NQN Seminar Series – Can Quantum Computers Speed up Gradient Descent?

Name: NQN Seminar Series – Can Quantum Computers Speed up Gradient Descent?
Uploaded: 2022-01-31T21:58:03.000Z
Duration: 1 h 58 min 46 s

Introduction and Project Overview

In this section, the speaker introduces themselves and the topic of the presentation. They mention that they will be discussing a recent project on whether quantum computers can speed up gradient descent.

Recent Project on Quantum Computers and Gradient Descent

The speaker worked on a project to investigate if quantum computers can accelerate gradient descent.

The work is now available on the archive with collaborators from Microsoft Research India.

The speaker highlights that they will not only discuss the results but also share the story behind choosing this problem and how research projects evolve.

Collaborators and Background

This section provides information about the collaborators involved in the project and sets the context for further discussion.

Collaborators

The project was a joint effort with collaborators from Microsoft Research India.

The names of the collaborators are Ankit Garg, Pranit Nitrapali, and Suhail Sheriff (who was an intern at Microsoft Research India during this work).

Exploring Research Questions

Here, the speaker explains how research projects often start with vague ideas or problems of interest that are then explored through various paths.

Starting Point for Research Projects

Research projects usually begin with a vague idea or problem that researchers would like to address.

It is common for researchers to explore different paths before reaching their final destination.

This particular project followed a similar trajectory, highlighting how research projects evolve over time.

High-Level Explanation for Non-Specialists

In this section, the speaker acknowledges that not all audience members may be familiar with quantum computing. They aim to provide a high-level explanation accessible to those interested in quantum computing.

Audience Familiarity with Quantum Computing

The speaker assumes that the audience is interested in quantum computing but may not be experts in the field.

They aim to provide a high-level explanation that can be understood by non-specialists.

Motivation for Studying Quantum Computing

The speaker explains their motivation for studying quantum computing, which is to understand which problems can be solved faster on quantum computers compared to classical computers.

Motivation: Identifying Problems with Quantum Speedup

The speaker's main motivation is to identify problems that can be solved more efficiently on quantum computers than on classical computers.

They emphasize the importance of finding applications where quantum speedup exists and avoiding wasting time on problems without such advantages.

Gradient Descent and Optimization

This section introduces gradient descent as an algorithm used for function minimization and optimization, providing context for the subsequent discussion.

Introduction to Gradient Descent

Gradient descent is an algorithm used for minimizing functions.

It takes a function as input and aims to find the minimum value of that function.

While commonly used in machine learning, it can be applied to various scenarios beyond that domain.

Gradient descent is often used as a heuristic algorithm or with provable guarantees depending on the specific application.

Application of Gradient Descent

Here, the speaker discusses how gradient descent is widely used in practice, particularly in training neural networks. They also mention its use in convex optimization.

Applications of Gradient Descent

One popular application of gradient descent is training neural networks, where it serves as a heuristic algorithm.

In this context, there isn't always mathematical proof that gradient descent finds the global minimum, but it works well in practice.

Gradient descent also finds optimal solutions with provable guarantees in convex optimization, which will be discussed later in the presentation.

Function of Two Parameters

The section discusses the concept of a function with two parameters, such as x and y coordinates. It uses the example of hiking in the mountains and trying to minimize a function that represents the height of the mountain at a given location.

A function with two parameters, like x and y coordinates, can represent various scenarios, such as determining the height of a mountain at a specific location.

Minimizing this function can be compared to hiking in the mountains and trying to reach the bottom or valley.

Local information is used in this scenario, where only nearby surroundings are visible, making it challenging to see the entire function at once.

The simplest approach is to walk downhill from your current position based on local information.

Intuition behind Gradient Descent

This section explains gradient descent intuitively using a one-dimensional function. It describes how iterations of gradient descent involve evaluating gradients and walking in the direction of steepest descent.

Gradient descent involves starting at an initial position (x0) and evaluating the gradient at that point.

The gradient represents the direction of steepest ascent or increase in value.

To move towards minimizing the function, you walk in the opposite direction of the gradient (minus sign).

Ada represents step length or how far you move in each iteration.

After taking steps, you re-evaluate your choice based on new information.

