💬 NLP

System 1&2 Synergy via Dynamic Model Interpolation

This paper proposes DAMI, a training-free framework that dynamically interpolates between intuitive System 1 and deliberative System 2 model checkpoints using query-specific reasoning intensity to achieve superior accuracy-efficiency trade-offs in mathematical reasoning.

Original authors: Chenxu Yang, Qingyi Si, Chong Tian, Xiyu Liu, Dingyu Yao, Chuanyu Qin, Zheng Lin, Weiping Wang, Jiaqi Wang

Published 2026-01-30

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CC BY 4.0

Original authors: Chenxu Yang, Qingyi Si, Chong Tian, Xiyu Liu, Dingyu Yao, Chuanyu Qin, Zheng Lin, Weiping Wang, Jiaqi Wang

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you have two different versions of a brilliant assistant:

The "Fast Thinker" (System 1): This assistant is like a seasoned barista. They can whip up a coffee (answer a simple question) in seconds. They are fast, efficient, and great for routine tasks. But if you ask them to solve a complex physics problem, they might just guess or give a shallow answer because they aren't used to slowing down.
The "Deep Thinker" (System 2): This assistant is like a professor in a library. They take their time, write out long proofs, double-check their work, and think deeply. They are incredibly accurate on hard problems, but they are slow and use up a lot of paper (computing power) even for simple questions.

The Problem:
Usually, you have to pick one or the other. If you want speed, you lose accuracy on hard stuff. If you want accuracy, you waste time and resources on easy stuff. Recent attempts to fix this tried to tell the "Deep Thinker" to "stop talking" or "write less" (Output Control). The authors of this paper say that's like telling a professor to stop thinking just to save paper; it cuts off their reasoning process and makes them fail.

The Solution: "DAMI" (Dynamic Model Interpolation)
The authors propose a new way to think about this. Instead of telling the model what to produce (how long to talk), they change how the model thinks.

They discovered that the "Fast Thinker" and the "Deep Thinker" are actually built on the same foundation. If you take the "brain" (the mathematical weights) of the Fast Thinker and the Deep Thinker and mix them together, you get a smooth spectrum of intelligence.

The Analogy: Imagine a dimmer switch for a lightbulb.
- 0% (Off/Fast): The light is just the Fast Thinker.
- 100% (Bright/Deep): The light is the Deep Thinker.
- 50% (Middle): You get a perfect blend.

The paper calls this Capability Control. Instead of cutting the Deep Thinker's sentence short, they simply turn the "dimmer switch" down for easy questions and turn it up for hard ones.

How Does It Know When to Switch?
The system, called DAMI, acts like a smart manager who looks at every question and decides exactly how much "thinking power" is needed. It uses two methods to make this decision:

The "Trainer" Method (DAMI-Pref): If you have some practice questions, you can teach a small helper model to look at a question and say, "This one needs 80% deep thinking," or "This one only needs 20%." It learns by comparing different levels of thinking to see which one gets the right answer with the least effort.
The "Instinct" Method (DAMI-Conf): If you don't have time to train anything, the system looks at how confident the Fast Thinker and Deep Thinker are. If the Fast Thinker is confused and the Deep Thinker is sure, the system knows it's a hard question and turns the "thinking dial" up. If both are confident, it keeps the dial low to save time.

The Results:
The authors tested this on math problems ranging from simple arithmetic to tough competition-level puzzles.

Speed: It was much faster than the full "Deep Thinker" because it didn't waste time overthinking easy questions.
Smarts: It was more accurate than the "Fast Thinker" because it knew when to slow down and think deeply for hard problems.
Efficiency: It used significantly fewer computer resources (tokens) while getting better results than trying to force the Deep Thinker to be short.

In a Nutshell:
The paper shows that you don't need to build a new, super-complex AI to be both fast and smart. You just need to take two existing AIs (one fast, one smart), mix their brains together, and use a smart switch to adjust the mix in real-time. This creates a single system that is efficient enough for a quick chat but deep enough to solve a math puzzle.

Technical Summary: System 1&2 Synergy via Dynamic Model Interpolation

Problem Statement

Large Language Models (LLMs) exhibit a functional dichotomy between "System 1" (intuitive, fast, concise) and "System 2" (deliberative, slow, reasoning-heavy) cognitive modes. While System 2 models (e.g., o1, DeepSeek-R1) achieve superior accuracy on complex tasks through extended Chain-of-Thought (CoT), they suffer from high computational costs and verbose outputs. Conversely, System 1 models are efficient but lack deep reasoning capabilities.

Current research attempts to unify these modes or optimize System 2 efficiency primarily through output control. These methods constrain what models produce by limiting token budgets, applying early-exit mechanisms, or compressing CoT. The authors argue this paradigm is misaligned: output length is a symptom of cognitive configuration, not the root cause of inefficiency. Furthermore, training a single unified model that seamlessly adapts between these modes remains challenging due to the "zero-sum nature" of training data, where the verbose style of System 2 conflicts with the conciseness of System 1, leading to mutual interference during joint optimization.

Methodology: DAMI (DynAmic Model Interpolation)

The paper proposes a paradigm shift from output control to capability control, modulating how models think rather than what they produce. This is achieved through Dynamic Model Interpolation, leveraging existing System 1 (Instruct) and System 2 (Thinking) checkpoints without additional training of the base models.

