Paper: On the Verification Complexity of Deterministic Nonsmooth Nonconvex Optimization (ICLR-2024)

Reviewer comment (Theory):
The interaction scheme between the verifier and the prover is confusing. Since the verifier is deterministic, any multi-round interaction protocol can be transformed into a single round interaction protocol with the same amount of communication complexity, i.e., the prover can send everything to the verifier at the very beginning of the protocol, since they can simulate the deterministic actions of the verifier.

Author response:
We thank the anonymous reviewer for their comments. 

In our paper we provide upper bounds for the verification complexity when one is provided with exact queries and oracles (Theorem 4). For this result, we require access to an omnipotent prover, so that the points $y_0,y_1,\dots,y_d$ for Theorem 4 can be generated by the prover. 

We focus on deterministic verifiers in order to provide lower bounds for the verification complexity (Theorems 2,3), since in the randomized setting, upper bounds are known. Our work is the first to provide an exponential separation between deterministic and randomized non-smooth non-convex optimization (Theorem 2).

Regarding the interactive protocol, we agree that it is equivalent to a single round of interaction, but this is only a presentational issue and we will consider simplifying the model accordingly for future versions of our paper.

Reviewer comment (Theory):
This paper only discussed the deterministic verification complexity of (uniform) Goldstein stationary point. However, from my vantage point, it appears that a randomized protocol may be a more general and intuitive choice, given that randomness is not a scarce resource for machine learning algorithms. Also, in a randomized protocol, the interaction scheme described in Definition 6 and 7 makes more sense, since in this case a multi-round interaction is indeed much powerful than a one-round interaction. The authors should provide further clarification and justification for their choice to focus on the deterministic model.

Author response:
We thank the anonymous reviewer for their comments. 

In our paper we provide upper bounds for the verification complexity when one is provided with exact queries and oracles (Theorem 4). For this result, we require access to an omnipotent prover, so that the points $y_0,y_1,\dots,y_d$ for Theorem 4 can be generated by the prover. 

We focus on deterministic verifiers in order to provide lower bounds for the verification complexity (Theorems 2,3), since in the randomized setting, upper bounds are known. Our work is the first to provide an exponential separation between deterministic and randomized non-smooth non-convex optimization (Theorem 2).

Regarding the interactive protocol, we agree that it is equivalent to a single round of interaction, but this is only a presentational issue and we will consider simplifying the model accordingly for future versions of our paper.

Reviewer comment (Theory):
Why did the authors introduce an interactive protocol in their deterministic setting?

Author response:
We thank the anonymous reviewer for their comments. 

In our paper we provide upper bounds for the verification complexity when one is provided with exact queries and oracles (Theorem 4). For this result, we require access to an omnipotent prover, so that the points $y_0,y_1,\dots,y_d$ for Theorem 4 can be generated by the prover. 

We focus on deterministic verifiers in order to provide lower bounds for the verification complexity (Theorems 2,3), since in the randomized setting, upper bounds are known. Our work is the first to provide an exponential separation between deterministic and randomized non-smooth non-convex optimization (Theorem 2).

Regarding the interactive protocol, we agree that it is equivalent to a single round of interaction, but this is only a presentational issue and we will consider simplifying the model accordingly for future versions of our paper.

Reviewer comment (Theory):
What is the randomized verification complexity, or what is the motivation of considering a deterministic setting instead of the randomized setting?

Author response:
We thank the anonymous reviewer for their comments. 

In our paper we provide upper bounds for the verification complexity when one is provided with exact queries and oracles (Theorem 4). For this result, we require access to an omnipotent prover, so that the points $y_0,y_1,\dots,y_d$ for Theorem 4 can be generated by the prover. 

We focus on deterministic verifiers in order to provide lower bounds for the verification complexity (Theorems 2,3), since in the randomized setting, upper bounds are known. Our work is the first to provide an exponential separation between deterministic and randomized non-smooth non-convex optimization (Theorem 2).

Regarding the interactive protocol, we agree that it is equivalent to a single round of interaction, but this is only a presentational issue and we will consider simplifying the model accordingly for future versions of our paper.

