It has been shown that Maximum Satisfiability (MaxSAT) problem instances can be effectively solved by partitioning the set of soft clauses into several disjoint sets. The partitioning methods can be based on clause weights (e.g., stratification) or based on graph representations of the formula. Afterwards, a merge procedure is applied to guarantee that an optimal solution is found. This paper proposes a new framework called UpMax that decouples the partitioning procedure from the MaxSAT solving algorithms. As a result, new partitioning procedures can be defined independently of the MaxSAT algorithm to be used. Moreover, this decoupling also allows users that build new MaxSAT formulas to propose partition schemes based on knowledge of the problem to be solved. We illustrate this approach using several problems and show that partitioning has a large impact on the performance of unsatisfiability-based MaxSAT algorithms.
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In many industrial applications, obtaining labeled observations is not straightforward as it often requires the intervention of human experts or the use of expensive testing equipment. In these circumstances, active learning can be highly beneficial in suggesting the most informative data points to be used when fitting a model. Reducing the number of observations needed for model development alleviates both the computational burden required for training and the operational expenses related to labeling. Online active learning, in particular, is useful in high-volume production processes where the decision about the acquisition of the label for a data point needs to be taken within an extremely short time frame. However, despite the recent efforts to develop online active learning strategies, the behavior of these methods in the presence of outliers has not been thoroughly examined. In this work, we investigate the performance of online active linear regression in contaminated data streams. Our study shows that the currently available query strategies are prone to sample outliers, whose inclusion in the training set eventually degrades the predictive performance of the models. To address this issue, we propose a solution that bounds the search area of a conditional D-optimal algorithm and uses a robust estimator. Our approach strikes a balance between exploring unseen regions of the input space and protecting against outliers. Through numerical simulations, we show that the proposed method is effective in improving the performance of online active learning in the presence of outliers, thus expanding the potential applications of this powerful tool.
Safety is one of the fundamental challenges in control theory. Recently, multi-step optimal control problems for discrete-time dynamical systems were formulated to enforce stability, while subject to input constraints as well as safety-critical requirements using discrete-time control barrier functions within a model predictive control (MPC) framework. Existing work usually focus on the feasibility or the safety for the optimization problem, and the majority of the existing work restrict the discussions to relative-degree one control barrier functions. Additionally, the real-time computation is challenging when a large horizon is considered in the MPC problem for relative-degree one or high-order control barrier functions. In this paper, we propose a framework that solves the safety-critical MPC problem in an iterative optimization, which is applicable for any relative-degree control barrier functions. In the proposed formulation, the nonlinear system dynamics as well as the safety constraints modeled as discrete-time high-order control barrier functions (DHOCBF) are linearized at each time step. Our formulation is generally valid for any control barrier function with an arbitrary relative-degree. The advantages of fast computational performance with safety guarantee are analyzed and validated with numerical results.
We consider the problem of query-efficient global max-cut on a weighted undirected graph in the value oracle model examined by [RSW18]. This model arises as a natural special case of submodular function maximization: on query $S \subseteq V$, the oracle returns the total weight of the cut between $S$ and $V \backslash S$. For most constants $c \in (0,1]$, we nail down the query complexity of achieving a $c$-approximation, for both deterministic and randomized algorithms (up to logarithmic factors). Analogously to general submodular function maximization in the same model, we observe a phase transition at $c = 1/2$: we design a deterministic algorithm for global $c$-approximate max-cut in $O(\log n)$ queries for any $c < 1/2$, and show that any randomized algorithm requires $\tilde{\Omega}(n)$ queries to find a $c$-approximate max-cut for any $c > 1/2$. Additionally, we show that any deterministic algorithm requires $\Omega(n^2)$ queries to find an exact max-cut (enough to learn the entire graph), and develop a $\tilde{O}(n)$-query randomized $c$-approximation for any $c < 1$. Our approach provides two technical contributions that may be of independent interest. One is a query-efficient sparsifier for undirected weighted graphs (prior work of [RSW18] holds only for unweighted graphs). Another is an extension of the cut dimension to rule out approximation (prior work of [GPRW20] introducing the cut dimension only rules out exact solutions).
