text:
\section{Introduction}
\IEEEPARstart{C}onditional Random Fields (CRFs) and Support
Vector Machines (SVMs) can be seen as representative of two
different approaches to classification problems. The former is
purely \emph{probabilistic} -- the conditional probability of
classes given each observation is explicitly modelled -- while the
latter is not -- classification is
performed without any attempt to model probabilities.
Both approaches have their strengths and weaknesses. CRFs~\cite{LafMcCPer01,ShaPer03}
are known to yield the Bayes optimal solution asymptomatically but do not have known tight generalisation bounds. In
contrast, SVMs have tighter generalisation bounds which typically shrink as the margin grows, and can easily incorporate interesting label-cost such as F1 score or hamming distance in structured cases. But SVMs could be inconsistent when there are
more than two classes~\cite{TewBar07,Liu07}.

Despite their differences, CRFs and SVMs appear very
similar when viewed as optimisation problems. The most salient
difference is the loss used by each: CRFs are trained using a log
loss while SVMs typically use a hinge loss.
In an attempt to capitalise on their relative strengths and avoid
their weaknesses, we propose a novel \emph{hybrid loss} which ``blends'' the two losses.
After some background (\S\ref{sec:background}) we provide the following analysis:
We argue that Fisher Consistency
for Classification (FCC) -- a.k.a.\ classification calibration --
is too coarse a notion and introduce a
distribution-dependent refinement called Conditional Fisher Consistency
for Classification (\S\ref{sec:consistency}).
We prove the hybrid loss is conditionally FCC and give a noise condition that relates
the hybrid loss's mixture parameter to a margin-like property of the data distribution
(\S\ref{sec:cfcc}).
We then show that, although FCC is effectively a non-parametric condition, it is
also a necessary condition for consistent risk minimisation using parametric models
(\S\ref{sec:parametric-fcc}).
Finally, we empirically test the hybrid loss on
various domains including multiclass classification, text chunking, human action recognition and show it consistently performs as least as well as -- and often better than -- both of its constituent losses (\S\ref{sec:experiments}).

\section{Losses}
\label{sec:background} In classification problems
\emph{observations} $x\in\Xcal$ are paired with \emph{labels}
$y\in\Ycal$ via some joint distribution $D$ over $\Xcal\times\Ycal$.
We will write $D(x,y)$ for the joint probability and $D(y|x)$ for
the conditional probability of $y$ given $x$. Since the labels $y$
are finite and discrete we will also use the notation $D_y(x)$ for
the conditional probability to emphasise that distributions over
$\Ycal$ can be thought of as vectors in $\RR^k$ for $k=|\Ycal|$.
We will also use $p$ and $q$ to denote distributions over $\Ycal$
but reserve their use for distributions generated by models.

\subsection{Multiclass Prediction}
\label{sec:multiclass}
When the number of possible labels is $k=|\Ycal|>2$ the
classification problem is known as a \emph{multiclass} classification problem.

Given $m$ training observations $S=\{(x_i,y_i)\}_{i=1}^m$ drawn i.i.d.\
from $D$, the aim of the learner is to produce a predictor $h : \Xcal
\to \Ycal$ that minimises the \emph{misclassification error} $e_D(h)
= \PP_D\left[ h(x) \ne y \right]$. Since the true distribution is
unknown, an approximate solution to this problem is typically found
by minimising a regularised empirical estimate of the risk for a
\emph{surrogate loss} $\ell$. Examples of surrogate losses will be
discussed below.

Once a loss is specified, a solution is found by solving
\begin{equation}\label{eq:erm}
    \min_{f} \frac{1}{m}\sum_{i=1}^m \ell(f(x_i),y_i) + \Omega(f)
\end{equation}
where each \emph{model} $f : \Xcal \to \RR^k$ assigns a vector of
scores $f(x)$ to each observation and the regulariser $\Omega(f)$
penalises overly complex functions. A model $f$ found in this way
can be transformed into a predictor by defining $h_f(x) =
\argmax_{y\in\Ycal} f_y(x)$ where ties are broken in some
arbitrary but deterministic way (see Section~\ref{sec:parametric-fcc} for
details).
We will overload the definition of
misclassification error and sometimes write $e_D(f)$ as shorthand
for $e_D(h_f)$.

A common surrogate loss for multiclass problems is a generalisation
of the binary class hinge loss used for SVMs~\cite{CraSin00}:
\begin{equation}\label{eq:hinge}
    \ell_H(f,y) = \left[1 - M(f,y)\right]_+
\end{equation}
where $[z]_+ = z$ for $z>0$ and is 0 otherwise, and $M(f,y) = f_y -
\max_{y'\ne y}f_{y'}$ is the \emph{margin} for the vector $f\in\RR^k$.
Intuitively, the hinge loss is minimised by models that not only
classify observations correctly but also maximise the difference
between the highest and second highest scores assigned to the
labels.

While there are other, consistent losses for SVMs~\cite{TewBar07,Liu07}, these cannot scale up to a large $k$. For example, the multiclass hinge loss $\sum_{j\neq y}[1+f_j(x)]_+$ is shown to be
consistent in \cite{Liu07}. However, it requires evaluating $f$ on all possible labels except the true $y$. This is intractable for labels where the possible assignments grow exponentially.
The other known and consistent multiclass hinge losses have similar intractability.

\subsection{Structured Prediction}
\label{sub:multiclass}
In the multiclass case $\{(x_i,y_i)\}_{i=1}^m$ are assumed i.i.d. However, in many cases, they are not i.i.d. Structured prediction \cite{BakHofSchSmoetal07} can deal with these cases by grouping correlated labels to form a structured label $y$. Here the structured label $y$ can be any object associated with $x$. For example, for the automated paragraph breaking problem, the input $x$ is a document, and the output $y$ is a sequence whose entries denote the beginning positions of the paragraphs. For image segmentation, the input $x$ is an $n_1$ by $n_2$ image, and the structured label $y$ is a 2-D lattice $\{y^{i,j}\}_{i = 1, \cdots, n_1;1= 1, \cdots, n_2}$. The framework of Probabilistic Graphical Models (PGMs) \cite{koller2009probabilistic} provides a principled way of modelling the dependencies of the components of $y$. For a $y$ with $L$ components \ie $y = (y^1,y^2, \cdots, y^L)$,  the graph of PGM $\Gcal = (\Vcal,\Ecal)$ consists of the node set $\Vcal = \{1,\cdots, L\}$ and the edge set $\Ecal$ that reflects the dependencies. Assuming each component $y^j \in \{1,\cdots,c\}$ for all $j$, there are $k = c^L$ many possible assignments for $y$. In other words, such a structured label $y$ can be seen as a multiclass problem with $k$ many classes in theory, although many multiclass algorithms will be intractable in the structured case.

Once the structured labels are formed, we can assume the structured input-output pairs $\{(x_i,y_i)\}_{i=1}^m$ are i.i.d. from some joint distribution. The predictors $h$ or the models $f$ can be learned in a fashion similar to \eq{eq:erm}.
The models are usually specified in terms
of a parameter vector $w\in\RR^n$ and a feature map $\phi :
\Xcal\times\Ycal \to \RR^n$ by defining $f_y(x;w) =
\inner{w}{\phi(x,y)}$ and in this case the regulariser is
$\Omega(f) = \frac{\lambda}{2}\|w\|^2$ for some choice of
$\lambda\in\RR$. This is the framework used to implement the SVMs
and CRFs used in the experiments described in Section~\ref{sec:experiments}.
Although much of our analysis does not assume any particular parametric model,
we explicitly discuss the implications of doing so in \S\ref{sec:parametric-fcc}.

\subsection{Probabilistic Interpretation and the Hybrid Loss}
\label{sec:background-pmodel}
CRFs are based on a model whereby
\begin{equation}\label{eq:prob}
    p_y(x;f) = \frac{\exp(f_y(x))}{\sum_{y\in\Ycal}{\exp(f_y(x))}},
\end{equation}
and use the log loss
\begin{equation*}
    \ell_L(p,y) = -\ln p_y.
\end{equation*}
This loss penalises models that assign low probability to likely labels and, implicitly, that assign high probability to
unlikely labels.

We can see that \eq{eq:prob} provides a probabilistic interpretation of the scores of $f_y(x)$.
It is easy to show that under this interpretation the hinge loss for $p = p(\cdot;f)$ is given by
\begin{equation*}
    \ell_H(p,y) = \left[1-\ln\frac{p_y}{\max_{y'\ne y}p_{y'}}\right]_+
\end{equation*}

We now propose a novel \emph{hybrid loss} that is a
combination of the hinge and log losses
\begin{equation}
    \ell_\alpha(p,y)
    = \alpha\ell_L(p,y) + (1-\alpha)\ell_H(p,y) \label{eq:hybrid}

\end{equation}
where the mixture of the two losses is controlled by a parameter
$\alpha\in[0,1]$. Setting $\alpha = 1$ or $\alpha = 0$ recovers the
log loss or hinge loss, respectively. The intention is that choosing
$\alpha$ close to 0 will emphasise having the maximum gap between the largest and
second largest label probabilities while an $\alpha$ close to 1 will
force models to prefer accurate probability assessments over strong
classification.

This family of hybrid losses is similar to a recent proposal
by Zhang et al \cite{Zhang:2009}. They also define a single parameter family of loss functions
called \emph{coherence functions} that interpolate between hinge loss and a loss that
is closely related to loss based on log-likelihood. Like the loss presented here, their
losses are surrogates for 0-1 loss and both families have the hinge loss as a limit point.
A key difference between the two proposals has to do with the consistency of losses in
each family: the coherence losses are all Fisher consistent for
probability estimation whereas the hybrid losses satisfy a weaker form of consistency
which we call conditional Fisher Consistency for Classification and analyse below.

Despite of the properties of the coherence functions, using them in structured cases is intractable. They require the evaluation of a function $\beta_j(x)$ for all classes \ie $  j = 1,\cdots, k$, see Algorithm 1 step 2 (c) in \cite{Zhang:2009}. Note that $k$ grows exponentially in structured cases. They encounter the same problem as other consistent multiclass SVMs. Our hybrid loss does not have this problem.

\section{Fisher Consistency For Classification}\label{sec:consistency}
A desirable property for a loss is that, given enough data, the models
obtained by minimising the loss at each observation will make predictions
that are consistent with the true label probabilities at each observation. We are mainly concerned with distributions
$D(x)$ over the set $\Ycal$ for some fixed (but irrelevant) $x$.
We will therefore overload $D$ and use it to denote a distribution
over $\Ycal$. Whether $D$ represents a distribution over labels or a distribution
over labels and observations should be clear from context.

We say a vector $f\in\RR^{\Ycal}$ is \emph{aligned} with
a distribution $D$ over $\Ycal$ whenever maximisers of $f$ are
also maximisers for $D$. That is, when
\(
    \argmax_{y\in\Ycal} f_y \subseteq \argmax_{y\in\Ycal} D_y.
\)
Since the probabilistic models described in \S\ref{sec:background-pmodel}
pass the components of a vector $f(x)$ through $\exp$ and rescale, it is clear
that a prediction $f(x)$ is aligned with $D$ if and only if $p_y(x;f)$ is
aligned with $D$.
Because of this correspondence, the following definitions of consistency are
equivalent regardless of whether general models and losses or their
probabilistic counterparts are used.

