A course designed around the unknown

AI4OPT was a semester-long, project-based seminar for BSc and MSc computer science students with limited or no prior training in optimization. In total, 18 BSc and 17 MSc students enrolled in Spring 2026. Students worked in teams of three to four on open problems drawn from active research in optimization. Throughout the semester, they used AI tools such as ChatGPT, Claude Code, and Gemini, and in some cases built their own agentic pipeline to learn background, generate ideas, implement experiments, test claims, and report their process, via both oral presentations and written report.

The course asked a deliberately ambitious question:

Can students with limited prior training in optimization use AI to learn enough, reason carefully enough, and verify rigorously enough to make meaningful progress on real open problems?

The inquiry pursued three intertwined objectives: (1) advancing students’ optimization training, (2) developing their capability to use AI and scrutinize AI output, and (3) cultivating authentic research practice.

Crucially, students were not graded on whether they solved the open problem. Evaluation focused instead on whether they could explain and defend their reasoning, distinguish known results from new or AI-suggested ideas, critically assess AI outputs, and make rigorous, meaningful progress. Partial results, counterexamples, clarified failure modes, and well-justified negative findings all counted.

Each project was paired with a mentor who had expertise in the underlying research question. The mentor provided targeted guidance and, importantly, helped verify whether the team’s final claims were correct and whether partial progress was genuinely meaningful.

The goal was not to replace systematic coursework or thesis supervision. Those remain essential. Instead, the course tested a complementary pathway: students begin with an authentic open problem, use AI to reconstruct the deep but narrow background they need, and learn through the process of problem solving.

This also changed the role of AI. It was not treated as an answer machine. Instead, it was treated as a fallible learning partner: useful for generating explanations, conjectures, analogies, proof attempts, code, counterexamples, and critiques, but never authoritative by itself.

What happened during the semester

Across nine open problems, three were solved and verified, two produced substantial partial progress, and four remained open.

Summary: 9 open problems. 3 solved and verified. 2 partial. 4 remained open.

These outcomes are exciting, but they should be interpreted carefully. Open-problem difficulty cannot be calibrated precisely in advance, so the distribution of solved, partial, and open projects should not be read as a ranking of teams or as a controlled measure of AI effectiveness.

For the purpose of education, the unresolved and partial projects mattered just as much. Students were assessed not only on whether they solved the original problem, but also on whether they could reconstruct the background, decompose the question, audit AI-generated arguments, explain failures, and defend verified claims. It is worthy noting that failure to solve was not failure to learn. In an open-problem course, an unresolved result can still represent substantial learning.

The student–AI learning loop

Across the projects, students developed variations of a common workflow, as depicted below. Students remained at the center of learning: they steered AI to generate ideas, challenged its reasoning, tested and verified claims, and recorded what survives. The key activity was not obtaining an AI output. It was understanding the output well enough to challenge it or defend it.

At the same time, we saw substantial variation in how students worked with AI. Some entered the course already comfortable building agentic workflows; others had little experience beyond web-based chat tools. Their attitudes also differed. Some used AI mainly to learn background while developing solutions themselves; others experimented extensively with multi-model or automated pipelines for generation, critique, and verification.

One group reflected:

“We were very fascinated by the other groups’ work in constructing architectures for automated verification and discovery… [but] it is exactly the very specific and purpose-built nature of these architectures that… will enable them to automate tasks efficiently.”

— Student group G2

These differences mattered. Progress appeared to depend not only on the difficulty of the open problem, but also on the interaction between subject knowledge, AI fluency, and student mindset. In this setting, the ability to use AI effectively became part of the learning itself—and also a new source of variation that future course designs will need to address.

How open problems + AI change the learning dynamic

Most courses are organized around known answers. Students first learn concepts and methods, then apply them to carefully designed exercises within their current knowledge map. This structure builds systematic foundations and allows predictable assessment.

Open problems change the learning dynamic. No solution is known in advance, uncertainty is unavoidable, and failure is expected. AI, meanwhile, helped make that uncertainty more navigable. Much of the learning happened in the space between the two.

From answer generation to productive struggle

The open-problem setting creates a natural testbed for what education researchers call productive failure: struggling with a difficult problem can itself become a source of learning, provided that failure is treated as useful information rather than a source of shame. With a genuine open problem, a failed attempt may yield something interesting: an assumption was missing, a proof route was too weak, an experiment revealed a barrier, or the original question needs to be reformulated, etc.

