pydata_london_2019.tex

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\title[The unreasonable effectiveness of feature hashing]%
      {The unreasonable effectiveness of\\feature hashing}
\author[Gianluca Campanella]%
       {Gianluca Campanella
        \fontspec{Gentium}\textcolor{gray}{[dʒanˈluːka kampaˈnɛlla]}}
\date{14\textsuperscript{th} July 2019}

\begin{document}

\maketitle

\begin{frame}{What I do nowadays}
    \LARGE%
    \only<1>{%
        \begin{center}
            I'm a Data Scientist at \\[\bigskipamount]
            \includegraphics[height=2.5em]{figures/microsoft} \\[\medskipamount]
            in \textbf{AzureCAT}
        \end{center}}
    \only<2>{%
        \begin{center}
            I also run my own company \\[\bigskipamount]
            \raisebox{-0.5\height}{\includegraphics[height=2.5em]{figures/estimand}}
            \raisebox{-0.5\height}{\huge Estimand} \\[\medskipamount]
            that provides \\
            \textbf{Data Science consulting and training}
        \end{center}}
\end{frame}

\begin{frame}{Contents}
    \tableofcontents[hideallsubsections]
\end{frame}

\begin{frame}{Acknowledgements}
    \begin{itemize}
        \setlength\itemsep{\bigskipamount}
        \item Sharat Chikkerur
        \item Markus Cozowicz
    \end{itemize}
\end{frame}

\section{Background}

\begin{frame}{Supervised ML}
    \begin{itemize}
        \item Observe $\mat{X}_{n \times p}$ and $\vec{y}_{n}$
              (typically $n \gg p$)
        \item Find the `best' estimator of some statistic of $\vec{y}$ given
              $\mat{X}$ \\
              (for example $\E{\vec{y} | \mat{X}}$)
    \end{itemize}
    \vfill\pause
    \begin{block}{Generalised linear models (GLMs)}
        \begin{itemize}
            \item $\E{\vec{y} | \mat{X}} = g^{-1} \left( \mat{X} \vec{\beta} \right)$
                  with $g$ given
            \item Find the `best' estimates for $\vec{\beta}$
        \end{itemize}
    \end{block}
\end{frame}

\begin{frame}{Features are often categorical}
    \begin{itemize}
        \setlength\itemsep{\bigskipamount}
        \item Discretised continuous features
        \item One\hyp{}hot encoding
        \item Bag\hyp{}of\hyp{}words representation
    \end{itemize}
\end{frame}

\begin{frame}{One-hot encoding}
    \centering
    \begin{tabular}{l}
        \toprule
        \texttt{country} \\
        \midrule
        England \\
        Wales \\
        Scotland \\
        Scotland \\
        Northern Ireland \\
        \bottomrule
    \end{tabular}
    $\to$
    $
        \kbordermatrix{
            & \texttt{ENG} & \texttt{NIR} & \texttt{SCT} & \texttt{WLS} \\
            & 1 & 0 & 0 & 0 \\
            & 0 & 0 & 0 & 1 \\
            & 0 & 0 & 1 & 0 \\
            & 0 & 0 & 1 & 0 \\
            & 0 & 1 & 0 & 0
        }
    $
\end{frame}

\begin{frame}{Bag-of-words representation}
    \centering
    \begin{tabular}{l}
        \toprule
        \texttt{document} \\
        \midrule
        Ashley loves cats. She also likes dogs. \\
        Barbara hates cats but she loves dogs. \\
        Carol loves cats and dogs. \\
        \bottomrule
    \end{tabular}
    \vfill
    $\downarrow$
    \vfill
    $
        \kbordermatrix{
            & \texttt{cat} & \texttt{dog} & \texttt{hate} & \texttt{like} & \texttt{love} \\
            & 1 & 1 & 0 & 1 & 1 \\
            & 1 & 1 & 1 & 0 & 1 \\
            & 1 & 1 & 0 & 0 & 1
        }
    $
\end{frame}

