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Variable importance plays a pivotal role in interpretable machine learning as it helps measure the impact of factors on the output of the prediction model. Model agnostic methods based on the generation of "null" features via permutation (or related approaches) can be applied. Such analysis is often utilized in pharmaceutical applications due to its ability to interpret black-box models, including tree-based ensembles. A major challenge and significant confounder in variable importance estimation however is the presence of between-feature correlation. Recently, several adjustments to marginal permutation utilizing feature knockoffs were proposed to address this issue, such as the variable importance measure known as conditional predictive impact (CPI). Assessment and evaluation of such approaches is the focus of our work. We first present a comprehensive simulation study investigating the impact of feature correlation on the assessment of variable importance. We then theoretically prove the limitation that highly correlated features pose for the CPI through the knockoff construction. While we expect that there is always no correlation between knockoff variables and its corresponding predictor variables, we prove that the correlation increases linearly beyond a certain correlation threshold between the predictor variables. Our findings emphasize the absence of free lunch when dealing with high feature correlation, as well as the necessity of understanding the utility and limitations behind methods in variable importance estimation.

This paper introduces an iterative algorithm for training additive models that enjoys favorable memory storage and computational requirements. The algorithm can be viewed as the functional counterpart of stochastic gradient descent, applied to the coefficients of a truncated basis expansion of the component functions. We show that the resulting estimator satisfies an oracle inequality that allows for model mis-specification. In the well-specified setting, by choosing the learning rate carefully across three distinct stages of training, we demonstrate that its risk is minimax optimal in terms of the dependence on the dimensionality of the data and the size of the training sample. We further illustrate the computational benefits by comparing the approach with traditional backfitting on two real-world datasets.

Random forests are popular methods for classification and regression, and many different variants have been proposed in recent years. One interesting example is the Mondrian random forest, in which the underlying trees are constructed according to a Mondrian process. In this paper we give a central limit theorem for the estimates made by a Mondrian random forest in the regression setting. When combined with a bias characterization and a consistent variance estimator, this allows one to perform asymptotically valid statistical inference, such as constructing confidence intervals, on the unknown regression function. We also provide a debiasing procedure for Mondrian random forests which allows them to achieve minimax-optimal estimation rates with $\beta$-H\"older regression functions, for all $\beta$ and in arbitrary dimension, assuming appropriate parameter tuning., Comment: 49 pages, 1 figure

This paper addresses challenges in robust transfer learning stemming from ambiguity in Bayes classifiers and weak transferable signals between the target and source distribution. We introduce a novel quantity called the ''ambiguity level'' that measures the discrepancy between the target and source regression functions, propose a simple transfer learning procedure, and establish a general theorem that shows how this new quantity is related to the transferability of learning in terms of risk improvements. Our proposed ''Transfer Around Boundary'' (TAB) model, with a threshold balancing the performance of target and source data, is shown to be both efficient and robust, improving classification while avoiding negative transfer. Moreover, we demonstrate the effectiveness of the TAB model on non-parametric classification and logistic regression tasks, achieving upper bounds which are optimal up to logarithmic factors. Simulation studies lend further support to the effectiveness of TAB. We also provide simple approaches to bound the excess misclassification error without the need for specialized knowledge in transfer learning., Comment: 86 pages, 4 figures

Stacking regressions is an ensemble technique that forms linear combinations of different regression estimators to enhance predictive accuracy. The conventional approach uses cross-validation data to generate predictions from the constituent estimators, and least-squares with nonnegativity constraints to learn the combination weights. In this paper, we learn these weights analogously by minimizing an estimate of the population risk subject to a nonnegativity constraint. When the constituent estimators are linear least-squares projections onto nested subspaces separated by at least three dimensions, we show that thanks to a shrinkage effect, the resulting stacked estimator has strictly smaller population risk than best single estimator among them. Here "best" refers to an estimator that minimizes a model selection criterion such as AIC or BIC. In other words, in this setting, the best single estimator is inadmissible. Because the optimization problem can be reformulated as isotonic regression, the stacked estimator requires the same order of computation as the best single estimator, making it an attractive alternative in terms of both performance and implementation.

