0.0.1 Introduction↩︎

Vision-language models have made substantial progress on multimodal reasoning [1], [2], but their behavior remains highly sensitive to how the information is presented [3]–[5]. As we also find in our experiments, a model may solve a problem when the relevant relations are written in text, yet fail when those relations must be recovered from an image; conversely, an image may make geometric structure salient in a way that a purely textual description does not. This inconsistency is especially concerning because the underlying reasoning problem has not changed, only the input view has. It suggests that current VLMs are not merely limited by problem difficulty [6], but also by view-dependent failures in grounding and reasoning. Our goal is to study and address these failures.

To do so, we focus on geometry problems that serve as a natural testbed for studying this phenomenon. Many geometry problems admit several views of the same underlying mathematical structure: a text-dominant view, where quantities and relations are enumerated in language; an image-dominant view, where the image carries essential geometric information and the accompanying text mainly specifies the question; and a combined image+text view, where both sources are available. Unlike many multimodal settings where visual inputs introduce information unavailable in language [7], [8], geometry allows the same problem to be expressed through text, diagrams, or both. This enables a controlled comparison of reasoning across views while keeping the underlying semantics unchanged. As we find in Table 1, this controlled comparison reveals a useful asymmetry: the same model can expose a valid reasoning path under one view even when it fails under another. This suggests that cross-view inconsistency is not just a robustness failure, but also a potential supervision signal.

Prior work on multimodal reasoning often improves models by using external teachers [9], curated rationales [10], or specialized training data that teaches the model how to combine visual and textual inputs [11], [12]. However, such supervision may not be available for every target setting, and distillation from external teachers can be fragile under distribution mismatch [13]. In contrast, we study whether paired views of the same problem can provide a form of self-supervision: when the model succeeds under one view but fails under another, can the successful view help improve the weaker one?

In this paper, we propose MIRROR (Modality-Informed Reciprocal Reasoning Optimization), a training framework that uses reasoning from stronger views of a problem to improve reasoning under weaker views. For each problem, we construct text-dominant, image-dominant, and combined image+text views (Figure 1), evaluate the current model under each view, and select the strongest-performing view as a problem-specific teacher. The student is trained under a restricted view with no information redundancy which is either text-dominant or image-dominant. The challenge is to transfer the teacher signal without forcing the student to imitate trajectories that depend on information unavailable from its own view. MIRROR addresses this with on-policy distillation: the student samples rollouts from its own view, while the selected teacher view only rescores the same student-generated trajectories. Combined with outcome-reward RL under the student view, this provides dense teacher guidance while keeping optimization on states actually visited by the restricted-view student.

To instantiate our approach, we construct ODA-Data, a paired geometry dataset with text-dominant, image-dominant, and combined image+text views of the same underlying problems. ODA-Data preserves problem identity across views, enabling us to measure both accuracy and cross-view consistency. We use ODA-Train for post-training and ODA-Val as a held-out benchmark to measure view-dependent solvability: whether a model solves the same problem consistently when presented through different views. Empirically, MIRROR outperforms both post-training on a mixture of prompts and recent open-source post-trained VLMs. On ODA-Val, MIRROR achieves the best performance across all image and text validation metrics, improving image pass@16 by 14.49% and text pass@16 by 5.88% over the base model. These gains transfer to external geometry benchmarks: on GeoInt, MIRROR improves over the base model by 7.59% in pass@16, and on MathVerse, it improves the mean judge score by 5.22%. Beyond accuracy, on ODA-Train, the fraction of problems solved under both image-dominant and text-dominant views increases from 42.5% to 60.7% with MIRROR training. MIRROR also expands the set of solvable examples, achieving the largest net solvability gain on ODA-Data at +8.95%. These results show that paired views provide an effective self-supervision signal for stronger and more consistent multimodal reasoning.

0.0.2 Related Work↩︎

Multimodal mathematical reasoning benchmarks and data scaling. Prior work shows that specialized multimodal math data can substantially improve MLLM reasoning. For example, Math-LLaVA fine-tunes LLaVA-1.5 on MathV360K [14], and MultiMath-7B trains on MultiMath-300K with image captions and step-by-step solutions [15]. Several benchmarks have also been introduced for visual mathematical reasoning, including MATH-Vision [16], MathVerse [4], and geometry-focused benchmarks such as LeanGeo-Bench [17], GeoInt [18], and GeoGoal [19]. More recently, CROSSMATH studies modality gaps using matched text-dominant, image-dominant, and the combined image+text versions of synthetic puzzles [20]. These works motivate our focus on geometry and provide evaluation settings that we build on, including MathVerse and GeoInt. Instead of focusing on scaling data or constructing benchmarks, our goal is algorithmic: how paired views of the same problem can be used during RL training to transfer reasoning behavior across modalities?

RL and cross-modal transfer for multimodal models. Recent work has explored RL and cross-modal transfer for multimodal reasoning. Vision-R1 uses synthetic chain-of-thought data and progressive RL [21]; R1-OneVision converts images into textual descriptions before SFT and RL [22]; and FAST uses difficulty-aware rewards with KL-regularized GRPO [23]. Closest to our setting, VOLD distills from a stronger text-only LLM teacher into a VLM student using on-policy distillation and GRPO [12]. These methods show that stronger teachers can improve multimodal reasoning. In contrast, we target self-supervised improvement: paired views of the same problem reveal cross-modal performance asymmetries that can serve as supervision for improving reasoning across modalities.

