myth of our American past. In Alligator Balls and Cotton Candy, Morgan explores how county fair images from across the country illuminate the great breadth and diversity of interests and aspirations of people in different regions, who nevertheless all represent what it means to be American.
An artist's reception at UAMA is scheduled for October 24 from 5:00 – 7:00 p.m. The reception is included with the price of admission. It is free to UAMA members and anyone with a CatCard.
Click here to read the exhibition review in Caliente!
TEACHERS, click here for helpful resources!


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Entrepreneurs Checklist
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Corinne Marie Pouliquen Elected to MSBA Agricultural Law Section
Corinne Marie Pouliquen of Berenato & White, LLC was elected to the Maryland State Bar Association (MSBA), Agricultural Law Section Council as a Member-at-Large, effective July 1, 2016. The Agricultural Law Section serves the interests of the agricultural and legal community in Maryland by providing a resource to legal professionals and others with regard to to agricultural matters, by monitoring applicable laws and proposed legislation that may impact agriculture, agribusiness, or aquaculture in Maryland, and by disseminating information about current developments in the area of agricultural law. As a former veterinary technician and long-time horse owner and farm owner in Maryland, Ms. Pouliquen intends to advance the interests of the agricultural community in Maryland, especially the horse industry and hobby farmers. Ms. Pouliquen is also a member of the Maryland Horse Industry Board, The Potomac Valley Dressage Association, Trail Riders of Today, Tuckahoe Equestrian Center, League of Maryland Horsemen, Chesapeake Plantation Walking Horse Club, United States Dressage Federation and the United States Equestrian Federation.
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Berenato & White, LLC is one of Maryland's most trusted names in Intellectual Property (IP) matters, providing counseling and litigation services to clients in the U.S. and around the world. Whether you are a start-up company, a spin-off, an individual inventor or a medium or large corporation, our attorneys are here to serve you. At Berenato & White, we tailor solutions to each individual client's specific needs and requirements. For more information about the firm, please explore our website or complete our Contact Us form.
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Fatma Al-Nabhani (árabe: فاطمة النبهاني ; nació el 20 de mayo de 1991 en Mascate) es una jugadora de tenis omaní.

Su mejor clasificación en la WTA fue la número 362 del mundo, que llegó el 4 de octubre de 2010. En dobles alcanzó número 238 del mundo, que llegó el 28 de abril de 2014. Hasta la fecha, ha ganado cuatro individuales y cinco títulos de dobles en el ITF tour.

Al-Nabhani hizo debut WTA Tour en los campeonatos del tenis de Dubái 2009, después de haber recibido un comodín con Magali de Lattre en el torneo de dobles, pero perdieron con Chan Yung-jan y Sun Tiantian en la primera ronda.

Jugadora por Omán en la Copa Federación, Al-Nabhani tiene un récord de ganados y perdidos de 9-6.

Títulos ITF

Individual (10)

Finalista (8)

Dobles (12)

Referencias

Enlaces externos
 Fatma Al-Nabhani en la Asociación de Tenis Femenino
 Fatma Al-Nabhani en la Federación Internacional de Tenis
 Fatma Al-Nabhani  en la Fed Cup

Personas de Mascate
Tenistas femeninas de Omán
Mujeres de Omán del siglo XXI

redpajama_set_name: RedPajamaWikipedia


1909, 13¢ blue green, o.g., never hinged, beautifully centered amid large well balanced margins, rich intense color and remarkable freshness, Extremely Fine to Superb, a wonderful gem, with only eight examples grading higher; with 1998 P.F. for block, 2010 P.F. (Graded XF 90 XQ) and 2010 P.S.E. certificates, the latter Graded (XF-Sup 95, SMQ $675).
1908, 1¢ green, vertical coil, line pair, o.g., never hinged, each stamp exceptionally centered amid evenly balanced margins, strong bright color and exceptional freshness, an Extremely Fine gem, an outstanding line pair.
1909, 5¢ blue, vertical coil, o.g., lightly hinged, perfect centering with beautifully balanced margins around, deep intense color, a Superb jumbo, the only example to receive this grade with only one higher; with 2017 P.S.E. certificate Graded (Superb 98J, SMQ $725 for 98 and $1,150 for 100).
1909, 4¢ orange brown, horizontal coil, line pair, o.g., lightly hinged, wide margins and choice centering, rich color, right stamp with small natural paper inclusion at bottom (cert calls perf indent), Very Fine; with 2018 P.S.E. certificate Graded (VF 80).


