"AutoGen offers a cost-effective hyperparameter optimization technique [EcoOptiGen](https://arxiv.org/abs/2303.04673) for tuning Large Language Models. The study finds that tuning hyperparameters can significantly improve the utility of LLMs.\n",
"In this notebook, we tune OpenAI ChatGPT (both GPT-3.5 and GPT-4) models for math problem solving. We use [the MATH benchmark](https://crfm.stanford.edu/helm/latest/?group=math_chain_of_thought) for measuring mathematical problem solving on competition math problems with chain-of-thoughts style reasoning.\n",
"AutoGen has provided an API for hyperparameter optimization of OpenAI ChatGPT models: `autogen.ChatCompletion.tune` and to make a request with the tuned config: `autogen.ChatCompletion.create`. First, we import autogen:"
"The [`config_list_openai_aoai`](https://microsoft.github.io/autogen/docs/reference/oai/openai_utils#config_list_openai_aoai) function tries to create a list of Azure OpenAI endpoints and OpenAI endpoints. It assumes the api keys and api bases are stored in the corresponding environment variables or local txt files:\n",
"- OpenAI API key: os.environ[\"OPENAI_API_KEY\"] or `openai_api_key_file=\"key_openai.txt\"`.\n",
"- Azure OpenAI API key: os.environ[\"AZURE_OPENAI_API_KEY\"] or `aoai_api_key_file=\"key_aoai.txt\"`. Multiple keys can be stored, one per line.\n",
"- Azure OpenAI API base: os.environ[\"AZURE_OPENAI_API_BASE\"] or `aoai_api_base_file=\"base_aoai.txt\"`. Multiple bases can be stored, one per line.\n",
" }, # only if the second Azure OpenAI API key is found\n",
"]\n",
"```\n",
"\n",
"You can directly override it if the above function returns an empty list, i.e., it doesn't find the keys in the specified locations."
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"## Load dataset\n",
"\n",
"We load the competition_math dataset. The dataset contains 201 \"Level 2\" Algebra examples. We use a random sample of 20 examples for tuning the generation hyperparameters and the remaining for evaluation."
"First we begin by solving the system of equations \\begin{align*}\n",
"3+a&=4-b, \\\\\n",
"4+b&=7+a.\n",
"\\end{align*}Adding the two equations, we get $3+a+4+b=4-b+7+a$, which simplifies to $7+a+b=11+a-b$. Cancelling $a$ from both sides, we get $7+b=11-b$. Solving for $b$, we find that $b=2$. Plugging this into the first equation above, we obtain $3+a=4-2$. Hence $a=-1$ and $3-a=\\boxed{4}$.\n"
"Before we start tuning, we must define the success metric we want to optimize. For each math task, we use voting to select a response with the most common answers out of all the generated responses. We consider the task successfully solved if it has an equivalent answer to the canonical solution. Then we can optimize the mean success rate of a collection of tasks."
"This will create a disk cache in \".cache/{seed}\". You can change `cache_path_root` from \".cache\" to a different path in `set_cache()`. The cache for different seeds are stored separately.\n",
"The tuning will take a while to finish, depending on the optimization budget. The tuning will be performed under the specified optimization budgets.\n",
"* `inference_budget` is the benchmark's target average inference budget per instance. For example, 0.004 means the target inference budget is 0.004 dollars, which translates to 2000 tokens (input + output combined) if the gpt-3.5-turbo model is used.\n",
"* `optimization_budget` is the total budget allowed for tuning. For example, 1 means 1 dollar is allowed in total, which translates to 500K tokens for the gpt-3.5-turbo model.\n",
"* `num_sumples` is the number of different hyperparameter configurations allowed to be tried. The tuning will stop after either num_samples trials are completed or optimization_budget dollars are spent, whichever happens first. -1 means no hard restriction in the number of trials and the actual number is decided by `optimization_budget`.\n",
"For example, the following code specifies a fixed prompt template. The default search space will be used for hyperparameters that don't appear in users' input."
