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<article xmlns:xlink="http://www.w3.org/1999/xlink">
  <front>
    <journal-meta />
    <article-meta>
      <title-group>
        <article-title>Collaborative Learning of Concept Representations for Video Data</article-title>
      </title-group>
      <contrib-group>
        <contrib contrib-type="author">
          <string-name>Francisco Torres</string-name>
          <email>torres@parc.com</email>
          <xref ref-type="aff" rid="aff0">0</xref>
          <xref ref-type="aff" rid="aff1">1</xref>
        </contrib>
        <contrib contrib-type="author">
          <string-name>Hoda Eldardiry</string-name>
          <xref ref-type="aff" rid="aff0">0</xref>
          <xref ref-type="aff" rid="aff1">1</xref>
        </contrib>
        <contrib contrib-type="author">
          <string-name>Gaurang Gavai</string-name>
          <xref ref-type="aff" rid="aff0">0</xref>
          <xref ref-type="aff" rid="aff1">1</xref>
        </contrib>
        <contrib contrib-type="author">
          <string-name>Chad Ramos</string-name>
          <xref ref-type="aff" rid="aff0">0</xref>
          <xref ref-type="aff" rid="aff1">1</xref>
        </contrib>
        <aff id="aff0">
          <label>0</label>
          <institution>Copyright held by the author(s). In A. Martin, K. Hinkelmann, A. Gerber</institution>
          ,
          <addr-line>D. Lenat, F. van Harmelen, P. Clark (Eds.)</addr-line>
          ,
          <institution>Proceedings of the AAAI 2019 Spring Symposium on Combining Machine Learning with Knowledge Engineering (AAAI-MAKE 2019). Stanford University</institution>
          ,
          <addr-line>Palo Alto, California</addr-line>
          ,
          <country country="US">USA</country>
        </aff>
        <aff id="aff1">
          <label>1</label>
          <institution>Palo Alto Research Center</institution>
          ,
          <addr-line>3333 Coyote Hill Road, Palo Alto, CA</addr-line>
          <country country="US">USA</country>
        </aff>
      </contrib-group>
      <abstract>
        <p>We present an approach for collaborative learning of representations for concepts embodied in unstructured datasets. Our approach learns both machine and expert interpretable representations, where ”expert” refers to expertise in the concept domain, as distinguished from a machine learning expert. In this paper, we focus on mining video data, but the approach is applicable to other types of data as well. Given a large video dataset and an expert-provided example clip that captures some notion of a desired concept, our proposed technology collaborates with the expert to discern the concept, even when it is not initially clear in the expert's mind. Since the expert may be exploring different possibilities, labeling a large set of data and then training a classifier to recognize the concept is not the right approach because it lacks the needed agility. For the learning to happen as the concept evolves, we use “learn-evolve-explain” cycles that generate (1) deep representations of the discerned concept, which can be used by a model for queries, and (2) visual representations that explains the discerned concept to the human expert. We summarize open source software developed to perform a collaborative video query and discuss our proposed road map for future work.</p>
      </abstract>
    </article-meta>
  </front>
  <body>
    <sec id="sec-1">
      <title>-</title>
      <p>Key to the impact of many emerging Artificial Intelligence
technologies is the collaborative nature of the learning.
Many systems leverage a variety of complementary
players, and the better these players can collaborate, the more
efficient the learning becomes. These collaborating players
include machine learning programs, humans, and physical
subsystems, each playing various roles. Machine learning
methods learn models, inferentially process data streams,
mine for interesting patterns, and sometimes generate
explanations of results. Physical subsystems sense the
environment and capture changing contexts. Human participants
manage high level activities and mission development by
creating demonstrations for teaching, providing feedback on
the output, and strategically reorganizing the machine
learning and physical subsystems when necessary. Ideally, human
users can delegate much of the more mundane work to
machine learning and physical subsystems, allowing the human
expert to focus on high level context and goals.
1.1</p>
    </sec>
    <sec id="sec-2">
      <title>Learning challenges</title>
      <p>This work focuses on two players in a collaborative learning
approach: (1) the machine learning program, and (2) a
human with expertise in the concept under investigation. Our
learning methodology enables an agile and exploratory
collaboration, while addressing the challenges listed below.</p>
      <p>Typical machine learning constraints that can be relaxed
by collaborative learning include:</p>
      <p>Up-front problem specification. Most learning
approaches require the expert to concretely specify the
problem they are trying to solve up front. This limits the
ability to conduct exploratory learning. Users who are not
machine learning practitioners would benefit from an
approach where the user presents one to a few instances of
some concept of interest and then engages in an intuitive
collaboration.</p>
    </sec>
    <sec id="sec-3">
      <title>Labeled data requirements. Supervised machine learn</title>
      <p>ing typically relies on a lot of labeled data for training.
