They do not study how the population loss decreases during the first stage.

In the course of preparing the supplementary materials we identified the following two mistakes. For the convenience of the reader we provide the full, corrected table below. C is an appropriatly chosen constant. Frei et al. (2022) Xu & Gu (2023) Theorem 3.1 Theorem 3.6 Theorem 3.8n C log null 1 δ null log null m δ null 1 δ 1 log null m δ null m C 1 log null n δ null log null n δ null log null n δ null log null n δ null γ 1 C 1 n 1 n 1 n 1 nd 1 k γ C 1 nd null log( The same mistake also means that the sentence starting on line 188 "Comparing In order to provide a convenient reference for the reader, we summarize our notation as follows. As such we typically resort to using a generically large enough constant C . For the reader's convenience we recap the data model studied in this work. We assume test data are drawn mutually i.i.d. In regard to the initialization of the network weights, for convenience we assume each neuron's To this end, we introduce the following notation, where p { 1, 1}. P(( B < κT) (T > 0) | w, v > 0) 1 P( T = 0 | w, v > 0) P( B κT | w, v > 0), therefore it suffices to upper bound the two probabilities on the right-hand-side. Using a variant of Hoeffding's bound for sampling without replacement (see Proposition Based on Lemma B.2, the following lemma bounds the probability that " on the counting functions: in particular we write P (i, l) + P (i, l) = P ( i, i) = 1 /2 and hence we conclude p + q = 1 / 2. As a result Observe by the data model, described in Section B.2, that We will often make use of the following similar but more pessimistic bounds on the activations.

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We give more discussions of our work in Appendix B. Appendix C provides missing technical details of Section 3 while Appendix D provides those of Section 4. In A.1, we present the details of the numerical experiments. A.4, we conduct experiments when the balanced initialization condition is not satisfied. We conduct over-parameterized regression with different linear networks. We now present the details of numerical experiments conducted in Section 5. ` GD and SGD when k (0) k is extremely large. Other settings are similar to that of rank-1 linear networks.

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Health and Human Services Secretary Robert F. Kennedy Jr. shares his journey to his official position and where his passion for health comes from on'My View with Lara Trump.' The White House revealed new details Wednesday regarding the Trump administration's efforts to advance healthcare technology and partnerships with private-sector technology companies. The "Make Health Tech Great Again" event was expected to provide more details on how the administration is advancing a "next-generation digital health ecosystem," after securing partnerships with companies including Amazon, Anthropic, Apple, Google, and OpenAI to better share information between patient and providers within Medicare and Medicaid services. U.S. Health and Human Services Secretary Robert F. Kennedy Jr., announced that the HHS will ban illegal immigrants from accessing taxpayer-funded programs. "For decades, bureaucrats and entrenched interests buried health data and blocked patients from taking control of their health," Department of Health and Human Services Secretary Robert F. Kennedy, Jr. said in a statement Wednesday ahead of the event.

California passes a new law aimed at shining a light on the growing number of crashes involving self-driving cars. Chances are, you have never actually ridden a scooter like this, zipping around corners, but you have definitely seen them weaving through city traffic. Just when you thought scooters were already a wild card on the road, imagine one that drives itself. That is exactly what the Omoway Omo X promises. Developed by a team of former Xpeng engineers, this scooter is not just electric, it is packed with smart features that push self-driving scooter tech to a whole new level, offering far more than you would ever expect from a two-wheeler.

A celebrated result of Pollard proves asymptotic consistency for $k$-means clustering when the population distribution has finite variance. In this work, we point out that the population-level $k$-means clustering problem is, in fact, well-posed under the weaker assumption of a finite expectation, and we investigate whether some form of asymptotic consistency holds in this setting. As we illustrate in a variety of negative results, the complete story is quite subtle; for example, the empirical $k$-means cluster centers may fail to converge even if there exists a unique set of population $k$-means cluster centers. A detailed analysis of our negative results reveals that inconsistency arises because of an extreme form of cluster imbalance, whereby the presence of outlying samples leads to some empirical $k$-means clusters possessing very few points. We then give a collection of positive results which show that some forms of asymptotic consistency, under only the assumption of finite expectation, may be recovered by imposing some a priori degree of balance among the empirical $k$-means clusters.