Reviewer comment (Theory):
The most questionable aspect, in my opinion, is the proof of Corollary 1. As previously discussed, the prover needs to communicate with the underlying function f to facilitate strong verification in Theorem 4. In the proof of Corollary 1, while I understand that $f_0$ and $f_1$ are consistent on the queried points {$x^{(i)}$}, I don't comprehend why an "omniscient" prover would provide exactly the same sequence {$x^{(i)}$} for both functions $f_0$ and $f_1$. As mentioned earlier, a legitimate prover should communicate with both $f_0$ and $f_1$ to provide the most useful information. This raises concerns about the correctness of Corollary 1, which is the main result addressing Q2.

Author response:
We thank the anonymous reviewer for their feedback. 
- We would like to clarify a technical misconception regarding Corollary 1. In particular, we note that the prover need not be honest and may provide adversarial input to the verifier (it is the verifier’s job to decide whether the prover is honest). As such, if we consider $(x^{(i)})$ to be the sequence that the honest prover provides when $f=f_0$, then another (malicious) prover may provide the same sequence to the verifier when $f=f_1$. The verifier needs to provide the same output in both cases (because it has the same input and is deterministic) and so has to fail in at least one of the scenarios above. We will clarify this point as well as our terminology about the prover in future versions of our paper and we thank the reviewer for pointing it out.
- As the reviewer points out, we need to add a technical clarification for Theorem 4, where the prover has to provide points where $f$ is differentiable. This will incur an arbitrarily small slackness in parameter $\delta$ (between completeness and soundness in Definition 6) and can, otherwise, be done without loss of generality. See also our response to Reviewer 6VnG.
- Regarding the use of Fact 2, we will add an appropriate clarification for future versions of our paper.
- Regarding Definition 7, we, once more, emphasize that the prover is not assumed to be honest and the verifier is asked to return the correct value of $b$ regardless of whether the prover is honest or malicious.

Reviewer comment (Theory):
On page 6, concerning the algorithm, why does the function $f$ need to be differentiable at points {y_i}? Consider the following example: let $\epsilon = 1/2$, $\delta = 1$, and f(x) = max{-1, min{ x, 1 }}. Consider $x=0$. In this case, $y_1 = -1$ and $y_2 = 1$ is the only correct choice, but the function is nondifferentiable at both points.

Author response:
We thank the anonymous reviewer for their feedback. 
- We would like to clarify a technical misconception regarding Corollary 1. In particular, we note that the prover need not be honest and may provide adversarial input to the verifier (it is the verifier’s job to decide whether the prover is honest). As such, if we consider $(x^{(i)})$ to be the sequence that the honest prover provides when $f=f_0$, then another (malicious) prover may provide the same sequence to the verifier when $f=f_1$. The verifier needs to provide the same output in both cases (because it has the same input and is deterministic) and so has to fail in at least one of the scenarios above. We will clarify this point as well as our terminology about the prover in future versions of our paper and we thank the reviewer for pointing it out.
- As the reviewer points out, we need to add a technical clarification for Theorem 4, where the prover has to provide points where $f$ is differentiable. This will incur an arbitrarily small slackness in parameter $\delta$ (between completeness and soundness in Definition 6) and can, otherwise, be done without loss of generality. See also our response to Reviewer 6VnG.
- Regarding the use of Fact 2, we will add an appropriate clarification for future versions of our paper.
- Regarding Definition 7, we, once more, emphasize that the prover is not assumed to be honest and the verifier is asked to return the correct value of $b$ regardless of whether the prover is honest or malicious.

Reviewer comment (Theory):
Another issue is that, in Fact 2, the result is stated for polytopes. But a subdifferential could be far from a polytope. Consider $\partial (x \mapsto ||x||)(0)$.

Author response:
We thank the anonymous reviewer for their feedback. 
- We would like to clarify a technical misconception regarding Corollary 1. In particular, we note that the prover need not be honest and may provide adversarial input to the verifier (it is the verifier’s job to decide whether the prover is honest). As such, if we consider $(x^{(i)})$ to be the sequence that the honest prover provides when $f=f_0$, then another (malicious) prover may provide the same sequence to the verifier when $f=f_1$. The verifier needs to provide the same output in both cases (because it has the same input and is deterministic) and so has to fail in at least one of the scenarios above. We will clarify this point as well as our terminology about the prover in future versions of our paper and we thank the reviewer for pointing it out.
- As the reviewer points out, we need to add a technical clarification for Theorem 4, where the prover has to provide points where $f$ is differentiable. This will incur an arbitrarily small slackness in parameter $\delta$ (between completeness and soundness in Definition 6) and can, otherwise, be done without loss of generality. See also our response to Reviewer 6VnG.
- Regarding the use of Fact 2, we will add an appropriate clarification for future versions of our paper.
- Regarding Definition 7, we, once more, emphasize that the prover is not assumed to be honest and the verifier is asked to return the correct value of $b$ regardless of whether the prover is honest or malicious.