This is expensive and poses a burden on the expert.</p>
    </sec>
    <sec id="sec-4">
      <title>Unexplained model behaviors. When an inference</title>
      <p>model generates an output without some explanation, an
expert may not understand why this output was generated.
Expecting the expert to fill in this understanding gap by
observing model behavior for a large enough set of
examples is not practicable at the scale and complexity of many
modern AI systems. In the absence of such
understanding, using the overall system outside of a narrow context
becomes risky, yet broader use may be necessary to
accomplish strategic goals or react to a changing context.
1.2</p>
    </sec>
    <sec id="sec-5">
      <title>Expert-guided collaborative learning of concepts</title>
      <p>Consider a domain expert studying a particular concept that
arises throughout a large video dataset. This expert wants
a technology that can locate video clips that capture the
concept, since manual review of all the video is
impractical. Our approach is to build a technology that uses
“learnevolve-explain” cycles; in this paper we report results for the
”learn” and ”evolve” steps and discuss our proposal on how
to architect the ”explain” step. In our approach, the expert
begins by providing an example video clip. This clip will
show some notion of the desired scenario, but it will have
activities and objects superfluous to the scope of the target
concept as well. Furthermore, the expert may evolve her
understanding of concept details and nuances as the algorithm
builds representations for recognizing the intended concept
and gets feedback from the expert. The learn-evolve-explain
cycle works as follows:</p>
    </sec>
    <sec id="sec-6">
      <title>LEARN: Learn feature representation of the concept depicted by the clip.</title>
      <p>1) Use an ensemble of deep neural networks pre-trained
on action recognition to extract deep feature
representations of the example clip.</p>
      <p>EVOLVE: Collaboratively evolve and clarify the
concept.
2) Search for similar clips using ensemble scoring.
3) Present proposed matching clips to the expert. Also
present “near misses” so the expert sees actions that the
algorithm considers to be outside the scope of the target
concept.
4) Get an expert’s feedback on the search results, and then
learn a better machine representation of the concept of
interest. By giving feedback, the expert also implicitly
clarifies ambiguities and uncertainties, both for the algorithm
and herself.
5) Return to step 1 for another learning cycle, until the
expert is satisfied.</p>
      <p>EXPLAIN: Generate a human-interpretable
representation.
6) After enough iterations of Learn + Evolve steps, a
generative algorithm generates an ”evaluation” video clip that
focuses on the target concept, suppressing and
minimizing additional content.
7) The expert either accepts this explanatory clip, or
engages in further learn-evolve-explain cycles.</p>
      <p>The learning algorithm and the expert jointly discern the
target concept of interest starting with an example clip,
without requiring a fixed concrete problem specification up front.
In steps 2 through 5, the learning of the concept of interest
evolves using an expert-guided ongoing collaborative
approach. As part of step 5, a target bootstrapping algorithm
takes its internal representations of all matching clips
validated by the expert and forms a consensus representation, all
in terms of machine feature representations. Finally, steps 6
and 7 aim to create a human-understandable representation
that explains the internal model representation of the target
concept.
1.3</p>
    </sec>
    <sec id="sec-7">
      <title>Technology elements</title>
      <p>The technology elements of this work can be summarized as
follows:</p>
      <p>A video clip representation approach (section 3)
A search algorithm (section 4)
A method to refine the search given expert user feedback
(section 5)
A method to refine the machine representation of the
target concept given expert user collaborative feedback
(section 6)
A method to generate a clip that captures only the concept,
while de-emphasizing other details (future work on our
proposed road map, section 9)
2</p>
      <sec id="sec-7-1">
        <title>Related work</title>
        <p>
          Our approach uses transfer learning, in that it uses
embeddings from pre-trained deep neural nets in a context other
than the original training objective. Using embeddings from
a pre-trained deep neural net in a context other than the
training objective often proves useful
          <xref ref-type="bibr" rid="ref4">(Goodfellow et al. 2016)</xref>
          .