On-policy Distillation. On-policy distillation has emerged as a way to learn from a teacher while reducing the distribution mismatch of standard behavioral cloning: instead of training only on teacher-generated traces, the student samples its own rollouts and receives token-level guidance from a teacher on the states it actually visits [24], [25]. More broadly, prior work on self-improvement has trained language-model agents to critique and iteratively refine their own responses through recursive self-feedback [26]. On-policy self-distillation provides a complementary mechanism for internal improvement, including self-distillation from demonstrations [27], feedback-conditioned self-teachers [28], and privileged-context teachers [29]. Recent multi-teacher or multi-model extensions [30]–[32] use specialized teachers, debate [33], or calibrated teacher signals to merge complementary capabilities into a single student [34]–[37]. MIRROR differs in both the source and structure of the teacher signal: rather than distilling from an external stronger model, a fixed teacher pool, or privileged solution traces, it selects the strongest on-policy generations for a certain view of the same problem as a problem-specific teacher, building a self-supervised approach for improving model.

Explicit intermediate structure for visual reasoning. A complementary line of work improves visual reasoning by either actively selecting informative visual evidence or introducing intermediate structures between perception and reasoning. Act2See learns to adaptively select and process visual observations for more effective video reasoning [38]. For visual mathematical reasoning, MathCoder-VL uses image-to-code supervision [39]. [40] introduces a geometry-specialized vision encoder for recognizing visual primitives. Related work on plane-geometry formalization [41] similarly exposes structured representations. These approaches introduce inductive biases through active perception, code, symbolic descriptions, formal proofs, or geometry priors. In contrast, MIRROR retains the original input views and remains end-to-end: rather than explicitly selecting evidence or constructing intermediate representations, it encourages the model to internalize transferable reasoning structure through cross-modal supervision and regularization.

0.0.3 Preliminaries and Notation↩︎

We study post-training of a multimodal language model, denoted by \(\pi_{\text{base}}\) with parameters \(\theta\), on problem instances that may be presented with different input views. For a problem \(\mathbf{x}\sim \rho\), let \(\mathcal{M}(\mathbf{x})\) denote the set of available views of the same underlying problem content. In this work, we focus on two views: a text-dominant view \(\mathbf{x}^{\text{text}}\) and an image-dominant view \(\mathbf{x}^{\text{img}}\). We can also readily construct a combined view \(\mathbf{x}^{comb} := (\mathbf{x}^{\text{text}}, \mathbf{x}^{\text{img}})\), which contains both views simultaneously. For any input view \(\tilde{\mathbf{x}} \in \mathcal{M}(\mathbf{x})\), a rollout \(\mathbf{y}\sim \pi(\cdot \mid \tilde{\mathbf{x}})\) attempts to solve the problem. As in standard outcome-reward RL, we assume that correctness can be evaluated from the final answer, and define a binary reward \(r(\mathbf{x},\mathbf{y}) \in \{0,1\}\). We study performance using pass@\(k\) [42] under each modality.

0.0.4 Understanding Differences in Reasoning Across Modalities↩︎

Before introducing our approach, we first analyze reasoning behaviors of the base model Qwen3-VL-4B-Instruct [43] when solving the same underlying problem from different input views. In this section, we characterize when text-dominant, image-dominant, or combined text-and-image inputs lead to different reasoning behaviors and performance. Concretely, we ask three questions: (1) how do the model’s reasoning trajectories differ when conditioned on different modalities? (2) when is reasoning from one modality systematically easier than reasoning from another? and (3) Can access to both text and images reveal supervision signals that are hidden from any single modality?

To compare reasoning traces for different modalities per problem, we construct ODA-Data, a geometry dataset derived from OpenDataArena/ODA-Math-460k [44]. Each example contains two views of the same problem: a “text-dominant” view and an “image-dominant” view. In our analysis and training, we also consider a third combined image+text view, which is constructed on the fly by concatenating the text-dominant prompt with the image from the image-dominant view. Starting from the pure text dataset ODA-Math-460k, we first filter out easy problems, generate and verify TikZ-based diagrams using Gemini-3-Pro-Preview [45], and rewrite the image-dominant prompt so that it no longer contains information present in the diagram. We then run modality-specific evaluations with the base model Qwen3-VL-4B-Instruct [43] and retain problems that are solved under one view but not the other, yielding approximately 2K curated examples. The dataset is randomly split 85:15 into ODA-Train for post-training and ODA-Val for evaluation. See Appendix 0.0.10 for full details. Because ODA-Data is filtered to retain modality-dependent examples, ODA-Val is a diagnostic benchmark for understanding cross-view performance asymmetry and transfer.

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0.0.4.1 How do Reasoning Traces Differ Across Modalities?

We first compare reasoning traces induced by the text-dominant view, the image-dominant view and the combined image+text view of the same underlying problem. Although all views contain sufficient information to solve the task, they often induce substantially different trajectories. Text-conditioned reasoning tends to follow relations explicitly stated in language, while image-conditioned reasoning first recovers objects, relations, and spatial structure from the diagram. This grounding step can introduce errors that are absent in the text view. However, diagrams can sometimes make geometric structure visually salient, leading to more reliable solution paths than the ones generated under the text-dominant view. All in all, we observe both pros and cons of reasoning with diagrams.

