--- title: Adaptive Proxy Prefix CoTs Distillation url: https://www.emergentmind.com/topics/proxy-prefix-cots type: topic --- # Adaptive Proxy Prefix CoTs Distillation Proxy Prefix CoTs refer to the adaptive truncation and utilization of teacher-generated chain-of-thought (CoT) reasoning prefixes for distillation into smaller student models, addressing the challenge of lengthy and structurally complex teacher trajectories in reasoning-intensive tasks such as mathematical problem solving. The central procedure—Prefix-ALIGNment distillation (P-ALIGN)—algorithmically locates the minimally sufficient prefix of reasoning steps necessary for effective student supervision, avoiding the negative impact of irrelevant or uncertain suffix information. This approach systematically surpasses prior distillation strategies in mathematical reasoning tasks, yielding improved downstream accuracy and more effective alignment between teacher signals and student learning capacity [2601.10064]. ## 1. Formal Definitions and Preliminaries For each query \(q\), a reasoning trajectory generated by a teacher language model is defined as \(R = \{z_1, z_2, \dots, z_T\}\), where each \(z_t\) is an atomic reasoning statement and \(T\) denotes trajectory length. Segmenting \(R\) as \(R = \{r_1,\ldots,r_m\}\), a prefix is given as \(P_t = \{r_1, \ldots, r_t\}\) and its complementary suffix \(S_t = \{r_{t+1}, \ldots, r_m\}\). The guiding models are denoted \(M_{\mathrm{teacher}}\) (e.g., DeepSeek-R1) and \(M_{\mathrm{student}}\) (e.g., Qwen2.5-7B, Qwen3-8B, Llama3.2-3B). ## 2. Adaptive Prefix Alignment Algorithm The methodological core of P-ALIGN is to locate the minimal prefix \(P_{t^*}\) that is sufficient for the student to solve \(q\). Sufficiency is operationalized via a "self-judging function": \[ S(P_t; q) = P(\text{ENOUGH} \mid q, P_t ; M_{\mathrm{student}}) \] where \(M_{\mathrm{student}}\), upon being prompted with \(P_t\), indicates sufficiency (ENOUGH) or insufficiency (NOT_ENOUGH). A sufficiency threshold \(\tau\) (typically \(\tau = 0.5\)) is applied such that a prefix is deemed sufficient iff \(S(P_t; q) \ge \tau\). A binary search procedure efficiently identifies \(t^*\), requiring only \(O(\log m)\) calls as opposed to the naive \(O(m)\) scan. The process proceeds by, for each candidate prefix length \(p\), invoking \(M_{\mathrm{student}}\) as a proxy judge, and converging to the minimal sufficient \(P_{t^*}\). ## 3. Training Objective and Dataset Construction Upon determination of each \(P_{i, t^*}\), a distillation dataset is constructed as follows: 1. The minimal prefix \(P_{i, t^*}\) for each sample \(i\) is passed under an alignment prompt to \(M_{\mathrm{student}}\) to generate the full CoT continuation \(y_i\). 2. Only those tuples \((q_i, P_{i, t^*}, y_i)\) that result in a final answer matching the ground truth are retained: \[ D_{\mathrm{align}} = \{(q_i, P_{i, t^*}: y_i) \mid \text{Ans}(y_i) = \text{gold}_i\} \] 3. The main objective is the cross-entropy alignment loss: \[ \mathcal{L}_{\mathrm{align}} = -\sum_{(q, P{:}y)\in D_{\mathrm{align}}}\sum_{t=1}^{|P{:}y|} \log P_\theta \left((P{:}y)_t \mid (P{:}y)_{