I Introduction
The effectiveness of many machine learning and data mining algorithms depends on an appropriate measure of pairwise distance between data points that accurately reflects the learning task, e.g., prediction, clustering or classification. The kNN classifier, Kmeans clustering, and the LaplacianSVM semisupervised classifier are examples of such
distancebasedmachine learning algorithms. In settings where there is clean, appropriatelyscaled spherical Gaussian data, standard Euclidean distance can be utilized. However, when the data is heavy tailed, multimodal, or contaminated by outliers, observation noise, or irrelevant or replicated features, use of Euclidean interpoint distance can be problematic, leading to bias or loss of discriminative power.
To reduce bias and loss of discriminative power of distancebased machine learning algorithms, datadriven approaches for optimizing the distance metric have been proposed. These methodologies, generally taking the form of dimensionality reduction or data “whitening,” aim to utilize the data itself to learn a transformation of the data that embeds it into a space where Euclidean distance is appropriate. Examples of such techniques include Principal Component Analysis
[1], Multidimensional Scaling [2], covariance estimation
[2, 1], and manifold learning [3]. Such unsupervised methods do not exploit human input on the distance metric, and they overly rely on prior assumptions, e.g., local linearity or smoothness.In distance metric learning one seeks to learn transformations of the data associated with a distance metric that is well matched to a particular task specified by the user. Pairwise labels or “edges” indicating point similarity or dissimilarity are used to learn a transformation of the data such that similar points are “close” to one another and dissimilar points are distant in the transformed space. Learning distance metrics in this manner allows a more precise notion of distance or similarity to be defined that is better related to the task at hand.
Figure 1 illustrates this notion. Data points, or nodes, have underlying similarities or distances between them. Absent an exhaustive label set, given an attribute distance function
it is possible to infer similarities between nodes as the distance between their attribute vectors. As an example, the kNN algorithm uses the Euclidean distance to infer similarity. However, the distance function must be specified a priori, and may not match the distance relevant to the task. Distance metric learning proposes a hybrid approach, where one is given a small number of pairwise labels, uses these to learn a distance function on the attribute space, and then uses this learned function to infer relationships between the rest of the nodes.
Many supervised and semisupervised distance metric learning approaches have been developed for machine learning and data mining [4]. This includes online algorithms [5] with regret guarantees for situations where similarity constraints are received sequentially.
This paper proposes a new distance metric tracking method that is applicable to the nonstationary time varying case of distance metric drift and has provably strongly adaptive tracking performance.
Specifically, we suppose the underlying groundtruth (or optimal) distance metric from which constraints are generated is evolving over time, in an unknown and potentially nonstationary way. In Figure 1, this corresponds to having the relationships between nodes change over time. This can, for example, be caused by changes in the set of features indicative of relations (e.g. polarizing buzzwords in collective discourse), changes in the underlying relationship structure (e.g. evolving communities), and/or changes in the nature of the relationships relevant to the problem or to the user. When any of these changes occur, it is imperative to be able to detect and adapt to them without casting aside previous knowledge.
We propose a strongly adaptive, online approach to track the underlying metric as the constraints are received. We introduce a framework called Online Convex Ensemble StrongLy Adaptive Dynamic Learning (OCELAD), which at every time step evaluates the recent performance of and optimally combines the outputs of an ensemble of online learners, each operating under a different driftrate assumption. We prove strong bounds on the dynamic regret of every subinterval, guaranteeing strong adaptivity and robustness to nonstationary metric drift such as discrete shifts, slow drift with a widelyvarying drift rate, and all combinations thereof. Applying OCELAD to the problem of nonstationary metric learning, we find that it gives excellent robustness and low regret when subjected to all forms of nonstationarity.
Social media provides some of the most dynamic, rapidly changing data sources available. Constant changes in world events, popular culture, memes, and other items of discussion mean that the words and concepts characteristic of subcultures, communities, and political persuasions are rapidly evolving in a highly nonstationary way. As this is exactly the situation our dynamic metric learning approach is designed to address, we will consider modeling political tweets in November 2015, during the early days of the United States presidential primary.
Ia Related Work
Linear Discriminant Analysis (LDA) and Principal Component Analysis (PCA) are classic examples of the use of linear transformations for projecting data into more interpretable low dimensional spaces. Unsupervised PCA seeks to identify a set of axes that best explain the variance contained in the data. LDA takes a supervised approach, minimizing the intraclass variance and maximizing the interclass variance given class labeled data points.
