Gaussian process optimization is a successful class of algorithms (e.g. GP-UCB) to optimize a black-box function through sequential evaluations. However, when the domain of the function is continuous, Gaussian process optimization has to either rely on a fixed discretization of the space, or solve a non-convex optimization subproblem at each evaluation. The first approach can negatively affect performance, while the second one puts a heavy computational burden on the algorithm. A third option, that only recently has been theoretically studied, is to adaptively discretize the function domain. Even though this approach avoids the extra non-convex optimization costs, the overall computational complexity is still prohibitive. An algorithm such as GP-UCB has a runtime of $O(T^4)$, where $T$ is the number of iterations. In this paper, we introduce Ada-BKB (Adaptive Budgeted Kernelized Bandit), a no-regret Gaussian process optimization algorithm for functions on continuous domains, that provably runs in $O(T^2 d_\text{eff}^2)$, where $d_\text{eff}$ is the effective dimension of the explored space, and which is typically much smaller than $T$. We corroborate our findings with experiments on synthetic non-convex functions and on the real-world problem of hyper-parameter optimization.
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A typical assumption in supervised machine learning is that the train (source) and test (target) datasets follow completely the same distribution. This assumption is, however, often violated in uncertain real-world applications, which motivates the study of learning under covariate shift. In this setting, the naive use of adaptive hyperparameter optimization methods such as Bayesian optimization does not work as desired since it does not address the distributional shift among different datasets. In this work, we consider a novel hyperparameter optimization problem under the multi-source covariate shift whose goal is to find the optimal hyperparameters for a target task of interest using only unlabeled data in a target task and labeled data in multiple source tasks. To conduct efficient hyperparameter optimization for the target task, it is essential to estimate the target objective using only the available information. To this end, we construct the variance reduced estimator that unbiasedly approximates the target objective with a desirable variance property. Building on the proposed estimator, we provide a general and tractable hyperparameter optimization procedure, which works preferably in our setting with a no-regret guarantee. The experiments demonstrate that the proposed framework broadens the applications of automated hyperparameter optimization.
Bregman proximal point algorithm (BPPA), as one of the centerpieces in the optimization toolbox, has been witnessing emerging applications. With simple and easy to implement update rule, the algorithm bears several compelling intuitions for empirical successes, yet rigorous justifications are still largely unexplored. We study the computational properties of BPPA through classification tasks with separable data, and demonstrate provable algorithmic regularization effects associated with BPPA. We show that BPPA attains non-trivial margin, which closely depends on the condition number of the distance generating function inducing the Bregman divergence. We further demonstrate that the dependence on the condition number is tight for a class of problems, thus showing the importance of divergence in affecting the quality of the obtained solutions. In addition, we extend our findings to mirror descent (MD), for which we establish similar connections between the margin and Bregman divergence. We demonstrate through a concrete example, and show BPPA/MD converges in direction to the maximal margin solution with respect to the Mahalanobis distance. Our theoretical findings are among the first to demonstrate the benign learning properties BPPA/MD, and also provide corroborations for a careful choice of divergence in the algorithmic design.
We study a parametric version of the Kannan-Lipton Orbit Problem for linear dynamical systems. We show decidability in the case of one parameter and Skolem-hardness with two or more parameters. More precisely, consider a $d$-dimensional square matrix $M$ whose entries are algebraic functions in one or more real variables. Given initial and target vectors $u,v\in \mathbb{Q}^d$, the parametric point-to-point orbit problem asks whether there exist values of the parameters giving rise to a concrete matrix $N \in \mathbb{R}^{d\times d}$, and a positive integer $n\in \mathbb{N}$, such that $N^nu = v$. We show decidability for the case in which $M$ depends only upon a single parameter, and we exhibit a reduction from the well-known Skolem Problem for linear recurrence sequences, suggesting intractability in the case of two or more parameters.
The generalization error of a classifier is related to the complexity of the set of functions among which the classifier is chosen. Roughly speaking, the more complex the family, the greater the potential disparity between the training error and the population error of the classifier. This principle is embodied in layman's terms by Occam's razor principle, which suggests favoring low-complexity hypotheses over complex ones. We study a family of low-complexity classifiers consisting of thresholding the one-dimensional feature obtained by projecting the data on a random line after embedding it into a higher dimensional space parametrized by monomials of order up to k. More specifically, the extended data is projected n-times and the best classifier among those n (based on its performance on training data) is chosen. We obtain a bound on the generalization error of these low-complexity classifiers. The bound is less than that of any classifier with a non-trivial VC dimension, and thus less than that of a linear classifier. We also show that, given full knowledge of the class conditional densities, the error of the classifiers would converge to the optimal (Bayes) error as k and n go to infinity; if only a training dataset is given, we show that the classifiers will perfectly classify all the training points as k and n go to infinity.