Table 1 summarizes three representative cases. In Example 1, the image-dominant input makes the geometric configuration of four points visually salient. The image-conditioned model finds the correct structural solution, while the text-conditioned model falls back to numerical search; after training with MIRROR, the text-conditioned model recovers the same structural reasoning pattern. In Example 2, the text-dominant input explicitly provides the side lengths of the \(13\)–\(14\)–\(15\) triangle, while the image-dominant input requires the model to infer them from the diagram. The image-conditioned model initially misreads the drawn semicircle diameter, but avoids this grounding error and recovers the correct text-supervised answer after training. In Example 3, neither restricted view is fully reliable: the text-dominant model cannot resolve the geometric configuration, while the image-dominant model misses the relation \(AP=AD\) from the diagram. The combined image+text view succeeds by using the text to verify relations while also grounding the reasoning in the diagram; after training, the image-dominant model identifies the missing condition and predicts the correct option. We include more detailed model responses for these examples in Appendix 0.0.11.

These examples highlight three complementary sources of cross-modal supervision. When the diagram reveals useful structure, image-conditioned reasoning can supervise text-conditioned reasoning. When the text avoids a perceptual bottleneck, text-conditioned reasoning can supervise image-conditioned reasoning. When neither restricted view is fully reliable, the combined view can provide a stronger teacher by jointly conditioning on visual structure and explicit textual relations. However, no single direction is uniformly reliable: the best teacher depends on whether the bottleneck is interpreting textual constraints, visual grounding, or missing cross-view context. This motivates our use of adaptive teacher selection over text-dominant, image-dominant, and combined views.

Quantitative analysis. We next quantify these view-dependent differences over ODA-Train using Qwen3-VL-4B-Instruct and examine pass@\(1\) metric. At the aggregate level, text-dominant achieves 29.58%, image-dominant achieves 10.54%, and combined image+text prompting achieves 26.84%. Across ODA-Train, the text-dominant view performs best on 51.14% of problems, the image-dominant view on 37.98%, and the combined image+text view on 10.88%. Thus, although text-dominant prompting is strongest on average, nearly half of the problems are best supervised by a non-text-dominant view. These results suggest that the qualitative behaviors in Table 1 are not isolated anecdotes, but recurring sources of modality asymmetry.

0.0.4.2 Does Mixed-Modality RL Transfer Across Views?

Given the asymmetries above, a natural question is whether standard RL can exploit multiple views simply by training on them together. That is, if the model observes text-dominant, and image-dominant views of the same problems during RL, will successful reasoning behavior automatically transfer from the more performant view to the less performant one? We answer this question by training standard GRPO [46] on such a mixed-modality single problem dataset, where the underlying problem appears under both text-dominant and image-dominant views. We also include MIRROR in Figure 2 as a preview of our method that will be introduced in Section 0.0.5; we defer the method details and main evaluation to later sections. Standard mixed-modality RL treats each view as an independent RL prompt and relies on shared policy updates to induce transfer across modalities. RL trained on a single view of the underlying problem removes this cross-view heterogeneity and optimizes only on a single student view.

As shown in Figure 2, naive mixed-modality RL (GRPO) achieves the lowest training reward on both text-dominant and image-dominant student views. Training GRPO on heterogeneous input views of the same questions can therefore inhibit performance gains on each view, rather than automatically transferring successful reasoning across views. In contrast, our approach MIRROR, introduced in the next section, achieves the highest training reward for both student views. This ordering indicates that incorporating multiple views of the same question in RL is not automatically beneficial. We next introduce MIRROR, which converts the strongest view of each problem into a teacher signal for training the student views.

0.0.5 Our Approach: MIRROR↩︎

Our goal is to improve reasoning under a student view, either the text-dominant view or the image-dominant view. Section 0.0.4 shows that no single view is uniformly best, and MIRROR converts this view-dependent heterogeneity into a self-supervision signal. For each problem, it selects the view on which the current model performs the best as a problem-specific teacher. The restricted-view student is trained with standard outcome-reward RL plus an auxiliary reverse-KL regularizer. Crucially, the student always generates rollouts from its own restricted view; the selected teacher view is used only to score those student-generated trajectories.

Best-modality teacher selection. Each problem has three candidate teacher views: text-dominant \(\mathbf{x}^{\text{text}}\), image-dominant \(\mathbf{x}^{\text{img}}\), and combined image+text \(\mathbf{x}^{\text{comb}}=(\mathbf{x}^{\text{text}},\mathbf{x}^{\text{img}})\). Let \(\mathcal{M}=\{\text{text},\text{img},\text{comb}\}\) denote the candidate teacher views. Teacher selection is performed online during training using the current policy model. For each problem \(\mathbf{x}\) and each candidate view \(j\in\mathcal{M}\), we sample \(K=16\) rollouts and estimate the view-specific success rate. We then select the best-performing view, breaking ties uniformly at random. The teacher distribution is then defined as a slow-moving copy of the model conditioned on the selected teacher view, \(\pi_{\text{teach}}(\cdot\mid\mathbf{x}) = \pi_{\bar\theta}(\cdot\mid \mathbf{x}^{(j^\star(\mathbf{x}))})\), where \(\bar\theta\) is an exponential moving average (EMA) of the policy parameters, described below. Thus, the teacher direction is selected independently for each problem and can adapt throughout training, rather than being fixed globally or precomputed from the base model. Although teacher selection considers three candidate views, the text-dominant and image-dominant rollouts are already generated as standard online GRPO rollouts; thus, the only additional rollout generation required by MIRROR is for the combined image+text view, which introduces negligible computational overhead. We provide a FLOPs comparison with mixed-modality GRPO in Appendix 0.0.8.

