Axiomatic Explanations for Visual Search, Retrieval, and Similarity Learning

1 INTRODUCTION . Search , recommendation , retrieval , and contrastive similarity learning powers many of today ’ s machine learning systems . These systems help us organize information at scales that no human could match . The recent surge in million and billion parameter contrastive learning architectures for vision and language underscore the growing need to understand these classes of systems ( Nayak , 2019 ; Chen et al. , 2020b ; a ; Radford et al. , 2021 ; Caron et al. , 2020 ) . At its core , these systems face the same challenges that classifiers and regressors face : richer models can often improve performance but hinder interpretability . For high-risk domains like medicine , incorrect search results can have serious consequences . In other domains , search engine bias can disproportionately hide certain voices and have systemic impact ( Mowshowitz & Kawaguchi , 2002 ; Diaz , 2008 ; Goldman , 2005 ) . Currently , there are several competing techniques practitioners can use to understand similarity model predictions ( Zhu et al. , 2019 ; Zheng et al. , 2020 ; Dong et al . ; Selvaraju et al. , 2017 ; Vaswani et al. , 2017 ) . However , it is often unclear when one approach should be chosen over another and there is not a formal theory describing an optimal search explanation method . To help solve this challenge we show that the theory of fair credit assignment provides a uniquely determined and axiomatic approach for “ explaining ” why a trained model judges two objects as similar . In many cases , existing approaches are special cases of this formalism . This allows us to design variants of these methods that better satisfy the axioms of fair credit assignment and can handle counterfactual or relative explanations . Though we explore this topic through the lens of visual search , we note that these techniques could also apply to text , tabular , or audio search systems . This work identifies two distinct classes of search engine explainability methods . “ First order ” approaches highlight the most important pixels that contribute to the similarity of objects and “ Second order ” explanations provide a full correspondence between the parts of query and retrieved image . We relate first order interpretations to existing theory on classifier explainability through a generic function transformation , as shown in the third column of Figure 1 . We find that second order explanations correspond to a uniquely specified generalization of the Shapley values ( Sundararajan et al. , 2020 ) and is equivalent to projecting Harsanyi Dividends onto low-order subsets ( Harsanyi , 1963 ) . Using this formalism , we present new second-order generalizations of Class Activation Maps ( Zhou et al. , 2016 ) , GradCAM ( Selvaraju et al. , 2017 ) , LIME ( Ribeiro et al. , 2016 ) , and SHAP ( Lundberg & Lee , 2017 ) . These contributions generalize several existing methods , illustrate a rich mathematical structure connecting model explainability and cooperative game theory , and allow practitioners to understand search engines with greater nuance and detail . We include a short video detailing the work at https : //tinyurl.com/2j3y2ffn . In summary we : • Present the first uniquely specified axiomatic framework for model-agnostic search , retrieval , and metric learning interpretability using the theory of Harsanyi dividends . • Show that our framework generalizes several existing model explanation methods ( Zhou et al. , 2016 ; Selvaraju et al. , 2017 ; Zhu et al. , 2019 ; Ribeiro et al. , 2016 ) and allows us to extend these methods to handle counterfactual explanations and extract dense pairwise correspondences between query and retrieved images . • Introduce a new kernel-based approximator for Shapley-Taylor indices that requires about 10× fewer function evaluations . • Show that our axiomatic approaches provide more faithful explanations of image similarity on the PascalVOC and MSCoCo datasets . 