arXiv subjects

Vasilis Kontonis

Publications and source records attributed to Vasilis Kontonis.

31 records · Page 2Linked to original sources

arXiv2022

Weighted Distillation with Unlabeled Examples

Distillation with unlabeled examples is a popular and powerful method for training deep neural networks in settings where the amount of labeled data is limited: A large ''teacher'' neural network is trained on the labeled data available, and then it is used to generate labels on an unlabeled dataset (typically much larger in size). These labels are then utilized to train the smaller ''student'' model which will actually be deployed. Naturally, the success of the approach depends on the quality of the teacher's labels, since the student could be confused if trained on inaccurate data. This paper proposes a principled approach for addressing this issue based on a ''debiasing'' reweighting of the student's loss function tailored to the distillation training paradigm. Our method is hyper-parameter free, data-agnostic, and simple to implement. We demonstrate significant improvements on popular academic datasets and we accompany our results with a theoretical analysis which rigorously justifies the performance of our method in certain settings.

cs.LG↗

arXiv2022

Learning a Single Neuron with Adversarial Label Noise via Gradient Descent

We study the fundamental problem of learning a single neuron, i.e., a function of the form $\mathbf{x}\mapstoσ(\mathbf{w}\cdot\mathbf{x})$ for monotone activations $σ:\mathbb{R}\mapsto\mathbb{R}$, with respect to the $L_2^2$-loss in the presence of adversarial label noise. Specifically, we are given labeled examples from a distribution $D$ on $(\mathbf{x}, y)\in\mathbb{R}^d \times \mathbb{R}$ such that there exists $\mathbf{w}^\ast\in\mathbb{R}^d$ achieving $F(\mathbf{w}^\ast)=ε$, where $F(\mathbf{w})=\mathbf{E}_{(\mathbf{x},y)\sim D}[(σ(\mathbf{w}\cdot \mathbf{x})-y)^2]$. The goal of the learner is to output a hypothesis vector $\mathbf{w}$ such that $F(\mathbb{w})=C\, ε$ with high probability, where $C>1$ is a universal constant. As our main contribution, we give efficient constant-factor approximate learners for a broad class of distributions (including log-concave distributions) and activation functions. Concretely, for the class of isotropic log-concave distributions, we obtain the following important corollaries: For the logistic activation, we obtain the first polynomial-time constant factor approximation (even under the Gaussian distribution). Our algorithm has sample complexity $\widetilde{O}(d/ε)$, which is tight within polylogarithmic factors. For the ReLU activation, we give an efficient algorithm with sample complexity $\tilde{O}(d\, \polylog(1/ε))$. Prior to our work, the best known constant-factor approximate learner had sample complexity $\tildeΩ(d/ε)$. In both of these settings, our algorithms are simple, performing gradient-descent on the (regularized) $L_2^2$-loss. The correctness of our algorithms relies on novel structural results that we establish, showing that (essentially all) stationary points of the underlying non-convex loss are approximately optimal.

cs.LG↗

arXiv2021

Learning General Halfspaces with General Massart Noise under the Gaussian Distribution

We study the problem of PAC learning halfspaces on $\mathbb{R}^d$ with Massart noise under the Gaussian distribution. In the Massart model, an adversary is allowed to flip the label of each point $\mathbf{x}$ with unknown probability $η(\mathbf{x}) \leq η$, for some parameter $η\in [0,1/2]$. The goal is to find a hypothesis with misclassification error of $\mathrm{OPT} + ε$, where $\mathrm{OPT}$ is the error of the target halfspace. This problem had been previously studied under two assumptions: (i) the target halfspace is homogeneous (i.e., the separating hyperplane goes through the origin), and (ii) the parameter $η$ is strictly smaller than $1/2$. Prior to this work, no nontrivial bounds were known when either of these assumptions is removed. We study the general problem and establish the following: For $η<1/2$, we give a learning algorithm for general halfspaces with sample and computational complexity $d^{O_η(\log(1/γ))}\mathrm{poly}(1/ε)$, where $γ=\max\{ε, \min\{\mathbf{Pr}[f(\mathbf{x}) = 1], \mathbf{Pr}[f(\mathbf{x}) = -1]\} \}$ is the bias of the target halfspace $f$. Prior efficient algorithms could only handle the special case of $γ= 1/2$. Interestingly, we establish a qualitatively matching lower bound of $d^{Ω(\log(1/γ))}$ on the complexity of any Statistical Query (SQ) algorithm. For $η= 1/2$, we give a learning algorithm for general halfspaces with sample and computational complexity $O_ε(1) d^{O(\log(1/ε))}$. This result is new even for the subclass of homogeneous halfspaces; prior algorithms for homogeneous Massart halfspaces provide vacuous guarantees for $η=1/2$. We complement our upper bound with a nearly-matching SQ lower bound of $d^{Ω(\log(1/ε))}$, which holds even for the special case of homogeneous halfspaces.

