Our goal is to $\textit{efficiently}$ discover a compact set of temporal logic rules to explain irregular events of interest. We introduce a neural-symbolic rule induction framework within the temporal point process model. The negative log-likelihood is the loss that guides the learning, where the explanatory logic rules and their weights are learned end-to-end in a $\textit{differentiable}$ way. Specifically, predicates and logic rules are represented as $\textit{vector embeddings}$, where the predicate embeddings are fixed and the rule embeddings are trained via gradient descent to obtain the most appropriate compositional representations of the predicate embeddings. To make the rule learning process more efficient and flexible, we adopt a $\textit{sequential covering algorithm}$, which progressively adds rules to the model and removes the event sequences that have been explained until all event sequences have been covered. All the found rules will be fed back to the models for a final rule embedding and weight refinement. Our approach showcases notable efficiency and accuracy across synthetic and real datasets, surpassing state-of-the-art baselines by a wide margin in terms of efficiency.

In recent years, solving optimization problems involving black-box simulators has become a point of focus for the machine learning community due to their ubiquity in science and engineering. The simulators describe a forward process $f_{\mathrm{sim}}: (\psi, x) \rightarrow y$ from simulation parameters $\psi$ and input data $x$ to observations $y$, and the goal of the optimization problem is to find parameters $\psi$ that minimize a desired loss function. Sophisticated optimization algorithms typically require gradient information regarding the forward process, $f_{\mathrm{sim}}$, with respect to the parameters $\psi$. However, obtaining gradients from black-box simulators can often be prohibitively expensive or, in some cases, impossible. Furthermore, in many applications, practitioners aim to solve a set of related problems. Thus, starting the optimization ``ab initio", i.e. from scratch, each time might be inefficient if the forward model is expensive to evaluate. To address those challenges, this paper introduces a novel method for solving classes of similar black-box optimization problems by learning an active learning policy that guides a differentiable surrogate's training and uses the surrogate's gradients to optimize the simulation parameters with gradient descent. After training the policy, downstream optimization of problems involving black-box simulators requires up to $\sim$90\% fewer expensive simulator calls compared to baselines such as local surrogate-based approaches, numerical optimization, and Bayesian methods.

Learning representations that generalize under distribution shifts is critical for building robust machine learning models. However, despite significant efforts in recent years, algorithmic advances in this direction have been limited. In this work, we seek to understand the fundamental difficulty of out-of-distribution generalization with deep neural networks. We first empirically show that perhaps surprisingly, even allowing a neural network to explicitly fit the representations obtained from a teacher network that can generalize out-of-distribution is insufficient for the generalization of the student network. Then, by a theoretical study of two-layer ReLU networks optimized by stochastic gradient descent (SGD) under a structured feature model, we identify a fundamental yet unexplored feature learning proclivity of neural networks, feature contamination: neural networks can learn uncorrelated features together with predictive features, resulting in generalization failure under distribution shifts. Notably, this mechanism essentially differs from the prevailing narrative in the literature that attributes the generalization failure to spurious correlations. Overall, our results offer new insights into the non-linear feature learning dynamics of neural networks and highlight the necessity of considering inductive biases in out-of-distribution generalization.

Constrained optimization offers a powerful framework to prescribe desired behaviors in neural network models. Typically, constrained problems are solved via their min-max Lagrangian formulations, which exhibit unstable oscillatory dynamics when optimized using gradient descent-ascent. The adoption of constrained optimization techniques in the machine learning community is currently limited by the lack of reliable, general-purpose update schemes for the Lagrange multipliers. This paper proposes the $\nu$PI algorithm and contributes an optimization perspective on Lagrange multiplier updates based on PI controllers, extending the work of Stooke, Achiam and Abbeel (2020). We provide theoretical and empirical insights explaining the inability of momentum methods to address the shortcomings of gradient descent-ascent, and contrast this with the empirical success of our proposed $\nu$PI controller. Moreover, we prove that $\nu$PI generalizes popular momentum methods for single-objective minimization. Our experiments demonstrate that $\nu$PI reliably stabilizes the multiplier dynamics and its hyperparameters enjoy robust and predictable behavior.

We demonstrate a substantial gap between the privacy guarantees of the Adaptive Batch Linear Queries (ABLQ) mechanism under different types of batch sampling: (i) Shuffling, and (ii) Poisson subsampling; the typical analysis of Differentially Private Stochastic Gradient Descent (DP-SGD) follows by interpreting it as a post-processing of ABLQ. While shuffling-based DP-SGD is more commonly used in practical implementations, it has not been amenable to easy privacy analysis, either analytically or even numerically. On the other hand, Poisson subsampling-based DP-SGD is challenging to scalably implement, but has a well-understood privacy analysis, with multiple open-source numerically tight privacy accountants available. This has led to a common practice of using shuffling-based DP-SGD in practice, but using the privacy analysis for the corresponding Poisson subsampling version. Our result shows that there can be a substantial gap between the privacy analysis when using the two types of batch sampling, and thus advises caution in reporting privacy parameters for DP-SGD.