Much of the recent work in Distance Metric Learning has focused on learning Mahalanobis distances on the basis of pairwise similarity/dissimilarity constraints. These methods have the same goals as LDA; pairs of points labeled “similar” should be close to one another while pairs labeled “dissimilar” should be distant. MMC [6], a method for identifying a Mahalanobis metric for clustering with side information, uses semidefinite programming to identify a metric that maximizes the sum of distances between points labeled with different classes subject to the constraint that the sum of distances between all points with similar labels be less than or equal to some constant.
Large Margin Nearest Neighbor (LMNN) [7] similarly uses semidefinite programming to identify a Mahalanobis distance. In this setting, the algorithm minimizes the sum of distances between a given point and its similarly labeled neighbors while forcing differently labeled neighbors outside of its neighborhood. This method has been shown to be computationally efficient [8] and, in contrast to the similarly motivated Neighborhood Component Analysis [9], is guaranteed to converge to a globally optimal solution. Information Theoretic Metric Learning (ITML) [10] is another popular Distance Metric Learning technique. ITML minimizes the KullbackLiebler divergence between an initial guess of the matrix that parameterizes the Mahalanobis distance and a solution that satisfies a set of constraints. For surveys of the metric learning literature, see [4, 11, 12].
In a dynamic environment, it is necessary to track the changing metric at different times, computing a sequence of estimates of the metric, and to be able to compute those estimates online. Online learning [13] meets these criteria by efficiently updating the estimate every time a new data point is obtained instead of minimizing an objective function formed from the entire dataset. Many online learning methods have regret guarantees, that is, the loss in performance relative to a batch method is provably small [13, 14]. In practice, however, the performance of an online learning method is strongly influenced by the learning rate, which may need to vary over time in a dynamic environment [15, 16, 17], especially one with changing drift rates.
Adaptive online learning methods attempt to address the learning rate problem by continuously updating the learning rate as new observations become available. For learning static parameters, AdaGradstyle methods [16, 17] perform gradient descent steps with the step size adapted based on the magnitude of recent gradients. Follow the regularized leader (FTRL) type algorithms adapt the regularization to the observations [18]. Recently, a method called Strongly Adaptive Online Learning (SAOL) has been proposed for learning parameters undergoing
discrete changes when the loss function is bounded between 0 and 1. SAOL maintains several learners with different learning rates and randomly selects the best one based on recent performance
[15]. Several of these adaptive methods have provable regret bounds [18, 19, 20]. These typically guarantee low total regret (i.e. regret from time 0 to time ) at every time [18]. SAOL, on the other hand, attempts to have low static regret on every subinterval, as well as low regret overall [15]. This allows tracking of discrete changes, but not slow drift. Our work improves upon the capabilities of SAOL by allowing for unbounded loss functions, using a convex combination of the ensemble instead of simple random selection, and providing guaranteed low regret when all forms of nonstationarity occur, not just discrete shifts. All of these additional capabilities are shown in Section VI to be critical for good metric learning performance.The remainder of this paper is structured as follows. In Section II we formalize the time varying distance metric tracking problem, and section III presents the basic COMID online learner and our RetroInitialized COMID Ensemble (RICE) of learners with dyadically scaled learning rates. Section IV presents our OCELAD algorithm, a method of adaptively combining learners with different learning rates. Strongly adaptive bounds on the dynamic regret of OCELAD and RICEOCELAD are presented in Section V, and results on both synthetic data and the Twitter dataset are presented in Section VI. Section VII concludes the paper.
Ii Nonstationary Metric Learning
Metric learning seeks to learn a metric that encourages data points marked as similar to be close and data points marked as different to be far apart. The timevarying Mahalanobis distance at time is parameterized by as
(1) 
where .
Suppose a temporal sequence of similarity constraints are given, where each constraint is the triplet , and are data points in , and the label if the points are similar at time and if they are dissimilar.