One group captured the productive struggle especially well:

“When evaluating a new route, our workflow was a sequence of deep single-path explorations with a human in the loop. We would interrogate the LLM iteratively, working to understand each idea ourselves through repeated questioning rather than accepting it directly. Once we judged a path worth pursuing, we followed it carefully until it either broke or produced a candidate proof. Failed paths informed rerouting, while candidate proofs advanced to a separate auditing stage.”

— Student group G3

AI’s fallibility appeared to make students more critical, not less

Across all groups, students reported substantial effort spent detecting flaws in AI-generated answers. Some described systematically examining conjectures and identifying assumptions that AI had left unstated. Others gradually took a more active steering role by proposing their own hypotheses and only asking AI to do more “concrete work” such as coding.

One group described the shift from using AI as an answer generator to using it as a critic:

“Directly asking for a proof was often unreliable. It was more useful to make different models discuss the same idea, criticize each other’s arguments, search for counterexamples, or identify missing assumptions. This made AI more useful as a source of pressure-testing than as a source of final results. AI also affected our critical thinking. Since many answers sounded convincing but were wrong, we had to become more skeptical and check every nontrivial mathematical claim ourselves. In a future AI-assisted research project, we would set up a structured pipeline from the beginning, with one place to store ideas, failed attempts, AI outputs, checks, and reasons why a construction worked or failed.”

— Student group G7

This distinction between AI-suggested and mathematically verified became central to the learning process, as another group summarized particularly:

“AI lowers the entry barrier to optimization research — it gets a newcomer to the frontier of a sub-question in days, not months. But it raises a new barrier: the ability to tell a real proof from a fluent one. Our project is in some sense about that second skill.”

— Student group G8

Engagement came from having something real at stake

Students showed a striking level of engagement and sustained curiosity. During the group work and the classroom presentations, students actively discussed how to improve their workflows, which prompts and agentic pipeline were more effective, and what role they themselves should play in the overall problem-solving process. Most groups devoted more time to the problem than was formally required.

One student reflection captured the experience simply: “struggled a lot but deeply interested in the problem.”

Taken together, these observations suggest that the AI-supported open problem projects may provide a particularly productive learning mode. The value of AI here is not that it helps students reach answers faster. Its educational value centers on students’ own efforts to evaluate, challenge, redirect, and learn from what a powerful but fallible AI system produces.

Lessons for AI-era mathematics and computer science education

The seminar suggests that AI for mathematics and computer science education should move beyond two familiar images: AI as tutor and AI as coding assistant. These roles are useful, but they are not the whole story.

For advanced MATH and CS education, AI can support more ambitious forms of project-based learning, even at a large scale. Students can approach authentic questions earlier, reconstruct the background they need on demand, and use AI to generate and stress-test possible routes. This creates a richer space between structured coursework and fully independent projects.

Several lessons stood out.

First, assessment should focus more on reasoning traces. In an AI-assisted environment, final answers are not enough. We need to see how students decomposed the problem, what alternatives they considered, why they rejected certain routes, how they verified key claims, and what they understood well enough to defend.

Second, instructors become designers of learning environments. Their role is not only to deliver content, but also to select problems, define milestones, create verification checkpoints, calibrate difficulty, and help students reflect on their AI workflows.

Third, AI literacy must be taught explicitly. Students differ widely in their ability to choose tools, structure interactions, build workflows, and audit outputs. If these skills are left implicit, AI may amplify existing differences between students. A fair course design should teach baseline AI-assisted learning practices directly.

The next bottleneck: good problems

Open problems have a built-in limitation: once solved, they are no longer open. Problem design may become the new bottleneck.

If AI continues to accelerate problem solving, this education model will need a renewable supply of problems that are open, meaningful, tractable, and independently verifiable. These problems must be hard enough to require real reasoning, but structured enough for students to make progress and receive feedback.

That may require new educational infrastructure: better problem-selection rubrics, and eventually AI-assisted systems for generating and maintaining suitable problem portfolios.

In this sense, open problems may become a new educational unit: richer than exercises, earlier and more scaffolded than thesis projects, and well matched to AI-supported active learning.

As answers become cheaper, good problems and sound judgement becomes more important. We believe this pilot seminar points toward a broader shift in what AI-era education should value.