If, for all label distributions $D$, minimising the conditional risk
$L(f,D) = \EE_{y\sim D}[\ell(f,y)]$ for a loss $\ell$ yields a vector $f^*$ aligned
with $D$ we will say $\ell$ is \emph{Fisher consistent for
classification} (FCC)~\footnote{%
    Note that the Fisher consistency for classification is weaker than Fisher
    consistency for density estimation. The former requires the same prediction
    only,
    while the latter requires the estimated density is the same as the true data
    distribution. In this paper, we focus on the former only.
    For an analysis of Fisher consistency for density estimation, we refer
    the reader to \cite{Reid:2009b}.}
-- or \emph{classification calibrated} \citep{TewBar07}.
This is an important property for losses since it is
related to the asymptotic consistency of the empirical risk
minimiser for that loss \citep[Theorem 2]{TewBar07}.

The standard multiclass hinge loss $\ell_H$ is known to be inconsistent for
classification when there are more than two
classes~\citep{Liu07,TewBar07}. The analysis in \citep{Liu07} shows
that the hinge loss is inconsistent whenever there is an instance $x$ with
a \emph{non-dominant} distribution -- that is, $D_y(x) <
1/2$ for all $y\in\Ycal$. Conversely, a distribution is
\emph{dominant} for an instance $x$ if there is some $y$ with
$D_y(x) \ge 1/2$.
In contrast, the log loss used to train non-parametric CRFs is
Fisher consistent for probability estimation -- that is, the
associated risk is minimised by the true conditional distribution --
and thus $\ell_C$ is FCC since the minimising distribution is equal
to $D(x)$ and thus aligned with $D(x)$.

\subsection{Conditional Consistency of the Hybrid Loss}
\label{sec:cfcc}

In order to analyse the consistency of the hybrid loss we require the
following refined notion of Fisher consistency.
If $D=(D_1,\ldots,D_k)$ is a
(conditional) distribution over the labels $\Ycal$ then we say the loss
$\ell$ is \emph{conditionally FCC with respect to $D$} whenever minimising the
conditional risk w.r.t. $D$, $L(f,D) = \EE_{y\sim
D}\left[\ell(f,y)\right]$ yields a predictor $f^*$ that is
aligned with $D$. Of course, if a loss $\ell$ is conditionally
FCC w.r.t. $D$ for \emph{all} $D$ it is, by definition,
(unconditionally) FCC.

The following theorem provides sufficient conditions on the hybrid parameter
$\alpha$ in terms of a label distribution $D$ so that the hybrid loss
$\ell_\alpha$ is conditionally FCC w.r.t. $D$.
\begin{theorem}\label{thm:fcc}
    Let $D=(D_1,\ldots,D_k)$ be a distribution over $\Ycal$ and let
    $D_{max} := \max_y D_y$ be the largest probability assigned to any
    $y\in\Ycal$. Also let $\Ycal_{max} := \{ y : D_y = D_{max} \}$ be the set of
    labels with maximal probability
    and $D_{next} := \max_{y\notin \Ycal_{max}} D_y$ be
	the second largest probability assigned to a label, or $D_{next} = \infty$
	if $\Ycal_{max} = \Ycal$.
    Then the hybrid loss $\ell_\alpha$ is conditionally FCC for $D$ whenever
    $D_{max} \ge \frac{1}{2}$ or
    \begin{equation}\label{eq:alpha}
        \alpha > 1 - \frac{D_{max} - D_{next}}{1-2D_{max}}.
    \end{equation}
\end{theorem}

The proof is by contradiction and proceeds at a high level by showing that if the
distribution $D$ satisfies $D_{max} \ge \frac{1}{2}$ or \eqref{eq:alpha} but the
minimiser $p$ of the risk $L_\alpha(p,D)$ is not aligned with $D$ we derive a
falsehood.
The argument is broken into two cases: when the risk minimising distribution $p$
has a unique maximum probability and when it does not. In both cases we show how to
construct an alternative distribution $q$ (that depends on $D$) such that
$L_\alpha(q,D) < L(p,D)$, yielding the required contradiction.
In the first case, $q$ is obtained by swapping the most probable label of $p$ with
that of $D$. In the second case (when $p$ has two or more most probable labels),
$q$ is obtained by perturbing $p$ slightly towards $D$.

\begin{proof}
    Since we are free to permute the labels within $\Ycal$, we will assume without loss
    of generality that there are $t$ ties for the most probable label and that
    $\Ycal_{max} = \{1, \ldots, t\}$ and so $D_1=\cdots=D_t$.
    Defining $L_\alpha(p,D) = \EE_{y\sim D}\left[\ell_\alpha(p,y)\right]$,
    the proof now proceeds by contradiction by assuming that there is some minimiser
    $p = \argmin_q L_\alpha(q,D)$ that is not aligned with
    $D$.
    For this to occur there must be some label $y^*>t$ such that $p_{y^*}$
    is as least as large as $p_1, \ldots, p_t$.
    For simplicity, and again without loss of generality, we will assume that
    $y^*$ is the label with the largest probability according
    to $p$ (that is, $y^* \in \argmax_y p_y$).
    We are also free to have permuted labels within $\Ycal_{max}$ to ensure
    $t \in \argmax_{y\in\Ycal_{max}} p_y$.

    The first case to consider is when $p_{y^*}$ is strictly larger than
    $p_t$.
    Here we construct a new distribution $q$ that swaps the
    values of $p_t$ and $p_{y^*}$ and leaves all the other values unchanged.
    That is, $q_{t} = p_{y^*}$, $q_{y^*} = p_t$ and $q_y = p_y$ for all
    $y\in\Ycal-\{t,y^*\}$.
    Intuitively, we will now show that this new point is ``closer'' to $D$ and
    therefore the CRF component of the loss will be reduced while the SVM component of the loss
    won't increase.
    To do so, we consider the difference in conditional risks:
    \begin{eqnarray}
        L_\alpha(p,D) - L_\alpha(q,D)
        &=& \sum_{y=1}^k D_y.(\ell_\alpha(p,y) - \ell_\alpha(q,y)) \nonumber \\
        &=& D_{t}.(\ell_\alpha(p,t) - \ell_\alpha(q,t)) \nonumber\\
        &&  + D_{y^*}.(\ell_\alpha(p,y^*) - \ell_\alpha(q,y^*)) \nonumber \\
        &=& (D_{t} - D_{y^*})(\ell_\alpha(q,y^*) - \ell_\alpha(q,t)) \nonumber
    \end{eqnarray}
    since $\ell_\alpha(p,t)=\ell_\alpha(q,y^*)$ and
    $\ell_\alpha(p,y^*) = \ell_\alpha(q,t)$ and the other terms cancel by
    construction.
    Since, by assumption $y^* > t$, we have $D_t - D_{y^*} > 0$, so all that
    is required now is to show that
    $\ell_\alpha(q,y^*) - \ell_\alpha(q,t) = \alpha\ln\frac{q_t}{q_{y^*}}
    +(1-\alpha)(\ell_H(q,y^*)-\ell_H(q,t))$
    is strictly positive.

    Since $p_{y^*} = q_t > q_y$ for $y\ne t$ we have $\ln\frac{q_t}{q_{y^*}} > 0$,
    $\ell_H(q,y^*) = \left[1-\ln\frac{q_{y^*}}{q_t}\right]_+ > 1$, and
    $\ell_H(q,t) = \left[1-\ln\frac{q_t}{q_{y^*}}\right]_+ < 1$, and so
    $\ell_H(q,y^*) - \ell_H(q,t) > 1 - 1 = 0$.
    Thus, $\ell_\alpha(q,y^*) - \ell_\alpha(q,t) > 0$ as required.
    This gives a contradiction and thus establishes the theorem in the case where
    $p_{y^*} > p_t$.

    Now suppose that $p_{y^*} = p_t$. That is, there is a tie for the maximum
    probability label in $p$ and at least one of these maximising labels
    coincides with the maximising labels of $D$.
    In this case we show that a slight perturbation of $p$ yields a new distribution
    $q$ with a strictly smaller loss.
    To define $q$ we let $\epsilon>0$ and set
    $q_{y^*} = p_{y^*} + \epsilon$, $q_t = p_t - \epsilon$, and $q_y = p_y$ for all
    other $y \ne t, y^*$.
    Now, for $y\ne t, y^*$ we have $\ell_L(p,y) - \ell_L(q,y) = 0$.
    Also for $y\ne t, y^*$ we have $p_t > p_y$ and $q_{y^*} > q_y$ thus
    $\ell_H(p,y) - \ell_H(q,y)
    = 1-\ln\frac{p_y}{p_t} - \left(1 -\ln\frac{q_y}{q_{y^*}}\right)
    = \ln\frac{p_t}{q_{y^*}}
    > 1-\frac{q_{y^*}}{p_t}
    = -\frac{\epsilon}{p_t}$
    since $-\ln x > 1 - x$ for $x > 0$ and
    $q_{y^*} = p_{y^*} + \epsilon = p_t + \epsilon$.
    By substituting this inequality into the definition of $\ell_\alpha$, we see
    that for all $y \ne t,y^*$
    \begin{equation}\label{eq:yne12}
        \ell_\alpha(p,y) - \ell_\alpha(q,y) > -\epsilon\frac{(1-\alpha)}{p_1}.
    \end{equation}

    For the label $y^*$ we see that the log loss component of $\ell_\alpha$
    satisfies
    $\ell_L(p,y^*) - \ell_L(q,y^*) = -\ln\frac{p_{y^*}}{q_{y^*}}
    > \frac{q_{y^*} - p_{y^*}}{q_{y^*}} = \frac{\epsilon}{q_{y^*}}$ and the
    difference between the hinge loss components becomes
    $\ell_H(p,y^*)-\ell_H(q,y^*)
    = (1-\ln\frac{p_{y^*}}{p_t}) - (1-\ln\frac{q_{y^*}}{q_t}) =
    \ln\frac{q_{y^*}}{q_t} = \ln\frac{p_{y^*}+\epsilon}{p_{y^*}-\epsilon}$ since
    $p_{y^*} = p_t$.
    Thus $\ell_H(p,y^*)-\ell_H(q,y^*)
    > 1 - \frac{p_{y^*} - \epsilon}{p_{y^*} + \epsilon}
    = \frac{2\epsilon}{p_{y^*}+\epsilon}$.
    And so
    \begin{align}
        \ell_\alpha(p,y^*) - \ell_\alpha(q,y^*)
        & > \epsilon\left[\frac{\alpha}{p_{y^*}+\epsilon}
            + \frac{2(1-\alpha)}{p_{y^*}+\epsilon}\right] \notag\\
        &= \epsilon\frac{2-\alpha}{p_{y^*}+\epsilon}
         > \epsilon\frac{2-\alpha}{p_{y^*}} \label{eq:y1}
    \end{align}
    since $\epsilon>0$ and $\alpha\le 1$.
    Finally, for the label $t$ we have
    $\ell_L(p,t)-\ell_L(q,t) = -\ln\frac{p_t}{q_t}
    > \frac{q_t - p_t}{q_t}
    = \frac{-\epsilon}{q_t} = \frac{-\epsilon}{p_{y^*}-\epsilon}$ since
    $q_t = p_t - \epsilon$ and $p_t = p_{y^*}$. Similarly,
    $\ell_H(p,t) - \ell_H(q,t) = (1-\ln\frac{p_t}{p_t})-(1-\ln\frac{q_t}{q_t})
    = \ln\frac{q_t}{q_{y^*}} > 1 - \frac{q_{y^*}}{q_t} =
    \frac{-2\epsilon}{p_{y^*}-\epsilon}$.
    Thus,
    \begin{align}
        \ell_\alpha(p,t) - \ell_\alpha(q,t)
        &> -\epsilon\left[
            \frac{\alpha}{p_{y^*}-\epsilon} + \frac{2(1-\alpha)}{p_{y^*}-\epsilon}
        \right] \notag\\
        &= -\epsilon\frac{2-\alpha}{p_{y^*}-\epsilon}
         > -\epsilon\frac{2-\alpha}{p_{y^*}} \label{eq:y2}
    \end{align}