cs.LG↗

arXiv2021

A Statistical Taylor Theorem and Extrapolation of Truncated Densities

We show a statistical version of Taylor's theorem and apply this result to non-parametric density estimation from truncated samples, which is a classical challenge in Statistics \cite{woodroofe1985estimating, stute1993almost}. The single-dimensional version of our theorem has the following implication: "For any distribution $P$ on $[0, 1]$ with a smooth log-density function, given samples from the conditional distribution of $P$ on $[a, a + \varepsilon] \subset [0, 1]$, we can efficiently identify an approximation to $P$ over the \emph{whole} interval $[0, 1]$, with quality of approximation that improves with the smoothness of $P$." To the best of knowledge, our result is the first in the area of non-parametric density estimation from truncated samples, which works under the hard truncation model, where the samples outside some survival set $S$ are never observed, and applies to multiple dimensions. In contrast, previous works assume single dimensional data where each sample has a different survival set $S$ so that samples from the whole support will ultimately be collected.

math.ST↗

arXiv2021

Agnostic Proper Learning of Halfspaces under Gaussian Marginals

We study the problem of agnostically learning halfspaces under the Gaussian distribution. Our main result is the {\em first proper} learning algorithm for this problem whose sample complexity and computational complexity qualitatively match those of the best known improper agnostic learner. Building on this result, we also obtain the first proper polynomial-time approximation scheme (PTAS) for agnostically learning homogeneous halfspaces. Our techniques naturally extend to agnostically learning linear models with respect to other non-linear activations, yielding in particular the first proper agnostic algorithm for ReLU regression.

cs.LG↗

arXiv2020

Convergence and Sample Complexity of SGD in GANs

We provide theoretical convergence guarantees on training Generative Adversarial Networks (GANs) via SGD. We consider learning a target distribution modeled by a 1-layer Generator network with a non-linear activation function $ϕ(\cdot)$ parametrized by a $d \times d$ weight matrix $\mathbf W_*$, i.e., $f_*(\mathbf x) = ϕ(\mathbf W_* \mathbf x)$. Our main result is that by training the Generator together with a Discriminator according to the Stochastic Gradient Descent-Ascent iteration proposed by Goodfellow et al. yields a Generator distribution that approaches the target distribution of $f_*$. Specifically, we can learn the target distribution within total-variation distance $ε$ using $\tilde O(d^2/ε^2)$ samples which is (near-)information theoretically optimal. Our results apply to a broad class of non-linear activation functions $ϕ$, including ReLUs and is enabled by a connection with truncated statistics and an appropriate design of the Discriminator network. Our approach relies on a bilevel optimization framework to show that vanilla SGDA works.

cs.LG↗

arXiv2020

A Polynomial Time Algorithm for Learning Halfspaces with Tsybakov Noise

We study the problem of PAC learning homogeneous halfspaces in the presence of Tsybakov noise. In the Tsybakov noise model, the label of every sample is independently flipped with an adversarially controlled probability that can be arbitrarily close to $1/2$ for a fraction of the samples. {\em We give the first polynomial-time algorithm for this fundamental learning problem.} Our algorithm learns the true halfspace within any desired accuracy $ε$ and succeeds under a broad family of well-behaved distributions including log-concave distributions. Prior to our work, the only previous algorithm for this problem required quasi-polynomial runtime in $1/ε$. Our algorithm employs a recently developed reduction \cite{DKTZ20b} from learning to certifying the non-optimality of a candidate halfspace. This prior work developed a quasi-polynomial time certificate algorithm based on polynomial regression. {\em The main technical contribution of the current paper is the first polynomial-time certificate algorithm.} Starting from a non-trivial warm-start, our algorithm performs a novel "win-win" iterative process which, at each step, either finds a valid certificate or improves the angle between the current halfspace and the true one. Our warm-start algorithm for isotropic log-concave distributions involves a number of analytic tools that may be of broader interest. These include a new efficient method for reweighting the distribution in order to recenter it and a novel characterization of the spectrum of the degree-$2$ Chow parameters.

cs.LG↗

arXiv2020

Algorithms and SQ Lower Bounds for PAC Learning One-Hidden-Layer ReLU Networks

We study the problem of PAC learning one-hidden-layer ReLU networks with $k$ hidden units on $\mathbb{R}^d$ under Gaussian marginals in the presence of additive label noise. For the case of positive coefficients, we give the first polynomial-time algorithm for this learning problem for $k$ up to $\tilde{O}(\sqrt{\log d})$. Previously, no polynomial time algorithm was known, even for $k=3$. This answers an open question posed by~\cite{Kliv17}. Importantly, our algorithm does not require any assumptions about the rank of the weight matrix and its complexity is independent of its condition number. On the negative side, for the more general task of PAC learning one-hidden-layer ReLU networks with arbitrary real coefficients, we prove a Statistical Query lower bound of $d^{Ω(k)}$. Thus, we provide a separation between the two classes in terms of efficient learnability. Our upper and lower bounds are general, extending to broader families of activation functions.