Small incorporated businesses have less than one-half of 1% chance of being audited, according to the IRS. But if you're the lucky one chosen, it helps to have your ducks in a row. Much like an earthquake preparedness kit -- which you hope you never need but are mighty glad you've got – audit-ready files are worth keeping, just in case.
Of course, audits might occur for other reasons. You might be the type of business that requires independent audits (we go through one every year as a venture-backed company!).
"Let me just say that I thank my lucky stars every day for Bill.com. It has really helped during audit time. Instead of running around like a chicken with no head, collecting invoices, making photocopies, and then scanning and emailing to auditors, I simply have to log in to Bill.com and pull the invoices, without getting up from my seat. Thank you!"
Below are widely recommended steps for preparing for an audit, followed by how Bill.com users are already prepared!
Get your paperwork in order. Accountants recommend you have all your expenses and payables organized and labeled by tax year.
Bill.com Users: your paperwork is already scanned, processed and searchable. No digging deep into filing cabinets or searching for misplaced papers.
Give your auditors access. Auditors will need access to your files, which could mean access to your office and your people, a potential slowdown for your entire team.
Bill.com Users: you can give the auditor access to your Bill.com account so they can do more work for themselves rather than keeping you or your team on standby. This "read-only" access lets you accommodate auditors without risk of any payments being made or data being altered.
Keep an audit trail. Auditors will want to know if something was approved prior to payment and if you followed documented processes.
Bill.com Users: Bill.com keeps an audit trail of all approvals, changes, and payments so auditors can get what they need without having to follow a paper trail or follow up with individuals in your company.
Ensure everything is reconciled. Accounting data that's in sync with bank records helps an audit move more quickly and smoothly.
Bill.com Users: Because payments are made from Bill.com and synced with your accounting package, there is no chance of any data not reconciling.
Have all your records handy. Auditors may ask to examine canceled checks, tax returns, deposits and other business records to ensure your filing was timely and accurate. They might also review state, federal and Social Security (FICA) withholding, Medicare taxes, advance earned-income credit, unemployment compensation, workers' compensation premiums, salaries, bonuses, and tax preparer or accountant records.
Bill.com Users: Bill.com is great for storing much more than bills. You can store all your financial documents with no disk space limitations, so thy are available to auditors with a few clicks. Want a physical copy? For just $34.99, you can order a DVD of your past year's documents, far easier to hand-off to an auditor than reams of paper.
Audits are never fun. But with Bill.com they can be much easier!


The Hangar is a multi-track recording studio in a large converted warehouse space that was operated for 12 years as Enharmonik Studios, a commercial facility. We closed the studio and business in 1998 and converted the room to a private use facility. The studio is owned by producer/engineer John Baccigaluppi, also the publisher of Tape Op Magazine.
The studio is now available for lease on a daily and weekly basis to qualified recording engineers and/or musicians with a working knowledge of the studio process. We have focused on making the studio a unique and atmospheric tracking room with quality gear and lots of musical instruments at an affordable price to encourage the creative recording process.
Check out our user guide at the top of the dial for detailed information on all of our equipment and how it can translate into great records.
This site was lovingly hand-coded by Dave Middleton. Thanks for visiting.


