Neural network pruning is a key technique towards engineering large yet scalable, interpretable, and generalizable models. Prior work on the subject has developed largely along two orthogonal directions: (1) differentiable pruning for efficiently and accurately scoring the importance of parameters, and (2) combinatorial optimization for efficiently searching over the space of sparse models. We unite the two approaches, both theoretically and empirically, to produce a coherent framework for structured neural network pruning in which differentiable pruning guides combinatorial optimization algorithms to select the most important sparse set of parameters. Theoretically, we show how many existing differentiable pruning techniques can be understood as nonconvex regularization for group sparse optimization, and prove that for a wide class of nonconvex regularizers, the global optimum is unique, group-sparse, and provably yields an approximate solution to a sparse convex optimization problem. The resulting algorithm that we propose, SequentialAttention++, advances the state of the art in large-scale neural network block-wise pruning tasks on the ImageNet and Criteo datasets.

Recent progress in Neural Causal Models (NCMs) showcased how identification and partial identification of causal effects can be automatically carried out via training of neural generative models that respect the constraints encoded in a given causal graph [Xia et al. 2022, Balazadeh et al. 2022]. However, formal consistency of these methods has only been proven for the case of discrete variables or only for linear causal models. In this work, we prove the consistency of partial identification via NCMs in a general setting with both continuous and categorical variables. Further, our results highlight the impact of the design of the underlying neural network architecture in terms of depth and connectivity as well as the importance of applying Lipschitz regularization in the training phase. In particular, we provide a counterexample showing that without Lipschitz regularization this method may not be asymptotically consistent. Our results are enabled by new results on the approximability of Structural Causal Models (SCMs) via neural generative models, together with an analysis of the sample complexity of the resulting architectures and how that translates into an error in the constrained optimization problem that defines the partial identification bounds.

We present a novel methodology for convex optimization algorithm design using ideas from electric RLC circuits. Given an optimization problem, the first stage of the methodology is to design an appropriate electric circuit whose continuous-time dynamics converge to the solution of the optimization problem at hand. Then, the second stage is an automated, computer-assisted discretization of the continuous-time dynamics, yielding a provably convergent discrete-time algorithm. Our methodology recovers many classical (distributed) optimization algorithms and enables users to quickly design and explore a wide range of new algorithms with convergence guarantees.

With the arrival of the Noisy Intermediate-Scale Quantum (NISQ) era, Variational Quantum Algorithms (VQAs) have emerged to obtain possible quantum advantage. In particular, how to effectively incorporate hard constraints in VQAs remains a critical and open question. In this paper, we manage to combine the Hamming Weight Preserving ansatz with a topological-aware parity check on physical qubits to enforce error mitigation and further hard constraints. We demonstrate the combination significantly outperforms peer VQA methods on both quantum chemistry problems and constrained combinatorial optimization problems e.g.

Current methods for training Binarized Neural Networks (BNNs) heavily rely on the heuristic straight-through estimator (STE), which crucially enables the application of SGD-based optimizers to the combinatorial training problem. Although the STE heuristics and their variants have led to significant improvements in BNN performance, their theoretical underpinnings remain unclear and relatively understudied. In this paper, we propose a theoretically motivated optimization framework for BNN training based on Gaussian variational inference. In its simplest form, our approach yields a non-convex linear programming formulation whose variables and associated gradients motivate the use of latent weights and STE gradients. More importantly, our framework allows us to formulate semidefinite programming (SDP) relaxations to the BNN training task. Such formulations are able to explicitly models pairwise correlations between weights during training, leading to a more accurate optimization characterization of the training problem. As the size of such formulations grows quadratically in the number of weights, quickly becoming intractable for large networks, we apply the Burer-Monteiro approach and only optimize over linear-size low-rank SDP solutions. Our empirical evaluation on CIFAR-10, CIFAR-100, Tiny-ImageNet and ImageNet datasets shows our method consistently outperforming all state-of-the-art algorithms for training BNNs.

The increasing size of language models necessitates a thorough analysis across multiple dimensions to assess trade-offs among crucial hardware metrics such as latency, energy consumption, GPU memory usage, and performance. Identifying optimal model configurations under specific hardware constraints is becoming essential but remains challenging due to the computational load of exhaustive training and evaluation on multiple devices. To address this, we introduce HW-GPT-Bench, a hardware-aware benchmark that utilizes surrogate predictions to approximate various hardware metrics across 13 devices of architectures in the GPT-2 family, with architectures containing up to 1.55B parameters. Our surrogates, via calibrated predictions and reliable uncertainty estimates, faithfully model the heteroscedastic noise inherent in the energy and latency measurements. To estimate perplexity, we employ weight-sharing techniques from Neural Architecture Search (NAS), inheriting pretrained weights from the largest GPT-2 model. Finally, we demonstrate the utility of HW-GPT-Bench by simulating optimization trajectories of various multi-objective optimization algorithms in just a few seconds.