Understanding Gradient

This section clarifies what gradients are and their role in gradient descent. It emphasizes that gradients indicate slopes and point towards directions of steepest descent.

Gradients represent vectors of partial derivatives but can be understood intuitively as slopes at specific points.

The gradient points towards the direction of steepest ascent, while minus the gradient points towards the downward direction or steepest descent.

Project Focus: Can Quantum Computers Speed Up Gradient Descent?

The section introduces the project's focus on whether quantum computers can accelerate gradient descent. It highlights the significance of this question due to the widespread use of gradient descent and its potential impact on various applications.

The project aims to investigate if quantum computers can provide a speedup for gradient descent.

Gradient descent is widely used in many applications, such as training neural networks.

If quantum computers can enhance gradient descent, it would have implications for all these applications.

Initially, it may seem unclear if this question is meaningful, but considering examples like factoring numbers using Shor's algorithm shows that different algorithms can have distinct approaches.

Comparing Shor's Algorithm with Classical Factorization Algorithms

This section compares Shor's algorithm (quantum) with classical factorization algorithms to highlight how different algorithms can have unique approaches despite achieving similar goals.

Shor's algorithm provides an exponential speedup over classical factorization algorithms for factoring numbers on quantum computers.

However, comparing Shor's algorithm with classical factorization algorithms reveals that they are fundamentally different and employ distinct techniques.

The transcript does not provide further sections or timestamps beyond this point.

Formalizing the Question

The speaker discusses the importance of formalizing the question in order to determine if a quantum algorithm can speed up gradient descent.

Choosing a Problem for Comparison

A quantum algorithm is considered to speed up gradient descent if it outputs the same sequence of points as gradient descent.

However, simply mimicking the classical algorithm would not result in a speedup.

Shor's algorithm provides a hint on how to formalize the question.

The goal is to compare a quantum computer's ability to solve a problem with gradient descent's ability.

Formalizing the Problem

Choose a problem that can be solved by gradient descent and where gradient descent is the optimal classical algorithm.

This problem should capture the power of gradient descent.

The question becomes whether a quantum computer can solve this problem faster than any classical algorithm.

Canonical Problem: Convex Optimization

Convex optimization involves minimizing a convex function over a convex set.

A convex function has certain properties, such as any segment joining two points lying above the function.

Working with convex sets and functions allows for approximate minimization within an epsilon range.

Crash Course on Convex Optimization

The speaker provides an overview of convex optimization and introduces the concept of minimizing a function over a convex set.

Definition of Convex Functions and Sets

A convex function has segments joining any two points lying above the function.

Examples are shown, including functions that are discontinuous or not smooth but still considered convex.

A convex set is one where segments joining any two points lie inside the set.

Minimizing Functions over Convex Sets

In convex optimization, one aims to minimize a function over a convex set.

Approximate minimization is allowed if the exact minimum cannot be determined.

Lipschitz constant is used to measure how quickly a function can change, affecting its minimization difficulty.

Summary

The transcript discusses the formalization of the question regarding whether a quantum algorithm can speed up gradient descent. It emphasizes the need to choose a problem that captures the power of gradient descent and where it is the optimal classical algorithm. The speaker introduces convex optimization as a canonical problem for comparison. A crash course on convex optimization is provided, explaining convex functions and sets, as well as the concept of minimizing functions over convex sets.

[t=16m38s] Lipschitz Constant and Minimization Problem

In this section, the concept of Lipschitz constant is introduced as a way to quantify how quickly a function can change. The minimization problem on a ball of radius r around the origin is defined, where the goal is to find an approximate minimizer x that is epsilon close to the minimum point x star.

Lipschitz Constant and Function Space Movement

The Lipschitz constant quantifies how quickly a function can change.

It ensures control over the rate at which the function changes for effective minimization.

If you move a fixed distance in the domain, the maximum distance you can move in the function space is at most g times that distance.

The Lipschitz constant captures how much the function value can change when moving in the domain.

Minimization Problem on Ball of Radius r

Given a function with Lipschitz constant g, we aim to minimize it on a ball of radius r around the origin.

The minimum point within this ball is denoted as x star.

Our goal is to find an approximate minimizer x that is epsilon close to x star.

Epsilon closeness means that the difference between the function values at x and x star should be at most epsilon.