Training under a restricted view. For a restricted view \(m\in\{\text{text},\text{img}\}\), the student receives the corresponding input \(\mathbf{x}^{(m)}\) and samples rollouts \(\mathbf{y}\sim\pi_\theta(\cdot\mid\mathbf{x}^{(m)})\). The selected teacher may condition on a different view \(\mathbf{x}^{(j^\star(\mathbf{x}))}\), including the combined image+text view. The challenge is therefore to use the stronger teacher view without training the student on off-policy teacher-generated trajectories or forcing it to rely on information unavailable from its restricted input. We address this by optimizing a weighted sum of outcome-reward RL and an on-policy reverse-KL regularizer: \[\begin{align} \label{eq:full95loss} \mathcal{L}(\theta;m) = \mathcal{L}_{\text{RL}}(\theta;m) + \lambda_{\text{KL}}\mathcal{L}_{\text{rKL}}(\theta;m), \end{align}\tag{1}\] where \(\lambda_{\text{KL}}\ge 0\) controls the strength of teacher guidance. The RL term is a GRPO-style on-policy objective that optimizes final-answer correctness under the restricted student view. The reverse-KL term, defined below, transfers token-level preferences from the selected teacher view while keeping rollouts on-policy with respect to the restricted-view student.

Reverse KL for self-supervised, on-policy distillation. The auxiliary loss distills the selected teacher’s reasoning preferences into the restricted-view student without sampling trajectories from the teacher. For a student view \(m\in\{\text{text},\text{img}\}\), we first sample a rollout from the policy being trained: \(\mathbf{y}\sim \pi_\theta(\cdot \mid \mathbf{x}^{(m)})\). We then keep this student-generated token sequence fixed and evaluate how likely each sampled token would be under the selected teacher view. Specifically, at each prefix \(y_{1:t}\), we compare the student’s next-token log-probability under the restricted input \(\mathbf{x}^{(m)}\) with the teacher-view log-probability of the same sampled token under \(\mathbf{x}^{(j^\star(\mathbf{x}))}\): \[\begin{align} \label{eq:rkl95obj} \mathcal{L}_{\text{rKL}}(\theta;m) = \mathbb{E}_{\mathbf{x},\mathbf{y}\sim \pi_\theta(\cdot \mid \mathbf{x}^{(m)})} \left[ \sum_{t=1}^{|\mathbf{y}|-1} \Big( \log \pi_\theta(y_{t+1}\mid \mathbf{x}^{(m)},y_{1:t}) - \log \pi_{\bar\theta}(y_{t+1}\mid \mathbf{x}^{(j^\star(\mathbf{x}))},y_{1:t}) \Big) \right]. \end{align}\tag{2}\] Because the expectation is over \(\mathbf{y}\sim\pi_\theta(\cdot\mid\mathbf{x}^{(m)})\), Equation 2 is an on-policy Monte Carlo estimate of a reverse-KL-style objective from the restricted-view student to the selected teacher view. Operationally, the teacher is used only to rescore the student’s own sampled trajectory: we ask how compatible the student’s tokens are with the stronger teacher view. This avoids unnecessary distribution shift because optimization occurs only on states induced by the policy being trained.

EMA teacher. Because the teacher in Equation 2 is derived from the same network being trained, the most direct choice is to score the teacher term with the current parameters \(\theta\). However, this makes the distillation target move every step: the student chases a target that its own updates displace. Empirically, this self-referential coupling induces a slow inflation of policy entropy and reference-policy KL that eventually destabilizes long-horizon training (Section 0.0.6). We therefore parameterize the teacher with an exponential moving average (Polyak average) of the policy parameters, \[\begin{align} \label{eq:ema95update} \bar\theta \;\leftarrow\; \alpha\,\bar\theta + (1-\alpha)\,\theta, \end{align}\tag{3}\] applied after every policy update, with \(\bar\theta\) initialized to \(\theta\) at begining. The EMA teacher is used only to rescore the student’s sampled tokens under the selected teacher view (forward-only, no gradients); teacher selection still uses the current policy’s rollouts. This mirrors target networks in value-based RL [47], [48] and weight-averaged teachers in self-supervised learning [49]–[51]: the teacher tracks the improving policy on a slower timescale, providing a slowly improving but stable target. Gradients are applied only through the student-view log-probabilities, while the teacher-view log-probabilities are treated as stop-gradient targets.

The two terms in Equation 1 play complementary roles: the GRPO term optimizes final-answer correctness under the restricted view, while the reverse-KL term provides dense token-level guidance from the strongest view on the student’s own trajectories. Unlike methods that sample from or imitate an external teacher distribution, MIRROR keeps rollouts on-policy with respect to the restricted-view student and uses the selected teacher view only as a supervision signal. Please see Algorithm  6 for the full pseudocode of MIRROR update.