Binary statistical decision making involves choosing between two states based on statistical evidence. The optimal decision strategy is typically formulated through a constrained optimization problem, where both the objective and constraints are expressed as integrals involving two Lebesgue measurable functions, one of which represents the strategy being optimized. In this work, we present a comprehensive formulation of the binary decision making problem and provide a detailed characterization of the optimal solution. Our framework encompasses a wide range of well-known and recently proposed decision making problems as specific cases. We demonstrate how our generic approach can be used to derive the optimal decision strategies for these diverse instances. Our results offer a robust mathematical tool that simplifies the process of solving both existing and novel formulations of binary decision making problems which are in the core of many Machine Learning algorithms.

This paper investigates a novel lossy compression framework operating under logarithmic loss, designed to handle situations where the reconstruction distribution diverges from the source distribution. This framework is especially relevant for applications that require joint compression and retrieval, and in scenarios involving distributional shifts due to processing. We show that the proposed formulation extends the classical minimum entropy coupling framework by integrating a bottleneck, allowing for controlled variability in the degree of stochasticity in the coupling.We explore the decomposition of the Minimum Entropy Coupling with Bottleneck (MEC-B) into two distinct optimization problems: Entropy-Bounded Information Maximization (EBIM) for the encoder, and Minimum Entropy Coupling (MEC) for the decoder. Through extensive analysis, we provide a greedy algorithm for EBIM with guaranteed performance, and characterize the optimal solution near functional mappings, yielding significant theoretical insights into the structural complexity of this problem.Furthermore, we illustrated the practical application of MEC-B through experiments in Markov Coding Games (MCGs) under rate limits. These games simulate a communication scenario within a Markov Decision Process, where an agent must transmit a compressed message from a sender to a receiver through its actions. Our experiments highlighted the trade-offs between MDP rewards and receiver accuracy across various compression rates, showcasing the efficacy of our method compared to conventional compression baseline.

The recent emergence of diffusion models has significantly advanced the precision of learnable priors, presenting innovative avenues for addressing inverse problems. Previous works have endeavored to integrate diffusion priors into the maximum a posteriori estimation (MAP) framework and design optimization methods to solve the inverse problem. However, prevailing optimization-based rithms primarily exploit the prior information within the diffusion models while neglecting their denoising capability. To bridge this gap, this work leverages the diffusion process to reframe noisy inverse problems as a two-variable constrained optimization task by introducing an auxiliary optimization variable that represents a'noisy' sample at an equivalent denoising step. The projection gradient descent method is efficiently utilized to solve the corresponding optimization problem by truncating the gradient through the $\mu$-predictor. The proposed algorithm, termed ProjDiff, effectively harnesses the prior information and the denoising capability of a pre-trained diffusion model within the optimization framework. Extensive experiments on the image restoration tasks and source separation and partial generation tasks demonstrate that ProjDiff exhibits superior performance across various linear and nonlinear inverse problems, highlighting its potential for practical applications.

In this paper, we focus on simple bilevel optimization problems, where we minimize a convex smooth objective function over the optimal solution set of another convex smooth constrained optimization problem. We present a novel bilevel optimization method that locally approximates the solution set of the lower-level problem using a cutting plane approach and employs an accelerated gradient-based update to reduce the upper-level objective function over the approximated solution set. We measure the performance of our method in terms of suboptimality and infeasibility errors and provide non-asymptotic convergence guarantees for both error criteria. Specifically, when the feasible set is compact, we show that our method requires at most $\mathcal{O}(\max\\{1/\sqrt{\epsilon_{f}}, 1/\epsilon_g\\})$ iterations to find a solution that is $\epsilon_f$-suboptimal and $\epsilon_g$-infeasible. Moreover, under the additional assumption that the lower-level objective satisfies the $r$-th Hölderian error bound, we show that our method achieves an iteration complexity of $\mathcal{O}(\max\\{\epsilon_{f}^{-\frac{2r-1}{2r}},\epsilon_{g}^{-\frac{2r-1}{2r}}\\})$, which matches the optimal complexity of single-level convex constrained optimization when $r=1$.