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Probability Tools for Sequential Random Projection

Yingru Li

TL;DR

The paper tackles sequential decision-making under uncertainty where dimensionality reduction must preserve geometry despite adaptive dependence between decisions and projection vectors. It develops a probabilistic framework built around a stopped martingale construction and the method of mixtures within a self-normalized process to obtain a non-asymptotic bound that extends the Johnson-Lindenstrauss lemma to sequential settings. Key contributions include introducing a stopped process, an exponential supermartingale, and an any-time concentration bound to derive a sequential JL inequality, with potential relaxations such as removing the unit-norm constraint. The framework provides a foundational tool for sequential, high-dimensional data processing in adaptive environments, with potential impact on reinforcement learning, bandit problems, and online dimensionality reduction.

Abstract

We introduce the first probabilistic framework tailored for sequential random projection, an approach rooted in the challenges of sequential decision-making under uncertainty. The analysis is complicated by the sequential dependence and high-dimensional nature of random variables, a byproduct of the adaptive mechanisms inherent in sequential decision processes. Our work features a novel construction of a stopped process, facilitating the analysis of a sequence of concentration events that are interconnected in a sequential manner. By employing the method of mixtures within a self-normalized process, derived from the stopped process, we achieve a desired non-asymptotic probability bound. This bound represents a non-trivial martingale extension of the Johnson-Lindenstrauss (JL) lemma, marking a pioneering contribution to the literature on random projection and sequential analysis.

Probability Tools for Sequential Random Projection

TL;DR

The paper tackles sequential decision-making under uncertainty where dimensionality reduction must preserve geometry despite adaptive dependence between decisions and projection vectors. It develops a probabilistic framework built around a stopped martingale construction and the method of mixtures within a self-normalized process to obtain a non-asymptotic bound that extends the Johnson-Lindenstrauss lemma to sequential settings. Key contributions include introducing a stopped process, an exponential supermartingale, and an any-time concentration bound to derive a sequential JL inequality, with potential relaxations such as removing the unit-norm constraint. The framework provides a foundational tool for sequential, high-dimensional data processing in adaptive environments, with potential impact on reinforcement learning, bandit problems, and online dimensionality reduction.

Abstract

We introduce the first probabilistic framework tailored for sequential random projection, an approach rooted in the challenges of sequential decision-making under uncertainty. The analysis is complicated by the sequential dependence and high-dimensional nature of random variables, a byproduct of the adaptive mechanisms inherent in sequential decision processes. Our work features a novel construction of a stopped process, facilitating the analysis of a sequence of concentration events that are interconnected in a sequential manner. By employing the method of mixtures within a self-normalized process, derived from the stopped process, we achieve a desired non-asymptotic probability bound. This bound represents a non-trivial martingale extension of the Johnson-Lindenstrauss (JL) lemma, marking a pioneering contribution to the literature on random projection and sequential analysis.
Paper Structure (13 sections, 6 theorems, 57 equations, 1 figure)

This paper contains 13 sections, 6 theorems, 57 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Sequential random projection
  3. Unpacking the Challenges.
  4. Innovations and Contributions.
  5. Probabilistic formalism & statements
  6. Probabilistic formalism
  7. Probability tools for sequential random projection
  8. Technical ideas & details
  9. Stopped process and exponential supermartingale
  10. Proof of \ref{['thm:seq-jl']}
  11. Conclusions
  12. Proof of \ref{['thm:anytime-bound']}: Method of mixtures
  13. Additional lemmas

Key Result

Lemma 1

For any $\alpha, \beta \in \mathbb{R}_+$ with $\alpha \ge \beta$, random variable $X \sim \operatorname{Beta}(\alpha, \beta)$ has variance ${ \operatorname{Var}_{}\left(X\right)} = \frac{\alpha \beta}{(\alpha + \beta)^2 (\alpha + \beta + 1)}$ and the centered MGF $\mathbb{E}_{}[\exp(\lambda(X - \mat

Figures (1)

  • Figure 1: Sequential dependence of high-dimensional random variables due to the adaptive nature of sequential decision-making.

Theorems & Definitions (26)

  • Definition 1: Adapted process
  • Definition 2: (Conditionally) $\sigma$-sub-Gaussian
  • Definition 3: Almost sure unit-norm
  • Lemma 1: MGF of Beta distribution li2024simple
  • Example 1: Uniform distribution over sphere $\mathcal{U}(\mathbb{S}^{M-1})$
  • Definition 4: Square-bounded process
  • Theorem 1: Sequential random projection in adaptive process
  • Remark 1
  • Remark 2
  • Example 2: Stylized stochastic process satisfying the condition in \ref{['thm:seq-jl']}.
  • ...and 16 more