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Generalization Bounds for Quantum Learning via Rényi Divergences

Naqueeb Ahmad Warsi, Ayanava Dasgupta, Masahito Hayashi

TL;DR

The paper extends classical generalization bounds to quantum learning by deriving upper bounds on the expected generalization error in terms of quantum and classical Rényi divergences, using a variational approach and a new definition of the expected true loss. It introduces the modified sandwich quantum Rényi divergence, demonstrates its advantages over the Petz form, and shows how these bounds recover Caro23 as a special case while offering improved tighter bounds both analytically and numerically. The work also provides probabilistic generalization bounds via Hölder-type inequalities and smooth max Rényi divergences, and analyzes the i.i.d. setting to connect with prior results. Overall, the framework deepens understanding of how quantum information measures govern generalization under measurement and disturbance, with clear links to the Caro et al. quantum learning model.

Abstract

This work advances the theoretical understanding of quantum learning by establishing a new family of upper bounds on the expected generalization error of quantum learning algorithms, leveraging the framework introduced by Caro et al. (2024) and a new definition for the expected true loss. Our primary contribution is the derivation of these bounds in terms of quantum and classical Rényi divergences, utilizing a variational approach for evaluating quantum Rényi divergences, specifically the Petz and a newly introduced modified sandwich quantum Rényi divergence. Analytically and numerically, we demonstrate the superior performance of the bounds derived using the modified sandwich quantum Rényi divergence compared to those based on the Petz divergence. Furthermore, we provide probabilistic generalization error bounds using two distinct techniques: one based on the modified sandwich quantum Rényi divergence and classical Rényi divergence, and another employing smooth max Rényi divergence.

Generalization Bounds for Quantum Learning via Rényi Divergences

TL;DR

The paper extends classical generalization bounds to quantum learning by deriving upper bounds on the expected generalization error in terms of quantum and classical Rényi divergences, using a variational approach and a new definition of the expected true loss. It introduces the modified sandwich quantum Rényi divergence, demonstrates its advantages over the Petz form, and shows how these bounds recover Caro23 as a special case while offering improved tighter bounds both analytically and numerically. The work also provides probabilistic generalization bounds via Hölder-type inequalities and smooth max Rényi divergences, and analyzes the i.i.d. setting to connect with prior results. Overall, the framework deepens understanding of how quantum information measures govern generalization under measurement and disturbance, with clear links to the Caro et al. quantum learning model.

Abstract

This work advances the theoretical understanding of quantum learning by establishing a new family of upper bounds on the expected generalization error of quantum learning algorithms, leveraging the framework introduced by Caro et al. (2024) and a new definition for the expected true loss. Our primary contribution is the derivation of these bounds in terms of quantum and classical Rényi divergences, utilizing a variational approach for evaluating quantum Rényi divergences, specifically the Petz and a newly introduced modified sandwich quantum Rényi divergence. Analytically and numerically, we demonstrate the superior performance of the bounds derived using the modified sandwich quantum Rényi divergence compared to those based on the Petz divergence. Furthermore, we provide probabilistic generalization error bounds using two distinct techniques: one based on the modified sandwich quantum Rényi divergence and classical Rényi divergence, and another employing smooth max Rényi divergence.
Paper Structure (22 sections, 21 theorems, 202 equations, 2 figures, 2 tables)

This paper contains 22 sections, 21 theorems, 202 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Notations, Facts and Definitions
  3. Discussion on Variational lower-bound on Divergence and its application
  4. Quantum learning framework proposed by Caro et al. Caro23
  5. Generalized Classical Learning Framework
  6. Quantum Learning Framework
  7. Justification for the validity of the proposed definition of expected true loss (Definition \ref{['exp_true_loss_def']})
  8. Discussion on previous works
  9. Classical learning paradigm
  10. Quantum learning paradigm
  11. Quantum Rényi Divergences based Bounds on Expected Generalization error
  12. Bounds on the expected quantum generalization error
  13. Bounds on the expected quantum generalization error under i.i.d assumption of quantum data
  14. Comparision between the results obtained in Theorems \ref{['Caro23_result_mod']}, \ref{['theo_exxp_gen_err_bound_renyi']} and Corollary \ref{['corr_exxp_gen_err_bound_renyi_weak']}
  15. Quantum Rényi Divergences based Bounds on Generalization error in probability
  16. ...and 7 more sections

Key Result

Lemma 1

Given a quantum state $\rho \in \mathcal{D}(\mathcal{H})$ and a self-adjoint operator $L \in \mathcal{B}(\mathcal{H})$ such that $a\mathbb{I} \preceq L \preceq b\mathbb{I}$ (where $a\geq b$ and $a,b \in \mathbb{R}$ and $\mathbb{I}$ denotes the projection over $\mathcal{H}$). Then, $\forall \lambda \ or equivalently,

Figures (2)

  • Figure 1: Quantum learning algorithm structure proposed by Caro23.
  • Figure 2: RHS of eq. \ref{['Caro23_result_mod_eq']} vs RHS of eq. \ref{['exp_gen_var_bound1']} vs RHS of eq. \ref{['exp_gen_var_bound_weak1']}.

Theorems & Definitions (65)

  • Definition 1
  • Definition 2: Rényi Divergence Renyi1961
  • Definition 3: Smooth max Rényi divergence Warsi2016
  • Definition 4: Convex conjugate of a function fenchel2014conjugate
  • Definition 5: sub-Gaussianity of random variables BLM_Concentration_2013
  • Definition 6: bhatia2013matrix
  • Definition 7
  • Definition 8: Measured Rényi Divergence fuchs1996distinguishabilityaccessibleinformationquantum
  • Definition 9: Petz Quantum Rényi Divergence Petz1986
  • Definition 10: Minimal/ Sandwiched Quantum Rényi Divergence MDSF_Renyi_2013, Wilde2014_Sandwich
  • ...and 55 more