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How we are speeding up the PhD admission process at Brunel
By Jeung Lee -
Postgraduate admissions at Brunel University London have plunged by 48% since 2013/14 so Brunel took action to change how prospective PhD students apply. Jeung Lee says their new process should cut application times by over 50%.
The College of Engineering, Design and Physical Sciences at Brunel conducted a preliminary study into the issue. They observed that the time for reviewing a PhD application was excessive and concluded that this might be contributing to a loss of qualified applicants to other universities, who may be able to provide offers more quickly.
In response, a team was assembled to undertake a 3-day Rapid Improvement Event (RIE) to redesign the process for PhD admission. The team comprised three college PhD programme managers, eight postgraduate research directors from all three colleges and members of the central admissions team.
Their brief was to deliver a standardised PhD admission process which would increase visibility and transparency using lean principles and reduce the number of days required to review an application and make an offer.
While three days was a large time commitment, this was definitely worthwhile.
Improvements made but still some problems
Once the processes were mapped out, the team identified that after the new application system came into place in October 2016, it improved the efficiency by reducing the times to two thirds for a successful offer to be made – but there were still problems.
For example, there were a number of variations of practices across academic departments, leading to an inconsistent approach to the admissions process, the system was very labour-intensive and lacked visibility – there was no technical capability to track an application, and monitoring was done manually and recorded in spreadsheets maintained by the college PhD programme offices.
Also, the central admissions team was 'in the dark' once an application was reviewed by a college. The college PhD programme offices also did not have an immediate access to the status of applications. This meant that no appropriate intervention could be made to expedite the process.
Read more: how to create a lean team
It became clear that the process needed to be redesigned, and the contribution from the academics was key to this. From the start, the academics suggested that the application system should be tailored to PhD admissions and provide 'fast track' services for applicants who have discussed their research projects with potential supervisors or whose applications are restricted by their funders.
They also emphasised that instead of rejecting qualified applications due to the mismatch of the research areas, the new process should enable them to refer applications to other department(s). Observing the various practices across the departments, they started to work together to develop a university-wide admissions process.
Time invested pays dividends
Reflecting on this experience, Dr Claire Donovan, reader in health economics, said: "While three days was a large time commitment, this was definitely worthwhile.
"The RIE allowed participants to gain the 'big picture' view of the systems and processes for PGR admissions across the whole university, and to learn the viewpoints and perspectives of the full range of stakeholders involved. We were also able to pick up on best practice in different areas, and reflect on how to improve our own approaches."
The central admissions team promptly responded to the requests to revise the application system, to ensure the right information is captured at the initial stage, thus speeding-up the process of screening applications and allocating potential supervisors.
After their brainstorming session, the team agreed to develop an internal tracking system, using eVision, a web-based interface managed by the student record system.
The internal tracking system is expected to replace the manual spreadsheets and academics' decision forms, providing easier access to applications. It will enable academics to refer applications within and outside a college. With its dashboard function, both the local college and central teams would be able to view the status of PhD applications and follow up pending tasks more efficiently.
The new process is expected to bring the following benefits:
Minimise delays and inefficiency in the application process
Manage expectations from all stakeholders with clear roles and responsibilities including keeping applicants engaged
Ensure fairness, transparency and compliance
Ensure better utilisation of resources: the tracking system can save up to £50,500 of the administrative cost per annum.
It should cut application times by over 50%.
Brunel hopes that by making the PhD application process clearer and more concise, they will see a steady increase in the number of applications they receive in the coming years.
Jeung Lee
Planning manager, Brunel University London


