On the Learning of Quasimetrics

1 INTRODUCTION . Learned symmetrical metrics have been proven useful for innumerable tasks including dimensionality reduction ( Tenenbaum et al. , 2000 ) , clustering ( Xing et al. , 2002 ) , classification ( Weinberger et al. , 2006 ; Hoffer & Ailon , 2015 ) , and information retrieval ( Wang et al. , 2014 ) . However , the real world is largely asymmetrical , and symmetrical metrics can only capture a small fraction of it . Generalizing metrics , quasimetrics ( Defn . 2.1 ) allow for asymmetrical distances and can be found in a wide range of domains ( see Fig . 1 ) . Ubiquitous physical forces , such as gravity and wind , as well as human-defined rules , such as one-way roads , make the traveling time between places a quasimetric . Furthermore , many of our social artifacts are directed graphs— genealogy charts , follow-relation on Twitter ( Leskovec & Krevl , 2014 ) , citation graphs ( Price , 2011 ) , hyperlinks over the Internet , etc . Shortest paths on these graphs naturally induce quasimetric spaces . In fact , we can generalize to Markov Decision Processes ( MDPs ) and observe that optimal goal-reaching plan costs ( i.e. , universal value/Q-functions ( Schaul et al. , 2015 ; Sutton et al. , 2011 ) ) always form a quasimetric ( Bertsekas & Tsitsiklis , 1991 ; Tian et al. , 2020 ) . Moving onto more abstract structures , quasimetrics can also be found as expected hitting times in Markov chains , and as conditional Shannon entropy H ( · | · ) in information theory . ( See the appendix for proofs and discussions of these quasimetrics . ) In this work , we study the task of quasimetric learning . Given a sampled training set of pairs and their quasimetric distances , we ask : how well can we learn a quasimetric that fits the training data ? We define quasimetric learning in analogy to metric learning : whereas metric learning is the problem of learning a metric function , quasimetric learning is the problem of learning a quasimetric function . This may involve searching over a hypothesis space constrained to only include quasimetric functions ( which is what our method does ) or it could involve searching for approximately quasimetric functions ( we compare to and analyze such approaches ) . Successful formulations have many potential applications , such as good structural priors in reinforcement learning ( Schaul et al. , 2015 ; Tian et al. , 2020 ) , graph learning ( Rizi et al. , 2018 ) and causal relation learning ( Balashankar & Subramanian , 2021 ) . Towards this goal , our contributions are • We study the quasimetric learning task with two goals : ( 1 ) fitting training data well and ( 2 ) respecting quasimetric constraints ( Sec . 3 ) ; • We prove that a large family of algorithms , including unconstrained networks trained in the Neural Tangent Kernel ( NTK ) regime ( Jacot et al. , 2018 ) , fail at this task , while a learned embedding into a latent quasimetric space can potentially succeed ( Sec . 4 ) ; • We propose Poisson Quasimetric Embeddings ( PQEs ) , the first quasimetric embedding formulation learnable with gradient-based optimization , with strong theoretical guarantees ( Sec . 5 ) ; • Our experiments complement the theory and demonstrate the benefits of PQEs on random graphs , social graphs and offline Q-learning ( Sec . 6 ) . 2 PRELIMINARIES ON QUASIMETRICS AND POISSON PROCESSES . Quasimetric space is a generalization of metric space where all requirements of metrics are satisfied , except that the distances can be asymmetrical . Definition 2.1 ( Quasimetric Space ) . A quasimetric space is a pair ( X , d ) , where X is a set of points and d : X × X → [ 0 , ∞ ] is the quasimetric , satisfying the following conditions : ∀x , y ∈ X , x = y ⇐⇒ d ( x , y ) = 0 , ( Identity of Indiscernibles ) ∀x , y , z ∈ X , d ( x , y ) + d ( y , z ) ≥ d ( x , z ) . ( Triangle Inequality ) Being asymmetric , quasimetrics are often thought of as ( shortest-path ) distances of some ( possibly infinite ) weighted directed graph . A natural way to quantify the complexity of a quasimetric is to consider that of its underlying graph . Quasimetric treewidth is an instantiation of this idea . Definition 2.2 ( Treewidth of Quasimetric Spaces ( Mémoli et al. , 2018 ) ) . Consider a quasimetric space M as shortest-path distances on a positively-weighted directed graph . Treewidth of M is the minimum over all such graphs ’ treewidths . Poisson processes are commonly used to model events ( or points ) randomly occurring across a set A ( Kingman , 2005 ) , e.g. , raindrops hitting a windshield , photons captured by a camera . The number of such events within a subset of A is modeled as Poisson distribution , whose mean is given by a measure µ of A that determines how “ frequently the events happen at each location ” . Definition 2.3 ( Poisson Process ) . For nonatomic measure µ on set A , a Poisson process on A with mean measure µ is a random countable subset P ⊂ A ( i.e. , the random events / points ) such that • for any disjoint measurable subsets A1 , . . . , An of A , the random variables N ( A1 ) , . . . , N ( An ) are independent , where N ( B ) , # { P ∩B } is the number of points of P in B , and • N ( B ) has the Poisson distribution with mean µ ( B ) , denoted as Pois ( µ ( B ) ) . Fact 2.4 ( Differentiability of P [ N ( A1 ) ≤ N ( A2 ) ] ) . For two measurable subsets A1 , A2 , P [ N ( A1 ) ≤ N ( A2 ) ] = P [ Pois ( µ ( A1 \A2 ) ) ≤ Pois ( µ ( A2 \A1 ) ) ︸ ︷︷ ︸ two independent Poissons ] . ( 1 ) Furthermore , for independent X ∼ Pois ( µ1 ) , Y ∼ Pois ( µ2 ) is differentiable w.r.t . µ1 and µ2 . In the special case where µ1 or µ2 is zero , we can simply compute P [ X ≤ Y ] = { P [ 0 ≤ Y ] = 1 if µ1 = 0 P [ X ≤ 0 ] = P [ X = 0 ] = e−µ1 if µ2 = 0 ( Pois ( 0 ) is always 0 ) = exp ( − ( µ1 − µ2 ) + ) , ( 2 ) where x+ , max ( 0 , x ) . In general , this probability and its gradients can be obtained via a connection to noncentral χ2 distribution ( Johnson , 1959 ) , which we overview and derive formulas in the appendix . Therefore , if A1 and A2 are parametrized by some θ such that µ ( A1 \ A2 ) and µ ( A2 \ A1 ) are differentiable w.r.t . θ , so is P [ N ( A1 ) ≤ N ( A2 ) ] . a b c : Train : Test Triangle inequality =⇒ ? ≤ d ( a , b ) + d ( b , c ) = 31 ? ≥ d ( a , b ) − d ( c , b ) = 28 0 0 029 2 1 11 ? 