Following [5], we introduce the following margin based constraints for all time points :
(2) 
where is a threshold that controls the margin between similar and dissimilar points. A diagram illustrating these constraints and their effect is shown in Figure 2. These constraints are softened by penalizing violation of the constraints with a convex loss function . This gives a combined loss function
(3)  
where , is the regularizer and the regularization parameter. Kunapuli and Shavlik [5] propose using nuclear norm regularization (
) to encourage projection of the data onto a low dimensional subspace (feature selection/dimensionality reduction), and we have also had success with the elementwise L1 norm (
). In what follows, we develop an adaptive online method to minimize the loss subject to nonstationary smoothness constraints on the sequence of metric estimates .Iii Retroinitialized COMID ensemble (RICE)
Viewing the acquisition of new data points as stochastic realizations of the underlying distribution [5] suggests the use of composite objective stochastic mirror descent techniques (COMID). For convenience, we set .
For the loss (3) and learning rate , application of COMID [14] gives the online learning update
(4)  
where is any Bregman divergence. As this is an online framework, the indexing directly corresponds to the received time series of pairwise constraints . In [5] a closedform algorithm for solving the minimization in (18) with
is developed for a variety of common losses and Bregman divergences, involving rank one updates and eigenvalue shrinkage.
The output of COMID depends strongly on the choice of . Critically, the optimal learning rate depends on the rate of change of [21], and thus will need to change with time to adapt to nonstationary drift. Choosing an optimal sequence for is clearly not practical in an online setting with nonstationary drift, since the drift rate is changing. We thus propose to maintain an ensemble of learners with a range of values, whose output we will adaptively combine for optimal nonstationary performance. If the range of is diverse enough, one of the learners in the ensemble should have good performance on every interval. Critically, the optimal learner in the ensemble may vary widely with time, since the drift rate and hence the optimal learning rate changes in time. For example, if a large discrete change occurs, the fast learners are optimal at first, followed by increasingly slow learners as the estimate of the new value improves. In other words, the optimal approach is fast reaction followed by increasing refinement, in a manner consistent with the attractive decay of the learning rate of optimal nonadaptive algorithms [21].
Define a set of intervals such that the lengths of the intervals are proportional to powers of two, i.e. , , with an arrangement that is a dyadic partition of the temporal axis, as in [15]. The first interval of length starts at (see Figure 3), and additional intervals of length exist such that the rest of time is covered.
Every interval is associated with a base COMID learner that operates on that interval. Each learner (18) has a constant learning rate proportional to the inverse square of the length of the interval, i.e. . Each learner (besides the coarsest) at level () is initialized to the last estimate of the next coarsest learner (level ) (see Figure 3). This strategy is equivalent to “backdating” the interval learners so as to ensure appropriate convergence has occurred before the interval of interest is reached, and is effectively a quantized square root decay of the learning rate. We call our method of forming an ensemble of COMID learners on dyadically nested intervals the RetroInitialized COMID Ensemble, or RICE, and summarize it in Figure 3.
At a given time , a set of intervals/COMID learners are active, running in parallel. Because the metric being learned is changing with time, learners designed for low regret at different scales (drift rates) will have different performance (analogous to the classical bias/variance tradeoff). In other words, there is a scale optimal at a given time.
To adaptively select and fuse the outputs of the ensemble, we introduce Online Convex Ensemble StrongLy Adaptive Dynamic Learning (OCELAD), a method that accepts an ensemble of blackbox learners and uses recent history to adaptively form an optimal weighted combination at each time.
Iv Ocelad
To maintain generality, in this section we assume the series of random loss functions is of the form where is the timevarying unknown parameters. We assume that an ensemble of online learners is provided on the dyadic interval set , each optimized for the appropriate scale. To select the appropriate scale, we compute weights that are updated based on the learner’s recent estimated regret. The weight update we use is inspired by the multiplicative weight (MW) literature [22], modified to allow for unbounded loss functions. At each step, we rescale the observed losses so they lie between 1 and 1, allowing for maximal weight differentiation while preventing negative weights.
(5)  
These hold for all , where , are the outputs at time of the learner on interval , and is called the estimated regret of the learner on interval at time . The initial value of is . Essentially, (5) is highly weighting low loss learners and lowly weighting high loss learners.
For any given time , the outputs of the learners of interval are combined to form the weighted ensemble estimate
(6) 
The weighted average of the ensemble is justified due to our use of a convex loss function (proven in the next section), as opposed to the possibly nonconvex losses of [22], necessitating a randomized selection approach. OCELAD is summarized in Algorithm 1, and the joint RICEOCELAD approach as applied to metric learning of is shown in Algorithm 2.