0.0.6 Experimental Evaluation↩︎

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The goal of our experiments is to evaluate whether MIRROR can turn asymmetries across views into an effective self-supervision signal for improving multimodal reasoning. We answer four questions in this section: (1) Does MIRROR improve restricted-view reasoning over single-view and mixed-modality GRPO baselines? (2) Does selecting the teacher view separately for each problem outperform using a fixed global teacher view? (3) Does MIRROR expand the set of solvable problem instances after training? (4) Are modality-asymmetric training examples more useful than random paired examples, and how sensitive is MIRROR to the cross-view KL coefficient \(\lambda_\text{KL}\) and to the parameterization of the teacher (current policy vs.slow-moving EMA)?

Experimental setup. We post-train Qwen3-VL-4B-Instruct and implement MIRROR in verl [52]. Models are trained with maximum response length \(16{,}384\), sampling temperature \(0.8\), GRPO clipping range \(\epsilon_{\mathrm{low}}=0.2,\epsilon_{\mathrm{high}}=0.26\), reference-policy KL coefficient \(0.001\), and entropy coefficient \(0.001\). MIRROR uses a reverse-KL coefficient of 0.01 and EMA weights \(\alpha=0.99\). Our in-distribution evaluation uses ODA-Val, where paired views preserve problem identity and allow us to measure pass@\(k\) under each view and track how training changes problem-level solvability across views. We additionally evaluate on public multimodal geometry benchmarks GeoInt [53] and MathVerse [4]. For GeoInt, we consider only answer-based question. We evaluate final-answer correctness with pass@1 and pass@16 estimated by 32 rollouts. For MathVerse, we use gpt-4o-2024-11-20 [54] as the judge through the LMMs-Eval suite [55] and report judge score [56].

Baselines and comparisons. We compare MIRROR against three groups of baselines. First, we evaluate the base Qwen3-VL-4B-Instruct model and existing multimodal reasoning models with different training objectives. Vision-R1-7B [21] is initialized from Qwen2.5-VL-7B-Instruct and targets visual mathematical reasoning with a two-stage cold-start and RL recipe. PAPO-7B [57] is initialized from Qwen2.5-VL-7B-Instruct and improves visual grounding during RL by contrasting original and masked-image rollouts with a perception-aware objective. Vero-8B [58] is initialized from Qwen3-VL-8B-Instruct and uses a broad general visual reasoning recipe with task-routed verifiers across diverse multimodal domains. Second, we compare against standard GRPO baselines trained on ODA-Train using a single view: text-dominant GRPO is trained only on text-dominant prompts, and image-dominant GRPO is trained only on image-dominant prompts. Third, we compare against mixed-modality GRPO, which is trained on the full ODA-Train mixture containing both text-dominant and image-dominant views. This baseline tests whether cross-view transfer emerges simply from standard RL when both modalities are present in the training prompts. MIRROR is trained on the same full mixture of student views. Finally, we include fixed-teacher variants of MIRROR, where the teacher is always the same view.

Result 1: MIRROR improves over standard and mixed-modality GRPO. Table 2 compares MIRROR with the base model, existing reasoning-based VLMs, and standard RL baselines. On ODA-Val, MIRROR achieves the best performance across all image and text validation metrics. Compared with the strongest single-view GRPO baseline, MIRROR improves image pass@16 from \(48.78\%\) to \(57.06\%\) and text pass@16 from \(83.16\%\) to \(86.10\%\), corresponding to absolute gains of \(+8.28\%\) and \(+2.94\%\), respectively. Compared with mixed-modality GRPO, the gains are larger: \(+11.38\%\) on image pass@16 and \(+4.44\%\) on text pass@16. These gains also transfer beyond ODA-Val. On GeoInt, MIRROR improves pass@1 over the base model from \(58.62\%\) to \(66.15\%\) and pass@16 from \(70.79\%\) to \(78.38\%\), yielding gains of \(+7.53\%\) and \(+7.59\%\). Notably, MIRROR is competitive with substantially larger-data post-trained VLMs: it outperforms Vision-R1-7B and PAPO-7B on all ODA-Val and GeoInt metrics, and exceeds Vero-8B on all ODA-Val image/text validation metrics, despite using only 2K training samples. In contrast, Vision-R1-7B uses 200K multimodal CoT cold-start samples and 10K multimodal math RL samples, PAPO-7B is trained on 38.9K multimodal reasoning samples, and Vero-8B is post-trained on roughly 600K samples. Vero-8B remains strongest on MathVerse, while MIRROR improves the Qwen3-VL-4B base from \(41.31\%\) to \(46.53\%\). Together, these results suggest that cross-view asymmetry provides a useful training signal beyond the in-distribution validation set and can substantially improve a smaller base model using orders of magnitude less post-training data than prior multimodal reasoning models.

