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Michael I. Jordan

Publications and source records attributed to Michael I. Jordan.

At least 127 records · Page 7Linked to original sources
arXiv2024

Improved Bayes Risk Can Yield Reduced Social Welfare Under Competition

As the scale of machine learning models increases, trends such as scaling laws anticipate consistent downstream improvements in predictive accuracy. However, these trends take the perspective of a single model-provider in isolation, while in reality providers often compete with each other for users. In this work, we demonstrate that competition can fundamentally alter the behavior of these scaling trends, even causing overall predictive accuracy across users to be non-monotonic or decreasing with scale. We define a model of competition for classification tasks, and use data representations as a lens for studying the impact of increases in scale. We find many settings where improving data representation quality (as measured by Bayes risk) decreases the overall predictive accuracy across users (i.e., social welfare) for a marketplace of competing model-providers. Our examples range from closed-form formulas in simple settings to simulations with pretrained representations on CIFAR-10. At a conceptual level, our work suggests that favorable scaling trends for individual model-providers need not translate to downstream improvements in social welfare in marketplaces with multiple model providers.

cs.GT↗
arXiv2024

Towards Optimal Statistical Watermarking

We study statistical watermarking by formulating it as a hypothesis testing problem, a general framework which subsumes all previous statistical watermarking methods. Key to our formulation is a coupling of the output tokens and the rejection region, realized by pseudo-random generators in practice, that allows non-trivial trade-offs between the Type I error and Type II error. We characterize the Uniformly Most Powerful (UMP) watermark in the general hypothesis testing setting and the minimax Type II error in the model-agnostic setting. In the common scenario where the output is a sequence of $n$ tokens, we establish nearly matching upper and lower bounds on the number of i.i.d. tokens required to guarantee small Type I and Type II errors. Our rate of $Θ(h^{-1} \log (1/h))$ with respect to the average entropy per token $h$ highlights potentials for improvement from the rate of $h^{-2}$ in the previous works. Moreover, we formulate the robust watermarking problem where the user is allowed to perform a class of perturbations on the generated texts, and characterize the optimal Type II error of robust UMP tests via a linear programming problem. To the best of our knowledge, this is the first systematic statistical treatment on the watermarking problem with near-optimal rates in the i.i.d. setting, which might be of interest for future works.

cs.LG↗
arXiv2024

A Specialized Semismooth Newton Method for Kernel-Based Optimal Transport

Kernel-based optimal transport (OT) estimators offer an alternative, functional estimation procedure to address OT problems from samples. Recent works suggest that these estimators are more statistically efficient than plug-in (linear programming-based) OT estimators when comparing probability measures in high-dimensions~\citep{Vacher-2021-Dimension}. Unfortunately, that statistical benefit comes at a very steep computational price: because their computation relies on the short-step interior-point method (SSIPM), which comes with a large iteration count in practice, these estimators quickly become intractable w.r.t. sample size $n$. To scale these estimators to larger $n$, we propose a nonsmooth fixed-point model for the kernel-based OT problem, and show that it can be efficiently solved via a specialized semismooth Newton (SSN) method: We show, exploring the problem's structure, that the per-iteration cost of performing one SSN step can be significantly reduced in practice. We prove that our SSN method achieves a global convergence rate of $O(1/\sqrt{k})$, and a local quadratic convergence rate under standard regularity conditions. We show substantial speedups over SSIPM on both synthetic and real datasets.

cs.LG↗
arXiv2024

Iterative Data Smoothing: Mitigating Reward Overfitting and Overoptimization in RLHF

Reinforcement Learning from Human Feedback (RLHF) is a pivotal technique that aligns language models closely with human-centric values. The initial phase of RLHF involves learning human values using a reward model from ranking data. It is observed that the performance of the reward model degrades after one epoch of training, and optimizing too much against the learned reward model eventually hinders the true objective. This paper delves into these issues, leveraging the theoretical insights to design improved reward learning algorithm termed 'Iterative Data Smoothing' (IDS). The core idea is that during each training epoch, we not only update the model with the data, but also update the date using the model, replacing hard labels with soft labels. Our empirical findings highlight the superior performance of this approach over the traditional methods.

