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Prospective Learning in Retrospect

Yuxin Bai, Cecelia Shuai, Ashwin De Silva, Siyu Yu, Pratik Chaudhari, Joshua T. Vogelstein

TL;DR

This work addresses the misalignment of PAC-based learning with nonstationary data and evolving objectives by adopting prospective learning, which treats data as drawn from a stochastic process and aims to minimize future risk via a time-indexed sequence of predictors. The authors extend the framework with practical instantiations: Prospective-ERM and Prospective-MLP using a time embedding, and nonparametric Prospective CART/GBTs, plus a foraging application that demonstrates sequential decision-making under a one-life constraint. Key findings show that (i) Prospective-MLPs remain robust under heterogeneous sampling and can handle infinite-task regimes with appropriate time embeddings, (ii) streaming-online training is feasible but slows convergence compared with batched training, (iii) prospective forests perform competitively with Prospective-MLPs and offer interpretability, and (iv) prospective foraging can surpass standard actor–critic RL in a one-life task when time embeddings are used. Collectively, these results validate prospective learning as a versatile framework for dynamic environments, with potential impact in supervised and reinforcement-learning contexts where task distributions drift over time.

Abstract

In most real-world applications of artificial intelligence, the distributions of the data and the goals of the learners tend to change over time. The Probably Approximately Correct (PAC) learning framework, which underpins most machine learning algorithms, fails to account for dynamic data distributions and evolving objectives, often resulting in suboptimal performance. Prospective learning is a recently introduced mathematical framework that overcomes some of these limitations. We build on this framework to present preliminary results that improve the algorithm and numerical results, and extend prospective learning to sequential decision-making scenarios, specifically foraging. Code is available at: https://github.com/neurodata/prolearn2.

Prospective Learning in Retrospect

TL;DR

This work addresses the misalignment of PAC-based learning with nonstationary data and evolving objectives by adopting prospective learning, which treats data as drawn from a stochastic process and aims to minimize future risk via a time-indexed sequence of predictors. The authors extend the framework with practical instantiations: Prospective-ERM and Prospective-MLP using a time embedding, and nonparametric Prospective CART/GBTs, plus a foraging application that demonstrates sequential decision-making under a one-life constraint. Key findings show that (i) Prospective-MLPs remain robust under heterogeneous sampling and can handle infinite-task regimes with appropriate time embeddings, (ii) streaming-online training is feasible but slows convergence compared with batched training, (iii) prospective forests perform competitively with Prospective-MLPs and offer interpretability, and (iv) prospective foraging can surpass standard actor–critic RL in a one-life task when time embeddings are used. Collectively, these results validate prospective learning as a versatile framework for dynamic environments, with potential impact in supervised and reinforcement-learning contexts where task distributions drift over time.

Abstract

In most real-world applications of artificial intelligence, the distributions of the data and the goals of the learners tend to change over time. The Probably Approximately Correct (PAC) learning framework, which underpins most machine learning algorithms, fails to account for dynamic data distributions and evolving objectives, often resulting in suboptimal performance. Prospective learning is a recently introduced mathematical framework that overcomes some of these limitations. We build on this framework to present preliminary results that improve the algorithm and numerical results, and extend prospective learning to sequential decision-making scenarios, specifically foraging. Code is available at: https://github.com/neurodata/prolearn2.

Paper Structure

This paper contains 20 sections, 7 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Prospective Learning Scenarios
  4. Example Stochastic Processes
  5. Relationship between Continual and Prospective Learning
  6. Training and Evaluating Learners
  7. Several Empirical Observations on Prospective-MLPs
  8. Prospective-MLP prevails under heterogeneous sampling.
  9. Prospective-MLP prevails in infinite task scenarios.
  10. Prospective-MLPs can be trained in a streaming or online manner.
  11. Prospective Forests
  12. Motivation and background
  13. Preliminary results
  14. Prospective Foraging
  15. Motivation and background
  16. ...and 5 more sections

Figures (7)

  • Figure 1: Pictorial depictions of 3 types of stochastic processes considered in our experiments. (Left) Periodic Process, (Middle) Linear process, and (Right) Hierarchical hidden Markov Process. The periodic and linear processes belong to scenario 2 whereas the hierarchical hidden Markov process is an instance of scenario 3.
  • Figure 2: Instantaneous (top) and prospective (bottom) risks of Follow-the-Leader (FTL, blue) and Prospective-MLP (P-MLP, red) trained on homogeneously (lighter shade) and heterogeneously (darker shade) sampled data from the periodic process. Homogeneous sampling is where you get exactly one sample each time step. In heterogeneous sampling, there can be missing samples and/or multiple samples available per time step.
  • Figure 3: Prospective risk of Follow-the-Leader (FTL), and Prospective-MLP with Fourier embeddings, and Prospective-MLP with monomial embeddings on periodic (Right top) and linear (Right bottom) processes. Prospective-MLP with Fourier embeddings performs best on the periodic process, whereas the variant with monomial embeddings achieves the best performance on the linear process.
  • Figure 4: Prospective risk of the learners that are trained in an online manner on data from the periodic process.
  • Figure 5: Prospective risk of Prospective-GBTs (red), Prospective-MLP (green) and Time-agnostic Gradient Boosted trees (Plain-GBTs, blue) across two scenarios where, (1) data is independent but not identically distributed (Left), and (2) data is neither independent nor identically distributed (Right). In both cases, the risk of Prospective-GBTs and Prospective-MLP approach the Bayes risk, with Prospective-GBTs converging faster. In contrast, the time-agnostic GBTs do not converge consistently. For comparison, the chance prospective risk is $0.5$ in the left panel and $0.3$ in the right panel.
  • ...and 2 more figures

Theorems & Definitions (3)

  • definition thmcounterdefinition: Prospective CART
  • definition thmcounterdefinition: Prospective Gradient Boosted Trees
  • definition thmcounterdefinition: Prospective Forager