Mixed-modality GRPO is the most direct test of whether paired views are sufficient by themselves. Although this baseline trains on both text-dominant and image-dominant views of the same underlying problems, it treats each view as an independent RL prompt and relies on transfer emerging implicitly through shared parameters. Figure 2 and  3 show that this is not reliable: mixed-view GRPO obtains lower training reward than single-view GRPO in both the controlled setting and full-dataset training. This behavior is also consistent with ray interference in heterogeneous on-policy RL mixtures [59]. In our setting, views differ in grounding requirements, and access to explicit relations. Optimizing all views with sparse outcome rewards does not guarantee that updates from an easier view teach the harder restricted view. MIRROR addresses this by specifying the direction of transfer: it selects the strongest-performing view to regularize the restricted-view student toward its teacher distribution. As detailed in Appendix 0.0.8, MIRROR incurs approximately \(37.5\%\) more FLOPs per update than standard mixed-modality GRPO. To control for this difference, we compare MIRROR at step 150 with GRPO at step 200, which correspond to roughly matched cumulative compute. Even under this compute-matched comparison, MIRROR achieves higher image reward (\(0.2625\) vs.\(0.2368\)) and text reward (\(0.4581\) vs.\(0.4387\)), indicating that its gains cannot be explained by additional computation alone.

Result 2: Adaptive teacher selection provides the most consistent gains. We next evaluate whether the teacher view should be fixed globally or selected adaptively per problem. Table 3 compares MIRROR with three fixed-teacher variants: always using a text-dominant teacher, always using an image-dominant teacher, and always using a combined image+text teacher. The fixed-teacher variants exhibit complementary strengths, but none is uniformly best. For image-dominant evaluation, the text-dominant teacher outperforms the image-dominant teacher; conversely, for text-dominant evaluation, the image-dominant teacher outperforms the text-dominant teacher. This pattern suggests that cross-view supervision can be more useful than simply reinforcing the student’s own view. However, the combined image+text teacher performs best on MathVerse, showing that the most useful teacher signal can also come from jointly conditioning on both modalities. These mixed preferences indicate that the optimal supervision source is problem-dependent. Rather than committing to a single global transfer direction, MIRROR selects the strongest-performing view per problem, matching the teacher to the observed modality asymmetry. This adaptive strategy yields the best result on most validation and GeoInt metrics. Together, these results support the central design principle of MIRROR: cross-view supervision is most effective when the teacher is selected adaptively rather than fixed globally.

Result 3: MIRROR expands solvability and improves cross-view consistency. Aggregate pass@\(k\) scores do not fully capture how post-training changes the set of problems a model can solve. We therefore measure solvability on ODA-Train before and after training. Net solvability gain is the before-to-after change in the percentage of sampled problems with at least one successful rollout. As shown in Figure 4, all GRPO variants improve net solvability, but MIRROR achieves the largest gain at \(+8.95\%\). Beyond expanding the set of solvable examples, MIRROR also improves cross-view consistency. On ODA-Train, only \(42.5\%\) of problems are solvable under both the text-dominant and image-dominant views by the base model. Standard GRPO improves this both-view solvability rate to \(53.6\%\), while MIRROR further increases it to \(60.7\%\). This shows that MIRROR not only improves marginal accuracy under each restricted view, but also makes the model more likely to solve the same underlying problem consistently across modalities.

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Ablation 1: Alignment is most effective with modality-asymmetric pairs. As described in Section [sec:data95introduction], ODA-Data contains problems that exhibit modality asymmetry. To test whether this asymmetry is important, we compare MIRROR with a control model trained on the same number of properly paired examples that do not exhibit clear modality asymmetry, using identical hyperparameters. Training on asymmetric examples improves text-dominant validation pass@1 by \(4.43\%\) relative to the paired-control set, while the control performs \(22.15\%\) worse on image-dominant validation. These results show that paired representations alone are insufficient. The useful supervision arises when a successful view can provide problem-specific guidance to a view that fails, supporting our hypothesis that modality asymmetry is the primary source of self-supervision in MIRROR.

Ablation 2: effect of the reverse-KL coefficient. Table 4 studies the strength of the reverse-KL regularizer while keeping all other training components fixed for 100 steps. We found that \(\lambda_{\text{KL}}=0.01\) achieves the best rank overall and therefore use it for continue training. Larger coefficients such as \(0.1\) caused training collapse, indicating that excessive alignment can overpower the RL objective.

Ablation 3: a slow-moving EMA teacher is necessary for long-horizon stability. Figure 5 compares MIRROR’s EMA teacher (\(\alpha=0.99\)) with a teacher evaluated using the current policy parameters \(\theta\), under otherwise identical settings (\(\lambda_{\text{KL}}=0.01\)). The runs behave similarly for the first \({\sim}50\) steps, after which the current-policy teacher exhibits self-reinforcing drift: entropy rises from \({\sim}0.3\) to \(3.9\), reference-policy KL reaches \(0.33\), response length falls from \({\sim}9{,}000\) to \({\sim}3{,}700\) tokens, and reward drops from \(0.34\) to \(0.21\) by step \({\sim}165\). In contrast, the EMA teacher keeps entropy and reference KL near \(0.29\) and \(0.02\), while reward continues improving to \(0.48\) at step \(290\). We interpret the failure of the current-policy teacher as a moving-target effect analogous to the instability addressed by target networks in value-based RL [47]: decoupling the target’s timescale from the student’s stabilizes optimization without sacrificing asymptotic performance.

