If you watched youtube live https://www.youtube.com/live/SKBG1sqdyIU, the most exciting part of the OpenAI o3 model release is the ability on the EpochAI Frontier Math Benchmark, which is new still in research stage Math problems, which requires professional Math scientist to spend hours or days to get the correct answer. If the these new research Math problem can be solved by AI models, certainly it already surpasses 99.9% human intelligence.

OpenAI o3 scored 25.2 accuracy on Epoch AI Frontier Math benchmark compared to o1 model' 2.0 score, which is a significant improvement over previous SOTA. Right now since it's still not open to public, the best I can think of it's the breakthrough in Reinforcement learning.

To test OpenAI o1 model's ability of graduate-level machine learning & math, I tried to ask OpenAI o1 to solve the problem of "Use Newton method to calculate the exact value of \sqrt{2}. Provide Detailed proof step by step." OpenAI o1 actually gives me a very long detailed step-by-step proof, which is much detailed and longer than GPT4o or GPT4o-mini. The complete response is very long and I just summarized the detailed steps of the reasoning here. Overall, I think OpenAI o1 can provide more accurate responses than previous GPT4o' answers. But the response usually is very long and there are a lot of top-down break-down, which makes the final result correct but sometimes hard to understand and follow. OpenAI o1 breaks down the machine learning math problem into these steps: ### Step 1: Define the Function and Its Derivative ### Step 2: Derive the Newton's Method Iteration Formula ### Step 3: Choose an Initial Approximation ### Step 4: Perform Iterations ### Step 5: Understanding the Result Each iteration yields a rational approximation that gets closer to ( \sqrt{2} ): - Iteration 1: ( x_1 = \dfrac{3}{2} = 1.5 ) - Iteration 2: ( x_2 = \dfrac{17}{12} \approx 1.4167 ) - Iteration 3: ( x_3 = \dfrac{577}{408} \approx 1.41422 ) - Iteration 4: ( x_4 = \dfrac{667,217}{471,632} \approx 1.41421 ) ### Conclusion Using Newton's method with the function ( f(x) = x^2 - 2 ) and starting approximation ( x_0 = 1 ), we generate a sequence ( { x_n } ) defined by: [ x_{n+1} = \frac{1}{2} \left( x_n + \frac{2}{x_n} \right) ] This sequence converges to ( \sqrt{2} ) as ( n ) approaches infinity. ### Final Answer By applying Newton's method to the equation ( x^2 - 2 = 0 ), we find that ( \sqrt{2} ) is the limit of the sequence: [ x_{n+1} = \dfrac{1}{2} \left( x_n + \dfrac{2}{x_n} \right) ] Starting from ( x_0 = 1 ), each iteration gives a rational approximation of ( \sqrt{2} ). Although we cannot express ( \sqrt{2} ) exactly using Newton's method (since it's irrational), the method provides increasingly accurate approximations.

Finally, OpenAI released o1 mdoel with stronger reasoning ability. And I looked through the detailed comparison of a math solving results on their website and the comparison between GPT4o vs OpenAI o1-preview on this Algebra problem. For the math question as the in the prompt, o1 uses a chain of thought when attempting to solve a problem, which is similar to how a human may think for a long time before responding to a difficult question. o1 response actually break down the question into a few steps: "Understanding the Given Information", "Defining a New Polynomial", "Properties of q(x)", "Constructing s(x)", "Matching Coefficients". Finding Additional Solutions and finally reaching Conclusion. The additional real numbers x satisfying p(1/x) = x^{2} are x=\frac{1}{n!} or -\frac{1}{n!}. Overall, the reasoning ability is quite complex compared to previous version, so it's helpful and the answers are correct.