cs.LG↗
arXiv2024

On the Stability of Nonlinear Receding Horizon Control: A Geometric Perspective

%!TEX root = LCSS_main_max.tex The widespread adoption of nonlinear Receding Horizon Control (RHC) strategies by industry has led to more than 30 years of intense research efforts to provide stability guarantees for these methods. However, current theoretical guarantees require that each (generally nonconvex) planning problem can be solved to (approximate) global optimality, which is an unrealistic requirement for the derivative-based local optimization methods generally used in practical implementations of RHC. This paper takes the first step towards understanding stability guarantees for nonlinear RHC when the inner planning problem is solved to first-order stationary points, but not necessarily global optima. Special attention is given to feedback linearizable systems, and a mixture of positive and negative results are provided. We establish that, under certain strong conditions, first-order solutions to RHC exponentially stabilize linearizable systems. Surprisingly, these conditions can hold even in situations where there may be \textit{spurious local minima.} Crucially, this guarantee requires that state costs applied to the planning problems are in a certain sense `compatible' with the global geometry of the system, and a simple counter-example demonstrates the necessity of this condition. These results highlight the need to rethink the role of global geometry in the context of optimization-based control.

math.OC↗
arXiv2023

Optimal Data Selection: An Online Distributed View

The blessing of ubiquitous data also comes with a curse: the communication, storage, and labeling of massive, mostly redundant datasets. We seek to solve this problem at its core, collecting only valuable data and throwing out the rest via submodular maximization. Specifically, we develop algorithms for the online and distributed version of the problem, where data selection occurs in an uncoordinated fashion across multiple data streams. We design a general and flexible core selection routine for our algorithms which, given any stream of data, any assessment of its value, and any formulation of its selection cost, extracts the most valuable subset of the stream up to a constant factor while using minimal memory. Notably, our methods have the same theoretical guarantees as their offline counterparts, and, as far as we know, provide the first guarantees for online distributed submodular optimization in the literature. Finally, in learning tasks on ImageNet and MNIST, we show that our selection methods outperform random selection by $5-20\%$.

cs.LG↗
arXiv2023

Operationalizing Counterfactual Metrics: Incentives, Ranking, and Information Asymmetry

From the social sciences to machine learning, it has been well documented that metrics to be optimized are not always aligned with social welfare. In healthcare, Dranove et al. (2003) showed that publishing surgery mortality metrics actually harmed the welfare of sicker patients by increasing provider selection behavior. We analyze the incentive misalignments that arise from such average treated outcome metrics, and show that the incentives driving treatment decisions would align with maximizing total patient welfare if the metrics (i) accounted for counterfactual untreated outcomes and (ii) considered total welfare instead of averaging over treated patients. Operationalizing this, we show how counterfactual metrics can be modified to behave reasonably in patient-facing ranking systems. Extending to realistic settings when providers observe more about patients than the regulatory agencies do, we bound the decay in performance by the degree of information asymmetry between principal and agent. In doing so, our model connects principal-agent information asymmetry with unobserved heterogeneity in causal inference.

cs.LG↗
arXiv2023

Prediction-Powered Inference

Prediction-powered inference is a framework for performing valid statistical inference when an experimental dataset is supplemented with predictions from a machine-learning system. The framework yields simple algorithms for computing provably valid confidence intervals for quantities such as means, quantiles, and linear and logistic regression coefficients, without making any assumptions on the machine-learning algorithm that supplies the predictions. Furthermore, more accurate predictions translate to smaller confidence intervals. Prediction-powered inference could enable researchers to draw valid and more data-efficient conclusions using machine learning. The benefits of prediction-powered inference are demonstrated with datasets from proteomics, astronomy, genomics, remote sensing, census analysis, and ecology.

stat.ML↗
arXiv2023

Contract Design With Safety Inspections

We study the role of regulatory inspections in a contract design problem in which a principal interacts separately with multiple agents. Each agent's hidden action includes a dimension that determines whether they undertake an extra costly step to adhere to safety protocols. The principal's objective is to use payments combined with a limited budget for random inspections to incentivize agents towards safety-compliant actions that maximize the principal's utility. We first focus on the single-agent setting with linear contracts and present an efficient algorithm that characterizes the optimal linear contract, which includes both payment and random inspection. We further investigate how the optimal contract changes as the inspection cost or the cost of adhering to safety protocols vary. Notably, we demonstrate that the agent's compensation increases if either of these costs escalates. However, while the probability of inspection decreases with rising inspection costs, it demonstrates nonmonotonic behavior as a function of the safety action costs. Lastly, we explore the multi-agent setting, where the principal's challenge is to determine the best distribution of inspection budgets among all agents. We propose an efficient approach based on dynamic programming to find an approximately optimal allocation of inspection budget across contracts. We also design a random sequential scheme to determine the inspector's assignments, ensuring each agent is inspected at most once and at the desired probability. Finally, we present a case study illustrating that a mere difference in the cost of inspection across various agents can drive the principal's decision to forego inspecting a significant fraction of them, concentrating its entire budget on those that are less costly to inspect.