0.0.7 Discussion and Perspectives on Future Work↩︎

In this work, we identify reasoning asymmetry across modalities as a source of self-supervision for multimodal reasoning. We introduce Modality-Informed Reciprocal Reasoning Optimization (MIRROR), which selects the strongest-performing view of each problem as an internal teacher and regularizing the student-view policy toward it during RL. Our results show that paired multimodal data is most effective when the training objective explicitly exploits the correspondence between views. Simply mixing text-dominant and image-dominant views into the RL distribution does not reliably transfer reasoning, because each view is still optimized as an independent prompt. In contrast, MIRROR makes the transfer direction explicit: the view that succeeds provides guidance for the view being trained. This yields stronger and more consistent reasoning, with the largest gains on modality-asymmetric examples where one view succeeds and another fails.

More broadly, our findings suggest a different way to use multimodal data. Rather than treating different input formats as independent examples, we can treat them as alternative views of the same latent problem and use their disagreements to expose missing reasoning capabilities. This perspective opens several directions for future work. Extending beyond geometry to scientific figures, charts, tables, robotics observations, or broader visual reasoning will require reliable ways to construct information-equivalent views. Teacher selection could also be improved with uncertainty-aware weighting or alignment at intermediate reasoning steps, rather than relying only on rollout rewards. Overall, our findings point to a broader opportunity: cross-modal inconsistency can be transformed from a robustness concern into a useful training signal for learning multimodal reasoners that are stronger and less sensitive to how information is represented.

Appendices↩︎

0.0.8 FLOPs Estimation↩︎

We estimate the computational overhead of MIRROR relative to standard mixed-modality GRPO. Following the common approximation used in language-model scaling analyses, we count one forward-only pass over \(D\) tokens as approximately \(2ND\) FLOPs [60], where \(N\) is the number of model parameters, and one training pass with forward and backward propagation as approximately \(6ND\) FLOPs [61]. This gives a coarse but useful estimate of the relative cost between rollout generation, teacher-logit evaluation, and gradient updates.

0.0.8.0.0.1 Setup.

Standard mixed-modality GRPO trains on two restricted student views: text-dominant and image-dominant. For each problem, it samples \(K\) rollouts from each view, giving \(2K\) online student rollouts. MIRROR uses the same two restricted views for student RL training, but performs teacher selection over three candidate views: \[\mathcal{M}=\{\text{text},\text{img},\text{comb}\}\]

The text-dominant and image-dominant candidate rollouts are exactly the online rollouts already generated for GRPO, so they do not need to be recomputed. Therefore, the only additional rollout generation required by MIRROR is the \(K\) rollouts from the combined image+text view.

0.0.8.0.0.2 Mixed-modality GRPO cost.

Let \(D\) denote the average rollout length. Standard mixed-modality GRPO generates \(2K\) rollouts per problem, so its forward-only rollout-generation cost is approximately \[\begin{align} C_{\mathrm{mixed\text{-}GRPO,gen}} &\approx 2K \cdot 2ND = 4KND. \end{align}\] It then performs one student-side policy update on these \(2K\) rollouts. Using the standard training approximation, the forward-backward update cost is \[\begin{align} C_{\mathrm{mixed\text{-}GRPO,train}} &\approx 2K \cdot 6ND = 12KND. \end{align}\] Thus, ignoring implementation-specific constants, the total cost is \[\begin{align} C_{\mathrm{mixed\text{-}GRPO}} &\approx 4KND + 12KND = 16KND. \end{align}\]

0.0.8.0.0.3 MIRROR cost.

MIRROR reuses the same \(2K\) text-dominant and image-dominant student rollouts as mixed-modality GRPO. It additionally samples \(K\) combined-view rollouts for online teacher selection. Therefore, its rollout-generation cost is \[\begin{align} C_{\texttt{MIRROR},\mathrm{gen}} &\approx 3K \cdot 2ND = 6KND. \end{align}\] After selecting the teacher view, MIRROR computes the reverse-KL regularizer on the \(2K\) student-view rollouts. This requires a teacher-forced forward pass under the selected teacher view to compute teacher log-probabilities on the same generated tokens using the EMA teacher weights. Since this is forward-only and stop-gradient, its cost is approximately \[\begin{align} C_{\texttt{MIRROR},\mathrm{KL\text{-}eval}} &\approx 2K \cdot 2ND = 4KND. \end{align}\] The student-side GRPO update is unchanged from mixed-modality GRPO: \[\begin{align} C_{\texttt{MIRROR},\mathrm{train}} &\approx 2K \cdot 6ND = 12KND. \end{align}\] Therefore, the total cost of MIRROR is approximately \[\begin{align} C_{\texttt{MIRROR}} &\approx 6KND + 4KND + 12KND = 22KND. \end{align}\]

0.0.8.0.0.4 Relative overhead.

Under this coarse FLOPs model, the relative cost of MIRROR compared with standard mixed-modality GRPO is \[\begin{align} \frac{C_{\texttt{MIRROR}}}{C_{\mathrm{mixed\text{-}GRPO}}} &\approx \frac{22KND}{16KND} = 1.375. \end{align}\] Thus, MIRROR adds roughly \(37.5\%\) FLOPs over standard mixed-modality GRPO under a single-update accounting. This overhead comes only from forward-only computation: \(K\) additional combined-view rollouts for teacher selection and a teacher-logit evaluation pass for the reverse-KL regularizer. MIRROR uses the same trainable model, the same \(2K\) student-view RL rollouts, and the same number of backward updates as mixed-modality GRPO. To examine whether standard GRPO would obtain similar optimization gains under comparable compute, we inspect the running-average training rewards at MIRROR step 150 and GRPO step 200 shown in figure  3, which correspond to approximately similar cumulative FLOPs. At these points in training, MIRROR has higher image reward (\(0.2625\) vs.\(0.2368\)) and text reward (\(0.4581\) vs.\(0.4387\)). This suggests that the advantage is not solely due to the additional per-update computation.

