We present a novel approach for adaptive, differentiable parameterization of large-scale random fields. If the approach is coupled with any gradient-based optimization algorithm, it can be applied to a variety of optimization problems, including history matching. The developed technique is based on principal component analysis (PCA) but modifies a purely data-driven basis of principal components considering objective function behavior. To define an efficient encoding, Gradient-Sensitive PCA uses an objective function gradient with respect to model parameters. We propose computationally efficient implementations of the technique, and two of them are based on stationary perturbation theory (SPT). Optimality, correctness, and low computational costs of the new encoding approach are tested, verified, and discussed. Three algorithms for optimal parameter decomposition are presented and applied to an objective of 2D synthetic history matching. The results demonstrate improvements in encoding quality regarding objective function minimization and distributional patterns of the desired field. Possible applications and extensions are proposed.

Combinatorial optimization has found applications in numerous fields, from aerospace to transportation planning and economics. The goal is to find an optimal solution among a finite set of possibilities. The well-known challenge one faces with combinatorial optimization is the state-space explosion problem: the number of possibilities grows exponentially with the problem size, which makes solving intractable for large problems. In the last years, deep reinforcement learning (DRL) has shown its promise for designing good heuristics dedicated to solve NP-hard combinatorial optimization problems. However, current approaches have two shortcomings: (1) they mainly focus on the standard travelling salesman problem and they cannot be easily extended to other problems, and (2) they only provide an approximate solution with no systematic ways to improve it or to prove optimality. In another context, constraint programming (CP) is a generic tool to solve combinatorial optimization problems. Based on a complete search procedure, it will always find the optimal solution if we allow an execution time large enough. A critical design choice, that makes CP non-trivial to use in practice, is the branching decision, directing how the search space is explored. In this work, we propose a general and hybrid approach, based on DRL and CP, for solving combinatorial optimization problems. The core of our approach is based on a dynamic programming formulation, that acts as a bridge between both techniques. We experimentally show that our solver is efficient to solve two challenging problems: the traveling salesman problem with time windows, and the 4-moments portfolio optimization problem. Results obtained show that the framework introduced outperforms the stand-alone RL and CP solutions, while being competitive with industrial solvers.

With the constant increase in demand for data connectivity, network service providers are faced with the task of reducing their capital and operational expenses while ensuring continual improvements to network performance. Although Network Function Virtualization (NFV) has been identified as a solution, several challenges must be addressed to ensure its feasibility. In this paper, we present a machine learning-based solution to the Virtual Network Function (VNF) placement problem. This paper proposes the Depth-Optimized Delay-Aware Tree (DO-DAT) model by using the particle swarm optimization technique to optimize decision tree hyper-parameters. Using the Evolved Packet Core (EPC) as a use case, we evaluate the performance of the model and compare it to a previously proposed model and a heuristic placement strategy.

To more flexibly balance between exploration and exploitation, a new meta-heuristic method based on Uncertainty Principle concepts is proposed in this paper. UP is is proved effective in multiple branches of science. In the branch of quantum mechanics, canonically conjugate observables such as position and momentum cannot both be distinctly determined in any quantum state. In the same manner, the branch of Spectral filtering design implies that a nonzero function and its Fourier transform cannot both be sharply localized. After delving into such concepts on Uncertainty Principle and their variations in quantum physics, Fourier analysis, and wavelet design, the proposed framework is described in terms of algorithm and flowchart. Our proposed optimizer's idea is based on an inherent uncertainty in performing local search versus global solution search. A set of compatible metrics for each part of the framework is proposed to derive preferred form of algorithm. Evaluations and comparisons at the end of paper show competency and distinct capability of the algorithm over some of the well-known and recently proposed metaheuristics.

To maximize cumulative user engagement (e.g. cumulative clicks) in sequential recommendation, it is often needed to tradeoff two potentially conflicting objectives, that is, pursuing higher immediate user engagement (e.g., click-through rate) and encouraging user browsing (i.e., more items exposured). Existing works often study these two tasks separately, thus tend to result in sub-optimal results. In this paper, we study this problem from an online optimization perspective, and propose a flexible and practical framework to explicitly tradeoff longer user browsing length and high immediate user engagement. Specifically, by considering items as actions, user's requests as states and user leaving as an absorbing state, we formulate each user's behavior as a personalized Markov decision process (MDP), and the problem of maximizing cumulative user engagement is reduced to a stochastic shortest path (SSP) problem. Meanwhile, with immediate user engagement and quit probability estimation, it is shown that the SSP problem can be efficiently solved via dynamic programming. Experiments on real-world datasets demonstrate the effectiveness of the proposed approach. Moreover, this approach is deployed at a large E-commerce platform, achieved over 7% improvement of cumulative clicks.

We propose a new technique to accelerate algorithms based on Gradient Descent using Carath\'eodory's Theorem. In the case of the standard Gradient Descent algorithm, we analyse the theoretical convergence of the approach under convexity assumptions and empirically display its ameliorations. As a core contribution, we then present an application of the acceleration technique to Block Coordinate Descent methods. Experimental comparisons on least squares regression with a LASSO regularisation term show remarkably improved performance on LASSO than the ADAM and SAG algorithms.

