1 Related Work
This work is concerned with the topics of data collection from noisy sources, learning what data to learn from, and Multi-Armed Bandit approaches. We present relevant work from each of these areas.
1.1 Data Collection
Collecting sufficient data for training classifiers is a classic problem in machine learning. The Web is the canonical example of a noisy source of vast amounts of data, and has provided many data sets on which to train models, e.g [Deng et al.2009, Philbin et al.2007, Fei-Fei, Fergus, and Perona2007]
. When labeling examples in these data sets, techniques like active learning[Settles2012] can help limit the labeling that needs to be done by humans, only asking for labels on inputs a model cannot predict confidently. Our work is similar to active learning in that we have an algorithm guiding the selection of training data, but differs in the criteria for data selection, and we assume all data has already been labeled by some process.
Techniques that handle noisy data with existing labels range from hand curating the data set [Welinder et al.2010], to adding model machinery intended to deal with noise [Sukhbaatar et al.2014], to including examples based on a dynamically calculated majority vote from a crowd [Deng et al.2009]. Most standard noise-handling techniques work at the instance level. Those that work at the label level [Welinder et al.2010, Garrigues et al.2017] usually remove data from a training set without explicitly being informed by the training process, but rather characteristics of the data (e.g. not having sufficient quality examples). Another approach is to create new categories that account for some noise [Torresani, Szummer, and Fitzgibbon2010, Li, Wu, and Tu2013]. Works like [Deng et al.2009, Krishna et al.2016] keep track of how difficult a label appears based on crowd agreement [Snow et al.2008] to include or exclude various examples. In contrast, our work attempts to automatically determine good labels based on model performance, not data set size or crowd factors.
1.2 Learning What to Learn
Previous methods that oversee the data selection process for training classifiers have had a variety of goals. Many models are interested in minimizing time to convergence. [Graves et al.2017] present a curriculum learner that generates an ordering of training tasks based on task difficulty, the intuition being that easy examples help models earlier in training, while harder examples are more appropriate later. Our work follows this same core idea, yet the problem we aim to solve is fundamentally different. While they train a model to complete a pre-determined set of tasks, we learn to focus on tasks, i.e classes, that are easiest to learn for a model. Like us, [Ruder and Plank2017]
are interested in manipulating the selection of training examples, though their goal is to choose examples based on suitability for transfer learning. In contrast our approach is more exploratory, as the only criteria for selecting classes is the resultant performance of a given model. Finally,[Fan et al.2017]
propose to supervise model training with a deep reinforcement learning algorithm. However, rather than actively select batches of training data, they train a filter to ignore certain examples within a given mini-batch at each training step.
1.3 Multi-Armed Bandits
In our approach, data is selected at each training step using a Multi-Armed Bandit algorithm, a problem first introduced in [Robbins1952]. The original formulation of the problem assumes that the reward for a given decision made by the algorithm is drawn from a fixed distribution every time that decision is made (i.e. a stochastic payoff function). In our setting, rewards are based on the state of a classifier at a given training step, which changes over time. Bandit algorithms developed for such scenarios are called adversarial MAB algorithms [Auer et al.1995], and make no assumptions on the payoff structure of decisions. We additionally assume that the class set we are choosing from is large, rendering naive selection strategies ineffective. For these settings, it is common to assume a structure on the decision-space, specifically that there exists a means of measuring similarity between decisions [Kleinberg, Slivkins, and Upfal2013]. One such approach is to assume that the reward function changes smoothly over the decision space, i.e that similar decisions will have similar payoffs [Srinivas et al.2010]. We use a time-varying version of this algorithm presented in [Bogunovic, Scarlett, and Cevher2016].
2 Exploiting Learnable Classes
Given an untrained model and a labeled data set, our work aims to develop an online algorithm that identifies the learnability of classes in the data set. In determining class learnability, we aim both to 1) avoid training on difficult/noisy classes thereby yielding a strong performance over a subset of data, and 2) to make an explicit ranking of the classes based on learnability. This ranking can be used to inform downstream processes of the model’s capabilities. In this section, we make precise the notion of a class learnability, present our online bandit algorithm, and develop a means of ranking classes in a data set. First, we begin with some preliminaries of Multi-Armed Bandit algorithms.
