The fluctuation effect of gradient expectation and variance caused by parameter update between consecutive iterations is neglected or confusing by current mainstream gradient optimization algorithms.Using this fluctuation effect, combined with the stratified sampling strategy, this paper designs a novel \underline{M}emory \underline{S}tochastic s\underline{T}ratified Gradient Descend(\underline{MST}GD) algorithm with an exponential convergence rate. Specifically, MSTGD uses two strategies for variance reduction: the first strategy is to perform variance reduction according to the proportion p of used historical gradient, which is estimated from the mean and variance of sample gradients before and after iteration, and the other strategy is stratified sampling by category. The statistic \ $\bar{G}_{mst}$\ designed under these two strategies can be adaptively unbiased, and its variance decays at a geometric rate. This enables MSTGD based on $\bar{G}_{mst}$ to obtain an exponential convergence rate of the form $\lambda^{2(k-k_0)}$($\lambda\in (0,1)$,k is the number of iteration steps,$\lambda$ is a variable related to proportion p).Unlike most other algorithms that claim to achieve an exponential convergence rate, the convergence rate is independent of parameters such as dataset size N, batch size n, etc., and can be achieved at a constant step size.Theoretical and experimental results show the effectiveness of MSTGD

The optimistic gradient method has seen increasing popularity as an efficient first-order method for solving convex-concave saddle point problems. To analyze its iteration complexity, a recent work [arXiv:1901.08511] proposed an interesting perspective that interprets the optimistic gradient method as an approximation to the proximal point method. In this paper, we follow this approach and distill the underlying idea of optimism to propose a generalized optimistic method, which encompasses the optimistic gradient method as a special case. Our general framework can handle constrained saddle point problems with composite objective functions and can work with arbitrary norms with compatible Bregman distances. Moreover, we also develop an adaptive line search scheme to select the stepsizes without knowledge of the smoothness coefficients. We instantiate our method with first-order, second-order and higher-order oracles and give sharp global iteration complexity bounds. When the objective function is convex-concave, we show that the averaged iterates of our $p$-th-order method ($p\geq 1$) converge at a rate of $\mathcal{O}(1/N^\frac{p+1}{2})$. When the objective function is further strongly-convex-strongly-concave, we prove a complexity bound of $\mathcal{O}(\frac{L_1}{\mu}\log\frac{1}{\epsilon})$ for our first-order method and a bound of $\mathcal{O}((L_p D^\frac{p-1}{2}/\mu)^{\frac{2}{p+1}}+\log\log\frac{1}{\epsilon})$ for our $p$-th-order method ($p\geq 2$) respectively, where $L_p$ ($p\geq 1$) is the Lipschitz constant of the $p$-th-order derivative, $\mu$ is the strongly-convex parameter, and $D$ is the initial Bregman distance to the saddle point. Moreover, our line search scheme provably only requires an almost constant number of calls to a subproblem solver per iteration on average, making our first-order and second-order methods particularly amenable to implementation.

We consider several hill-climbing approaches to clustering as formulated by Fukunaga and Hostetler in the 1970's. We study both continuous-space and discrete-space (i.e., medoid) variants and establish their consistency.

Distance measures provide the foundation for many popular algorithms in Machine Learning and Pattern Recognition. Different notions of distance can be used depending on the types of the data the algorithm is working on. For graph-shaped data, an important notion is the Graph Edit Distance (GED) that measures the degree of (dis)similarity between two graphs in terms of the operations needed to make them identical. As the complexity of computing GED is the same as NP-hard problems, it is reasonable to consider approximate solutions. In this paper we present a QUBO formulation of the GED problem. This allows us to implement two different approaches, namely quantum annealing and variational quantum algorithms that run on the two types of quantum hardware currently available: quantum annealer and gate-based quantum computer, respectively. Considering the current state of noisy intermediate-scale quantum computers, we base our study on proof-of-principle tests of their performance.

Multi-Object Tracking (MOT) is most often approached in the tracking-by-detection paradigm, where object detections are associated through time. The association step naturally leads to discrete optimization problems. As these optimization problems are often NP-hard, they can only be solved exactly for small instances on current hardware. Adiabatic quantum computing (AQC) offers a solution for this, as it has the potential to provide a considerable speedup on a range of NP-hard optimization problems in the near future. However, current MOT formulations are unsuitable for quantum computing due to their scaling properties. In this work, we therefore propose the first MOT formulation designed to be solved with AQC. We employ an Ising model that represents the quantum mechanical system implemented on the AQC. We show that our approach is competitive compared with state-of-the-art optimization-based approaches, even when using of-the-shelf integer programming solvers. Finally, we demonstrate that our MOT problem is already solvable on the current generation of real quantum computers for small examples, and analyze the properties of the measured solutions.