    Putting the inequalities (\ref{eq:yne12}), (\ref{eq:y1}) and (\ref{eq:y2})
    together yields
    \begin{eqnarray*}
        \lefteqn{L_\alpha(p,D) - L_\alpha(q,D)} \\
        &>& D_{y^*} \epsilon \left[\frac{2-\alpha}{p_{y^*}}\right]
            - D_t \epsilon \left[\frac{2-\alpha}{p_{y^*}}\right]
            - \sum_{y\ne y^*,t} D_y \epsilon \frac{1-\alpha}{p_{y^*}} \\
        &=& \frac{\epsilon}{p_{y^*}}
            \left[
                (D_{y^*}-D_t)(2-\alpha) - (1-D_{y^*}-D_t)(1-\alpha)
            \right] \\
        &=& \frac{\epsilon}{p_{y^*}}\left[D_{y^*} - D_t + (1-\alpha)(2D_{y^*} - 1)\right].
    \end{eqnarray*}
    Since $D_{y^*} > D_t$,
    when $D_{y^*} \ge \frac{1}{2}$ the final term is non-negative without any
	additional constraint on $\alpha\in[0,1]$ and since $D_{y^*} > D_t$, the
	difference in risks is thus positive.
    When $D_{y^*} < \frac{1}{2}$ the difference in risks is positive
    whenever
    \begin{equation}
        \alpha > 1 - \frac{D_{y^*} - D_t}{1 - 2D_{y^*}}.
    \end{equation}
    Observing that $D_{max} = D_{y^*}$ and $D_{next} = D_t$
    completes the proof.
\end{proof}

Theorem~\ref{thm:fcc} can be inverted and interpreted as a
constraint on the conditional distributions of some data distribution $D$ such that a
hybrid loss with parameter $\alpha$ will yield consistent predictions. Specifically,
the hybrid loss will be consistent if, for all $x\in\Xcal$ such that
$q = D(x)$ has no dominant label (\emph{i.e.}, $D_y(x) \le
\frac{1}{2}$ for all $y\in\Ycal$), the gap $D_{y_1}(x) -
D_{y_2}(x)$ between the top two probabilities is larger than
$(1-\alpha)(1-2D_{y_1}(x))$. When this is not the case for some $x$,
the classification problem for that instance is, in some sense, too
difficult to disambiguate. In this sense, the bound can be seen as a property
on distributions akin to Tsybakov's noise condition
\citep{Che06}. Both conditions are non-constructive as they depend on the
unknown distribution but provide some guidance as to the effect of parameter
choices (i.e., $\alpha$ for the hybrid loss and regularisation constants for SVMs).
Exploring the relationship between conditional FCC and the Tsybakov noise condition
is the focus of ongoing work.

\subsection{Parametric Consistency}\label{sec:parametric-fcc}

Since Fisher consistency is defined point-wise on observations, it is not directly
applicable to parametric models as these enforce inter-observational constraints
(\emph{e.g.} smoothness).
Abstractly, assuming parametric hypotheses can be seen as a restriction over
the space of allowable scoring functions. When learning parametric models, risks
are minimised over some subset $\Fcal$ of functions from $\Xcal \to \RR^{\Ycal}$
instead of all possible functions.
We now show that, given some weak assumptions on the hypothesis class $\Fcal$, a
loss being FCC is a necessary condition if the loss is also to be
$\Fcal$-consistent.

We say a loss $\ell$ is $\Fcal$-consistent if, for any distribution, minimising its
associated risk over $\Fcal$ yields a classifier with minimal
0-1 loss in $\Fcal$.\footnote{%
    While this is simpler and stronger than the usual asymptotic notation of
    consistency \cite{LugVay04} it most readily relates to FCC and suffices for our
    discussion since we are only establishing that FCC is a necessary condition.}
Recall from Section~\ref{sub:multiclass} that the risk of a hypothesis $f\in\Fcal$ associated with a loss $\ell$ and
distribution $D$ over $\Xcal\times\Ycal$ is $L_D(f) = \EE_D\left[\ell(y,f(x))\right]$
and the 0-1 risk or misclassification error for $f$ is
$e_D(f) = \PP_D\left[h(x) \neq y \right]$ where $h$ is some
classifier deterministically derived by some tie-breaking procedure on $f$.
More precisely, we will say a \emph{tie-breaker} $T$ is a function from the power set
of $\Ycal$ to $\Ycal$ that guarantees $T(Y) \in Y$ for all non-empty $Y \subset \Ycal$.
Finally, we define $h_f(x) = T(\argmax_{y\in\Ycal} f_y(x))$ to be
the classifier derived from $f$ using $T$.

Given a function class $\Fcal$ we say \emph{$\ell$ is $\Fcal$-consistent} if, for all
distributions $D$ and all tie-breakers $T$ defining the classifiers $h_f$,
\begin{equation}\label{eq:F-consistency}
    L_D(f^*) = \inf_{f\in\Fcal} L_D(f)
    \implies
	e_D(h_{f^*}) = \inf_{f\in\Fcal} e_D(h_f).
\end{equation}

We need a relatively weak condition on function classes $\Fcal$ to state our theorem.
We say a class $\Fcal$ is \emph{regular} if the follow two properties hold: 1) For any
$g\in\RR^{\Ycal}$ there exists an $x\in\Xcal$ and an $f\in\Fcal$ so that $f(x) = g$;
and 2) For any $x\in\Xcal$ and $y\in\Ycal$ there exists an $f\in\Fcal$ so that there
is a \emph{unique} $y \in \Ycal$ which maximises $f_{y}(x)$.

Intuitively, the first condition says that for any distribution over labels
there must be a function in the class which models it perfectly on some point in the
input space.
The second condition requires that any mode can be modelled on any input by a function
that has no ties for its maximum value.
Importantly, these properties are fairly weak in that they do not say anything about the
constraints a function class might put on relationships between distributions modelled
on different inputs.

\begin{theorem}\label{thm:fcc-necessary}
    For regular function classes $\Fcal$ any loss that is $\Fcal$-consistent is
    necessarily also Fisher Consistent for Classification (FCC).
\end{theorem}

\begin{proof}
    The proof is by contradiction.
    We assume we have a regular function class $\Fcal$ and a loss $\ell$ which is
    $\Fcal$-consistent but not FCC.
    That is, (\ref{eq:F-consistency}) holds for $\ell$ but there exists a
    distribution $p$ over $\Ycal$
    such that there is a $g \in \RR^{\Ycal}$ which minimises the conditional risk
	$L_{p}(g)$ but $g$ is not aligned with $p$
	(i.e., $\argmax_{y\in\Ycal} g_y \not\subset \argmax_{y\in\Ycal} p_y$).

    By the assumption of the regularity of $\Fcal$, property 1 means there is
	an $x\in\Xcal$ and a
    $f\in\Fcal$ so that $f(x) = g$.
    We now define a distribution $D$ over $\Xcal\times\Ycal$ that puts all its mass
    on the set $\{x\}\times\Ycal$ so that $D(x,y) = p_y$.
    Since this distribution is concentrated on a single $x$ its full risk and
    conditional risk on $x$ are the same. That is, $L_D(\cdot) = L_p(\cdot)$.
    Thus,
    \[
        L_D(f) = L_p(f) = \inf_{f'\in\Fcal} L_p(f') = \inf_{f'\in\Fcal} L_D(f')
    \]
    By the assumption of $\Fcal$-consistency, since $f$ is a minimiser of $L_D$ the
	classifier $h_f$ must also minimise $e_D$ for any choice of tie-breaker $T$
	used to define $h_f$.
	Because $g = f(x)$, the construction of $D$ implies that
	$e_D(h_f)   = e_p(h_g)
	 			= \PP_{y\sim p}\left[y \neq h_{g}(x)\right]
				= 1 - p_{y_g}$
	where $y_g = h_g(x)$ is the label predicted by $h_g$.
	However, since $g$ is not aligned with $p$ by assumption and
	\eqref{eq:F-consistency} holds for any $T$, we are free to
	choose the tie-breaker $T$ defining $h_g$ so that
	$h_g(x) = T(\argmax_{y} g_y(x)) \notin \argmax_y p_y$.
	Thus
	\begin{equation}\label{eq:eDbigger}
        e_D(h_f) = e_p(h_g) = 1-p_{y_g} > 1 - p_{y^*}
	\end{equation}
    since $y_g \neq y_* \in \argmax_y p_y$.

    By the second regularity property, there must also be an $\hat{f}\in\Fcal$ such
    that $y^*$ is the unique maximiser of $\hat{f}_y(x)$ for $y \in \Ycal$.
	Since $y^*$ is a unique maximiser, any choice of tie-breaker $T$ will result
	in a classifier satisfying $h_{\hat{f}}(x) = y^*$ as any $T$ must guarantee
	$T(\{ y \}) \in \{ y \}$ for all $y \in \Ycal$.
	Therefore, we arrive at the contradiction
	\[
		1 - p_{y^*} = e_D(h_{\hat{f}}) \geq e_D(h_f) > 1 - p_{y^*}
	\]
	since $h_f$ is a minimiser of $e_D$ and $e_D(h_f) > 1 - p_{y^*}$ by
	\eqref{eq:eDbigger}.
    Thus, we have shown that there exists a distribution $D$ so $f\in\Fcal$ is a
    minimiser of the risk $L_D$ but $h_f$ is not a minimiser of the
    misclassification rate $e_D$, contradicting the assumption of the
    $\Fcal$-consistency of $\ell$. Therefore, $\ell$ must be FCC.

\end{proof}

The above analysis of the hybrid loss suggests it should outperform the hinge loss due to its improved consistency on distributions with
non-dominant labels. Furthermore, it should also make more efficient use of
data than log loss on distributions with dominant labels.
These hypotheses are confirmed in the next section by applying the hybrid, log and
hinge losses to a
number of synthetic multiclass data sets in which the data set size and proportion
of examples with non-dominant labels are carefully controlled.

We also compare the hybrid loss with the log and hinge losses on
several real structured estimation problems and observe that the hybrid loss
regularly outperforms the other losses and consistently performs at least as well
as either of the other losses on any problem.

\section{Multiclass Classification}
Two types of multiclass simulations were performed. The first examined the
performances of the hybrid, log and hinge losses when no observation had a
dominant label. That is all observations were drawn from a $D$ with $D_y(x) <1/2$
for all labels $y$. The second experiment considered distributions with a
controlled mixture of observations with dominant and non-dominant labels.