In previous literature, backward error analysis was used to find ordinary differential equations (ODEs) approximating the gradient descent trajectory. It was found that finite step sizes implicitly regularize solutions because terms appearing in the ODEs penalize the two-norm of the loss gradients. We prove that the existence of similar implicit regularization in RMSProp and Adam depends on their hyperparameters and the training stage, but with a different "norm" involved: the corresponding ODE terms either penalize the (perturbed) one-norm of the loss gradients or, conversely, impede its reduction (the latter case being typical). We also conduct numerical experiments and discuss how the proven facts can influence generalization.

We study the fundamental limits of matching pursuit, or the pure greedy algorithm, for approximating a target function $ f $ by a linear combination $f_n$ of $n$ elements from a dictionary. When the target function is contained in the variation space corresponding to the dictionary, many impressive works over the past few decades have obtained upper and lower bounds on the error $\|f-f_n\|$ of matching pursuit, but they do not match. The main contribution of this paper is to close this gap and obtain a sharp characterization of the decay rate, $n^{-\alpha}$, of matching pursuit. Specifically, we construct a worst case dictionary which shows that the existing best upper bound cannot be significantly improved. It turns out that, unlike other greedy algorithm variants which converge at the optimal rate $ n^{-1/2}$, the convergence rate $n^{-\alpha}$ is suboptimal. Here, $\alpha \approx 0.182$ is determined by the solution to a certain non-linear equation.

Decision tree learning is increasingly being used for pointwise inference. Important applications include causal heterogenous treatment effects and dynamic policy decisions, as well as conditional quantile regression and design of experiments, where tree estimation and inference is conducted at specific values of the covariates. In this paper, we call into question the use of decision trees (trained by adaptive recursive partitioning) for such purposes by demonstrating that they can fail to achieve polynomial rates of convergence in uniform norm with non-vanishing probability, even with pruning. Instead, the convergence may be arbitrarily slow or, in some important special cases, such as honest regression trees, fail completely. We show that random forests can remedy the situation, turning poor performing trees into nearly optimal procedures, at the cost of losing interpretability and introducing two additional tuning parameters. The two hallmarks of random forests, subsampling and the random feature selection mechanism, are seen to each distinctively contribute to achieving nearly optimal performance for the model class considered.

We develop a theoretical framework for the analysis of oblique decision trees, where the splits at each decision node occur at linear combinations of the covariates (as opposed to conventional tree constructions that force axis-aligned splits involving only a single covariate). While this methodology has garnered significant attention from the computer science and optimization communities since the mid-80s, the advantages they offer over their axis-aligned counterparts remain only empirically justified, and explanations for their success are largely based on heuristics. Filling this long-standing gap between theory and practice, we show that oblique regression trees (constructed by recursively minimizing squared error) satisfy a type of oracle inequality and can adapt to a rich library of regression models consisting of linear combinations of ridge functions and their limit points. This provides a quantitative baseline to compare and contrast decision trees with other less interpretable methods, such as projection pursuit regression and neural networks, which target similar model forms. Contrary to popular belief, one need not always trade-off interpretability with accuracy. Specifically, we show that, under suitable conditions, oblique decision trees achieve similar predictive accuracy as neural networks for the same library of regression models. To address the combinatorial complexity of finding the optimal splitting hyperplane at each decision node, our proposed theoretical framework can accommodate many existing computational tools in the literature. Our results rely on (arguably surprising) connections between recursive adaptive partitioning and sequential greedy approximation algorithms for convex optimization problems (e.g., orthogonal greedy algorithms), which may be of independent theoretical interest. Using our theory and methods, we also study oblique random forests.

This paper shows that decision trees constructed with Classification and Regression Trees (CART) and C4.5 methodology are consistent for regression and classification tasks, even when the number of predictor variables grows sub-exponentially with the sample size, under natural 0-norm and 1-norm sparsity constraints. The theory applies to a wide range of models, including (ordinary or logistic) additive regression models with component functions that are continuous, of bounded variation, or, more generally, Borel measurable. Consistency holds for arbitrary joint distributions of the predictor variables, thereby accommodating continuous, discrete, and/or dependent data. Finally, we show that these qualitative properties of individual trees are inherited by Breiman's random forests. A key step in the analysis is the establishment of an oracle inequality, which allows for a precise characterization of the goodness-of-fit and complexity tradeoff for a mis-specified model.

