IF-MAP: Influence-Function-Inspired Agent Prioritization in Multi-Agent Systems

IF-MAP. Influence-Function-Inspired Agent Prioritization in Multi-Agent Systems. By Jie Mao and eight others, from the Chinese University of Hong Kong, Shenzhen, the Harbin Institute of Technology, and one other institution. Published on the seventeenth of October. Its code repository is public and real: about twenty-five source files.

In a multi-agent system built from language models, improving every agent can be prohibitively expensive. With limited resources, the practical question becomes which agent to prioritise to get the biggest gain for the system as a whole. Failure attribution looks like it should answer that, but it answers a different question. Knowing which agent is responsible for failures does not directly tell you how much the system would gain by improving that agent. Blame and potential are not the same quantity.

IF-MAP is a method inspired by influence functions, built to identify the agents with high potential for system-level improvement. Rather than asking who caused a failure, it asks how sensitive the end-to-end result is to each agent, and uses that sensitivity to rank them.

For each agent, IF-MAP applies controlled semantic perturbations to its prompt and measures how the end-to-end loss changes. It fits a regularised local response model to those changes, in a low-dimensional subspace of prompt embeddings, and takes the norm of the estimated response as that agent's priority score. The procedure is black-box. It needs no access to model parameters or gradients, and the priorities it produces can guide whatever improvement effort comes next.

The evaluation covers six multi-agent workflows spanning code generation, question answering and mathematical reasoning, and the same priorities are used for two kinds of improvement: prompt optimization and model upgrades. Optimizing the prompts of the selected agents gives a macro-average gain of three point eight one percentage points, against one point eight five when agents are chosen uniformly.

Upgrading models is where the gap widens. Across five target models, upgrading the selected agents yields macro-average gains ranging from seven point six two up to sixteen point four two percentage points. Uniform selection gives a range from three point eight one up to eight point six two.

Probing has a cost of its own, yet IF-MAP still saves tokens overall. Relative to optimizing every agent, it cuts token use by seventy-four point two percent with Qwen3-8B, and by thirty-five point two percent with DeepSeek-V3.2. Relative to upgrading every agent, it reduces target-model tokens by roughly sixty-four percent, with a reported range from sixty-three point six to sixty-four point nine.

The distinction the paper rests on is between responsibility and leverage. An agent can be implicated in failures without being the one whose improvement would move the system most. Measuring how the whole pipeline responds when each agent is nudged is what lets IF-MAP spend a limited budget where it buys the most end-to-end performance.

IF-MAP. Influence-Function-Inspired Agent Prioritization in Multi-Agent Systems. By Jie Mao and eight others, from the Chinese University of Hong Kong, Shenzhen, the Harbin Institute of Technology, and one other institution. Published on the seventeenth of October. This is the long version: the whole paper rather than the abstract.

A multi-agent system built from language models splits a task across roles. One agent writes code, another tests it, another merges candidate answers. When the system falls short and there is budget to improve only one agent, which one should it be? Upgrading every agent to a stronger model raises the price of every single run. The natural move is to ask which agent caused the failures. The argument here is that this answers a different question. An agent can contribute to errors often and still offer little room for improvement, while another admits a change that lifts everything downstream. The target, then, is not responsibility. It is the end-to-end gain you would actually get from improving that agent.

The setting is narrow and precise. The workflow is fixed: its roles, connections and assigned models stay put. Each task example gets a system score between zero and one, and the loss is one minus that score. What a prioritizer wants is the expected drop in loss from applying some improvement procedure to one agent alone, with every other agent held fixed. Measuring that directly means running the full improvement on every agent, which is exactly the cost being avoided. The inspiration is influence functions, a classical tool for estimating how perturbing training data changes a learned model's predictions. Only the perturb-and-measure perspective is borrowed: no parameters, gradients or Hessians are touched. Run the workflow, read the loss.

Probing comes first. For each agent, starting from its original prompt, the method builds at most ten perturbed prompts: clarify what the downstream agent needs, resolve an ambiguous instruction, add a verification step, clarify edge cases, and so on, plus adaptive slots that revise earlier candidates using their measured effects. Every variant keeps the agent's role and output interface, and structural checks reject broken or duplicate prompts. Each probe changes only that one agent's prompt, and then the whole workflow runs on the probing set, so the measured loss includes whatever the change sets off downstream.

Scoring comes next. A fixed embedding model, Qwen3-Embedding-8B, maps each prompt to a normalized vector. Each probe then yields two things: the direction in which the prompt moved, as a unit vector, and the change in loss divided by how far it moved, so a loss change per unit of embedding distance. A linear model is fitted to predict that normalized response from the direction. With at most ten probes in a large embedding space, an unrestricted fit would be unreliable, so two restraints apply. The fit is confined to a small basis built from a singular value decomposition of the probe directions themselves, and ridge regularization shrinks large, unstable coefficients. The fitted vector acts as a local slope, and its length is the agent's priority score, called G-norm. That length has a clean reading: for a common step size within the fitted subspace, it is proportional to the largest first-order loss reduction the model predicts. Rank the agents by it, and improve the top one.

There are six workflows, all derived from AFlow: HumanEval and MBPP for code, NQ, HotpotQA and DROP for question answering, and MATH. Qwen3-8B runs the workflows, and DeepSeek-V3.2 rewrites prompts. The same priorities then feed two quite different interventions. Under prompt optimization, improving the agent IF-MAP picks gives a macro-average gain of three point eight one percentage points, against one point eight five for an agent chosen uniformly at random, and above all six failure-attribution baselines. Then, with no re-probing, the selected agent is handed a stronger model instead. Averaged over the three Qwen targets, the gain is twelve point five two points, against six point five seven for a uniform choice. Across all five targets, the gain is between one point eight six and two point zero five times the uniform one.

The comparison that tests the thesis most directly is the scoring ablation. Same probes, same test outcomes, only the rule that turns probes into a score changes. G-norm beats both magnitude-only scores on selected gain, by about three points on model upgrades. The geometry of the prompt changes is doing real work. Probe count matters too: cut the pool to three probes and the model-upgrade gain roughly halves.

There is no limitations section. The conclusion carries one sentence of caveats: realized gains depend on the embedding model, on local linearity, on probe coverage, and on whether the available interventions line up with the fitted loss-reducing direction. The rest sits in the appendix. Uncertainty is the standard error across three prioritization runs, and the authors say it describes variability rather than serving as a significance test. Every workflow is AFlow-derived, and five are simple chains; independently built workflows with different interaction graphs are left as future work. Probing costs more base-model executions than the attribution baselines, though far fewer calls to the diagnostic model. And the probe-budget effect is not monotonic per workflow: on HumanEval, three probes did better than the full pool for prompt optimization.

Blame and leverage are different quantities. Rather than asking a judge model to read failed traces and name a culprit, IF-MAP nudges each agent with a handful of meaningful prompt edits, watches the end-to-end loss, and fits a slope. A steep slope marks an agent where change pays off. The decision also transfers: priorities found by editing prompts picked the right agent to hand a bigger model, and upgrading that one agent instead of all of them cut target-model tokens by sixty-three point six to sixty-four point nine percent. The score says where the loss responds, not yet what edit to make.