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Legendary Musician George Benson Returns with Guitar Man on October 4
(Los Angeles) – On October 4, 2011, National Endowment of the Arts Jazz Master and Grammy-winning legend George Benson brings his guitar to the forefront in his newest album Guitar Man. The 12-song collection includes a mix of jazz and pop standards – some in a combo setting and some solo, but all of them tied together seamlessly by Benson's soulful and exploratory signature sound. Lending a hand on this recording is a solid team made up of veterans and newcomers alike – pianist Joe Sample, keyboardist and musical director David Garfield, bassist Ben Williams and drummer Harvey Mason. Williams is one of the hottest up-and-coming new artists on the jazz scene. Mason is a regular member of Fourplay, and a studio collaborator with Benson all the way back to Benson's 1976 blockbuster album, Breezin'.
In a career that spans five decades, more than 30 recordings as a leader and ten Grammy Awards, Benson has used his jazz roots as the foundation for an engaging mix of pop, R&B and other shades that add up to a style that appeals to a broad mainstream audience. Along the way, he has also established himself as a formidable jazz singer – one whose biggest career hits have showcased his vocals. But Guitar Man is just what the title implies – an album that highlights Benson's unparalleled guitar playing, perhaps more than any other album he has released in decades.
For the new project, the crew came together in the studio with a minimum of prior rehearsal time but an eagerness to jump in and lay down tracks in something very close to the live experience – what Benson describes as an "old school" approach. The impromptu sensibility comes across in the final product, much of which came together with minimal takes in a single day of recording.
"We figured that we would get the best energy if we went into the studio with some live musicians who are savvy and flexible," says Benson, "and boy, did we accomplish that."
On Guitar Man, Benson's mastery of the guitar is demonstrated in a variety of styles and settings, all with legendary jazz roots. The opening track "Tenderly" is a solo guitar track that serves as a reminder that Benson is one guitar man with sufficient technical and interpretive skills to be a band unto himself. The second song is an intriguing rendition of Lennon and McCartney's "I Want To Hold Your Hand." Along with Benson on nylon-string guitar and Garfield on piano are guitarists Paul Jackson Jr. and Ray Fuller; bassist Freddie Washington; drummer Oscar Seaton, Jr. (who regularly tours with Benson); violinist Charlie Bisharat; and flutist/clarinetist Dan Higgins. All come together to create a fully orchestrated sound that casts one of the most simplistic of the Beatles' early love ballads into something full-bodied and engaging.
The remainder of the set consists of either solo guitar tracks or Benson backed by the aforementioned five-man team, which lays down an easygoing rendition of Stevie Wonder's "My Cherie Amour." Benson delivers the lyrics in his own engaging vocal style that includes a healthy dose of his trademark scatting guitar accompaniment.
Other highlights include a rollicking version of the Champs' 1958 instrumental hit, "Tequila," followed a couple tracks later by "My One and Only Love," which opens with a 16-bar solo jazz guitar intro that segues into a sweet vocal ballad. Benson delivers a playful straightahead rendition of "Paper Moon" with the quartet, followed by a solo guitar rendition of "Danny Boy" (one of the few times, if ever, that a guitar sounds like bagpipes). In the final stretch, Benson and Garfield set up a lush guitar-and-piano arrangement of the smoky standard, "Since I Fell For You," with Benson once again stepping up to the mic for an emotional delivery of the song's impassioned lyrics.
Benson has never been one to shy away from innovation or experimentation. For this guitar man, putting a jazz spin on pop standards – not just on this recording but throughout his career – is less of a taboo when you're willing to dispense with labels and the limitations that come with them.
"People categorize things because they need to find someplace to put them on their shelf," he says. "It's all music to me. I think a lot of pop tunes that were very big in the United States many years ago were recorded by jazz musicians playing in the background. Most of the Motown records were recorded that way. Those guys were jazz musicians who were living in Detroit and were called to do a job, and they did it very well…I try to do the same thing. I try to make it sound like it's natural, because to me it is. There are only two kinds of music, good and bad. There are a lot of things in between, but they're eventually going to fall on one side or the other of that equation."
George Benson will be touring in support of Guitar Man. The live shows will feature Benson's masterful guitar playing, and Benson and his band will be performing classic George Benson hits along with an acoustic presentation of a few songs from Guitar Man.
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International Star and Italian Crooner Patrizio Buanne Releases PATRIZIO on Concord Jazz
International Star and Italian Crooner Patrizio Buanne Releases PATRIZIO on Concord Jazz Concord Jazz is proud to announce the release of Italian singer Patrizio Buanne's newest album, PATRIZIO. Having sold over one million albums worldwid …
Aug 1, 2011Musicians: News, Recordings, & FilmConcord Music Group, Inc.Patrizio Buanne
Acoustic Alchemy Brings Its Trademark Innovation And Optimism To Heads Up Debut
Roseland marks the guitar group's maiden voyage on new label The story of Acoustic Alchemy has always been about innovation and adaptability. In the nearly 25 years since their first recording, the UK-based group has crafted a sound that m …
Aug 10, 2011Musicians: News, Recordings, & FilmHeads Up InternationalAcoustic Alchemy