Decision trees and their ensembles are endowed with a rich set of diagnostic tools for ranking and screening variables in a predictive model. Despite the widespread use of tree based variable importance measures, pinning down their theoretical properties has been challenging and therefore largely unexplored. To address this gap between theory and practice, we derive finite sample performance guarantees for variable selection in nonparametric models using a single-level CART decision tree (a decision stump). Under standard operating assumptions in variable screening literature, we find that the marginal signal strength of each variable and ambient dimensionality can be considerably weaker and higher, respectively, than state-of-the-art nonparametric variable selection methods. Furthermore, unlike previous marginal screening methods that attempt to directly estimate each marginal projection via a truncated basis expansion, the fitted model used here is a simple, parsimonious decision stump, thereby eliminating the need for tuning the number of basis terms. Thus, surprisingly, even though decision stumps are highly inaccurate for estimation purposes, they can still be used to perform consistent model selection.

Within the machine learning community, the widely-used uniform convergence framework has been used to answer the question of how complex, over-parameterized models can generalize well to new data. This approach bounds the test error of the worst-case model one could have fit to the data, but it has fundamental limitations. Inspired by the statistical mechanics approach to learning, we formally define and develop a methodology to compute precisely the full distribution of test errors among interpolating classifiers from several model classes. We apply our method to compute this distribution for several real and synthetic datasets, with both linear and random feature classification models. We find that test errors tend to concentrate around a small typical value $\varepsilon^*$, which deviates substantially from the test error of the worst-case interpolating model on the same datasets, indicating that "bad" classifiers are extremely rare. We provide theoretical results in a simple setting in which we characterize the full asymptotic distribution of test errors, and we show that these indeed concentrate around a value $\varepsilon^*$, which we also identify exactly. We then formalize a more general conjecture supported by our empirical findings. Our results show that the usual style of analysis in statistical learning theory may not be fine-grained enough to capture the good generalization performance observed in practice, and that approaches based on the statistical mechanics of learning may offer a promising alternative.

Decision trees with binary splits are popularly constructed using Classification and Regression Trees (CART) methodology. For regression models, this approach recursively divides the data into two near-homogenous daughter nodes according to a split point that maximizes the reduction in sum of squares error (the impurity) along a particular variable. This paper aims to study the statistical properties of regression trees constructed with CART methodology. In doing so, we find that the training error is governed by the Pearson correlation between the optimal decision stump and response data in each node, which we bound by constructing a prior distribution on the split points and solving a nonlinear optimization problem. We leverage this connection between the training error and Pearson correlation to show that CART with cost-complexity pruning achieves an optimal complexity/goodness-of-fit tradeoff when the depth scales with the logarithm of the sample size. Data dependent quantities, which adapt to the dimensionality and latent structure of the regression model, are seen to govern the rates of convergence of the prediction error.

Classical results on the statistical complexity of linear models have commonly identified the norm of the weights $\|w\|$ as a fundamental capacity measure. Generalizations of this measure to the setting of deep networks have been varied, though a frequently identified quantity is the product of weight norms of each layer. In this work, we show that for a large class of networks possessing a positive homogeneity property, similar bounds may be obtained instead in terms of the norm of the product of weights. Our proof technique generalizes a recently proposed sampling argument, which allows us to demonstrate the existence of sparse approximants of positive homogeneous networks. This yields covering number bounds, which can be converted to generalization bounds for multi-class classification that are comparable to, and in certain cases improve upon, existing results in the literature. Finally, we investigate our sampling procedure empirically, which yields results consistent with our theory.