        </p>
        <p>
          Also, active deep learning is similar in some respects
to the expert-guided collaborative learning we are
describing in this paper; both aim to make best use of the
expert’s time. For example, Gal et al
          <xref ref-type="bibr" rid="ref2">(Gal, Islam, and
Ghahramani 2017)</xref>
          take advantage of specialized models such as
Bayesian neural networks to construct efficient deep active
learning paradigms. Interesting strategies such as deep
adversarial active learning
          <xref ref-type="bibr" rid="ref1">(Ducoffe and Precioso 2018)</xref>
          also
reduce the amount of expert input required. While active
learning approaches focus on efficient use of expert input,
they typically assume that the concept of interest has already
been discerned and aim to achieve more efficient labeling.
Our approach could be used as an active learning tool when
the classes are well understood beforehand, but our primary
focus in on collaborative learning of a concept that is not
specified up front, starting with as little as one example from
an expert. We do see potential value in integrating modern
active learning algorithms like
          <xref ref-type="bibr" rid="ref2">(Gal, Islam, and Ghahramani
2017)</xref>
          and
          <xref ref-type="bibr" rid="ref1">(Ducoffe and Precioso 2018)</xref>
          with the
collaborative learning discussed in this paper.
        </p>
        <p>3</p>
      </sec>
      <sec id="sec-7-2">
        <title>Video clip representation</title>
        <p>To prepare data for use, we divide a video dataset into clips,
and we compute a signature for each clip using an
ensemble of deep learning neural networks. Currently, we define
a clip to be 10 seconds long, although exploring this
parameter and overlap of clips is on our road map. Below we
discuss our ensemble model approach for feature
representation. Our ensemble design captures two properties of the
content in each video clip: appearance and motion. We
pretrain our ensemble using a publicly available video dataset
that contains a broad set of human actions.
3.1</p>
      </sec>
    </sec>
    <sec id="sec-8">
      <title>Definitions</title>
      <p>Video clip: a short section of video, e.g., a 10 second clip.
Video clip signature: a set of deep embedded feature
vectors that encode clip characteristics.</p>
      <p>Feature vector: a vector of embedded features for a video
clip computed using a neural network; in general a
function of the clip and the neural network.</p>
      <p>
        Stream: a deep neural network model that uses video data
processed in multiple ways, as part of a collection of
streams in a multi-stream architecture.
The ensemble we have been studying comprises three RGB
and three warped optical flow deep nets adopted from the
Temporal Segment Networks (TSN) work by Wang et al
(2016). For each mode, the three networks were trained on
the three published splits of UCF-101 data
        <xref ref-type="bibr" rid="ref7">(Soomro,
Zamir, and Shah 2012)</xref>
        . The results reported here correspond
to the six 1024-element global pool embedded feature
vectors in the six deep nets, which are the last feature vectors
before the final layers that classify outputs into the 101 UCF
classes. Once computed, we store video clip signatures as
structured data in a database, enabling structured queries for
comparing signatures.
      </p>
      <p>
        The TSN approach takes a short video, divides it into a
specified number of snippets, analyzes each snippet using
both a spatial (RGB) convolution neural network and a
temporal (optical flow) neural network, and then applies
consensus functions to arrive at a final determination of the
action. Optical flow shows the velocity of pixels, and warped
optical flow attempts to suppress background motion, such
as effects of camera, rather than actor, movement. Whereas
        <xref ref-type="bibr" rid="ref9">Wang et al. (2016)</xref>
        report that an optical flow neural net
performed nearly as well as a warped optical flow neural net in
their work and takes less computation time to prepare, we
have been using warped optical flow neural nets in order to
increase robustness to camera motion. Our use cases have
focused on cameras mounted on moving vehicles, unlike the
UCF-101 dataset, and our initial tests suggested that warped
optical flow could perform better than optical flow alone.
      </p>
      <p>We chose TSN modeling because the UCF-101 dataset
used in the TSN work has a mean clip length of 7.2 sec,
comparable to the clip lengths of interest to us. Action
concepts like ”walking with a dog”, ”band marching”, ”riding
a bike down a path”, and ”walking across the street” are the
focus in our approach, as opposed to longer, more
complicated activities, like a video of someone going through all
the steps to bake a cake or build a piece of furniture. TSN
emphasizes analyzing snippets of video for short actions,
rather than creating a longer term memory of an evolving
activity. Our methodology is meant to be used for tasks like
identifying a test vehicle stopping for pedestrians crossing
in front of it, and we would not expect it to work well in
deducing whether a series of actions in a video corresponds
to someone doing some shopping before picking up a child.
