Version: Next

Ask-tell Optimization with Ax

Complex optimization problems where we wish to tune multiple parameters to improve metric performance, but the inter-parameter interactions are not fully understood, are common across various fields including machine learning, robotics, materials science, and chemistry. This category of problem is known as "black-box" optimization. The complexity of black-box optimization problems further increases if evaluations are expensive to conduct, time-consuming, or noisy.

We can use Ax to efficiently conduct an experiment in which we "ask" for candidate points to evaluate, "tell" Ax the results, and repeat. We'll uses Ax's Client, a tool for managing the state of our experiment, and we'll learn how to define an optimization problem, configure an experiment, run trials, analyze results, and persist the experiment for later use using the Client.

Because Ax is a black box optimizer, we can use it to optimize any arbitrary function. In this example we will minimize the Hartmann6 function, a complicated 6-dimensional function with multiple local minima. Hartmann6 is a challenging benchmark for optimization algorithms commonly used in the global optimization literature -- it tests the algorithm's ability to identify the true global minimum, rather than mistakenly converging on a local minimum. Looking at its analytic form we can see that it would be incredibly challenging to efficiently find the global minimum either by manual trial-and-error or traditional design of experiments like grid-search or random-search.

f(\mathbf{x})=-\sum_{i=1}^4 \alpha_i \exp \left(-\sum_{j=1}^6 A_{i j}\left(x_j-P_{i j}\right)^2\right)

where

\alpha=(1.0,1.2,3.0,3.2)^T

\mathbf{A}=\left(\begin{array}{cccccc}10 & 3 & 17 & 3.50 & 1.7 & 8 \\ 0.05 & 10 & 17 & 0.1 & 8 & 14 \\ 3 & 3.5 & 1.7 & 10 & 17 & 8 \\ 17 & 8 & 0.05 & 10 & 0.1 & 14\end{array}\right)

\mathbf{P}=10^{-4}\left(\begin{array}{cccccc}1312 & 1696 & 5569 & 124 & 8283 & 5886 \\ 2329 & 4135 & 8307 & 3736 & 1004 & 9991 \\ 2348 & 1451 & 3522 & 2883 & 3047 & 6650 \\ 4047 & 8828 & 8732 & 5743 & 1091 & 381\end{array}\right)

Learning Objectives

Understand the basic concepts of black box optimization
Learn how to define an optimization problem using Ax
Configure and run an experiment using Ax's Client
Analyze the results of the optimization

Prerequisites

Familiarity with Python and basic programming concepts
Understanding of adaptive experimentation and Bayesian optimization

Step 1: Import Necessary Modules

First, ensure you have all the necessary imports:

import numpy as np
from ax.api.client import Client
from ax.api.configs import (
    ExperimentConfig,
    RangeParameterConfig,
    ParameterType,
)

Output:
/opt/hostedtoolcache/Python/3.12.10/x64/lib/python3.12/site-packages/pyro/ops/stats.py:527: SyntaxWarning: invalid escape sequence 'g'
we have :math:ES^{*}(P,Q) ge ES^{*}(Q,Q) with equality holding if and only if :math:P=Q, i.e.

Step 2: Initialize the Client

Create an instance of the Client to manage the state of your experiment.

client = Client()

Step 3: Configure the Experiment

The Client instance can be configured with a series of Configs that define how the experiment will be run.

The Hartmann6 problem is usually evaluated on the hypercube $x_i \in (0, 1)$ , so we will define six identical RangeParameterConfigs with these bounds and add these to an ExperimentConfig along with other metadata about the experiment.

You may specify additional features like parameter constraints to further refine the search space and parameter scaling to help navigate parameters with nonuniform effects. For more on configuring experiments, see this recipe.

# Define six float parameters x1, x2, x3, ... for the Hartmann6 function
parameters = [
    RangeParameterConfig(
        name=f"x{i + 1}", parameter_type=ParameterType.FLOAT, bounds=(0, 1)
    )
    for i in range(6)
]

# Create an experiment configuration
experiment_config = ExperimentConfig(
    name="hartmann6_experiment",
    parameters=parameters,
    # The following arguments are optional
    description="Optimization of the Hartmann6 function",
    owner="developer",
)

# Apply the experiment configuration to the client
client.configure_experiment(experiment_config=experiment_config)

Step 4: Configure Optimization

Now, we must configure the objective for this optimization, which we do using Client.configure_optimization. This method expects a string objective, an expression containing either a single metric to maximize, a linear combination of metrics to maximize, or a tuple of multiple metrics to jointly maximize. These expressions are parsed using SymPy. For example:

"score" would direct Ax to maximize a metric named score
"-loss" would direct Ax to Ax to minimize a metric named loss
"task_0 + 0.5 * task_1" would direct Ax to maximize the sum of two task scores, downweighting task_1 by a factor of 0.5
"score, -flops" would direct Ax to simultaneously maximize score while minimizing flops

For more information on configuring objectives and outcome constraints, see this recipe.

metric_name = "hartmann6" # this name is used during the optimization loop in Step 5
objective = f"-{metric_name}" # minimization is specified by the negative sign

client.configure_optimization(objective=objective)

Step 5: Run Trials

Here, we will configure the ask-tell loop.

