Solving olympiad geometry without human demonstrations.

Proving mathematical theorems at the olympiad level represents a notable milestone in human-level automated reasoning1-4, owing to their reputed difficulty among the world's best talents in pre-university mathematics. Current machine-learning approaches, however, are not applicable to most mathematical domains owing to the high cost of translating human proofs into machine-verifiable format. The problem is even worse for geometry because of its unique translation challenges1,5, resulting in severe scarcity of training data. We propose AlphaGeometry, a theorem prover for Euclidean plane geometry that sidesteps the need for human demonstrations by synthesizing millions of theorems and proofs across different levels of complexity. AlphaGeometry is a neuro-symbolic system that uses a neural language model, trained from scratch on our large-scale synthetic data, to guide a symbolic deduction engine through infinite branching points in challenging problems. On a test set of 30 latest olympiad-level problems, AlphaGeometry solves 25, outperforming the previous best method that only solves ten problems and approaching the performance of an average International Mathematical Olympiad (IMO) gold medallist. Notably, AlphaGeometry produces human-readable proofs, solves all geometry problems in the IMO 2000 and 2015 under human expert evaluation and discovers a generalized version of a translated IMO theorem in 2004.

A graph placement methodology for fast chip design.

Chip floorplanning is the engineering task of designing the physical layout of a computer chip. Despite five decades of research1, chip floorplanning has defied automation, requiring months of intense effort by physical design engineers to produce manufacturable layouts. Here we present a deep reinforcement learning approach to chip floorplanning. In under six hours, our method automatically generates chip floorplans that are superior or comparable to those produced by humans in all key metrics, including power consumption, performance and chip area. To achieve this, we pose chip floorplanning as a reinforcement learning problem, and develop an edge-based graph convolutional neural network architecture capable of learning rich and transferable representations of the chip. As a result, our method utilizes past experience to become better and faster at solving new instances of the problem, allowing chip design to be performed by artificial agents with more experience than any human designer. Our method was used to design the next generation of Google's artificial intelligence (AI) accelerators, and has the potential to save thousands of hours of human effort for each new generation. Finally, we believe that more powerful AI-designed hardware will fuel advances in AI, creating a symbiotic relationship between the two fields.

Modified natural cycle for embryo transfer using frozen-thawed blastocysts: A satisfactory option.

OBJECTIVE: To describe pregnancy outcomes of frozen-thawed blastocysts cycles using modified natural cycle frozen embryo transfers (NC-FET) and down-regulated hormonally controlled frozen embryo transfers (HC-FET) protocols. STUDY DESIGN: This retrospective cohort study included all patients undergoing either modified NC-FET or down-regulated HC-FET using frozen-thawed day 5 embryos. Cycles with donor blastocysts were excluded. Four hundred twenty eight patients underwent a total of 493 FET cycles. Patients with regular menses and evidence of ovulation underwent modified NC-FET. These patients were given hCG 10,000 IU IM on the day of LH-surge. Vaginal progesterone (P4) was started two days later and blastocyst transfer was planned seven days after detecting the LH surge. Anovulatory patients and some ovulatory patients underwent down-regulated HC-FET. These patients were placed on medroxy-progesterone acetate (10mg) for 10days to bring on menses and were also given a half-dose of GnRH-agonist (GnRH-a) on the third day of medroxy-progesterone acetate. Exogenous estradiol was initiated on the third day of menses. Once serum E2 levels reached >500pg/mL and endometrial lining reached >8mm, intramuscular (IM) P4 in oil was administered. Blastocyst FET was planned 6days after initiating P4. The primary outcomes included clinical pregnancy and delivery rates. RESULTS: There were 197 patients in the modified NC-FET protocol and 181 in the down-regulated HC-FET protocol. Mean age (years), day-3 FSH levels (mIU/mL) and percentage of patients with male factor infertility were significantly higher and mean BMI (kg/m2) was significantly lower in modified NC-FET compared to HC-FET, respectively. Analysis of the first cycle pregnancy outcomes revealed no significant differences in clinical pregnancy rate (54.3% vs. 52.5%) and delivery rate (47.2% vs. 43.6%) between modified NC-FET and HC-FET. Logistic regression analysis showed age (OR=0.939, 95% CI 0.894-0.989, p=0.011), number of blastocysts transferred (OR=1.414, 95% CI 1.046-1.909, p=0.024), and the year of FET (OR=1.127, 95% CI 1.029-1.234, p=0.010) were significant factors impacting clinical pregnancy. An age analysis within three age groups (≤35, 36-39, ≥40) was performed, but no significant difference in clinical pregnancy was observed. CONCLUSION: Our data suggests that modified NC-FET protocol has comparable pregnancy outcomes to down-regulated HC-FET when utilizing frozen-thawed day 5 embryos.

Learning graph matching.

As a fundamental problem in pattern recognition, graph matching has applications in a variety of fields, from computer vision to computational biology. In graph matching, patterns are modeled as graphs and pattern recognition amounts to finding a correspondence between the nodes of different graphs. Many formulations of this problem can be cast in general as a quadratic assignment problem, where a linear term in the objective function encodes node compatibility and a quadratic term encodes edge compatibility. The main research focus in this theme is about designing efficient algorithms for approximately solving the quadratic assignment problem, since it is NP-hard. In this paper we turn our attention to a different question: how to estimate compatibility functions such that the solution of the resulting graph matching problem best matches the expected solution that a human would manually provide. We present a method for learning graph matching: the training examples are pairs of graphs and the 'labels' are matches between them. Our experimental results reveal that learning can substantially improve the performance of standard graph matching algorithms. In particular, we find that simple linear assignment with such a learning scheme outperforms Graduated Assignment with bistochastic normalisation, a state-of-the-art quadratic assignment relaxation algorithm.