2.1 MAB Preliminaries
Given a collection of potential decisions (called arms in MAB nomenclature) and a number of rounds to choose (“pull”) an arm, an MAB algorithm selects a sequence of arms . After each round, the algorithm observes a reward associated with the arm choice . The goal of a bandit algorithm is to maximize , or equivalently to minimize cumulative regret. At each round , regret is defined as the difference between the reward for the best arm and the reward for the arm that was picked . The cumulative regret is defined as the sum of regret at each round: . Bandit algorithms are designed to minimize . The challenge lies in not knowing the distributions generating . In order to balance exploring promising arms with exploiting historically high reward arms, [Auer2002] propose picking arms with high upper confidence bounds (UCB). The UCB formula is given by
with being the mean reward of arm ,
being the standard deviation of rewards of, and
being a hyperparameter that weights the exploration term. Intuitively, when an arm is largely unexplored, the uncertainty termwill dominate the UCB, leading to an exploratory arm pull. In contrast, a high average reward leads to exploitation. The balance of exploration vs exploitation can be viewed as balancing learning a reward function with using said function to make decisions. Algorithms focused only on exploitation will never learn the reward function, while algorithms focus only on exploration may not make rewarding decisions.
Finding how learnable a class is for a given model in an online fashion fits naturally into the MAB problem setting. Our proposed method proceeds in conjunction with with the stochastic training of a classifier. We treat the selection of examples during training as an arm pull, picking examples that are all labeled with a particular class, and calculate a reward for those examples intended to measure their learnability. In the next section, we formalize our MAB problem setting, define our notion of class learnability, and introduce a reward function for our MAB algorithm based on this definition.
2.2 Problem Formulation
In this work, we consider the multi-label classification setting where single instances may be members of multiple classes. Let be a multi-label classifier parameterized by . Let (not to be confused with ) be a training algorithm that iteratively updates the parameters
with respect to some loss function. We assume that we have access to a training set and a validation set . Let be the set of classes reflected in the training and validation labels. We denote training and validation subsets whose instances are all labelled as a class by and .
We now formalize the notion of class learnability given , , , , and . We base our concept of learnability on model generalization. In the case of a learnable class , we expect that can be trained by on to produce a parameterization such that generalizes to out-of-sample instances. In our work, we measure loss on the validation set to approximate how well the model generalizes for . One central assumption of this work is that certain classes in are more learnable than others, that is will vary over different . This implies an ordering over classes in defined as follows:
For given classes , the goal of our algorithm is to guide into spending more training iterations on than . We expect that doing so will not only help improve potential generalization on highly ranked classes, but also avoid negative effects that noisy classes may have on a classifier.
In order to effectively find the above ordering on classes, it becomes important to avoid parameterizations wherein even though , since this would lead to the class selection algorithm choosing more than . Therefore we aim to train our selection algorithm online in lockstep with , receiving feedback at every iteration in order to find a good ranking as quickly as possible. In order to train our selection algorithm, we measure how the model is progressing one step at a time, using only the information that can be gleaned from the validation subset . One way to measure learning progress is via self prediction gain [Graves et al.2017, Oudeyer, Kaplan, and Hafner2007]:
where are the model’s parameters before updates them to using . Note that in practice, many training algorithms will use stochastically selected batches from a subset and/or , but for ease of notation we may refer to the whole set in place of a batch. Self prediction gain can be thought of as measuring how much the loss improves on the validation data after a training update. Intuitively, for a very noisy class, e.g a class with randomly assigned training and validation inputs, in expectation SPG should be zero since is unlikely to learn anything generalizable from training examples . For classes with more signal, it is likely that SPG will yield a positive value if the model can be trained by to generalize. Therefore at each training step , we select a class such that and yield a high expected value of SPG. We now outline the specific bandit algorithm used to select classes per round.
2.3 Bandit Supervision
At a high level, the job of the bandit algorithm at a given training step is to sample a class that maximizes the SPG of (1). At step , a history of arm pulls and their resultant SPGs are available for the bandit to approximate the true, unknown SPG function. To use this history, we require that our bandit model the uncertainty of the SPG per class; measures of uncertainty are required to balance exploration and exploitation. Further, we allow for large sets of classes (i.e arms), and assume that SPG rewards will change with time, since they depend on the changing model . A changing SPG function limits our ability to exploit historically successful arms. In order to handle large class sets, our bandit assumes a structure on the classes, specifically that there exists a measure of similarity between them. To handle the changing SPG function, we apply a discounting factor that limits the impact of observations made early on in training. Specifically, we utilize the bandit of [Bogunovic, Scarlett, and Cevher2016], an approach that models via a time varying Gaussian Process.