Basic Convex Optimization Problem

The problem discussed here is a basic convex optimization problem.

The objective is to minimize a given function within a specified region or ball.

[t=17m30s] Input Specification and Black Box Assumption

This section focuses on input specification and assumptions made regarding access to information about the function. Different ways of specifying input are discussed, including zeroth order convex optimization and first order convex optimization. The concept of black box assumption is introduced.

Input Specification for Convex Optimization Problems

Input specification plays a crucial role in determining the complexity of a problem.

Zeroth order convex optimization involves only having access to the function f, allowing evaluation at any point.

First order convex optimization allows evaluation of both the function f and its gradient at any chosen point.

Black Box Assumption

The black box assumption refers to having access to a subroutine that computes the function f and its gradient.

It is similar to being able to observe one's location and the slope around it while hiking in mountains.

The function is not understood globally, but can be evaluated at specific points using the provided subroutine.

[t=21m03s] Problem Statement and Evaluation Metric

This section restates the minimization problem defined earlier and introduces the evaluation metric for assessing algorithm performance. The goal is to output an approximate minimizer x while minimizing the number of evaluations of f and grad f.

Problem Statement Recap

The problem involves minimizing a given function within a ball of radius r around the origin.

An approximate minimizer x that is epsilon close to x star needs to be found.

Evaluation Metric: Number of Function Evaluations

The performance of an algorithm is measured by how many times it evaluates f and grad f.

Minimizing this count is crucial for efficient algorithm design.

Timestamps are provided in [HH:MM:SS] format.

New Section

In this section, the speaker discusses the concept of scaling the input and output spaces in a function and introduces the parameters involved in solving the problem.

Scaling Input and Output Spaces

The function f can be multiplied by a constant alpha to scale the output space.

Evaluating f on an input x can be replaced with evaluating f on beta times x for some constant beta, allowing freedom in scaling the input space.

This approach provides flexibility in scaling both input and output spaces.

New Section

The speaker explains that there are two types of algorithms for solving the problem discussed, depending on the regime being considered.

Classical Algorithms for Solving the Problem

There are two optimal algorithms: dimension-independent algorithms and dimension-dependent algorithms.

Gradient descent is an example of a dimension-independent algorithm with complexity g times r over epsilon squared.

Dimension-independent algorithms have no dependence on n (the dimension) and are suitable for applications with large dimensional spaces.

New Section

The speaker focuses on gradient descent as a dimension-independent algorithm and its advantages in terms of complexity.

Gradient Descent as a Dimension-Independent Algorithm

Gradient descent is an example of a dimension-independent algorithm used to solve the problem efficiently.

It has a complexity of g times r over epsilon squared, without any dependence on n (the dimension).

This makes it particularly useful in machine learning and optimization applications where high-dimensional spaces are involved.

New Section

The speaker emphasizes their focus on gradient descent as a dimension-independent algorithm and its relevance to solving the problem effectively.

Focus on Dimension Independent Algorithms

Today's talk will primarily focus on gradient descent as an example of a dimension-independent algorithm.

The goal is to find a solution that optimally solves the problem without any classical algorithm performing better.

The desired outcome is to have a complexity dependent on g, r, and epsilon, without any additional dependence on the dimension n.

New Section

The speaker explains the steps involved in gradient descent and how it achieves its complexity.

Steps of Gradient Descent

Start at the origin (center of the search space) since the problem involves minimizing a ball of unit radius around zero.

Repeat a loop t times, where t represents the complexity gr over epsilon squared.

In each iteration:

Take one step in the direction of the gradient from your current point xi.

Set the step size (magnitude) as epsilon, which is the parameter being minimized.

Use a projection function p to ensure you stay within the domain (ball of radius r) being minimized.

New Section

The speaker explains how gradient descent ensures staying within the domain being minimized and outputs an average of all iterates.

Projection onto Domain

To stay within the domain being minimized (a ball of radius r), use a projection function p.

If a step takes you outside this ball, bring yourself back to distance r from the origin using p.

This ensures that you remain within the domain and do not end up with points outside it.

New Section

The speaker highlights that gradient descent achieves its complexity by making limited queries to subroutines or black boxes.

Complexity Analysis

Gradient descent has a complexity of gr over epsilon squared due to its loop running for t times (complexity factor).