Stochastic gradient descent with momentum, also known as Stochastic Heavy Ball method (SHB), is one of the most popular algorithms for solving large-scale stochastic optimization problems in various machine learning tasks. In practical scenarios, tuning the step-size and momentum parameters of the method is a prohibitively expensive and time-consuming process. In this work, inspired by the recent advantages of stochastic Polyak step-size in the performance of stochastic gradient descent (SGD), we propose and explore new Polyak-type variants suitable for the update rule of the SHB method. In particular, using the Iterate Moving Average (IMA) viewpoint of SHB, we propose and analyze three novel step-size selections: MomSPS$_{\max}$, MomDecSPS, and MomAdaSPS. For MomSPS$_{\max}$, we provide convergence guarantees for SHB to a neighborhood of the solution for convex and smooth problems (without assuming interpolation). If interpolation is also satisfied, then using MomSPS$_{\max}$, SHB converges to the true solution at a fast rate matching the deterministic HB. The other two variants, MomDecSPS and MomAdaSPS, are the first adaptive step-sizes for SHB that guarantee convergence to the exact minimizer without prior knowledge of the problem parameters and without assuming interpolation. The convergence analysis of SHB is tight and obtains the convergence guarantees of SGD with stochastic Polyak step-sizes as a special case. We supplement our analysis with experiments that validate the theory and demonstrate the effectiveness and robustness of the new algorithms.

Scaling hyperparameter optimisation to very large datasets remains an open problem in the Gaussian process community. This paper focuses on iterative methods, which use linear system solvers, like conjugate gradients, alternating projections or stochastic gradient descent, to construct an estimate of the marginal likelihood gradient. We discuss three key improvements which are applicable across solvers: (i) a pathwise gradient estimator, which reduces the required number of solver iterations and amortises the computational cost of making predictions, (ii) warm starting linear system solvers with the solution from the previous step, which leads to faster solver convergence at the cost of negligible bias, (iii) early stopping linear system solvers after a limited computational budget, which synergises with warm starting, allowing solver progress to accumulate over multiple marginal likelihood steps. These techniques provide speed-ups of up to $72\times$ when solving to tolerance, and decrease the average residual norm by up to $7\times$ when stopping early.

Adaptive gradient methods are arguably the most successful optimization algorithms for neural network training. While it is well-known that adaptive gradient methods can achieve better dimensional dependence than stochastic gradient descent (SGD) under favorable geometry for stochastic convex optimization, the theoretical justification for their success in stochastic non-convex optimization remains elusive. In this paper, we aim to close this gap by analyzing the convergence rates of AdaGrad measured by the $\ell_1$-norm of the gradient. Specifically, when the objective has $L$-Lipschitz gradient and the stochastic gradient variance is bounded by $\sigma^2$, we prove a worst-case convergence rate of $\tilde{\mathcal{O}}(\frac{\sqrt{d}L}{\sqrt{T}} + \frac{\sqrt{d} \sigma}{T^{1/4}})$, where $d$ is the dimension of the problem.We also present a lower bound of ${\Omega}(\frac{\sqrt{d}}{\sqrt{T}})$ for minimizing the gradient $\ell_1$-norm in the deterministic setting, showing the tightness of our upper bound in the noiseless case. Moreover, under more fine-grained assumptions on the smoothness structure of the objective and the gradient noise and under favorable gradient $\ell_1/\ell_2$ geometry, we show that AdaGrad can potentially shave a factor of $\sqrt{d}$ compared to SGD. To the best of our knowledge, this is the first result for adaptive gradient methods that demonstrates a provable gain over SGD in the non-convex setting.

A fundamental problem in machine learning is understanding the effect of early stopping on the parameters obtained and the generalization capabilities of the model. Even for linear models, the effect is not fully understood for arbitrary learning rates and data. In this paper, we analyze the dynamics of discrete full batch gradient descent for linear regression. With minimal assumptions, we characterize the trajectory of the parameters and the expected excess risk. Using this characterization, we show that when training with a learning rate schedule $\eta_k$, and a finite time horizon $T$, the early stopped solution $\beta_T$ is equivalent to the minimum norm solution for a generalized ridge regularized problem. We also prove that early stopping is beneficial for generic data with arbitrary spectrum and for a wide variety of learning rate schedules. We provide an estimate for the optimal stopping time and empirically demonstrate the accuracy of our estimate.

This paper addresses the complex issue of resource-constrained scheduling, an NP-hard problem that spans critical areas including chip design and high-performance computing. Traditional scheduling methods often stumble over scalability and applicability challenges. We propose a novel approach using a differentiable combinatorial scheduling framework, utilizing Gumbel-Softmax differentiable sampling technique. This new technical allows for a fully differentiable formulation of linear programming (LP) based scheduling, extending its application to a broader range of LP formulations. To encode inequality constraints for scheduling tasks, we introduce \textit{constrained Gumbel Trick}, which adeptly encodes arbitrary inequality constraints. Consequently, our method facilitates an efficient and scalable scheduling via gradient descent without the need for training data. Comparative evaluations on both synthetic and real-world benchmarks highlight our capability to significantly improve the optimization efficiency of scheduling, surpassing state-of-the-art solutions offered by commercial and open-source solvers such as CPLEX, Gurobi, and CP-SAT in the majority of the designs.