A Gaussian Process (GP) defines a Gaussian distribution of the target function’s value at each point in the function’s domain. The distribution at a pointis parameterized by a mean (x) and standard deviation , with representing the expected value of the function at that point. GPs satisfy the need to represent the uncertainty of points in a domain via . Further, a GP captures the relationship of a target function at different points via a covariance , with the covariance between two points and given by for some positive definite kernel function . Intuitively, the kernel measures similarity between two points in the domain, e.g two classes, influencing the GP to predict similar function values for similar points. By modeling the relationship between classes, we alleviate the burden of exhaustively sampling rewards for each class. In our work, we use the word2vec embedding [Mikolov et al.2016] of a given class (denoted ), combined with the Matern kernel [Matérn2013] to measure similarity between classes. The Matern kernel is a generalization of the Gaussian (squared exponential) covariance function, often used because its smoothness can be controlled via a hyperparameter. Tunable smoothness is potentially valuable to our work, since natural language similarity can vary significantly depending on the domain.
At a training step , our GP defines a joint Gaussian distribution over the rewards of classes known as a prior distribution. This prior is used to forecast the reward for selecting a training class . After selection, the SPG for this class is calculated via (1
), and this value is added to the history of rewards. Our bandit updates its joint distribution by incorporating the newly observed SPG, thus producing aposterior distribution. Calculation of the posterior is influenced by two different aspects of each entry in the SPG history: one is the inter-class similarity as defined by the kernel, the other is the time between a historical entry and the most recent observation. We now describe the selection and update steps in detail.
The selection mechanism then follows directly from the concept of upper confidence bounds and estimated priors forand . At time , we select a class according to the following UCB equation
is the word vector associated with class. Observing the reward for the chosen class is as simple as calculating the loss on a validation set as in (1). After observing a new reward, and can be updated as in [Bogunovic, Scarlett, and Cevher2016]:
In order to perform this update, we have access to the history of class selections up to time , the corresponding vector history , and the associated rewards for these arm pulls . We then create a matrix to be ( being the Hadamard product). Intuitively, this matrix compares the vectors and of previously selected classes via the kernel , while discounting the similarity based upon how far apart in time the two arms were pulled. The time discounting is captured via a hyperparameter , applied to , that quantifies how much rewards are expected to change at each step; indicates that rewards are independent between timesteps, and means there is no variation. is joined with a term that models noise produced when sampling from the underlying reward function, and is then used in conjunction with a function that compares historical arms to a given arm , again discounting old arm pulls by .
To select a class in the next round, the updated mean and variance are fed into (2). When our algorithm selects a class to learn based on (2), exploration will dominate if a class has not been recently selected and no similar classes have been recently selected, since will surpass the other terms, yielding a high variance. Otherwise, the algorithm will select classes that the model has done well on in the past, as will become significant. We outline how this process is coupled with the training of our model in Alg. 1.
2.4 Ranking Classes
We use class rankings to determine when bandit supervised models have converged as follows. We calculate a ranking of classes every iterations, with our experiments herein using . We define convergence to be a sufficiently small change in ranking as measured via the normalized Kendall tau distance [Kendall1955] given in (5). This distance measures the number of pairwise differences between two rankings and , computed at rounds and respectively, the set of which is given by:
where are the ranks of class at rounds and . The associated distance counts the number of differences (numerator) and normalizes by the total number of pairs (denominator).
A value of zero indicates identical rankings, while a value of one indicates rankings in the opposite order. We consider our models to have converged when .
There are many situations in which identifying the ordering on classes can be useful, for example in order to inform downstream processes of a classifier’s capabilities. While recovering the underlying ground truth ordering is difficult since it requires identifying an ideal set of parameters for each class, we can use the bandit history to give a good approximation, the idea being that the number of times a class was selected for training is a good indicator of the model’s propensity to generalize on said class. For evaluation purposes, we are interested limiting the variability that results from the stochasticity in Alg. 1, and therefore produce an average ranking with information from multiple runs. This ranking is calculated by sorting the classes given the average amount each class was pulled, i.e
for runs of Alg. 1 and pulls of arm in run .
From a data exploration perspective, the average rankings could help in finding potentially noisy labels, or give insights into the limitations of . From an evaluation perspective, rankings can indicate how well the algorithm is working. Ideally, there would not be significant variations in the ranking list between runs, as this could indicate that is too susceptible to small changes in initial conditions, or simply that our algorithm fails to find a proper ordering of classes. With this definition of an average ranking, we can now evaluate the quality of our bandit algorithm.
