Learning to optimize: A tutorial for continuous and mixed-integer optimization

Learning to Optimize (L2O) stands at the intersection of traditional optimization and machine learning, utilizing the capabilities of machine learning to enhance conventional optimization techniques. As real-world optimization problems frequently share common structures, L2O provides a tool to exploit these structures for better or faster solutions. This tutorial dives deep into L2O techniques, introducing how to accelerate optimization algorithms, promptly estimate the solutions, or even reshape the optimization problem itself, making it more adaptive to real-world applications. By considering the prerequisites for successful applications of L2O and the structure of the optimization problems at hand, this tutorial provides a comprehensive guide for practitioners and researchers alike.

Bagging Improves Generalization Exponentially

Bagging is a popular ensemble technique to improve the accuracy of machine learning models. It hinges on the well-established rationale that, by repeatedly retraining on resampled data, the aggregated model exhibits lower variance and hence higher stability, especially for discontinuous base learners. In this paper, we provide a new perspective on bagging: By suitably aggregating the base learners at the parametrization instead of the output level, bagging improves generalization performances exponentially, a strength that is significantly more powerful than variance reduction. More precisely, we show that for general stochastic optimization problems that suffer from slowly (i.e., polynomially) decaying generalization errors, bagging can effectively reduce these errors to an exponential decay. Moreover, this power of bagging is agnostic to the solution schemes, including common empirical risk minimization, distributionally robust optimization, and various regularizations. We demonstrate how bagging can substantially improve generalization performances in a range of examples involving heavy-tailed data that suffer from intrinsically slow rates.

Revisiting Zeroth-Order Optimization for Memory-Efficient LLM Fine-Tuning: A Benchmark

In the evolving landscape of natural language processing (NLP), fine-tuning pre-trained Large Language Models (LLMs) with first-order (FO) optimizers like SGD and Adam has become standard. Yet, as LLMs grow {in size}, the substantial memory overhead from back-propagation (BP) for FO gradient computation presents a significant challenge. Addressing this issue is crucial, especially for applications like on-device training where memory efficiency is paramount. This paper proposes a shift towards BP-free, zeroth-order (ZO) optimization as a solution for reducing memory costs during LLM fine-tuning, building on the initial concept introduced by MeZO. Unlike traditional ZO-SGD methods, our work expands the exploration to a wider array of ZO optimization techniques, through a comprehensive, first-of-its-kind benchmarking study across five LLM families (Roberta, OPT, LLaMA, Vicuna, Mistral), three task complexities, and five fine-tuning schemes. Our study unveils previously overlooked optimization principles, highlighting the importance of task alignment, the role of the forward gradient method, and the balance between algorithm complexity and fine-tuning performance. We further introduce novel enhancements to ZO optimization, including block-wise descent, hybrid training, and gradient sparsity. Our study offers a promising direction for achieving further memory-efficient LLM fine-tuning. Codes to reproduce all our experiments are at https://github.com/ZO-Bench/ZO-LLM .

Rethinking the Capacity of Graph Neural Networks for Branching Strategy

Graph neural networks (GNNs) have been widely used to predict properties and heuristics of mixed-integer linear programs (MILPs) and hence accelerate MILP solvers. This paper investigates the capacity of GNNs to represent strong branching (SB) scores that provide an efficient strategy in the branch-and-bound algorithm. Although message-passing GNN (MP-GNN), as the simplest GNN structure, is frequently employed in the existing literature to learn SB scores, we prove a fundamental limitation in its expressive power -- there exist two MILP instances with different SB scores that cannot be distinguished by any MP-GNN, regardless of the number of parameters. In addition, we establish a universal approximation theorem for another GNN structure called the second-order folklore GNN (2-FGNN). We show that for any data distribution over MILPs, there always exists a 2-FGNN that can approximate the SB score with arbitrarily high accuracy and arbitrarily high probability. A small-scale numerical experiment is conducted to directly validate our theoretical findings.

DIG-MILP: a Deep Instance Generator for Mixed-Integer Linear Programming with Feasibility Guarantee

Mixed-integer linear programming (MILP) stands as a notable NP-hard problem pivotal to numerous crucial industrial applications. The development of effective algorithms, the tuning of solvers, and the training of machine learning models for MILP resolution all hinge on access to extensive, diverse, and representative data. Yet compared to the abundant naturally occurring data in image and text realms, MILP is markedly data deficient, underscoring the vital role of synthetic MILP generation. We present DIG-MILP, a deep generative framework based on variational auto-encoder (VAE), adept at extracting deep-level structural features from highly limited MILP data and producing instances that closely mirror the target data. Notably, by leveraging the MILP duality, DIG-MILP guarantees a correct and complete generation space as well as ensures the boundedness and feasibility of the generated instances. Our empirical study highlights the novelty and quality of the instances generated by DIG-MILP through two distinct downstream tasks: (S1) Data sharing, where solver solution times correlate highly positive between original and DIG-MILP-generated instances, allowing data sharing for solver tuning without publishing the original data; (S2) Data Augmentation, wherein the DIG-MILP-generated instances bolster the generalization performance of machine learning models tasked with resolving MILP problems.

