Bilevel Optimization without Lower-Level Strong Convexity from the Hyper-Objective Perspective

Bilevel optimization reveals the inner structure of otherwise oblique optimization problems, such as hyperparameter tuning and meta-learning. A common goal in bilevel optimization is to find stationary points of the hyper-objective function. Although this hyper-objective approach is widely used, its theoretical properties have not been thoroughly investigated in cases where the lower-level functions lack strong convexity. In this work, we take a step forward and study the hyper-objective approach without the typical lower-level strong convexity assumption. Our hardness results show that the hyper-objective of general convex lower-level functions can be intractable either to evaluate or to optimize. To tackle this challenge, we introduce the gradient dominant condition, which strictly relaxes the strong convexity assumption by allowing the lower-level solution set to be non-singleton. Under the gradient dominant condition, we propose the Inexact Gradient-Free Method (IGFM), which uses the Switching Gradient Method (SGM) as the zeroth order oracle, to find an approximate stationary point of the hyper-objective. We also extend our results to nonsmooth lower-level functions under the weak sharp minimum condition

On Implicit Bias in Overparameterized Bilevel Optimization

Many problems in machine learning involve bilevel optimization (BLO), including hyperparameter optimization, meta-learning, and dataset distillation. Bilevel problems consist of two nested sub-problems, called the outer and inner problems, respectively. In practice, often at least one of these sub-problems is overparameterized. In this case, there are many ways to choose among optima that achieve equivalent objective values. Inspired by recent studies of the implicit bias induced by optimization algorithms in single-level optimization, we investigate the implicit bias of gradient-based algorithms for bilevel optimization. We delineate two standard BLO methods -- cold-start and warm-start -- and show that the converged solution or long-run behavior depends to a large degree on these and other algorithmic choices, such as the hypergradient approximation. We also show that the inner solutions obtained by warm-start BLO can encode a surprising amount of information about the outer objective, even when the outer parameters are low-dimensional. We believe that implicit bias deserves as central a role in the study of bilevel optimization as it has attained in the study of single-level neural net optimization.Comment: ICML 202

Blockwise Stochastic Variance-Reduced Methods with Parallel Speedup for Multi-Block Bilevel Optimization

In this paper, we consider non-convex multi-block bilevel optimization (MBBO) problems, which involve $m\gg 1$ lower level problems and have important applications in machine learning. Designing a stochastic gradient and controlling its variance is more intricate due to the hierarchical sampling of blocks and data and the unique challenge of estimating hyper-gradient. We aim to achieve three nice properties for our algorithm: (a) matching the state-of-the-art complexity of standard BO problems with a single block; (b) achieving parallel speedup by sampling $I$ blocks and sampling $B$ samples for each sampled block per-iteration; (c) avoiding the computation of the inverse of a high-dimensional Hessian matrix estimator. However, it is non-trivial to achieve all of these by observing that existing works only achieve one or two of these properties. To address the involved challenges for achieving (a, b, c), we propose two stochastic algorithms by using advanced blockwise variance-reduction techniques for tracking the Hessian matrices (for low-dimensional problems) or the Hessian-vector products (for high-dimensional problems), and prove an iteration complexity of $O(\frac{m\epsilon^{-3}\mathbb{I}(I<m)}{I\sqrt{I}} + \frac{m\epsilon^{-3}}{I\sqrt{B}})$ for finding an $\epsilon$-stationary point under appropriate conditions. We also conduct experiments to verify the effectiveness of the proposed algorithms comparing with existing MBBO algorithms

BiERL: A Meta Evolutionary Reinforcement Learning Framework via Bilevel Optimization

Evolutionary reinforcement learning (ERL) algorithms recently raise attention in tackling complex reinforcement learning (RL) problems due to high parallelism, while they are prone to insufficient exploration or model collapse without carefully tuning hyperparameters (aka meta-parameters). In the paper, we propose a general meta ERL framework via bilevel optimization (BiERL) to jointly update hyperparameters in parallel to training the ERL model within a single agent, which relieves the need for prior domain knowledge or costly optimization procedure before model deployment. We design an elegant meta-level architecture that embeds the inner-level's evolving experience into an informative population representation and introduce a simple and feasible evaluation of the meta-level fitness function to facilitate learning efficiency. We perform extensive experiments in MuJoCo and Box2D tasks to verify that as a general framework, BiERL outperforms various baselines and consistently improves the learning performance for a diversity of ERL algorithms.Comment: Published as a conference paper at European Conference on Artificial Intelligence (ECAI) 202