"optimized config {'max_tokens': 375, 'n': 44, 'prompt': '{problem} Solve the problem carefully. Simplify your answer as much as possible. Put the final answer in \\\\boxed{{}}.', 'model': 'gpt-3.5-turbo', 'allow_format_str_template': True, 'temperature': 0.7466815201029384}\n",
"best result on tuning data {'expected_success': 0.9818164607828072, 'success': 1.0, 'success_vote': 0.95, 'voted_answer': 'To find the number of integers in the sequence, we need to find when each term becomes less than 1. \\n\\nStarting with 6075, we divide by 3 to get $\\\\frac{6075}{3} = 2025$. Since 2025 is an integer, it is included in the sequence.\\n\\nDividing 2025 by 3, we get $\\\\frac{2025}{3} = 675$. Again, 675 is an integer, so it is included in the sequence.\\n\\nIf we divide 675 by 3, we get $\\\\frac{675}{3} = 225$. 225 is an integer, so it is included in the sequence.\\n\\nDividing 225 by 3, we get $\\\\frac{225}{3} = 75$. 75 is an integer, so it is included in the sequence.\\n\\nDividing 75 by 3, we get $\\\\frac{75}{3} = 25$. 25 is an integer, so it is included in the sequence.\\n\\nIf we divide 25 by 3, we get $\\\\frac{25}{3} \\\\approx 8.3333$, which is not an integer. Thus, 25 is the last integer in the sequence.\\n\\nThere are a total of $\\\\boxed{6}$ integers in the sequence.', 'votes': 34.85, 'total_cost': 0.463802, 'cost': 0.27552199999999993, 'inference_cost': 0.01310685, 'training_iteration': 0, 'config': {'temperature_or_top_p': {'temperature': 0.7466815201029384}, 'max_tokens': 375, 'n': 44, 'prompt': 0, 'model': 'gpt-3.5-turbo', 'allow_format_str_template': True}, 'config/temperature_or_top_p': {'temperature': 0.7466815201029384}, 'config/max_tokens': 375, 'config/n': 44, 'config/prompt': 0, 'config/model': 'gpt-3.5-turbo', 'config/allow_format_str_template': True, 'experiment_tag': 'exp', 'time_total_s': 201.2768588066101}\n"
" \"content\": \"We start by solving the first equation for $a$: $$3+a=4-b.$$Adding $-3$ to both sides gives $a=1-b$. Substituting this expression for $a$ into the second equation gives $$4+b=7+(1-b).$$Simplifying this expression, we find that $b=2$. Substituting $b=2$ into the first equation to solve for $a$, we find that $a=1-2=-1$. Finally, we have $3-a=3-(-1)=3+1=\\\\boxed{4}$.\"\n",
" \"content\": \"Adding $a$ to both sides of the first equation gives $3+a+a=4-b+a$, which simplifies to $3+2a=4+a-b$. Adding $b$ to both sides of the second equation gives $4+b+b=7+a+b$, which simplifies to $4+2b=7+a+b$. Rearranging the equations gives $2a-b=1$ and $2b-a=3$. Adding these two equations gives $3a=4$, so $a=\\\\frac{4}{3}$. Substituting this into the first equation gives $2\\\\left(\\\\frac{4}{3}\\\\right)-b=1$, so $b=\\\\frac{5}{3}$. Thus, $3-a=3-\\\\frac{4}{3}=\\\\frac{9}{3}-\\\\frac{4}{3}=\\\\boxed{\\\\frac{5}{3}}$.\"\n",
" \"content\": \"We can first rewrite the given equations as $a+b=1$ and $a-b=3$. Adding these equations, we get $(a+b)+(a-b)=1+3$, which simplifies to $2a=4$. Dividing both sides by 2, we find $a=2$. Substituting this value of $a$ into the first equation, we have $2+b=1$, so $b=-1$. Finally, we can compute $3-a=3-2=\\\\boxed{1}$.\"\n",
" \"content\": \"From the first equation, we have $a=4-b-3$. Substituting this into the second equation, we get $4+b=7+4-b-3$. Simplifying the right side gives us $4+b=8-b$. Adding $b$ to both sides gives $4+2b=8$. Subtracting 4 from both sides gives $2b=4$ and dividing both sides by 2 gives $b=2$. Substituting this into the first equation gives $3+a=4-2$, so $a=-1$. Finally, $3-a=3-(-1)=3+1=\\\\boxed{4}$.\"\n",
" \"content\": \"Adding the two equations, we have $(3+a)+(4+b)=(4-b)+(7+a)$. Expanding both sides gives $7+a+b=11+a-b$. Subtracting $a$ and $b$ from both sides gives $7=11$, which is impossible. Thus, there is no solution to the given equations.\"\n",