Learning PHP is usually the first step in web programming. It is the largest and the most popular web programming language. It's been around since the 90's and has powered the majority of popular websites including Digg, Facebook, Yahoo and every WordPress blog. Among all the server-side language or programming language used in developing a website and web application; PHP and MySQL are the preferred choice and used by most people around the world.
PHP is a widespread HTML – also an embedded scripting language which is broadly used to create dynamic web pages rapidly and effortlessly. Full form of PHP is Hypertext Preprocessor, earlier it was known as Personal Home Page. It is open source software invented by Rasmus Lerdorf in the year 1995. It's usage can also be as general purpose programming language.
What's Eligibility to Learn PHP?
Are you looking to make a career in PHP or want to learn fundamentals or in-depth of PHP, then you have landed at right place. In my counseling career I have came various clients who want to learn PHP but are confused as to where from start. So through this write up, I would like to clarify this question. Here I have tried to render the eligibility criteria depending upon varied requirements of PHP.
First of all I would like to clarify what is PHP. PHP is the popular open source server-side scripting language widely used for producing web pages. If we put it simply," PHP language is for controlling the backend administration of a website." In-fact various open source applications like Drupal, Joomla, WordPress, Os Commerce, Magento and so on are developed in PHP MySQL itself.
Read more about What's Eligibility to Learn PHP?
How to Use PHP Filters?
I am sure you have learnt about PHP Filters in your PHP course. PHP filters are very useful in dealing with data security and data management. Data security and data management is a major concern in all PHP applications. Data management refers to the handling of data and maintaining it for use and processing of the application in PHP. Data for an application in PHP programs can have two sources, one could be the internal organization which runs the application and the other could be the external data. The data from the internal organization is easy to handle as it is pre-organized in the required formats. The main challenge faced while working with the external data. The external data could include the data from the user input, cookies, web services, server variables and database query results.
Read more about How to Use PHP Filters?


In July 1991 four adventurous souls completed a three week canoe trip on the Horton River in the Northwest Territories. Among them was long term local and President of the Whistler Museum board, John Hetherington, as well as Whistler Ski Patrolman Pat "Dago" Coulter.
The Horton River is one of the most remote rivers in North America, though it deserves great renown. The river has several distinct features: it empties into the Arctic Ocean at a point further north than any other mainland river in Canada, it runs alongside the Smoking Hills (it broke through the Smoking Hills to Franklin Bay around 1800AD, cutting off the last 120 kilometres of river), and it is now in the process of creating a new delta.
As part of the Whistler Museum Speaker Series, John Hetherington will be sharing photographs and stories from this remarkable adventure. The event, "Canoeing the Horton River," will take place on Wednesday, February 18th, from 7 to 9pm (doors are at 6pm) at the museum. Tickets are $7 each ($5 for museum members) and can be purchased by telephone or in person from Whistler Museum. There will be a cash bar and complimentary tea and coffee.
Anyone interested in the Canadian Arctic, canoeing, adventure sports, or anything to do with the Canadian wilderness should not miss this one.


