Isn't it great to solve the Multi-Objective optimization problem of Goal Programming using "R"? If yes, then here is something which you can vouch for. In this write-up, I am going to deep dive into how can we solve the Multi-Objective Optimization Problem when we have the model already developed at hand? I will make use of the'R' language to find the optimal solutions given the developed model, its objectives, constraints, and deviational variables. I will further elaborate on the solution to make you understand every aspect of it.

We propose a new approach to SAT solving which solves SAT problems in vector spaces as a cost minimization problem of a non-negative differentiable cost function J^sat. In our approach, a solution, i.e., satisfying assignment, for a SAT problem in n variables is represented by a binary vector u in {0,1}^n that makes J^sat(u) zero. We search for such u in a vector space R^n by cost minimization, i.e., starting from an initial u_0 and minimizing J to zero while iteratively updating u by Newton's method. We implemented our approach as a matrix-based differential SAT solver MatSat. Although existing main-stream SAT solvers decide each bit of a solution assignment one by one, be they of conflict driven clause learning (CDCL) type or of stochastic local search (SLS) type, MatSat fundamentally differs from them in that it continuously approach a solution in a vector space. We conducted an experiment to measure the scalability of MatSat with random 3-SAT problems in which MatSat could find a solution up to n=10^5 variables. We also compared MatSat with four state-of-the-art SAT solvers including winners of SAT competition 2018 and SAT Race 2019 in terms of time for finding a solution, using a random benchmark set from SAT 2018 competition and an artificial random 3-SAT instance set. The result shows that MatSat comes in second in both test sets and outperforms all the CDCL type solvers.

Many scientific conferences employ a two-phase paper review process, where some papers are assigned additional reviewers after the initial reviews are submitted. Many conferences also design and run experiments on their paper review process, where some papers are assigned reviewers who provide reviews under an experimental condition. In this paper, we consider the question: how should reviewers be divided between phases or conditions in order to maximize total assignment similarity? We make several contributions towards answering this question. First, we prove that when the set of papers requiring additional review is unknown, a simplified variant of this problem is NP-hard. Second, we empirically show that across several datasets pertaining to real conference data, dividing reviewers between phases/conditions uniformly at random allows an assignment that is nearly as good as the oracle optimal assignment. This uniformly random choice is practical for both the two-phase and conference experiment design settings. Third, we provide explanations of this phenomenon by providing theoretical bounds on the suboptimality of this random strategy under certain natural conditions. From these easily-interpretable conditions, we provide actionable insights to conference program chairs about whether a random reviewer split is suitable for their conference.

An easy to follow tutorial with beginners concepts of optimization, constrained optimization, feasible region and its importance in machine learning.

In this letter, we study the problem of autonomously placing a teddy bear on a previously unseen chair for sitting. To achieve this goal, we present a novel method for robots to imagine the sitting pose of the bear by physically simulating a virtual humanoid agent sitting on the chair. We also develop a robotic system which leverages motion planning to plan SE(2) motions for a humanoid robot to walk to the chair and whole-body motions to put the bear on it, respectively. Furthermore, to cope with the cases where the chair is not in an accessible pose for placing the bear, a human-robot interaction (HRI) framework is introduced in which a human follows language instructions given by the robot to rotate the chair and help make the chair accessible. We implement our method with a robot arm and a humanoid robot. We calibrate the proposed system with 3 chairs and test on 12 previously unseen chairs in both accessible and inaccessible poses extensively. Results show that our method enables the robot to autonomously put the teddy bear on the 12 unseen chairs with a very high success rate. The HRI framework is also shown to be very effective in changing the accessibility of the chair. Source code will be available. Video demos are available at https://chirikjianlab.github.io/putbearonchair/.

Bayesian optimization (BO) is a flexible and powerful framework that is suitable for computationally expensive simulation-based applications and guarantees statistical convergence to the global optimum. While remaining as one of the most popular optimization methods, its capability is hindered by the size of data, the dimensionality of the considered problem, and the nature of sequential optimization. These scalability issues are intertwined with each other and must be tackled simultaneously. In this work, we propose the Scalable$^3$-BO framework, which employs sparse GP as the underlying surrogate model to scope with Big Data and is equipped with a random embedding to efficiently optimize high-dimensional problems with low effective dimensionality. The Scalable$^3$-BO framework is further leveraged with asynchronous parallelization feature, which fully exploits the computational resource on HPC within a computational budget. As a result, the proposed Scalable$^3$-BO framework is scalable in three independent perspectives: with respect to data size, dimensionality, and computational resource on HPC. The goal of this work is to push the frontiers of BO beyond its well-known scalability issues and minimize the wall-clock waiting time for optimizing high-dimensional computationally expensive applications. We demonstrate the capability of Scalable$^3$-BO with 1 million data points, 10,000-dimensional problems, with 20 concurrent workers in an HPC environment.

