In this work we consider active {\em local learning}: given a query point $x$, and active access to an unlabeled training set $S$, output the prediction $h(x)$ of a near-optimal $h \in H$ using significantly fewer labels than would be needed to actually learn $h$ fully. In particular, the number of label queries should be independent of the complexity of $H$, and the function $h$ should be well-defined, independent of $x$. This immediately also implies an algorithm for {\em distance estimation}: estimating the value $opt(H)$ from many fewer labels than needed to actually learn a near-optimal $h \in H$, by running local learning on a few random query points and computing the average error. For the hypothesis class consisting of functions supported on the interval $[0,1]$ with Lipschitz constant bounded by $L$, we present an algorithm that makes $O(({1 / \epsilon^6}) \log(1/\epsilon))$ label queries from an unlabeled pool of $O(({L / \epsilon^4})\log(1/\epsilon))$ samples. It estimates the distance to the best hypothesis in the class to an additive error of $\epsilon$ for an arbitrary underlying distribution. We further generalize our algorithm to more than one dimensions. We emphasize that the number of labels used is independent of the complexity of the hypothesis class which is linear in $L$ in the one-dimensional case. Furthermore, we give an algorithm to locally estimate the values of a near-optimal function at a few query points of interest with number of labels independent of $L$. We also consider the related problem of approximating the minimum error that can be achieved by the Nadaraya-Watson estimator under a linear diagonal transformation with eigenvalues coming from a small range. For a $d$-dimensional pointset of size $N$, our algorithm achieves an additive approximation of $\epsilon$, makes $\tilde{O}({d}/{\epsilon^2})$ queries and runs in $\tilde{O}({d^2}/{\epsilon^{d+4}}+{dN}/{\epsilon^2})$ time.

We consider networks, trained via stochastic gradient descent to minimize $\ell_2$ loss, with the training labels perturbed by independent noise at each iteration. We characterize the behavior of the training dynamics near any parameter vector that achieves zero training error, in terms of an implicit regularization term corresponding to the sum over the data points, of the squared $\ell_2$ norm of the gradient of the model with respect to the parameter vector, evaluated at each data point. This holds for networks of any connectivity, width, depth, and choice of activation function. We interpret this implicit regularization term for three simple settings: matrix sensing, two layer ReLU networks trained on one-dimensional data, and two layer networks with sigmoid activations trained on a single datapoint. For these settings, we show why this new and general implicit regularization effect drives the networks towards “simple” models.

We show that top-down decision tree learning heuristics (such as ID3, C4.5, and CART) are amenable to highly efficient {\sl learnability estimation}: for monotone target functions, the error of the decision tree hypothesis constructed by these heuristics can be estimated with {\sl polylogarithmically} many labeled examples, exponentially smaller than the number necessary to run these heuristics, and indeed, exponentially smaller than information-theoretic minimum required to learn a good decision tree. This adds to a small but growing list of fundamental learning algorithms that have been shown to be amenable to learnability estimation. En route to this result, we design and analyze sample-efficient {\sl minibatch} versions of top-down decision tree learning heuristics and show that they achieve the same provable guarantees as the full-batch versions. We further give ``active local'' versions of these heuristics: given a test point $x^\star$, we show how the label $T(x^\star)$ of the decision tree hypothesis $T$ can be computed with polylogarithmically many labeled examples, exponentially smaller than the number necessary to learn~$T$.

We propose a simple extension of {\sl top-down decision tree learning heuristics} such as ID3, C4.5, and CART. Our algorithm achieves provable guarantees for all target functions $f: \{-1,1\}^n \to \{-1,1\}$ with respect to the uniform distribution, circumventing impossibility results showing that existing heuristics fare poorly even for simple target functions. The crux of our extension is a new splitting criterion that takes into account the correlations between $f$ and {\sl small subsets} of its attributes. The splitting criteria of existing heuristics (e.g. Gini impurity and information gain), in contrast, are based solely on the correlations between $f$ and its {\sl individual} attributes. Our algorithm satisfies the following guarantee: for all target functions $f : \{-1,1\}^n \to \{-1,1\}$, sizes $s\in \N$, and error parameters $\eps$, it constructs a decision tree of size $s^{\tilde{O}((\log s)^2/\eps^2)}$ that achieves error $\le O(\opt_s) + \eps$, where $\opt_s$ denotes the error of the optimal size-$s$ decision tree for $f$. A key technical notion that drives our analysis is the {\sl noise stability} of $f$, a well-studied smoothness measure of $f$.

In this paper, we obtain improved running times for regression and top eigenvector computation for numerically sparse matrices. Given a data matrix $\mat{A} \in \R^{n \times d}$ where every row $a \in \R^d$ has $\|a\|_2^2 \leq L$ and numerical sparsity $\leq s$, i.e. $\|a\|_1^2 / \|a\|_2^2 \leq s$, we provide faster algorithms for these problems for many parameter settings. For top eigenvector computation, when $\gap > 0$ is the relative gap between the top two eigenvectors of $\mat{A}^\top \mat{A}$ and $r$ is the stable rank of $\mat{A}$ we obtain a running time of $\otilde(nd + r(s + \sqrt{r s}) / \gap^2)$ improving upon the previous best unaccelerated running time of $O(nd + r d / \gap^2)$. As $r \leq d$ and $s \leq d$ our algorithm everywhere improves or matches the previous bounds for all parameter settings. For regression, when $\mu > 0$ is the smallest eigenvalue of $\mat{A}^\top \mat{A}$ we obtain a running time of $\otilde(nd + (nL / \mu) \sqrt{s nL / \mu})$ improving upon the previous best unaccelerated running time of $\otilde(nd + n L d / \mu)$. This result expands when regression can be solved in nearly linear time from when $L/\mu = \otilde(1)$ to when $L / \mu = \otilde(d^{2/3} / (sn)^{1/3})$. Furthermore, we obtain similar improvements even when row norms and numerical sparsities are non-uniform and we show how to achieve even faster running times by accelerating using approximate proximal point \cite{frostig2015regularizing} / catalyst \cite{lin2015universal}. Our running times depend only on the size of the input and natural numerical measures of the matrix, i.e. eigenvalues and $\ell_p$ norms, making progress on a key open problem regarding optimal running times for efficient large-scale learning.