These notes aim to shed light on the recently proposed structured projected intermediate gradient optimization technique (SPIGOT, Peng et al., 2018). SPIGOT is a variant of the straight-through estimator (Bengio et al., 2013) which bypasses gradients of the argmax function by back-propagating a surrogate "gradient." We provide a new interpretation to the proposed gradient and put this technique into perspective, linking it to other methods for training neural networks with discrete latent variables. As a by-product, we suggest alternate variants of SPIGOT which will be further explored in future work.

Recently, a higher competition in logistics business introduces new challenges to the vehicle routing problem (VRP). Re-route planning, also known as dynamic VRP, is one of the important challenges. The re-route planning has to be performed when new customers request for deliveries while the delivery vehicles, i.e., trucks, are serving other customers. While the re-route planning has been studied in the literature, most of the existing works do not consider different uncertainties. Therefore, in this paper, we propose two systems, i.e., (i) an offline package pickup and delivery planning with stochastic demands (PDPSD) and (ii) a re-route package pickup and delivery planning with stochastic demands (Re-route PDPSD). Accordingly, we formulate the PDPSD system as a two-stage stochastic optimization. We then extend the PDPSD system to the Re-route PDPSD system with a re-route algorithm. Furthermore, we evaluate performance of the proposed systems by using the dataset from Solomon Benchmark suite and a real data from a Singapore logistics 1company. The results show that the PDPSD system can achieve the lower cost than that of the baseline model. In addition, the Re-route PDPSD system can help the supplier efficiently and successfully to serve more customers while the trucks are already on the road.

This paper proposes a method for learning a trajectory-conditioned policy to imitate diverse demonstrations from the agent's own past experiences. We demonstrate that such self-imitation drives exploration in diverse directions and increases the chance of finding a globally optimal solution in reinforcement learning problems, especially when the reward is sparse and deceptive. Our method significantly outperforms existing self-imitation learning and count-based exploration methods on various sparse-reward reinforcement learning tasks with local optima. In particular, we report a state-of-the-art score of more than 25,000 points on Montezuma's Revenge without using expert demonstrations or resetting to arbitrary states.

Estimating the perfect times for drivers to pick up food delivery orders for a number of different restaurants can be one of the most difficult of computational problems. Think of it like the Traveling Salesperson NP-Hard combinatorial optimization problem: The customer wants food delivered in a timely manner, and the delivery person wants to food ready when they roll-up. If the estimates are off by even a tiny bit, then customers are unhappy and delivery people will work elsewhere. Yet, car-sharing service Uber is building a global service, called Uber Eats, that will rely on accurate predictions to succeed. The secret to its success will be machine learning, built from the company's in-house ML platform, nicknamed Michelangelo.

It is well known that the historical logs are used for evaluating and learning policies in interactive systems, e.g. recommendation, search, and online advertising. Since direct online policy learning usually harms user experiences, it is more crucial to apply off-policy learning in real-world applications instead. Though there have been some existing works, most are focusing on learning with one single historical policy. However, in practice, usually a number of parallel experiments, e.g. multiple AB tests, are performed simultaneously. To make full use of such historical data, learning policies from multiple loggers becomes necessary. Motivated by this, in this paper, we investigate off-policy learning when the training data coming from multiple historical policies. Specifically, policies, e.g. neural networks, can be learned directly from multi-logger data, with counterfactual estimators. In order to understand the generalization ability of such estimator better, we conduct generalization error analysis for the empirical risk minimization problem. We then introduce the generalization error bound as the new risk function, which can be reduced to a constrained optimization problem. Finally, we give the corresponding learning algorithm for the new constrained problem, where we can appeal to the minimax problems to control the constraints. Extensive experiments on benchmark datasets demonstrate that the proposed methods achieve better performances than the state-of-the-arts.

Summarizing large-scaled directed graphs into small-scale representations is a useful but less studied problem setting. Conventional clustering approaches, which based on "Min-Cut"-style criteria, compress both the vertices and edges of the graph into the communities, that lead to a loss of directed edge information. On the other hand, compressing the vertices while preserving the directed edge information provides a way to learn the small-scale representation of a directed graph. The reconstruction error, which measures the edge information preserved by the summarized graph, can be used to learn such representation. Compared to the original graphs, the summarized graphs are easier to analyze and are capable of extracting group-level features which is useful for efficient interventions of population behavior. In this paper, we present a model, based on minimizing reconstruction error with non-negative constraints, which relates to a "Max-Cut" criterion that simultaneously identifies the compressed nodes and the directed compressed relations between these nodes. A multiplicative update algorithm with column-wise normalization is proposed. We further provide theoretical results on the identifiability of the model and on the convergence of the proposed algorithms. Experiments are conducted to demonstrate the accuracy and robustness of the proposed method.