\subsection{Non-dominant Distributions}

\begin{figure}[tb]
 \centering
     \includegraphics[width=0.9\linewidth]{trainError}
  \caption{Training Error with various number of classes. $\alpha =
  0.5$ for the hybrid loss. Fisher consistency analyses the behaviour of a loss observing the entire data population. Thus the training data are the entire data, so are the testing data. Consequently, the training error is the testing error.
} \label{fig:multiclass}
\end{figure}

\begin{figure*}[tb]
 \centering
 \subfigure[Hybrid v.s. Hinge (31/15)]{
     \includegraphics[width=0.3\linewidth]{hybrid_hinge}
     \label{fig:hybrid_svm}
     }
      \subfigure[Hybrid v.s. Log (34/15)]{
     \includegraphics[width=0.3\linewidth]{hybrid_log}
     \label{fig:hybrid_crf}
     }
      \subfigure[Hinge v.s. Log (30/23)]{
     \includegraphics[width=0.3\linewidth]{hinge_log}
     \label{fig:svm_crf}
     }
  \caption{Performance of the hybrid, hinge, and log losses on
  non-dominant/dominant mixtures.
  Points denote pairs of test accuracies for models trained on one of 60 data sets
  using the losses named on the axes.
  Score $(a/b)$ denotes the vertical loss with $a$ wins and $b$ losses (ties
  not counted).}
\label{fig:multiclass_mix}
\end{figure*}

To make the experiment as simple as possible, we considered an observation space of
size $|\Xcal|=1$ and focused on varying the number of labels and their
probabilities.

Fisher consistency analyses the behaviour of losses while observing the entire data population. To mimic seeing the entire data population and the dominant/non-dominant class case, we use a constant vector in $\RR^2$ as features, and learn the parameter vectors $w_y
\in\RR^2$ for $y\in\Ycal$. The labels $y$ take different values proportionally as follows:
The label set $\Ycal$ took the sizes $|\Ycal|=3,4,5,\dots,10$.
One label $y^*\in\Ycal$ is assigned probability $D_{y^*}(x) = 0.46$ and the
remainder are given an equal portion of 0.54 (\emph{e.g.}, in the 3 class case the
other labels each have probability 0.27, and in the 10 class case, 0.06).
Note that this means for all the label set sizes, the gap
$D_{y^*}(x) - D_{y}(x)$ is at least 0.19 which is always greater than
$(1-\alpha)(1-2D_{y^*}(x)) = 0.04$ so the hybrid consistency condition
(\ref{eq:alpha}) is always met. This way, the size of training data does not affect the training error as long as the proportions of $y$ values are not altered. In the same vein, the proportions of $y$ values in the test data are the same as that in the training data, thus test errors and training errors should be the same in this case. Thus we plot the resulting training errors (and the test errors) for hinge, log and hybrid losses in Figure~\ref{fig:multiclass} as a function of the number of labels.
As we can clearly see, the hinge loss error increases as the number
of classes increases, whereas the errors for the log and the hybrid
losses remain a constant ($1-D_{y^*}(x)$), in concordance with the consistency
analysis.

Models were found using LBFGS \cite{Byretal94} with inexact line search, thus landing on the hinge point almost never happens. In theory, it is a problem --- it may not converge for the non-smooth optimisation problem. But in practice, it works well.

\begin{figure*}[tb]
 \centering
 \subfigure[the testing
  set ]{
     \includegraphics[width=0.4\linewidth]{conll2000_test_prob}
     \label{fig:chunk_test}
     }
      \subfigure[the training set]{
     \includegraphics[width=0.4\linewidth]{conll2000_train_prob}
     \label{fig:chunk_train}
     }
  \caption{Estimated probabilities of the true label $D_{y_i}(x_i)$ and most likely label $D_{y_i^*}(x_i)$. Sentences are sorted according to $D_{y_i}(x_i)$ and $D_{y_i^*}(x_i)$ respectively in ascending
  order. $D=1/2$ is shown as the straight black dot line. About 700
  sentences out of 2012 in the testing set and 2000 sentences out of 8936 in the training set
  have no dominant label.
} \label{fig:chunking}
\end{figure*}

{\Large
\begin{table*}[t]
    \centering {\caption{Accuracy, precision, recall and F1 Score on the
    CONLL2000 text chunking task.} \label{table:conll2000}}
    \begin{tabular}{c|c|c|c|c|c}

    Train Portion &Loss &Accuracy &Precision & Recall &F1 Score\\
    &Hinge&91.14 &85.31 &85.52 &85.41\\
    0.1&Log&92.05 &87.04 &87.01 &87.02\\
    &Hybrid&\bf{92.07} &\bf{87.17} &\bf{86.93} &\bf{87.05}\\
    \hline
    &Hinge&94.61 &91.23 &91.37 &91.30\\
    1&Log&95.10 &92.32 &91.97 &92.15\\
    &Hybrid&\bf{95.11} &\bf{92.35} &\bf{92.00} &\bf{92.17}\\
    \end{tabular}
\end{table*}
}

\subsection{Mix of Non-dominant and Dominant Distributions}

The second synthetic experiment examined how the three losses performed
given various training set sizes (denoted by $m$) and various proportions of
instances with non-dominant distributions (denoted by $\rho$).

We generated 60 different data sets, all with $\Ycal=\{1,2,3,4,5\}$,
in the following manner:
Instances came from either a non-dominant class distribution
or a dominant class distribution. In the non-dominant class case,
$x\in\RR^{100}$ is set to a predefined, constant, non-zero vector
and its label distribution is $D_{1}(x) = 0.4$ and $D_y(x) = 0.15$ for
$y > 1$.
In the dominant case, each dimension $x_i$ was drawn from a
normal distribution $N(\mu = 1+y,\sigma = 0.6)$ depending on the class
$y = 1,\dots, 5$.
The proportion $\rho$ ranged over 10 values $\rho=0.1, 0.2, 0.3,\dots,1$ and for
each $\rho$, test and validation sets of size 1000 were generated.
Training set sizes of $m = 30, 60, 100, 300, 600, 1000$ were used for each
$\rho$ value for a total of 60 training sets.
The optimal regularisation parameter $\lambda$ and hybrid loss parameter $\alpha$
were selected using the validation set for each loss on each training set.
Then models with parameters $w_y\in\RR^{100}$ for $y\in\Ycal$ were found using
LBFGS \cite{Byretal94} for each of the three losses on each of the 60 training sets and then
assessed using the test set.

The results are summarised in Figure~\ref{fig:multiclass_mix}.
Each point shows the test accuracy for a pair of losses.
The predominance of points above the diagonal lines in a) and b) show that the
hybrid loss outperforms the hinge loss and the log loss in most of the data sets.
while the log and hinge losses perform competitively against each other.

\begin{table*}[t]
\centering {\caption{Accuracy, precision, recall and F1 Score on the
baseNP chunking task for training on increasing portions of training
set.} \label{table:basenp}
\begin{tabular}{c|c|c|c|c|c}
Train Portion &Loss &Accuracy &Precision & Recall &F1 Score\\
&Hinge&88.48 &71.70 &75.96 &73.77\\
0.1&Log&90.86 &81.09 &78.96 &80.01\\
&Hybrid&\bf{90.90} &\bf{81.23} &\bf{79.09} &\bf{80.15}\\
\hline
&Hinge&94.64 &87.58 &88.30 &87.94\\%epochs = 50, Fold2/
1& Log&95.21 &90.07 &88.89 &89.48\\
&Hybrid&\bf{95.24} &\bf{90.12} &\bf{88.98} &\bf{89.55}\\
\end{tabular}
}\\
\end{table*}

\section{Structured Estimation}\label{sec:experiments}
Unlike the general multiclass case, structured estimation problems
have a higher chance of non-dominant distributions because of the
very large number of labels as well as ties or ambiguity regarding
those labels. For example, in text chunking,
changing the tag of one phrase while leaving the rest unchanged
should not drastically change the probability predictions --
especially when there are ambiguities.
Due to the prevalence of non-dominant distributions, we expect
models trained using the hinge loss to perform poorly on these
problems relative to those trained with hybrid or log losses.
We emphasise that our main motivation for investigating structured
prediction problems is that, as multiclass problems,
they tend to have non-dominant distributions.

\subsection{CONLL2000 Text Chunking} Our first structured estimation experiment
is carried out on the CONLL2000 text chunking task \cite{conll2000}.
The data set has 8936 training sentences and 2012 test sentences
with 106978 and 23852 phrases (a.k.a. chunks), respectively. The
task is to divide a text into syntactically correlated parts of words
such as noun phrases, verb phrases, and so on. For a sentence with
$L$ chunks, its label consists of the tagging sequence of all its
chunks, \emph{i.e.} $y = (y^1,y^2,\dots,y^L)$, where $y^j$ is the chunking
tag for chunk $j$. As is common in this task, the label $y$ is
modelled as a chain-structured graphical model to account for the dependency between
adjacent chunking tags $(y_i^j,y_i^{j+1})$ given observation $x_i$.
Clearly, the model has exponentially many possible labels, which
suggests the absence of a dominant class.

Since the true underlying distribution is unknown, we train a
CRF
on the training set and then apply
the trained model to both testing and training datasets to obtain an
estimate of the conditional distributions for each instance. We sort
the sentences $x_i$ from highest to lowest estimated probability on
the true chunking label $y_i$ given $x_i$. The result is plotted in
Figure~\ref{fig:chunking}, from which we observe the existence of
many non-dominant distributions --- about 1/3 of the testing
sentences and about 1/4 of the training sentences.

We use the feature template from the CRF++ toolkit \cite{crfplusplus}, and
    the CRF code from Leon Bottou \cite{crfsgd}. Stochastic Gradient Descent (SGD) \cite{crfsgd} is used for training. During training, dynamic programming (\ie Viterbi algorithm) for inference is used.
We split the data into 3 parts: training ($20\%$), testing ($40\%$)
and validation ($40\%$).
The regularisation parameter $\lambda$ and
the weight $\alpha$ were determined via parameter selection using the
validation set. To see the performance with different training
sizes, we took part of the training data to learn the model and gathered statistics
on the test set. The accuracy, precision, recall
and F1 Score on the test set are reported in Table~\ref{table:conll2000}
when using 10\% and 100\% of the training set.
The hybrid loss marginally outperforms both the hinge loss and the log loss.

\subsection{baseNP Chunking}
A similar methodology to the previous experiment is applied to the BaseNP data set
\cite{crfplusplus}. It has 900 sentences in
total and the task is to automatically classify a chunking phrase as
baseNP or not.

Again SGD \cite{crfsgd} is used for training. During training, dynamic programming (\ie Viterbi algorithm) for inference is used.
We split the data into 3 parts: training ($20\%$),
testing ($40\%$) and validation ($40\%$). Once again, $\lambda$ and $\alpha$ are
determined via model selection on the validation set. We report the
test accuracy, precision, recall and F1 Score in
Table~\ref{table:basenp} for training on increasing proportions of
the training set. The hybrid marginally outperforms the other two losses on all
measures.