We begin by defining the Hartmann6 function as written above. Remember, this is just an example problem and any Python function can be substituted here.

# Hartmann6 function
def hartmann6(x1, x2, x3, x4, x5, x6):
    alpha = np.array([1.0, 1.2, 3.0, 3.2])
    A = np.array([
        [10, 3, 17, 3.5, 1.7, 8],
        [0.05, 10, 17, 0.1, 8, 14],
        [3, 3.5, 1.7, 10, 17, 8],
        [17, 8, 0.05, 10, 0.1, 14]
    ])
    P = 10**-4 * np.array([
        [1312, 1696, 5569, 124, 8283, 5886],
        [2329, 4135, 8307, 3736, 1004, 9991],
        [2348, 1451, 3522, 2883, 3047, 6650],
        [4047, 8828, 8732, 5743, 1091, 381]
    ])

    outer = 0.0
    for i in range(4):
        inner = 0.0
        for j, x in enumerate([x1, x2, x3, x4, x5, x6]):
            inner += A[i, j] * (x - P[i, j])**2
        outer += alpha[i] * np.exp(-inner)
    return -outer

hartmann6(0.1, 0.45, 0.8, 0.25, 0.552, 1.0)

Output:
np.float64(-0.4878737485613134)

Optimization Loop

We will iteratively call client.get_next_trials to "ask" Ax for a parameterization to evaluate, then call hartmann6 using those parameters, and finally "tell" Ax the result using client.complete_trial.

This loop will run multiple trials to optimize the function.

# Number of trials to run
num_trials = 30

# Run trials
for _ in range(num_trials):
    trials = client.get_next_trials(
        maximum_trials=1
    )  # We will request just one trial at a time in this example
    for trial_index, parameters in trials.items():
        x1 = parameters["x1"]
        x2 = parameters["x2"]
        x3 = parameters["x3"]
        x4 = parameters["x4"]
        x5 = parameters["x5"]
        x6 = parameters["x6"]

        result = hartmann6(x1, x2, x3, x4, x5, x6)

        # Set raw_data as a dictionary with metric names as keys and results as values

        raw_data = {metric_name: result}

        # Complete the trial with the result

        client.complete_trial(trial_index=trial_index, raw_data=raw_data)
        print(f"Completed trial {trial_index} with {raw_data=}")

Output:
Completed trial 0 with raw_data={'hartmann6': np.float64(-0.5019255343509779)}
Completed trial 1 with raw_data={'hartmann6': np.float64(-0.022448780563003885)}
Completed trial 2 with raw_data={'hartmann6': np.float64(-0.14519093834066654)}
Completed trial 3 with raw_data={'hartmann6': np.float64(-0.8604693413805216)}
Completed trial 4 with raw_data={'hartmann6': np.float64(-0.04463788653477111)}
Completed trial 5 with raw_data={'hartmann6': np.float64(-1.53747420257552)}
Completed trial 6 with raw_data={'hartmann6': np.float64(-0.7746285626008993)}
Completed trial 7 with raw_data={'hartmann6': np.float64(-0.5958143119835694)}
Completed trial 8 with raw_data={'hartmann6': np.float64(-0.9180207256799746)}
Completed trial 9 with raw_data={'hartmann6': np.float64(-1.4571186548903279)}
Completed trial 10 with raw_data={'hartmann6': np.float64(-0.5652020510015644)}
Completed trial 11 with raw_data={'hartmann6': np.float64(-1.9025798308645765)}
Completed trial 12 with raw_data={'hartmann6': np.float64(-2.4023918423159127)}
Completed trial 13 with raw_data={'hartmann6': np.float64(-2.667845217834489)}
Completed trial 14 with raw_data={'hartmann6': np.float64(-1.734596106152918)}
Completed trial 15 with raw_data={'hartmann6': np.float64(-2.203282961813707)}
Completed trial 16 with raw_data={'hartmann6': np.float64(-2.9661849718642617)}
Completed trial 17 with raw_data={'hartmann6': np.float64(-2.204289273292293)}
Completed trial 18 with raw_data={'hartmann6': np.float64(-2.453902532362328)}
Completed trial 19 with raw_data={'hartmann6': np.float64(-2.5879260020462516)}
Completed trial 20 with raw_data={'hartmann6': np.float64(-3.034029178842838)}
Completed trial 21 with raw_data={'hartmann6': np.float64(-0.5740518442706214)}
Completed trial 22 with raw_data={'hartmann6': np.float64(-3.253454881380015)}
Completed trial 23 with raw_data={'hartmann6': np.float64(-2.8468072715625494)}
Completed trial 24 with raw_data={'hartmann6': np.float64(-3.213619091963154)}
Completed trial 25 with raw_data={'hartmann6': np.float64(-3.2123294457451803)}
Completed trial 26 with raw_data={'hartmann6': np.float64(-3.2824154473972658)}
Completed trial 27 with raw_data={'hartmann6': np.float64(-3.1691049255409105)}
Completed trial 28 with raw_data={'hartmann6': np.float64(-3.299233158827855)}
Completed trial 29 with raw_data={'hartmann6': np.float64(-3.177834274334158)}