Stochastic (sub)gradient methods require step size schedule tuning to perform well in practice. Classical tuning strategies decay the step size polynomially and lead to optimal sublinear rates on (strongly) convex problems. An alternative schedule, popular in nonconvex optimization, is called \emph{geometric step decay} and proceeds by halving the step size after every few epochs. In recent work, geometric step decay was shown to improve exponentially upon classical sublinear rates for the class of \emph{sharp} convex functions. In this work, we ask whether geometric step decay similarly improves stochastic algorithms for the class of sharp nonconvex problems. Such losses feature in modern statistical recovery problems and lead to a new challenge not present in the convex setting: the region of convergence is local, so one must bound the probability of escape. Our main result shows that for a large class of stochastic, sharp, nonsmooth, and nonconvex problems a geometric step decay schedule endows well-known algorithms with a local linear rate of convergence to global minimizers. This guarantee applies to the stochastic projected subgradient, proximal point, and prox-linear algorithms. As an application of our main result, we analyze two statistical recovery tasks---phase retrieval and blind deconvolution---and match the best known guarantees under Gaussian measurement models and establish new guarantees under heavy-tailed distributions.

This paper deals with robust optimization applied to network flows. Two robust variants of the minimum-cost integer flow problem are considered. Thereby, uncertainty in problem formulation is limited to arc unit costs and expressed by a finite set of explicitly given scenarios. It is shown that both problem variants are NPhard. To solve the consid ered variants, several heuristics based on local search or evolutionar y computing are proposed. The heuristics are experimentally evaluated on appr opriate problem instances.

In this two-part work, we propose an algorithmic framework for solving non-convex problems whose objective function is the sum of a number of smooth component functions plus a convex (possibly non-smooth) or/and smooth (possibly non-convex) regularization function. The proposed algorithm incorporates ideas from several existing approaches such as alternate direction method of multipliers (ADMM), successive convex approximation (SCA), distributed and asynchronous algorithms, and inexact gradient methods. Different from a number of existing approaches, however, the proposed framework is flexible enough to incorporate a class of non-convex objective functions, allow distributed operation with and without a fusion center, and include variance reduced methods as special cases. Remarkably, the proposed algorithms are robust to uncertainties arising from random, deterministic, and adversarial sources. The part I of the paper develops two variants of the algorithm under very mild assumptions and establishes first-order convergence rate guarantees. The proof developed here allows for generic errors and delays, paving the way for different variance-reduced, asynchronous, and stochastic implementations, outlined and evaluated in part II.

Experimental design is a process of obtaining a product with target property via experimentation. Bayesian optimization offers a sample-efficient tool for experimental design when experiments are expensive. Often, expert experimenters have 'hunches' about the behavior of the experimental system, offering potentials to further improve the efficiency. In this paper, we consider per-variable monotonic trend in the underlying property that results in a unimodal trend in those variables for a target value optimization. For example, sweetness of a candy is monotonic to the sugar content. However, to obtain a target sweetness, the utility of the sugar content becomes a unimodal function, which peaks at the value giving the target sweetness and falls off both ways. In this paper, we propose a novel method to solve such problems that achieves two main objectives: a) the monotonicity information is used to the fullest extent possible, whilst ensuring that b) the convergence guarantee remains intact. This is achieved by a two-stage Gaussian process modeling, where the first stage uses the monotonicity trend to model the underlying property, and the second stage uses `virtual' samples, sampled from the first, to model the target value optimization function. The process is made theoretically consistent by adding appropriate adjustment factor in the posterior computation, necessitated because of using the `virtual' samples. The proposed method is evaluated through both simulations and real world experimental design problems of a) new short polymer fiber with the target length, and b) designing of a new three dimensional porous scaffolding with a target porosity. In all scenarios our method demonstrates faster convergence than the basic Bayesian optimization approach not using such `hunches'.