Data Distillation: A Survey

The popularity of deep learning has led to the curation of a vast number of massive and multifarious datasets. Despite having close-to-human performance on individual tasks, training parameter-hungry models on large datasets poses multi-faceted problems such as (a) high model-training time; (b) slow research iteration; and (c) poor eco-sustainability. As an alternative, data distillation approaches aim to synthesize terse data summaries, which can serve as effective drop-in replacements of the original dataset for scenarios like model training, inference, architecture search, etc. In this survey, we present a formal framework for data distillation, along with providing a detailed taxonomy of existing approaches. Additionally, we cover data distillation approaches for different data modalities, namely images, graphs, and user-item interactions (recommender systems), while also identifying current challenges and future research directions.Comment: Accepted at TMLR '23. 21 pages, 4 figure

Principled and Efficient Bilevel Optimization for Machine Learning

Automatic differentiation (AD) is a core element of most modern machine learning libraries that allows to efficiently compute derivatives of a function from the corresponding program. Thanks to AD, machine learning practitioners have tackled increasingly complex learning models, such as deep neural networks with up to hundreds of billions of parameters, which are learned using the derivative (or gradient) of a loss function with respect to those parameters. While in most cases gradients can be computed exactly and relatively cheaply, in others the exact computation is either impossible or too expensive and AD must be used in combination with approximation methods. Some of these challenging scenarios arising for example in meta-learning or hyperparameter optimization, can be framed as bilevel optimization problems, where the goal is to minimize an objective function that is evaluated by first solving another optimization problem, the lower-level problem. In this work, we study efficient gradient-based bilevel optimization algorithms for machine learning problems. In particular, we establish convergence rates for some simple approaches to approximate the gradient of the bilevel objective, namely the hypergradient, when the objective is smooth and the lower-level problem consists in finding the fixed point of a contraction map. Leveraging such results, we also prove that the projected inexact hypergradient method achieves a (near) optimal rate of convergence. We establish these results for both the deterministic and stochastic settings. Additionally, we provide an efficient implementation of the methods studied and perform several numerical experiments on hyperparameter optimization, meta-learning, datapoisoning and equilibrium models, which show that our theoretical results are good indicators of the performance in practice

BOME! Bilevel Optimization Made Easy: A Simple First-Order Approach

Bilevel optimization (BO) is useful for solving a variety of important machine learning problems including but not limited to hyperparameter optimization, meta-learning, continual learning, and reinforcement learning. Conventional BO methods need to differentiate through the low-level optimization process with implicit differentiation, which requires expensive calculations related to the Hessian matrix. There has been a recent quest for first-order methods for BO, but the methods proposed to date tend to be complicated and impractical for large-scale deep learning applications. In this work, we propose a simple first-order BO algorithm that depends only on first-order gradient information, requires no implicit differentiation, and is practical and efficient for large-scale non-convex functions in deep learning. We provide non-asymptotic convergence analysis of the proposed method to stationary points for non-convex objectives and present empirical results that show its superior practical performance

Contextual Stochastic Bilevel Optimization

We introduce contextual stochastic bilevel optimization (CSBO) -- a stochastic bilevel optimization framework with the lower-level problem minimizing an expectation conditioned on some contextual information and the upper-level decision variable. This framework extends classical stochastic bilevel optimization when the lower-level decision maker responds optimally not only to the decision of the upper-level decision maker but also to some side information and when there are multiple or even infinite many followers. It captures important applications such as meta-learning, personalized federated learning, end-to-end learning, and Wasserstein distributionally robust optimization with side information (WDRO-SI). Due to the presence of contextual information, existing single-loop methods for classical stochastic bilevel optimization are unable to converge. To overcome this challenge, we introduce an efficient double-loop gradient method based on the Multilevel Monte-Carlo (MLMC) technique and establish its sample and computational complexities. When specialized to stochastic nonconvex optimization, our method matches existing lower bounds. For meta-learning, the complexity of our method does not depend on the number of tasks. Numerical experiments further validate our theoretical results.Comment: The paper is accepted by NeurIPS 202

A Lower Bound and a Near-Optimal Algorithm for Bilevel Empirical Risk Minimization

Bilevel optimization problems, which are problems where two optimization problems are nested, have more and more applications in machine learning. In many practical cases, the upper and the lower objectives correspond to empirical risk minimization problems and therefore have a sum structure. In this context, we propose a bilevel extension of the celebrated SARAH algorithm. We demonstrate that the algorithm requires $\mathcal{O}((n+m)^{\frac12}\varepsilon^{-1})$ gradient computations to achieve $\varepsilon$-stationarity with $n+m$ the total number of samples, which improves over all previous bilevel algorithms. Moreover, we provide a lower bound on the number of oracle calls required to get an approximate stationary point of the objective function of the bilevel problem. This lower bound is attained by our algorithm, which is therefore optimal in terms of sample complexity