" \"content\": \"We are given the equations $3+a=4-b$ and $4+b=7+a$. We can solve for $a$ and $b$ by using elimination.\\n\\nWe can eliminate $a$ by adding the two equations together. This gives us $(3+a)+(4+b)=(4-b)+(7+a)$. Simplifying both sides gives us $7+a+b=11+a-b$. We can then eliminate $a$ by subtracting $a$ from both sides, which gives us $7+b=11-b$. Adding $b$ to both sides gives us $7+2b=11$. Subtracting $7$ from both sides gives us $2b=4$. Dividing both sides by $2$, we get $b=2$.\\n\\nNow that we know the value of $b$, we can substitute it back into one of the original equations to solve for $a$. Let's use the equation $4+b=7+a$. Since we found that $b=2$, we have $4+2=7+a$. Simplifying gives us $6=7+a$. Subtracting $7$ from both sides gives us $-1=a$.\\n\\nFinally, we can find $3-a$ by substituting the value of $a$ that we found back into the expression. We have $3-a=3-(-1)$. Simplifying gives us $3-(-1)=3+1=\\\\boxed{4}$.\"\n",
" \"content\": \"We can start by solving the given equations separately. From $3+a=4-b$, we can add $b$ to both sides to find $3+a+b=4$. Similarly, from $4+b=7+a$, we can subtract $a$ from both sides to find $4+b-a=7$. Adding these two equations together, we have $3+a+b+4+b-a=4+7$, which simplifies to $7+b=11$. Subtracting $7$ from both sides gives us $b=4$. Substituting this value of $b$ into the equation $4+b-a=7$, we find $4+4-a=7$. Simplifying this equation further, we have $8-a=7$. Subtracting $8$ from both sides gives us $-a=-1$, or $a=1$. Finally, we can substitute the values of $a$ and $b$ into the expression $3-a$, which gives us $3-1=\\\\boxed{2}$.\"\n",
" \"content\": \"Starting with the first equation, we have $3+a=4-b$. Rearranging, we get $a=-1-b$. Substituting this into the second equation, we have $4+b=7+(-1-b)$. Simplifying, we get $b=-2$. Substituting this value of $b$ back into the first equation, we have $3+a=4-(-2)$. Simplifying further, we have $3+a=6$. Subtracting $3$ from both sides, we get $a=3$. Finally, we can find $3-a=3-3=\\\\boxed{0}$.\"\n",
" \"content\": \"Adding the two equations gives $(3+a)+(4+b)=(4-b)+(7+a)$. Simplifying both sides gives $7+a+b=11+a-b$. Subtracting $a$ from both sides gives $7+b=11-b$. Adding $b$ to both sides gives $b+b=11-7$, so $2b=4$ and $b=2$. Substituting this value back into either equation gives $4+2=7+a$, so $a=4$. Therefore, $3-a=3-4=\\\\boxed{-1}$.\"\n",
" \"content\": \"Rearranging the first equation, we have $a=4-b-3$ and rearranging the second equation, we have $b=7+a-4$. Substituting these expressions for $a$ and $b$ into the equation $3-a$, we have $3-(4-b-3)$. Simplifying the expression inside the parentheses, we have $3-(4-b-3)=3-4+b+3=9+b$. Substituting the expression for $b$ into $9+b$, we have $9+(7+a-4)=9+7+a-4=12+a$. Finally, substituting the expression for $a$ into $12+a$, we have $12+(4-b-3)=12+4-b-3=\\\\boxed{10-b}$.\"\n",
" \"content\": \"We have the system of equations \\\\begin{align*}\\n3+a&=4-b\\\\\\\\\\n4+b&=7+a\\n\\\\end{align*} Rearranging the first equation, we have $a+b=1$. Substituting this into the second equation, we get $4+1=7+a$, so $a=-4$. Thus, $3-a=\\\\boxed{7}$.\"\n",
" \"content\": \"Simplifying the first equation, we have $a=1-b$. Substituting this into the second equation, we have $4+b=7+(1-b)$. Expanding the right side gives $4+b=7+1-b$. Combining like terms gives $2b=4$, so $b=2$. Substituting this back into $a=1-b$, we find that $a=-1$. Thus, $3-a=3-(-1)=3+1=\\\\boxed{4}$.\"\n",
" \"content\": \"From the first equation, we have $a=4-b-3$. Substituting this into the second equation, we have $4+b=7+(4-b-3)$. Simplifying the right side of the equation gives $4+b=8-b$. Adding $b$ to both sides gives $4+2b=8$. Subtracting 4 from both sides gives $2b=4$. Dividing both sides by 2 gives $b=2$. Substituting this value back into the first equation gives $3+a=4-2$. Simplifying the right side gives $3+a=2$. Subtracting 3 from both sides gives $a=-1$. Finally, we have $3-a=3-(-1)=3+1=\\\\boxed{4}$.\"\n",