Decision trees with binary splits are popularly constructed using Classification and Regression Trees (CART) methodology. For binary classification and regression models, this approach recursively divides the data into two near-homogenous daughter nodes according to a split point that maximizes the reduction in sum of squares error (the impurity) along a particular variable. This paper aims to study the bias and adaptive properties of regression trees constructed with CART. In doing so, we derive an interesting connection between the bias and the mean decrease in impurity (MDI) measure of variable importance---a tool widely used for model interpretability---defined as the sum of impurity reductions over all non-terminal nodes in the tree. In particular, we show that the probability content of a terminal subnode for a variable is small when the MDI for that variable is large and that this relationship is exponential---confirming theoretically that decision trees with CART have small bias and are adaptive to signal strength and direction. Finally, we apply these individual tree bounds to tree ensembles and show consistency of Breiman's random forests. The context is surprisingly general and applies to a wide variety of multivariable data generating distributions and regression functions. The main technical tool is an exact characterization of the conditional probability content of the daughter nodes arising from an optimal split, in terms of the partial dependence function and reduction in impurity.

For any ReLU network there is a representation in which the sum of the absolute values of the weights into each node is exactly $1$, and the input layer variables are multiplied by a value $V$ coinciding with the total variation of the path weights. Implications are given for Gaussian complexity, Rademacher complexity, statistical risk, and metric entropy, all of which are shown to be proportional to $V$. There is no dependence on the number of nodes per layer, except for the number of inputs $d$. For estimation with sub-Gaussian noise, the mean square generalization error bounds that can be obtained are of order $V \sqrt{L + \log d}/\sqrt{n}$, where $L$ is the number of layers and $n$ is the sample size.

It has been experimentally observed in recent years that multi-layer artificial neural networks have a surprising ability to generalize, even when trained with far more parameters than observations. Is there a theoretical basis for this? The best available bounds on their metric entropy and associated complexity measures are essentially linear in the number of parameters, which is inadequate to explain this phenomenon. Here we examine the statistical risk (mean squared predictive error) of multi-layer networks with $\ell^1$-type controls on their parameters and with ramp activation functions (also called lower-rectified linear units). In this setting, the risk is shown to be upper bounded by $[(L^3 \log d)/n]^{1/2}$, where $d$ is the input dimension to each layer, $L$ is the number of layers, and $n$ is the sample size. In this way, the input dimension can be much larger than the sample size and the estimator can still be accurate, provided the target function has such $\ell^1$ controls and that the sample size is at least moderately large compared to $L^3\log d$. The heart of the analysis is the development of a sampling strategy that demonstrates the accuracy of a sparse covering of deep ramp networks. Lower bounds show that the identified risk is close to being optimal.

Random forests have become an important tool for improving accuracy in regression and classification problems since their inception by Leo Breiman in 2001. In this paper, we revisit a historically important random forest model originally proposed by Breiman in 2004 and later studied by G\'erard Biau in 2012, where a feature is selected at random and the splits occurs at the midpoint of the node along the chosen feature. If the regression function is Lipschitz and depends only on a small subset of $ S $ out of $ d $ features, we show that, given access to $ n $ observations and properly tuned split probabilities, the mean-squared prediction error is $ O((n(\log n)^{(S-1)/2})^{-\frac{1}{S\log2+1}}) $. This positively answers an outstanding question of Biau about whether the rate of convergence for this random forest model could be improved. Furthermore, by a refined analysis of the approximation and estimation errors for linear models, we show that this rate cannot be improved in general. Finally, we generalize our analysis and improve extant prediction error bounds for another random forest model in which each tree is constructed from subsampled data and the splits are performed at the empirical median along a chosen feature.

A popular class of problem in statistics deals with estimating the support of a density from $n$ observations drawn at random from a $d$-dimensional distribution. The one-dimensional case reduces to estimating the end points of a univariate density. In practice, an experimenter may only have access to a noisy version of the original data. Therefore, a more realistic model allows for the observations to be contaminated with additive noise. In this paper, we consider estimation of convex bodies when the additive noise is distributed according to a multivariate Gaussian distribution, even though our techniques could easily be adapted to other noise distributions. Unlike standard methods in deconvolution that are implemented by thresholding a kernel density estimate, our method avoids tuning parameters and Fourier transforms altogether. We show that our estimator, computable in $(O(\ln n))^{(d-1)/2}$ time, converges at a rate of $ O_d(\log\log n/\sqrt{\log n}) $ in Hausdorff distance, in accordance with the polylogarithmic rates encountered in Gaussian deconvolution problems. Part of our analysis also involves the optimality of the proposed estimator. We provide a lower bound for the minimax rate of estimation in Hausdorff distance that is $\Omega_d(1/\log^2 n)$.