Additional materials

Interested readers can find more details below:

• Invited Talk at ICML Workshop on AI as a Tool for Mathematics, Computer Science, and Machine Learning: From Learning to Discovery: Lessons from a Semester of AI-Assisted Optimization Seminar [Slides] [Video]
• AI4OPT seminar report by Niao He, Zebang Shen, and Fan Wang [Report]
• AI4OPT final report template [Template]

1. Problem

We consider unconstrained minimization

for differentiable functions \(f:\mathbb R^d\to\mathbb R\). For fixed step size \(\gamma\) and momentum \(\beta\), the heavy-ball method (HB) is

For any \(L>\mu>0\) and any \(d\in\mathbb{N}\), we let \(\mathcal F_{\mu,L}(\mathbb{R}^d)\) denote the function class of all \(L\)-smooth \(\mu\)-strongly convex functions \(f\) on \(\mathbb{R}^d\), that is, \(f:\mathbb{R}^d\to\mathbb{R}\) satisfying

for all \(x,y\in\mathbb{R}^d\). We denote the set of \(L\)-smooth \(\mu\)-strongly convex quadratic functions by \(\mathcal Q_{\mu,L}(\mathbb{R}^d)\). We have \(\mathcal Q_{\mu,L}(\mathbb{R}^d) \subsetneq \mathcal F_{\mu,L}(\mathbb{R}^d)\).

Let \(\kappa=L/\mu\) denote the condition number. On the class \(\mathcal Q_{\mu,L}(\mathbb{R}^d)\) with any \(d\in\mathbb{N}\), it is known that HB with some properly chosen \((\gamma,\beta)\) achieves the accelerated rate: HB returns a point \(\hat{x}\) satisfying \(\lVert \hat{x}-x^\star \rVert\leq\varepsilon\) within \(O(\sqrt{\kappa}\log\frac{1}{\varepsilon})\) iterations. This rate is asymptotically optimal among all first-order methods. On the other hand, for the more general class of \(\mathcal{F}_{\mu,L}(\mathbb{R}^d)\), HB is known to achieve the non-accelerated rate \(O(\kappa\log\frac{1}{\varepsilon})\), and it remained open for a long time whether this rate can be improved to the accelerated rate. Recently, Goujaud et al. (2025a) showed that the answer is no for \(\mathcal{F}_{\mu,L}(\mathbb{R}^d)\) with any \(d\geq 2\). Informally speaking, they showed that for any parameter \((\gamma,\beta)\), if HB achieves the accelerated rate on \(\mathcal Q_{\mu,L}(\mathbb{R}^d)\), then there exists a function in \(\mathcal F_{\mu,L}(\mathbb{R}^2)\) on which HB cycles and does not converge. This implies the non-acceleration of HB in dimension at least two, and the possibility of acceleration on \(\mathcal F_{\mu,L}(\mathbb{R})\) was later posed as an open problem [Goujaud et al., 2025b].

In this blog post, we outline our approach to resolve the open problem. We show that the answer is no by constructing cycles in one dimension. A preprint will be released soon. We learned through personal communication that Dieuleveut and his collaborators have also solved the problem.

3. Main Result

The following theorem is our main result:

Construction. We let the cycle be the projection of the rotated \(m\)-jump \(K\)-th root-of-unity cycle onto the real line:

where the angle of rotation \(\varphi_\ast\) is chosen as

If HB cycles over \(\{ y_t \}\), then the gradient at \(y_t\) must be

By verifying the interpolation conditions for smooth strongly convex functions [Taylor et al., 2017], there exists a function \(f\in\mathcal F_{\mu,L}(\mathbb R)\) satisfying \(f'(y_t)=g_t\) for all \(t\in\{0,1,\ldots,K-1\}\). By construction, HB cycles over \(y_0,\ldots,y_{K-1}\), meaning root-of-unity cycling implies cycling in dimension one, from which it follows that:

4. Proof Idea

We let \(\varphi\) be the angle of rotation and \(y_t=\cos(mt\theta_K+\varphi)\) for \(t=0,1,\ldots,K-1\). If HB cycles over \(\{y_t\}\), then by the update rule, the gradients at \(y_t\) must be

Step 1: Rotate slightly so that the projected roots have a stable sort order

If \(\varphi = 0\), the roots of unity may be projected onto the same point but with different gradients, i.e., \(y_s=y_t\) but \(g_s\neq g_t\). To overcome this issue, we show that by taking

all \(\{y_t\}\) are distinct, and their order is fixed. Let \(y_{\sigma(0)} < y_{\sigma(1)} < \cdots < y_{\sigma(K-1)}\) denote the sorted order.

