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Can Mamba Learn In Context with Outliers? A Theoretical Generalization Analysis

Hongkang Li, Songtao Lu, Xiaodong Cui, Pin-Yu Chen, Meng Wang

TL;DR

This work analyzes a one-layer Mamba model with a linear attention component followed by a nonlinear gating layer to study in-context learning (ICL) under distribution-shifted prompts with additive outliers. It establishes convergence and generalization guarantees as a function of training prompt length $l_{tr}$, testing prompt length $l_{ts}$, outlier fraction $p_a$, and test-time outlier fraction $\alpha$, showing robust ICL up to $\alpha$ near 1 when training prompts include outliers. In comparison to linear Transformers, Mamba yields stronger robustness to outlier density at the cost of longer training and more context, due to the gating mechanism that suppresses corrupted examples and a linear attention that emphasizes same-pattern context. Empirical results on synthetic data support the theory, illustrating that Mamba's ICL mechanism relies on both the attention selecting informative context and the gating providing locality-based outlier suppression, with multi-layer Mamba experiments confirming similar dynamics across layers and sensitivity to outlier placement. These findings advance the theoretical understanding of efficient linear-attention architectures for robust ICL and guide the design of Mamba-based language and multi-modal models.

Abstract

The Mamba model has gained significant attention for its computational advantages over Transformer-based models, while achieving comparable performance across a wide range of language tasks. Like Transformers, Mamba exhibits in-context learning (ICL) capabilities, i.e., making predictions for new tasks based on a prompt containing input-label pairs and a query, without requiring fine-tuning. Despite its empirical success, the theoretical understanding of Mamba remains limited, largely due to the nonlinearity introduced by its gating mechanism. To the best of our knowledge, this paper presents the first theoretical analysis of the training dynamics of a one-layer Mamba model, which consists of a linear attention component followed by a nonlinear gating layer, and its ICL generalization on unseen binary classification tasks, even when the prompt includes additive outliers. Our analysis shows that Mamba leverages the linear attention layer to select informative context examples and uses the nonlinear gating layer to suppress the influence of outliers. By establishing and comparing to the analysis of linear Transformers under the same setting, we show that although Mamba may require more training iterations to converge, it maintains accurate predictions even when the proportion of outliers exceeds the threshold that a linear Transformer can tolerate. These theoretical findings are supported by empirical experiments.

Can Mamba Learn In Context with Outliers? A Theoretical Generalization Analysis

TL;DR

This work analyzes a one-layer Mamba model with a linear attention component followed by a nonlinear gating layer to study in-context learning (ICL) under distribution-shifted prompts with additive outliers. It establishes convergence and generalization guarantees as a function of training prompt length , testing prompt length , outlier fraction , and test-time outlier fraction , showing robust ICL up to near 1 when training prompts include outliers. In comparison to linear Transformers, Mamba yields stronger robustness to outlier density at the cost of longer training and more context, due to the gating mechanism that suppresses corrupted examples and a linear attention that emphasizes same-pattern context. Empirical results on synthetic data support the theory, illustrating that Mamba's ICL mechanism relies on both the attention selecting informative context and the gating providing locality-based outlier suppression, with multi-layer Mamba experiments confirming similar dynamics across layers and sensitivity to outlier placement. These findings advance the theoretical understanding of efficient linear-attention architectures for robust ICL and guide the design of Mamba-based language and multi-modal models.

Abstract

The Mamba model has gained significant attention for its computational advantages over Transformer-based models, while achieving comparable performance across a wide range of language tasks. Like Transformers, Mamba exhibits in-context learning (ICL) capabilities, i.e., making predictions for new tasks based on a prompt containing input-label pairs and a query, without requiring fine-tuning. Despite its empirical success, the theoretical understanding of Mamba remains limited, largely due to the nonlinearity introduced by its gating mechanism. To the best of our knowledge, this paper presents the first theoretical analysis of the training dynamics of a one-layer Mamba model, which consists of a linear attention component followed by a nonlinear gating layer, and its ICL generalization on unseen binary classification tasks, even when the prompt includes additive outliers. Our analysis shows that Mamba leverages the linear attention layer to select informative context examples and uses the nonlinear gating layer to suppress the influence of outliers. By establishing and comparing to the analysis of linear Transformers under the same setting, we show that although Mamba may require more training iterations to converge, it maintains accurate predictions even when the proportion of outliers exceeds the threshold that a linear Transformer can tolerate. These theoretical findings are supported by empirical experiments.

Paper Structure

This paper contains 31 sections, 12 theorems, 185 equations, 6 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Major Contributions
  3. Related Works
  4. Problem Formulation
  5. Main Theoretical Results
  6. Main Theoretical Insights
  7. Data and Tasks Modeling
  8. Learning, Generalization, and Sample Complexity Analysis of Mamba
  9. A Theoretical Comparison between Linear Transformers and Mamba
  10. The Mechanism of Mamba in implementing ICL
  11. Experiment
  12. Comparison between One-Layer Mamba and Linear Transformer Models on ICL with Outliers
  13. The ICL Mechanism of Multi-Layer Mamba
  14. Conclusion, Limitations, and Future Works
  15. Proof Sketch of Main Theorems
  16. ...and 16 more sections

Key Result

Theorem 1

(Convergence and Sample Complexity of Mamba) For any $\epsilon>0$, of (i) $B\gtrsim B_M:=\max\{B_T, \beta^{-4}V^2\kappa_a^{-2}(1-p_a)^{-2}\log\epsilon^{-1}\}$, (ii) $V\beta^{-4}\lesssim \kappa_a\lesssim V\beta(1-p_a)p_a^{-1}\epsilon^{-1}$, and (iii) then (iv) after iterations with $\eta\leq 1$ and using $N=BT$ samples, we have that

Figures (6)

  • Figure 1: An example of outliers in context inputs.
  • Figure 2: ICL classification error of Mamba and linear Transformer against $\alpha$ with different prompt outliers. (A) Label flipping. (B) Targeted labeling. (C) Random labeling.
  • Figure 3: The summation of 1st-layer attention scores on examples with the same or a different relevant pattern as the query.
  • Figure 4: The 1st-layer gating value of examples with (red) or without (green) additive outliers.
  • Figure 5: The summation of attention scores in the 2nd and 3rd layers.
  • ...and 1 more figures

Theorems & Definitions (34)

  • Definition 1
  • Definition 2
  • Example 1
  • Theorem 1
  • Remark 1
  • Remark 2
  • Theorem 2
  • Remark 3
  • Theorem 3
  • Remark 4
  • ...and 24 more