Applied researchers often construct a network from a random sample of nodes in order to infer properties of the parent network. Two of the most widely used sampling schemes are subgraph sampling, where we sample each vertex independently with probability $p$ and observe the subgraph induced by the sampled vertices, and neighborhood sampling, where we additionally observe the edges between the sampled vertices and their neighbors. In this paper, we study the problem of estimating the number of motifs as induced subgraphs under both models from a statistical perspective. We show that: for any connected $h$ on $k$ vertices, to estimate $s=\mathsf{s}(h,G)$, the number of copies of $h$ in the parent graph $G$ of maximum degree $d$, with a multiplicative error of $\epsilon$, (a) For subgraph sampling, the optimal sampling ratio $p$ is $\Theta_{k}(\max\{ (s\epsilon^2)^{-\frac{1}{k}}, \; \frac{d^{k-1}}{s\epsilon^{2}} \})$, achieved by Horvitz-Thompson type of estimators. (b) For neighborhood sampling, we propose a family of estimators, encompassing and outperforming the Horvitz-Thompson estimator and achieving the sampling ratio $O_{k}(\min\{ (\frac{d}{s\epsilon^2})^{\frac{1}{k-1}}, \; \sqrt{\frac{d^{k-2}}{s\epsilon^2}}\})$. This is shown to be optimal for all motifs with at most $4$ vertices and cliques of all sizes. The matching minimax lower bounds are established using certain algebraic properties of subgraph counts. These results quantify how much more informative neighborhood sampling is than subgraph sampling, as empirically verified by experiments on both synthetic and real-world data. We also address the issue of adaptation to the unknown maximum degree, and study specific problems for parent graphs with additional structures, e.g., trees or planar graphs.

Learning properties of large graphs from samples has been an important problem in statistical network analysis since the early work of Goodman \cite{Goodman1949} and Frank \cite{Frank1978}. We revisit a problem formulated by Frank \cite{Frank1978} of estimating the number of connected components in a large graph based on the subgraph sampling model, in which we randomly sample a subset of the vertices and observe the induced subgraph. The key question is whether accurate estimation is achievable in the \emph{sublinear} regime where only a vanishing fraction of the vertices are sampled. We show that it is impossible if the parent graph is allowed to contain high-degree vertices or long induced cycles. For the class of chordal graphs, where induced cycles of length four or above are forbidden, we characterize the optimal sample complexity within constant factors and construct linear-time estimators that provably achieve these bounds. This significantly expands the scope of previous results which have focused on unbiased estimators and special classes of graphs such as forests or cliques. Both the construction and the analysis of the proposed methodology rely on combinatorial properties of chordal graphs and identities of induced subgraph counts. They, in turn, also play a key role in proving minimax lower bounds based on construction of random instances of graphs with matching structures of small subgraphs.

The MDL two-part coding $ \textit{index of resolvability} $ provides a finite-sample upper bound on the statistical risk of penalized likelihood estimators over countable models. However, the bound does not apply to unpenalized maximum likelihood estimation or procedures with exceedingly small penalties. In this paper, we point out a more general inequality that holds for arbitrary penalties. In addition, this approach makes it possible to derive exact risk bounds of order $1/n$ for iid parametric models, which improves on the order $(\log n)/n$ resolvability bounds. We conclude by discussing implications for adaptive estimation., Comment: To appear in IEEE Transactions on Information Theory, 2018

We give convergence guarantees for estimating the coefficients of a symmetric mixture of two linear regressions by expectation maximization (EM). In particular, we show that the empirical EM iterates converge to the target parameter vector at the parametric rate, provided the algorithm is initialized in an unbounded cone. In particular, if the initial guess has a sufficiently large cosine angle with the target parameter vector, a sample-splitting version of the EM algorithm converges to the true coefficient vector with high probability. Interestingly, our analysis borrows from tools used in the problem of estimating the centers of a symmetric mixture of two Gaussians by EM. We also show that the population EM operator for mixtures of two regressions is anti-contractive from the target parameter vector if the cosine angle between the input vector and the target parameter vector is too small, thereby establishing the necessity of our conic condition. Finally, we give empirical evidence supporting this theoretical observation, which suggests that the sample based EM algorithm performs poorly when initial guesses are drawn accordingly. Our simulation study also suggests that the EM algorithm performs well even under model misspecification (i.e., when the covariate and error distributions violate the model assumptions).

