GenTac: Diffusion-Based Soccer Tactics

• GenTac is a diffusion-based generative framework that models open-play soccer tactics as coupled sequences of multi-agent trajectories and discrete tactical events.
• It employs a conditional discrete diffusion process to forecast plausible future states by integrating context such as opponent behavior, team identity, and tactical objectives.
• Evaluations on TacBench demonstrate GenTac’s ability to achieve low displacement errors and maintain coherent team structures, with potential applications across various sports.

Searching arXiv for the specified paper and closely related context. GenTac is a diffusion-based generative framework for modeling and forecasting open-play soccer tactics as a stochastic process over continuous multi-player trajectories and discrete semantic events (Rao et al., 13 Apr 2026). It is designed for the setting in which match evolution is inherently multi-agent, stochastic, and branching, rather than adequately represented by a single deterministic future. In this formulation, a possession generates both stacked 2D positions for players and the ball and a fine-grained tactical event label at each time step, while optionally conditioning on contextual variables such as opponent behavior, team identity, league identity, or tactical objectives.

1. Formal problem statement

GenTac views an open-play soccer possession through two coupled sequences. The trajectory sequence is

$$x_{0:T} = \bigl\{x_0,x_1,\dots,x_T\bigr\},\quad x_t\in\mathbb R^{(2N+1)\times 2},$$

where each frame contains the 2D positions $(x,y)$ of $N$ players on each of two teams plus the ball. The event sequence is

$$e_{0:T} = \bigl\{e_0,e_1,\dots,e_T\bigr\},\quad e_t\in\{1,\dots,15\},$$

with each $e_t$ denoting one of 15 tactical labels. The optional context variable satisfies

$$\mathcal C=\{\varnothing,\;c^{\rm opp},\;c^{\rm team},\;c^{\rm league},\;c^{\rm obj}\},$$

corresponding respectively to no extra context, opponent’s full trajectory, a particular team identity, a league identity, or a tactical objective such as offense or defense. The joint task is to learn

$p_\theta\bigl(x_{0:T},\,e_{0:T}\mid c\bigr).$

This formulation places GenTac in a broader line of multi-agent sequence modeling, but with an explicit emphasis on open-play possessions rather than highly structured restarts. A common misconception is that tactical forecasting is adequately captured by a single best-guess rollout. GenTac is instead defined around the distribution of plausible futures, so sampling diversity is intrinsic rather than incidental (Rao et al., 13 Apr 2026).

2. Diffusion formulation and training objective

Trajectory generation is implemented as a conditional discrete diffusion process over each causal window of length $w$. A clean history $x_{\rm h}$ of length $H$ conditions a future segment $(x,y)$0 of length $(x,y)$1. In the forward process, Gaussian corruption is applied only to the future segment, while the history is kept exact. With linear noise schedule $(x,y)$2, and

$(x,y)$3

the noising kernel is

$(x,y)$4

which yields

$(x,y)$5

The reverse process learns a neural network

$(x,y)$6

that predicts the corruption noise. The reverse kernel is written as

$(x,y)$7

with

$(x,y)$8

The paper notes that $(x,y)$9 may equivalently be viewed as a score network $N$0 in the standard score-matching diffusion interpretation. Training minimizes the denoising objective

$N$1

where $N$2 is sampled uniformly from $N$3 (Rao et al., 13 Apr 2026).

Methodologically, this makes GenTac a conditional generative forecaster rather than a deterministic regressor. This suggests that its principal modeling commitment is to recover a distribution over future tactical states while maintaining causal dependence on observed history.

3. Representation of trajectories and tactical events

At each frame, GenTac concatenates all player and ball coordinates into a token grid of shape $N$4. Over a sequence of length $N$5, the tensor is

$N$6

Each $N$7 pair is projected via a learnable linear map into $N$8, after which three embeddings are added: a temporal embedding shared by all entities at the same time step, a group embedding distinguishing team 1, team 2, and the ball, and an entity embedding assigning a unique slot to each player and the ball.

The tactical event taxonomy grounds trajectories into 5 coarse types and 15 subtypes. The coarse types are Build-up, Transition, Threat, Set Piece, and Interruption (Stoppage). The listed subtypes are Ball Win and Progression under Transition; Goal, Shot Off Target, Shot Saved, Clearance, and Defended under Threat; and Corner, Free Kick, Penalty, Throw-in, Kick-off, and Goal-Kick under Set Piece. Build-up has 1 subtype.