The latter involves a series of actions and an abstract sense
of intent, which is a different type of video machine learning
task. The methodology proposed here could be a component
of a larger machine learning technology that comprehends
the latter activity, but it could not achieve that goal alone.
3.3</p>
    </sec>
    <sec id="sec-9">
      <title>Choice of stream types</title>
      <p>Any appropriate deep nets can be used, and different types
of problems will do better with different deep net models.
For example, adding an Image Net stream could help if
concepts having more to do with images than video action are
important, e.g. if someone is interested in finding all
examples of crossing a street at a stop sign versus a stop light. The
differences between a stop sign and a stop light are image
differences, not action differences. In our user studies so far,
we have found that users often want to include such ”image”
features in their search. Another example where a different
neural net could be useful is the case of studying facial
expressions and head motions. Neural nets trained on facial
expressions rather than the actions in the UCF-101 dataset
will probably do better, e.g., for videos capturing people’s
faces while driving cars.
3.4</p>
    </sec>
    <sec id="sec-10">
      <title>Choice of deep network embeddings</title>
      <p>Depending on the concept of interest, embeddings from
different layers other than the final hidden layer in a deep net
may be more useful to integrate into an ensemble model. For
example, if one is looking for more basic motions, like
veering left vs. right, lower layers may better distinguish such
actions. In contrast, our currently UCF-101 trained TSN
networks may not have that differentiation of left vs. right
motion present by the time higher layers are reached, since
these networks were trained to predict the correct action
regardless of left vs. right motion.</p>
      <p>In future work, we plan to investigate including more
layers and let the human-algorithm collaboration discern how
much to weigh lower vs higher layers in the ensemble.
4</p>
      <sec id="sec-10-1">
        <title>Searching for similar clips</title>
        <p>As described above, the search algorithm uses an
ensemble of deep neural net embeddings. Similarities of clips are
quantified by computing dot products of embedded feature
vectors for the example clips and possible matches. The
individual dot products are then combined into an ensemble
score.
4.1</p>
      </sec>
    </sec>
    <sec id="sec-11">
      <title>Choice of similarity measure for a single embedding</title>
      <p>To compute the similarity of an embedded feature for a
reference clip and the same embedded feature for a second clip
i, we use
f D(iN)TN
f DreNf N</p>
      <p>2
f DreNf N 2
(1)
where f are the feature vectors, T denotes transpose, and
DNN represents the deep neural net type (e.g., RGB or
warped optical flow). Similarity corresponds to how close
the result is to one. In experiments so far, we find this
similarity measure works well in an ensemble model, whereas
the results for single neural nets alone seem to suffer
significantly higher variance.</p>
      <p>Note that Eqn 1 differs from a cosine similarity. (The
denominator is the square of the L2 norm of the embedded
feature for the reference frame, not the product of the norms for
both features.) We use this “dot product” similarity because
it emphasizes what is similar along the hyper-dimensional
direction of the reference embedding, not what is different
in orthogonal hyper-dimensional directions. Note that two
embedded feature vectors can be far apart in their
hyperdimensional space (e.g., according to a Euclidean or other
distance metric) and still have a good similarity score.
Since the signature of a video clip is the set of features
computed using multiple types of deep neural nets (DNN)
trained on multiple splits of data, we need to specify how
we are ensembling all of the similarity metrics.</p>
      <p>Ensemble over data splits For each type of DNN and
each candidate clip i, we compute the similarity
'(Di)NN =</p>
      <p>j=1
1 X3 f D(iN)TN;j f DreNf N;j
3</p>
      <p>2
f DreNf N;j 2
where f (i)</p>
      <p>DNN;j is the feature for split j of the given DNN
type, computed for clip i, and f ref
DNN;j is the corresponding
feature for the reference clip. When clip i is also the
reference frame, '(Di)NN
the projection of f (i) DNN;j equals the squared L2</p>
      <p>DNN;j on f ref</p>
      <p>DNN;j , the summand equals one, even if f D(iN)N;j
norm of f ref
and f ref</p>
      <p>DNN;j are not equal. This is the desired behavior we
discussed in section 3.1.</p>
      <p>= 1, and more generally, whenever</p>
    </sec>
    <sec id="sec-12">
      <title>Ensemble over DNN streams The next step is to deter</title>
      <p>mine an overall similarity score. To do so, we use a
Euclidean space in which each DNN corresponds to a
dimension along which '(Di)NN is measured. In this space, a value
of '(Di)NN = 1 for each DNN dimension is the best
possible similarity. We do not necessarily want to weigh each
DNN dimension the same, but instead want to learn optimal
weights. Thus, we compute the overall similarity of clip i to
the reference as
i = DNN</p>
      <p>P
wD2NN
1
'(Di)NN</p>
      <p>2</p>
      <p>P
DNN
wD2NN
where wDNN are the weights for each type of DNN.
