Skill-Based Ordering Score

• Skill-based ordering score is a quantitative metric that ranks agents, models, or data samples by assessing mastery of interdependent skills.
• It employs empirical training outcomes and directed skills graphs to optimize sampling strategies and performance evaluations.
• Applications include competitive game rating adjustments, curriculum learning for language models, and perceptual skill assessments in complex tasks.

A skill-based ordering score is a quantitative metric or functional designed to induce or recover a global ranking on agents, systems, or data samples, grounded in the relative mastery of defined skills. This concept plays a pivotal role in the evaluation, training, and comparison of entities (players, models, videos, datasets) where performance can be decomposed into skill components, which may exhibit interdependencies or be confounded by stochasticity and hidden information. Skill-based ordering scores trace their origins to both the ranking of human and artificial agents in games with structured outcomes and to the emergent ordering of skills in the context of large-scale machine learning, including LLMs and procedural task assessment.

1. Formal Definitions and Mathematical Frameworks

A skill, in the context of machine learning and data-driven modeling, is defined as a unit of behavior linked to an associated dataset $X_s\subseteq X$, where $X$ is the space of all task instances. A skill-based ordering, as formulated in “Skill-it! A Data-Driven Skills Framework for Understanding and Training LLMs” (Chen et al., 2023), hinges on linking these skills to observable training outcomes. Explicitly, for a model $f\in \mathcal{F}$ and evaluation metric $L$, skill $s$ satisfies:

• For any $D_s \subseteq X_s$, training on $D_s$ yields $\mathbb{E}[L(f, X_s \setminus D_s)]$ decreasing on average.

Given a collection $S = \{s_1, \ldots, s_k\}$, the skill-ordering score involves constructing a directed skills graph $G = (S, E)$, where an edge $(s_i \to s_j) \in E$ indicates that $s_i$ acts as a learning prerequisite for $s_j$. Let $A \in \mathbb{R}^{k \times k}$ be the adjacency matrix; the critical ordering score is estimated by measuring whether joint training on $D_{s_i} \cup D_{s_j}$ accelerates the acquisition of $s_j$ versus $D_{s_j}$ alone. Specifically, $A_{ij} > 0$ if:

$$\#\text{tokens to reach}\ L_j(f) \leq \ell^\star\ \text{with}\ D_{s_i} \cup D_{s_j} \leq \#\text{tokens with}\ D_{s_j}$$

This ordering score is directly operationalized in sampling strategies and in the optimization of multi-skill learning regimens (Chen et al., 2023).

In the context of game rating, such as in Rummy, the skill-based ordering score is integrated into a modified Elo framework by replacing the binary outcome function $S_i$ with a continuous, score-based deviation from a skill- and luck-adjusted benchmark, thereby yielding a rating update more sensitive to both performance margin and confounding factors (Chakraborty et al., 21 Dec 2025).

2. Estimation Procedures and Algorithms

The construction of skill-based ordering scores requires robust estimation algorithms to identify prerequisites and relative rankings. In “Skill-it!” two primary routines are advanced (Chen et al., 2023):

• Brute-force skill graph estimation: For $k$ skills, (1) train the model on each $X_{s_j}$ and evaluate loss reduction, (2) train for each $(s_i,s_j)$ on $X_{s_i} \cup X_{s_j}$ and re-evaluate, (3) set $A_{ij} > 0$ if the paired reduction exceeds the baseline.
• Linear approximation: Use an abbreviated training horizon $h$ and set $A_{ij}>0$ if the reduction in loss for $s_j$ from training on $s_i$ is positive, i.e., $L_j(f) - L_j(f_{h,i}) > 0$.

The resulting adjacency $A$ quantifies the empirical skill ordering, forming the basis of the Skill-It online sampling strategy, where a probability vector $p_t$ over skills is updated using a multiplicative-weights rule:

$$p_{t+1}^i = p_t^i \exp\left( \eta \sum_{j=1}^{m} A_{ij} L_j(f_t) \right)$$

Here, $\eta$ is a learning rate and $L_j(f_t)$ the current loss on evaluation skill $j$.

For perceptual skill assessment, a skill-based ordering score emerges as a learned scalar $f(p)$ assigned to each trial or video instance $p$ via a supervised deep ranking architecture. Pairwise judgments $E(p_i, p_j) \in \{-1, 0, +1\}$ are aggregated via a Siamese Temporal Segment Network and a combined margin+similarity loss, incentivizing $f(p_i) > f(p_j)$ for $E(p_i, p_j)=+1$ and $|f(p_i) - f(p_j)| < m$ for indistinguishable pairs. The global ordering is induced by ranking all items via $f(p)$ (Doughty et al., 2017).