" \"content\": \"From the first equation, subtracting $a$ and adding $4$ to both sides gives $7=b-a$. Substituting this into the second equation gives $4+(b-a)=7+a$, so $4+7=b+a$. Combining these equations gives $3+b+a=11+a$, so $b=\\\\boxed{8}$. Substituting into the first equation gives $3+a=4-8$ which gives $a=\\\\boxed{-9}$. Finally, $3-a=3-(-9)=3+9=\\\\boxed{12}$.\"\n",
" \"content\": \"We can start by solving the first equation for $a$ in terms of $b$. Subtracting $3$ from both sides of the equation $3+a=4-b$ gives $a=1-b$. We can substitute this expression for $a$ in the second equation to solve for $b$: \\\\begin{align*}\\n4+b&=7+a\\\\\\\\\\n4+b&=7+(1-b)\\\\\\\\\\n4+b&=8-b\\\\\\\\\\n2b&=4\\\\\\\\\\nb&=2.\\n\\\\end{align*}Substituting this value of $b$ back into the first equation to solve for $a$, we have $3+a=4-2$, so $a=-1$. Finally, we can find $3-a=3-(-1)=\\\\boxed{4}$.\"\n",
" \"content\": \"From the first equation, we have $a=1-b$. Substituting this into the second equation gives $4+b=7+1-b$. Simplifying gives $2b=4$, so $b=2$. Substituting this back into the first equation gives $3+a=4-2$, so $a=-1$. Therefore, $3-a=3-(-1)=4$. Simplifying gives $\\\\boxed{4}$.\"\n",
" \"content\": \"Adding the two given equations, we have $(3+a)+(4+b)=(4-b)+(7+a)$. Simplifying both sides gives $7+a+b=11+a-b$. We can subtract $a$ from both sides to get $7+b=11-b$. Adding $b$ to both sides gives $7+2b=11$. Subtracting 7 from both sides gives $2b=4$. Dividing by 2 gives $b=2$.\\n\\nWe can substitute $b=2$ into the first equation $3+a=4-b$ to solve for $a$. We have $3+a=4-2$ which simplifies to $3+a=2$. Subtracting 3 from both sides gives $a=-1$.\\n\\nFinally, we can substitute $a=-1$ into $3-a$ to find $3-a=3-(-1)$. Simplifying gives $3-a=3+1=\\\\boxed{4}$.\"\n",
" \"content\": \"Adding the two given equations, we have $(3+a)+(4+b)=(4-b)+(7+a)$. Simplifying both sides gives $7+a+b=11+a-b$. Subtracting $a$ from both sides gives $7+b=11-b$. Adding $b$ to both sides gives $7+2b=11$. Subtracting $7$ from both sides gives $2b=4$. Finally, dividing both sides by $2$ gives $b=2$. Substituting this value for $b$ into the second given equation, we have $4+2=7+a$. Simplifying gives $a=-1$. Therefore, $3-a=3-(-1)=4$. Thus, the final answer is $\\\\boxed{4}$.\"\n",
" \"content\": \"Let's start by simplifying the given equations. We have $3+a=4-b$, which we can rearrange to get $a=-b+1$. Similarly, we have $4+b=7+a$, which rearranges to $b=a+3$. \\n\\nWe can substitute the value of $b$ from the second equation into the first equation to get $a=(-a-3)+1$. Simplifying this equation gives $2a=-2$, so $a=-1$. \\n\\nSubstituting this value of $a$ into the second equation gives $b=(-1)+3$, so $b=2$. \\n\\nFinally, we can find $3-a$ by substituting $a=-1$ into $3-a$. This gives $3-(-1)=3+1=\\\\boxed{4}$.\"\n",
" \"content\": \"We have the equations $3+a=4-b$ and $4+b=7+a$. We can solve these equations using substitution or elimination. Let's solve it using elimination.\\n\\nTo eliminate $a$, we can add the first equation to the second equation. This gives us $(3+a)+(4+b)=(4-b)+(7+a)$. Simplifying both sides, we have $7 + a + b = 11 + a - b$.\\n\\nNow, let's isolate $b$ by subtracting $a$ from both sides: $7 + b = 11 - b$.\\n\\nTo isolate $b$ on one side, we can add $b$ to both sides: $7 + 2b = 11$.\\n\\nSubtracting $7$ from both sides gives $2b= 4$. Dividing both sides by $2$, we find $b=2$.\\n\\nNow, we can substitute $b=2$ into the second equation $4+b=7+a$. This gives $4+2=7+a$, or $6=7+a$. Subtracting $7$ from both sides gives $-1=a$.\\n\\nFinally, we can find $3-a$ by substituting $a=-1$ into $3-a$. This gives $3-(-1)=3+1=\\\\boxed{4}$.\"\n",