Conversion rate (CVR) prediction plays an important role in advertising systems. Recently, supervised deep neural network-based models have shown promising performance in CVR prediction. However, they are data hungry and require an enormous amount of training data. In online advertising systems, although there are millions to billions of ads, users tend to click only a small set of them and to convert on an even smaller set. This data sparsity issue restricts the power of these deep models. In this paper, we propose the Contrastive Learning for CVR prediction (CL4CVR) framework. It associates the supervised CVR prediction task with a contrastive learning task, which can learn better data representations exploiting abundant unlabeled data and improve the CVR prediction performance. To tailor the contrastive learning task to the CVR prediction problem, we propose embedding masking (EM), rather than feature masking, to create two views of augmented samples. We also propose a false negative elimination (FNE) component to eliminate samples with the same feature as the anchor sample, to account for the natural property in user behavior data. We further propose a supervised positive inclusion (SPI) component to include additional positive samples for each anchor sample, in order to make full use of sparse but precious user conversion events. Experimental results on two real-world conversion datasets demonstrate the superior performance of CL4CVR. The source code is available at //github.com/DongRuiHust/CL4CVR.
Recent years have seen many insights on deep learning optimisation being brought forward by finding implicit regularisation effects of commonly used gradient-based optimisers. Understanding implicit regularisation can not only shed light on optimisation dynamics, but it can also be used to improve performance and stability across problem domains, from supervised learning to two-player games such as Generative Adversarial Networks. An avenue for finding such implicit regularisation effects has been quantifying the discretisation errors of discrete optimisers via continuous-time flows constructed by backward error analysis (BEA). The current usage of BEA is not without limitations, since not all the vector fields of continuous-time flows obtained using BEA can be written as a gradient, hindering the construction of modified losses revealing implicit regularisers. In this work, we provide a novel approach to use BEA, and show how our approach can be used to construct continuous-time flows with vector fields that can be written as gradients. We then use this to find previously unknown implicit regularisation effects, such as those induced by multiple stochastic gradient descent steps while accounting for the exact data batches used in the updates, and in generally differentiable two-player games.
This study investigates a class of initial-boundary value problems pertaining to the time-fractional mixed sub-diffusion and diffusion-wave equation (SDDWE). To facilitate the development of a numerical method and analysis, the original problem is transformed into a new integro-differential model which includes the Caputo derivatives and the Riemann-Liouville fractional integrals with orders belonging to (0,1). By providing an a priori estimate of the solution, we have established the existence and uniqueness of a numerical solution for the problem. We propose a second-order method to approximate the fractional Riemann-Liouville integral and employ an L2 type formula to approximate the Caputo derivative. This results in a method with a temporal accuracy of second-order for approximating the considered model. The proof of the unconditional stability of the proposed difference scheme is established. Moreover, we demonstrate the proposed method's potential to construct and analyze a second-order L2-type numerical scheme for a broader class of the time-fractional mixed SDDWEs with multi-term time-fractional derivatives. Numerical results are presented to assess the accuracy of the method and validate the theoretical findings.
The state of the art related to parameter correlation in two-parameter models has been reviewed in this paper. The apparent contradictions between the different authors regarding the ability of D--optimality to simultaneously reduce the correlation and the area of the confidence ellipse in two-parameter models were analyzed. Two main approaches were found: 1) those who consider that the optimality criteria simultaneously control the precision and correlation of the parameter estimators; and 2) those that consider a combination of criteria to achieve the same objective. An analytical criterion combining in its structure both the optimality of the precision of the estimators of the parameters and the reduction of the correlation between their estimators is provided. The criterion was tested both in a simple linear regression model, considering all possible design spaces, and in a non-linear model with strong correlation of the estimators of the parameters (Michaelis--Menten) to show its performance. This criterion showed a superior behavior to all the strategies and criteria to control at the same time the precision and the correlation.