\begin{frame}{High-dimensional feature space}
    \only<1>{%
        \begin{block}{What's going on?}
            \begin{itemize}
                \item $n$ is fixed
                \item $p$ depends on the cardinality of features
            \end{itemize}
        \end{block}}
    \only<2>{%
        \begin{block}{Why is this a problem?}
            \begin{itemize}
                \item Online learning
                \item $p \to n$ breaks GLMs (though regularisation helps)
            \end{itemize}
        \end{block}}
    \only<3>{%
        \centering
        \raisebox{-0.5\height}{\includegraphics[height=0.5\textheight]{figures/cat}}
        \quad
        $\to$
        \quad
        \raisebox{-0.5\height}{\includegraphics[height=0.2\textheight]{figures/ai}}
        \quad
        $\to$
        \quad
        $
            \kbordermatrix{
                & \texttt{cat} & \texttt{dog} \\
                & 0.99         & 0.01         \\
            }
        $}
    \only<4>{%
        \begin{block}{Observations in $\mat{X}$ are high\hyp{}dimensional but
                      \alert{sparse}}
            \begin{itemize}
                \item Ad tech
                \item E\hyp{}commerce
                \item Social networks
            \end{itemize}
        \end{block}}
    \only<5>{%
        \begin{block}{\alert{Can we find a `smaller' $\mat{X}$?}}
            \begin{itemize}
                \item Time\hyp{} and space\hyp{}efficient to compute
                \item Fixed $p$
                \item Sparse
            \end{itemize}
        \end{block}}
\end{frame}

\begin{frame}{Recap}
    \begin{itemize}
        \setlength\itemsep{\bigskipamount}
        \item Categorical and textual features are common
              \begin{itemize}
                  \item[$\to$] $p \propto$ feature cardinality
                               (often very large)
                  \item[$\to$] Sparse $\mat{X}$
              \end{itemize}
        \item Want an efficiently computed, fixed\hyp{}$p$, sparse
              representation
    \end{itemize}
\end{frame}

\section{Feature hashing}

\begin{frame}[fragile]{Feature hashing}
    \only<1>{%
        \begin{center}
            \texttt{country} = `England'
            \vfill
            $\downarrow$
            \vfill
            Hash value \\
            {\footnotesize%
             $h \left( \texttt{country=England} \right) = 1462978411$}
            \vfill
            $\downarrow$
            \vfill
            Feature index \\
            {\footnotesize%
             $i = h \left( \texttt{country=England} \right) \operatorname{mod} 100 = 11$}
        \end{center}}
    \only<2>{%
        \centering
        \begin{tabular}{l}
            \toprule
            \texttt{country} \\
            \midrule
            England \\
            Wales \\
            Scotland \\
            Scotland \\
            Northern Ireland \\
            \bottomrule
        \end{tabular}
        $\to$
        $
            \kbordermatrix{
                & i \\
                & 11 \\
                & 62 \\
                & 54 \\
                & 54 \\
                & 8 \\
            }
        $
        $\to$
        $
            \kbordermatrix{
                & \ldots & \vec{x}_{8} & \ldots & \vec{x}_{11} & \ldots & \vec{x}_{54} & \ldots & \vec{x}_{62} & \ldots \\
                & & & & 1 & & & & & \\
                & & & & & & & & 1 & \\
                & & & & & & 1 & & & \\
                & & & & & & 1 & & & \\
                & & 1 & & & & & & &
            }
        $}
    \only<3>{%
        \begin{itemize}
            \item Time\hyp{} (\texttt{\%timeit}: 468 ns ± 7.2 ns) and
                  space\hyp{}efficient
            \item Fixed $p$
            \item Sparse
        \end{itemize}}
\end{frame}

\begin{frame}{Hash function}
    \begin{center}
        $h$ maps \alert{any} integer input onto integers in some range
        (e.g.\ \texttt{int32})
    \end{center}
    \vfill
    \begin{block}{Important properties}
        \begin{itemize}
            \item Uniform output
            \item Avalanche effect
                  \begin{itemize}
                      \item $h \left( \texttt{a} \right) \to 3001393763$
                      \item $h \left( \texttt{c} \right) \to 1701913768$
                  \end{itemize}
        \end{itemize}
    \end{block}
\end{frame}

\begin{frame}[fragile]{Feature hashing in Python}
    \begin{py3}
        import mmh3

        s = "country=England"