We recall that if $A$ is an invertible and symmetric real $m\times m$ matrix, then it is diagonalisable. Therefore, if we denote by $\mathcal{E}^{+}(A)\subset \mathbb{R}^m$ (respectively $\mathcal{E}^{-}(A)\subset \mathbb{R}^m$) to be the vector subspace generated by eigenvectors with positive eigenvalues of $A$ (correspondingly the vector subspace generated by eigenvectors with negative eigenvalues of $A$), then we have an orthogonal decomposition $\mathbb{R}^m=\mathcal{E}^{+}(A)\oplus \mathcal{E}^{-}(A)$. Hence, every $x\in \mathbb{R}^m$ can be written uniquely as $x=pr_{A,+}(x)+pr_{A,-}(x)$ with $pr_{A,+}(x)\in \mathcal{E}^{+}(A)$ and $pr_{A,-}(x)\in \mathcal{E}^{-}(A)$. We propose the following simple new modification of quasi-Newton's methods. {\bf New Q-Newton's method.} Let $\Delta =\{\delta _0,\delta _1,\delta _2,\ldots \}$ be a countable set of real numbers which has at least $m+1$ elements. Let $f:\mathbb{R}^m\rightarrow \mathbb{R}$ be a $C^2$ function. Let $\alpha >0$. For each $x\in \mathbb{R}^m$ such that $\nabla f(x)\not=0$, let $\delta (x)=\delta _j$, where $j$ is the smallest number so that $\nabla ^2f(x)+\delta _j||\nabla f(x)||^{1+\alpha}Id$ is invertible. (If $\nabla f(x)=0$, then we choose $\delta (x)=\delta _0$.) Let $x_0\in \mathbb{R}^m$ be an initial point. We define a sequence of $x_n\in \mathbb{R}^m$ and invertible and symmetric $m\times m$ matrices $A_n$ as follows: $A_n=\nabla ^2f(x_n)+\delta (x_n) ||\nabla f(x_n)||^{1+\alpha}Id$ and $x_{n+1}=x_n-w_n$, where $w_n=pr_{A_n,+}(v_n)-pr_{A_n,-}(v_n)$ and $v_n=A_n^{-1}\nabla f(x_n)$. The main result of this paper roughly says that if $f$ is $C^3$ and a sequence $\{x_n\}$, constructed by the New Q-Newton's method from a random initial point $x_0$, {\bf converges}, then the limit point is not a saddle point, and the convergence rate is the same as that of Newton's method.

Stochastic optimization algorithms have become indispensable in modern machine learning. An unresolved foundational question in this area is the difference between with-replacement sampling and without-replacement sampling -- does the latter have superior convergence rate compared to the former? A groundbreaking result of Recht and R\'e reduces the problem to a noncommutative analogue of the arithmetic-geometric mean inequality where $n$ positive numbers are replaced by $n$ positive definite matrices. If this inequality holds for all $n$, then without-replacement sampling indeed outperforms with-replacement sampling. The conjectured Recht-R\'e inequality has so far only been established for $n = 2$ and a special case of $n = 3$. We will show that the Recht-R\'e conjecture is false for general $n$. Our approach relies on the noncommutative Positivstellensatz, which allows us to reduce the conjectured inequality to a semidefinite program and the validity of the conjecture to certain bounds for the optimum values, which we show are false as soon as $n = 5$.

We introduce AdaHessian, a second order stochastic optimization algorithm which dynamically incorporates the curvature of the loss function via ADAptive estimates of the Hessian. Second order algorithms are among the most powerful optimization algorithms with superior convergence properties as compared to first order methods such as SGD and ADAM. The main disadvantage of traditional second order methods is their heavier per-iteration computation and poor accuracy as compared to first order methods. To address these, we incorporate several novel approaches in AdaHessian, including: (i) a new variance reduction estimate of the Hessian diagonal with low computational overhead; (ii) a root-mean-square exponential moving average to smooth out variations of the Hessian diagonal across different iterations; and (iii) a block diagonal averaging to reduce the variance of Hessian diagonal elements. We show that AdaHessian achieves new state-of-the-art results by a large margin as compared to other adaptive optimization methods, including variants of ADAM. In particular, we perform extensive tests on CV, NLP, and recommendation system tasks and find that AdaHessian: (i) achieves 1.80\%/1.45\% higher accuracy on ResNets20/32 on Cifar10, and 5.55\% higher accuracy on ImageNet as compared to ADAM; (ii) outperforms ADAMW for transformers by 0.27/0.33 BLEU score on IWSLT14/WMT14 and 1.8/1.0 PPL on PTB/Wikitext-103; and (iii) achieves 0.032\% better score than AdaGrad for DLRM on the Criteo Ad Kaggle dataset. Importantly, we show that the cost per iteration of AdaHessian is comparable to first-order methods, and that it exhibits robustness towards its hyperparameters. The code for AdaHessian is open-sourced and publicly available.

Real-Time Bidding (RTB) display advertising is a method for purchasing display advertising inventory in auctions that occur within milliseconds. The performance of RTB campaigns is generally measured with a series of Key Performance Indicators (KPIs) - measurements used to ensure that the campaign is cost-effective and that it is purchasing valuable inventory. While an RTB campaign should ideally meet all KPIs, simultaneous improvement tends to be very challenging, as an improvement to any one KPI risks a detrimental effect toward the others. Here we present an approach to simultaneously controlling multiple KPIs with a PID-based feedback-control system. This method generates a control score for each KPI, based on both the output of a PID controller module and a metric that quantifies the importance of each KPI for internal business needs. On regular intervals, this algorithm - Sequential Control - will choose the KPI with the greatest overall need for improvement. In this way, our algorithm is able to continually seek the greatest marginal improvements to its current state. Multiple methods of control can be associated with each KPI, and can be triggered either simultaneously or chosen stochastically, in order to avoid local optima. In both offline ad bidding simulations and testing on live traffic, our methods proved to be effective in simultaneously controlling multiple KPIs, and bringing them toward their respective goals.