Lei Wu

Xianbang Chen, Yikui Liu, Lei Wu · pku

Generally, day-ahead unit commitment (UC) is conducted in a predict-then-optimize process: it starts by predicting the renewable energy source (RES) availability and system reserve requirements; given the predictions, the UC model is then optimized to determine the economic operation plans. In fact, predictions within the process are raw. In other words, if the predictions are further tailored to assist UC in making the economic operation plans against realizations of the RES and reserve requirements, UC economics will benefit significantly. To this end, this paper presents a cost-oriented tailor of RES-and-reserve predictions for UC, deployed as an add-on to the predict-then-optimize process. The RES-and-reserve tailor is trained by solving a bi-level mixed-integer programming model: the upper level trains the tailor based on its induced operating cost; the lower level, given tailored predictions, mimics the system operation process and feeds the induced operating cost back to the upper level; finally, the upper level evaluates the training quality according to the fed-back cost. Through this training, the tailor learns to customize the raw predictions into cost-oriented predictions. Moreover, the tailor can be embedded into the existing predict-then-optimize process as an add-on, improving the UC economics. Lastly, the presented method is compared to traditional, binary-relaxation, neural network-based, stochastic, and robust methods.

Hongrui Chen, Jihao Long, Lei Wu

In this work, we analyze the learnability of reproducing kernel Hilbert spaces (RKHS) under the $L^\infty$ norm, which is critical for understanding the performance of kernel methods and random feature models in safety- and security-critical applications. Specifically, we relate the $L^\infty$ learnability of a RKHS to the spectrum decay of the associate kernel and both lower bounds and upper bounds of the sample complexity are established. In particular, for dot-product kernels on the sphere, we identify conditions when the $L^\infty$ learning can be achieved with polynomial samples. Let $d$ denote the input dimension and assume the kernel spectrum roughly decays as $λ_k\sim k^{-1-β}$ with $β>0$. We prove that if $β$ is independent of the input dimension $d$, then functions in the RKHS can be learned efficiently under the $L^\infty$ norm, i.e., the sample complexity depends polynomially on $d$. In contrast, if $β=1/\mathrm{poly}(d)$, then the $L^\infty$ learning requires exponentially many samples.

Mingze Wang, Ruoxi Yu, Weinan E et al.

Transformers have demonstrated exceptional in-context learning capabilities, yet the theoretical understanding of the underlying mechanisms remains limited. A recent work (Elhage et al., 2021) identified a ``rich'' in-context mechanism known as induction head, contrasting with ``lazy'' $n$-gram models that overlook long-range dependencies. In this work, we provide both approximation and dynamics analyses of how transformers implement induction heads. In the {\em approximation} analysis, we formalize both standard and generalized induction head mechanisms, and examine how transformers can efficiently implement them, with an emphasis on the distinct role of each transformer submodule. For the {\em dynamics} analysis, we study the training dynamics on a synthetic mixed target, composed of a 4-gram and an in-context 2-gram component. This controlled setting allows us to precisely characterize the entire training process and uncover an {\em abrupt transition} from lazy (4-gram) to rich (induction head) mechanisms as training progresses.