In this paper, we provide a novel framework for the analysis of generalization error of first-order optimization algorithms for statistical learning when the gradient can only be accessed through partial observations given by an oracle. Our analysis relies on the regularity of the gradient w.r.t. the data samples, and allows to derive near matching upper and lower bounds for the generalization error of multiple learning problems, including supervised learning, transfer learning, robust learning, distributed learning and communication efficient learning using gradient quantization. These results hold for smooth and strongly-convex optimization problems, as well as smooth non-convex optimization problems verifying a Polyak-Lojasiewicz assumption. In particular, our upper and lower bounds depend on a novel quantity that extends the notion of conditional standard deviation, and is a measure of the extent to which the gradient can be approximated by having access to the oracle. As a consequence, our analysis provides a precise meaning to the intuition that optimization of the statistical learning objective is as hard as the estimation of its gradient. Finally, we show that, in the case of standard supervised learning, mini-batch gradient descent with increasing batch sizes and a warm start can reach a generalization error that is optimal up to a multiplicative factor, thus motivating the use of this optimization scheme in practical applications.
We consider the problem of latent bandits with cluster structure where there are multiple users, each with an associated multi-armed bandit problem. These users are grouped into \emph{latent} clusters such that the mean reward vectors of users within the same cluster are identical. At each round, a user, selected uniformly at random, pulls an arm and observes a corresponding noisy reward. The goal of the users is to maximize their cumulative rewards. This problem is central to practical recommendation systems and has received wide attention of late \cite{gentile2014online, maillard2014latent}. Now, if each user acts independently, then they would have to explore each arm independently and a regret of $\Omega(\sqrt{\mathsf{MNT}})$ is unavoidable, where $\mathsf{M}, \mathsf{N}$ are the number of arms and users, respectively. Instead, we propose LATTICE (Latent bAndiTs via maTrIx ComplEtion) which allows exploitation of the latent cluster structure to provide the minimax optimal regret of $\widetilde{O}(\sqrt{(\mathsf{M}+\mathsf{N})\mathsf{T}})$, when the number of clusters is $\widetilde{O}(1)$. This is the first algorithm to guarantee such strong regret bound. LATTICE is based on a careful exploitation of arm information within a cluster while simultaneously clustering users. Furthermore, it is computationally efficient and requires only $O(\log{\mathsf{T}})$ calls to an offline matrix completion oracle across all $\mathsf{T}$ rounds.
A mainstream type of current self-supervised learning methods pursues a general-purpose representation that can be well transferred to downstream tasks, typically by optimizing on a given pretext task such as instance discrimination. In this work, we argue that existing pretext tasks inevitably introduce biases into the learned representation, which in turn leads to biased transfer performance on various downstream tasks. To cope with this issue, we propose Maximum Entropy Coding (MEC), a more principled objective that explicitly optimizes on the structure of the representation, so that the learned representation is less biased and thus generalizes better to unseen downstream tasks. Inspired by the principle of maximum entropy in information theory, we hypothesize that a generalizable representation should be the one that admits the maximum entropy among all plausible representations. To make the objective end-to-end trainable, we propose to leverage the minimal coding length in lossy data coding as a computationally tractable surrogate for the entropy, and further derive a scalable reformulation of the objective that allows fast computation. Extensive experiments demonstrate that MEC learns a more generalizable representation than previous methods based on specific pretext tasks. It achieves state-of-the-art performance consistently on various downstream tasks, including not only ImageNet linear probe, but also semi-supervised classification, object detection, instance segmentation, and object tracking. Interestingly, we show that existing batch-wise and feature-wise self-supervised objectives could be seen equivalent to low-order approximations of MEC. Code and pre-trained models are available at //github.com/xinliu20/MEC.
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