" \"content\": \"We can start by combining like terms in both equations. From the first equation, we have $a+3=b-4$, and from the second equation, we have $b+4=a-7$. \\n\\nNow, we can substitute $b-4$ for $a+3$ in the second equation, since they are equal. This gives us $b+4=(b-4)-7$. \\n\\nSimplifying, we have $b+4=b-11$. \\n\\nSubtracting $b$ from both sides, we get $4=-11$. \\n\\nThis is a contradiction, since $4$ does not equal $-11$. \\n\\nTherefore, the solution to this system of equations does not exist, and we cannot find the value of $3-a$. Thus, the answer is $\\\\boxed{\\\\text{DNE}}$.\"\n",
" \"content\": \"We can start by solving the first equation, $3+a=4-b$, for $a$ in terms of $b$ by subtracting $3$ from both sides and then adding $b$ to both sides. This gives us $a = 1-b$.\\n\\nWe can substitute this expression for $a$ into the second equation, $4+b=7+a$, to solve for $b$ in terms of $a$. After simplifying, we have $b=4-a$.\\n\\nTo find $3-a$, we substitute $b=4-a$ into the first equation $3+a=4-b$. This gives us $3+a=4-(4-a)$.\\n\\nSimplifying this equation gives $3+a=4-4+a$, so $3+a=a$.\\n\\nTherefore, $3-a = \\\\boxed{3}$.\"\n",
" \"content\": \"To solve this problem, we can start by solving the first equation $3+a=4-b$ for $b$. Subtracting $3$ from both sides gives $a=1-b$. We can substitute this into the second equation $4+b=7+a$ to get $4+b=7+(1-b)$. Expanding the right side gives $4+b=7+1-b$. Combining like terms gives $b+b=7+1-4$. Simplifying the right side gives $2b=4$. Dividing both sides by $2$ gives $b=2$. Now we can substitute this back into the first equation to solve for $a$. We have $3+a=4-2$, so $3+a=2$. Subtracting $3$ from both sides gives $a=-1$. Finally, we can find $3-a$ by subtracting $a$ from $3$. We have $3-a=3-(-1)=3+1=\\\\boxed{4}$.\"\n",
" \"content\": \"Starting with the first equation, we have $3+a=4-b$. Rearranging this equation, we get $a=-b+1$. \\n\\nSubstituting this expression for $a$ into the second equation, we have $4+b=7+(-b+1)$. Simplifying this equation gives $b=-2$. \\n\\nSubstituting this value of $b$ into the equation $a=-b+1$, we find $a=3$. \\n\\nFinally, we can evaluate $3-a$ using the value we found for $a$. We have $3-a=3-3=\\\\boxed{0}$.\"\n",
" \"content\": \"Adding the two given equations, we have $(3+a)+(4+b)=(4-b)+(7+a)$. Simplifying both sides gives $7+a+b=11+a-b$. Subtracting $a$ from both sides gives $7+b=11-b$. Adding $b$ to both sides gives $7+2b=11$. Subtracting $7$ from both sides gives $2b=4$. Dividing both sides by $2$ gives $b=2$. Substituting this value of $b$ into the first given equation, we have $3+a=4-2$, or $a=-1$. Finally, substituting these values into $3-a$, we have $3-(-1)=3+1=\\\\boxed{4}$.\"\n",
" \"content\": \"From the first equation, $3+a=4-b$, we can subtract $3$ from both sides to find that $a=1-b$. Substituting this into the second equation, we have $4+b=7+(1-b)$. Expanding the right side gives $4+b=7+1-b$. We can then simplify this to $b+4=8-b$ by combining like terms. Adding $b$ to both sides gives $2b+4=8$. Subtracting $4$ from both sides gives $2b=4$. Finally, dividing both sides by $2$ gives $b=2$. \\n\\nNow that we have found the value of $b$, we can substitute it back into the first equation to find $3+a=4-2$. Simplifying gives $3+a=2$. Subtracting $3$ from both sides gives $a=-1$. \\n\\nFinally, we can find $3-a$ by subtracting $-1$ from $3$. This gives $3-(-1)=3+1=\\\\boxed{4}$.\"\n",
" \"content\": \"We can start by solving the first equation, $3+a=4-b$, for $a$. Subtracting 3 from both sides gives $a=1-b$. \\n\\nWe can substitute this value for $a$ into the second equation $4+b=7+a$. Substituting gives $4+b=7+(1-b)$. Expanding the right side gives $4+b=7+1-b$. Combining like terms gives $4+b=8-b$. \\n\\nAdding $b$ to both sides gives $4=8-2b$. Subtracting 8 from both sides gives $-4=-2b$. Dividing both sides by $-2$ gives $2=b$.\\n\\nNow that we know $b=2$, we can substitute this value back into the first equation to find $a$. Substituting gives $3+a=4-2$. Simplifying gives $3+a=2$. Subtracting 3 from both sides gives $a=-1$.\\n\\nFinally, we can find $3-a$ by substituting the value of $a$ we found into $3-a$. Substituting gives $3-(-1)$. Simplifying gives $3+1=4$.\\n\\nThus, $3-a=\\\\boxed{4}$.\"\n",