3 Empirical Evaluation
Loosely stated, our bandit supervision has two objectives. The first is to influence the training of a model such that the attention payed to a class reflects the learnability of said class. The second, a result of the first, is to produce a classifier that performs well on learnable classes. The goal of this section is to evaluate the performance of our bandit on these two objectives. We quantitatively evaluate performance on the first objective by using a clean data set that we pollute with label noise. For the second, we train classifiers on naturally noisier, unaltered data sets and evaluate the classification performance of our model relative to the produced class ordering.
3.1 Evaluating Bandit Ordering
Given an existing data set, it is unlikely that the ordering on classes will be known a priori. This poses a challenge from an evaluation perspective, since a ranking like that presented in (6) cannot be compared directly to some ground truth. We therefore create a synthetic scenario for testing our bandit algorithm. We start with the Cifar100 data set [Krizhevsky and Hinton2009], a collection of 50k training images (10k of which we set aside for validation) and 10k test images, each of which are labeled as belonging to one of a hundred classes. We then sample from training images uniformly at random, constructing 100 new subsets of and assign to each of these subsets a new, out of vocabulary label. The out of vocabulary classes are chosen from a list of the 100 most common English words that are disjoint from the original classes. In total, our newly constructed data set contains 400 training examples per original class, and 400 examples per new, noisy class.
In running our bandit algorithm on this data set, we hope to show that the produced ranking of classes favors the original classes over the new noisy classes. That is, for any original class and any noisy class , we want our bandit to recover , yielding a ranking with the original classes in the top 100, and noise classes in the bottom 100. In Tab. 1, we show the frequency with which an original class appears in the top 100 classes, ranked as in (6), averaged across ten runs. Additionally, the goal of bandit supervision is to mitigate negative effects that noisy classes may have on model performance, specifically accuracy on learnable classes. As such, we compare the accuracy of a model trained without bandit supervision (ELU-N) to the same model trained with bandit supervision (MAB-S). Test set accuracy is evaluated on the original classes only, and is presented in Tab. 1
. For our classifier, we use the convolutional neural network architecture presented in[Clevert, Unterthiner, and Hochreiter2016], which has been shown to achieve state-of-the-art performance on Cifar100. We compare to their results that were generated by training only on the original Cifar data (ELU).
|MAB-S||67.14||92||Bed, Plain, Bridge,|
|Train, Cattle, Crab,|
|Forest, Baby, Telephone,|
In training the network with bandit supervision, we notice an interesting trade-off with respect to running time. The online selection mechanism prohibits time saving techniques that are common to training CNNs, such as batch pre-fetching. Therefore, each iteration of our bandit supervised training is slower than an iteration of standard mini-batch stochastic gradient descent. However, we also notice our bandit supervised network tends to converge in fewer iterations than the baselines. The aggregate result is a minimal wall-clock overhead associated with bandit supervision.
Tab. 1 shows results in keeping with our intuition. The accuracy of our approach is superior to that of standard mini-batch selection when trained on noisy data. However, we do not fully recover the accuracy of the architecture trained on the original, noise free data. Our bandit is also able to correctly rank 92 of the original classes in the top 100. The eight noise classes ranked in the top 100, “write”, “hot”, “word”, “water”, “call”, “sound”, “your”, and “thing”, may have been chosen for a few reasons. Most likely, some are close to original classes in the word vector space, e.g “water” is very similar to the many fish and nature related classes in Cifar100. Depending on the kernel and vector representations, sufficiently high rewards on original classes may artificially prop up a related noisy class in the eyes of the bandit. For a more naturally constructed data set, we suspect classes that are considered similar by the kernel are more likely to share similar rewards.
3.2 Exploring Large Tag Sets
In this section we intend to gauge the performance of bandit supervision in the face of more realistic noise. The first “real-world” data set that we investigate is a subset of the YFCC 100M data set [Thomee et al.2016], a collection of 100 million tagged images and videos gathered from Flickr. In order to create a manageable but informative subset, we first gather tags of interest from the work of [Garrigues et al.2017], who identify tags commonly searched for by Flickr users. We then construct our data subset by gathering the first million images of YFCC 100M that are labeled with at least one of the tags in our set. In total, our data has 1 million images labeled with 4566 tags.
