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Induced Model Matching: Restricted Models Help Train Full-Featured Models

Usama Muneeb, Mesrob I. Ohannessian

TL;DR

The IMM principle is generally applicable in common scenarios where restricted data is cheaper to collect or restricted models are easier to learn.

Abstract

We consider scenarios where a very accurate (often small) predictive model using restricted features is available when training a full-featured (often larger) model. This restricted model may be thought of as side-information'', and can come either from an auxiliary dataset or from the same dataset by forcing the restriction. How can the restricted model be useful to the full model? To answer this, we introduce a methodology called Induced Model Matching (IMM). IMM aligns the context-restricted, or induced, version of the large model with the restricted model. We relate IMM to approaches such as noising, which is implicit in addressing the problem, and reverse knowledge distillation from weak teachers, which is explicit but does not exploit restriction being the nature of the weakness. We show that these prior methods can be thought of as approximations to IMM and can be problematic in terms of consistency. Experimentally, we first motivate IMM using logistic regression as a toy example. We then explore it in language modeling, the application that initially inspired it, and demonstrate it on both LSTM and transformer full models, using bigrams as restricted models. We lastly give a simple RL example, which shows that POMDP policies can help learn better MDP policies. The IMM principle is thus generally applicable in common scenarios where restricted data is cheaper to collect or restricted models are easier to learn.

Induced Model Matching: Restricted Models Help Train Full-Featured Models

TL;DR

The IMM principle is generally applicable in common scenarios where restricted data is cheaper to collect or restricted models are easier to learn.

Abstract

We consider scenarios where a very accurate (often small) predictive model using restricted features is available when training a full-featured (often larger) model. This restricted model may be thought of as side-information'', and can come either from an auxiliary dataset or from the same dataset by forcing the restriction. How can the restricted model be useful to the full model? To answer this, we introduce a methodology called Induced Model Matching (IMM). IMM aligns the context-restricted, or induced, version of the large model with the restricted model. We relate IMM to approaches such as noising, which is implicit in addressing the problem, and reverse knowledge distillation from weak teachers, which is explicit but does not exploit restriction being the nature of the weakness. We show that these prior methods can be thought of as approximations to IMM and can be problematic in terms of consistency. Experimentally, we first motivate IMM using logistic regression as a toy example. We then explore it in language modeling, the application that initially inspired it, and demonstrate it on both LSTM and transformer full models, using bigrams as restricted models. We lastly give a simple RL example, which shows that POMDP policies can help learn better MDP policies. The IMM principle is thus generally applicable in common scenarios where restricted data is cheaper to collect or restricted models are easier to learn.
Paper Structure (63 sections, 1 theorem, 38 equations, 10 figures, 5 tables, 2 algorithms)

This paper contains 63 sections, 1 theorem, 38 equations, 10 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Works
  3. Noising and Data Augmentation in Language Modeling
  4. $N$-grams and their Merits
  5. Knowledge Distillation
  6. Problem Description
  7. Problem Setting
  8. Induced Models
  9. Problem Statement
  10. Induced Model Matching (IMM)
  11. Construction of the IMM risk
  12. IMM as a regularizer
  13. Separating IMM into components
  14. Computation
  15. Sampled IMM
  16. ...and 48 more sections

Key Result

Proposition 6.1

Assume that we are optimizing the idealized noising objective of Eq. eq:noising --- i.e., we are operating in the infinite-data regime --- and let $Q^\star$ be its global minimizer. Assume further that the model class for $Q$ contains the true model $P$ --- i.e., we are in the realizable case. Then,

Figures (10)

  • Figure 1: Comparing test accuracy of logistic model, trained using interpolation, noising, and IMM (with a Bayes Optimal $\overline P$). Bars are $10^\textrm{th}$ to $90^\textrm{th}$ percentiles of $300$ runs.
  • Figure 2: Average reward of MDP trained without and with IMM incorporating POMDP. Details in Appendix \ref{['appendix:additional-experimental-details']}.
  • Figure 3: Schematic overview of IMM.
  • Figure 4: Performance on restricted task, i.e. $\textsf{IMM}(Q)$ measured on models $Q$ trained using Eq. \ref{['eq:main-objective-logreg']} as the objective with varying $\tfrac{\lambda}{1+\lambda}$ ratio (refer to Appendix \ref{['appendix:lambda']}). We stop at $\tfrac{\lambda}{1+\lambda}=0.9$ because $\tfrac{\lambda}{1+\lambda}=1$ would zero out the contribution of the main objective (and replacing the main objective completely is never the intention).
  • Figure 5: A visualization of the inductive bias brought upon by IMM in the logistic regression example.
  • ...and 5 more figures

Theorems & Definitions (2)

  • Proposition 6.1
  • proof : Proof