The agnostic PAC learning model consists of: a Hypothesis Space $\mathcal{H}$, a probability distribution $P$, a sample complexity function $m_{\mathcal{H}}(\epsilon,\delta): [0,1]^{2} \mapsto \mathbb{Z}_{+}$ of precision $\epsilon$ and confidence $1 - \delta$, a finite i.i.d. sample $\mathcal{D}_{N}$, a cost function $\ell$ and a learning algorithm $\mathbb{A}(\mathcal{H},\mathcal{D}_{N})$, which estimates $\hat{h} \in \mathcal{H}$ that approximates a target function $h^{\star} \in \mathcal{H}$ seeking to minimize out-of-sample error. In this model, prior information is represented by $\mathcal{H}$ and $\ell$, while problem solution is performed through their instantiation in several applied learning models, with specific algebraic structures for $\mathcal{H}$ and corresponding learning algorithms. However, these applied models use additional important concepts not covered by the classic PAC learning theory: model selection and regularization. This paper presents an extension of this model which covers these concepts. The main principle added is the selection, based solely on data, of a subspace of $\mathcal{H}$ with a VC-dimension compatible with the available sample. In order to formalize this principle, the concept of Learning Space $\mathbb{L}(\mathcal{H})$, which is a poset of subsets of $\mathcal{H}$ that covers $\mathcal{H}$ and satisfies a property regarding the VC dimension of related subspaces, is presented as the natural search space for model selection algorithms. A remarkable result obtained on this new framework are conditions on $\mathbb{L}(\mathcal{H})$ and $\ell$ that lead to estimated out-of-sample error surfaces, which are true U-curves on $\mathbb{L}(\mathcal{H})$ chains, enabling a more efficient search on $\mathbb{L}(\mathcal{H})$. Hence, in this new framework, the U-curve optimization problem becomes a natural component of model selection algorithms.

The robust nature of MCTS, versus a traditional approach like depth-first search in alpha-beta pruning, has not only enabled a leapfrog in performance in computer Go, but has also led to its utilization in other games where it is difficult to evaluate states, as well as in other domains (Browne et al., 2012). However, MCTS is known to suffer from slow convergence in certain situations (Coquelin and Munos, 2007), in particular when the precise calculation of a narrow tactical sequence is critical for success. For example in boardgames, (Ramanujan et al., 2010) defines a level-k search trap for player p after a move m as a state of the game where the opponent of p has a guaranteed k -move winning strategy . More relevantly, they show through a series of experiments that MCTS performs poorly even in shallow traps, in contrast to regular minimax search; see also (Ramanujan et al., 2011; Ramanujan and Sel-man, 2011). T o better understand this phenomenon, we take a closer look at the update rule Q n Q n 1 R n 1 Q n 1 n (1) which is performed during the backpropagation phase of MCTS. Here, the current estimate of the value of a state is taken to be the simple average of all previous returns accrued upon visiting that state. Proceeding, we discuss various methods which seek to improve backpropagation by challenging the basic assumptions implied by (1): (i) Value estimation by averaging returns: Instead of updating a parent node's value with that of its MAX (MIN) child as in minimax search, backpropagation in MCTS averages all returns to obtain a good signal in noisy environments (this is 1 arXiv:2001.09325v1

The boolean satisfiability problem is a famous NP-complete problem in computer science. An effective way for this problem is the stochastic local search (SLS). However, in this method, the initialization is assigned in a random manner, which impacts the effectiveness of SLS solvers. To address this problem, we propose NLocalSAT. NLocalSAT combines SLS with a solution prediction model, which boosts SLS by changing initialization assignments with a neural network. We evaluated NLocalSAT on five SLS solvers (CCAnr, Sparrow, CPSparrow, YalSAT, and probSAT) with problems in the random track of SAT Competition 2018. The experimental results show that solvers with NLocalSAT achieve 27%~62% improvement over the original SLS solvers.

Nowadays, we are immersed in tens of newly-proposed evolutionary and swam-intelligence metaheuristics, which makes it very difficult to choose a proper one to be applied on a specific optimization problem at hand. On the other hand, most of these metaheuristics are nothing but slightly modified variants of the basic metaheuristics. For example, Differential Evolution (DE) or Shuffled Frog Leaping (SFL) are just Genetic Algorithms (GA) with a specialized operator or an extra local search, respectively. Therefore, what comes to the mind is whether the behavior of such newly-proposed metaheuristics can be investigated on the basis of studying the specifications and characteristics of their ancestors. In this paper, a comprehensive evaluation study on some basic metaheuristics i.e. Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Artificial Bee Colony (ABC), Teaching-Learning-Based Optimization (TLBO), and Cuckoo Optimization algorithm (COA) is conducted, which give us a deeper insight into the performance of them so that we will be able to better estimate the performance and applicability of all other variations originated from them. A large number of experiments have been conducted on 20 different combinatorial optimization benchmark functions with different characteristics, and the results reveal to us some fundamental conclusions besides the following ranking order among these metaheuristics, {ABC, PSO, TLBO, GA, COA} i.e. ABC and COA are the best and the worst methods from the performance point of view, respectively. In addition, from the convergence perspective, PSO and ABC have significant better convergence for unimodal and multimodal functions, respectively, while GA and COA have premature convergence to local optima in many cases needing alternative mutation mechanisms to enhance diversification and global search.

