# Search

### Clues for Which I Search and Choose

Before we leave these model-free chronicles behind, let me turn to the converse of the Linearization Principle. We have seen that random search works well on simple linear problems and appears better than some RL methods like policy gradient. Does random search break down as we move to harder problems? Let's apply random search to problems that are of interest to the RL community. The deep RL community has been spending a lot of time and energy on a suite of benchmarks, maintained by OpenAI and based on the MuJoCo simulator.

Modern supervised learning techniques, particularly those using so called deep nets, involve fitting high dimensional labelled data sets with functions containing very large numbers of parameters. Much of this work is empirical, and interesting phenomena have been observed that require theoretical explanations, however the non-convexity of the loss functions complicates the analysis. Recently it has been proposed that some of the success of these techniques resides in the effectiveness of the simple stochastic gradient descent algorithm in the so called interpolation limit in which all labels are fit perfectly. This analysis is made possible since the SGD algorithm reduces to a stochastic linear system near the interpolating minimum of the loss function. Here we exploit this insight by analyzing a distributed algorithm for gradient descent, also in the interpolating limit. The algorithm corresponds to gradient descent applied to a simple penalized distributed loss function, $L({\bf w}_1,...,{\bf w}_n) = \Sigma_i l_i({\bf w}_i) + \mu \sum_{}|{\bf w}_i-{\bf w}_j|^2$. Here each node is allowed its own parameter vector and  denotes edges of a connected graph defining the communication links between nodes. It is shown that this distributed algorithm converges linearly (ie the error reduces exponentially with iteration number), with a rate $1-\frac{\eta}{n}\lambda_{min}(H) ### Attention Solves Your TSP We propose a framework for solving combinatorial optimization problems of which the output can be represented as a sequence of input elements. As an alternative to the Pointer Network, we parameterize a policy by a model based entirely on (graph) attention layers, and train it efficiently using REINFORCE with a simple and robust baseline based on a deterministic (greedy) rollout of the best policy found during training. We significantly improve over state-of-the-art results for learning algorithms for the 2D Euclidean TSP, reducing the optimality gap for a single tour construction by more than 75% (to 0.33%) and 50% (to 2.28%) for instances with 20 and 50 nodes respectively. ### Gradient Descent Quantizes ReLU Network Features Deep neural networks are often trained in the over-parametrized regime (i.e. with far more parameters than training examples), and understanding why the training converges to solutions that generalize remains an open problem. Several studies have highlighted the fact that the training procedure, i.e. mini-batch Stochastic Gradient Descent (SGD) leads to solutions that have specific properties in the loss landscape. However, even with plain Gradient Descent (GD) the solutions found in the over-parametrized regime are pretty good and this phenomenon is poorly understood. We propose an analysis of this behavior for feedforward networks with a ReLU activation function under the assumption of small initialization and learning rate and uncover a quantization effect: The weight vectors tend to concentrate at a small number of directions determined by the input data. As a consequence, we show that for given input data there are only finitely many, "simple" functions that can be obtained, independent of the network size. This puts these functions in analogy to linear interpolations (for given input data there are finitely many triangulations, which each determine a function by linear interpolation). We ask whether this analogy extends to the generalization properties - while the usual distribution-independent generalization property does not hold, it could be that for e.g. smooth functions with bounded second derivative an approximation property holds which could "explain" generalization of networks (of unbounded size) to unseen inputs. ### Deep Reinforcement Learning with Model Learning and Monte Carlo Tree Search in Minecraft Deep reinforcement learning has been successfully applied to several visual-input tasks using model-free methods. In this paper, we propose a model-based approach that combines learning a DNN-based transition model with Monte Carlo tree search to solve a block-placing task in Minecraft. Our learned transition model predicts the next frame and the rewards one step ahead given the last four frames of the agent's first-person-view image and the current action. Then a Monte Carlo tree search algorithm uses this model to plan the best sequence of actions for the agent to perform. On the proposed task in Minecraft, our model-based approach reaches the performance comparable to the Deep Q-Network's, but learns faster and, thus, is more training sample efficient. ### Gradient Descent with Random Initialization: Fast Global Convergence for Nonconvex Phase Retrieval This paper considers the problem of solving systems of quadratic equations, namely, recovering an object of interest$\mathbf{x}^{\natural}\in\mathbb{R}^{n}$from$m$quadratic equations/samples$y_{i}=(\mathbf{a}_{i}^{\top}\mathbf{x}^{\natural})^{2}$,$1\leq i\leq m$. This problem, also dubbed as phase retrieval, spans multiple domains including physical sciences and machine learning. We investigate the efficiency of gradient descent (or Wirtinger flow) designed for the nonconvex least squares problem. We prove that under Gaussian designs, gradient descent --- when randomly initialized --- yields an$\epsilon$-accurate solution in$O\big(\log n+\log(1/\epsilon)\big)$iterations given nearly minimal samples, thus achieving near-optimal computational and sample complexities at once. This provides the first global convergence guarantee concerning vanilla gradient descent for phase retrieval, without the need of (i) carefully-designed initialization, (ii) sample splitting, or (iii) sophisticated saddle-point escaping schemes. All of these are achieved by exploiting the statistical models in analyzing optimization algorithms, via a leave-one-out approach that enables the decoupling of certain statistical dependency between the gradient descent iterates and the data. ### Updates on Policy Gradients I've been swamped with a bit of a travel binge and am hopelessly behind on blogging. After my last post on nominal control, I received an email from Pavel Christof pointing out that if we switch from stochastic gradient descent to Adam, policy gradient works much better. Indeed, I implemented this myself, and he's totally right. Let's revisit the last post with a revised Jupyter notebook. First, I coded up Adam in pure python to avoid introducing any deep learning package dependencies (it's only 4 lines of python, after all). ### Heuristic Search and Information Visualization Methods for School Redistricting Heuristic Search and Information Visualization Methods for School Redistricting ### Expressive Commerce and Its Application to Sourcing: How We Conducted$35 Billion of Generalized Combinatorial Auctions

Expressive Commerce and Its Application to Sourcing: How We Conducted \$35 Billion of Generalized Combinatorial Auctions

### Algorithm Selection for Combinatorial Search Problems: A Survey

Algorithm Selection for Combinatorial Search Problems: A Survey