Learning to Abstain in the Presence of Uninformative Data

1 INTRODUCTION . Despite the success of machine learning in computer vision ( Deng et al. , 2009 ; Krizhevsky et al. , 2009 ; He et al. , 2016a ; Huang et al. , 2017 ) and natural language processing ( Vaswani et al. , 2017 ; Devlin et al. , 2018 ) , the power of ML is yet to make significant impact in other areas . One major challenge is the inherently high noise-to-signal ratio in certain domains . For example , in Finance , while stock prices generally reflect the information about financial health of their companies , over the short term , their fluctuations most closely resemble random walks - which are naturally unpredictable - and are usually modeled as such ( Tsay , 2005 ) . In biomedical research , the underlying phenomena are often highly complex and are affected by unobservable factors . The outcome may appear highly random to the measurements ( gene expression , medical histories ) , if the true causing factor is not included or is overwhelmed by others . We are interested in dealing with datasets that may contain large fraction of noisy/uninformative/not learnable data in both training and testing stages . Direct application of standard supervised learning methods to such datasets is both challenging and unwarranted . At the training stage , the uninformative data can significantly bias the model or even completely overwhelm the true signal ( Nettleton et al. , 2010 ) . Therefore , naïve forecasts in majority-uninformative datasets are doomed to be unreliable . Compared to other learning methods , deep neural networks are even more affected by the presence of noise , due to their strong memorization power ( Zhang et al. , 2017 ) : they are likely to overfit the noise and make overly confident predictions where no real structure exists . In this paper , we propose a novel method for learning on datasets where a significant portion of content is pure noise . Instead of forcing the classifier to make predictions for every sample , we learn to decide whether a datapoint is informative or not . If successful , the method abstains from making decisions where no structure exists , and predominantly learns from the remaining predictable data . Our idea is inspired by the classic selective prediction problem ( Chow , 1957 ) , in which one learns to select a subset of data and only predict on that subset . However , the goal of selective prediction is very different from ours . A selective prediction method considers all data relevant . It pursues a balance between coverage ( i.e . proportion of the data selected ) and conditional accuracy on the selected data . In our problem , we assume that uninformative data is an integral part of the data generative process . No learning method , no matter how powerful , can be successful on such data . Our goal is to identify these uninformative samples as well as possible , and at the same time , to train a classifier by minimizing conditional risk on the remaining informative data . Our method learns both a predictor , f , that classifies samples , and a selector , g , that selects learnable data for the predictor and rejects/abstains from the uninformative data . We are using g to approximate the ground truth indicator function of structured/informative data , g∗ . We assume that g∗ exists as a part of the data generation process , but it is never revealed to us , even during training . Instead of direct supervision , we therefore must rely on the predictor ’ s mistakes to train the selector . To achieve this goal , we propose a novel selector loss enforcing that ( 1 ) the selected data best fits the predictor , and ( 2 ) the portion of the data where we abstain from forecasting , does not contain many correct predictions . This loss function is quite different from the loss in classic selective prediction , which penalizes all unselected data equally . A major contribution of this paper is the derivation of theoretical guarantees for the empirical minimizer of our loss . We analyze the proposed selector loss function and provide sample complexity for learning a nearly optimal selector . We show that optimizing such loss can recover nearly all the structured/informative data in a PAC fashion ( Valiant , 1984 ; Kearns et al. , 1994 ; Valiant , 2013 ) , i.e . one can approximate the ground truth selector function g∗ well with high probability , given sufficient samples . What may be surprising is that this guarantee holds even in a challenging setting where the uninformative data represents the majority of the training set . This theoretical guarantee lets us expand to a more challenging and realistic setting . When the sample size is limited and the initially learned predictor is not sufficiently close to the ground truth , we extend our method to an iterative algorithm , in which we progressively optimize both the predictor and the selector . The selector is improved by optimizing our novel selector loss . Meanwhile , the predictor is improved by optimizing the empirical risk , reweighted based on the selector ’ s output ; uninformative or nearly-uninformative samples identified by the selector will be down-weighed . Experiments on real-world datasets demonstrate superiority of our method to existing baselines . Note that in this paper , we assume that each sample is either informative or not . Extending our method to a more general setting with continuous transitions between informative and uninformative is non-trivial and is left as future work . 