The central problem of supervised inference is to predict labels of unseen data points from a set of labeled training data. The literature on this subject is vast, ranging from classical parametric and non-parametric statistics [47, 48] to more recent machine learning methods, such as kernel machines , boosting , random forests 
, and deep neural networks
. There is a wealth of theoretical analyses for these methods based on a spectrum of techniques including non-parametric estimation, capacity control such as VC-dimension or Rademacher complexity , and regularization theory 
. In nearly all of these results, theoretical analysis of generalization requires “what you see is what you get” setup, where prediction performance on unseen test data is close to the performance on the training data, achieved by carefully managing the bias-variance trade-off. Furthermore, it is widely accepted in the literature that interpolation has poor statistical properties and should be dismissed out-of-hand. For example, in their book on non-parametric statistics,Györfi et al. [25, page 21] say that a certain procedure “may lead to a function which interpolates the data and hence is not a reasonable estimate”.
Yet, this is not how many modern machine learning methods are used in practice. For instance, the best practice for training deep neural networks is to first perfectly fit the training data . The resulting (zero training loss) neural networks after this first step can already have good performance on test data . Similar observations about models that perfectly fit training data have been made for other machine learning methods, including boosting , random forests , and kernel machines . These methods return good classifiers even when the training data have high levels of label noise [50, 52, 11].
An important effort to show that fitting the training data exactly can under certain conditions be theoretically justified is the margins theory for boosting  and other margin-based methods [27, 6, 23, 33, 28]. However, this theory lacks explanatory power for the performance of classifiers that perfectly fit noisy labels, when it is known that no margin is present in the data [50, 11]. Moreover, margins theory does not apply to regression and to functions (for regression or classification) that interpolate the data in the classical sense .
In this paper, we identify the challenge of providing a rigorous understanding of generalization in machine learning models that interpolate training data. We take first steps towards such a theory by proposing and analyzing interpolating methods for classification and regression with non-trivial risk and consistency guarantees.
Many existing forms of generalization analyses face significant analytical and conceptual barriers to being able to explain the success of interpolating methods.
- Capacity control.
Existing capacity-based bounds (e.g., VC dimension, fat-shattering dimension, Rademacher complexity) for empirical risk minimization [3, 7, 4, 36, 27] do not give useful risk bounds for functions with zero empirical risk whenever there is non-negligible label noise. This is because function classes rich enough to perfectly fit noisy training labels generally have capacity measures that grow quickly with the number of training data, at least with the existing notions of capacity . Note that since the training risk is zero for the functions of interest, the generalization bound must bound their true risk, as it equals the generalization gap (difference between the true and empirical risk). Whether such capacity-based generalization bounds exist is open for debate.
Generalization analyses based on algorithmic stability [13, 8] control the difference between the true risk and the training risk, assuming bounded sensitivity of an algorithm’s output to small changes in training data. Like standard uses of capacity-based bounds, these approaches are not well-suited to settings when training risk is identically zero but true risk is non-zero.
Many analyses are available for regularization approaches to statistical inverse problems, ranging from Tikhonov regularization to early stopping [15, 41, 51, 9]. To obtain a risk bound, these analyses require the regularization parameter (or some analogous quantity) to approach zero as the number of data tends to infinity. However, to get (the minimum norm) interpolation, we need while is fixed, causing the bounds to diverge.
Recently, Wyner et al.  proposed a thought-provoking explanation for the performance of AdaBoost and random forests in the interpolation regime, based on ideas related to “self-averaging” and localization. However, a theoretical basis for these ideas is not developed in their work.
There are two important exceptions to the aforementioned discussion of non-parametric methods. First, the nearest neighbor rule (also called -nearest neighbor, in the context of the general family of -nearest neighbor rules) is a well-known interpolating classification method, though it is not generally consistent for classification (and is not useful for regression when there is significant amount of label noise). Nevertheless, its asymptotic risk can be shown to be bounded above by twice the Bayes risk .111More precisely, the expected risk of the nearest neighbor rule converges to , where is the regression function; this quantity can be bounded above by , where is the Bayes risk. A second important (though perhaps less well-known) exception is the non-parametric smoothing method of Devroye et al.  based on a singular kernel called the Hilbert kernel (which is related to Shepard’s method ). The resulting estimate of the regression function interpolates the training data, yet is proved to be consistent for classification and regression.