" \"content\": \"Starting with the first equation, we can subtract $a$ from both sides to isolate $3$:\\n\\n\\\\[3+a-a=4-b-a\\\\qquad\\\\Rightarrow\\\\qquad 3=4-b-a.\\\\]\\n\\nRearranging the terms, we have $b+a=4-3=1.$ Similarly, starting with the second equation, we can subtract $b$ from both sides to obtain $a+b=7-4=3.$ Adding these two equations, we have $2a+2b=4,$ so $a+b=2.$ Subtracting this equation from $b+a=1,$ we get $b-a=\\\\boxed{-1}.$\"\n",
" \"content\": \"From the first equation, we have $a=4-b-3=-b+1$. Substituting this into the second equation, we get $4+b=7+(-b+1)$, which simplifies to $4+b=8-b$. Solving for $b$, we find $2b=4$, so $b=2$. Substituting this back into $a=-b+1$, we find $a=-2+1=-1$. Finally, $3-a=3-(-1)=3+1=\\\\boxed{4}$.\"\n",
" \"content\": \"We start by solving the first equation for $a$ by subtracting 3 from both sides to find $a=1-b$. Substituting this into the second equation gives $4+b=7+(1-b)$. Expanding the brackets gives $4+b=7+1-b$. Simplifying gives $2b=4$, so $b=2$. Substituting this into $a=1-b$ gives $a=1-2=-1$. Finally, substituting this into $3-a$ gives $3-(-1)=4$, so our final answer is $\\\\boxed{4}$.\"\n",
" \"content\": \"Adding the two given equations, we have $(3+a)+(4+b)=(4-b)+(7+a)$. Applying the commutative property of addition, we can rearrange the terms to get $(3+4)+(a+b)=(4+7)+(-b+a)$. Simplifying both sides gives $7+(a+b)=11+(a-b)$. We can rewrite this equation as $a+b+7=a-b+11$. Subtracting $a+b+4$ from both sides yields $7-4=a-b+11-(a+b+4)$. Simplifying gives $3=-4-b$. Adding $b$ to both sides gives $3+b=-4$. Subtracting 4 from both sides yields $b-1=-4$. Then, adding 1 to both sides gives $b=-3$. Substituting this into the first equation $3+a=4-b$, we can substitute $-3$ for $b$ to get $3+a=4-(-3)$. Simplifying gives $3+a=4+3$. Subtracting 3 from both sides yields $a=4$. Finally, substituting this into $3-a$, we can substitute $4$ for $a$ to get $3-4=\\\\boxed{-1}$.\"\n",
" \"content\": \"From the first equation, we have $a=4-b-3=1-b$. Substituting this into the second equation gives $4+b=7+(1-b)$. Expanding the right side gives $4+b=7+1-b$. Simplifying the right side gives $4+b=8-b$. Adding $b$ to both sides gives $4+2b=8$. Subtracting 4 from both sides gives $2b=4$. Dividing both sides by 2 gives $b=2$. Substituting this into the first equation gives $3+a=4-2$. Simplifying the right side gives $3+a=2$. Subtracting 3 from both sides gives $a=-1$. Finally, we have $3-a=3-(-1)=\\\\boxed{4}$.\"\n",
" \"content\": \"We start by solving the first equation for $a$: \\\\begin{align*}\\n3+a&=4-b \\\\\\\\\\na&=1-b.\\n\\\\end{align*}We substitute this expression for $a$ into the second equation: \\\\begin{align*}\\n4+b&=7+a \\\\\\\\\\n4+b&=7+(1-b) \\\\\\\\\\n4+b&=8-b.\\n\\\\end{align*}Adding $b$ to both sides gives $4+2b=8$, so $2b=4$ and $b=2$. Substituting this value into $a=1-b$ gives $a=1-2=-1$. Therefore, $3-a=3-(-1)=3+1=4$. Thus, $3-a=\\\\boxed{4}$.\"\n",
" \"content\": \"We start by solving the first equation, $3+a=4-b$, for $a$. Subtracting $3$ from both sides gives $a=1-b$.\\n\\nWe can substitute this expression for $a$ into the second equation, $4+b=7+a$. Substituting $1-b$ for $a$ gives $4+b=7+(1-b)$.\\n\\nExpanding the parentheses gives $4+b=7+1-b$. Simplifying the right side gives $4+b=8-b$.\\n\\nAdding $b$ to both sides gives $4+2b=8$. Subtracting $4$ from both sides gives $2b=4$.\\n\\nFinally, dividing both sides by $2$ gives $b=2$. We can substitute this value back into the equation $a=1-b$ to find $a=1-2=-1$.\\n\\nNow, we can find $3-a$ by subtracting $(-1)$ from $3$. This gives $3-(-1)=3+1=\\\\boxed{4}$.\"\n",