Bayesian optimization (BO) is a sample-efficient approach for tuning design parameters to optimize expensive-to-evaluate, black-box performance metrics. In many manufacturing processes, the design parameters are subject to random input noise, resulting in a product that is often less performant than expected. Although BO methods have been proposed for optimizing a single objective under input noise, no existing method addresses the practical scenario where there are multiple objectives that are sensitive to input perturbations. In this work, we propose the first multi-objective BO method that is robust to input noise. We formalize our goal as optimizing the multivariate value-at-risk (MVaR), a risk measure of the uncertain objectives. Since directly optimizing MVaR is computationally infeasible in many settings, we propose a scalable, theoretically-grounded approach for optimizing MVaR using random scalarizations. Empirically, we find that our approach significantly outperforms alternative methods and efficiently identifies optimal robust designs that will satisfy specifications across multiple metrics with high probability.

This article was published as a part of the Data Science Blogathon. "True optimization is the revolutionary contribution of modern research to decision processes" – George Dantzig. We will find an optimal value for a linear equation with different linear constraints. We can solve linear programming using the PuLP library, and other Python libraries that can perform similar tasks are SciPy, Pyomo, CVXOPT. Let's discuss associated concepts and problem statements in detail.

This paper proposes a new regularization technique for reinforcement learning (RL) towards making policy and value functions smooth and stable. RL is known for the instability of the learning process and the sensitivity of the acquired policy to noise. Several methods have been proposed to resolve these problems, and in summary, the smoothness of policy and value functions learned mainly in RL contributes to these problems. However, if these functions are extremely smooth, their expressiveness would be lost, resulting in not obtaining the global optimal solution. This paper therefore considers RL under local Lipschitz continuity constraint, so-called L2C2. By designing the spatio-temporal locally compact space for L2C2 from the state transition at each time step, the moderate smoothness can be achieved without loss of expressiveness. Numerical noisy simulations verified that the proposed L2C2 outperforms the task performance while smoothing out the robot action generated from the learned policy.

Residential consumers have become active participants in the power distribution network after being equipped with residential EV charging provisions. This creates a challenge for the network operator tasked with dispatching electric power to the residential consumers through the existing distribution network infrastructure in a reliable manner. In this paper, we address the problem of scheduling residential EV charging for multiple consumers while maintaining network reliability. An additional challenge is the restricted exchange of information: where the consumers do not have access to network information and the network operator does not have access to consumer load parameters. We propose a distributed framework which generates an optimal EV charging schedule for individual residential consumers based on their preferences and iteratively updates it until the network reliability constraints set by the operator are satisfied. We validate the proposed approach for different EV adoption levels in a synthetically created digital twin of an actual power distribution network. The results demonstrate that the new approach can achieve a higher level of network reliability compared to the case where residential consumers charge EVs based solely on their individual preferences, thus providing a solution for the existing grid to keep up with increased adoption rates without significant investments in increasing grid capacity.

This work shows that a diverse collection of linear optimization methods, when run on general data, fail to overfit, despite lacking any explicit constraints or regularization: with high probability, their trajectories stay near the curve of optimal constrained solutions over the population distribution. This analysis is powered by an elementary but flexible proof scheme which can handle many settings, summarized as follows. Firstly, the data can be general: unlike other implicit bias works, it need not satisfy large margin or other structural conditions, and moreover can arrive sequentially IID, sequentially following a Markov chain, as a batch, and lastly it can have heavy tails. Secondly, while the main analysis is for mirror descent, rates are also provided for the Temporal-Difference fixed-point method from reinforcement learning; all prior high probability analyses in these settings required bounded iterates, bounded updates, bounded noise, or some equivalent. Thirdly, the losses are general, and for instance the logistic and squared losses can be handled simultaneously, unlike other implicit bias works. In all of these settings, not only is low population error guaranteed with high probability, but moreover low sample complexity is guaranteed so long as there exists any low-complexity near-optimal solution, even if the global problem structure and in particular global optima have high complexity.