Understanding the role of importance weighting for deep learning

1 INTRODUCTION . Importance weighting is a standard tool for estimating a quantity under a target distribution while only the samples from some source distribution is accessible . It has been drawing extensive attention in the communities of statistics and machine learning . Causal inference for deep learning investigates heavily on the propensity score weighting method that applies the off-policy optimization with counterfactual estimator ( Gilotte et al. , 2018 ; Jiang & Li , 2016 ) , modelling with observational feedback ( Schnabel et al. , 2016 ; Xu et al. , 2020 ) and learning from controlled intervention ( Swaminathan & Joachims , 2015 ) . The importance weighting methods are also applied to characterize distribution shifts for deep learning models ( Fang et al. , 2020 ) , with modern applications in such as the domain adaptation ( Azizzadenesheli et al. , 2019 ; Lipton et al. , 2018 ) and learning from noisy labels ( Song et al. , 2020 ) . Other usages include curriculum learning ( Bengio et al. , 2009 ) and knowledge distillation ( Hinton et al. , 2015 ) , where the weights characterize the model confidence on each sample . To reduce the discrepancy between the source and target distribution for model training , a standard routine is to minimize a weighted risk ( Rubinstein & Kroese , 2016 ) . Many techniques have been developed to this end , and the common strategy is re-weighting the classes proportionally to the inverse of their frequencies ( Huang et al. , 2016 ; 2019 ; Wang et al. , 2017 ) . For example , Cui et al . ∗The work was done when the author was with Walmart Labs . ( 2019 ) proposes re-weighting by the inverse of effective number of samples . The focal loss ( Lin et al. , 2017 ) down-weights the well-classified examples , and the work by Li et al . ( 2019 ) suggests an improved technique which down-weights examples based on the magnitude of the gradients . Despite the empirical successes of various re-weighting methods , it is ultimately not clear how importance weighting lays influence from the theoretical standpoint . The recent study of Byrd & Lipton ( 2019 ) observes from experiments that there is little impact of importance weights on the converged deep neural network , if the data can be separated by the model using gradient descent . They connect this phenomenon to the implicit bias of gradient descent ( Soudry et al. , 2018 ) - a novel topic that studies why over-parameterized models trained on separable data is biased toward solutions that generalize well . Implicit bias of gradient descent has been observed and studied for linear model ( Soudry et al. , 2018 ; Ji & Telgarsky , 2018b ) , linear neural network ( Ji & Telgarsky , 2018a ; Gunasekar et al. , 2018 ) , two-layer neural network with homogeneous activation ( Chizat & Bach , 2020 ) and smooth neural networks ( Nacson et al. , 2019 ; Lyu & Li , 2019 ) . To summarize , those work reveals that the direction of the parameters ( for linear predictor ) and the normalized margin ( for nonlinear predictor ) , regardless of the initialization , respectively converge to those of a max-margin solution . The pivotal role of margin for deep learning models has been explored actively after the long journey of understanding the generalization of over-parameterized neural networks ( Bartlett et al. , 2017 ; Golowich et al. , 2018 ; Neyshabur et al. , 2018 ) . For instance , Wei et al . ( 2019 ) studies the margin of the neural networks for separable data under weak regularization . They show that the normalized margin also converges to the max-margin solution , and provide a generalization bound for a neural network that hinges on its margin . Although there are rich understandings for the implicit bias of gradient descent and the margin-based generalization , very few efforts are dedicated to studying how they adjust to the weighted empiricalrisk minimization ( ERM ) setting . The established results do not directly transfer since importance weighting can change both the optimization geometry and how the generalization is measured . In this paper , we fill in the gap by showing the impact of importance weighting on the implicit bias of gradient descent as well as the generalization performance . By studying the optimization dynamics of linear models , we first reveal the effect of importance weighting on the convergence speed under linearly separable data . When the data is not linearly separable , we characterize the unique role of importance weighting on defining the intercept term upon the implicit bias . We then investigate the non-linear neural network under a weak regularization as Wei et al . ( 2019 ) . We provide a novel