\subsection{Human action recognition}
Here we consider recognising human actions in TV episodes, where each contains one or more persons and may interact with each other. We evaluate our method on the TVHI dataset \citep{patron:2010high}, which contains 300 short videos collected from TV episodes and includes five action classes: \emph{handshake} (HS), \emph{hug} (HG), \emph{high-five} (HF), \emph{kiss} (KS) and \emph{others} (OT). Here a person labelled the action \emph{others} means that there is no interaction between this person and any other persons in the image. Each video contains a number (up to seven) of people performing one of the five action classes. The ground-truth provided with the dataset includes upper body bounding boxes, discrete body poses, the action labels and the interaction status between any pair of persons (\ie a binary variable indicating whether there is an interaction). We manually choose 2,188 images from this dataset and divide these examples into three sets without intersection: the training set (400 frames), the validating set (894 frames) and the testing set (894 frames). Here $\alpha$ is determined via model selection on the validation set. Note in \cite{patron:2010high} their task is to predict both interactions and actions, whereas here our task is to predict actions given interaction status. More specifically, our goal is to solve the estimation problem of finding the actions $y\in \Ycal$ of all subjects in an observation image $x \in \Xcal$, given pairwise interaction status.

We use PGMs to model the dependency of the actions in the same image. Consider a graph $G=(\Vcal,\Ecal)$ with each node $i \in \Vcal$ representing an action variable $y^i$ and each edge $(i,j) \in \Ecal$ reflecting the dependency of the two action variables. The edge set $\Ecal$ is constructed according to the annotated interaction status. If there is an interaction between two persons in the annotation, then an edge between two corresponding nodes is added to the edge set $\Ecal$.

We cast this estimation problem as finding an energy function $E(x,y)$ such that for an observation image $x \in \Xcal$, we assign the actions that receive the smallest energy with respect to $E$, that is
\begin{equation}
y ^*= \argmin_{y\in \Ycal} E(x,y;w).
\label{eq:infer}
\end{equation}
Here we use the energy function $E$ with unary terms $U$ and pairwise terms $S$ as follows,
\begin{equation}
E(x,y;w)=\sum_{i\in \Vcal}U(y^i,x;w')+\sum_{(i,j)\in \Ecal}S(y^i,y^j,x;w'').
\end{equation}
where $w=[w';w'']$, and
\begin{subequations}
\begin{align}
&U(y^i,x;w')=\inner{\phi_1(x^i,y^i)}{w'},\\
&S(y^i,y^j,x;w'')=\inner{\phi_2(x^i,x^j,y^i,y^j)}{w''}.
\end{align}
\label{eq:energy}
\end{subequations}
Here $\phi_1$ and $\phi_2$ are node and edge features (which we will define later),  and $x^i$ is the sub-image of the bounding box on the $i$-th subject. The model parameter $w$ will be learned during training.

Our feature representation is a combination of several visual cues including multiclass SVM action classification scores, human body poses and the relative position between two individuals, which have been exploited to distinguish different actions in \cite{patron:2010high, patron:2012structured}. Here we combine these visual cues in a similar way as \cite{patron:2010high}. To be specific, let $\eb_1(y^i)\in \{0,1\}^5$ represent an unit vector with the $y^i$-th dimension equals 1 (0 elsewhere). Similarly, let $\eb_2(r^{ij})\in \{0,1\}^6$ denote another unit vector with the $r^{ij}$-th dimension equals 1 (0 elsewhere). Here $r^{ij}$ denotes the relative position of person $j$ to $i$. To compute $r^{ij}$, we employ a simple method in \cite{patron:2010high} which only requires the bounding boxes of person $i$ and $j$. Each $r^{ij}$ value represents a relative position in the set $\{overlap, adjacent-left, adjacent-right, near-left, near-right, far\}$. Let $\otimes$ denote the Kronecker product. The features are defined as
\begin{subequations}
\begin{align}
&\phi_1(x^i,y^i)=\db^i\otimes \eb_1(y^i),\label{eq:node_feat}\\
&\phi_2(x^i,x^j,y^i,y^j)=\eb_1(y^i)\otimes\eb_1(y^j)\otimes\eb_2(r^{ij})\otimes\ob^i\otimes\ob^j\notag\\
&\qquad\qquad+\eb_1(y^j)\otimes\eb_1(y^i)\otimes\eb_2(r^{ij})\otimes\ob^j\otimes\ob^i,\label{eq:edge_feat}
\end{align}
\label{eq:feat}
\end{subequations}
where $\db^i\in \RR^5$ is a score vector contains the action classification scores obtained by applying a multiclass SVM classifier to the histograms of gradients (HoG) descriptor \cite{dalal2005histograms} extracted from the bounding box area of person $i$. Similarly, $\ob^i\in \RR^5$ represents another score vector of the pose classification scores (the descriptors for pose classification is the same as that used for action classification). Here we consider five body pose classes $\{profile-left, profile-right, frontal-left, frontal-right, backwards\}$. To extract the HoG features, we superimpose an $8\times 8$ grid on the bounding box area and accumulate HoG for each grid cell using five orientation bins. The final descriptor is a concatenation of the sub-descriptors of all cells.

Intuitively, the node term $U$ reflects the confidence of assigning person $i$ the action label $y^i$ observing $x^i$. The edge term $S$ encodes the correlation between actions $y^i$ and $y^j$ observing $x^i,x^j$.

Implementing Kronecker product naively to compute energies could be memory and time consuming. Fortunately, we see that the feature vectors in \eq{eq:feat} are highly sparse (especially the edge feature). Thus we only need to multiply the non-zero components of the feature vectors with corresponding components of the $w$ without using Kronecker product.

\begin{figure*}
\centering
\subfigure[hinge loss]{\includegraphics[width=.32\textwidth]{cm_hinge.pdf}}
\subfigure[log loss]{\includegraphics[width=.32\textwidth]{cm_log.pdf}}
\subfigure[hybrid loss]{\includegraphics[width=.32\textwidth]{cm_hybrid.pdf}}
\caption{Confusion matrices on the TVHI dataset. For each class the best are highlighted by green rectangles. The hybrid loss achieves the best classification accuracy on three out of five action classes, \ie \emph{other, hug} and \emph{kiss}.}
\label{fig:conf_mat_tvhi}
\end{figure*}

\begin{figure*}
\centering
\subfigure{\includegraphics[width=44mm]{img_hs_0026.jpg}}
\subfigure{\includegraphics[width=44mm]{hinge_hs_0026.pdf}}
\subfigure{\includegraphics[width=44mm]{log_hs_0026.pdf}}
\subfigure{\includegraphics[width=44mm]{hybrid_hs_0026.pdf}}
\subfigure{\includegraphics[width=44mm]{img_hs_11_42.jpg}}
\subfigure{\includegraphics[width=44mm]{hinge_hs_11_42.pdf}}
\subfigure{\includegraphics[width=44mm]{log_hs_11_42.pdf}}
\subfigure{\includegraphics[width=44mm]{hybrid_hs_11_42.pdf}}
\subfigure{\includegraphics[width=44mm]{img_hf_0025.jpg}}
\subfigure{\includegraphics[width=44mm]{hinge_hf_0025.pdf}}
\subfigure{\includegraphics[width=44mm]{log_hf_0025.pdf}}
\subfigure{\includegraphics[width=44mm]{hybrid_hf_0025.pdf}}
\subfigure{\includegraphics[width=44mm]{img_hg_0066.jpg}}
\subfigure{\includegraphics[width=44mm]{hinge_hg_0066.pdf}}
\subfigure{\includegraphics[width=44mm]{log_hg_0066.pdf}}
\subfigure{\includegraphics[width=44mm]{hybrid_hg_0066.pdf}}
\subfigure{\includegraphics[width=44mm]{img_ks_0029.jpg}}
\subfigure{\includegraphics[width=44mm]{hinge_ks_0029.pdf}}
\subfigure{\includegraphics[width=44mm]{log_ks_0029.pdf}}
\subfigure{\includegraphics[width=44mm]{hybrid_ks_0029.pdf}}
\caption{Visualisation of some action predictions using different losses. {\bf FIRST COLUMN}:  the input images; {\bf SECOND COLUMN}: the hinge loss results; {\bf THIRD COLUMN}: the log loss results; {\bf FOURTH COLUMN}: the hybrid loss results. The PGMs are superimposed on the images with green node and red node indicate correct and incorrect predictions respectively. Note OT, HS, HF, HG, KS denote five action classes: \emph{others, handshake, highfive, hug, kiss}. The edges (line segments in pink) come from the interaction annotation in the dataset, and are used to model the dependency between two action variables of subjects.}
\label{fig:tvhi_pred_vis}
\end{figure*}

Following the work of SVMs for structured prediction \cite{TsoJoaHofAlt05}, the hinge loss in this case is
\begin{align}
\ell_{H}(x,y,w) =[\Delta(y^{\dag},y) + E(x,y;w) -  E(x,y^{\dag};w)]_+.
\label{eq:ghinge}
\end{align}
where
\begin{align}
y^{\dag}_{i} & =  \argmin_{y}(E(y,x_i;w)-\Delta_{\text{}}(y,y_i)).
\label{eq:sub-sup}
\end{align}
Here $\Delta$ is a label cost function (or distance) that describes the discrepancy of two labels.
We use the popular Hamming distance, which is defined as
\begin{align}
\label{eq:hamming_loss}
\Delta_{\text{Ham}}(y,y') = \frac{1}{L}\sum_{j=1}^L \delta(y^{j}\ne y'^{j}),
\end{align}
where $y = (y^{j})_{j=1}^{L}$ and
$y' =(y'^{j})_{j=1}^{L}$.

For the log loss, let $E(x,y;w) = -f_{y}(x)$. Thus, according to \eq{eq:prob} we have
\begin{align}
p(y|x;w) = \frac{\exp(-E(x,y;w))}{\sum_{y' \in \Ycal} \exp(-E(x,y';w))}.
\end{align}
So the log loss is
\begin{align}
&\ell_{L}(x,y,w) =  -\log\Big(\frac{\exp(-E(x,y;w))}{\sum_{y' \in \Ycal} \exp(-E(x,y';w))}\Big)\\
& = E(x,y;w) + \log\Big({\sum_{y' \in \Ycal} \exp(-E(x,y';w))}\Big).
\label{eq:log-loss-z}
\end{align}

According to \eq{eq:hybrid} the hybrid loss is
\begin{align}
\ell_{\alpha}(x,y,w) = \alpha \ell_L(x,y,w)\!+\!(1-\alpha)\ell_H(x,y,w).
\label{eq:hybrid-har}
\end{align}
The sub-gradient of the hybrid loss is simply a convex combination of the sub-gradient of the hinge loss and the gradient of the log loss. It is known that the sub-gradient of the hinge loss can be computed via standard MAP inference techniques, and the gradient of the log loss can be computed via standard marginal inference techniques. We use the max-product algorithm for the hinge loss and the sum-product algorithm for the log loss.

To accelerate the training, we apply the stochastic sub-gradient method from \citep{shalev2007pegasos} to the hybrid loss. Here the maximum number of iterations is set to be 30 and the min-batch size is set to be 10. Once the parameters are learned, we use the standard max-product algorithm to make prediction on testing data.

In order to evaluate the recognition performance of different losses, we show the confusion matrices in Figure~\ref{fig:conf_mat_tvhi}. It can be seen that the hybrid loss achieves the best true positive rates on 3 classes (OT, HG and KS) out of 5 action classes, while the log loss and the hinge loss perform best on the HS class and the HF class respectively. Note all losses perform much worse on the HF class than the rest classes. This is because the training set is highly biased as the number of persons performing the \emph{high-five} action in the training set is much less than other classes.

We also give some recognition examples as that shown in Figure~\ref{fig:tvhi_pred_vis}. The first column shows four input images, each containing multiple persons with occlusions making the recognition task difficult. As we can see, hinge loss performs worst with 10 persons out of 18 mislabelled. The log loss outperforms the hinge loss in general as 5 persons are misclassified. For the hybrid loss, persons in all images are perfectly classified except for the third image, where all 3 persons are misclassified.