The joint factorization is

$N$9

where $$e_{0:T} = \bigl\{e_0,e_1,\dots,e_T\bigr\},\quad e_t\in\{1,\dots,15\},$$0 is implemented by the diffusion forecaster, optionally conditioned on past events, and $$e_{0:T} = \bigl\{e_0,e_1,\dots,e_T\bigr\},\quad e_t\in\{1,\dots,15\},$$1 by a small classifier on the encoded trajectory (Rao et al., 13 Apr 2026).

This coupling of continuous trajectories with discrete semantics is central to the framework’s scope. It avoids treating event labels as mere post hoc annotations and instead makes them part of the tactical state space. A plausible implication is that the event layer serves both interpretability and downstream tactical summarization.

4. Contextual conditioning and counterfactual control

GenTac injects context in two distinct ways. For opponent-conditioning, the opponent’s future trajectory is appended to the condition channels, while only the target team’s future is masked as noise. For style- or objective-conditioning, a learnable vector of dimension $$e_{0:T} = \bigl\{e_0,e_1,\dots,e_T\bigr\},\quad e_t\in\{1,\dots,15\},$$2 is embedded, concatenated or added after the spatiotemporal encoder, and allowed to modulate subsequent attention layers, analogously to FiLM layers or cross-attention keys and values. In this contextualized form, the reverse kernel can be written as

$$e_{0:T} = \bigl\{e_0,e_1,\dots,e_T\bigr\},\quad e_t\in\{1,\dots,15\},$$3

with $$e_{0:T} = \bigl\{e_0,e_1,\dots,e_T\bigr\},\quad e_t\in\{1,\dots,15\},$$4 taking $$e_{0:T} = \bigl\{e_0,e_1,\dots,e_T\bigr\},\quad e_t\in\{1,\dots,15\},$$5, and optionally the last event $$e_{0:T} = \bigl\{e_0,e_1,\dots,e_T\bigr\},\quad e_t\in\{1,\dots,15\},$$6, as input (Rao et al., 13 Apr 2026).

Full-horizon generation proceeds autoregressively in windows of length $$e_{0:T} = \bigl\{e_0,e_1,\dots,e_T\bigr\},\quad e_t\in\{1,\dots,15\},$$7. For each sample, GenTac initializes noisy futures, denoises them through $$e_{0:T} = \bigl\{e_0,e_1,\dots,e_T\bigr\},\quad e_t\in\{1,\dots,15\},$$8 reverse steps, appends each generated window to the current history, and repeats until the horizon is covered. Repeating this procedure $$e_{0:T} = \bigl\{e_0,e_1,\dots,e_T\bigr\},\quad e_t\in\{1,\dots,15\},$$9 times produces a distribution of rollouts rather than a single path.

Counterfactual simulation is realized by swapping the context embedding to an “offensive” or “defensive” embedding and then re-sampling. According to the description, this shifts the learned diffusion vector field and yields rollouts with systematically higher expected threat under offense or more compact spatial structures under defense. This operationalizes tactical “what-if” analysis in a generative setting rather than through deterministic perturbation.

A frequent misunderstanding in tactical simulation is to treat conditioning as a simple metadata tag. In GenTac, conditioning is part of the denoising dynamics themselves, so style, opposition, and objectives influence the geometry of generated futures rather than merely labeling outputs.

5. Evaluation on TacBench

Evaluation is organized around geometric accuracy, collective structural consistency, and tactical outcomes. Geometric accuracy uses Average Displacement Error (ADE) and Final Displacement Error (FDE): $e_t$0

Collective structural metrics operate on team positions $e_t$1 with centroid $e_t$2. The reported metrics are stretch index, surface area, team width, team length, Frobenius norm, centroid displacement, and Kuramoto order $e_t$3. Errors are reported as the absolute deviation $e_t$4 from ground truth. Tactical outcome metrics use a precomputed EPV grid and include off-ball expected threat (OBET), depth/width threat, defensive disruption, and defensive dominant region.