generalization bound that reflects how importance weighting leads to the interplay between the empirical risk and a compounding term that consists of the model complexity as well as the deviation between the source target distribution . Based on our theoretical results , we discuss several exploratory developments on importance weighting that are worthy of further investigations . • A good set of weights for learning can be inversely proportional to the hard-to-classify extent . For example , a sample that is close to ( far from ) the oracle decision boundary should have a large ( small ) weight . • If the importance weights are jointly trained according to a weighting model , the impact of the weighting model eventually diminishes after showing strong correlation with the hard-to-classify extent such as margin . • The usefulness of explicit regularization on weighted ERM can be studied , via their impact on the margin , on balancing the empirical loss and the distribution divergence . In summary , our contribution are three folds . • We characterize the impact of importance weighting on the implicit bias of gradient descent . • We find a generalization bound that hinges on the importance weights . For finite-step training , the role of importance weighting on the generalization bound is reflected in how the margin is affected , and how it balances the source and target distribution . • We propose several exploratory topics for importance weighting that worth further investigating from both the application and theoretical perspective . The rest of the paper is organized as follows . In Section 2 , we introduce the background , preliminary results and the experimental setup . In Section 3 and 4 , we demonstrate the influence of the importance weighting for linear and non-linear models in terms of the implicit bias of gradient descent and the generalization performance . We then discuss the extended investigations in Section 5 . 2 PRELIMINARIES . We use bold-font letters for vectors and matrices , uppercase letters for random variables and distributions , and ‖ · ‖ to denote ` 2 norm when no confusion arises . We denote the training data by D = { wi , xi , yi } ni=1 where xi ∈ X denotes the features , yi is binary or categorical , and the importance weight is bounded such that : wi ∈ [ 1/M , M ] for someM > 1 . We mention that the importance weights are often defined with respect to the source distribution Ps from which the training data is drawn , and the target distribution Pt . We do not make this assumption here because importance weighting is often applied for more general purposes . Therefore , wi can be defined arbitrarily . We use f ( θ , x ) to denote the predictor and define F = { f ( θ , · ) | θ ∈ Θ ⊂ Rd } . For the sake of notation , we focus on the binary setting : yi ∈ { −1 , +1 } with f ( θ , x ) ∈ R. However , it will become clear later that our results can be easily extended to the multi-class setting . Consider the weighted empirical risk minimization ( ERM ) task with the risk given byL ( θ ; w ) = 1/n ∑n i=1 wi ` ( yif ( θ , xi ) ) for some non-negative loss function ` ( · ) . The weight-agnostic counterpart is denoted by : L ( θ ) = 1/n ∑n i=1 ` ( yif ( θ , xi ) ) . We focus particularly on the exponential loss ` ( u ) = exp ( −u ) and log loss ` ( u ) = log ( 1 + exp ( −u ) ) . For the multi-class problem where yi ∈ [ k ] , we extend our setup using the softmax function where the logits are now given by { fj ( θ , x ) } kj=1 . For optimization , we consider using gradient descent to minimize the total loss : θ ( t+1 ) ( w ) = θ ( t ) ( w ) − ηt∇L ( θ ; w ) ∣∣ θ=θ ( t ) ( w ) , where the learning rate ηt can be constant or step-dependent . From parameter norm divergence to support vectors . Suppose D is separated by f ( θ ( t ) , x ) after some point during training . The key factor that contributes to the implicit bias for both linear and non-linear predictor under a weak regularization 1 is that the norm of the parameters diverges after separation , i.e . limt→∞ ‖θ ( t ) ‖2 = ∞ , as a consequence of using gradient descent . Now we examine ‖θ ( t ) ( w ) ‖2 . The heuristic is that if ` ( · ) is exponential-like , multiplying by wi only changes its tail property up to a constant while the asymptotic behavior is not affected . In particular , the necessary conditions for norm divergence under gradient descent can be summarized by : • C1 . The loss function ` ( · ) has a exponential tail behavior ( that we formalize in Appendix A.1 ) such that limu→∞ ` ( −u ) = limu→∞∇ ` ( −u ) = 0 ; • C2 . The predictor f ( θ , x ) is α-homogeneous such that f ( c · θ , x ) = cαf ( θ , x ) , ∀c > 0 . In addition , we need certain regularities from f ( θ , x ) to ensure the