\section{Conclusion and Discussion}\label{sec:conclusions}

We have provided theoretical and empirical motivation for the use of
a novel hybrid loss for multiclass and structured prediction
problems which can be used in place of the more common log loss or
multiclass hinge loss. This new loss attempts to blend the strength
of purely discriminative approaches to classification, such as
Support Vector machines, with probabilistic approaches, such as
Conditional Random Fields. Theoretically, the hybrid loss enjoys
better consistency guarantees than the hinge loss while
experimentally we have seen that the addition of a purely
discriminative component can improve accuracy when data is less
prevalent.

In general the consistency condition may not hold if $\alpha$ is selected by cross-validation. For example, when the selected $\alpha$ is very small. However, we observe the selected $\alpha$ values in our experiments are always very close to 1.

\subsection{Future Work}
Theoretically, we expect that some stronger sufficient conditions on $\alpha$ are
possible since the bounds used to establish Theorem~\ref{thm:fcc}
are not tight. Our conjecture is that a necessary and sufficient
condition would include a dependency on the number of classes.
We are also investigating connections between $\alpha$ and the multiclass Tsybakov
noise condition \citep{Che06}.

To our knowledge, the notion of a regular function class for the purposes of
consistency analysis is novel.
Characterisations of this property for existing parametric models would make
testing for regularity easier.

In structured prediction, there is still a big gap between the analysis and the practice. For example, in structured prediction, we know the parametric hinge loss is not consistent for binary label cost function, but we don't know whether the parametric hybrid loss is. Moreover, we don't have theoretical results for general label cost functions. To better connect our theory with actual practice on structured prediction problems, we plan to investigate consistency for general cost functions (e.g. Hamming loss) that are more commonly used in these problems.

\ifCLASSOPTIONcompsoc

  \section*{Acknowledgments}
\else

  \section*{Acknowledgment}
\fi

The bulk of this research was performed while Q. Shi was with NICTA. NICTA is funded by the Australian Government as represented by the Department of Broadband, Communications and the Digital Economy and the Australian Research Council through the ICT Centre of Excellence program.

This research was partly supported under Australian Research Council Discovery Projects funding scheme (DP0988439 and DP140102270) and Australian Research Council Discovery Early Career Researcher Award funding scheme (DE120101161) and supported by The Australian Centre for Visual Technologies and The Computer Vision group of The University of Adelaide.

We would like to thank the anonymous reviewers of an earlier version of this work
for their constructive feedback, particularly regarding the Theorems 1 and 2.

\ifCLASSOPTIONcaptionsoff
  \newpage
\fi

\bibliographystyle{plain}

redpajama_set_name: RedPajamaArXiv


text:
Personalized travel
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Mykonos, the world's leading gay destination
Pearl of the Cyclades and temple of the gay parties
Mykonos is probably the best known gay destination in the world. The island is located in the Aegean Sea, in the far south of Europe. It's part of the Cyclades archipelago, which includes more than 200 islands and islets, among which the island of Delos, listed as a UNESCO World Heritage Site. You can actually visit this ancient Greek gem from Mykonos. Although Mykonos is a gay destination to party and enjoy the turquoise sea, ancient history lovers will love it, as the island is also the departure point for many excursions that will take you back to Greek mythology. Apollo, Dionysus and Heracles will have no more secrets for you!
General information about Mykonos
An integral part of Greece, Mykonos is divided between legend and mythology. His name would come from the hero Mycon, grandson of Apollo. With its Mediterranean climate and postcard landscapes, the island really became a tourist destination during the 1950s. In 1970, it became the main destination for the gay clientele. Moreover, it's also the favourite vacation spot of the famous fashion designer Jean-Paul Gaultier, whom you can meet in the alleys of the city while shopping (personal anecdote of the author of these lines and I assure you that he has a very laid-back attitude). The population of Mykonos is just over 10,000, but its population increases fivefold during summer.
Why is Mykonos so LGBTQ+ friendly?
For some 50 years now, Mykonos has been unanimously recognized as a favourite destination for LGBTQ+ clientele. When gay-friendly destinations were still being counted on the fingers of one hand, Mykonos was already one of them. This shows just how welcoming the island has become to the community. We can even talk about a "straight-friendly" island.
LGBTQ+ Human Rights in Greece
There is still some work to do in terms of adoption (allowed only for one member of the couple, not both) and acceptance in the most secluded areas, but in the case of Mykonos, you have nothing to worry about. In Greece, since 2010, transgender people are protected by law and from any discrimination on the basis of their gender identity. Civil partnership exists since 2015. Gay marriage is being legalised, but at the moment, only foreign marriages are recognised in law.
Currency, visa, useful advices...
The island of Mykonos is part of Greece, which is a member country of the European Union. The currency is the Euro.
For citizens of the European Union, Switzerland or a member country of the European Economic Area, a national identity card or passport is sufficient, as Greece is a member of the Schengen Area.
For travellers from Canada, the United States and Australia, there is an agreement with the EU that allows you to be exempt from visa requirements for stays of less than 90 days. However, please note that in 2022, you will have to obtain your ETIAS before your departure. This is an electronic travel authorisation that must be done online before the trip, which will cost €7 and will give access to the Schengen area. Here is the link for more information: ETIAS.
All your gay places in Mykonos
Gay clubs and parties
Gay Beach
Gay Pride 2021
Gay Apps
Gay Map
Elysium Sunset Bar @Elysium Hotel (right next to the School of Fine Arts). Very famous trendy bar of the gay Elysium Hotel. Open to non-hotel guests. Magnificent view of Mykonos and the sunset. There are shows of gogos, djs, etc… To get there, you have to go up the very steep hill from the centre of Mykonos towards the School of Fine Arts. To visit at least once during your stay just for the atmosphere and the view.
Babylon Bar (harbour area). Gay bar in Mykonos in the harbour area, great view, quiet and with excellent cocktails.
JackieO'. (harbour area). Right next to Babylon. Gay bar with also a very nice view and authentic brick decoration.
Lola (Zanni Pitaraki, 4). Longstanding gay bar with a very nice and welcoming staff. Go at least once during your stay.
Kastro's (harbour area). It is a restaurant, but its ideal position in front of the sea makes it an ideal place to have a glass of wine or a cocktail.
Porta Bar (center). Famous gay bar-disco in Mykonos. Go after dinner to start, and why not finish your evening.
54 Cocktail Bar & Sunset Lounge (Manto Square) Lounge bar with excellent cocktails.
Montparnasse – the Piano Bar (harbour area). Announced as permanently closed (Covid-19). Checking.
Important note: in the case of Mykonos, the above list represents bars with a 100% gay history and reputation. This is a non-exhaustive listing, as Mykonos is a predominantly LGBTQ+ destination, all the bars on the island are deemed gay-friendly.
JackieO' Beach Club @beach of Super Paradise. This club is located in the south of the island, on the gay beach of Super Paradise. You can get there by taxi or bus from the central bus station in Mykonos (there are regular rotations). Club atmosphere, swimming pool, music, djs, all in front of the sea. You can come during the day and in the evening.
XLSIOR Mykonos @any number of places. Very well known international gay festival. Next programming from 18 to 25 August 2021. See details on XLSIOR MYKONOS.
Cavo Paradiso Club (Paradise Beach). Well-known show nights and international djs. gay friendly.
This will become your routine every morning: take the scooter, the bus (from the central station in town), or a taxi to go to one of these beaches during your stay.
Elia Beach. In the south of the island, superb gay beach with naturist area. The sand is fine, the water is clear and there is music. Really nice spot.
Agrari Beach. Located just on the west side of Elia beach, walk a little past the naturist area. The deckchair rental is a little cheaper than on Elia.
Super Paradise Beach. Lively gay beach with beautiful crystal clear water, but very crowded. There's the Super Paradise Beach Gay Club nearby.
You can also go to Paradise Beach (located right next to the well known Cavo Paradiso club) but don't be mistaken between Paradise Beach and Super Paradise Beach if you have to join your friends in either place.
Kalafatis Beach. Located on the other side of the island, very beautiful beach of fine sand and crystal clear water.
Other beaches in Mykonos are also considered "gay friendly".
Unlike many gay destinations in the world, there is no Gay Cruising Bar or Gay Sauna in Mykonos. Of course, some hotels have their own sauna and spa area.
Nevertheless, you can always try to take a walk around the Orthodox church of Paraportiani after dark… Who knows what you might find there.
August 18 to 25, 2021.
Awaiting final confirmation.
It's an international event not to be missed in Mykonos. All Mykonos gay pride events take place during the XLSIOR Mykonos Festival: Beach Party, WE Party, Deep Party, Vicious Party and of course the parade.
Grindr and Romeo will be your best friends to meet new people in Greece. Even if during the peak season, almost all gay dating applications are used there as Mykonos is an international gay meeting point.
Map of Mykonos
XLSIOR Mykonos Festival
Video source: YouTube – A K Official (XLSIOR) – July 28, 2020. Photo credit: licensed by Shutterstock.com / photographer: Sven Hansche.
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text:
Q: How to design my code so that it can use alternative information to help create an object I have a class named Product, which takes a product model number and number of wheels for that product as part of its creation information, and then it instantiates the product by doing a computation on the number of wheels and on various base parameters
For example, Product(5, 22) creates a model 5 product with 22 wheels on it, where that particular product's weight and dimensions are computed from the number of wheels and from the base specifications for that model number.  Right now base specifications are part of the class itself and are hardcoded into the class, along with the computation formulas.
My Issue:  in the (legacy) code I am working with, I cannot always use the model number directly.  It may not be available.  Instead, an unrelated product_id is given, where the link between product_id and model_number is in the database.
I want to keep my class as simple as possible.  As such I do not want to introduce another constructor that allows product_id as an input parameter, in addition to the existing model_number, as it will be duplication of code.  Also, I am not too keen on putting database logic inside the class, but maybe I can, as thinking about it now, this may be a good candidate for a database-wrapper class, i.e. active record (where my base specifications can be moved off from the code and into the database).
Question:  How do I create and return the object while following good object oriented principles when the model_number I typically use for its creation is not available, but another parameter is available instead (product_id in this case), which links 1:1 to model_number?
Sample solutions that I don't quite  like:

*

*Do not involve the database -- since I am working with just a few Products at this time, and Products do not change often, I can create a "conversion function" that serves as a map between product_id and model_number and not touch the database.  Similarly, create a Convert object that has same functionality and use it before creating the object.  Problem:  duplicating DB functionality in the code.

*Put DB functionality inside code and add alternate constructor.  Problem:  multiple entry-points of Product creation for the class create code duplication.

Update:
There basically are 3 aspects here:

*

*base data (whether hardcoded or part of the database)

*finding base data by various parameters be it model number or other identifying information

*doing computation using base data as per model number

A: Maybe create a ProductFactory, with methods to create a Product given a number of wheels and either a product ID or a model number.  Given a model number it defers to the class' constructor, and if given a product ID it performs the DB lookup, gets the model number, then defers to the class constructor.