(2)
(3)</p>
      <sec id="sec-12-1">
        <title>Refining search</title>
        <p>To decide which clips to present to the user for review,
similarities i are computed for the latest best guess of the
values of wDNN (wRGB and wwarped optical flow in the examples
presented here). A small number of clips with similarities
better than the current estimate of the threshold are selected,
as well as a small number of clips with similarities close
to but below the threshold. The user then gives feedback,
and the algorithm computes new estimates of wDNN and
the threshold for i. In our experiments we tended to see
better outcomes when both matches and “near misses” were
presented for review. This is perhaps similar to people
establishing common understandings by agreeing both on
positive and negative examples of concepts.</p>
        <p>6</p>
      </sec>
      <sec id="sec-12-2">
        <title>Refining machine representation: Target bootstrapping</title>
        <p>In this section, we discuss refining the machine
representation of the target concept given expert feedback. The goal is
to replace f DreNf N;j with a new bootstrapped set f DbNN;j that
is most consistent with all user validated matches, capturing
what is similar with all of them and ignoring what is
different. Referring back to equation 3, in mathematical terms we
want to find f DbNN;j such that i 1 for all user-validated
matching clips. Any set of f DbNN;j for which</p>
        <p>b b 2
f D(iN)TN;j fDNN;j = fDNN;j 2
(4)
for all clips i in the set of user validated matches, all splits j
and all DNN types corresponds to a bootstrapped reference
that has i = 1. Any such f DbNN;j corresponds to a
hyperplane perpendicular to it that contains all the endpoints of
f D(iN)N;j for all user-validated matches i.</p>
        <p>There are an infinite number of such hyper-planes, since
the dimensions of the fDNN;j are much higher (1024 in this
work) than the number of matches a user will have
validated. As a starting point for the bootstrapping, we compute
the largest (in the L2 norm sense) bootstrapped f DbNN;j that
satisfies equation 4 because it is the least restrictive in terms
of reducing the impact of any one hyper-dimension on the
similarity metric i. In this way, we do not force more
restrictions on the user’s intent than is necessary for inferring
a unifying machine representation. To reduce variance, we
use bagging, choosing samples with replacement for each
bag and averaging over three bags. Information about
nonmatches can also be incorporated. The mathematical details
for the base version of target bootstrapping are provided in
the appendix.</p>
        <p>7</p>
      </sec>
      <sec id="sec-12-3">
        <title>Software implementation</title>
        <p>We have built Agile Video Query software
implementing agile discovery for video datasets and made it
available open source at https://github.com/PARC-projects/
video-query-home, free for non-commercial use. The
software components are (1) a Django API, (2) an Angular
browser client, (3) back end python algorithms, and (4) a
Postgres database. We are continuing to develop the
software and welcome others to participate.
Using video from moving vehicles, we have begun
performing user studies of the Agile Video Query software.
Scenarios studied so far include: (1) vehicles interacting with
pedestrians at intersections, (2) vehicles driving underneath
an overpass, and (3) vehicles taking a noticeable left turn.
See Figure 1 for a snapshot of the first scenario. (Copyright
permission to show snapshots from the video for the second
and third scenarios is pending.)</p>
        <p>Not surprisingly, different users interact differently with
the software. For some users, the software motivates
focusing tightly on a particular concept, thereby helping to study
a well-defined hypothesis in a disciplined manner. We have
also observed cases where the software motivates a user to
expand the scope of interest and perform curiosity-driven
data exploration, in contrast with the former user type.</p>
        <p>Figure 2 shows results for queries of vehicles
interacting with pedestrians crossing a street, for two users of the
first type discussed above. For Figure 2a, the user accepted
any clip where the vehicle interacted with or was stopped
for a pedestrian as a valid match. As illustrated, 80% of the
matches are correctly located above the learned threshold ,
and 22.5% of the clips above the threshold are false
positives (i.e., red x). For Figure 2b, the user only validated clips
in which both the vehicle and the pedestrian were moving,
rejecting clips in which the vehicle remained stopped.
