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The informativeness of the gradient revisited

Rustem Takhanov

TL;DR

The paper investigates the limits of gradient-based learning when targets come from almost pairwise independent function classes. It introduces an Integral Probability Metric-based measure of almost pairwise independence and proves a general bound on the gradient variance Var_h[∇C_h(w)] = Ō̃(ε + e^{- rac{1}{2}\mathcal{E}_c}) that intertwines target independence, input collision entropy, and model regularity. Applying the bound to Learning with Errors (LWE) and high-frequency functions reveals that uniform input distributions render gradient-based attacks ineffective due to exponentially small variance, while non-uniform inputs with low collision entropy can render such attacks more feasible; this is corroborated by empirical analysis of sparse secret LWE variants. The work also analyzes high-frequency targets, showing the informativeness of gradients decays with the frequency parameter R, yielding barren plateaus unless inputs are tuned to boost informative signals. Overall, the framework provides both theoretical limits and practical guidance for evaluating cryptographic primitives against gradient-based techniques and highlights open questions about constructing favorable input distributions from less informative samples.

Abstract

In the past decade gradient-based deep learning has revolutionized several applications. However, this rapid advancement has highlighted the need for a deeper theoretical understanding of its limitations. Research has shown that, in many practical learning tasks, the information contained in the gradient is so minimal that gradient-based methods require an exceedingly large number of iterations to achieve success. The informativeness of the gradient is typically measured by its variance with respect to the random selection of a target function from a hypothesis class. We use this framework and give a general bound on the variance in terms of a parameter related to the pairwise independence of the target function class and the collision entropy of the input distribution. Our bound scales as $ \tilde{\mathcal{O}}(\varepsilon+e^{-\frac{1}{2}\mathcal{E}_c}) $, where $ \tilde{\mathcal{O}} $ hides factors related to the regularity of the learning model and the loss function, $ \varepsilon $ measures the pairwise independence of the target function class and $\mathcal{E}_c$ is the collision entropy of the input distribution. To demonstrate the practical utility of our bound, we apply it to the class of Learning with Errors (LWE) mappings and high-frequency functions. In addition to the theoretical analysis, we present experiments to understand better the nature of recent deep learning-based attacks on LWE.

The informativeness of the gradient revisited

TL;DR

The paper investigates the limits of gradient-based learning when targets come from almost pairwise independent function classes. It introduces an Integral Probability Metric-based measure of almost pairwise independence and proves a general bound on the gradient variance Var_h[∇C_h(w)] = Ō̃(ε + e^{- rac{1}{2}\mathcal{E}_c}) that intertwines target independence, input collision entropy, and model regularity. Applying the bound to Learning with Errors (LWE) and high-frequency functions reveals that uniform input distributions render gradient-based attacks ineffective due to exponentially small variance, while non-uniform inputs with low collision entropy can render such attacks more feasible; this is corroborated by empirical analysis of sparse secret LWE variants. The work also analyzes high-frequency targets, showing the informativeness of gradients decays with the frequency parameter R, yielding barren plateaus unless inputs are tuned to boost informative signals. Overall, the framework provides both theoretical limits and practical guidance for evaluating cryptographic primitives against gradient-based techniques and highlights open questions about constructing favorable input distributions from less informative samples.

Abstract

In the past decade gradient-based deep learning has revolutionized several applications. However, this rapid advancement has highlighted the need for a deeper theoretical understanding of its limitations. Research has shown that, in many practical learning tasks, the information contained in the gradient is so minimal that gradient-based methods require an exceedingly large number of iterations to achieve success. The informativeness of the gradient is typically measured by its variance with respect to the random selection of a target function from a hypothesis class. We use this framework and give a general bound on the variance in terms of a parameter related to the pairwise independence of the target function class and the collision entropy of the input distribution. Our bound scales as , where hides factors related to the regularity of the learning model and the loss function, measures the pairwise independence of the target function class and is the collision entropy of the input distribution. To demonstrate the practical utility of our bound, we apply it to the class of Learning with Errors (LWE) mappings and high-frequency functions. In addition to the theoretical analysis, we present experiments to understand better the nature of recent deep learning-based attacks on LWE.

Paper Structure

This paper contains 18 sections, 15 theorems, 113 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Examples of almost pairwise independent classes
  3. Measuring almost pairwise independence by the Integral Probability Metric
  4. Gradient-based optimization
  5. Upper bound on the variance
  6. Application of Theorem \ref{['aes-bound']} to attacks on LWE
  7. Experiments with binary and ternary secrets
  8. Application of Theorem \ref{['aes-bound']} to high-frequency functions
  9. Nature of the barren plateau phenomenon: example
  10. Applications
  11. Conclusions and open problems
  12. Proof of Theorem \ref{['aes-bound']}
  13. Proof of Lemma \ref{['aes-rough']}
  14. Proofs for Section \ref{['LWE-case']}
  15. Proofs for Section \ref{['waves']}
  16. ...and 3 more sections

Key Result

Theorem 1

Suppose that $\delta > 0$ is chosen such that ${\rm Var}_{h \sim \chi}[\nabla C_h(\mathbf{w})] \leq \delta^3$ for any $\mathbf{w} \in O$. Then, a $\delta$-accurate gradient oracle can be defined, guaranteeing that for any algorithm of the specified type and any probability $p \in (0, 1)$, the algori

Figures (3)

  • Figure 1: Scatter and linear regression plots for "$-\log(\varepsilon)$ vs $\log(|\mathcal{H}|)$" for different $a=2,\cdots,q$.
  • Figure 2: The objective function $C_h(\omega)$ for $\psi(x) = \{x\}$, where $\{x\}$ is a fractional part of $x$, and for frequencies $w=10$ and $w=40$.
  • Figure 3: We trained a 3-layer fully connected neural network with ReLU activations to approximate a high-frequency wave $\psi(ax)$ where $\psi(x) = \{x\}$. For each value of $A$, $a$ was randomly sampled 5 times from the set $\{0,1,\cdots, A-1\}$. The plot shows the mean squared error (MSE) on a test set averaged over these trials as a function of training epochs. The horizontal asymptote represents the MSE of the random guessing, ${\rm MSE} = \frac{1}{12}$. For the first picture and the second pictures the number of neurons on layers are $[1,64,128,1]$ and $[1,640,1280,1]$ (over-parameterized model) respectively. As can be seen there is not much gain in adding more parameters to a model.

Theorems & Definitions (36)

  • Example 1
  • Example 2
  • Theorem 1: DBLP:journals/jmlr/Shamir18
  • Theorem 2
  • Remark 1
  • Theorem 3
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 4
  • ...and 26 more