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Claude 4 Opus (T) via @hdevalence.bsky.social
Thinking Summary

The user is asking about Algorithm 1 in the paper "Stealing Part of a Production Language Model". Let me look at Algorithm 1. Algorithm 1 is "Hidden-Dimension Extraction Attack" and appears in the paper. Looking at line 5: ``` \State $\Q_i \gets \api(\x_i)$ ``` The user is asking whether they're setting the i-th row or column of Q to the oracle output. Looking at the context, they initialize Q as an empty matrix $\Q = \mathbf{0}^{n \times l}$ where n is the number of queries and l is the vocabulary size. Since $\api(\x_i)$ returns a logit vector of dimension l, and Q has dimensions n × l, they must be setting the i-th row of Q to be the logit vector returned by the API. This makes sense because they're collecting n different logit vectors (one for each random prompt), and stacking them as rows in the matrix Q.

Looking at Algorithm 1 (Hidden-Dimension Extraction Attack), line 5 states:

\State $\Q_i \gets \api(\x_i)$

They are setting the i-th row of Q to the oracle output.

This is clear from the context:

  • Q is initialized as a n×ln \times l matrix (line 2)
  • The API returns a logit vector of dimension ll
  • They iterate from i=1i = 1 to nn (line 3)

So each query response \api(\xi)\api(\x_i) is a vector of length ll (the vocabulary size), and they're stacking these as rows to build the matrix Q. This is confirmed later in Lemma 1 where they define:

Let \Q\br\x1,…\xn∈Rl×n\Q\br{\x_1, \ldots \x_{n}} \in \R^{l \times n} denote the matrix with columns \api\br\x1,…,\api\br\xn\api\br{\x_1}, \ldots, \api\br{\x_{n}}

Wait, this seems inconsistent! The lemma says the API responses are columns, but the algorithm initialization suggests rows. Looking more carefully at the formalization section, they write \Q = \Et \cdot \H where \Et\Et is l×hl \times h and \H is h×nh \times n, which gives \Q\Q as l×nl \times n.

So actually, they're using column notation - \Qi\Q_i refers to the i-th column of Q, not row. Each API response is stored as a column vector in the l×nl \times n matrix Q.

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Claude 4 Opus (T) via @hdevalence.bsky.social
Thinking Summary

The user is asking me to redo my response without KaTeX syntax errors. Let me check my previous response for any KaTeX issues. I see that I used some notation that might not render properly. Let me rewrite it more carefully.

Looking at Algorithm 1 (Hidden-Dimension Extraction Attack), line 5 states:

State Q_i ← O(x_i)

They are setting the i-th column of Q to the oracle output.

This is clear from the context:

  • Q is initialized as an n×ln \times l matrix (line 2)
  • The API returns a logit vector of dimension ll
  • They iterate from i=1i = 1 to nn (line 3)

Initially this seems confusing because the algorithm initializes Q as n×ln \times l, but looking at the formalization section provides clarity. In Lemma 1 they define:

Let Q(x1,…,xn)∈Rl×n\mathbf{Q}(x_1, \ldots, x_n) \in \mathbb{R}^{l \times n} denote the matrix with columns O(x1),…,O(xn)\mathcal{O}(x_1), \ldots, \mathcal{O}(x_n)

So there's a notational inconsistency in the paper. The key insight is that they write Q=ET⋅H\mathbf{Q} = \mathbf{E}^T \cdot \mathbf{H} where:

  • ET\mathbf{E}^T is l×hl \times h (the embedding projection matrix)
  • H\mathbf{H} is h×nh \times n (hidden states)
  • This gives Q\mathbf{Q} as l×nl \times n

Therefore, QiQ_i refers to the i-th column of Q. Each API response (a logit vector of length ll) is stored as a column in the l×nl \times n matrix Q.

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Claude 4 Opus (T) via @hdevalence.bsky.social
Thinking Summary

The user is pointing out that there's likely a typo on line 2 of Algorithm 1. Let me check: Line 2 says: "Initialize an empty matrix Q = 0^{n × l}" But based on the formalization and Lemma 1, Q should be l × n (vocabulary size × number of queries), not n × l. So yes, this appears to be a typo in the algorithm.

Yes, exactly! Line 2 appears to have a typo.

Line 2 states:

Initialize an empty matrix Q = 0^{n×l}

But it should be:

Initialize an empty matrix Q = 0^{l×n}

This is a clear typo because:

  1. The formalization shows Q=ET⋅H\mathbf{Q} = \mathbf{E}^T \cdot \mathbf{H} where the result must be l×nl \times n
  2. Lemma 1 explicitly states: "Let Q(x1,…,xn)∈Rl×n\mathbf{Q}(x_1, \ldots, x_n) \in \mathbb{R}^{l \times n} denote the matrix with columns O(x1),…,O(xn)\mathcal{O}(x_1), \ldots, \mathcal{O}(x_n)"
  3. Each API call returns a logit vector of length ll, and they're collecting nn of these as columns

So the matrix Q should have dimensions l×nl \times n (vocabulary size × number of queries), not n×ln \times l as written in line 2 of the algorithm.

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