Estimation of functions of $ d $ variables is considered using ridge combinations of the form $ \textstyle\sum_{k=1}^m c_{1,k} \phi(\textstyle\sum_{j=1}^d c_{0,j,k}x_j-b_k) $ where the activation function $ \phi $ is a function with bounded value and derivative. These include single-hidden layer neural networks, polynomials, and sinusoidal models. From a sample of size $ n $ of possibly noisy values at random sites $ X \in B = [-1,1]^d $, the minimax mean square error is examined for functions in the closure of the $ \ell_1 $ hull of ridge functions with activation $ \phi $. It is shown to be of order $ d/n $ to a fractional power (when $ d $ is of smaller order than $ n $), and to be of order $ (\log d)/n $ to a fractional power (when $ d $ is of larger order than $ n $). Dependence on constraints $ v_0 $ and $ v_1 $ on the $ \ell_1 $ norms of inner parameter $ c_0 $ and outer parameter $ c_1 $, respectively, is also examined. Also, lower and upper bounds on the fractional power are given. The heart of the analysis is development of information-theoretic packing numbers for these classes of functions.

Recently, a general method for analyzing the statistical accuracy of the EM algorithm has been developed and applied to some simple latent variable models [Balakrishnan et al. 2016]. In that method, the basin of attraction for valid initialization is required to be a ball around the truth. Using Stein's Lemma, we extend these results in the case of estimating the centers of a two-component Gaussian mixture in $d$ dimensions. In particular, we significantly expand the basin of attraction to be the intersection of a half space and a ball around the origin. If the signal-to-noise ratio is at least a constant multiple of $ \sqrt{d\log d} $, we show that a random initialization strategy is feasible.

We establish $ L^{\infty} $ and $ L^2 $ error bounds for functions of many variables that are approximated by linear combinations of ReLU (rectified linear unit) and squared ReLU ridge functions with $ \ell^1 $ and $ \ell^0 $ controls on their inner and outer parameters. With the squared ReLU ridge function, we show that the $ L^2 $ approximation error is inversely proportional to the inner layer $ \ell^0 $ sparsity and it need only be sublinear in the outer layer $ \ell^0 $ sparsity. Our constructions are obtained using a variant of the Jones-Barron probabilistic method, which can be interpreted as either stratified sampling with proportionate allocation or two-stage cluster sampling. We also provide companion error lower bounds that reveal near optimality of our constructions. Despite the sparsity assumptions, we showcase the richness and flexibility of these ridge combinations by defining a large family of functions, in terms of certain spectral conditions, that are particularly well approximated by them.

Let $ f^{\star} $ be a function on $ \mathbb{R}^d $ with an assumption of a spectral norm $ v_{f^{\star}} $. For various noise settings, we show that $ \mathbb{E}\|\hat{f} - f^{\star} \|^2 \leq \left(v^4_{f^{\star}}\frac{\log d}{n}\right)^{1/3} $, where $ n $ is the sample size and $ \hat{f} $ is either a penalized least squares estimator or a greedily obtained version of such using linear combinations of sinusoidal, sigmoidal, ramp, ramp-squared or other smooth ridge functions. The candidate fits may be chosen from a continuum of functions, thus avoiding the rigidity of discretizations of the parameter space. On the other hand, if the candidate fits are chosen from a discretization, we show that $ \mathbb{E}\|\hat{f} - f^{\star} \|^2 \leq \left(v^3_{f^{\star}}\frac{\log d}{n}\right)^{2/5} $. This work bridges non-linear and non-parametric function estimation and includes single-hidden layer nets. Unlike past theory for such settings, our bound shows that the risk is small even when the input dimension $ d $ of an infinite-dimensional parameterized dictionary is much larger than the available sample size. When the dimension is larger than the cube root of the sample size, this quantity is seen to improve the more familiar risk bound of $ v_{f^{\star}}\left(\frac{d\log (n/d)}{n}\right)^{1/2} $, also investigated here., Comment: Submitted to Annals of Statistics