        h = mmh3.hash(s, seed=42, signed=False)
        print(h)  # => 1462978411

        i = h % 100
        print(i)  # => 11
    \end{py3}
\end{frame}

\begin{frame}[fragile]{Hashing of Unicode strings}
    \begin{itemize}
        \item Strings are just (very large) integers
        \item Be careful when handling Unicode
    \end{itemize}
    \vfill
    \begin{py3}
        s1 = "sch\u00f6n"
        s2 = "scho\u0308n"

        print(s1)        # => schön
        print(s2)        # => schön
        print(s1 == s2)  # => False
    \end{py3}
\end{frame}

\begin{frame}[fragile]{Unicode normalisation}
    \begin{py3}
        import unicodedata

        s1 = unicodedata.normalize("NFKD", "sch\u00f6n")
        s2 = unicodedata.normalize("NFKD", "scho\u0308n")

        print(s1)        # => schön
        print(s2)        # => schön
        print(s1 == s2)  # => True
    \end{py3}
\end{frame}

\begin{frame}[fragile]{Projection}
    \begin{itemize}
        \item The modulo operator is expensive
        \item If the hash size is a power of two, we can do better
    \end{itemize}
    \vfill
    \begin{py3}
        %timeit h % 128
        # => 81.3 ns ± 1.92 ns per loop

        %timeit h & 127
        # => 56.9 ns ± 1.4 ns per loop
    \end{py3}
\end{frame}

\begin{frame}[fragile]{Collisions}
    \begin{py3}
        def h(x):
            return mmh3.hash(x, seed=42, signed=False) & 127

        print(h("country=United Kingdom"))  # => 6
        print(h("country=Bulgaria"))        # => 6
    \end{py3}
    \vfill
    \begin{itemize}
        \item Smaller hash size $\to$ more collisions
        \item Impact on statistical performance and interpretability
    \end{itemize}
\end{frame}

\begin{frame}[fragile]{Sign function $\xi$}
    \begin{itemize}
        \item Use another function $\xi$ to determine the sign
        \item Collisions cancel out (in expectation)
    \end{itemize}
    \vfill
    \begin{py3}
        def h(x):
            hash_ = mmh3.hash(x, seed=42, signed=True)
            return abs(hash_) & 127, (hash_ >= 0) * 2 - 1

        print(h("country=United Kingdom"))  # => (  6,  1)
        print(h("country=Bulgaria"))        # => (122, -1)
    \end{py3}
\end{frame}

\begin{frame}{Recap}
    \begin{itemize}
        \setlength\itemsep{\bigskipamount}
        \item $h$ maps any integer input onto integers in some range
        \item $h$ deterministically scrambles arbitrary\hyp{}length bitmaps,
              producing fixed\hyp{}length bitmaps
        \item The impact of collisions on statistical performance is small
    \end{itemize}
\end{frame}

\subsection{Example}

\begin{frame}{Original data}
    \centering
    \begin{tabular}{llrrr}
        \toprule
        \texttt{user} & \texttt{movie}    & \texttt{has\_cats} & \texttt{has\_dogs} & \texttt{rating} \\
        \midrule
        Ashley        & A Dog's Journey   & 0                  & 1                  & 4               \\
        Ashley        & Catwoman          & 1                  & 0                  & 5               \\
        Ashley        & The Aristocats    & 1                  & 1                  & 5               \\
        Ashley        & The Queen's Corgi & 0                  & 1                  & 3               \\
        Barbara       & A Dog's Journey   & 0                  & 1                  & 5               \\
        Barbara       & The Aristocats    & 1                  & 1                  & 2               \\
        Barbara       & The Queen's Corgi & 0                  & 1                  & 5               \\
        Carol         & Catwoman          & 1                  & 0                  & 5               \\
        Carol         & The Aristocats    & 1                  & 1                  & 5               \\
        Carol         & The Queen's Corgi & 0                  & 1                  & 4               \\
        \bottomrule
    \end{tabular}
\end{frame}