cs.LG↗

arXiv2020

Learning Halfspaces with Tsybakov Noise

We study the efficient PAC learnability of halfspaces in the presence of Tsybakov noise. In the Tsybakov noise model, each label is independently flipped with some probability which is controlled by an adversary. This noise model significantly generalizes the Massart noise model, by allowing the flipping probabilities to be arbitrarily close to $1/2$ for a fraction of the samples. Our main result is the first non-trivial PAC learning algorithm for this problem under a broad family of structured distributions -- satisfying certain concentration and (anti-)anti-concentration properties -- including log-concave distributions. Specifically, we given an algorithm that achieves misclassification error $ε$ with respect to the true halfspace, with quasi-polynomial runtime dependence in $1/\epsilin$. The only previous upper bound for this problem -- even for the special case of log-concave distributions -- was doubly exponential in $1/ε$ (and follows via the naive reduction to agnostic learning). Our approach relies on a novel computationally efficient procedure to certify whether a candidate solution is near-optimal, based on semi-definite programming. We use this certificate procedure as a black-box and turn it into an efficient learning algorithm by searching over the space of halfspaces via online convex optimization.

cs.LG↗

arXiv2020

Non-Convex SGD Learns Halfspaces with Adversarial Label Noise

We study the problem of agnostically learning homogeneous halfspaces in the distribution-specific PAC model. For a broad family of structured distributions, including log-concave distributions, we show that non-convex SGD efficiently converges to a solution with misclassification error $O(\opt)+\eps$, where $\opt$ is the misclassification error of the best-fitting halfspace. In sharp contrast, we show that optimizing any convex surrogate inherently leads to misclassification error of $ω(\opt)$, even under Gaussian marginals.

cs.LG↗

arXiv2020

Learning Halfspaces with Massart Noise Under Structured Distributions

We study the problem of learning halfspaces with Massart noise in the distribution-specific PAC model. We give the first computationally efficient algorithm for this problem with respect to a broad family of distributions, including log-concave distributions. This resolves an open question posed in a number of prior works. Our approach is extremely simple: We identify a smooth {\em non-convex} surrogate loss with the property that any approximate stationary point of this loss defines a halfspace that is close to the target halfspace. Given this structural result, we can use SGD to solve the underlying learning problem.

cs.LG↗

arXiv2019

Efficient Truncated Statistics with Unknown Truncation

We study the problem of estimating the parameters of a Gaussian distribution when samples are only shown if they fall in some (unknown) subset $S \subseteq \R^d$. This core problem in truncated statistics has long history going back to Galton, Lee, Pearson and Fisher. Recent work by Daskalakis et al. (FOCS'18), provides the first efficient algorithm that works for arbitrary sets in high dimension when the set is known, but leaves as an open problem the more challenging and relevant case of unknown truncation set. Our main result is a computationally and sample efficient algorithm for estimating the parameters of the Gaussian under arbitrary unknown truncation sets whose performance decays with a natural measure of complexity of the set, namely its Gaussian surface area. Notably, this algorithm works for large families of sets including intersections of halfspaces, polynomial threshold functions and general convex sets. We show that our algorithm closely captures the tradeoff between the complexity of the set and the number of samples needed to learn the parameters by exhibiting a set with small Gaussian surface area for which it is information theoretically impossible to learn the true Gaussian with few samples.

math.ST↗

arXiv2017

Learning Powers of Poisson Binomial Distributions

We introduce the problem of simultaneously learning all powers of a Poisson Binomial Distribution (PBD). A PBD of order $n$ is the distribution of a sum of $n$ mutually independent Bernoulli random variables $X_i$, where $\mathbb{E}[X_i] = p_i$. The $k$'th power of this distribution, for $k$ in a range $[m]$, is the distribution of $P_k = \sum_{i=1}^n X_i^{(k)}$, where each Bernoulli random variable $X_i^{(k)}$ has $\mathbb{E}[X_i^{(k)}] = (p_i)^k$. The learning algorithm can query any power $P_k$ several times and succeeds in learning all powers in the range, if with probability at least $1- δ$: given any $k \in [m]$, it returns a probability distribution $Q_k$ with total variation distance from $P_k$ at most $ε$. We provide almost matching lower and upper bounds on query complexity for this problem. We first show a lower bound on the query complexity on PBD powers instances with many distinct parameters $p_i$ which are separated, and we almost match this lower bound by examining the query complexity of simultaneously learning all the powers of a special class of PBD's resembling the PBD's of our lower bound. We study the fundamental setting of a Binomial distribution, and provide an optimal algorithm which uses $O(1/ε^2)$ samples. Diakonikolas, Kane and Stewart [COLT'16] showed a lower bound of $Ω(2^{1/ε})$ samples to learn the $p_i$'s within error $ε$. The question whether sampling from powers of PBDs can reduce this sampling complexity, has a negative answer since we show that the exponential number of samples is inevitable. Having sampling access to the powers of a PBD we then give a nearly optimal algorithm that learns its $p_i$'s. To prove our two last lower bounds we extend the classical minimax risk definition from statistics to estimating functions of sequences of distributions.

cs.DS↗