Reviewer comment (Theory):
I do not understand why, in Definition 7, completeness and soundness are stated separately and the verifier needs to return a boolean $b \in$ {0,1}. As well-noted by the authors, the "if" condition in the completeness part is always true, so the "verifier" will always return $b=1$, and the "if" condition in the soundness part is also always true.

Author response:
We thank the anonymous reviewer for their feedback. 
- We would like to clarify a technical misconception regarding Corollary 1. In particular, we note that the prover need not be honest and may provide adversarial input to the verifier (it is the verifier’s job to decide whether the prover is honest). As such, if we consider $(x^{(i)})$ to be the sequence that the honest prover provides when $f=f_0$, then another (malicious) prover may provide the same sequence to the verifier when $f=f_1$. The verifier needs to provide the same output in both cases (because it has the same input and is deterministic) and so has to fail in at least one of the scenarios above. We will clarify this point as well as our terminology about the prover in future versions of our paper and we thank the reviewer for pointing it out.
- As the reviewer points out, we need to add a technical clarification for Theorem 4, where the prover has to provide points where $f$ is differentiable. This will incur an arbitrarily small slackness in parameter $\delta$ (between completeness and soundness in Definition 6) and can, otherwise, be done without loss of generality. See also our response to Reviewer 6VnG.
- Regarding the use of Fact 2, we will add an appropriate clarification for future versions of our paper.
- Regarding Definition 7, we, once more, emphasize that the prover is not assumed to be honest and the verifier is asked to return the correct value of $b$ regardless of whether the prover is honest or malicious.

Reviewer comment (Presentation):
The authors are expected to discuss future work as it has enough space. I believe this could help the reader understand the paper's contributions and the potential impact in future.

Author response:
We wish to thank the reviewer for their constructive feedback. We will add a discussion about open directions in future versions of our paper.

Reviewer comment (Theory):
The authors claim that they "prove that any deterministic zeroth-order verifier that is restricted to queries in a lattice needs a number of queries that is exponential in the dimension". I wonder what happens in the case of smooth nonconvex optimization. Are there any upper or lower bounds for the queries in a lattice to find an approximate stationary point of a smooth nonconvex function with Lipschitz gradients?

Author response:
In the smooth setting, there are upper bounds which demonstrate that queries on a lattice with $\mathrm{poly}(\epsilon)$ accuracy are sufficient to find a Clarke stationary point, but we show that in the non-smooth setting, a dependence on $\delta$ is necessary. We refer the reviewer to the discussion below Theorem 3 (pages 3,4).

Reviewer comment (Theory):
The authors say "The main idea behind Theorem 7 is to reduce the construction of f to a geometric question in two dimensions, taking advantage of the symmetry of the lattice". I don't quite understand what this "geometric question" is and what is the "advantage of the symmetry of the lattice". Can the authors explain more about the intuition behind the main idea of the construction in this theorem?

Author response:
Regarding the construction of Theorem 7, the main idea is to project the lattice to a two-dimensional subspace, forming a two-dimensional grid. Then, one can create a high-dimensional function for which the center of the grid is not Goldstein stationary by forming a line that passes through the points of the grid (the line corresponds to the zeros of the function) and ensuring that no angle of the line is too sharp.

Reviewer comment (Theory):
Question 1a — In Theorem 4, why is it valid to assume that we get a gradient at the query point instead of a subgradient, given that the function \( f \) is assumed to **not** be differentiable everywhere?

Author response:
We wish to thank the anonymous reviewer for their comments and for alerting us of a missing technical clarification on Theorem 4. 

In particular, while it is not explicitly stated in Theorem 4,  the honest prover may without loss of generality only provide points $y_0,y_1,\dots,y_d$ where $f$ is differentiable. This follows from Definitions 1 and 3, where the generalized gradient is defined itself as the convex hull of certain points. These points are limits of convergent sequences of gradient values and each of them can therefore be approximated arbitrarily well by some given point of the sequence. We can therefore define the set $S$ in Fact 1 in terms of points where $f$ is differentiable.

A simpler approach is to assume that $f$ is differentiable, but the gradient is allowed to change arbitrarily quickly within small neighborhoods of points.

We will add this clarification in future versions of the paper.