  Efficient fine-tuning of large language models for task-specific applications
is imperative, yet the vast number of parameters in these models makes their
training increasingly challenging. Despite numerous proposals for effective
methods, a substantial memory overhead remains for gradient computations during
updates. \thm{Can we fine-tune a series of task-specific small models and
transfer their knowledge directly to a much larger model without additional
training?} In this paper, we explore weak-to-strong specialization using logit
arithmetic, facilitating a direct answer to this question. Existing
weak-to-strong methods often employ a static knowledge transfer ratio and a
single small model for transferring complex knowledge, which leads to
suboptimal performance. % To address this, To surmount these limitations, we
propose a dynamic logit fusion approach that works with a series of
task-specific small models, each specialized in a different task. This method
adaptively allocates weights among these models at each decoding step, learning
the weights through Kullback-Leibler divergence constrained optimization
problems. We conduct extensive experiments across various benchmarks in both
single-task and multi-task settings, achieving leading results. By transferring
expertise from the 7B model to the 13B model, our method closes the performance
gap by 96.4\% in single-task scenarios and by 86.3\% in multi-task scenarios
compared to full fine-tuning of the 13B model. Notably, we achieve surpassing
performance on unseen tasks. Moreover, we further demonstrate that our method
can effortlessly integrate in-context learning for single tasks and task
arithmetic for multi-task scenarios. (Our implementation is available in
https://github.com/Facico/Dynamic-Logit-Fusion.)