existence of critical points and the convergence of gradient descent : • C3 . for any x ∈ X , f ( · , x ) is β-smooth and l-Lipschitz on Rd . C1 can be satisfied by the exponential loss , log loss and cross entropy loss under the multi-class setting . For standard deep learning models such as multilayer perceptron ( MLP ) , C2 implies that the activation functions are homogeneous such as ReLU and LeakyReLU , and bias terms are disallowed . C3 is a common technical assumptions whose practical implications are discussed in Appendix A.1 . Among the three necessary conditions , importance weighting only affects C1 up to a constant , so its impact on the norm divergence diminishes in the asymptotic regime . The formal statement is provided as below . Claim 1 . There exists a constant learning rate for gradient descent , such that for any w ∈ [ 1/M , M ] n , with a weak regularization , limt→∞ ∥∥θ ( t ) ( w ) ∥∥ =∞ under C1-C3 . Compared with the previous work , we extend the norm divergence result not only to weighted ERM but a more general setting where a weak regularization is considered . We defer the proof to Appendix A.1 . A direct consequence of parameter norm divergence is that both the risk and the gradient are dominated by the terms with the smallest margin , i.e . arg mini yif ( θ , xi ) , which are also referred to as the `` support vectors '' . To make sense of this point , notice that both the risk and the gradient have the form of : ∑ i Ci exp ( − yif ( θ , xi ) ) , where Ci are low-order terms . Since f ( θ , xi ) = ‖θ‖α2 f ( θ/‖θ‖2 , xi ) due to the homogeneous assumption in C2 , it holds that : 1The regularized loss is given by Lλ ( θ ; w ) = L ( θ ; w ) + λ‖θ‖r for a fixed r > 0 . The weak regularization refers to the case where λ→ 0. limt→∞ exp ( − yif ( θ ( t ) ( w ) , xi ) ) → 0 . Therefore , the decision boundaries may share certain characteristics with the support vector machine ( SVM ) since they rely on the same support vectors . As a matter of fact , the current understandings on the implicit bias of gradient descent are mostly established on the connection with hard-margin SVM : min θ∈Rd ‖θ‖2 s.t . yif ( θ , xi ) ≥ 1 ∀i = 1 , 2 , . . . , n , ( 1 ) whose optimization path coincides with the max-margin problem : max‖θ‖2≤1 mini=1 , ... , n yif ( θ , xi ) , as shown by Nacson et al . ( 2019 ) . Define γ ( θ ) : = mini yif ( θ , xi ) . We use θ∗ to denote the optimal solution and γ∗ = γ ( θ∗ ) : = mini yif ( θ∗ , xi ) to denote the corresponding margin . Implicit bias of gradient descent . We start by considering the weight-agnostic setting . When D is linear separable , it is reasonable to conjecture that the separating hyperplane under a linear f ( θ , · ) overlaps with the solution of hard-margin SVM . Soudry et al . ( 2018 ) and Ji & Telgarsky ( 2018b ) first show that ‖θ ( t ) ‖ converges in direction to θ∗ , i.e . limt→∞ θ ( t ) /‖θ ( t ) ‖2 = θ∗ . For nonlinear predictors , however , the parameter direction is less meaningful . Instead , it has been pointed out that neural networks often achieve perfect separation of the training data ( Zhang et al. , 2016 ) . Therefore , we are more interested in the margin whose pivoting role for the generalization of neural networks is studied extensively ( Neyshabur et al. , 2017 ; Bartlett et al. , 2017 ; Golowich et al. , 2018 ) . Specifically , it has been show in Nacson et al . ( 2019 ) and Lyu & Li ( 2019 ) that the normalized margin , defined by γ̃ ( θ ( t ) ) : = γ ( θ ( t ) /‖θ ( t ) ‖2 ) , converges to the maximum margin γ∗ without regularization . It becomes clear at this point that to understand the role of importance weighting for deep learning , we must characterize the impact of weights on the implicit bias since they reveal the optimization geometry and generalization performance . Formally , we address the following critical questions . • Q1 . Does importance weighting modify the convergence results ( convergence in direction for linear predictor and in normalized margin for nonlinear predictor ) ? • If the convergence results remain unchanged , then : – Q2 . in what way is importance weighting affecting the optimization process ; – Q3 . how does importance weighting influence the generalization from the source distribution to the target distribution ?

Summary: It is now well-understood that when the data are linearly separable,  gradient descent over the linear class of functions converges toward the hard margin solution. It highlights the implicit bias of gradient descent. Among all solutions interpolating the dataset, gradient descent selects the one with larger margin, partly explaining why over-parametrized models may generalize.  The picture for non-linear class is a little bit more complicated.