V Strongly Adaptive Dynamic Regret
The standard static regret of an online learning algorithm generating an estimate sequence is defined as
(7) 
where is a loss with parameter . Since in our case the optimal parameter value is changing, the static regret of an algorithm on an interval is not useful. Instead, let be an arbitrary sequence of parameters. Then, the dynamic regret of an algorithm relative to any comparator sequence on the interval is defined as
(8) 
where are generated by . This allows for comparison to any possible dynamically changing batch estimate .
In [21] the authors derive dynamic regret bounds that hold over all possible sequences such that , i.e. bounding the total amount of variation in the estimated parameter. Without this temporal regularization, minimizing the loss would cause to grossly overfit. In this sense, setting the comparator sequence to the “ground truth sequence” or “batch optimal sequence” both provide meaningful intuitive bounds.
Strongly adaptive regret bounds [15] can provide guarantees that static regret is low on every subinterval, instead of only low in the aggregate. We use the notion of dynamic regret to introduce strongly adaptive dynamic regret bounds, proving that dynamic regret is low on every subinterval simultaneously. The following result is proved in the appendix. Suppose there are a sequence of random loss functions . The goal is to estimate a sequence that minimizes the dynamic regret.
Theorem 1 (General OCELAD Regret Framework).
Let be an arbitrary sequence of parameters and define as a function of and an interval . Choose an ensemble of learners such that given an interval the learner creates an output sequence satisfying the dynamic regret bound
(9) 
for some constant . Then the strongly adaptive dynamic learner using as the ensemble creates an estimation sequence satisfying
on every interval .
In other words, the regret of OCELAD on any finite interval is sublinear in the length of that interval (), and scales with the amount of variation in true/optimal batch parameter estimates. The logarithmic term in exists because of the logarithmically increasing number of learners active at time , required to achieve guaranteed regret on intervals for which can be up to the order of .
In a dynamic setting, bounds of this type are particularly desirable because they allow for changing drift rate and guarantee quick recovery from discrete changes. For instance, suppose a number of discrete switches (large parameter changes or changes in drift rate) occur at times satisfying . Then since , this implies that the total expected dynamic regret on remains low (), while simultaneously guaranteeing that an appropriate learning rate is achieved on each subinterval .
Now, reconsider the dynamic metric learning problem of Section II. It is reasonable to assume that the transformed distance between any two points is bounded, implying and that . Thus the loss (and the gradient) are bounded. We can then show the COMID learners in the RICE ensemble have low dynamic regret. The proof of the following result is given in the appendix.
Corollary 1 (Dynamic Regret: Metric Learning COMID).
Let the sequence be generated by (18), and let be an arbitrary sequence with . Then using gives
(10) 
and setting ,
(11)  
(12) 
Since the COMID learners have low dynamic regret on the metric learning problem, we can apply the OCELAD framework to the RICE ensemble.
Theorem 2 (Strongly Adaptive Dynamic Regret of RICEOCELAD applied to metric learning).
Let be any sequence of metrics with on the interval , and define . Let be the RICE ensemble with . Then the RICEOCELAD metric learning algorithm (Algorithm 2) satisfies
(13)  
for every subinterval simultaneously. is a constant.
Vi Results
Via Synthetic Data
We run our metric learning algorithms on a synthetic dataset undergoing different types of simulated metric drift. We create a synthetic 2000 point dataset with 2 independent threeway clusterings (denoted as clusterings A and B) of the points when projected onto orthogonal 3dimensional subspaces of . The clusterings are formed as 3D Gaussian blobs with cluster assignment probabilities .5, .3, and .2. The remaining 19 coordinates are filled with isotropic Gaussian noise. Specifically, datapoints are generated as
where , are independent,
is the standard deviation of the noise dimensions, and the
are the means and covariances associated with each blob. The label of under clustering A is , and the label of under clustering B is .We create a scenario exhibiting nonstationary drift, combining continuous drifts and shifts between the two clusterings (A and B). To simulate continuous drift, at each time step we perform a random rotation of the dataset, i.e.
where is a random walk (analogous to Brownian motion) on the 25D sphere of rotation matrices in , with chosen uniformly at random. The timevarying rate of change (random walk stepsize) chosen for is shown in Figure 5, with the small changes in at each time step accumulating to major changes over longer intervals. For the first interval, partition A is used and the dataset is static, no drift occurs (). Then, the partition is changed to B, followed by an interval of first moderate, then fast, and then moderate drift. Finally, the partition reverts back to A, followed by slow drift. The similarity labels are dictated by the partition active at time . In order to achieve good performance, the online metric learners must be able to track both large discrete changes (change in clustering) as well as the nonstationary gradual drift in .