2 BACKGROUND . This work focuses on search , retrieval , metric learning , and recommendation architectures . Often , these systems use similarity between objects or learned features ( Bengio et al. , 2013 ) to rank , retrieve , or suggest content ( Bing , 2017 ; Su & Khoshgoftaar , 2009 ; Chopra et al. , 2005 ; Radford et al. , 2021 ) . More formally , we refer to systems that use a distance , relevance , or similarity function of the form : d : X × Y → R to quantify the relationship between items from sets X and Y . In search and retrieval , X represents the space of search queries and Y represents the space of results , the function d assigns a relevance to each query result pair . Without loss of generality , we consider d as a “ distance-like ” function where smaller values indicate more relevance . The expression arg miny∈Y d ( x , y ) yields the most relevant result for a query x ∈ X . Specializing this notion yields a variety of different kinds of ML systems . IfX = Y = Range ( N ( · ) ) whereN is an image featurization network such as ResNet50 ( He et al. , 2016 ) , the formalism yields a visual search engine or “ reverse image search ” . Though this work focuses on visual search , we note that ifX is the space of character sequences and Y is the space of webpages , this represents web search . If X are users and Y are the space of items , such as songs or news articles , the formalism represents a recommendation problem . In this work we aim to extract meaningful “ interpretations ” or “ explanations ” of the function d . 2.1 MODEL INTERPRETABILITY . The Bias-Variance trade-off ( Kohavi et al. , 1996 ) affects all machine learning systems and governs the relationship between a model ’ s expressiveness and generalization ability . In data-rich scenarios , a model ’ s bias dominates generalization error and increasing the size of the model class can improve performance . However , increasing model complexity can severely degrade its interpretability because added parameters can lose their connection to physically meaningful quantities . This is a challenge not just for classification and regression systems , but for search and recommendation architectures as well . For example , the best performing competitors on the Netflix prize dataset ( Bennett et al. , 2007 ) involve significantly more complex models than earlier baselines ( Rendle et al. , 2019 ) . Famously , the Netflix grand prize winner , the “ BellKor ” algorithm ( Koren , 2009 ) , boosts and ensembles several different underling methods making it difficult to interpret the model by inspecting its parameters . To tackle these challenges , some works introduce model classes that are naturally interpretable ( Nori et al. , 2019 ; Hastie & Tibshirani , 1990 ) . Alternatively , other works propose model-agnostic methods to explain the predictions of classifiers and regressors . Many of these approaches explain the local structure around a specific prediction . Lundberg & Lee ( 2017 ) show that the Shapley value ( Shapley , 1951 ) , a measure of fair credit assignment , provides a unique and axiomatically characterized solution to classifier interpretability ( SHAP ) . Furthermore , the authors show that Shapley values generalize LIME , DeepLIFT ( Shrikumar et al. , 2017 ) , Layer-Wise Relevance Propagation ( Bach et al. , 2015 ) , and several other methods ( Štrumbelj & Kononenko , 2014 ; Datta et al. , 2016 ; Lipovetsky & Conklin , 2001 ; Saabas , 2014 ) . Many works in computer vision use an alternative approach called Class Activation Maps ( CAMs ) . CAM projects the predicted class of a deep global average pooled ( GAP ) convolutional network onto the feature space to create a low resolution heatmap of class-specific network attention . GradCAM ( Selvaraju et al. , 2017 ) generalizes CAM to architectures other than GAP and yields robust interpretations with only a single network evaluation . In Section 4 we show that CAM , GradCAM , and their analogue for search engine interpretability , Zhu et al . ( 2019 ) , are also unified by the Shapley value and its second order generalization , the ShapleyTaylor index . 2.2 FAIR CREDIT ASSIGNMENT AND THE SHAPLEY VALUE . Shapley values provide a principled and axiomatic framework for classifier interpretation . We provide a brief overview of Shapley values but point readers to Molnar ( 2020 ) for more details . Shapley values originally arose in cooperative game theory as the only fair way to allocate the profit of a company to its employees based on their contributions . They have since become a standard method for explaining predictions of machine learning models . In particular , Shapley values measure of how much “ credit ” a model ’ s features receive for the prediction under a fair credit assignment . More formally , a “ coalition game ” is a set N of |N | players and a “ value ” function v : 2N → R. In cooperative game theory , this function v represents the expected payoff for a given cooperating coalition of players . Shapley ( 1951 ) show that the unique , fair credit assignment to each player , φv ( i ∈ N ) , can be calculated as : φv ( i ) : = ∑ S⊆N\ { i } |S| ! ( |N | − |S| − 1 ) ! |N | ! ( v ( S ∪ { i } ) − v ( S ) ) ( 1 ) Informally , this equation measures the average increase in value that a player i brings to a coalition S by weighting each increase , v ( S ∪ { i } ) − v ( S ) , by the number of ways this event could have happened during the formation of the “ grand coalition ” N . We note that this assignment , φv , is the unique assignment that satisfies four reasonable properties : symmetry under player re-labeling , no credit assignment to dummy players , linearity ( or it ’ s alternative monotonicity ) , and efficiency which states that Shapley values should sum to v ( N ) − v ( ∅ ) ( Young , 1985 ) . Intuitively , these axioms require that a fair explanation should treat every feature equally ( Symmetry ) , should not assign importance to features that are not used ( Dummy ) , should behave linearly when the value function is transformed ( Linear ) , and should sum to the function ’ s value ( Efficiency ) . To connect this work to model interpretability , we can identify the “ features ” used in a model as the “ players ” and interpret the value function , v ( S ) , as the expected prediction of the model when featuresN \S are replaced by values from a “ background ” distribution . This background distribution allows for “ counterfactual ” or relative explanations ( Goyal et al. , 2019 ) . 3 RELATED WORK . The field of model interpretability has a considerable body of literature , and we mention just a handful of the works that are particularly related . One of our baseline methods , Dong et al. , was one of the first to present a generic visual search engine explanation reminiscent of a Parzen-Window based estimator . Fong & Vedaldi ( 2017 ) introduce a method for explaining classifiers based on meaningful pertubation and Chefer et al . ( 2021 ) introduce a method for improving interpretation for transformer-based classifiers . Zhu et al . ( 2019 ) lifted CAM to search engines and we find that our Shapley-Taylor based method aligns with their approach for GAP architectures . Singh & Anand ( 2019 ) and Fernando et al . ( 2019 ) use LIME and DeepSHAP to provide first-order interpretations of text but do not apply their methods to images . Ancona et al . ( 2019 ) introduce a distribution propagation approach for improving the estimation of Shapley Values for deep models and can be combined with our approach . Many works implicitly use components that align with ShapleyTaylor indices for particular functions . Works such as Fischer et al . ( 2015a ) ; Sun et al . ( 2020 ) ; Wang et al . ( 2020a ) ; Chen et al . ( 2020c ) ; Hou et al . ( 2019 ) use feature correlation layers to estimate and utilize correspondences between images . We show these layers are equivalent to Shapley-Taylor indices on the GAP architecture , and this allows create a correlation layer that handles counterfactual backgrounds . Other recent works have used learned co-attention within transformer architectures to help pool and share information across multiple domain types ( Wei et al. , 2020 ) . Fu et al . ( 2020 ) attempt to learn a variant of GradCAM that better aligns with axioms similar to Shapley Values by adding efficiency regularizers . The method is not guaranteed to satisfy the axioms but is more “ efficient ” . We rely on several works to extend Shapley values to more complex interactions . Harsanyi ( 1963 ) generalized the Shapely value by introducing a “ dividend ” that , when split and distributed among players , yields the Shapley values . Owen ( 1972 ) introduces an equivalent way to extend Shapley values using a multi-linear extension of the game ’ s characteristic function . Sundararajan et al . ( 2020 ) introduce the Shapley-Taylor index and show is equivalent to the Lagrangian remainder of Owen ’ s multi-linear extension . Integrated Hessians ( Janizek et al. , 2020 ) enable estimation of a secondorder variant of the Aumann-Shapley values and we use this approach to create a more principled second-order interpretation method for differentiable search engines .

Summary: This paper proposes an approach for explaining visual similarity models.  They provide a general framework that enables them to adapt existing explanation methods to the similarity learning task, and then discuss how they apply their framework to generalize multiple methods.  The authors evaluate their work on PASVAL VOC and COCO, where they use automatic metrics to argue that their approach accurately captures model performance.