\begin{frame}{After feature hashing}
    \only<1>{%
        \centering
        \begin{tabular}{rrrr}
            \toprule
            \texttt{user} & \texttt{movie} & \texttt{has\_cats} & \texttt{has\_dogs} \\
            \midrule
            $2^{-}$       & $5^{-}$        &                    & $110^{-}$          \\
            $2^{-}$       & $1^{-}$        & $9^{+}$            &                    \\
            $2^{-}$       & $104^{-}$      & $9^{+}$            & $110^{-}$          \\
            $2^{-}$       & $67^{+}$       &                    & $110^{-}$          \\
            $19^{+}$      & $5^{-}$        &                    & $110^{-}$          \\
            $19^{+}$      & $104^{-}$      & $9^{+}$            & $110^{-}$          \\
            $19^{+}$      & $67^{+}$       &                    & $110^{-}$          \\
            $77^{+}$      & $1^{-}$        & $9^{+}$            &                    \\
            $77^{+}$      & $104^{-}$      & $9^{+}$            & $110^{-}$          \\
            $77^{+}$      & $67^{+}$       &                    & $110^{-}$          \\
            \bottomrule
        \end{tabular}}
    \only<2>{%
        \centering
        $
            \kbordermatrix{
                & \ldots & \vec{x}_{1} & \ldots & \vec{x}_{2} & \ldots & \vec{x}_{5} & \ldots & \vec{x}_{9} & \ldots & \vec{x}_{19} & \ldots & \vec{x}_{67} & \ldots & \vec{x}_{77} & \ldots & \vec{x}_{104} & \ldots & \vec{x}_{110} & \ldots \\
                & &    & & -1 & & -1 & &    & &    & &    & &    & &    & & -1 & \\
                & & -1 & & -1 & &    & &  1 & &    & &    & &    & &    & &    & \\
                & &    & & -1 & &    & &  1 & &    & &    & &    & & -1 & & -1 & \\
                & &    & & -1 & &    & &    & &    & &  1 & &    & &    & & -1 & \\
                & &    & &    & & -1 & &    & &  1 & &    & &    & &    & & -1 & \\
                & &    & &    & &    & &  1 & &  1 & &    & &    & & -1 & & -1 & \\
                & &    & &    & &    & &    & &  1 & &  1 & &    & &    & & -1 & \\
                & & -1 & &    & &    & &  1 & &    & &    & &  1 & &    & &    & \\
                & &    & &    & &    & &  1 & &    & &    & &  1 & & -1 & & -1 & \\
                & &    & &    & &    & &    & &    & &  1 & &  1 & &    & & -1 & \\
            }
        $}
\end{frame}

\begin{frame}[t]{Model}
    \[
        \texttt{rating} \sim \alert<2>{\texttt{user}} + \alert<3>{\texttt{movie}} + \alert<4>{\texttt{has\_cats}} + \alert<4>{\texttt{has\_dogs}}
    \]
    \vfill
    \begin{itemize}
        \item \alert<2>{Average effect for users}
        \item \alert<3>{Average effect for movies}
        \item \alert<4>{Average effect for movie\hyp{}related features}
    \end{itemize}
\end{frame}

\begin{frame}[t]{Interactions}
    \[
        \texttt{rating} \sim \texttt{user} + \texttt{movie} + \alert{\texttt{user} \times \left( \texttt{has\_cats} + \texttt{has\_dogs} \right)}
    \]
    \vfill
    \begin{itemize}
        \item Average effect for users
        \item Average effect for movies
        \item Average effect for movie\hyp{}related features
        \item \alert{User correction}
    \end{itemize}
\end{frame}

\begin{frame}[fragile]{Interactions with feature hashing}
    \centering
    \begin{columns}
        \begin{column}{0.5\textwidth}
            \centering
            $h \left( \texttt{user=Ashley} \right) = 2^{-}$ \\
            \vspace{0.5em}
            $\searrow$
        \end{column}
        \begin{column}{0.5\textwidth}
            \centering
            $h \left( \texttt{has\_cats} \right) = 9^{+}$ \\
            \vspace{0.5em}
            $\swarrow$
        \end{column}
    \end{columns}
    \vspace{1em}
    \begin{py3}
        def interact(h1, h2):
            i1, s1 = h1
            i2, s2 = h2
            return ((i1 ^ i2) * 16777619) & 127, s1 * s2
    \end{py3}
    \vspace{0.5em}
    $\downarrow$ \\
    \vspace{0.5em}
    $h \left( \texttt{user=Ashley} \times \texttt{has\_cats} \right) = 81^{-}$
\end{frame}

