COLT 2020 Papers

We show a simple perceptron-like algorithm to learn origin-centered halfspaces in $\mathbb{R}^n$ with accuracy $1-\epsilon$ and confidence $1-\delta$ in time $\mathcal{O}\left(\frac{n^2}{\epsilon}\left(\log \frac{1}{\epsilon}+\log \frac{1}{\delta}\right)\right)$ using $\mathcal{O}\left(\frac{n}{\epsilon}\left(\log \frac{1}{\epsilon}+\log \frac{1}{\delta}\right)\right)$ labeled examples drawn uniformly from the unit $n$-sphere. This improves upon algorithms given in Baum(1990), Long(1994) and Servedio(1999). The time and sample complexity of our algorithm match the lower bounds given in Long(1995) up to logarithmic factors.

The taxing computational effort that is involved in solving some high-dimensional statistical problems, in particular problems involving non-convex optimization, has popularized the development and analysis of algorithms that run efficiently (polynomial-time) but with no general guarantee on statistical consistency. In light of the ever-increasing compute power and decreasing costs, a more useful characterization of algorithms is by their ability to calibrate the invested computational effort with various characteristics of the input at hand and with the available computational resources. We propose a new greedy algorithm for the $\ell_0$-sparse PCA problem which supports the calibration principle. We provide both a rigorous analysis of our algorithm in the spiked covariance model, as well as simulation results and comparison with other existing methods. Our findings show that our algorithm recovers the spike in SNR regimes where all polynomial-time algorithms fail while running in a reasonable parallel-time on a cluster.

We study the algorithmic problem of estimating the mean of a heavy-tailed random vector in R^d, given n i.i.d. samples. The goal is to design an efficient estimator that attains the optimal sub-gaussian error bound, only assuming that the random vector has bounded mean and covariance. Polynomial-time solutions to this problem are known but have high runtime due to their use of semi-definite programming (SDP). Moreover, conceptually, it remains open whether convex relaxation is truly necessary for this problem. In this work, we show that it is possible to go beyond SDP and achieve better computational efficiency. In particular, we provide a spectral algorithm that achieves the optimal statistical performance and runs in time O ( n^2 d ), improving upon the previous fastest runtime O( n^{3.5}+ n^2 d ) by Cherapanamjeri et.al. (COLT ’19). Our algorithm is spectral in that it only requires (approximate) eigenvector computations, which can be implemented very efficiently by, for example, power iteration or the Lanczos method. At the core of our algorithm is a novel connection between the furthest hyperplane problem introduced by Karnin et. al. (COLT ’12) and a structural lemma on heavy-tailed distributions by Lugosi and Mendelson (Ann. Stat. ’19). This allows us to iteratively reduce the estimation error at a geometric rate using only the information derived from the top singular vector of the data matrix, leading to a significantly faster running time.

We develop a \emph{corrective mechanism} for neural network approximation: the total available non-linear units are divided into multiple groups and the first group approximates the function under consideration, the second approximates the error in approximation produced by the first group and corrects it, the third group approximates the error produced by the first and second groups together and so on. This technique yields several new representation and learning results for neural networks: 1. Two-layer neural networks in the random features regime (RF) can memorize arbitrary labels for $n$ arbitrary points in $\mathbb{R}^d$ with $\tilde{O}(\tfrac{n}{\theta^4})$ ReLUs, where $\theta$ is the minimum distance between two different points. This bound can be shown to be optimal in $n$ up to logarithmic factors. 2. Two-layer neural networks with ReLUs and smoothed ReLUs can represent functions with an error of at most $\epsilon$ with $O(C(a,d)\epsilon^{-1/(a+1)})$ units for $a \in \mathbb{N}\cup\{0\}$ when the function has $\Theta(ad)$ bounded derivatives. In certain cases $d$ can be replaced with effective dimension $q \ll d$. Our results indicate that neural networks with only a single nonlinear layer are surprisingly powerful with regards to representation, and show that in contrast to what is suggested in recent work, depth is not needed in order to represent highly smooth functions. 3. Gradient Descent on the recombination weights of a two-layer random features network with ReLUs and smoothed ReLUs can learn low degree polynomials up to squared error $\epsilon$ with $\mathrm{subpoly}(1/\epsilon)$ units. Even though deep networks can approximate these polynomials with $\mathrm{polylog}(1/\epsilon)$ units, existing \emph{learning} bounds for this problem require $\mathrm{poly}(1/\epsilon)$ units. To the best of our knowledge, our results give the first sub-polynomial learning guarantees for this problem.

We study {\it small-loss} bounds for adversarial multi-armed bandits with graph feedback, that is, adaptive regret bounds that depend on the loss of the best arm or related quantities, instead of the total number of rounds. We derive the first small-loss bound for general strongly observable graphs, resolving an open problem of Lykouris et al. (2018). Specifically, we develop an algorithm with regret $\mathcal{\tilde{O}}(\sqrt{\kappa L_*})$ where $\kappa$ is the clique partition number and $L_*$ is the loss of the best arm, and for the special case of self-aware graphs where every arm has a self-loop, we improve the regret to $\mathcal{\tilde{O}}(\min\{\sqrt{\alpha T}, \sqrt{\kappa L_*}\})$ where $\alpha \leq \kappa$ is the independence number. Our results significantly improve and extend those by Lykouris et al. (2018) who only consider self-aware undirected graphs. Furthermore, we also take the first attempt at deriving small-loss bounds for weakly observable graphs. We first prove that no typical small-loss bounds are achievable in this case, and then propose algorithms with alternative small-loss bounds in terms of the loss of some specific subset of arms. A surprising side result is that $\mathcal{\tilde{O}}(\sqrt{T})$ regret is achievable even for weakly observable graphs as long as the best arm has a self-loop. Our algorithms are based on the Online Mirror Descent framework but require a suite of novel techniques that might be of independent interest. Moreover, all our algorithms can be made parameter-free without the knowledge of the environment.