The analyses of the nearest neighbor rule and Hilbert kernel regression estimate are not based on bounding generalization gap, the difference between the true risk and the empirical risk. Rather, the true risk is analyzed directly by exploiting locality properties of the prediction rules. In particular, the prediction at a point depends primarily or entirely on the values of the function at nearby points. This inductive bias favors functions where local information in a neighborhood can be aggregated to give an accurate representation of the underlying regression function.
What we do.
Our approach to understanding the generalization properties of interpolation methods is to understand and isolate the key properties of local classification, particularly the nearest neighbor rule.
First, we construct and analyze an interpolating function based on multivariate triangulation and linear interpolation on each simplex (Section 3), which results in a geometrically intuitive and theoretically tractable prediction rule. Like nearest neighbor, this method is not statistically consistent, but, unlike nearest neighbor, its asymptotic risk approaches the Bayes risk as the dimension becomes large, even when the Bayes risk is far from zero—a kind of “blessing of dimensionality”222 This does not remove the usual curse of dimensionality, which is similar to the standard analyses of
This does not remove the usual curse of dimensionality, which is similar to the standard analyses of-NN and other non-parametric methods.. Moreover, under an additional margin condition the difference between the Bayes risk and our classifier is exponentially small in the dimension.
A similar finding holds for regression, as the method is nearly consistent when the dimension is high.
Next, we propose a weighted & interpolated nearest neighbor (wiNN) scheme based on singular weight functions (Section 4). The resulting function is somewhat less natural than that obtained by simplicial interpolation, but like the Hilbert kernel regression estimate, the prediction rule is statistically consistent in any dimension. Interestingly, conditions on the weights to ensure consistency become less restrictive in higher dimension—another “blessing of dimensionality”. Our analysis provides the first known non-asymptotic rates of convergence to the Bayes risk for an interpolated predictor, as well as tighter bounds under margin conditions for classification. In fact, the rate achieved by wiNN regression is statistically optimal under a standard minimax setting333An earlier version of this article paper contained a bound with a worse rate of convergence based on a loose analysis. The subsequent work  found that a different Nadaraya-Watson kernel regression estimate (with a singular kernel) could achieve the optimal convergence rate; this inspired us to seek a tighter analysis of our wiNN scheme..
Our results also suggest an explanation for the phenomenon of adversarial examples , which are seemingly ubiquitous in modern machine learning. In Section 5, we argue that interpolation inevitably results in adversarial examples in the presence of any amount of label noise. When these schemes are consistent or nearly consistent, the set of adversarial examples (where the interpolating classifier disagrees with the Bayes optimal) has small measure but is asymptotically dense. Our analysis is consistent with the empirical observations that such examples are difficult to find by random sampling , but are easily discovered using targeted optimization procedures, such as Projected Gradient Descent .
Finally, we discuss the difference between direct and inverse interpolation schemes; and make some connections to kernel machines, and random forests in (Section 6).
The goal of regression and classification is to construct a predictor given labeled training data , that performs well on unseen test data, which are typically assumed to be sampled from the same distribution as the training data. In this work, we focus on interpolating methods that construct predictors satisfying for all .
Algorithms that perfectly fit training data are not common in statistical and machine learning literature. The prominent exception is the nearest neighbor rule, which is among of the oldest and best-understood classification methods. Given a training set of labeled example, the nearest neighbor rule predicts the label of a new point to be the same as that of the nearest point to within the training set. Mathematically, the predicted label of is , where . (Here, always denotes the Euclidean norm.) As discussed above, the classification risk of the nearest neighbor rule is asymptotically bounded by twice the Bayes (optimal) risk . The nearest neighbor rule provides an important intuition that such classifiers can (and perhaps should) be constructed using local information in the feature space.
In this paper, we analyze two interpolating schemes, one based on triangulating and constructing the simplicial interpolant for the data, and another, based on weighted nearest neighbors with singular weight function.