Step 6: Analyze Results

After running trials, you can analyze the results. Most commonly this means extracting the parameterization from the best performing trial you conducted.

Hartmann6 has a known global minimum of $f(x*) = -3.322$ at $x* = (0.201, 0.150, 0.477, 0.273, 0.312, 0.657)$ . Ax is able to identify a point very near to this true optimum using just 30 evaluations. This is possible due to the sample-efficiency of Bayesian optimization, the optimization method we use under the hood in Ax.

best_parameters, prediction, index, name = client.get_best_parameterization()
print("Best Parameters:", best_parameters)
print("Prediction (mean, variance):", prediction)

Output:
Best Parameters: {'x1': 0.21188346451138274, 'x2': 0.16428117335904663, 'x3': 0.4341952205574083, 'x4': 0.25907492071763133, 'x5': 0.29839289904146926, 'x6': 0.6470518604430703}
Prediction (mean, variance): {'hartmann6': (np.float64(-3.2758915184784447), np.float64(0.00119121385765093))}

Step 7: Compute Analyses

Ax can also produce a number of analyses to help interpret the results of the experiment via client.compute_analyses. Users can manually select which analyses to run, or can allow Ax to select which would be most relevant. In this case Ax selects the following:

Parrellel Coordinates Plot shows which parameterizations were evaluated and what metric values were observed -- this is useful for getting a high level overview of how thoroughly the search space was explored and which regions tend to produce which outcomes
Interaction Analysis Plot shows which parameters have the largest affect on the function and plots the most important parameters as 1 or 2 dimensional surfaces
Summary lists all trials generated along with their parameterizations, observations, and miscellaneous metadata

# display=True instructs Ax to sort then render the resulting analyses
cards = client.compute_analyses(display=True)

Parallel Coordinates for hartmann6

The parallel coordinates plot displays multi-dimensional data by representing each parameter as a parallel axis. This plot helps in assessing how thoroughly the search space has been explored and in identifying patterns or clusterings associated with high-performing (good) or low-performing (bad) arms. By tracing lines across the axes, one can observe correlations and interactions between parameters, gaining insights into the relationships that contribute to the success or failure of different configurations within the experiment.

