We present a data-driven framework for incorporating side information in dynamic optimization under uncertainty. Specifically, our approach uses predictive machine learning methods (such as k-nearest neighbors, kernel regression, and random forests) to weight the relative importance of various data-driven uncertainty sets in a robust optimization formulation. Through a novel measure concentration result for local machine learning methods, we prove that the proposed framework is asymptotically optimal for stochastic dynamic optimization with covariates. We also describe a general-purpose approximation for the proposed framework, based on overlapping linear decision rules, which is computationally tractable and produces high-quality solutions for dynamic problems with many stages. Across a variety of examples in shipment planning, inventory management, and finance, our method achieves improvements of up to 15 over alternatives and requires less than one minute of computation time on problems with twelve stages.
As discussed previously, the methodology in this paper also follows recent work on incorporating covariates in optimization under uncertainty using local predictive methods (such as @math -nearest neighbor regression, kernel regression, and random forests). In particular, the asymptotic optimality justification of @cite_11 in single-stage settings relies on the strong universal consistency for local predictive models (, @cite_6 ). Our proof of asymptotic optimality instead relies on convergence guarantees rooted in distributionally robust optimization. The reason we use a different approach is that the arguments for the convergence for local predictive models from @cite_11 require finite dimensional decision variables. In contrast, the convergence guarantees in this paper apply for dynamic optimization over general spaces of policies.
@cite_6 @cite_11
2211648829 1507350577
Prediction markets are economic mechanisms for aggregating information about future events through sequential interactions with traders. The pricing mechanisms in these markets are known to be related to optimization algorithms in machine learning and through these connections we have some understanding of how equilibrium market prices relate to the beliefs of the traders in a market. However, little is known about rates and guarantees for the convergence of these sequential mechanisms, and two recent papers cite this as an important open question. In this paper we show how some previously studied prediction market trading models can be understood as a natural generalization of randomized coordinate descent which we call randomized subspace descent (RSD). We establish convergence rates for RSD and leverage them to prove rates for the two prediction market models above, answering the open questions. Our results extend beyond standard centralized markets to arbitrary trade networks. Discriminative latent-variable models are typically learned using EM or gradient-based optimization, which suffer from local optima. In this paper, we develop a new computationally efficient and provably consistent estimator for a mixture of linear regressions, a simple instance of a discriminative latent-variable model. Our approach relies on a low-rank linear regression to recover a symmetric tensor, which can be factorized into the parameters using a tensor power method. We prove rates of convergence for our estimator and provide an empirical evaluation illustrating its strengths relative to local optimization (EM). Guided policy search algorithms can be used to optimize complex nonlinear policies, such as deep neural networks, without directly computing policy gradients in the high-dimensional parameter space. Instead, these methods use supervised learning to train the policy to mimic a “teacher” algorithm, such as a trajectory optimizer or a trajectory-centric reinforcement learning method. Guided policy search methods provide asymptotic local convergence guarantees by construction, but it is not clear how much the policy improves within a small, finite number of iterations. We show that guided policy search algorithms can be interpreted as an approximate variant of mirror descent, where the projection onto the constraint manifold is not exact. We derive a new guided policy search algorithm that is simpler and provides appealing improvement and convergence guarantees in simplified convex and linear settings, and show that in the more general nonlinear setting, the error in the projection step can be bounded. We provide empirical results on several simulated robotic manipulation tasks that show that our method is stable and achieves similar or better performance when compared to prior guided policy search methods, with a simpler formulation and fewer hyperparameters. Distributed averaging describes a class of network algorithms for the decentralized computation of aggregate statistics. Initially, each node has a scalar data value, and the goal is to compute the average of these values at every node (the so-called average consensus problem). Nodes iteratively exchange information with their neighbors and perform local updates until the value at every node converges to the initial network average. Much previous work has focused on algorithms where each node maintains and updates a single value; every time an update is performed, the previous value is forgotten. Convergence to the average consensus is achieved asymptotically. The convergence rate is fundamentally limited by network connectivity, and it can be prohibitively slow on topologies such as grids and random geometric graphs, even if the update rules are optimized. In this paper, we provide the first theoretical demonstration that adding a local prediction component to the update rule can significantly improve the convergence rate of distributed averaging algorithms. We focus on the case where the local predictor is a linear combination of the node's current and previous values (i.e., two memory taps), and our update rule computes a combination of the predictor and the usual weighted linear combination of values received from neighboring nodes. We derive the optimal mixing parameter for combining the predictor with the neighbors' values, and conduct a theoretical analysis of the improvement in convergence rate that can be achieved using this acceleration methodology. For a chain topology on N nodes, this leads to a factor of N improvement over standard consensus, and for a two-dimensional grid, our approach achieves a factor of ?N improvement.