2.1 Statistical model and notations
We assume are iid labeled examples from . Here, are the iid training data, and is an independent test example from the same distribution. Let denote the marginal distribution of , with support denoted by ; and let denote the conditional mean of given , i.e., the function given by . For (binary) classification, we assume the range of is (so ), and we let denote the Bayes optimal classifier, which is defined by . This classifier minimizes the risk under zero-one loss, while the conditional mean function minimizes the risk under squared loss.
The goal of our analyses will be to establish excess risk bounds for empirical predictors ( and , based on training data) in terms of their agreement with for classification and with for regression. For classification, the expected risk can be bounded as , while for regression, the expected mean squared error is precisely . Our analyses thus mostly focus on and
(where the probability and expectations are with respect to both the training data and the test example).
2.2 Smoothness, margin, and regularity conditions
Below we list some standard conditions needed for further development.
The regularity condition from Audibert and Tsybakov  is not very restrictive. For example, if , then and .
Uniform distribution condition.
In what follows, we mostly assume uniform marginal distribution over a certain domain. This is done for the sake of simplicity and is not an essential condition. For example, in every statement the uniform measure can be substituted (with a potential change of constants) by an arbitrary measure with density bounded from below.
3 Interpolating scheme based on multivariate triangulation
In this section, we describe and analyze an interpolating scheme based on multivariate triangulation. Our main interest in this scheme is in its natural geometric properties and the risk bounds for regression and classification which compare favorably to those of the original nearest neighbor rule (despite the fact that neither is statistically consistent in general).
|Nearest neighbor||Simplicial interpolation|
3.1 Definition and basic properties
We define an interpolating function based on training data from and a (multivariate) triangulation scheme . This function is simplicial interpolation [26, 19]. We assume without loss of generality that the (unlabeled) examples span . The triangulation scheme partitions the convex hull of the unlabeled examples into non-degenerate simplices444We say a simplex in is non-degenerate if it has non-zero -dimensional Lebesgue measure. with vertices at the unlabeled examples; these simplices intersect only at -dimensional faces. Each is contained in at least one of these simplices; let denote the set of unlabeled examples that are vertices for a simplex containing . Let be the corresponding set of labeled examples .555Of course, some points have more than one containing simplex; we will see that the ambiguity in defining and for such points is not important. For any point , we define to be the unique linear interpolation of at (defined below). For points , we arbitrarily assert , and define .
Recall that a linear (affine) interpolation of at a new point is given by the system of equations , where are (unique) solutions to the system of equations
The predictions of the plug-in classifier based on simplicial interpolation are qualitatively very different from those of the nearest neighbor rule. This is true even when restricting attention to a single simplex. Suppose, for example, that for all , so the Bayes classifier predicts for all in the simplex. On the other hand, due to label noise, we may have some . Suppose in fact that only , while for all . In this scenario (depicted in Figure 1 for ), the nearest neighbor rule (erroneously) predicts on a larger fraction of the simplex than the plug-in classifier based on . The difference can be striking in high dimensions: for nearest neighbor versus for simplicial interpolation in -dimensional version of Figure 1. This provides an intuition why, in contrast to the nearest neighbor rule, simplicial interpolation can yield to classifiers that are nearly optimal in high dimensions.
Suppose are vertices of a non-degenerate simplex in , and is in their convex hull with barycentric coordinates —i.e., , , and . The linear interpolation of at is given by .
One consequence of Proposition 3.1 for is that if is contained in two adjacent simplices (that share a -dimensional face), then it does not matter which simplex is used to define ; the value of is the same in any case. Geometrically, we see that the restriction of the interpolating linear function to a face of the simplex coincides with the interpolating linear function constructed on a sub-simplex formed by that face. Therefore, we deduce that is a piecewise linear and continuous interpolation of the data on .
We note that our prediction rule requires only locating the vertices of the simplex containing a given point, rather than the considerably harder problem of constructing a full triangulation. In fact, locating the containing simplex in a Delaunay triangulation reduces to solving polynomial-size linear programs; in contrast, computing the full Delaunay triangulation has complexity exponential in the (intrinsic) dimension .
3.2 Mean squared error
We first illustrate the behavior of simplicial interpolation in a simple regression setting. Here, are iid labeled examples from . For simplicity, we assume that
is the uniform distribution on a full-dimensional compact and convex subset of.