We generate a series of constraints from random pairs of points in the dataset (, ) running each experiment with 3000 random trials. For each experiment conducted in this section, we evaluate performance using two metrics. We plot the Knearest neighbor error rate, using the learned embedding at each time point, averaging over all trials. We quantify the clustering performance by plotting the empirical probability that the normalized mutual information (NMI) of the Kmeans clustering of the unlabeled data points in the learned embedding at each time point exceeds 0.8 (out of a possible 1). Clustering NMI, rather than kNN classification performance, is a more intuitive and realistic indicator of metric learning performance, particularly when finding a relevant embedding in which the clusters are wellseparated is the primary goal.
In our results, we consider RICEOCELAD, SAOL with COMID [15], nonadaptive COMID [5], LMNN (batch) [7], and online ITML [10].
For RICEOCELAD, we set the base interval length time step throughout, and set via crossvalidation in a separate scenario with no drift, emphasizing that the parameters do not need to be tuned for different drift rates. All parameters for the other algorithms were set via cross validation, so as to err on the side of optimism in a truly online scenario. For nonadaptive COMID, we set the high learning rate using cross validation for moderate drift, and we set the low learning rate via cross validation in the case of no drift. The results are shown in Figure 5. Online ITML fails due to its bias agains lowrank solutions [10], and the batch method and low learning rate COMID fail due to an inability to adapt. The high learning rate COMID does well at first, but as it is optimized for slow drift it cannot adapt to the changes in drift rate as well or recover quickly from the two partition changes. SAOL, as it is designed for mildlyvarying bounded loss functions without slow drift and does not use retroinitialized learners, completely fails in this setting (zero probability of NMI throughout). RICEOCELAD, on the other hand, adapts well throughout the entire interval, as predicted by the theory.
ViB Tracking Metrics on Twitter
As noted in the introduction, social media represents a type of highly nonstationary, high dimensional and richly clustered data. We consider political tweets in November 2015, during the early days of the United States presidential primary, and attempt to learn timevarying metrics on the TFIDF features.
We first extracted all available tweets containing the hashtags #trump2016, #cruz2016, #bernie2016, #hillary2016, representing the two most successful primary candidates from each of the two major parties. We then removed all hashtags from the tweets, and extracted 194 term frequency  inverse document frequency (TFIDF) stemmed word features. TFIDF features have been applied to various problems in Twitter data [23, 24, 25]. This provided us with a time series of hashtaglabeled 194dimensional TFIDF feature vectors. We chose to generate pairwise comparisons by considering timeadjacent tweets and labeling them as similar if they shared the same candidate hashtag, and dissimilar if they had different candidate hashtags. This created a time series of 13600 pairwise comparisons, with the time intervals between comparisons highly nonstationary, strongly depending on time of day, day of the week, and various other factors.
We ran RICEOCELAD metric learning on this time series of pairwise comparisons, with the base interval set at length 1 and base learning rate set at 1. This emphasizes RICEOCELAD’s complete freedom from tuning parameters. To illustrate the learned embedding on the TFIDF stems, Figure 7 shows the projection of tweets from the last week of the month onto the first two principal components of the learned metric from the midpoint of the last week. Note the clear separation into clusters by political hashtag as desired, with a LOOkNN error rate of 7.8% in the learned embedding. The standard PCA embedding, on the other hand, is highly disorganized, and suffers a 60.6% LOOkNN error rate in the same scenario.
Having confirmed that our approach successfully learns the relevant embedding, we illustrate how the learned metric evolves throughout the month in response to changing events. For each metric , we computed the first two principle component vectors and . For each feature stem, we found the corresponding entries in , and used these as coordinates in a scatter plot, creating word/stem scatter plots (Figure 8). By way of interpretation, the scatter plot location of a word/stem is the point in the 2D embedding to which a tweet containing only that word would be mapped, and quantifies the contribution of each word/stem to the metric.
