Bayesian optimization is a powerful tool for global optimization of expensive functions. One of its key components is the underlying probabilistic model used for the objective function f. In practice, however, it is often unclear how one should appropriately choose a model, especially when gathering data is expensive. In this work, we introduce a novel automated Bayesian optimization approach that dynamically selects promising models for explaining the observed data using Bayesian Optimization in the model space. Crucially, we account for the uncertainty in the choice of model; our method is capable of using multiple models to represent its current belief about f and subsequently using this information for decision making. We argue, and demonstrate empirically, that our approach automatically finds suitable models for the objective function, which ultimately results in more-efficient optimization.

Bayesian optimization (BO) is a model-based approach for gradient-free black-box function optimization, such as hyperparameter optimization. Typically, BO relies on conventional Gaussian process (GP) regression, whose algorithmic complexity is cubic in the number of evaluations. As a result, GP-based BO cannot leverage large numbers of past function evaluations, for example, to warm-start related BO runs. We propose a multi-task adaptive Bayesian linear regression model for transfer learning in BO, whose complexity is linear in the function evaluations: one Bayesian linear regression model is associated to each black-box function optimization problem (or task), while transfer learning is achieved by coupling the models through a shared deep neural net. Experiments show that the neural net learns a representation suitable for warm-starting the black-box optimization problems and that BO runs can be accelerated when the target black-box function (e.g., validation loss) is learned together with other related signals (e.g., training loss). The proposed method was found to be at least one order of magnitude faster that methods recently published in the literature.

The \emph{Cheap Gradient Principle}~\citep{Griewank:2008:EDP:1455489} --- the computational cost of computing a $d$-dimensional vector of partial derivatives of a scalar function is nearly the same (often within a factor of $5$) as that of simply computing the scalar function itself --- is of central importance in optimization; it allows us to quickly obtain (high-dimensional) gradients of scalar loss functions which are subsequently used in black box gradient-based optimization procedures. The current state of affairs is markedly different with regards to computing sub-derivatives: widely used ML libraries, including TensorFlow and PyTorch, do \emph{not} correctly compute (generalized) sub-derivatives even on simple differentiable examples. This work considers the question: is there a \emph{Cheap Sub-gradient Principle}? Our main result shows that, under certain restrictions on our library of non-smooth functions (standard in non-linear programming), provably correct generalized sub-derivatives can be computed at a computational cost that is within a (dimension-free) factor of $6$ of the cost of computing the scalar function itself.

In this paper, we study the problem of escaping from saddle points in smooth nonconvex optimization problems subject to a convex set $\mathcal{C}$. We propose a generic framework that yields convergence to a second-order stationary point of the problem, if the convex set $\mathcal{C}$ is simple for a quadratic objective function.

We consider semidefinite programs (SDPs) of size nwith equality constraints. In order to overcome scalability issues, Burer and Monteiro proposed a factorized approach based on optimizing over a matrix Y of size n ksuch that X = YY is the SDP variable. The advantages of such formulation are twofold: the dimension of the optimization variable is reduced, and positive semidefiniteness is naturally enforced. However, optimization in Y is non-convex. In prior work, it has been shown that, when the constraints on the factorized variable regularly define a smooth manifold, provided k is large enough, for almost all cost matrices, all second-order stationary points (SOSPs) are optimal. Importantly, in practice, one can only compute points which approximately satisfy necessary optimality conditions, leading to the question: are such points also approximately optimal? To answer it, under similar assumptions, we use smoothed analysis to show that approximate SOSPs for a randomly perturbed objective function are approximate global optima, with k scaling like the square root of the number of constraints (up to log factors). Moreover, we bound the optimality gap at the approximate solution of the perturbed problem with respect to the original problem.

Existing convergence of distributed optimization methods in non-Euclidean geometries typically rely on kernel assumptions: (i) global Lipschitz smoothness and (ii) bi-convexity of the associated Bregman divergence function. Unfortunately, these conditions are violated by nearly all kernels used in practice, leaving a huge theory-practice gap. This work closes this gap by developing a unified analytical tool that guarantees convergence under mild conditions. Specifically, we introduce Hessian relative uniform continuity (HRUC), a regularity satisfied by nearly all standard kernels. Importantly, HRUC is closed under concatenation, positive scaling, composition, and various kernel combinations. Leveraging the geometric structure induced by HRUC, we derive convergence guarantees for mirror descent-based gradient tracking without imposing any restrictive assumptions. More broadly, our analysis techniques extend seamlessly to other decentralized optimization methods in genuinely non-Euclidean and non-Lipschitz settings.

Neural Information Processing Systems http://nips.cc/

In this paper, we propose a novel maximum causal Tsallis entropy (MCTE) framework for imitation learning which can efficiently learn a sparse multi-modal policy distribution from demonstrations. We provide the full mathematical analysis of the proposed framework. First, the optimal solution of an MCTE problem is shown to be a sparsemax distribution, whose supporting set can be adjusted. The proposed method has advantages over a softmax distribution in that it can exclude unnecessary actions by assigning zero probability. Second, we prove that an MCTE problem is equivalent to robust Bayes estimation in the sense of the Brier score. Third, we propose a maximum causal Tsallis entropy imitation learning (MCTEIL) algorithm with a sparse mixture density network (sparse MDN) by modeling mixture weights using a sparsemax distribution. In particular, we show that the causal Tsallis entropy of an MDN encourages exploration and efficient mixture utilization while Shannon entropy is less effective.

Neural Information Processing Systems http://nips.cc/