The classes present in Flickr data vary in learnability because they can be subjectively assigned to examples. We are also interested in learnability variation that is caused by the domain of images itself, i.e images that may be objectively and correctly labeled, yet still yield certain classes that are harder to learn than others. Recipe1m [Salvador et al.2017] is a data set that fits this description, as it contains over 800k images paired with recipes and ingredients. We use this data to form the task of identifying ingredients given an image of a completed recipe. Many ingredients may be very difficult to identify directly, thereby producing a range of learnability across ingredients. As an example, salt is present in a significant portion of recipes across many cuisines, and yet is rarely identifiable in an image.
For these data sets, we focus on the performance of models trained by bandit supervision vs. those trained by baseline approaches. The simplest baseline to which we compare is standard mini-batch gradient descent, where batches are drawn without regard for the labels they contain. A more sophisticated baseline is to choose the target classes given the number of training examples per class, limiting training to the top most frequently occurring classes. Conventional machine learning wisdom dictates that generalizability improves as training data set size increases [Abu-Mostafa, Magdon-Ismail, and Lin2012]
. We hope to see the bandit select classes that do not exactly match those selected by this training-example-count-based heuristic. Instead, the bandit’s selections should yield a trained modelwith better quantitative performance on its chosen labels. In order to measure the performance of , we use the per class F1-Score. Specifically, we calculate true and false positives and negatives for every class across test examples. Given the true and false positives (, ) and false negatives () for a class , the recall , precision , and F1-Score are given by:
We turn to F1-Score for evaluation because it does not rely on true negatives present in the data, and because it captures both recall and precision. In the data sets we use, true negatives far outweigh any other statistic, rendering measures like accuracy uninformative. For fairly comparing our method to baselines, we analyze the per-class performance on the top classes for each of the training methods, for various values of . In training on the YFCC 100M subset, , while for the Recipe1m data set, . In both sets of experiments, is the deep convolutional architecture of [Garrigues et al.2017], an architecture specifically developed for tagging images with low latency during inference. Figure 2 presents a quantitative analysis of the F1-Score across the top classes for each value of and the bandit, while Fig. 3 qualitatively compares results between methods.
Quantitatively, our bandit algorithm performs very well relative to the baselines for a small subset of frequently selected classes, as intended, achieving a much higher F-1 Score for top ranked classes. There is a severe dropoff after a fairly small amount of classes, though the performance of the other models suggests that both image tagging tasks are difficult. It should be noted that after the bandit-trained-model’s F-1 score drops off, it generally performs worse than the other models. This simply confirms that use cases wherein performance must be above some minimum threshold for all classes are not well suited for our approach. After observing the significant divide between high and low performing classes, one question that arises is whether or not the bandit algorithm can be tuned to do well on more classes. We leave this question for future work. Qualitatively, Fig. 3 shows interesting distinctions between the classes chosen by the bandit algorithm and those associated with the most training data. For the Flickr tags, we were surprised to see a concept like “folly” ranked so highly by the bandit. Yet randomly sampled images do show similarities in this tag (3/5 displayed are images of water sports). This suggests that strong priors, i.e indications before training of what classes ought to be learnable, may be difficult to come by.
4 Conclusion and Future Work
In this paper, we motivated and formalized the concept of class learnability, defining it to be an ordering over classes based on potential class generalization error for a given data set. We then developed an online algorithm using existing Multi-Armed Bandit techniques that directs the training of a given classifier, with the intention of focusing on classes as their learnability dictates. We showed that this bandit algorithm is capable of identifying learnable classes in a noisy data set, and that focusing on learnable classes yields models with low generalization error relative to simple baselines and data-science inspired heuristics alike. These results were consistent across a variety of data sets, each with a unique kind of noise in the data’s labeling.
In the future, we are interested in applying our algorithm to data sets created with a variety of labeling processes. As an example, one could imagine re-labeling a data set given synonyms of existing classes, thus creating new data partitions that may facilitate better classifier performance. We believe that our bandit algorithm would be well suited to explore such a class space, identifying which concepts in a synset produce the best data partitions. Similarly, we suspect that our bandit may be appropriate for exploring fine grained recognition data sets, where there is a hierarchy on classes. In such a scenario, our bandit could act in a curriculum learning capacity, starting training by identifying coarse concepts and progressing towards more fine-grained classes if the model is ready. We also intend to investigate various other bandit formulations. As an example, there exist Combinatorial Multi-Armed Bandit algorithms that are capable of selecting a combination of arms at a given iteration, a technique that may allow for more sophisticated batch selection and more successful optimization of a classifier.
Matthew Klawonn is sponsored by an LMI SMART scholarship. Professor Hendler’s work is supported in part by the IBM Corporation through the IBM-Rensselaer AI Research Collaboration.
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