Over-parameterized deep networks trained using gradient-based optimizers are a popular choice for solving classification and ranking problems. Without appropriately tuned $\ell_2$ regularization or weight decay, such networks have the tendency to make output scores (logits) and network weights large, causing training loss to become too small and the network to lose its adaptivity (ability to move around) in the parameter space. Although regularization is typically understood from an overfitting perspective, we highlight its role in making the network more adaptive and enabling it to escape more easily from weights that generalize poorly. To provide such a capability, we propose a method called Logit Attenuating Weight Normalization (LAWN), that can be stacked onto any gradient-based optimizer. LAWN controls the logits by constraining the weight norms of layers in the final homogeneous sub-network. Empirically, we show that the resulting LAWN variant of the optimizer makes a deep network more adaptive to finding minimas with superior generalization performance on large-scale image classification and recommender systems. While LAWN is particularly impressive in improving Adam, it greatly improves all optimizers when used with large batch sizes

We use functional mirror ascent to propose a general framework (referred to as FMA-PG) for designing policy gradient methods. The functional perspective distinguishes between a policy's functional representation (what are its sufficient statistics) and its parameterization (how are these statistics represented) and naturally results in computationally efficient off-policy updates. For simple policy parameterizations, the FMA-PG framework ensures that the optimal policy is a fixed point of the updates. It also allows us to handle complex policy parameterizations (e.g., neural networks) while guaranteeing policy improvement. Our framework unifies several PG methods and opens the way for designing sample-efficient variants of existing methods. Moreover, it recovers important implementation heuristics (e.g., using forward vs reverse KL divergence) in a principled way. With a softmax functional representation, FMA-PG results in a variant of TRPO with additional desirable properties. It also suggests an improved variant of PPO, whose robustness and efficiency we empirically demonstrate on MuJoCo. Via experiments on simple reinforcement learning problems, we evaluate algorithms instantiated by FMA-PG.

We introduce Pathfinder, a variational method for approximately sampling from differentiable log densities. Starting from a random initialization, Pathfinder locates normal approximations to the target density along a quasi-Newton optimization path, with local covariance estimated using the inverse Hessian estimates produced by the optimizer. Pathfinder returns draws from the approximation with the lowest estimated Kullback-Leibler (KL) divergence to the true posterior. We evaluate Pathfinder on a wide range of posterior distributions, demonstrating that its approximate draws are better than those from automatic differentiation variational inference (ADVI) and comparable to those produced by short chains of dynamic Hamiltonian Monte Carlo (HMC), as measured by 1-Wasserstein distance. Compared to ADVI and short dynamic HMC runs, Pathfinder requires one to two orders of magnitude fewer log density and gradient evaluations, with greater reductions for more challenging posteriors. Importance resampling over multiple runs of Pathfinder improves the diversity of approximate draws, reducing 1-Wasserstein distance further and providing a measure of robustness to optimization failures on plateaus, saddle points, or in minor modes. The Monte Carlo KL-divergence estimates are embarrassingly parallelizable in the core Pathfinder algorithm, as are multiple runs in the resampling version, further increasing Pathfinder's speed advantage with multiple cores.

Gradient descent (GD) type optimization methods are the standard instrument to train artificial neural networks (ANNs) with rectified linear unit (ReLU) activation. Despite the great success of GD type optimization methods in numerical simulations for the training of ANNs with ReLU activation, it remains - even in the simplest situation of the plain vanilla GD optimization method with random initializations and ANNs with one hidden layer - an open problem to prove (or disprove) the conjecture that the risk of the GD optimization method converges in the training of such ANNs to zero as the width of the ANNs, the number of independent random initializations, and the number of GD steps increase to infinity. In this article we prove this conjecture in the situation where the probability distribution of the input data is equivalent to the continuous uniform distribution on a compact interval, where the probability distributions for the random initializations of the ANN parameters are standard normal distributions, and where the target function under consideration is continuous and piecewise affine linear. Roughly speaking, the key ingredients in our mathematical convergence analysis are (i) to prove that suitable sets of global minima of the risk functions are \emph{twice continuously differentiable submanifolds of the ANN parameter spaces}, (ii) to prove that the Hessians of the risk functions on these sets of global minima satisfy an appropriate \emph{maximal rank condition}, and, thereafter, (iii) to apply the machinery in [Fehrman, B., Gess, B., Jentzen, A., Convergence rates for the stochastic gradient descent method for non-convex objective functions. J. Mach. Learn. Res. 21(136): 1--48, 2020] to establish convergence of the GD optimization method with random initializations.