Each iteration makes one access to subroutines or black boxes containing gradients, resulting in the number of queries being made.

New Section

The speaker claims that gradient descent is an optimal algorithm for the problem and cannot be improved upon by any classical algorithm.

Optimality of Gradient Descent

Gradient descent outputs a correct answer for the problem.

Any classical algorithm, even if randomized, cannot perform better than gradient descent in terms of solving the problem optimally.

Optimality of Gradient Descent

The speaker discusses the optimality of gradient descent among classical algorithms and introduces the concept of subgradients for non-differentiable points.

Optimality of Gradient Descent

Gradient descent is optimal among classical algorithms.

It must be at least O(1/ε^2) or asymptotically at least O(1/ε^2).

Subgradients can be used instead of gradients for non-differentiable points.

Subgradients satisfy similar properties as gradients for convex functions.

Subgradients are always defined for convex functions, unlike gradients.

The only issue with subgradients is that they may not be unique.

Can Quantum Computers Speed Up Gradient Descent?

The speaker formalizes the question of whether quantum computers can speed up gradient descent and explains the process of formalizing a problem in theoretical computer science.

Formalizing the Question

The question is whether quantum computers can solve a problem that gradient descent optimally solves.

Gradient descent makes O(1/ε^2) queries, which no classical algorithm can improve upon.

Even if classical algorithms are allowed to make errors and random choices, they still cannot outperform gradient descent.

This problem is a basic one in convex optimization and commonly found in textbooks.

The natural question to consider when exploring quantum speedup for gradient descent.

Art of Formalization

Starting with an informal question and then formalizing it into a mathematical question is common in theoretical computer science.

There may be multiple ways to formalize a problem, but this particular formulation was chosen.

Formalization requires rigorous mathematical definitions to prove results.

Quantum Access to Functions

The speaker explains what it means for a quantum algorithm to have access to a function and provides an example using boolean functions.

Quantum Access to Functions

Classical access to a function means having a black box that can evaluate the function on any input.

Quantum access allows for analog evaluation of the function using quantum operations.

Quantum access is a standard notion in quantum algorithms.

Example: Boolean Functions

Boolean functions take n bits as input and output one bit.

Classical access to a boolean function involves submitting inputs to a black box that evaluates the function and returns the output.

Quantum access allows for analog evaluation of boolean functions using quantum operations.

Conclusion

The transcript discusses the optimality of gradient descent among classical algorithms and introduces the concept of subgradients for non-differentiable points. It then explores the question of whether quantum computers can speed up gradient descent, formalizes the problem, and explains what it means for a quantum algorithm to have access to a function.

Introduction to Quantum Notation

In this section, the speaker introduces quantum notation and explains its significance in quantum algorithms. The notation involves submitting a classical basis state corresponding to x and a dummy bit b to a quantum black box.

Quantum Notation Explanation

The quantum notation involves submitting a classical basis state corresponding to x and a dummy bit b to the quantum black box.

The quantum black box leaves the input untouched and copies the answer f(x) into the second register, XORing it with b.

This XOR operation ensures that all subroutines in quantum algorithms are reversible, as required by unitary processes or matrices.

Reversibility means that given an input x and output f(x), one should be able to recover the input from the output. However, this is not possible with just f(x), so XORing it with b allows for recovery of both x and b using the operation uf again.

This standard definition of the quantum version ensures a level playing field between classical and quantum access assumptions.

Reversible Subroutines in Quantum Algorithms

This section discusses why subroutines in quantum algorithms need to be reversible and how they can be implemented using unitary processes or matrices.

Reversibility Requirement

All subroutines in quantum algorithms must be reversible.

Reversibility means that once an input is put into a subroutine, such as obtaining f(x), it should be possible to recover the input from the output.

In contrast, classical operations like obtaining f(x) directly do not allow for easy recovery of the input.

Implementing Reversible Subroutines

The operation uf used in the previous section is its own reverse or inverse.

This operation uf can be used to recover the input x and b by applying it again to the answer obtained.

The standard definition of the quantum version ensures that classical source code for f can be mechanically transformed into quantum source code for uf.

This allows for a fair comparison between classical and quantum access assumptions, ensuring that the power of quantum access is not greater than classical access.

