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Daron Anderson

Publications and source records attributed to Daron Anderson.

10 recordsLinked to original sources
arXiv2025

Humanity's Last Exam

Benchmarks are important tools for tracking the rapid advancements in large language model (LLM) capabilities. However, benchmarks are not keeping pace in difficulty: LLMs now achieve over 90\% accuracy on popular benchmarks like MMLU, limiting informed measurement of state-of-the-art LLM capabilities. In response, we introduce Humanity's Last Exam (HLE), a multi-modal benchmark at the frontier of human knowledge, designed to be the final closed-ended academic benchmark of its kind with broad subject coverage. HLE consists of 2,500 questions across dozens of subjects, including mathematics, humanities, and the natural sciences. HLE is developed globally by subject-matter experts and consists of multiple-choice and short-answer questions suitable for automated grading. Each question has a known solution that is unambiguous and easily verifiable, but cannot be quickly answered via internet retrieval. State-of-the-art LLMs demonstrate low accuracy and calibration on HLE, highlighting a significant gap between current LLM capabilities and the expert human frontier on closed-ended academic questions. To inform research and policymaking upon a clear understanding of model capabilities, we publicly release HLE at https://lastexam.ai.

cs.LG↗
arXiv2022

Lazy Lagrangians with Predictions for Online Learning

We consider the general problem of online convex optimization with time-varying additive constraints in the presence of predictions for the next cost and constraint functions. A novel primal-dual algorithm is designed by combining a Follow-The-Regularized-Leader iteration with prediction-adaptive dynamic steps. The algorithm achieves $\mathcal O(T^{\frac{3-\beta}{4}})$ regret and $\mathcal O(T^{\frac{1+\beta}{2}})$ constraint violation bounds that are tunable via parameter $\beta\!\in\![1/2,1)$ and have constant factors that shrink with the predictions quality, achieving eventually $\mathcal O(1)$ regret for perfect predictions. Our work extends the FTRL framework for this constrained OCO setting and outperforms the respective state-of-the-art greedy-based solutions, without imposing conditions on the quality of predictions, the cost functions or the geometry of constraints, beyond convexity.

cs.LG↗
arXiv2020

Indecomposable Continuum with a Strong Non-Cut Point

We construct an indecomposable continuum with exactly one strong non-cut point. The method is an adaptation of Bellamy $[1]$. We start with an $\omega_1$-chain of indecomposable metric continua and retractions. The inverse limit is an indecomposable continuum with exactly two composants. Our example is formed by identifying a point in each composant.

math.GN↗
arXiv2020

Continuum Without Non-Block Points

For any composant $E \subset \mathbb H^*$ and corresponding near-coherence class $\mathscr E \subset \omega^*$ we prove the following are equivalent : (1) $E$ properly contains a dense semicontinuum. (2) Each countable subset of $E$ is contained in a dense proper semicontinuum of $E$. (3) Each countable subset of $E$ is disjoint from some dense proper semicontinuum of $E$. (4) $\mathscr E $ has a minimal element in the finite-to-one monotone order of ultrafilters. (5) $\mathscr E $ has a $Q$-point. A consequence is that NCF is equivalent to $\mathbb H^*$ containing no proper dense semicontinuum and no non-block points. This gives an axiom-contingent answer to a question of the author. Thus every known continuum has either a proper dense semicontinuum at every point or at no points. We examine the structure of indecomposable continua for which this fails, and deduce they contain a maximum semicontinuum with dense interior.

math.GN↗
arXiv2020

The Shore Point Existence Problem is Equivalent to the Non-Block Point Existence Problem

We prove the three propositions are equivalent: $(a)$ Every Hausdorff continuum has two or more shore points. $(b)$ Every Hausdorff continuum has two or more non-block points. $(c)$ Every Hausdorff continuum is coastal at each point. Thus it is consistent that all three properties fail. We also give the following characterisation of shore points: The point $p$ of the continuum $X$ is a shore point if and only if there is a net of subcontinua in $\{K \in C(X): K \subset \kappa(p) - p\}$ tending to $X$ in the Vietoris topology. This contrasts with the standard characterisation which only demands the net elements be contained in $X-p$. In addition we prove every point of an indecomposable continuum is a shore point.

math.GN↗
arXiv2020

Shore and Non-Block Points in Hausdorff Continua

We study the shore and non-block points of non-metric continua. We reduce the problem of showing a continuum to have non-block points to that of showing an indecomposable continuum to have non-block points. As a corollary we prove that separable continua have at least two non-block points -- and moreover are irreducible about their set of non-block points.

math.GN↗
arXiv2020

Lazy Online Gradient Descent is Universal on Polytopes

We prove the familiar Lazy Online Gradient Descent algorithm is universal on polytope domains. That means it gets $O(1)$ pseudo-regret against i.i.d opponents, while simultaneously achieving the well-known $O(\sqrt N)$ worst-case regret bound. For comparison the bulk of the literature focuses on variants of the Hedge (exponential weights) algorithm on the simplex. These can in principle be lifted to general polytopes; however the process is computationally unfeasible for many important classes where the number of vertices grows quickly with the dimension. The lifting procedure also ignores any Euclidean bounds on the cost vectors, and can create extra factors of dimension in the pseudo-regret bound. Gradient Descent is simpler than the handful of purpose-built algorithms for polytopes in the literature, and works in a broader setting. In particular existing algorithms assume the optimiser is unique, while our bound allows for several optimal vertices.

cs.LG↗
arXiv2019

Learning The Best Expert Efficiently

We consider online learning problems where the aim is to achieve regret which is efficient in the sense that it is the same order as the lowest regret amongst K experts. This is a substantially stronger requirement that achieving $O(\sqrt{n})$ or $O(\log n)$ regret with respect to the best expert and standard algorithms are insufficient, even in easy cases where the regrets of the available actions are very different from one another. We show that a particular lazy form of the online subgradient algorithm can be used to achieve minimal regret in a number of "easy" regimes while retaining an $O(\sqrt{n})$ worst-case regret guarantee. We also show that for certain classes of problem minimal regret strategies exist for some of the remaining "hard" regimes.

cs.LG↗
arXiv2019

Optimality of the Subgradient Algorithm in the Stochastic Setting

We show that the Subgradient algorithm is universal for online learning on the simplex in the sense that it simultaneously achieves $O(\sqrt N)$ regret for adversarial costs and $O(1)$ pseudo-regret for i.i.d costs. To the best of our knowledge this is the first demonstration of a universal algorithm on the simplex that is not a variant of Hedge. Since Subgradient is a popular and widely used algorithm our results have immediate broad application.

math.ST↗