1.1 RELATED WORK . Learning with untrusted data aims to recover the ground truth model from a partially corrupted dataset . Different noise models for untrusted data have been studied , including random label noise ( Bylander , 1994 ; Natarajan et al. , 2013 ; Han et al. , 2018 ; Yu et al. , 2019 ; Zheng et al. , 2020 ; Zhang et al. , 2020 ) , bounded label noise ( Massart & Nédélec , 2006 ; Awasthi et al. , 2015 ; Diakonikolas et al. , 2019 ; 2020 ) and adversarial noise ( Kearns & Li , 1993 ; Kearns et al. , 1994 ; Kalai et al. , 2008 ; Klivans et al. , 2009 ; Awasthi et al. , 2017 ) . In the most pessimistic setting , if the majority data is corrupted by arbitrary adversarial noise , even mean estimation may be impossible . This is known as List-Decodable problem ( Balcan et al. , 2008 ; Charikar et al. , 2017 ; Diakonikolas et al. , 2018 ) , where the best one can do when the proportion of trusted data α is less than 0.5 , is to return 1α many hypotheses of the mean , knowing one of them is promising . We , on the other hand , aim to produce a single accurate model even in a setting where the majority of the data is uninformative ( the noise , of course , must be of a certain type , see next section ) . While above works assume the presence of noisy data only in the training stage , we study the case where noise is an integral part of the generative process and thus will appear during inference as well , where it must be detected and discarded once more . Selective learning is an active research area . It extends the classic selective prediction problem and studies how to select a subset of data for different learning tasks . We can summarize existing methods into 4 categories : Monte Carlo sampling based methods ( Gal & Ghahramani , 2016 ; Kendall & Gal , 2017 ; Pearce et al. , 2020 ) , margin based methods ( Fumera & Roli , 2002 ; Bartlett & Wegkamp , 2008 ; Grandvalet et al. , 2008 ; Wegkamp et al. , 2011 ; Zhang et al. , 2018 ) , confidence based methods ( Wiener & El-Yaniv , 2011 ; Geifman & El-Yaniv , 2017 ; Jiang et al. , 2018 ) and customized selective loss ( Cortes et al. , 2016 ; Geifman & El-Yaniv , 2019 ; Liu et al. , 2019 ) . Notably , several works propose customized losses , and incorporate them into neural networks . In ( Geifman & El-Yaniv , 2019 ) , the network maintains an extra output neuron to indicate rejection of datapoints . Liu et al . ( 2019 ) use Gambler loss where a cost term is associated with each output neuron and a doubling-rate-like loss function is used to balance rejections and predictions . Cortes et al . ( 2016 ) perform data selection with an extra model and introduce a selective loss that helps maximize the coverage ratio , thus trading off a small fraction of data for a better precision . Existing works on selective prediction are all motivated from the coverage perspective - i.e . one wants to make safe prediction to achieve higher precision while maintain a reasonable recall ( El-Yaniv et al. , 2010 ) . Whereas our paper is the first to investigate the case where some ( or even majority ) of the data is uninformative , and thus must be discarded at prediction time . Unlike with the selective prediction , there is a latent ground truth indicator function of whether a datapoint should be selected or not . Our method is guaranteed to identify those uninformative samples . 2 PROBLEM FORMULATION . In this section , we describe the inherently-noisy data generation process that we aim to study . The model has three important features : 1 ) the uninformative portion of the data has labels generated by coin flipping ; 2 ) the informative datapoints are labeled with a latent ground truth function ; and 3 ) the uninformative data has a distinguishable support from the informative data . Formally : Definition 1 ( Noisy Generative Process ) . We define Noisy Generative Process by the following notation x ∼ Dα where Dα ≡ { x ∼ DU with prob . 1− α ( Uninformative/Noisy Data ) x ∼ DI with prob . α ( Informative/Structured Data ) . ( 1 ) Let ΩD ⊆ Rd be the support of Dα . Suppose { ΩU , ΩI } is a partition of ΩD , the ground truth labeling function f∗ : X → { +1 , −1 } is in hypothesis class F and data is sampled according to : x ∼ Dα ; y ≡ { Bernoulli ( 0.5 ) , if x ∈ ΩU f∗ ( x ) , if x ∈ ΩI . ( 2 ) Note that the f∗ is defined on the whole domain , although only the part within ΩI is relevant . α represents the fraction of informative or structured data in the population . For the rest of the paper , we abuse the notation and use ( x , y ) ∼ Dα to refer to samples generated from this process . Next definition describes a separability condition between informative and uninformative data . It allows to distinguish noise from structure and enables us to approximate the ground truth selector via empirical minimization . Definition 2 ( H-Separable ) . Given compact set Ω and its partition { ΩU , ΩI } , { ΩU , ΩI } satisfies H-Separable condition if there exists g∗ ∈ H satisfying g∗ ( x ) : X → { +1 , −1 } : g∗ ( x ) ≡ { −1 , if x ∈ ΩU 1 , if x ∈ ΩI . ( 3 ) One can view g∗ ( · ) in definition 2 as the target selector we wish to recover . Having introduced the data generation