Linear models have shown great effectiveness and flexibility in many fields such as machine learning, signal processing and statistics. They can represent rich spaces of functions while preserving the convexity of the optimization problems where they are used, and are simple to evaluate, differentiate and integrate. However, for modeling non-negative functions, which are crucial for unsupervised learning, density estimation, or non-parametric Bayesian methods, linear models are not applicable directly. Moreover, current state-of-the-art models like generalized linear models either lead to non-convex optimization problems, or cannot be easily integrated. In this paper we provide the first model for non-negative functions which benefits from the same good properties of linear models.

In the arena of privacy-preserving machine learning, differentially private stochastic gradient descent (DP-SGD) has outstripped the objective perturbation mechanism in popularity and interest. Though unrivaled in versatility, DP-SGD requires a non-trivial privacy overhead (for privately tuning the model's hyperparameters) and a computational complexity which might be extravagant for simple models such as linear and logistic regression. This paper revamps the objective perturbation mechanism with tighter privacy analyses and new computational tools that boost it to perform competitively with DP-SGD on unconstrained convex generalized linear problems.

We revisit the classic randomized sketch of a tensor product of q vectors x_i\in\mathbb{R} n . However, in their analysis C_{\Omega} 2 can be as large as \Theta(n {2q}), even for a set \Omega of O(1) vectors x . We give a new analysis of this sketch, providing nearly optimal bounds. For the important case of q 2 and \delta 1/\poly(n), this shows that m \Theta(\epsilon {-2} \log(n) \epsilon {-1} \log 2(n)), demonstrating that the \epsilon {-2} and \log 2(n) terms do not multiply each other. In a number of applications, one has \Omega \poly(n) and in this case our bounds are optimal up to a constant factor.

The scalability of Distributed Stochastic Gradient Descent (SGD) is today limited by communication bottlenecks. The key idea in CSER is first a new technique called error reset'' that adapts arbitrary compressors for SGD, producing bifurcated local models with periodic reset of resulting local residual errors. Second we introduce partial synchronization for both the gradients and the models, leveraging advantages from them. We prove the convergence of CSER for smooth non-convex problems. Empirical results show that when combined with highly aggressive compressors, the CSER algorithms accelerate the distributed training by nearly 10\times for CIFAR-100, and by 4.5\times for ImageNet.

We study the problem of fitting an ultrametric distance to a dissimilarity graph in the context of hierarchical cluster analysis. Standard hierarchical clustering methods are specified procedurally, rather than in terms of the cost function to be optimized. We aim to overcome this limitation by presenting a general optimization framework for ultrametric fitting. Our approach consists of modeling the latter as a constrained optimization problem over the continuous space of ultrametrics. So doing, we can leverage the simple, yet effective, idea of replacing the ultrametric constraint with a min-max operation injected directly into the cost function.

Variance reduction has emerged in recent years as a strong competitor to stochastic gradient descent in non-convex problems, providing the first algorithms to improve upon the converge rate of stochastic gradient descent for finding first-order critical points. However, variance reduction techniques typically require carefully tuned learning rates and willingness to use excessively large "mega-batches" in order to achieve their improved results. We present a new algorithm, STORM, that does not require any batches and makes use of adaptive learning rates, enabling simpler implementation and less hyperparameter tuning. Our technique for removing the batches uses a variant of momentum to achieve variance reduction in non-convex optimization. On smooth losses F, STORM finds a point x with \mathbb{E}[\ abla F(x)\ ]\le O(1/\sqrt{T} \sigma {1/3}/T {1/3}) in T iterations with \sigma 2 variance in the gradients, matching the best-known rate but without requiring knowledge of \sigma .

The k-Nearest Neighbors (kNN) classifier is a fundamental non-parametric machine learning algorithm. However, it is well known that it suffers from the curse of dimensionality, which is why in practice one often applies a kNN classifier on top of a (pre-trained) feature transformation. From a theoretical perspective, most, if not all theoretical results aimed at understanding the kNN classifier are derived for the raw feature space. This leads to an emerging gap between our theoretical understanding of kNN and its practical applications. In this paper, we take a first step towards bridging this gap.

In this work, we consider the problem of model selection for deep reinforcement learning (RL) in real-world environments. Typically, the performance of deep RL algorithms is evaluated via on-policy interactions with the target environment. However, comparing models in a real-world environment for the purposes of early stopping or hyperparameter tuning is costly and often practically infeasible. This leads us to examine off-policy policy evaluation (OPE) in such settings. We focus on OPE of value-based methods, which are of particular interest in deep RL with applications like robotics, where off-policy algorithms based on Q-function estimation can often attain better sample complexity than direct policy optimization. Furthermore, existing OPE metrics either rely on a model of the environment, or the use of importance sampling (IS) to correct for the data being off-policy.

This work focuses on off-policy evaluation (OPE) with function approximation in infinite-horizon undiscounted Markov decision processes (MDPs). For MDPs that are ergodic and linear (i.e. In a more general setting, when the feature dynamics are approximately linear and for arbitrary rewards, we propose a new approach for estimating stationary distributions with function approximation. We formulate this problem as finding the maximum-entropy distribution subject to matching feature expectations under empirical dynamics. We show that this results in an exponential-family distribution whose sufficient statistics are the features, paralleling maximum-entropy approaches in supervised learning.

The likelihood function is a fundamental component in Bayesian statistics. However, evaluating the likelihood of an observation is computationally intractable in many applications. In this paper, we propose a non-parametric approximation of the likelihood that identifies a probability measure which lies in the neighborhood of the nominal measure and that maximizes the probability of observing the given sample point. We show that when the neighborhood is constructed by the Kullback-Leibler divergence, by moment conditions or by the Wasserstein distance, then our optimistic likelihood can be determined through the solution of a convex optimization problem, and it admits an analytical expression in particular cases. We also show that the posterior inference problem with our optimistic likelihood approximation enjoys strong theoretical performance guarantees, and it performs competitively in a probabilistic classification task.