Industry is a term that in general people have relatively negative associations with. It is implied by the fact that, first and foremost, it is related to being harmful for the environment, as for example diverse mines as well as factories tend to release plenty gases and other elements that minimize the standard of water and the air.
Thus, although in this area significantly has changed and more and more enterprises introduce diverse alternatives and strategies in order to decrease this impact, still plenty people find this field of economy be something that is perceived negatively.
In order to reconsider this, we ought to ask ourselves regards what the world would look like if there would be no roads, no this kind solid buildings and miscellaneous new technologies. Moreover, development of thetechnology, which is frequently considered to be harmful for Earth, in fact gives us hope that in future we would be able to more effectively protect our planet from diverse problems.
That's the reason why, if we belong to people criticizing industry and thinking that it is only a source of something harmful, bad and negative, we ought to rethink it and ask ourselves what our life would look like if there would be no cars, no motorways, no roads, no electricity etc. In such situation we might observe that the comfort of life we have these days is mostly available due to pretty visible impact of the previously analyzed topic. Nonetheless, pointing out its harmful influence on the Earth and environment is also meaningful.
In the light of the points mentioned above, the best attitude in the field of industry is to have an appropriate balance between requiring from it to be less harmful for our planet and getting to know how crucial role it has played for making our life be more comfortable. Its role is pretty visible as thanks to it we have been offered with many innovations and options that help us a lot in fulfilling different duties we have every day.
Different economists have found out that although contemporarily the most popular trend is towards increasing role of the services, a great percentage of the GDP of the significant percentage of countries on the Earth is referred to the industry. It is indicated by the fact that, above all, this area of economy is quite likely to be an important element of the existence of the mankind even in the farthest predictions for the future. It is implied by the fact that due to it we have been given with different inventions like for instance motorways, skyscrapers, cars, airplanes etc.


Born in 1987 in Karlsruhe, Chris has shown a creative talent since early age. Already at the age of 14 Chris sacrifices most of his free time for his life passion- DANCE. Chris starts getting interested in different dance styles and soon has an impressive repertoire to offer. He learns Latin dance at the Dance Sport Club-TSC Astoria Karlsruhe; hip hop at the Dance School Vollrath-Toennies; rock'n'roll at the Golden Fifties Karlsruhe. Chris has also a basic education in classic dance which he obtained at the Lagunilla & Reijerink taught by Carlos Lagunilla in Karlsruhe.
Since 2005 Chris has engrossed his skills and knowledge in Salsa, Bachata and other Afro-Latino dance styles at the Juan-Dance-Factory School in Karlsruhe. Chris had the possibility to be trained by teacher such as Eddie Albaron (Wagner), Tanja Wagner, Israel Gutierez, Anne and Anichi, Nina Uszkureit, Luis Saburini and some more. While performing with the performance team of San-Juan-Dance-Factory Chris collected stage experience. They have been performing at congresses, galas and events in different European cities Amsterdam, Stuttgart, Freiburg, Frankfurt and Karlsruhe. His dance performance has been enriched by international congresses and one on one coaching with Seo Fernandez, Fernando Sosa (Tropical Gem), Juan Matos and Adrian y Antia.
Since 2005 Chris has been regularly giving Salsa classes for groups and single students of different levels. He has been coaching the Salsa Group at the Spot Dance Club- Astoria in Karlsruhe. Chris has also given workshops in different cities across Europe and especially Germany such as Salsonic and Conecxion in Frankfurt. Currently he is in charge of La Danza in City of Cologne.
Chris likes to challenge himself and his skills ,which is of high priority to him, that is why he has founded a successful Salsa Master-class Group in Frankfurt. Together with Kris his dance partner, Chris is working currently on similar project in Cologne. He has also been involved in charity work and projects such as dance classes for criminal offenders at the JVA Frankfurt.
Since September 2008 Chris has been studying at the University of Applied Science for Music and Art at Frankfurt am Main. His daily training consists of ballet classes, modern as well as contemporary dance. Since October 2010 Chris continues his study at the University of Applied Science for Music and Dance in Cologne.
One of his new projects is the CDC (Chris Dance Company). CDC is a group of young professional dancers under the responsibility of Chris. The new project of CDC is created 2012 and shows a mixture of the musical Chicago, swing elements and contemporary salsa on2.