Equivalence of Classical and Quantum Circuits/Algorithms

This section explains how classical circuits/algorithms computing a function f can be converted into equivalent quantum circuits/algorithms computing uf.

Conversion from Classical to Quantum

A classical circuit with g gates or a classical algorithm running in time t to compute f can be converted into a quantum circuit uf using roughly the same number of gates or time.

The conversion process is straightforward and results in a quantum circuit/algorithm that computes uf.

There may be some constant factor overhead, but it does not significantly impact the overall computation time.

Extension to Real Numbers

This section briefly mentions how the model extends to real numbers, where precision becomes important.

Handling Real Numbers

For real numbers, the same model applies, but they are specified with arbitrarily large precision rather than infinite precision.

Instead of actual real numbers, they are represented by providing polynomially many bits for their specification.

Non-covex optimization and other topics mentioned in questions are not covered in this transcript.

New Section

In this section, the speaker discusses the properties of convex and non-convex functions in optimization problems. They explain how convex functions have no local minima and that if a local minimum exists, it is also a global minimum. The difficulty in formalizing the question for non-convex functions is highlighted, as well as the possibility that gradient descent may not be the best classical algorithm for non-convex optimization.

Properties of Convex and Non-Convex Functions

Convex functions do not have local minima to get trapped in.

If there is a local minimum, it is also a global minimum.

Finding a point where the derivative is zero and everything around it is larger than the current value indicates finding the minimum.

Non-convex functions lack these properties, making it challenging to define what one wants to happen in optimization problems.

Formalizing Questions for Non-Convex Problems

It can be difficult to formalize what one wants gradient descent to achieve in non-convex settings.

Gradient descent may not be the correct classical algorithm for non-convex optimization.

Challenges with Non-Convex Optimization

Choosing a problem where gradient descent is the best classical algorithm becomes crucial.

It might turn out that other algorithms are more suitable for non-convex optimization problems.

Formalizing questions for non-convex problems remains an ongoing challenge.

Conclusion

The speaker acknowledges that they currently have no further updates or reports on addressing these challenges but encourages further questions from the audience.

Understanding Subset Queries and Quantum Speedup

In this section, the speaker discusses the concept of subset queries and their relevance in evaluating functions. They explain how sending an input vector of all ones can determine if there exists a specific value within a set. The speaker also introduces the idea of indicator vectors and how they can be used to evaluate subsets. Additionally, they mention that classical algorithms require one over epsilon squared queries, while quantum algorithms can solve these problems quadratically faster.

Evaluating Functions with Subset Queries

Sending an input vector of all ones helps evaluate the function f at any chosen input.

If all xi values are one, then the function returns the maximum value of zi (either plus one or minus one).

The output is plus one only if there exists a zi equal to plus one; otherwise, it is minus one.

Indicator Vectors for Subset Queries

Instead of using an all ones vector as input, an indicator vector can be used for a subset S.

An indicator vector tells us if the answer evaluates to one when there is a zi within indices i and s where zi equals one.

Subset queries allow selecting a subset of n bits in the input string to check if any contain a one.

Quantum Speedup with Subset Queries

Alexander's Belovs showed that subset queries on an n-bit string can be learned with only square root of n quantum queries.

Learning all z values allows determining the optimal point x* since x* is expressed in terms of z.

Quantum algorithms provide quadratic speedup over classical algorithms for solving this family of functions.

Investigating Quantum Speedup

In this section, the speaker reflects on their research project and explores whether quantum algorithms consistently offer quadratic speedup compared to classical algorithms. They discuss their findings regarding a different family of hard functions for which quantum algorithms still require one over epsilon squared queries, indicating no quantum speedup in this setting.

Reflection on Quantum Speedup

The speaker questions whether the quadratic speedup observed in their research is a general phenomenon or just a coincidence.

They worked on the problem extensively, examining various functions and potential limitations of quantum algorithms.

No Quantum Speedup in Certain Cases

After months of investigation, it was concluded that there exists a different family of hard functions where even quantum algorithms require one over epsilon squared queries.

In the specific setting of first-order con black box convex optimization, there is no quantum speedup compared to gradient descent.