An online resource scheduling framework is proposed for minimizing the sum of weighted task latency for all the mobile users, by optimizing offloading decision, transmission power, and resource allocation in the mobile edge computing (MEC) system. Towards this end, a deep reinforcement learning (DRL) method is proposed to obtain an online resource scheduling policy. Firstly, a related and regularized stacked auto encoder (2r-SAE) with unsupervised learning is proposed to perform data compression and representation for high dimensional channel quality information (CQI) data, which can reduce the state space for DRL. Secondly, we present an adaptive simulated annealing based approach (ASA) as the action search method of DRL, in which an adaptive h-mutation is used to guide the search direction and an adaptive iteration is proposed to enhance the search efficiency during the DRL process. Thirdly, a preserved and prioritized experience replay (2p-ER) is introduced to assist the DRL to train the policy network and find the optimal offloading policy. Numerical results are provided to demonstrate that the proposed algorithm can achieve near-optimal performance while significantly decreasing the computational time compared with existing benchmarks. It also shows that the proposed framework is suitable for resource scheduling problem in large-scale MEC networks, especially in the dynamic environment.

This paper studies a classic maximum entropy sampling problem (MESP), which aims to select the most informative principal submatrix of a prespecified size from a covariance matrix. MESP has been widely applied to many areas, including healthcare, power system, manufacturing and data science. By investigating its Lagrangian dual and primal characterization, we derive a novel convex integer program for MESP and show that its continuous relaxation yields a near-optimal solution. The results motivate us to study an efficient sampling algorithm and develop its approximation bound for MESP, which improves the best-known bound in literature. We then provide an efficient deterministic implementation of the sampling algorithm with the same approximation bound. By developing new mathematical tools for the singular matrices and analyzing the Lagrangian dual of the proposed convex integer program, we investigate the widely-used local search algorithm and prove its first-known approximation bound for MESP. The proof techniques further inspire us with an efficient implementation of the local search algorithm. Our numerical experiments demonstrate that these approximation algorithms can efficiently solve medium-sized and large-scale instances to near-optimality. Our proposed algorithms are coded and released as open-source software. Finally, we extend the analyses to the A-Optimal MESP (A-MESP), where the objective is to minimize the trace of the inverse of the selected principal submatrix.

This problem is well-known to be NPhard [3]. For our purposes, an edge version is important: Given a subset of edges (instead of vertices), find a connected subgraph of G containing all prescribed edges with the minimal weight. This edge version of Steiner tree problem makes our problem hard even if the limit L is sufficiently large to ensure that only one inert (or chemical) car can maintain all inert roads. In this case, the set of all inert roads may be disconnected so a set of chemical roads has to be added to the inert car's plan to ensure connectivity. Finding such a set of chemical roads with the minimal weight is exactly the edge version of the Steiner tree problem. This kind of reasoning is used in Section 5 to obtain lower bounds on a minimal deadhead.

In bipartite networks, community structures are restricted to being disassortative, in that nodes of one type are grouped according to common patterns of connection with nodes of the other type. This makes the stochastic block model (SBM), a highly flexible generative model for networks with block structure, an intuitive choice for bipartite community detection. However, typical formulations of the SBM do not make use of the special structure of bipartite networks. In this work, we introduce a Bayesian nonparametric formulation of the SBM and a corresponding algorithm to efficiently find communities in bipartite networks without overfitting. The biSBM improves community detection results over general SBMs when data are noisy, improves the model resolution limit by a factor of $\sqrt{2}$, and expands our understanding of the complicated optimization landscape associated with community detection tasks. A direct comparison of certain terms of the prior distributions in the biSBM and a related high-resolution hierarchical SBM also reveals a counterintuitive regime of community detection problems, populated by smaller and sparser networks, where non-hierarchical models outperform their more flexible counterpart.

January 23, 2020 Abstract The optimal binning is the optimal discretization of a variable into bins given a discrete or continuous numeric target. We present a rigorous and extensible mathematical programming formulation to solving the optimal binning problem for a binary, continuous and multi-class target type, incorporating constraints not previously addressed. For all three target types, we introduce a convex mixed-integer programming formulation. Several algorithmic enhancements such as automatic determination of the most suitable monotonic trend via a Machine-Learning-based classifier and implementation aspects are thoughtfully discussed. The new mathematical programming formulations are carefully implemented in the open-source python library OptBinning. 1 Introduction Binning (grouping or bucketing) is a technique to discretize the values of a continuous variable into bins (groups or buckets). From a modeling perspective, the binning technique may address prevalent data issues such as the handling of missing values, the presence of outliers and statistical noise, and data scaling. Furthermore, the binning process is a valuable interpretable tool to enhance the understanding of the nonlinear dependence between a variable and a given target while reducing the model complexity. Ultimately, resulting bins can be used to perform data transformations. Binning techniques are extensively used in machine learning applications, exploratory data analysis and as an algorithm to speed up learning tasks; recently, binning has been applied to accelerate learning in gradient boosting decision tree [12].

The expressiveness of search space is a key concern in neural architecture search (NAS). Previous approaches are mainly limited to searching for single-path networks. Incorporating multi-path search space with the current one-shot doctrine remains untackled. In this paper, we investigate the supernet behavior under multi-path's setting. We show that a trivial generalization from single-path to multi-path incurs severe feature inconsistency, which deteriorates both supernet training stability and model ranking ability. To remedy this degradation, we employ what we term as shadow batch normalizations (SBN) to catch changing statistics when activating different sets of paths. Extensive experiments on a common NAS benchmark, NAS-bench-101, show that SBN can boost ranking performance at neglectable cost. It breaks the Kendall Tau's record with a clear margin, reaching 0.597. Moreover, we take advantage of feature similarities on activated paths to largely reduce the number of needed SBNs. We call our method MixPath. When proxylessly searching on ImageNet, we obtain several lightweight models that outperform EfficientNet-B0 with fewer FLOPs, parameters and 300x fewer searching resources. Our code will be available https://github.com/xiaomi-automl/MixPath.git .