" \"content\": \"Combining the two equations, we have $3+a+4+b=4-b+7+a$. Simplifying both sides, we have $7+a+b=11+a-b$. Subtracting $a$ from both sides, we have $7+b=11-b$. Adding $b$ to both sides, we have $7+2b=11$. Subtracting 7 from both sides, we have $2b=4$. Dividing both sides by 2, we have $b=2$. Substituting this value back into the first equation, we have $3+a=4-2$, so $3+a=2$. Subtracting 3 from both sides, we have $a=-1$. Therefore, $3-a=3-(-1)=3+1=\\\\boxed{4}$.\"\n",
" \"content\": \"We can start by subtracting $a$ from both sides of the first equation to find that $3 = 4 - b - a$. Similarly, we can subtract $a$ from both sides of the second equation to find that $4 = 7 + a - b$. Adding these two equations gives $3 + 4 = 4 - b - a + 7 + a - b$, which simplifies to $7 = 11 - 2b$. Solving for $b$ gives $b = 2$. Substituting this value into the first equation gives $3 + a = 4 - 2$, so $a = 3$. Then $3 - a = 3 - 3 = \\\\boxed{0}$.\"\n",
" \"content\": \"Adding the two given equations, we have $$(3+a)+(4+b)=(4-b)+(7+a).$$Expanding both sides gives $$7+a+b=11+a-b.$$We can then cancel out the $a$ term on both sides to get $$7+b=11-b.$$Adding $b$ to both sides gives $$7+2b=11.$$Subtracting $7$ from both sides gives $$2b=4.$$Dividing both sides by $2$ gives $$b=2.$$Plugging this value of $b$ into either of the original equations, we can solve for $a$. Using the first equation, we have $$3+a=4-2 \\\\Rightarrow a=-1.$$Finally, we can find $3-a$ as $$3-a=3-(-1)=3+1=\\\\boxed{4}.$$\"\n",
" \"content\": \"We can start by adding $a$ to both sides of the first equation and subtracting $b$ from both sides of the second equation to obtain \\\\begin{align*}\\na+b&=1, \\\\\\\\\\na-b&=-3.\\n\\\\end{align*} We can then add these equations to eliminate $b$: $$2a=1+(-3)=-2.$$Dividing both sides by $2$ gives $a=-1$. Substituting into the second equation gives $-1-b=-3$, so $b=2$. Finally, we find that $3-a=3-(-1)=\\\\boxed{4}$.\"\n",
" \"content\": \"We can start by subtracting $a$ from both sides of the first equation and subtracting $b$ from both sides of the second equation to obtain \\\\begin{align*}\\n3&=4-b-a,\\\\\\\\\\n4&=7+a-b.\\n\\\\end{align*}We can rearrange the first equation to get $b+a=4-3=1$. Similarly, we can rearrange the second equation to get $a-b=4-7=-3$. Adding these equations, we find that $(b+a)+(a-b)=1+(-3)$, which implies $2a= -2$. Hence, $a=-1$. We can substitute this value of $a$ into $a-b=-3$ to find that $-1-b=-3$, so $b=-1-(-3)=2$. Finally, we have \\\\begin{align*}\\n3-a&=3-(-1)=3+1=\\\\boxed{4}.\\n\\\\end{align*}\"\n",
" \"content\": \"Adding the two given equations, we have $$(3+a)+(4+b)=(4-b)+(7+a).$$Simplifying both sides gives $7+a+b=11+a-b$. Subtracting $a$ and $b$ from both sides gives $7=11$, which is a contradiction. Therefore, there are no solutions to the given equations, and the value of $3-a$ is undefined. So we have $3-a=\\\\boxed{ \\\\text{undefined}}$.\"\n",
" \"content\": \"To solve this problem, we can start by isolating $a$ in both equations. \\n\\nFrom the first equation, $3+a=4-b$, we can subtract 3 from both sides to get $a=1-b$. \\n\\nFrom the second equation, $4+b=7+a$, we can subtract 4 from both sides to get $b=3+a$. \\n\\nNow, we can substitute $1-b$ for $a$ in the second equation to get $b=3+1-b$. Simplifying this equation gives $2b=4$, so $b=2$. \\n\\nSubstituting $b=2$ into the equation $a=1-b$, we find that $a=1-2=-1$. \\n\\nFinally, we can find $3-a$ by subtracting $(-1)$ from 3, which gives us $3-(-1)=\\\\boxed{4}$.\"\n",
" \"content\": \"We start by solving the first equation for $a$ in terms of $b$. Subtracting $3$ from both sides, we have $a=1-b$. Substituting this into the second equation, we get $4+b=7+(1-b)$. Simplifying, we have $4+b=8-b$. Adding $b$ to both sides, we have $4+2b=8$. Subtracting $4$ from both sides, we have $2b=4$. Dividing both sides by $2$, we have $b=2$. Substituting this into the equation $a=1-b$, we have $a=1-2=-1$. Thus, $3-a=3-(-1)=3+1=\\\\boxed{4}$.\"\n",