cs.GT↗
arXiv2023

Fine-Tuning Language Models with Advantage-Induced Policy Alignment

Reinforcement learning from human feedback (RLHF) has emerged as a reliable approach to aligning large language models (LLMs) to human preferences. Among the plethora of RLHF techniques, proximal policy optimization (PPO) is of the most widely used methods. Despite its popularity, however, PPO may suffer from mode collapse, instability, and poor sample efficiency. We show that these issues can be alleviated by a novel algorithm that we refer to as Advantage-Induced Policy Alignment (APA), which leverages a squared error loss function based on the estimated advantages. We demonstrate empirically that APA consistently outperforms PPO in language tasks by a large margin, when a separate reward model is employed as the evaluator. In addition, compared with PPO, APA offers a more stable form of control over the deviation from the model's initial policy, ensuring that the model improves its performance without collapsing to deterministic output. In addition to empirical results, we also provide a theoretical justification supporting the design of our loss function.

cs.CL↗
arXiv2023

Class-Conditional Conformal Prediction with Many Classes

Standard conformal prediction methods provide a marginal coverage guarantee, which means that for a random test point, the conformal prediction set contains the true label with a user-specified probability. In many classification problems, we would like to obtain a stronger guarantee--that for test points of a specific class, the prediction set contains the true label with the same user-chosen probability. For the latter goal, existing conformal prediction methods do not work well when there is a limited amount of labeled data per class, as is often the case in real applications where the number of classes is large. We propose a method called clustered conformal prediction that clusters together classes having "similar" conformal scores and performs conformal prediction at the cluster level. Based on empirical evaluation across four image data sets with many (up to 1000) classes, we find that clustered conformal typically outperforms existing methods in terms of class-conditional coverage and set size metrics.

stat.ML↗
arXiv2023

A Unifying Perspective on Multi-Calibration: Game Dynamics for Multi-Objective Learning

We provide a unifying framework for the design and analysis of multicalibrated predictors. By placing the multicalibration problem in the general setting of multi-objective learning -- where learning guarantees must hold simultaneously over a set of distributions and loss functions -- we exploit connections to game dynamics to achieve state-of-the-art guarantees for a diverse set of multicalibration learning problems. In addition to shedding light on existing multicalibration guarantees and greatly simplifying their analysis, our approach also yields improved guarantees, such as obtaining stronger multicalibration conditions that scale with the square-root of group size and improving the complexity of $k$-class multicalibration by an exponential factor of $k$. Beyond multicalibration, we use these game dynamics to address emerging considerations in the study of group fairness and multi-distribution learning.

cs.LG↗
arXiv2023

On Optimal Caching and Model Multiplexing for Large Model Inference

Large Language Models (LLMs) and other large foundation models have achieved noteworthy success, but their size exacerbates existing resource consumption and latency challenges. In particular, the large-scale deployment of these models is hindered by the significant resource requirements during inference. In this paper, we study two approaches for mitigating these challenges: employing a cache to store previous queries and learning a model multiplexer to choose from an ensemble of models for query processing. Theoretically, we provide an optimal algorithm for jointly optimizing both approaches to reduce the inference cost in both offline and online tabular settings. By combining a caching algorithm, namely Greedy Dual Size with Frequency (GDSF) or Least Expected Cost (LEC), with a model multiplexer, we achieve optimal rates in both offline and online settings. Empirically, simulations show that the combination of our caching and model multiplexing algorithms greatly improves over the baselines, with up to $50\times$ improvement over the baseline when the ratio between the maximum cost and minimum cost is $100$. Experiments on real datasets show a $4.3\times$ improvement in FLOPs over the baseline when the ratio for FLOPs is $10$, and a $1.8\times$ improvement in latency when the ratio for average latency is $1.85$.