\begin{frame}{After feature hashing with interactions}
    \only<1>{%
        \centering
        \begin{tabular}{rrrrrr}
            \toprule
            \texttt{user} & \texttt{movie} & \texttt{has\_cats} & \texttt{has\_dogs} & \texttt{likes\_cats} & \texttt{likes\_dogs} \\
            \midrule
            $2^{-}$       & $5^{-}$        &                    & $110^{-}$          &                      & $4^{+}$              \\
            $2^{-}$       & $1^{-}$        & $9^{+}$            &                    & $81^{-}$             &                      \\
            $2^{-}$       & $104^{-}$      & $9^{+}$            & $110^{-}$          & $81^{-}$             & $4^{+}$              \\
            $2^{-}$       & $67^{+}$       &                    & $110^{-}$          &                      & $4^{+}$              \\
            $19^{+}$      & $5^{-}$        &                    & $110^{-}$          &                      & $71^{-}$             \\
            $19^{+}$      & $104^{-}$      & $9^{+}$            & \alert{$110^{-}$}  & \alert{$110^{+}$}    & $71^{-}$             \\
            $19^{+}$      & $67^{+}$       &                    & $110^{-}$          &                      & $71^{-}$             \\
            $77^{+}$      & $1^{-}$        & $9^{+}$            &                    & $12^{+}$             &                      \\
            $77^{+}$      & $104^{-}$      & $9^{+}$            & $110^{-}$          & $12^{+}$             & $25^{-}$             \\
            $77^{+}$      & $67^{+}$       &                    & $110^{-}$          &                      & $25^{-}$             \\
            \bottomrule
        \end{tabular}}
    \only<2>{%
        \centering
        $
            \kbordermatrix{
                & \vec{x}_{1} & \vec{x}_{2} & \vec{x}_{4} & \vec{x}_{5} & \vec{x}_{9} & \vec{x}_{12} & \vec{x}_{19} & \vec{x}_{25} & \vec{x}_{67} & \vec{x}_{71} & \vec{x}_{77} & \vec{x}_{81} & \vec{x}_{104} & \vec{x}_{110} \\
                &    & -1 &  1 & -1 &    &    &    &    &    &    &    &    &    & -1        \\
                & -1 & -1 &    &    &  1 &    &    &    &    &    &    & -1 &    &           \\
                &    & -1 &  1 &    &  1 &    &    &    &    &    &    & -1 & -1 & -1        \\
                &    & -1 &  1 &    &    &    &    &    &  1 &    &    &    &    & -1        \\
                &    &    &    & -1 &    &    &  1 &    &    & -1 &    &    &    & -1        \\
                &    &    &    &    &  1 &    &  1 &    &    & -1 &    &    & -1 & \alert{0} \\
                &    &    &    &    &    &    &  1 &    &  1 & -1 &    &    &    & -1        \\
                & -1 &    &    &    &  1 &  1 &    &    &    &    &  1 &    &    &           \\
                &    &    &    &    &  1 &  1 &    & -1 &    &    &  1 &    & -1 & -1        \\
                &    &    &    &    &    &    &    & -1 &  1 &    &  1 &    &    & -1        \\
            }
        $}
\end{frame}

\begin{frame}{Pros and cons}
    \begin{block}{Pros}
        \begin{itemize}
            \item Time\hyp{} and space\hyp{}efficient $\to$ online learning
            \item Sparsity\hyp{}preserving
            \item Implicit handling of missing data
        \end{itemize}
    \end{block}
    \vfill
    \begin{block}{Cons}
        \begin{itemize}
            \item Collisions $\to$ statistical performance
            \item Inverse mapping $\to$ interpretability
        \end{itemize}
    \end{block}
\end{frame}