" \"content\": \"From the first equation, we have $3+a=4-b$. Rearranging this equation, we get $a=-1-b$. Substituting this value of $a$ into the second equation, we have $4+b=7+(-1-b)$. Simplifying this equation, we get $b+1=-b+6$. Adding $b$ to both sides and subtracting $1$ from both sides, we have $2b=5$. Therefore, $b=\\\\frac{5}{2}$. Substituting this value of $b$ into the first equation, we have $3+a=4-\\\\frac{5}{2}$. Simplifying this equation, we get $a=\\\\frac{3}{2}$. Finally, we have $3-a=3-\\\\frac{3}{2}=\\\\boxed{\\\\frac{3}{2}}$.\"\n",
" \"content\": \"Manipulating the given equations, we have $a=1-b$ and $b=3+a$. Substituting $a=1-b$ into the second equation, we get $b=3+1-b$, which implies $2b=4$, or $b=2$. Substituting $b=2$ into the first equation, we get $a=1-2=-1$. Finally, evaluating $3-a$, we find that $3-a=3-(-1)=\\\\boxed{4}$.\"\n",
"metric_results on the example data instance: {'expected_success': 1.0, 'success': True, 'success_vote': 1.0, 'voted_answer': 'We start by solving the first equation for $a$: $$3+a=4-b.$$Adding $-3$ to both sides gives $a=1-b$. Substituting this expression for $a$ into the second equation gives $$4+b=7+(1-b).$$Simplifying this expression, we find that $b=2$. Substituting $b=2$ into the first equation to solve for $a$, we find that $a=1-2=-1$. Finally, we have $3-a=3-(-1)=3+1=\\\\boxed{4}$.', 'votes': 27}\n"
"You can use `autogen.ChatCompletion.test` to evaluate the performance of an entire dataset with the tuned config. The following code will take a while (30 mins to 1 hour) to evaluate all the test data instances if uncommented and run. It will cost roughly $3. "
"# print(\"performance on test data with the tuned config:\", result)"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"What about the default, untuned gpt-4 config (with the same prompt as the tuned config)? We can evaluate it and compare:"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"performance on test data from gpt-4 with a default config: {'expected_success': 0.6965174129353234, 'success': 0.6965174129353234, 'success_vote': 0.6965174129353234, 'votes': 1.0, 'cost': 1.9264799999999993, 'inference_cost': 0.009584477611940295}\n"
]
}
],
"source": [
"# the following code will cost roughly $2 if uncommented and run.\n",
"# print(\"performance on test data from gpt-4 with a default config:\", default_result)"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"tuned config succeeds in 90.5% test cases\n",
"untuned config succeeds in 69.7% test cases\n"
]
}
],
"source": [
"# print(\"tuned config succeeds in {:.1f}% test cases\".format(result[\"success_vote\"] * 100))\n",
"# print(\"untuned config succeeds in {:.1f}% test cases\".format(default_result[\"success_vote\"] * 100))"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"The default use of GPT-4 has a much lower accuracy. Note that the default config has a lower inference cost. What if we heuristically increase the number of responses n?"
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {},
"outputs": [],
"source": [
"# The following evaluation costs $3 and longer than one hour if you uncomment it and run it.\n",
"The inference cost is doubled and matches the tuned config. But the success rate doesn't improve much. What if we further increase the number of responses n to 5?"
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {},
"outputs": [],
"source": [
"# The following evaluation costs $8 and longer than one hour if you uncomment it and run it.\n",
"# print(\"performance on test data from gpt-4 with a default config and n=5:\", result_n5)"
]
},
{
"attachments": {},
"cell_type": "markdown",
"metadata": {},
"source": [
"We find that the 'success_vote' metric is increased at the cost of exceeding the inference budget. But the tuned configuration has both higher 'success_vote' (91% vs. 87%) and lower average inference cost ($0.015 vs. $0.037 per instance).\n",