Proximal Policy Optimization (PPO) is a highly popular model-free reinforcement learning (RL) approach. However, in continuous state and actions spaces and a Gaussian policy -- common in computer animation and robotics -- PPO is prone to getting stuck in local optima. In this paper, we observe a tendency of PPO to prematurely shrink the exploration variance, which naturally leads to slow progress. Motivated by this, we borrow ideas from CMA-ES, a black-box optimization method designed for intelligent adaptive Gaussian exploration, to derive PPO-CMA, a novel proximal policy optimization approach that can expand the exploration variance on objective function slopes and shrink the variance when close to the optimum. This is implemented by using separate neural networks for policy mean and variance and training the mean and variance in separate passes. Our experiments demonstrate a clear improvement over vanilla PPO in many difficult OpenAI Gym MuJoCo tasks.
Stochastic gradient Markov chain Monte Carlo (SGMCMC) has become a popular method for scalable Bayesian inference. These methods are based on sampling a discrete-time approximation to a continuous time process, such as the Langevin diffusion. When applied to distributions defined on a constrained space, such as the simplex, the time-discretisation error can dominate when we are near the boundary of the space. We demonstrate that while current SGMCMC methods for the simplex perform well in certain cases, they struggle with sparse simplex spaces; when many of the components are close to zero. However, most popular large-scale applications of Bayesian inference on simplex spaces, such as network or topic models, are sparse. We argue that this poor performance is due to the biases of SGMCMC caused by the discretization error. To get around this, we propose the stochastic CIR process, which removes all discretization error and we prove that samples from the stochastic CIR process are asymptotically unbiased. Use of the stochastic CIR process within a SGMCMC algorithm is shown to give substantially better performance for a topic model and a Dirichlet process mixture model than existing SGMCMC approaches.
In this work, we consider the distributed optimization of non-smooth convex functions using a network of computing units. We investigate this problem under two regularity assumptions: (1) the Lipschitz continuity of the global objective function, and (2) the Lipschitz continuity of local individual functions. Under the local regularity assumption, we provide the first optimal first-order decentralized algorithm called multi-step primal-dual (MSPD) and its corresponding optimal convergence rate. A notable aspect of this result is that, for non-smooth functions, while the dominant term of the error is in $O(1/\sqrt{t})$, the structure of the communication network only impacts a second-order term in $O(1/t)$, where $t$ is time. In other words, the error due to limits in communication resources decreases at a fast rate even in the case of non-strongly-convex objective functions. Under the global regularity assumption, we provide a simple yet efficient algorithm called distributed randomized smoothing (DRS) based on a local smoothing of the objective function, and show that DRS is within a $d^{1/4}$ multiplicative factor of the optimal convergence rate, where $d$ is the underlying dimension.
We develop an approach to risk minimization and stochastic optimization that provides a convex surrogate for variance, allowing near-optimal and computationally efficient trading between approximation and estimation error. Our approach builds off of techniques for distributionally robust optimization and Owen's empirical likelihood, and we provide a number of finite-sample and asymptotic results characterizing the theoretical performance of the estimator. In particular, we show that our procedure comes with certificates of optimality, achieving (in some scenarios) faster rates of convergence than empirical risk minimization by virtue of automatically balancing bias and variance. We give corroborating empirical evidence showing that in practice, the estimator indeed trades between variance and absolute performance on a training sample, improving out-of-sample (test) performance over standard empirical risk minimization for a number of classification problems.
Robust estimation is much more challenging in high dimensions than it is in one dimension: Most techniques either lead to intractable optimization problems or estimators that can tolerate only a tiny fraction of errors. Recent work in theoretical computer science has shown that, in appropriate distributional models, it is possible to robustly estimate the mean and covariance with polynomial time algorithms that can tolerate a constant fraction of corruptions, independent of the dimension. However, the sample and time complexity of these algorithms is prohibitively large for high-dimensional applications. In this work, we address both of these issues by establishing sample complexity bounds that are optimal, up to logarithmic factors, as well as giving various refinements that allow the algorithms to tolerate a much larger fraction of corruptions. Finally, we show on both synthetic and real data that our algorithms have state-of-the-art performance and suddenly make high-dimensional robust estimation a realistic possibility.
We propose a new approach to inverse reinforcement learning (IRL) based on the deep Gaussian process (deep GP) model, which is capable of learning complicated reward structures with few demonstrations. Our model stacks multiple latent GP layers to learn abstract representations of the state feature space, which is linked to the demonstrations through the Maximum Entropy learning framework. Incorporating the IRL engine into the nonlinear latent structure renders existing deep GP inference approaches intractable. To tackle this, we develop a non-standard variational approximation framework which extends previous inference schemes. This allows for approximate Bayesian treatment of the feature space and guards against overfitting. Carrying out representation and inverse reinforcement learning simultaneously within our model outperforms state-of-the-art approaches, as we demonstrate with experiments on standard benchmarks ("object world","highway driving") and a new benchmark ("binary world").
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