0 20 40 60 80 100 Count Unconstrained Network ( Training MSE = 0.02 ± 0.06 ) 0 10 20 30 Valid Range 0 20 40 60 80 100 Euclidean Space Embedding ( Training MSE = 58.83 ± 0.00 ) 0 10 20 30 0 20 40 60 80 100 Poisson Quasimetric Embedding ( Training MSE = 0.02 ± 0.07 ) 0 10 20 30 Figure 2 : Quasimetric learning on a 3-element space . Leftmost : Training set contains all pairs except for ( a , c ) . Arrow labels show quasimetric distances ( rather than edge weights ) . A quasimetric d̂ should predict d̂ ( a , c ) ∈ [ 28 , 30 ] . Right three : Different formulations are trained to fit training pairs distances , and then predict on the test pair . Plots show distribution of the prediction over 100 runs . 3 QUASIMETRIC LEARNING . Consider a quasimetric space ( X , d ) . The quasimetric learning task aims to infer a quasimetric from observing a training set { ( xi , yi , d ( xi , yi ) ) } i ⊂ X × X × [ 0 , ∞ ] . Naturally , our goals for a learned predictor d̂ : X × X → R are : respecting the quasimetric constraints and fitting training distances . Crucially , we are not simply aiming for the usual sense of generalization , i.e. , low population error . Knowing that true distances have a quasimetric structure , we can better evaluate predictors and desire ones that fit the training data and are ( approximately ) quasimetrics . These objectives also indirectly capture generalization because a predictor failing either requirement must have large error on some pairs , whose true distances follow quasimetric constraints . We formalize this relation in Thm . 4.6 . 3.1 LEARNING ALGORITHMS AND MODEL CLASSES . Ideally , the learning should scale well with data , potentially generalize to unseen samples , and support integration with other deep learning systems ( e.g. , via differentiation ) . One option is to simply learn a generic function approximator that takes in ( x , y ) ∈ X × X ( as a concatenated input ) and outputs a scalar as the prediction of the pair ’ s distance , or some transformed version of it ( e.g. , log distance ) . This approach has been adopted by the community in learning graph distances ( Rizi et al. , 2018 ) and plan costs in MDPs ( Tian et al. , 2020 ) . When the function approximator is a deep neural network , we refer to such formulations as unconstrained ( deep neural ) networks . While deep networks are known to fit training data well ( Jacot et al. , 2018 ) , in this paper we also investigate whether they learn to be ( approximately ) quasimetrics . Alternatively , we can encode each input to a latent space , where a bivariate function outputs the distance prediction . When the function is a latent metric , we obtain metric embeddings . Our proposed Poisson Quasimetric Embedding ( PQE ) ( specified in Sec . 5 ) is also of this kind , where the function is a differentiable latent quasimetric that can approximate arbitrary quasimetrics in data space . 3.2 A TOY EXAMPLE . To build up intuition on how various models perform according to our two goals , we consider a toy quasimetric space with only 3 elements in Fig . 2 . The space has a total of 9 pairs , from which we use 8 as the training set . Due to quasimetric requirements ( esp . triangle inequality ) , knowing distances of these 8 pairs restricts valid values for the heldout pair to a particular range ( which is [ 28 , 31 ] in this case ) . If a model approximates 8 training pairs well and respects quasimetric constraints well , its prediction on that heldout pair should fall into this range . We train three models w.r.t . mean squared error ( MSE ) over the training set using gradient descent : • Unconstrained deep network that predicts distance , • Metric embedding into a latent Euclidean space with a deep encoder , • Quasimetric embedding into a latent PQE space with a deep encoder ( our method ) . ( Details of Poisson Quasimetric Embedding ( PQE ) are explained later in Sec . 5 . ) The three approaches exhibit interesting qualitative differences . Euclidean embedding , unable to model asymmetries in training data , fails to attain a low training error . While both other methods approximate training distances well , unconstrained networks greatly violate quasimetric constraints ; only PQEs respect the constraints and consistently predicts within the valid range . Here , the structural prior of embedding into a quasimetric latent space appears important to successful learning . Without any such prior , unconstrained networks fail badly . In the next section , we present a rigorous theoretical study of the quasimetric learning task , which confirms this intuition .

Summary: This paper studies the problem of learning of quasimetrics. It first proves that an orthogonal equivariant model cannot learn quasimetrics reliably, and then proposes the Poisson Quasimetric Embedding model which is both universal and differentiable. To be more precise, the Poisson process (or just a soft modification of Order Embedding) is differentiable and can be used to approximate quasipartitions, and quasipartitions can be used to approximate arbitrary quasimetrics.  The paper also conducts extensive experiments and indeed PQE has better performance for many tasks.