Although 79% of the matches are correctly located above the
learned threshold, the false positive fraction is worse at 38%.
9</p>
      </sec>
      <sec id="sec-12-4">
        <title>Technology road map</title>
        <p>By analyzing for diverging sample distributions in the
embedded feature space, future enhancements will help the
human-machine collaboration discern when the user’s
interest is broadening or conflicted, and act by handling multiple
concepts separately. We expect these enhancements to take
the form of further development of target bootstrapping. Our
road map also includes adding a larger set of deep neural
net types, starting with an image-centric deep net to address
non-action image recognition needs that have come up
repeatedly in our user studies.</p>
        <p>
          We are also working toward a generative algorithm that
will use bootstrapped targets to produce video clips that
humans can easily and intuitively interpret and evaluate. The
goal of this capability is to provide a human-interpretable
view of the machine representations. For our intended
purposes, the generated video should highlight the actions of
interest while also obscuring or subduing irrelevant features.
Our current approach is to leverage both LIME
          <xref ref-type="bibr" rid="ref6">(Ribeiro,
Singh, and Guestrin 2016)</xref>
          and neural style transfer
techniques
          <xref ref-type="bibr" rid="ref3">(Gatys, Ecker, and Bethge 2016)</xref>
          .
        </p>
        <p>LIME computes which pixels are important for any one
prediction made by a DNN image classifier, presenting a
modified image with those pixels replaced with a highlight
color. With a modified version of LIME, we expect to be able
to show if the machine representation is locking into
something unexpected, such as irrelevant trees in the background.
If a user sees this happening, then the user can provide
further examples without those features in the next round of
feedback to the algorithm. Further along the roadmap, we
will research adding an explicit ability for users to directly
remove irrelevant things revealed by visualizations of the
machine representation.</p>
        <p>Highlighting important pixels is only a partial solution,
however, because it will not reveal what relations among
pixels are important versus unimportant. Consider, for
example, video clips of pedestrians walking in front of a car
at an intersection. Highlighting the pedestrians reveals some
information, but it does not tell the user whether the machine
representation prioritizes, say, the pattern on a pedestrian’s
shirt or the reflective stripes on the safety vest of a jogger.
We are investigating using neural style imaging as a way to
alter a clip in order to subdues patterns and details that are
In target boostrapping, we want to choose the least
restrictive bootstrapped f ref</p>
        <p>DNN;j that satisfies equation 4, in order
to not force more restrictions on the user’s intent than is
justified. Accordingly, we choose the target bootstrapping to be</p>
        <p>In terms of this scaled target, equations 5 and 6 become,
for each choice of DNN and j
unimportant for the similarity score. Our hypothesis is this
approach can provide a richer human-interpretable version
of the machine representation; for example, the modified
clip may hypothetically show dull, monotone clothing for
queries that are not focused on clothing, or conversely, show
the details of a jogger’s safety vest if the user is intending
the query to focus on joggers with such vests.</p>
        <p>10</p>
      </sec>
      <sec id="sec-12-5">
        <title>Acknowledgement</title>
        <p>We thank the U.S. Department of Transportation Federal
Highway Administration and Dr. Ana Maria Eigen for
support and funding under EAR grant ID DTFH6115H00006.
(5)
(7)
(8)
(9)
(10)
(11)
(12)
(13)
(14)
(15)
max 1 f bT
2</p>
        <p>f b
such that f (i)T f b = f b 22 (6)
for all clips i that the user has validated to be a match. When
using bagging, the set of all clips is replaced by a set
randomly chosen from the entire set with replacement.</p>
        <p>As written, this maximization problem is in a form that is
difficult to handle. To put it in a nicer form, we introduce the
scaled target
t =</p>
        <p>f b
min 1 tT t</p>
        <p>2
such that f (i)T t = 1
This is a straightforward quadratic minimization problem
with linear equality constraints.</p>
        <p>Using the method of Lagrange multipliers, the Lagrangian
for equations 8 and 9 is</p>
        <p>L = 1 tT t + T (F t 1)</p>
        <p>2
where is a vector of Lagrange multipliers, and
2
=</p>
        <p>F F T
= 0;
1
1J
Substituting back into equation 12, we derive the solution
for t:
t = F T</p>
        <p>F F T
1
1J</p>
      </sec>
    </sec>
  </body>
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