In general, each may deviate from its conditional mean by a non-negligible amount, and hence any function that interpolates the training data is “fitting noise”. Nevertheless, in high dimension, the mean squared error of such a function will be quite close to that of the (optimal) conditional mean function.
Assume is the uniform distribution on a full-dimensional compact and convex subset of ; satisfies the -smoothness condition; and the conditional variance function satisfies the -smoothness condition. Let denote the maximum diameter of any simplex in the triangulation derived from . Then
In addition to the assumptions in Theorem 3.2, assume is a simple polytope in and is constructed using Delaunay triangulation. Then
3.3 Classification risk
We now analyze the statistical risk of the plug-in classifier based on , given by
As in Section 3.2, we assume that is the uniform distribution on a full-dimensional compact and convex subset of .
We first state an easy consequence of Corollary 3.3 using known properties of plug-in classifiers.
Under the same conditions as Corollary 3.3,
When the conditional mean function satisfies a margin condition, the in Corollary 3.4 can be replaced with a quantity that is exponentially small in , as we show next.
Suppose satisfies the -hard margin condition. As above, assume is the uniform distribution on a simple polytope in , and is constructed using Delaunay triangulation. Furthermore, assume is Lipschitz away from the class boundary (i.e., on ) and that the class boundary has finite -dimensional volume666Specifically, , where “” denotes the Minkowski sum, i.e., the -neighborhood of .. Then, for some absolute constants (which may depend on ),
Both Corollary 3.4 and Theorem 3.5 show that the risk of can be very close to the Bayes risk in high dimensions, thus exhibiting a certain “blessing of dimensionality". This stands in contrast to the nearest neighbor rule, whose asymptotic risk does not diminish with the dimension and is bounded by twice the Bayes risk, .
4 Interpolating nearest neighbor schemes
In this section, we describe a weighted nearest neighbor scheme that, like the -nearest neighbor rule, interpolates the training data, but is similar to the classical (unweighted) -nearest neighbor rule in terms of other properties, including convergence and consistency. (The classical -nearest neighbor rule is not generally an interpolating method except when .)
4.1 Weighted & interpolated nearest neighbors
For a given , let be the -th nearest neighbor of among the training data from , and let be the corresponding label. Let be a function . A weighted nearest neighbor scheme is simply a function of the form
In what follows, we investigate the properties of interpolating schemes of this type.
We will need two key observations for the analyses of these algorithms.
- Conditional independence.
The first key observation is that, under the usual iid sampling assumptions on the data, the first nearest neighbors of are conditionally independent given . That implies that
is a sum of conditionally iid random variables777Note that these variables are not independent in the ordering given by the distance to , but a random permutation makes them independent.. Hence, under a mild condition on , we expect them to concentrate around their expected value. Assuming some smoothness of , that value is closely related to , thus allowing us to establish bounds and rates.
- Interpolation and singular weight functions.
The second key point is that is an interpolating scheme, provided that has a singularity when . Indeed, it is easily seen that if , then . Extending continuously to the data points yields a weighted & interpolated nearest neighbor (wiNN) scheme.
We restrict attention to singular weight functions of the following radial type. Fix a positive integer and a decreasing function with a singularity at zero, . We take
Concretely, we will consider that diverge near as or , .
The denominator in the argument of is not strictly necessary, but it allows for convenient normalization in view of the conditional independence of -nearest neighbors given . Note that the weights depend on the sample and are thus data-adaptive.
are unbounded for singular weight functions, concentration only requires certain bounded moments. Geometrically, the volume of the region around the singularity needs to be small enough. For radial weight functions that we consider, this condition is more easily satisfied in high dimension. Indeed, the volume around the singularity becomes exponentially small in high dimension.
Our wiNN schemes are related to Nadaraya-Watson kernel regression [32, 49]. The use of singular kernels in the context of interpolation was originally proposed by Shepard ; they do not appear to be commonly used in machine learning and statistics, perhaps due to a view that interpolating schemes are unlikely to generalize or even be consistent; the non-adaptive Hilbert kernel regression estimate  (essentially, and ) is the only exception we know of.
4.2 Mean squared error
We first state a risk bound for wiNN schemes in a regression setting. Here, are iid labeled examples from .