\subsection{Use case}

\begin{frame}{Scenario}
    \begin{itemize}
        \item Business directory service (think Yelp)
        \item Want to improve relevance of and personalise search results
    \end{itemize}
    \vfill\pause
    \begin{columns}
        \begin{column}{0.5\textwidth}
            \begin{center}
                \large\bf%
                CTR optimisation
            \end{center}
            Maximise probability of click on top results
        \end{column}
        \begin{column}{0.5\textwidth}
            \begin{center}
                \large\bf%
                Personalisation
            \end{center}
            Take into account past user interactions
        \end{column}
    \end{columns}
\end{frame}

\begin{frame}{Data}
    \begin{center}
        \begin{tabular}{lllr}
            \toprule
            \texttt{user} & \texttt{query}     & \texttt{business}      & \texttt{click} \\
            \midrule
            A             & Italian restaurant & Bocca di Lupo          & 1              \\
            A             & Italian restaurant & Brasserie Zédel        & 0              \\
            A             & Italian restaurant & Emilia's Crafted Pasta & 0              \\
            A             & Italian restaurant & Fucina                 & 1              \\
            A             & Italian restaurant & Trullo                 & 0              \\
            \bottomrule
        \end{tabular}
    \end{center}
    \vfill
    \begin{itemize}
        \item $3.5 \times 10^{8}$ observations ($250$ days)
        \item $4.5 \times 10^{6}$ businesses with many (sparse) attributes
    \end{itemize}
\end{frame}

\begin{frame}{Model}
    \only<1-4>{%
        \[
            \begin{split}
                \texttt{click} \sim&\;\alert<2>{\texttt{user}} + \alert<2>{\texttt{query}} + \alert<2>{\texttt{business}} + \\
                                    &\;\alert<3>{\texttt{query} \times \texttt{business}} + \alert<4>{\texttt{user} \times \texttt{business}}
            \end{split}
        \]
        \vfill
        \begin{itemize}
            \item \alert<2>{Average effects}
            \item \alert<3>{Relevance correction}
            \item \alert<4>{User correction}
        \end{itemize}}
    \only<5>{%
        \begin{itemize}
            \setlength\itemsep{\bigskipamount}
            \item Fixed $p = 2^{23} \approx 8.4 \times 10^{6}$
            \item Regularised logistic regression (Spark ML)
            \item Grid search over elastic net hyperparameters
        \end{itemize}}
\end{frame}

\begin{frame}{Results}
    \begin{itemize}
        \setlength\itemsep{\bigskipamount}
        \item Estimated $\approx 5\%$ increase in CTR based on train/test split
        \item A/B testing showed $\approx 10\%$ increase in CTR
    \end{itemize}
\end{frame}

\section{Library support}

\begin{frame}{Vowpal Wabbit (VW)}
    \begin{itemize}
        \setlength\itemsep{\bigskipamount}
        \item C++ with Python bindings
        \item Linear, logistic and Poisson regression with interactions
        \item Extremely fast and scalable
    \end{itemize}
\end{frame}

\begin{frame}{\texttt{scikit-learn}}
    \begin{itemize}
        \setlength\itemsep{\bigskipamount}
        \item Some support for feature hashing:
              \texttt{feature\_extraction.FeatureHasher} and
              \texttt{feature\_extraction.text.HashingVectorizer}
        \item No interactions
    \end{itemize}
\end{frame}

\begin{frame}{Apache Spark}
    \begin{itemize}
        \setlength\itemsep{\bigskipamount}
        \item Some support for feature hashing in MLlib:
              \texttt{ml.feature.FeatureHasher} and
              \texttt{ml.feature.HashingTF}
        \item PySpark transformers (hashing and interactions)
        \item VW bindings coming soon to MMLSpark
    \end{itemize}
\end{frame}

\begin{frame}{Recap}
    \begin{block}{Feature hashing is\ldots}
        \begin{itemize}
            \item Great for online training on lots of data
            \item Well\hyp{}suited for high\hyp{}cardinality features
                  (sparse $\mat{X}$)
            \item Even better with interactions
        \end{itemize}
    \end{block}
\end{frame}

\begin{frame}
    \begin{center}
        {\LARGE%
         Thank you!}
        \vfill
        If you want to keep in touch\ldots \\[\medskipamount]
        \includegraphics[height=1ex]{figures/mail}~gianluca@campanella.org
        \hspace{1em}
        \includegraphics[height=1ex]{figures/github}~gcampanella
        \hspace{1em}
        \includegraphics[height=1ex]{figures/linkedin}~gcampanella
    \end{center}
\end{frame}

\end{document}