Summary for hartmann6_experiment

High-level summary of the Trial-s in this Experiment

	trial_index	arm_name	trial_status	generation_node	hartmann6	x1	x2	x3	x4	x5	x6
0	0	0_0	COMPLETED	Sobol	-0.501926	0.221352	0.505657	0.791436	0.888981	0.406419	0.956151
1	1	1_0	COMPLETED	Sobol	-0.022449	0.704186	0.250006	0.379711	0.458323	0.961675	0.123115
2	2	2_0	COMPLETED	Sobol	-0.145191	0.771154	0.755537	0.534144	0.589093	0.701101	0.255063
3	3	3_0	COMPLETED	Sobol	-0.860469	0.287191	0.00037	0.137009	0.000216	0.176761	0.664214
4	4	4_0	COMPLETED	Sobol	-0.044638	0.407901	0.976937	0.3023	0.195248	0.006709	0.822607
5	5	5_0	COMPLETED	MBM	-1.53747	0.073155	0	0.25316	0.148463	0.20774	0.816667
6	6	6_0	COMPLETED	MBM	-0.774629	0	0	0	0	0.371688	0.665134
7	7	7_0	COMPLETED	MBM	-0.595814	0.196423	0	0.434822	0.166626	0.037395	0.885145
8	8	8_0	COMPLETED	MBM	-0.918021	0.03533	0	0.405542	0.163929	0.234641	0.990631
9	9	9_0	COMPLETED	MBM	-1.45712	0	0	0.218451	0.244235	0.313667	0.90227
10	10	10_0	COMPLETED	MBM	-0.565202	0	0	0.240638	0	0.107605	0.607965
11	11	11_0	COMPLETED	MBM	-1.90258	0	0	0.249127	0.236354	0.284169	0.819713
12	12	12_0	COMPLETED	MBM	-2.40239	0.025407	0	0.336605	0.305614	0.29196	0.756491
13	13	13_0	COMPLETED	MBM	-2.66784	0.119091	0	0.439416	0.379654	0.296443	0.695929
14	14	14_0	COMPLETED	MBM	-1.7346	0	0	0.25926	0.45492	0.254297	0.652627
15	15	15_0	COMPLETED	MBM	-2.20328	0.178457	0	0.505581	0.437471	0.369049	0.725332
16	16	16_0	COMPLETED	MBM	-2.96618	0.257104	0	0.529468	0.302724	0.290807	0.647679
17	17	17_0	COMPLETED	MBM	-2.20429	0	0	0.67244	0.317315	0.275811	0.636231
18	18	18_0	COMPLETED	MBM	-2.4539	0.45743	0	0.43786	0.285296	0.308876	0.619494
19	19	19_0	COMPLETED	MBM	-2.58793	0.240049	0	0.139718	0.307023	0.300896	0.655223
20	20	20_0	COMPLETED	MBM	-3.03403	0.235173	0.299635	0.477795	0.299152	0.296647	0.676215
21	21	21_0	COMPLETED	MBM	-0.574052	0.217013	0.86203	0.450361	0.284583	0.322264	0.614581
22	22	22_0	COMPLETED	MBM	-3.25346	0.238937	0.161156	0.475236	0.299304	0.288184	0.678222
23	23	23_0	COMPLETED	MBM	-2.84681	0.286002	0.17272	0.520515	0.332247	0.247339	0.725811
24	24	24_0	COMPLETED	MBM	-3.21362	0.203698	0.155476	0.41977	0.273532	0.325993	0.606466
25	25	25_0	COMPLETED	MBM	-3.21233	0.225211	0.15683	0.464645	0.253081	0.350249	0.679414
26	26	26_0	COMPLETED	MBM	-3.28241	0.211883	0.164281	0.434195	0.259075	0.298393	0.647052
27	27	27_0	COMPLETED	MBM	-3.16911	0.227873	0.18671	0.538931	0.231134	0.302479	0.621384
28	28	28_0	COMPLETED	MBM	-3.29923	0.169304	0.174556	0.466938	0.270202	0.316357	0.657593
29	29	29_0	COMPLETED	MBM	-3.17783	0.165182	0.209001	0.399035	0.246805	0.303844	0.662241

Sensitivity Analysis for hartmann6

Understand how each parameter affects hartmann6 according to a second-order sensitivity analysis.

x1 vs. hartmann6

The slice plot provides a one-dimensional view of predicted outcomes for hartmann6 as a function of a single parameter, while keeping all other parameters fixed at their status_quo value (or mean value if status_quo is unavailable). This visualization helps in understanding the sensitivity and impact of changes in the selected parameter on the predicted metric outcomes.

x4 vs. hartmann6

x2, x4 vs. hartmann6

The contour plot visualizes the predicted outcomes for hartmann6 across a two-dimensional parameter space, with other parameters held fixed at their status_quo value (or mean value if status_quo is unavailable). This plot helps in identifying regions of optimal performance and understanding how changes in the selected parameters influence the predicted outcomes. Contour lines represent levels of constant predicted values, providing insights into the gradient and potential optima within the parameter space.

x5, x6 vs. hartmann6

x2, x5 vs. hartmann6

Cross Validation for hartmann6

The cross-validation plot displays the model fit for each metric in the experiment. It employs a leave-one-out approach, where the model is trained on all data except one sample, which is used for validation. The plot shows the predicted outcome for the validation set on the y-axis against its actual value on the x-axis. Points that align closely with the dotted diagonal line indicate a strong model fit, signifying accurate predictions. Additionally, the plot includes 95% confidence intervals that provide insight into the noise in observations and the uncertainty in model predictions. A horizontal, flat line of predictions indicates that the model has not picked up on sufficient signal in the data, and instead is just predicting the mean.

Conclusion

This tutorial demonstrates how to use Ax's Client for ask-tell optimization of Python functions using the Hartmann6 function as an example. You can adjust the function and parameters to suit your specific optimization problem.

Learning Objectives​

Prerequisites​

Step 1: Import Necessary Modules​

Step 2: Initialize the Client​

Step 3: Configure the Experiment​

Step 4: Configure Optimization​

Step 5: Run Trials​

Optimization Loop​

Step 6: Analyze Results​

Step 7: Compute Analyses​

Conclusion​