1. Theoretical Foundation

The authors establish that linear parameter interpolation between Instruct ( $\Theta^{(I)}$ ) and Thinking ( $\Theta^{(T)}$ ) models yields predictable behavior, grounded in three empirical and theoretical findings:

Performance Monotonicity: Increasing the interpolation coefficient $\lambda$ (where $\lambda=0$ is Instruct and $\lambda=1$ is Thinking) produces a smooth, convex Pareto frontier between accuracy and efficiency. Intermediate models often outperform both endpoints, achieving higher accuracy than Instruct with fewer tokens than Thinking.
Representation Continuity: Principal Component Analysis (PCA) reveals that interpolated models form a continuous trajectory in representation space, with the first principal component strongly correlating with $\lambda$ ( $r=0.974$ ).
Structural Connectivity: The Instruct and Thinking models exhibit high cosine similarity (>0.99) across all transformer layers, satisfying Linear Mode Connectivity (LMC). This indicates both models reside within the same optimization basin, ensuring the interpolation path avoids high-loss regions.

2. The DAMI Framework

The core innovation is DAMI, which dynamically estimates a query-specific Reasoning Intensity $\lambda(q) \in [0, 1]$ to configure the model's cognitive depth for each input:
$\Theta^{(M)}(q) = \lambda(q)\Theta^{(T)} + (1 - \lambda(q))\Theta^{(I)}$

Two complementary methods are proposed for estimating $\lambda(q)$ :

DAMI-Pref (Training-Based): Designed for scenarios with in-domain training data. It frames $\lambda$ estimation as a preference learning problem rather than regression. A lightweight reward model is trained to score pairs of interpolation coefficients based on a hierarchical criterion: prioritize accuracy, then use efficiency (token count) as a tiebreaker. This approach is robust to label noise and requires less data than regression-based methods.
DAMI-Conf (Training-Free): Designed for zero-shot deployment. It estimates $\lambda(q)$ $λ (q)$ using intrinsic confidence signals from the base models. It fuses two signals:
1. Holistic Ambiguity: Measures overall system uncertainty (low confidence in both models implies high difficulty).
2. Cognitive Discrepancy: Measures the disagreement in confidence between the Instruct and Thinking models (high discrepancy implies the query requires distinct strategies).
  These signals are fused and passed through a calibrated sigmoid function to determine $\lambda(q)$ .

Key Contributions

Paradigm Shift: The paper advocates moving from output control (limiting generation) to capability control (modulating cognitive configuration via parameter interpolation).
DAMI Framework: A novel framework that enables query-adaptive reasoning intensity without training a unified model from scratch, leveraging existing checkpoints.
Dual Estimation Strategies: The introduction of DAMI-Pref (preference-based) and DAMI-Conf (confidence-based) to address different deployment constraints (data availability vs. zero-shot needs).
Theoretical Validation: Formal proofs and empirical evidence demonstrating that linear interpolation between co-originated models satisfies LMC, ensuring valid, monotonic, and continuous transitions in performance and representation.

Experimental Results

The authors evaluated DAMI on five mathematical reasoning benchmarks (GSM8K, MATH-500, AMC, AIME 2024/25) using two model families (Qwen3-4B and DeepSeek-R1/Qwen2.5-7B).

Performance vs. Efficiency: DAMI consistently establishes superior Pareto frontiers compared to static merging, early-exit (DEER), and routing baselines.
- On Qwen3-4B, DAMI-Pref achieved 91.6% accuracy (a +3.4% absolute gain over the Thinking model) while reducing token consumption by 29%.
- DAMI-Conf reduced token consumption by 40% while maintaining 90.7% accuracy (+2.5% over Thinking).
Adaptive Behavior: The "Thinking Ratio" (percentage of queries triggering deep reasoning) adapts to difficulty. For simple tasks (GSM8K), DAMI activates deep thinking for only ~25-41% of queries, whereas for complex tasks (AIME), this rises to 79-95%.
Multimodal Generalization: The framework was tested on multimodal tasks (video and image-text reasoning) using Qwen3-VL-8B. DAMI-Conf outperformed baselines, achieving a 2.5% absolute improvement in video understanding and 1.6% in image-text reasoning.
Case Study: In a geometry problem, the Thinking model generated 2,846 tokens with redundant verification. DAMI generated a correct solution in 351 tokens by directly formulating the problem and calculating, demonstrating that capability control produces inherently efficient reasoning rather than truncated output.

Significance and Claims

The paper claims that the dichotomy between fast and slow thinking need not be a binary choice resolved at training time. By establishing that reasoning depth exists on a smooth spectrum accessible through simple linear operations in parameter space, DAMI offers a flexible cognitive architecture.

The authors emphasize that their approach bypasses the difficulty of training unified System 1/2 models and avoids the performance degradation associated with output-constrained methods (which risk truncating reasoning chains). Instead, DAMI achieves a "sweet spot" by dynamically configuring the model's cognitive depth, effectively combining the efficiency of System 1 with the reasoning depth of System 2. The work suggests that reasoning resource allocation can be treated as a well-posed prediction problem, where estimating the optimal $\lambda(q)$ directly determines model behavior.