Let be a wiNN scheme with singular weight function . Assume the following conditions:
is the uniform distribution on a compact subset of and satisfies the regularity condition for some and .
satisfies the -smoothness for some and .
for some .
Let , and assume . For any , let be the distance from to its st nearest neighbor among . Then
The bound in Theorem 4.3 is stated in terms of the expected distance to the st nearest neighbor raised to the power; this is typically bounded by . Choosing leads to a convergence rate of , which is minimax optimal.
4.3 Classification risk
We now analyze the statistical risk of the plug-in classifier based on .
Under the same conditions as Theorem 4.3,
Choosing leads to a convergence rate of .
We now give a more direct analysis, largely based on that of Chaudhuri and Dasgupta  for the standard -nearest neighbor rule, that leads to improved rates under favorable conditions.
For , define the effective interiors of the two classes by
and define the effective boundary by
Points away from the boundary, i.e., in or for , are likely to have nearest neighbors in or , respectively, so that interpolating their labels yields accurate predictions.
Let be a wiNN scheme with singular weight function , and let be the corresponding plug-in classifier. Fix any and . Then
Assume the following conditions:
is the uniform distribution on a compact subset of and satisfies the regularity condition for some and .
satisfies the -smoothness and -margin conditions for some , , , .
for some .
Let , and assume
Then for any ,
For consistency, we set , and in the bound, we plug-in and . This leads to a convergence rate of .
The factor in the final term in Corollary 4.6 results from an application of Chebyshev inequality. Under additional moment conditions, which are satisfied for certain functions (e.g., ) with better-behaved singularity at zero than , it can be replaced by . Additionally, while the condition is convenient for analysis, it is sufficient to assume that approaches infinity no faster than .
5 Ubiquity of adversarial examples in interpolated learning
The recently observed phenomenon of adversarial examples  in modern machine learning has drawn a significant degree of interest. It turns out that by introducing a small perturbation to the features of a correctly classified example (e.g., by changing an image in a visually imperceptible way or even by modifying a single pixel ) it is nearly always possible to induce neural networks to mis-classify a given input in a seemingly arbitrary and often bewildering way.
We will now discuss how our analyses, showing that Bayes optimality is compatible with interpolating the data, provide a possible mechanism for these adversarial examples to arise. Indeed, such examples are seemingly unavoidable in interpolated learning and, thus, in much of the modern practice. As we show below, any interpolating inferential procedure must have abundant adversarial examples in the presence of any amount of label noise. In particular, in consistent on nearly consistent schemes, like those considered in this paper, while the predictor agrees with the Bayes classifier on the bulk
of the probability distribution, every “incorrectly labeled” training example (i.e., an example whose label is different from the output of the Bayes optimal classifier) has a small “basin of attraction” with every point in the basin misclassified by the predictor. The total probability mass of these “adversarial” basins is negligible given enough training data, so that a probability of misclassifying a randomly chosen point is low. However, assuming non-zero label noise, the union of these adversarial basins asymptotically is a dense subset of the support for the underlying probability measure and hence there are misclassified examples in every open set. This is indeed consistent with the extensive empirical evidence for neural networks. While their output is observed to be robust to random feature noise, adversarial examples turn out to be quite difficult to avoid and can be easily found by targeted optimization methods such as PCG . We conjecture that it may be a general property or perhaps a weakness of interpolating methods, as some non-interpolating local classification rules can be robust against certain forms of adversarial examples .
To substantiate this discussion, we now provide a formal mathematical statement. For simplicity, let us consider a binary classification setting. Let be a probability distribution with non-zero density defined on a compact domain and assume non-zero label noise everywhere, i.e., for all , , or equivalently, . Let be a consistent interpolating classifier constructed from iid sampled data points (e.g., the classifier constructed in Section 4.3).
Let be the set of points at which disagrees with the Bayes optimal classifier ; in other words, is the set of “adversarial examples” for . Consistency of implies that, with probability one, or, equivalently, . On the other hand, the following result shows that the sets are asymptotically dense in , so that there is an adversarial example arbitrarily close to any .
For any and , there exists , such that for all , with probability , every point in is within distance of the set .