Figure 8 shows word stem scatter plots for the learned metrics at the beginning and end of the month, and the day of and the day after the televised November 10 Republican debate. Only the top 60 terms most relevant to the metric are shown for clarity. Observe the changing structure of the term embeddings, with new terms arising and leaving as the discussion evolves. An alternate view of this experiment is shown in Figure 9, showing the changing relevance of selected individual terms throughout the month. In the captions, we have mentioned explanatory contextual information that can be found in news articles from the period. In both figures, timevarying structure is evident, with Figure 8 emphasizing how similar embeddings of words indicate similar meaning/relevance to a candidate, and with Figure 9 emphasizing the nonstationary emergence and recession of clusteringrelevant terms as the discussion evolves in response to news events.
The ability of RICEOCELAD metric learning, without parameter tuning or specialized feature extraction, to successfully adapt the embedding and identify terms and their relevance to the discussion in this highly nonstationary environment confirms the power of our proposed methodology. RICEOCELAD allows significant insight into complex, nonstationary data sources to be gleaned by tracking a taskrelevant, adaptive, timevarying metric/low dimensional embedding of the data.
Vii Conclusion
Learning a metric on a complex dataset enables both unsupervised methods and/or a user to home in on the problem of interest while deemphasizing extraneous information. When the problem of interest or the data distribution is nonstationary, however, the optimal metric can be timevarying. We considered the problem of tracking a nonstationary metric and presented an efficient, strongly adaptive online algorithm (OCELAD), that combines the outputs of any black box learning ensemble (such as RICE), and has strong theoretical regret guarantees. Performance of our algorithm was evaluated both on synthetic and real datasets, demonstrating its ability to learn and adapt quickly in the presence of changes both in the clustering of interest and in the underlying data distribution.
Potential directions for future work include the learning of more expressive metrics beyond the Mahalanobis metric, the incorporation of unlabeled data points in a semisupervised learning framework
[26], and the incorporation of an active learning framework to select which pairs of data points to obtain labels for at any given time
[27].Appendix A OCELAD  Strongly Adaptive Dynamic Regret
We will prove Theorem 1, giving strongly adaptive dynamic regret bounds. The bound for RICEOCELAD applied to metric learning follows by combining this general result with Corollary 1.
Define as a function of
(14) 
and set
(15) 
Note that where is the indicator function for the interval , and assume that , i.e. the estimated regret is bounded, where the bound need not be known.
Recall our definition of the set of intervals such that the lengths of the intervals are proportional to powers of two, i.e. , , with an arrangement that is a dyadic partition of the temporal axis. The first interval of length starts at (see Figure 3), and additional intervals of length exist such that the rest of the time axis is covered.
We first prove a pair of lemmas.
Lemma 1.
for all .
Proof.
For all , by the definition of the set of dyadic intervals , we have that the number of intervals in with endpoint is given by , where indicates cardinality. Thus summing over all intervals in the dyadic set of intervals ,
Then
Suppose that . Furthermore, note that
since is convex. Thus
Since , the lemma follows by induction.
∎
Lemma 2.
for every .
Proof.
Define the restriction of to an interval as . Note the following lemma from [15]:
Lemma 3.
Consider the arbitrary interval . Then, the interval can be partitioned into two finite sequences of disjoint and consecutive intervals, given by and , such that
This enables us to extend the bounds to every arbitrary interval and thus complete the proof.
Appendix B Online DML Dynamic Regret
In this section, we derive the dynamic regret of the COMID metric learning algorithm. Recall that the COMID algorithm is given by
(18)  
where is any Bregman divergence and is the learning rate parameter. From [21] we have:
Theorem 3.
Let the sequence , be generated via the COMID algorithm, and let be an arbitrary sequence in . Then using gives a dynamic regret
(19) 
Using a nonincreasing learning rate , we can then prove a bound on the dynamic regret for a quite general set of stochastic optimization problems.
Applying this to our problem, we have
For being the hinge loss and ,
The other two quantities are guaranteed to exist and depend on the choice of Bregman divergence and . Thus,
Corollary 2 (Dynamic Regret: Metric Learning COMID).
Let the sequence be generated by (18), and let be an arbitrary sequence with . Then using gives
(20) 
and setting ,
(21)  
(22) 
Corollary 2 is a bound on the regret relative to the batch estimate of that minimizes the total batch loss subject to a bounded variation . Also note that setting gives the same bound as (22).
In other words, we pay a linear penalty on the total amount of variation in the underlying parameter sequence. From (22), it can be seen that the boundminimizing increases with increasing , indicating the need for an adaptive learning rate.
For comparison, if the metric is in fact static then by standard stochastic mirror descent results [21]
Theorem 4 (Static Regret).
If and , then
(23) 
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