Conclusion

The transcript discusses subset queries and their significance in evaluating functions. It explains how indicator vectors can be used to determine if certain values exist within subsets. The transcript also explores the concept of quantum speedup and reflects on whether it consistently provides quadratic speedup compared to classical algorithms. Ultimately, it concludes that while there are cases where quantum algorithms offer significant speedup, there are also scenarios where they do not outperform classical methods.

New Section

In this section, the speaker discusses the setting of first-order black box convex optimization and its implications for speed-ups in other settings. They also mention the use of orthonormal random vectors and additive terms in the quantum algorithm.

Focusing on First-Order Black Box Convex Optimization

The speaker explains that while there may be speed-ups in other settings, in the exact setting of first-order black box convex optimization, it has been determined that there are no speed-ups.

They mention that orthonormal random vectors and an additive term dependent on the norm of the function x and index are used in the quantum algorithm.

Standard techniques for proving quantum lower bounds do not seem to work in this non-standard setting, but a technique called the hybrid argument is still applicable.

The speaker refers to a recent classical paper on parallel lore bars for gradient descent as a source of ideas for establishing lower bounds.

New Section

In this section, the speaker speculates about variations of the problem that might give quantum speed-up. They discuss different settings such as non-convex optimization and higher order derivatives.

Speculating About Variations of the Problem

The speaker mentions that variations of the problem could include non-convex optimization or different levels of access to information (e.g., second or third order derivatives).

They explain that even in smooth convex functions, gradient descent is not always the best classical algorithm; accelerated gradient descent is optimal.

There are many different settings to explore where one could potentially find quantum speed-up, but currently, there is limited knowledge about which specific sub-families of functions would exhibit such speed-up.

New Section

In this section, the speaker addresses a question about sub-families of functions that might have quantum speed-up. They discuss the specific property used in their research and the difficulty in formulating it into a natural class of functions.

Sub-Families of Functions with Quantum Speed-Up

The speaker acknowledges that they do not know the answer to which sub-families of functions might have quantum speed-up.

They mention that the property used in their research was somewhat specific, involving subset or queries, but it has not been formulated into an interesting class of functions.

One possibility they considered was if the function could always be described as the maximum of a set of linear functions, but further exploration is needed to determine if this holds true.

The transcript provided does not contain enough information for additional sections.

The Search for Classes with Quantum Speedups

In this section, the speaker discusses the search for classes of functions that can achieve quantum speedups and highlights the challenges in finding such classes.

The Quest for Natural Classes with Quantum Speedups

The speaker mentions that they have not been able to find a class of functions that includes the class where a quadratic speedup is possible.

It is mentioned that there could be other classes where quantum speedups are possible, but further research is needed.

Open Problems and Questions

This section covers some open problems and questions related to optimization algorithms and quantum speedups.

Open Problem 1: Dimension Independent Setting

Gradient descent is optimal in a dimension-independent setting, but there are classical cutting plane algorithms like the ellipsoid method that perform better.

The best known quantum lower bounds for this scenario are only square root of n, while classical algorithms can achieve order n queries.

Closing this gap between classical and quantum algorithms in optimization remains an open problem.

Open Problem 2: Non-gradient Descent Optimization

There are scenarios in non-convex optimization where gradient descent may not be the right algorithm to use.

Classical cutting plane algorithms like the ellipsoid method can outperform gradient descent in certain cases.

Quantum algorithms currently have a quadratic speedup limit, but it is possible to achieve more than that in non-gradient descent optimization problems.

Other Situations Where Quantum Speedup May Be Possible

Gradient descent is commonly used in various settings where its effectiveness cannot be proven or guaranteed, such as non-convex optimization.

While it has been shown that gradient descent cannot be sped up using quantum computers in certain basic settings, there may still be other applications or variations of gradient descent worth exploring.

Accelerated gradient descent and stochastic gradient descent are examples of variations that could potentially benefit from quantum speedups.

Many optimization problems, including these variations, remain unsolved and continue to be interesting areas of research.

Conclusion and Next Steps

The speaker concludes the talk and provides information about upcoming events in the Northwest Quantum Nexus seminar series.

The speaker thanks the audience for attending and concludes the third installment of the Northwest Quantum Nexus seminar series.

Attendees are directed to visit the nwquantum.com website for more information on future events, including the upcoming December seminar.

The session ends with a wish for everyone to have a great Thanksgiving.