cs.LG↗
arXiv2023

Nesterov Meets Optimism: Rate-Optimal Separable Minimax Optimization

We propose a new first-order optimization algorithm -- AcceleratedGradient-OptimisticGradient (AG-OG) Descent Ascent -- for separable convex-concave minimax optimization. The main idea of our algorithm is to carefully leverage the structure of the minimax problem, performing Nesterov acceleration on the individual component and optimistic gradient on the coupling component. Equipped with proper restarting, we show that AG-OG achieves the optimal convergence rate (up to a constant) for a variety of settings, including bilinearly coupled strongly convex-strongly concave minimax optimization (bi-SC-SC), bilinearly coupled convex-strongly concave minimax optimization (bi-C-SC), and bilinear games. We also extend our algorithm to the stochastic setting and achieve the optimal convergence rate in both bi-SC-SC and bi-C-SC settings. AG-OG is the first single-call algorithm with optimal convergence rates in both deterministic and stochastic settings for bilinearly coupled minimax optimization problems.

math.OC↗
arXiv2023

Last-Iterate Convergence of Saddle-Point Optimizers via High-Resolution Differential Equations

Several widely-used first-order saddle-point optimization methods yield an identical continuous-time ordinary differential equation (ODE) that is identical to that of the Gradient Descent Ascent (GDA) method when derived naively. However, the convergence properties of these methods are qualitatively different, even on simple bilinear games. Thus the ODE perspective, which has proved powerful in analyzing single-objective optimization methods, has not played a similar role in saddle-point optimization. We adopt a framework studied in fluid dynamics -- known as High-Resolution Differential Equations (HRDEs) -- to design differential equation models for several saddle-point optimization methods. Critically, these HRDEs are distinct for various saddle-point optimization methods. Moreover, in bilinear games, the convergence properties of the HRDEs match the qualitative features of the corresponding discrete methods. Additionally, we show that the HRDE of Optimistic Gradient Descent Ascent (OGDA) exhibits \emph{last-iterate convergence} for general monotone variational inequalities. Finally, we provide rates of convergence for the \emph{best-iterate convergence} of the OGDA method, relying solely on the first-order smoothness of the monotone operator.

math.OC↗
arXiv2023

Scaff-PD: Communication Efficient Fair and Robust Federated Learning

We present Scaff-PD, a fast and communication-efficient algorithm for distributionally robust federated learning. Our approach improves fairness by optimizing a family of distributionally robust objectives tailored to heterogeneous clients. We leverage the special structure of these objectives, and design an accelerated primal dual (APD) algorithm which uses bias corrected local steps (as in Scaffold) to achieve significant gains in communication efficiency and convergence speed. We evaluate Scaff-PD on several benchmark datasets and demonstrate its effectiveness in improving fairness and robustness while maintaining competitive accuracy. Our results suggest that Scaff-PD is a promising approach for federated learning in resource-constrained and heterogeneous settings.

cs.LG↗
arXiv2023

Modeling Content Creator Incentives on Algorithm-Curated Platforms

Content creators compete for user attention. Their reach crucially depends on algorithmic choices made by developers on online platforms. To maximize exposure, many creators adapt strategically, as evidenced by examples like the sprawling search engine optimization industry. This begets competition for the finite user attention pool. We formalize these dynamics in what we call an exposure game, a model of incentives induced by algorithms, including modern factorization and (deep) two-tower architectures. We prove that seemingly innocuous algorithmic choices, e.g., non-negative vs. unconstrained factorization, significantly affect the existence and character of (Nash) equilibria in exposure games. We proffer use of creator behavior models, like exposure games, for an (ex-ante) pre-deployment audit. Such an audit can identify misalignment between desirable and incentivized content, and thus complement post-hoc measures like content filtering and moderation. To this end, we propose tools for numerically finding equilibria in exposure games, and illustrate results of an audit on the MovieLens and LastFM datasets. Among else, we find that the strategically produced content exhibits strong dependence between algorithmic exploration and content diversity, and between model expressivity and bias towards gender-based user and creator groups.

cs.GT↗
arXiv2023

Optimization on manifolds: A symplectic approach

Optimization tasks are crucial in statistical machine learning. Recently, there has been great interest in leveraging tools from dynamical systems to derive accelerated and robust optimization methods via suitable discretizations of continuous-time systems. However, these ideas have mostly been limited to Euclidean spaces and unconstrained settings, or to Riemannian gradient flows. In this work, we propose a dissipative extension of Dirac's theory of constrained Hamiltonian systems as a general framework for solving optimization problems over smooth manifolds, including problems with nonlinear constraints. We develop geometric/symplectic numerical integrators on manifolds that are "rate-matching," i.e., preserve the continuous-time rates of convergence. In particular, we introduce a dissipative RATTLE integrator able to achieve optimal convergence rate locally. Our class of (accelerated) algorithms are not only simple and efficient but also applicable to a broad range of contexts.

cond-mat.stat-mech↗