0.0.8.0.0.5 EMA teacher overhead.

The EMA teacher does not change the FLOPs accounting above: the teacher-logit evaluation pass (\(C_{\texttt{MIRROR},\mathrm{KL\text{-}eval}}\approx 4KND\)) is simply executed with the EMA weights \(\bar\theta\) instead of the current weights \(\theta\), and the Polyak update in Equation 3 is a single \(O(N)\) elementwise operation per training step (under one second per step in our runs, versus roughly \(20\) minutes per full training step). The main cost is memory: one additional forward-only copy of the policy parameters, kept in fp32 for numerically stable high-decay averaging and offloaded to CPU between teacher evaluations. Checkpoints additionally store the EMA parameters so that training resumes exactly.

0.0.9 Training Details↩︎

We train all models using GRPO with verl [52]. Unless otherwise specified, experiments use Qwen3-VL-4B-Instruct as the base model, vLLM as the rollout engine, and \(8\) nodes with \(8\) H200 GPUs per node. Each training run uses a maximum prompt length of \(4096\) tokens and a maximum response length of \(16{,}384\) tokens. We use a sampling temperature of \(0.8\) and generate \(n=16\) rollouts per prompt. The training batch size is \(256\), with PPO mini-batch size \(128\) and micro-batch size \(1\) per GPU. We use GRPO with clipping thresholds \(\epsilon_{\mathrm{low}}=0.2\) and \(\epsilon_{\mathrm{high}}=0.26\). We train with learning rate \(1\times 10^{-6}\), entropy coefficient \(0.001\), and a reference-policy KL loss coefficient of \(0.001\) using the low-variance KL estimator. For MIRROR, we additionally enable the cross-modal reverse-KL regularizer with coefficient \(\lambda_{\mathrm{KL}}=0.01\) by default, and compute the teacher target log-probabilities from an EMA copy of the policy with decay \(\alpha=0.99\), updated after every actor step. The EMA copy is kept in fp32 and is forward-only (no optimizer state); it is saved with each checkpoint so training resumes exactly.

Each RL training job uses \(8\) nodes with \(8\) H200 GPUs per node, runs at approximately \(20\) minutes per training step, and is trained for at least \(200\) steps, corresponding to roughly \(67\) wall-clock hours or about \(4{,}267\) H200 GPU-hours per model.

7pt

0.0.10 ODA-Data Construction↩︎

To study modality-dependent reasoning while holding the underlying problem fixed, we construct ODA-Data, a paired multimodal geometry dataset in which each problem is represented in two aligned views: a text-dominant view and an image-dominant view. Starting from 97K geometry-style problems from OpenDataArena/ODA-Math-460k [44], we first apply a difficulty filter that removes easy questions with annotated pass@5=1, yielding 16K candidate questions. For each candidate, we prompt an LLM to generate a diagram as TikZ code, a standard LaTeX package for programmatic vector graphics. We retain only diagrams whose TikZ code compiles successfully and use an LLM judge to verify that the generated diagram is semantically aligned with the original problem statement.

To construct the image-dominant view, we further prompt the judge to remove from the text any information already presented in the diagram, so that the remaining text and image provide complementary information. All LLM-based generation and judging steps use Gemini-3-Pro-Preview. We then perform modality-specific rollout evaluations using the base model and remove questions that the base model reliably solves under both the text-dominant and image-dominant views, so that the final dataset emphasizes problems whose solvability depends on the input representation. This process yields approximately 2K curated paired multimodal examples.

We split these examples into ODA-Train and ODA-Val using an 85:15 split, with 15% of the dataset held out for validation. ODA-Train is used for cross-modal post-training, while ODA-Val serves as a held-out benchmark for evaluating modality-dependent reasoning, cross-modal consistency, and one-sided solvability. Each example contains the original text problem, generated diagram, image-dominant prompt, text-dominant prompt, answer annotations, and modality-specific rollout/evaluation metadata. This structure makes ODA-Data useful not only for training MIRROR, but also for studying whether VLMs solve the same mathematical problem consistently across textual and visual representations.

For Tikz code generation, the prompt is:

For verifying the tikz compiled image against the question, the prompt is:

For rewriting image questions to remove information present in the diagram from the question, the prompt is:

0.0.11 Examples↩︎

0.0.11.1 Example 1

0.0.11.2 Example 2

0.0.11.3 Example 3

0.0.11.4 Example 4

0.0.11.5 Example 5

0.0.12 Pseudocode for MIRROR↩︎

Algorithm 6 outlines the training loop of MIRROR.

References↩︎

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1. Equal contribution. Corresponding Author: yewen@usc.edu, yuxiaoq@andrew.cmu.edu↩︎