Optimization problems involving mixed variables, i.e., variables of numerical and categorical nature, can be challenging to solve, especially in the presence of complex constraints. Moreover, when the objective function is the result of a complicated simulation or experiment, it may be expensive to evaluate. This paper proposes a novel surrogate-based global optimization algorithm to solve linearly constrained mixed-variable problems up to medium-large size (around 100 variables after encoding and 20 constraints) based on constructing a piecewise affine surrogate of the objective function over feasible samples. We introduce two types of exploration functions to efficiently search the feasible domain via mixed-integer linear programming solvers. We also provide a preference-based version of the algorithm, which can be used when only pairwise comparisons between samples can be acquired while the underlying objective function to minimize remains unquantified. The two algorithms are tested on mixed-variable benchmark problems with and without constraints. The results show that, within a small number of acquisitions, the proposed algorithms can often achieve better or comparable results than other existing methods.

To train image-caption retrieval (ICR) methods, contrastive loss functions are a common choice for optimization functions. Unfortunately, contrastive ICR methods are vulnerable to predictive feature suppression. Predictive features are features that correctly indicate the similarity between a query and a candidate item. However, in the presence of multiple predictive features during training, encoder models tend to suppress redundant predictive features, since these features are not needed to learn to discriminate between positive and negative pairs. While some predictive features are redundant during training, these features might be relevant during evaluation. We introduce an approach to reduce predictive feature suppression for resource-constrained ICR methods: latent target decoding (LTD). We add an additional decoder to the contrastive ICR framework, to reconstruct the input caption in a latent space of a general-purpose sentence encoder, which prevents the image and caption encoder from suppressing predictive features. We implement the LTD objective as an optimization constraint, to ensure that the reconstruction loss is below a bound value while primarily optimizing for the contrastive loss. Importantly, LTD does not depend on additional training data or expensive (hard) negative mining strategies. Our experiments show that, unlike reconstructing the input caption in the input space, LTD reduces predictive feature suppression, measured by obtaining higher recall@k, r-precision, and nDCG scores than a contrastive ICR baseline. Moreover, we show that LTD should be implemented as an optimization constraint instead of a dual optimization objective. Finally, we show that LTD can be used with different contrastive learning losses and a wide variety of resource-constrained ICR methods.

Neural QCFG is a grammar-based sequence-tosequence (seq2seq) model with strong inductive biases on hierarchical structures. It excels in interpretability and generalization but suffers from expensive inference. In this paper, we study two low-rank variants of Neural QCFG for faster inference with different trade-offs between efficiency and expressiveness. Furthermore, utilizing the symbolic interface provided by the grammar, we introduce two soft constraints over tree hierarchy and source coverage. We experiment with various datasets and find that our models outperform vanilla Neural QCFG in most settings.

Data pipelines are widely employed in modern enterprises to power a variety of Machine-Learning (ML) and Business-Intelligence (BI) applications. Crucially, these pipelines are \emph{recurring} (e.g., daily or hourly) in production settings to keep data updated so that ML models can be re-trained regularly, and BI dashboards refreshed frequently. However, data quality (DQ) issues can often creep into recurring pipelines because of upstream schema and data drift over time. As modern enterprises operate thousands of recurring pipelines, today data engineers have to spend substantial efforts to \emph{manually} monitor and resolve DQ issues, as part of their DataOps and MLOps practices. Given the high human cost of managing large-scale pipeline operations, it is imperative that we can \emph{automate} as much as possible. In this work, we propose Auto-Validate-by-History (AVH) that can automatically detect DQ issues in recurring pipelines, leveraging rich statistics from historical executions. We formalize this as an optimization problem, and develop constant-factor approximation algorithms with provable precision guarantees. Extensive evaluations using 2000 production data pipelines at Microsoft demonstrate the effectiveness and efficiency of AVH.

We present a unified hard-constraint framework for solving geometrically complex PDEs with neural networks, where the most commonly used Dirichlet, Neumann, and Robin boundary conditions (BCs) are considered. Specifically, we first introduce the "extra fields" from the mixed finite element method to reformulate the PDEs so as to equivalently transform the three types of BCs into linear equations. Based on the reformulation, we derive the general solutions of the BCs analytically, which are employed to construct an ansatz that automatically satisfies the BCs. With such a framework, we can train the neural networks without adding extra loss terms and thus efficiently handle geometrically complex PDEs, alleviating the unbalanced competition between the loss terms corresponding to the BCs and PDEs. We theoretically demonstrate that the "extra fields" can stabilize the training process. Experimental results on real-world geometrically complex PDEs showcase the effectiveness of our method compared with state-of-the-art baselines.

Many practical settings call for the reconstruction of temporal signals from corrupted or missing data. Classic examples include decoding, tracking, signal enhancement and denoising. Since the reconstructed signals are ultimately viewed by humans, it is desirable to achieve reconstructions that are pleasing to human perception. Mathematically, perfect perceptual-quality is achieved when the distribution of restored signals is the same as that of natural signals, a requirement which has been heavily researched in static estimation settings (i.e. when a whole signal is processed at once). Here, we study the problem of optimal causal filtering under a perfect perceptual-quality constraint, which is a task of fundamentally different nature. Specifically, we analyze a Gaussian Markov signal observed through a linear noisy transformation. In the absence of perceptual constraints, the Kalman filter is known to be optimal in the MSE sense for this setting. Here, we show that adding the perfect perceptual quality constraint (i.e. the requirement of temporal consistency), introduces a fundamental dilemma whereby the filter may have to "knowingly" ignore new information revealed by the observations in order to conform to its past decisions. This often comes at the cost of a significant increase in the MSE (beyond that encountered in static settings). Our analysis goes beyond the classic innovation process of the Kalman filter, and introduces the novel concept of an unutilized information process. Using this tool, we present a recursive formula for perceptual filters, and demonstrate the qualitative effects of perfect perceptual-quality estimation on a video reconstruction problem.