In luck-driven, hidden-information games, the score-based ordering is embedded in a rating update:

1. For each round, compute observed scores $A_i$, hand-quality metrics $H_i$, and rating/hand-quality differentials $D_R$, $D_H$.
2. Calculate a post-hoc benchmark $B_i$ based on a logistic of ($D_R$, $D_H$).
3. Determine the rating update as

$R_i' = R_i + K_i \cdot (A_i - B_i)$

but only when the observed deviation aligns with the player's win/loss outcome, preserving monotonic skill advancement (Chakraborty et al., 21 Dec 2025).

3. Applications Across Domains

Skill-based ordering scores have significant practical impact in domains where randomization, hidden information, or the compositionality of tasks obscures direct skill attribution.

• Competitive environments with randomness and hidden information: The score-based Elo variant for Rummy enables discrimination of agent skill amidst stochastic effects, correcting for initial-hand variation and separating luck from skill in rating updates (Chakraborty et al., 21 Dec 2025).
• Data-driven skill curricula for LLMs: The skills graph and ordering score drive efficient data sampling, accelerating downstream acquisition of advanced abilities (e.g., compositional arithmetic, cross-lingual transfer) via curricularized training schedules (Chen et al., 2023).
• Automated skill assessment from video: The Siamese deep ranking approach enables objective quantification and global ordering of human performances in complex tasks (surgery, drawing, manipulation), useful for both analytics and automated how-to guidance (Doughty et al., 2017).

4. Evaluation Metrics and Empirical Results

Empirical evaluation of ordering scores includes both quantitative and qualitative metrics:

• Pairwise accuracy: In perceptual skill assessment, the “percentage of correctly ordered pairs” measures the extent to which the induced ranking $f(p)$ matches expert/crowdsourced labels. Two-stream TSN ranking achieved 70–83% accuracy across distinct skill domains (Doughty et al., 2017).
• Convergence and discriminative power: The modified score-based Elo for Rummy demonstrated rating stabilization within 4,000–5,000 games/strategy, with low coefficient of variation (<4%) for top strategies and clear stratification between naïve, mid-tier, and advanced agents. Predictive accuracy assessed via F1 score peaked at 0.7927 under optimal luck adjustment (Chakraborty et al., 21 Dec 2025).
• Cross-skill benefit visualization: Ordering matrix $A$ reveals task structure, with sparsity/density reflecting mutual interdependence of skills. In language modeling, dense graphs (e.g., Alpaca) show little advantage for ordered skill sampling, while intermediate-density graphs (Natural Instructions) showed substantial improvements in data efficiency and validation loss (Chen et al., 2023).
• Ablations: Disabling ordering (i.e., random or uniform sampling, or identity skill graph) yields slower convergence and poorer performance, highlighting the value of skill-based ordering in curriculum learning (Chen et al., 2023).

5. Significance, Limitations, and Theoretical Context

Skill-based ordering scores contribute to the precise characterization and ordering of agent or model capabilities in scenarios marked by entangled prerequisites, stochasticity, or ambiguous feedback. Their principled construction, rooted in empirical loss reduction or performance benchmarking, allows for data-driven discovery of learning dependencies, automated curriculum design, and fair skill rating in competitive environments.

Key limitations include:

• Skill identifiability and graph density: In some settings (e.g., Alpaca instruction types), the skills graph is too dense (near-complete) for ordering to yield meaningful curriculum differentiation; where data sources are too unrelated (nearly empty graphs), ordering offers minimal benefit (Chen et al., 2023).
• Robustness of empirical estimation: Recovery of true underlying skills via cluster analysis of pointwise loss is nontrivial; embedding-based approaches failed to accurately reconstitute true skill classes (<40%), while loss-based clustering achieved moderate success (61%) in synthetic settings (Chen et al., 2023).
• Confounding factors and volatility: Classical binary-outcome rating systems underperform in high-variance or score-margined domains, supporting the need for continuous, context-aware ordering mechanisms (Chakraborty et al., 21 Dec 2025).

6. Connections to Broader Research and Methodologies

Skill-based ordering scores intersect with curriculum learning, competence-based sampling, and meta-metrics in both reinforcement learning and supervised learning. The utilization of empirical skill graphs and adaptive sampling links to minimax and bandit-based strategies, while perceptual ordering with deep ranking aligns with advances in metric learning and ranking SVMs. The transition from static, outcome-based assessment (e.g., classical Elo) to score- and context-margined updates reflects broader movement toward integrating fine-grained, continuous, and interpretable performance measures in both AI model evaluation and human skill analytics.

7. Future Prospects and Open Challenges

Advancing skill-based ordering scores involves:

• Enhancing robustness of skill discovery in settings with limited or noisy supervision.
• Generalizing ordering score estimation to dynamic, open-world sets of skills and agents.
• Integrating explicit uncertainty quantification for ordering under high stochasticity.
• Extending ordering frameworks to multi-agent, multi-domain, and lifelong learning scenarios where skill dependencies evolve dynamically.

A plausible implication is that as AI systems and evaluation scenarios grow in complexity and heterogeneity, principled skill-based ordering scores will become foundational in both practical assessment and theoretical understanding of adaptive, skill-conditioned learning systems.