Step 2: Simplify the interpolation conditions

To show the existence of functions in \(\mathcal{F}_{\mu,L}(\mathbb{R})\), we invoke the interpolation conditions [Lessard et al., 2016, Taylor et al., 2017]. In one dimension, the conditions reduce to checking slopes of the sorted points. Namely, it is enough to ensure that each adjacent sorted pair \((\sigma(i), \sigma(i+1))\) satisfies

Write \(y_{\sigma(i+1)}-y_{\sigma(i)} = \operatorname{Re}(w_i)>0\) where \(w_{i}=e^{i\varphi}(\omega_K^{m\sigma(i+1)}-\omega_K^{m\sigma(i)})\). By subsituting the formula for \(g_t\) into the interpolation conditions, the conditions can be expressed as

Further simplification yields

Step 3: Choose the angle of rotation

The remaining thing is to show that there exists \(\varphi\) that fulfills the above conditions on \(\arg w_i\).

By studying the properties of the permutation \(\sigma\), the angle \(\arg w_i\) can only take two possible values:

Therefore, it suffices to ensure

That is,

Thus a valid \(\varphi\) exists if and only if

This is precisely the condition for the existence of root-of-unity cycles. In other words, if \((\gamma,\beta)\in\textrm{int}(\Omega^{\text{rou}}_{K,m})\), then the above interval is non-empty, and we can choose \(\varphi\) to be the midpoint of the interval

This concludes the proof.

5. Additional Results

We let the augmented root-of-unity region be

This augmented region is strictly larger than the ordered one that considers only \(m=1\). For example, there are parameters \((\gamma,\beta)\) that lie in \(\Omega^{\mathrm{rou}}_{5,2}\) but not in \(\Omega^{\mathrm{rou}}_{K,1}\) for any \(K\).

The same calculation also explains the fractional regions observed in Figure 8 of Goujaud et al., (2025b). For \(m\in U_K\), checking only the residual \(\phi_{K,m}(1)\) is equivalent, after a change of variables, to evaluating the ordered polynomial condition at a fractional period \(K/m\) (see Theorem 3.5 of Goujaud et al., (2025a)). However, \(\phi_{K,m}(\Delta)\) is not necessarily maximized by \(\Delta=1\); the maximizer is

Thus the fractional condition is generally only a relaxation, and the corresponding region can be strictly larger than the true cycling region.

6. Use of AI and Formal Verification

AI tools were useful for algebraic exploration, for reorganizing rough proof attempts, and for translating parts of the argument into Lean. The final mathematical claims were not accepted directly from model output: the role of AI was to accelerate iteration, not to replace verification.

The one-dimensional cycling theorem was formalized in Lean 4 with Mathlib (see here). The formalization checks the core angle argument, the angle choice, the projected slope bounds, and the resulting cycle-existence statement, modulo two external mathematical inputs: the one-dimensional smooth strongly convex interpolation criterion and the previously known containment of the acceleration-relevant parameters in the roots-of-unity cycling region.

In practice, Lean was most valuable for controlling details that are easy to get wrong on paper: signs of arguments, boundary cases, modular inverses, and the exact interval in which the phase must lie.

References

1. [Polyak, 1964] Some methods of speeding up the convergence of iteration methods. USSR Comput. Math. Math. Phys., 1964.
2. [Nesterov, 1983] A method for solving a convex programming problem with convergence rate $O(1/k^2)$. Dokl. Akad. Nauk SSSR, 1983.
3. [Ghadimi et al., 2015] Global convergence of the heavy-ball method for convex optimization. Eur. Control. Conf., 2015.
4. [Lessard et al., 2016] Analysis and design of optimization algorithms via integral quadratic constraints. SIAM J. Optim., 2016.
5. [Taylor et al., 2017] Smooth strongly convex interpolation and exact worst-case performance of first-order methods. Math. Program., 2017.
6. [Goujaud et al., 2025a] Provable non-accelerations of the heavy-ball method. Math. Program., 2025.
7. [Goujaud et al., 2025b] Open problem: Two riddles in heavy-ball dynamics. arXiv preprint, 2025.