Applying deep learning to solve real-life instances of hard combinatorial problems has tremendous potential. Research in this direction has focused on the Boolean satisfiability (SAT) problem, both because of its theoretical centrality and practical importance. A major roadblock faced, though, is that training sets are restricted to random formulas of size several orders of magnitude smaller than formulas of practical interest, raising serious concerns about generalization. This is because labeling random formulas of increasing size rapidly becomes intractable. By exploiting the probabilistic method in a fundamental way, we remove this roadblock entirely: we show how to generate correctly labeled random formulas of any desired size, without having to solve the underlying decision problem. Moreover, the difficulty of the classification task for the formulas produced by our generator is tunable by varying a simple scalar parameter. This opens up an entirely new level of sophistication for the machine learning methods that can be brought to bear on Satisfiability. Using our generator, we train existing state-of-the-art models for the task of predicting satisfiability on formulas with 10,000 variables. We find that they do no better than random guessing. As a first indication of what can be achieved with the new generator, we present a novel classifier that performs significantly better than random guessing 99% on the same datasets, for most difficulty levels. Crucially, unlike past approaches that learn based on syntactic features of a formula, our classifier performs its learning on a short prefix of a solver's computation, an approach that we expect to be of independent interest.

Given independent standard Gaussian points $v_1, \ldots, v_n$ in dimension $d$, for what values of $(n, d)$ does there exist with high probability an origin-symmetric ellipsoid that simultaneously passes through all of the points? This basic problem of fitting an ellipsoid to random points has connections to low-rank matrix decompositions, independent component analysis, and principal component analysis. Based on strong numerical evidence, Saunderson, Parrilo, and Willsky [Proc. of Conference on Decision and Control, pp. 6031-6036, 2013] conjecture that the ellipsoid fitting problem transitions from feasible to infeasible as the number of points $n$ increases, with a sharp threshold at $n \sim d^2/4$. We resolve this conjecture up to logarithmic factors by constructing a fitting ellipsoid for some $n = \Omega( \, d^2/\mathrm{polylog}(d) \,)$, improving prior work of Ghosh et al. [Proc. of Symposium on Foundations of Computer Science, pp. 954-965, 2020] that requires $n = o(d^{3/2})$. Our proof demonstrates feasibility of the least squares construction of Saunderson et al. using a convenient decomposition of a certain non-standard random matrix and a careful analysis of its Neumann expansion via the theory of graph matrices.

Many important problems in AI, among them #SAT, parameter learning and probabilistic inference go beyond the classical satisfiability problem. Here, instead of finding a solution we are interested in a quantity associated with the set of solutions, such as the number of solutions, the optimal solution or the probability that a query holds in a solution. To model such quantitative problems in a uniform manner, a number of frameworks, e.g. Algebraic Model Counting and Semiring-based Constraint Satisfaction Problems, employ what we call the semiring paradigm. In the latter the abstract algebraic structure of the semiring serves as a means of parameterizing the problem definition, thus allowing for different modes of quantitative computations by choosing different semirings. While efficiently solvable cases have been widely studied, a systematic study of the computational complexity of such problems depending on the semiring parameter is missing. In this work, we characterize the latter by NP(R), a novel generalization of NP over semiring R, and obtain NP(R)-completeness results for a selection of semiring frameworks. To obtain more tangible insights into the hardness of NP(R), we link it to well-known complexity classes from the literature. Interestingly, we manage to connect the computational hardness to properties of the semiring. Using this insight, we see that, on the one hand, NP(R) is always at least as hard as NP or ModpP depending on the semiring R and in general unlikely to be in FPSPACEpoly. On the other hand, for broad subclasses of semirings relevant in practice we can employ reductions to NP, ModpP and #P. These results show that in many cases solutions are only mildly harder to compute than functions in NP, ModpP and #P, give us new insights into how problems that involve counting on semirings can be approached, and provide a means of assessing whether an algorithm is appropriate for a given class of problems.

This paper considers distributed online convex optimization with adversarial constraints. In this setting, a network of agents makes decisions at each round, and then only a portion of the loss function and a coordinate block of the constraint function are privately revealed to each agent. The loss and constraint functions are convex and can vary arbitrarily across rounds. The agents collaborate to minimize network regret and cumulative constraint violation. A novel distributed online algorithm is proposed and it achieves an $\mathcal{O}(T^{\max\{c,1-c\}})$ network regret bound and an $\mathcal{O}(T^{1-c/2})$ network cumulative constraint violation bound, where $T$ is the number of rounds and $c\in(0,1)$ is a user-defined trade-off parameter. When Slater's condition holds (i.e, there is a point that strictly satisfies the inequality constraints), the network cumulative constraint violation bound is reduced to $\mathcal{O}(T^{1-c})$. Moreover, if the loss functions are strongly convex, then the network regret bound is reduced to $\mathcal{O}(\log(T))$, and the network cumulative constraint violation bound is reduced to $\mathcal{O}(\sqrt{\log(T)T})$ and $\mathcal{O}(\log(T))$ without and with Slater's condition, respectively. To the best of our knowledge, this paper is the first to achieve reduced (network) cumulative constraint violation bounds for (distributed) online convex optimization with adversarial constraints under Slater's condition. Finally, the theoretical results are verified through numerical simulations.