A Human-on-the-Loop Optimization Autoformalism Approach for Sustainability

This paper outlines a natural conversational approach to solving personalized energy-related problems using large language models (LLMs). We focus on customizable optimization problems that necessitate repeated solving with slight variations in modeling and are user-specific, hence posing a challenge to devising a one-size-fits-all model. We put forward a strategy that augments an LLM with an optimization solver, enhancing its proficiency in understanding and responding to user specifications and preferences while providing nonlinear reasoning capabilities. Our approach pioneers the novel concept of human-guided optimization autoformalism, translating a natural language task specification automatically into an optimization instance. This enables LLMs to analyze, explain, and tackle a variety of instance-specific energy-related problems, pushing beyond the limits of current prompt-based techniques. Our research encompasses various commonplace tasks in the energy sector, from electric vehicle charging and Heating, Ventilation, and Air Conditioning (HVAC) control to long-term planning problems such as cost-benefit evaluations for installing rooftop solar photovoltaics (PVs) or heat pumps. This pilot study marks an essential stride towards the context-based formulation of optimization using LLMs, with the potential to democratize optimization processes. As a result, stakeholders are empowered to optimize their energy consumption, promoting sustainable energy practices customized to personal needs and preferences.

DSGD-CECA: Decentralized SGD with Communication-Optimal Exact Consensus Algorithm

Decentralized Stochastic Gradient Descent (SGD) is an emerging neural network training approach that enables multiple agents to train a model collaboratively and simultaneously. Rather than using a central parameter server to collect gradients from all the agents, each agent keeps a copy of the model parameters and communicates with a small number of other agents to exchange model updates. Their communication, governed by the communication topology and gossip weight matrices, facilitates the exchange of model updates. The state-of-the-art approach uses the dynamic one-peer exponential-2 topology, achieving faster training times and improved scalability than the ring, grid, torus, and hypercube topologies. However, this approach requires a power-of-2 number of agents, which is impractical at scale. In this paper, we remove this restriction and propose \underline{D}ecentralized \underline{SGD} with \underline{C}ommunication-optimal \underline{E}xact \underline{C}onsensus \underline{A}lgorithm (DSGD-CECA), which works for any number of agents while still achieving state-of-the-art properties. In particular, DSGD-CECA incurs a unit per-iteration communication overhead and an $\tilde{O}(n^3)$ transient iteration complexity. Our proof is based on newly discovered properties of gossip weight matrices and a novel approach to combine them with DSGD's convergence analysis. Numerical experiments show the efficiency of DSGD-CECA.

Towards Constituting Mathematical Structures for Learning to Optimize

Learning to Optimize (L2O), a technique that utilizes machine learning to learn an optimization algorithm automatically from data, has gained arising attention in recent years. A generic L2O approach parameterizes the iterative update rule and learns the update direction as a black-box network. While the generic approach is widely applicable, the learned model can overfit and may not generalize well to out-of-distribution test sets. In this paper, we derive the basic mathematical conditions that successful update rules commonly satisfy. Consequently, we propose a novel L2O model with a mathematics-inspired structure that is broadly applicable and generalized well to out-of-distribution problems. Numerical simulations validate our theoretical findings and demonstrate the superior empirical performance of the proposed L2O model.

Lower Bounds and Accelerated Algorithms in Distributed Stochastic Optimization with Communication Compression

Communication compression is an essential strategy for alleviating communication overhead by reducing the volume of information exchanged between computing nodes in large-scale distributed stochastic optimization. Although numerous algorithms with convergence guarantees have been obtained, the optimal performance limit under communication compression remains unclear. In this paper, we investigate the performance limit of distributed stochastic optimization algorithms employing communication compression. We focus on two main types of compressors, unbiased and contractive, and address the best-possible convergence rates one can obtain with these compressors. We establish the lower bounds for the convergence rates of distributed stochastic optimization in six different settings, combining strongly-convex, generally-convex, or non-convex functions with unbiased or contractive compressor types. To bridge the gap between lower bounds and existing algorithms' rates, we propose NEOLITHIC, a nearly optimal algorithm with compression that achieves the established lower bounds up to logarithmic factors under mild conditions. Extensive experimental results support our theoretical findings. This work provides insights into the theoretical limitations of existing compressors and motivates further research into fundamentally new compressor properties.

An Improved Analysis of Policy Gradient and Natural Policy Gradient Methods

In this paper, we revisit and improve the convergence of policy gradient (PG), natural PG (NPG) methods, and their variance-reduced variants, under general smooth policy parametrizations. More specifically, with the Fisher information matrix of the policy being positive definite: i) we show that a state-of-the-art variance-reduced PG method, which has only been shown to converge to stationary points, converges to the globally optimal value up to some inherent function approximation error due to policy parametrization; ii) we show that NPG enjoys a lower sample complexity; iii) we propose SRVR-NPG, which incorporates variance-reduction into the NPG update. Our improvements follow from an observation that the convergence of (variance-reduced) PG and NPG methods can improve each other: the stationary convergence analysis of PG can be applied to NPG as well, and the global convergence analysis of NPG can help to establish the global convergence of (variance-reduced) PG methods. Our analysis carefully integrates the advantages of these two lines of works. Thanks to this improvement, we have also made variance-reduction for NPG possible, with both global convergence and an efficient finite-sample complexity.