A: You don't want to duplicate the DB mapping product_id -> model_number by hardcoding the logic, that makes sense. So the first thing you need is a function  like
int mapProductIdToModelNumber(int product_id)

which does this mapping by getting the information from the database (consider caching when this function is used very often).
What remains is how to make this function available to your Product class without tight coupling. You have the following alternatives:

*

*pass the function directly to the code blocks whereever Product objects are created (depends on your programming language how to accomplish that)

*implement the function in a ProductFactory class. Derive ProductFactory from an interface IProductFactory and pass an object of this type to the code areas where you currently want a new Product(). Use the factory the way @TMN suggested.
That way, your legacy code stays decoupled from the database, since it won't depend directly from a database-bound ProductFactory, only from IProductFactory, which can be easily mocked out (for example, for testing purposes).

A: Maybe this -- give up the idea of a Product class and create a DB-wrapper where I will say $pf = ProductFinder(), then do $pf->findById() and $pf->findByModel() as needed.  Problem:  Where do I put my computation code?  It can't be in the same place, as it it will violate Single Responsibility Principle.  aka, computation and data handling should be separate.  Then maybe Product class can be the computational part, and ProductFinder be the database pulling part.

A: I'd like to build upon the solution @TMN is presenting and take it a little further to answer your SRP problem. Putting database code inside the ProductFactory would be non-sensical so to encapsulate this responsibility you can create a separate class ProductModelNumberProvider with just one method findByProductId(int productId).
The ProductFactory should provide a setter setProductModelNumberProvider(ProductModelNumberProvider lookupProvider) which will assign the provider internally and call findByProductId() when appropriate.
You will have a ProductFactory which is responsible for building the product and a ProductModelNumberProvider which will do the database interaction if needed.
Have a look at your StackOverflow question for a more concrete example.

A: What I actually ended up doing:
At first I was wondering if I am better off with such complexity.
Simpler way was:
There is another pattern here that I could use that keeps things in one class:  https://stackoverflow.com/a/2175213/2883328.  Breaks SRP.
$product = Product::fromProductId($productId, $wheels);

SRP way (chosen)

*

*ProductData class that just has the data (can be swapped by database later)

*ProductFactory class that uses Provider class to look up the model number

*ProductNumberProvider class that provides model number when given product id

*Product class that does loads ProductData and does computations

Code:
$provider = new ProductModelNumberProvider();
$factory = new ProductFactory($provider);
$product = $factory->constructFromProductId($productId, $wheels);

