Search for: All records

Langevin algorithms are gradient descent methods with additive noise. They have been used for decades in Markov Chain Monte Carlo (MCMC) sampling, optimization, and learning. Their convergence properties for unconstrained non-convex optimization and learning problems have been studied widely in the last few years. Other work has examined projected Langevin algorithms for sampling from log-concave distributions restricted to convex compact sets. For learning and optimization, log-concave distributions correspond to convex losses. In this paper, we analyze the case of non-convex losses with compact convex constraint sets and IID external data variables. We term the resulting method the projected stochastic gradient Langevin algorithm (PSGLA). We show the algorithm achieves a deviation of 𝑂(𝑇−1/4(𝑙𝑜𝑔𝑇)1/2) from its target distribution in 1-Wasserstein distance. For optimization and learning, we show that the algorithm achieves 𝜖-suboptimal solutions, on average, provided that it is run for a time that is polynomial in 𝜖 and slightly super-exponential in the problem dimension. 

We give an algorithm for source identification of a mixture of k product distributions on n bits. This is a fundamental problem in machine learning with many applications. Our algorithm identifies the source parameters of an identifiable mixture, given, as input, approximate values of multilinear moments (derived, for instance, from a sufficiently large sample), using 2^O(k^2) n^O(k) arithmetic operations. Our result is the first explicit bound on the computational complexity of source identification of such mixtures. The running time improves previous results by Feldman, O’Donnell, and Servedio (FOCS 2005) and Chen and Moitra (STOC 2019) that guaranteed only learning the mixture (without parametric identification of the source). Our analysis gives a quantitative version of a qualitative characterization of identifiable sources that is due to Tahmasebi, Motahari, and Maddah-Ali (ISIT 2018). 

We study the problem of robust learning under clean-label data-poisoning attacks, where the at-tacker injects (an arbitrary set of) correctly-labeled examples to the training set to fool the algorithm into making mistakes on specific test instances at test time. The learning goal is to minimize the attackable rate (the probability mass of attackable test instances), which is more difficult than optimal PAC learning. As we show, any robust algorithm with diminishing attackable rate can achieve the optimal dependence on ε in its PAC sample complexity, i.e., O(1/ε). On the other hand, the attackable rate might be large even for some optimal PAC learners, e.g., SVM for linear classifiers. Furthermore, we show that the class of linear hypotheses is not robustly learnable when the data distribution has zero margin and is robustly learnable in the case of positive margin but requires sample complexity exponential in the dimension. For a general hypothesis class with bounded VC dimension, if the attacker is limited to add at most t >0 poison examples, the optimal robust learning sample complexity grows almost linearly with t.