Let be the training data used to construct . Fix a finite -cover of with respect to the Euclidean distance. Since is interpolating and is never zero nor one, for every , there is a non-zero probability (over the outcome of the label ) that ; in this case, the training point is an adversarial example for . By choosing large enough, we can ensure that with probability at least over the random draw of the training data, every element of the cover is within distance of at least one adversarial example, upon which every point in is within distance (by triangle inequality) of the same. ∎
A similar argument for regression shows that while an interpolating may converge to in , it is generally impossible for it to converge in unless there is no label noise. An even more striking result is that for the Hilbert scheme of Devroye et al., the regression estimator almost surely does not converge at any fixed point, even for the simple case of a constant function corrupted by label noise . This means that with increasing sample size , at any given point misclassification will occur an infinite number of times with probability one. We expect similar behavior to hold for the interpolation schemes presented in this paper.
6 Discussion and connections
In this paper, we considered two types of algorithms, one based on simplicial interpolation and another based on interpolation by weighted nearest neighbor schemes. It may be useful to think of nearest neighbor schemes as direct methods, not requiring optimization, while our simplicial scheme is a simple example of an inverse method, using (local) matrix inversion to fit the data. Most popular machine learning methods, such as kernel machines, neural networks, and boosting, are inverse schemes. While nearest neighbor and Nadaraya-Watson methods often show adequate performance, they are rarely best-performing algorithms in practice. We conjecture that the simplicial interpolation scheme may provide insights into the properties of interpolating kernel machines and neural networks.
To provide some evidence for this line of thought, we show that in one dimension simplicial interpolation is indeed a special case of interpolating kernel machine. We will briefly sketch the argument without going into the details. Consider the space of real-valued functions with the norm . This space is a reproducing kernel Hilbert Space corresponding to the Laplace kernel . It can be seen that as the minimum norm interpolant is simply linear interpolation between adjacent points on the line. Note that this is the same as our simplicial interpolating method.
Interestingly, a version of random forests similar to PERT  also produces linear interpolation in one dimension (in the limit, when infinitely many trees are sampled). For simplicity assume that we have only two data points with labels and respectively. A tree that correctly classifies those points is simply a function of the form , where . Choosing a random uniformly from , we observe that is simply the linear function interpolating between the two data points. The extension of this argument to more than two data points in dimension one is straightforward. It would be interesting to investigate the properties of such methods in higher dimension. We note that it is unclear whether a random forest method of this type should be considered a direct or inverse method. While there is no explicit optimization involved, sampling is often used instead of optimization in methods like simulated annealing.
Finally, we note that while kernel machines (which can be viewed as two-layer neural networks) are much more theoretically tractable than general neural networks, none of the current theory applies in the interpolated regime in the presence of label noise . We hope that simplicial interpolation can shed light on their properties and lead to better understanding of modern inferential methods.
We would like to thank Raef Bassily, Luis Rademacher, Sasha Rakhlin, and Yusu Wang for conversations and valuable comments. We acknowledge funding from NSF. DH acknowledges support from NSF grants CCF-1740833 and DMR-1534910. PPM acknowledges support from the Crick-Clay Professorship (CSHL) and H N Mahabala Chair (IITM). This work grew out of discussions originating at the Simons Institute for the Theory of Computing in 2017, and we thank the Institute for the hospitality. PPM and MB thank ICTS (Bangalore) for their hospitality at the 2017 workshop on Statistical Physics Methods in Machine Learning.
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Appendix A Proofs
a.1 Proof of Proposition 3.1
We can lift the simplex into with the mapping . Since has non-zero -dimensional volume , it follows that the cone has -dimensional volume . This implies that are linearly independent. So, letting and , we can write as
where we have used the linear independence of to ensure the invertibility of . Therefore, since for , we have
a.2 Proof of Theorem 3.2
Proof of Theorem 3.2.
Throughout we condition on , and write
For the first term, observe that if , then and hence .
We now consider the second term, conditional on and . Let . Since , its barycentric coordinates in are distributed as . Let and for . Also, let be the conditional variance function. By the smoothness assumptions, we have
By Proposition 3.1, we have
For the first term,
by Jensen’s inequality and the bound on . For the second term, we have