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April 24, 2018. A former NFL player gets in trouble for what he was holding in his daughter's prom picture, Fast and Furious to get its own Netflix animated series. Plus, scientists have finally discovered the true scent of a certain planet we all know and love.
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text:
Tag: women
Categories : Global Political Economy WomenTags : backpagechildrenpolitical economyprostitutionsalesforcesex traffickingsexual exploitationwomen1 Comment
On March 30, 2019 March 30, 2019 By Sara
I recently attended a workshop aimed at preparing a submission for a UN consultation on business & human rights on the role of the so-called 'sex industry' in Australia. One of the key observations made by scholars, practitioners, and activists was that this legal, regulated industry in Australia enjoys a special, exempted status from the scrutiny of business ethics such as the UN Working Group on Business and Human Rights. While there is much to be said about the incongruity of the 'job like any other' discourse regarding the sex industry and mechanisms to ensure corporate responsibility for upholding basic human rights, one aspect I find far less considered is the way in which both legal and illegal forms of sexual exploitation seem to undergird so many aspects of our regular, licit economy.
It struck me from socializing with men and women in the banking and IT sectors how normal it is for workplace socializing to be conducted in strip clubs. A number of my friends and acquaintances admitted the regularity with which their 'team' would wind up after Thursday night drinks at a strip club, while others relayed rumors of their company holding corporate accounts with such establishments. It has made me wonder just how integral is sexual exploitation to lubricating our core economic sectors in an advanced, capitalist country?
With this 'feminist curiosity' in mind, I came across an article about 50 formerly prostituted women suing Salesforce for facilitating their being sexually trafficked. Salesforce is an American Fortune 500 tech company perhaps best known for its 'customer relationship management' (CRM) software, used by millions. This week, fifty "Jane Does" have filed a lawsuit in San Francisco accusing Salesforce of "sex trafficking, negligence, and conspiracy" for providing CRM services to the notorious classified advertising site Backpage.com.
Last year, Backpage made international news when its CEO pleaded guilty to an array of charges associated with sex trafficking and money laundering as a result of publishing ads relating to prostitution, sex trafficking, and the sexual exploitation of minors. This new lawsuit against Salesforce alleges that the company and CEO, Marc Benioff:
engaged in nefarious activities while claiming to be fighting trafficking. "Behind the scenes … Salesforce kept taking Backpage's money and supporting it with the CRM database of pimps, johns, and traffickers that Backpage needed to operate."
Most perniciously, at the same time the company was attempting to paint itself as 'good guys' concerned with global corporate responsibility,
"Salesforce knew the scourge of sex trafficking because it sought publicity for trying to stop it. But at the same time, this publicly traded company was, in actuality, among the vilest of rogue companies, concerned only with their bottom line," the suit alleges. "And human beings—many more than just these 50—were raped and abused because of it."
"The Jane Does were forced, coerced, and made victims of sex trafficking by means of force, fear, fraud, deceit, coercion, violence, duress, menace, or threat of unlawful injury to themselves and others, including family members," the lawsuit claims. "Salesforce committed acts at issue with malice, oppression, fraud, and duress."
The claim of Salesforce's complicity rests on the recognition that the industry of sexually exploiting women and children cannot operate alone, but requires all the same support services, technical services, and financial services as any other. Thus, the claim lodged by the 50 women accuses Salesforce of complicity in actively maximizing "customer acquisition" of johns/punters and aggressive marketing towards this acquisition. As such, the suit claims that not only did Salesforce's technology provide "the backbone of Backpage's exponential growth," but it was also under Salesforce's guidance that Backpage came to dominate the illicit industry in the trafficking and trade in women's and children's bodies.
Such recognition of the complicity of giants in our economy in the sex industry is an important first step. For as research has shown, the sexual exploitation of women and children isn't done just by a handful of deviant, lonely men. Rather, it is sustained by both a widespread culture of acceptance that women's bodies can be bought and used as commodities, as well as by an economic system that enables the trade.
Categories : Peace and Security TerrorismTags : countering violent extremismcounterterrosimcveIraqISISjihadi bridespeace and securitysexual jihadSyriaterrorismUNunited nationswomenWPS1 Comment
Instrumentalizing Women's Security in the Counterterrorism Agenda
On July 17, 2018 July 29, 2018 By Sara
The UN Women, Peace and Security (WPS) agenda has become the dominant international legal and discursive framework for addressing gendered violence in international relations. The United Nations made its first great strides towards recognizing violence against women as an issue of international security in its landmark Security Council Resolution 1325 on Women, Peace and Security. For many, the securitization of gendered violence through 1325 and the subsequent resolutions that now make up the WPS agenda has been a decisive win, elevating to the 'high politics' of security the long-expressed concerns that the effects of war, armed conflict, mass violence, and other forms of international insecurity have distinctly gendered effects on men and women. However, as the WPS agenda has developed, the increasingly instrumentalist and reductive view of 'gender' and of 'violence' employed therein has had unintended impacts on structural, cultural, and economic forms of violence against women.
Since the adoption of SCR 1325, the development of the WPS agenda has been divergent and highly contested. Many feminists have noted with concern the practical bifurcation of the agenda into two parallel, and rarely integrated, concerns: protection versus participation, with 'protection' afforded greater attention and institutional support within the UN and other agents of global governance. This prioritization of 'protecting' women in times of conflict and insecurity has problematically both relied upon and reinforced gendered logics of protection that reproduce stereotypical ideas of women as passive, weak, and victims, in need of saving by rational, heroic, and militarized men.
When, in 2015, the UN Security Council passed a new WPS resolution, 2242, on countering terrorism and violent extremism, some optimistic that it finally represented a shift in perspective from seeing women only as helpless victims of men's violence in armed conflict to also being active agents whose inclusion in political peace processes is central to advancing our peace efforts. Yet, the move to incorporate countering violent extremism (CVE) into the WPS agenda has largely focused on women's roles in preventing radicalization and extremism, further instrumentalizing women's security towards conventionally statist priorities. Key stakeholders in the implementation of both CVE and WPS measures have tended to operate in a way that seeks to fit women and their concerns into existing militarized prerogatives for addressing international security, rather than considering the social conditions that make such insecurities possible. In this way, the incorporation of gender into the CVE agenda may further represent how gender and gender-based violence has become securitized through the WPS agenda.
What has not been systematically addressed in this 'gender mainstreaming' paradigm for CVE are the multiple ways that terrorist groups use highly gendered narratives in their recruitment campaigns to bolster their appeal both to men and women, and how such narratives resonate with their target audiences. For example, ISIS' recruitment narrative relies on hyper-masculinized and violently militarized motifs, portraying their jihadists as 'real men' who are rewarded for their service with promises of a home, monthly allowance, and a wife. The reliance on conventional gender norms and exploitation of gendered anxieties regarding the capacity to fulfill said norms are critical to understanding the appeal of extremist groups.
Yet, within the CVE frameworks, gender remains narrowly understood as relevant only to 'women,' and the interest in gender instrumental to the security of states. Rather than rendering the global security regime developed to combat the threat of terrorism a "gender-friendly" space, global policy on terrorism and counterterrorism "show the continued dominance of a masculine paradigm in those arenas central to international security."
How the WPS-CVE Nexus is Failing Women
In July 2018, thousands of women are being held in Iraq and Syria on terrorism related charges on accusations of their links to Islamic State. Known as "jihadi brides" or "sexual jihadis," these women are being denied basic provisions of human rights and facing punishment on the mere suspicion of their links to IS militants. Facing death sentences for their links to terrorism, the implementation of CVE measures in this case has exacerbated gendered abuses.
Captured in 2017, many of the women are being held in detention camps in legal and political limbo, as their home countries refuse to repatriate them, fearing the spread of radical Islamism. While many of the women are from neighbouring Gulf States, a number also come from Western countries including Germany, France, Russia, and the United States. Others have already been charged with terrorism-related offences and are now facing 10-minute death sentence trials under the Iraqi judicial system. Accused of entering the country illegally and supporting ISIS by living in the caliphate, the thousands of women are facing punishments to the full extent of counter-terror laws in Iraq. Iraqi officials speaking to the New York Times explained:
"These Islamic State criminals committed crimes against humanity and against our people in Iraq, in Mosul and Salahuddin and Anbar, everywhere," said Gen. Yahya Rasool, the spokesman for the Iraqi joint operations command. "To be loyal to the blood of the victims and to be loyal to the Iraqi people, criminals must receive the death penalty, a punishment that would deter them and those who sympathize with them."
According to critics, one of the most egregious outcomes of these women's being swept up in the counter-terror proceedings is that it has also impeded political will to investigate gender-based crimes perpetrated by ISIS, including the systematic use of forced marriage, sexual slavery, and strict curtailments of women's rights. News reports suggest that the lives of 'jihadi brides' were strictly monitored and controlled, with women facing harsh punishments if they behaved in a way considered un-Islamic.
The example of jihadi brides underscores the tensions between the two available subject positions of women in armed conflict. The narratives of these women as dangerous terrorists is premised on the fact that their active participation in violence runs counter to the idealised feminine role we expect of women, ultimately characterizing them as gender deviant. On the other hand, to characterize them solely as unwitting or unwilling victims, subject to the will of their male protectors and guardians, reinforces gendered stereotypes and deny any agency or attachment to political ideals that the women may hold. The explicitly sexual connotation associated with 'sexual jihadis' implies a sexual deviancy and suggests that the women may be getting what they deserve for allowing themselves to be 'seduced' or 'lured' into ISIS in the first place by men more powerful or more clever than themselves.
Yet, even where attitudinal shift has taken place towards women who have been associated with ISIS from seeing them only as victims to also seeing them as active agents of terrorism, the focus has remained on how to instrumentalize women towards operational effectiveness in preventing and countering terrorism and violent extremism. For example, in March 2016 at an event on Gender and CVE, the US Under-Secretary of State Sarah Sewall stated that "empowered women provide powerful antidotes to violent extremism. They are able to refute extremist narratives and nihilistic visions with independence and authenticity. Societies that respect the rights of all and fully engage the participation of all have no room for violent extremism. So women's empowerment is not only essential for defeating violent extremism; defeating violent extremism is essential for women's empowerment. The two go hand-in-hand" (qtd in Chowdhury Fink and Davidian 2018, p. 163). These sentiments are echoed in SCR 2242 itself, which calls for 'the participation and leadership of women and women's organizations in developing strategies to counter terrorism and violent extremism." Such sentiments have sparked concern amongst scholars and civil society activists regarding the co-optation of the WPS agenda in service of counter-terrorism policy and the retention of a problematically narrow scope of concern for what constitutes 'violence' of relevance to international peace and security. Women's unfeminine, unruly behaviour is interpreted as a warning sign and thus efforts to address women's participation in terrorism, or to empower them to join efforts in countering terrorism and violent extremism, still stem from assumptions that women are not independent, political agents.
Gender in the CVE Agenda
Both narratives, while exploiting stereotypical assumptions about 'femininity' and appropriate roles and behaviours of women in relation to violence and armed conflict, work to downplay the significance that politics may have in women's participation in organized violence, including terrorism. Much like the label of 'terrorist' has succeeded in securitizing, and thus de-politicizing, the social and political grievances that lead to violence, or that make participation in a terrorist group attractive. A 2017 report issued by the Royal United Services Institute for Defence and Security Studies (RUSI) showed how reductive the 'jihadi bride' narrative has been, and how far it went towards constructing women in terrorist groups as passive, manipulated, and brainwashed. Contrary to this prevailing narrative, the study found that many women, particularly those coming from Western countries, were drawn by a sense of 'empowerment' offered by ISIS, and were thus "deliberately seeking to challenge both traditional and Western-imposed gender norms, by seeking a new identity for themselves". The report also suggests that exclusion and marginalization from wider Western society were compelling 'push' factors for women to join ISIS. Yet, prevailing efforts to account for women's role in terrorism and counter-terrorism strategies ignore the structural social, political and economic roots of their own involvement, as well as the unique ways that women's social positions may present alternative reasons or pathways for radicalization than their male counterparts.
Thus, while SCR 2242 and the focus on women in combatting violent extremism does, for the first time, shift the prevailing discourse of the WPS agenda from one of protection to one of participation, the scope for participation is restricted. Women's agency and capacity for empowerment is discussed in complete isolation from the underlying social dynamics that both shape their social, economic, and political positions and that underlie the political economic dimensions of violence, armed conflict, and terrorism in the first place. Much like other global initiatives aimed at 'empowering' women, the discourse is designed to 'sell' women's empowerment as good for the economy (or, in this case, good for political stability) rather than as a good in and of itself.
Because international security regimes enabled through processes of securitization enable states to use extraordinary means of force to eliminate threats outside of the oversight of democratic processes and civil society, the marrying of the WPS agenda to national security prerogatives, including the securitization of terrorism and violent extremism, may have quite serious implications for gender and gendered forms of violence. As feminists, we ought to be critically interrogating the extent to which gender-based violence and gendered vulnerabilities are best resolved through their formulation as acute existential threats that require such exceptional responses.
Categories : Masculinity TerrorismTags : AustraliaCanadadomestic violencelawMinnasianpoliceterrorismTorontoVictoriaviolence against womenwomenLeave a comment
When is Terrorism Not Terrorism?
When the political motivations are misogyny, of course.
This week, Canada was rocked by a devastating mass killing, when 25 year-old Alek Minnasian drove a rented van into dozens of people on a busy Toronto street. By the end of the 'van rampage,' 10 people were killed and at least 15 more injured. And although the means and mode of attack very closely resembled some recent 'van rampages' in Europe that have been connected to terrorism, in Canada it shall not be named so.
Minassian's 10th grade yearbook photo, (c) the Toronto Sun.
The carnage was reminiscent of deadly attacks by Islamic State supporters using vehicles that have shaken up Nice, France, Berlin, Barcelona, London and New York. But late Monday, Canada's public safety minister, Ralph Goodale, said this time appeared to be different.
"The events that happened on the street behind us are horrendous," he said, "but they do not appear to be connected in any way to national security based on the information at this time."
It did not take long, though, for media reports to begin digging into Minnasian and discovered the likely motive behind his 'rampage':
While the police did not disclose a motive for the rampage, interviews with former acquaintances of Mr. Minassian, witnesses and others, and his now-deleted Facebook account, portray a troubled young man who harbored resentments toward women, had a penchant for computer programming, served briefly in the military last year, and appeared determined to die.
In a Facebook post made minutes before embarking on this rampage, Minnasian apparently wrote:
"The Incel Rebellion has already begun!" the posting stated. "We will overthrow all the Chads and Stacys! All hail Supreme Gentleman Elliot Rodger!"
While Toronto Police declined to comment on the sex of the victims, it has now been reported that the majority were women.
According to internet culture, 'incels' are 'involuntary celibates', a group of extremely misogynistic men's right's activists who rabidly hate women and blame feminism for, essentially, making women not want to have sex with them. According to researcher Arshy Mann, self-described incels are:
"almost entirely men who are laser-focused on their inability to have sex & blame women. Of the manosphere communities, incels are the most virulently misogynistic."
Amongst this subculture, Elliot Rodger is worshipped as an almost deity. Rodger's own manifesto, the insight into his motive for killing six people on a college campus in California in 2014, expressed frustration over not being able to find a girlfriend, his hatred of women, his contempt for racial minorities and interracial couples, and his plans for what he described as "retribution".
And again, as we saw in the aftermath of Rodger's 'rampage,' this violence is not being called 'terrorism,' but rather the unfortunate effects of one man's 'mental illness.'
Yet, by individualizing the problem of men's violence – especially, in such overt and extreme forms – we lose focus of the way in which through its everydayness, the persistent threat of violence against women is in and of itself a means of terrorizing women. In their provocative book Loving to Survive, Dee L. R. Graham, Roberta Rigsby and Edna Rawlings argue that men's violence fosters in women an omnipresent, and therefore often unrecognized, terror. This terror manifests in protective measures women take against the potential for rape, represented in any strange man she encounters, as well as strategies to reduce their risk of angering men. This omnipresent threat of violence can be theorized as a form of patriarchal terrorism.
Unfortunately, within our existing legal and humanitarian frameworks, we haven't the capacity to even conceive of men's violence as terrorism, namely because this jurisprudence rests on the assumption of 1) an ideological agenda and 2) a community targeted as such. In both domestic and international law, women's experiences have been largely trivialized, overlooked, or relegated to the private sphere of concern because the law has been grounded in the experiences of men.
As with the laws governing crimes against humanity and genocide, a core component of labelling an act of violence as 'terrorism' is the ability to show the offence to be directed against a community and not an individual. That is – is the violence discriminate or indiscriminate? Strangely, though, for all of international legal history, 'women' are not considered to be a community or recognizable group in and of themselves. Rather 'community' is defined strictly in ethnic, racial, religious, or political terms.
As MacKinnon once argued with regards to 'genocidal rape':
The acts of sexual violence perpetrated as an act of genocide are "routinely done to women everywhere every day on the basis of their sex. All the sexual atrocities that become genocidal in genocides are inflicted on women every day under conditions of sex inequality" and are inflicted upon them as women because they are women (MacKinnon 2006: 225).
The exclusion of sex as a 'community' against whom the rape may be used instrumentally may be a deliberate choice by the international community as it would open the door to understanding all rape as political and instrumental. As women are not considered a people, sex has not been included in the legal definition of a group that can be destroyed.
Recognizing this absence, Andrea Dworkin coined the term gynocide to "designate the relentless violence perpetrated by the gender class men against the gender class women" (1976: 16) to express the terrorizing of women through gender-based violence committed world wide at times of war and times of peace. However, it is only when the very same acts perpetrated against women daily are directed against a group of people on the basis of race, religion, ethnicity, or nationality, it is recognized as destructive.
Thus, despite overwhelming evidence that not only was Minassian's motive ideological, and that his violence was used in the pursuit of a political aim (read: textbook definition of 'terrorism'), and that his target was women, as such, the rhetoric remains one of maladjustment, personal trauma, and mental health issues. Despite overwhelming evidence we now have in the West that these 'lone-wolf' mass murderers nearly all have a history of violence against women, we are loathe to consider how that violence may be, in and of itself, political.
David Futrelle has been researching this virulently misogynist online movement through his site We Hunted the Mammoth, and explains both the breadth and depth of the 'Incel' phenomenon. In his post on Minassian, Futrelle includes screenshots from one forum user, BlkPillPres, who advocates more effective mass violence in pursuit of the 'Incel Rebellion.' To him, mass shootings are ineffective. He says:
"What I can't wait for, the one I know is really going to fuck with normies, really punish society, is when the first incel mass rape/serial rape takes place, when a guy leaves a manifesto after killing himself detailing all the rapes he's done, that will be the best ER ever, because his victims don't just get to die, and their families don't just get to 'move on'…"
Futrelle warns that these are not isolated sentiments, but warnings of a larger subcultural movement that could have very serious social repercussions. In a piece written for Elle, he warns:
"[The Incel movement] has transformed young men dealing with depression — or simply the ordinary unhappiness of life — into a veritable underground army of angry, bitter misogynists who feel they have nothing to live for and have no hope of improving their lives in what they see as our "gynocentric" society.
If these young men aren't stopped, there will be more horrors like what we saw this week in Toronto, if not worse. In the forums on incel hangout Incels.me, some are already hailing Minassian as a hero, and looking forward to the next wave of incel terror attacks."
In Australia, recent receptivity on the parts of the Victorian Police, for example, towards feminist arguments about the sociopolitical basis of men's violence may be bucking this trend. Late last year, the Victorian Police announced that they would be treating domestic violence perpetrators like terrorists. In a statement, Acting Chief Commissioner Shane Patton, argued:
"The ramifications are the same in the long run. We have death, we have serious trauma, we have serious injury and we have people impacted for the rest of their lives."
Intimate terrorism will now be investigated as major crimes by specialized police units. One of their priorities will be to target repeat offenders and work to predict violence in order to intervene before women and children are injured or killed.
Yet, we should temper our enthusiasm, since only a couple of short months after this initiative was announced, the Victorian government also announced a new A$31.6 million centre to prevent and combat terrorist and lone-actor attacks. The centre, problematically, includes no experts on domestic violence or violence against women. This suggests we have a long way to go towards politicizing men's violence against women not only as acts of terror in and of themselves, but also as 'warning signs' of the potential for public forms of large-scale violence.