process and the separability condition , we now describe our main assumption for the remainder of the paper . Assumption 1 . Data Sn = { xi , yi } Ni=1 is i.i.d generated according to the Noisy Generative Process ( Definition 1 ) , with f∗ ∈ F , support ΩDα = ΩU ∪̇ΩI where ΩU , ΩI areH-Separable . Throughout this paper , we are interested in the following learning task : Problem 1 ( Abstain from Uninformative Data ) . Under Assumption 1 with enough i.i.d observations from Dα : 1 ) given hypothesis class F , we aim to learn the underlying ground truth classifier f∗ ( x ) ; and 2 ) given the hypothesis classH , we aim to learn the selector g∗ ( x ) . Evaluation metrics . We define metrics to evaluate the quality of both the prediction and the selection . For the prediction , we borrow the selective risk definition from selective learning ( El-Yaniv et al. , 2010 ; Geifman & El-Yaniv , 2019 ; 2017 ) using our own notation . Definition 3 ( Selective Risk ) . Given a predictor f ∈ F and a selector g ∈ H , we define the selective risk as : CR ( f , g ) = E ( x , y ) ∼Dα [ 1 { f ( x ) 6= y } |g ( x ) ≥ 0 ] and its empirical version : CRS ( f , g ) = ∑n i=1 1 { f ( xi ) 6=yi } 1 { g ( xi ) ≥0 } ∑n i=1 1 { g ( xi ) ≥0 } . Selective risk measures the average risk conditioned on instances that are picked by the selector g ( x ) . Note here when combined with the ground truth selector g∗ , the selective risk of the ground truth predictor f∗ goes to zero . Without a selector , however , f∗ has classical classification risk ( which will be formally defined in Definition 5 later ) of more than 12 ( 1− α ) . The metric used to evaluate the quality of a learned selector g is its false positive/negative rate . When g ( x ) ≥ 0 and g∗ ( x ) < 0 , the selector g accepts an uninformative datapoint and thus commits a false positive error . When g ( x ) < 0 and g∗ ( x ) ≥ 0 , g rejects/abstains from an informative datapoint resulting in a false negative error . Definition 4 ( Evaluation Metric for the Selector ) . Given distribution Dα as defined in Defnition 1 and g∗ as target selector , we denote the false positive and false negative of a selector g as False Positive : P [ g ( x ) ≥ 0|g∗ ( x ) < 0 ] ; False Negative : P [ g ( x ) < 0|g∗ ( x ) ≥ 0 ] ( 4 )

Summary: The paper is concerned with selective classification in a stylised 'realisably noisy' data model, wherein the support of the input distribution is partitioned into two chunks, the "informative" $\Omega_I$ and the "uninformative" $\Omega_U,$ such that  - The labels are completely noisy ($\mathrm{Bern}(1/2)$) if the input lies in $\Omega_U$. - The labels are completely clean if the input lies in $\Omega_I$. - The learner has access to hypothesis classes $\mathcal{F}, \mathcal{H}$ such that      - There exists an $f^* \in \mathcal{F}$ with $f^*(x) = y$ on $\Omega_I$.     - There exists a $g^* \in \mathcal{H}$ with $g^*(x) = 2\mathbf{1}\{x \in \Omega_I\} - 1.$  The paper studies the problem of learning a pair $(f,g),$ with $g$ serving as a selector, and $f$ as a predictor, such that $g \approx g^*$ and $f \approx y$ on $\{g \ge 0\}$ (and $\approx f^*$ on $\Omega_I$ if $g$ is learnt well), with the concrete goal of attaining both small selective risk ($P(f^*(x) \neq y|g(x) \ge 0)$) and both low false alarm and low missed detection for $g$ with respect to $g^*$. As is standard in selective classification, supervision of the value of $g^*$ is not given, and instead only $(x,y)$ pairs are provided.  This is approached by designing a new loss for learning a selector given a predictor $f$, denoted $W(g;f, \theta)$, that basically uses $\mathbf{1}\{f(x) = y\}$ as proxy supervision for when $g < 0$ or $g > 0$ is preferred, but weighted by a factor to account for the prevalence of uninformative versus informative data. It is first shown, using uniform convergence techniques, that if the indicator ERM problem can be solved, then given  if $f^*_{S_n}$ is taken to be a minimiser of standard empirical risk, and $g^*_{S_n}$ to be the minimiser of an empirical version of $W(g;f^*_{S_n}, \theta)$ for appropriately chosen $\theta$ (in a way which depends on the prevalence of informative data $\alpha$), then these goals can be attained in a PAC sense, with sample complexity scaling as $\tilde\{O\}( \frac{d_\{\mathrm{vc}\}(\mathcal{F}) + d_\{\mathrm{vc}\}(\mathcal{H}) }{\epsilon^2 \alpha^2})$.  The paper then switches gears, and a alternating minimisation based heuristic method for practically learning $(f,g)$ is proposed. The idea is to first learn (soft functions) $f$ and then $g$ using relaxed versions of the above losses, and then multiplicatively increase the weight of points that are selected by the $g$ in their contribution to the loss for $f$, and repeat. The remainder of the paper is devoted to empirical evaluation of this method in two situations - firstly, uniformative (randomised labels) MNIST data is given along with varying amounts of clean Fashion MNIST data; and secondly, part of the classes from the SVHN dataset have their labels randomised to represent uninformative data, while the rest remain clean. While the scheme is observed to have similar performance in how well the selector identifies the underlying $g^*$, the proposed method has remarkably better selective risk compared to baselines from the recent selective classification literature.