The reported quantitative findings are as follows (Rao et al., 13 Apr 2026):

• Unconditioned forecasts with 4 s history and 0.2 s window achieve $e_t$5 m and $e_t$6 m.
• Opponent-conditioning reduces $e_t$7 to $e_t$8 m.
• Team-conditioning on Auckland FC preserves structural consistency; for example, surface area error drops by $e_t$9 at 5 s, at the expense of slight geometric loss.
• League-conditioning improves short-horizon ADE by up to $$\mathcal C=\{\varnothing,\;c^{\rm opp},\;c^{\rm team},\;c^{\rm league},\;c^{\rm obj}\},$$0 for A-League versus German leagues.
• Objective-conditioning steers OBET by $$\mathcal C=\{\varnothing,\;c^{\rm opp},\;c^{\rm team},\;c^{\rm league},\;c^{\rm obj}\},$$1, depth/width threat by $$\mathcal C=\{\varnothing,\;c^{\rm opp},\;c^{\rm team},\;c^{\rm league},\;c^{\rm obj}\},$$2 units, and defensive metrics in the expected direction.
• Event grounding from real trajectories reaches type@1 $$\mathcal C=\{\varnothing,\;c^{\rm opp},\;c^{\rm team},\;c^{\rm league},\;c^{\rm obj}\},$$3 and subtype@1 $$\mathcal C=\{\varnothing,\;c^{\rm opp},\;c^{\rm team},\;c^{\rm league},\;c^{\rm obj}\},$$4.
• Event forecasting via generated rollouts yields top-3 type recall $$\mathcal C=\{\varnothing,\;c^{\rm opp},\;c^{\rm team},\;c^{\rm league},\;c^{\rm obj}\},$$5 and top-5 subtype recall $$\mathcal C=\{\varnothing,\;c^{\rm opp},\;c^{\rm team},\;c^{\rm league},\;c^{\rm obj}\},$$6.

Taken together, these results support four capabilities highlighted in the paper: geometric fidelity with collective structural consistency, simulation of team and league style, controllable counterfactual tactical shifts, and anticipation of future tactical outcomes. The emphasis on structural metrics is significant because low displacement error alone does not establish that a team shape remains tactically coherent.

6. Transfer to other sports, limitations, and extensions

GenTac is reported to generalize to other dynamic team sports by adapting only $$\mathcal C=\{\varnothing,\;c^{\rm opp},\;c^{\rm team},\;c^{\rm league},\;c^{\rm obj}\},$$7 and the frame rate, with all other modeling choices remaining identical. The reported settings are basketball at 5 fps with $$\mathcal C=\{\varnothing,\;c^{\rm opp},\;c^{\rm team},\;c^{\rm league},\;c^{\rm obj}\},$$8, American football at 10 fps with $$\mathcal C=\{\varnothing,\;c^{\rm opp},\;c^{\rm team},\;c^{\rm league},\;c^{\rm obj}\},$$9, and ice hockey at 30 fps with $p_\theta\bigl(x_{0:T},\,e_{0:T}\mid c\bigr).$0. The corresponding opponent-conditioned values are $p_\theta\bigl(x_{0:T},\,e_{0:T}\mid c\bigr).$1 m for basketball, $p_\theta\bigl(x_{0:T},\,e_{0:T}\mid c\bigr).$2 m for American football, and $p_\theta\bigl(x_{0:T},\,e_{0:T}\mid c\bigr).$3 m for ice hockey (Rao et al., 13 Apr 2026).

The stated strengths are that GenTac captures the full distribution of plausible futures rather than only a mean trajectory, unifies continuous trajectories and discrete events in one framework, supports rich context including opponent, club style, league style, and strategic objectives, enables counterfactual “what-if” tactical simulations, and generalizes across ball sports with minimal adaptation. The stated limitations are dependence on high-fidelity tracking data, the low-dimensionality of context $p_\theta\bigl(x_{0:T},\,e_{0:T}\mid c\bigr).$4, forecast quality decline at very long horizons as with all autoregressive models, and the use of coarse tactical-objective labels. Proposed extensions include learning continuous latent objectives via inverse RL rather than discrete labels, incorporating real-time broadcast video input for on-the-fly trajectory forecasting, integrating player-specific latent skill embeddings or fatigue models, and extending to mixed continuous/discrete controls in robotics or traffic forecasting.

These limitations help delimit the method’s present scope. In particular, the explicit note that latent cognitive factors are unobserved restricts any interpretation of the model as a complete theory of tactical intent. The proposed extensions suggest a broader research direction in which multi-agent diffusion forecasting becomes a general-purpose tool for structured dynamical systems. In the paper’s own summary, GenTac pioneers a diffusion-based generative paradigm for decoding and guiding the stochastic, multi-agent choreography at the heart of soccer and related team sports (Rao et al., 13 Apr 2026).