The ever-increasing computational demands of large language models (LLMs) make efficient inference a central challenge. While recent advances leverage specialized architectures or selective activation, they typically require (re)training or architectural modifications, limiting their broad applicability. Training-free sparse activation, in contrast, offers a plug-and-play pathway to efficiency; however, existing methods often rely solely on hidden state magnitudes, leading to significant approximation error and performance degradation. To address this, we introduce WINA (Weight-Informed Neuron Activation): a simple framework for training-free sparse activation that incorporates both hidden state magnitudes and weight matrix structure. By also leveraging the ℓ2-norm of the model’s weight matrices, WINA yields a principled sparsification strategy with provably optimal approximation error bounds, offering better and tighter theoretical guarantees than prior state-of-the-art approaches. Overall, WINA also empirically outperforms many previous training-free methods across diverse LLM architectures and datasets: not only matching or exceeding their accuracy at comparable sparsity levels, but also sustaining performance better at more extreme sparsity levels. Together, these results position WINA as a practical, theoretically grounded, and broadly deployable solution for efficient inference. Our source code is available at https://github.com/microsoft/wina.