**Victor Sanches Portella**

September 2023

cs.ubc.ca/~victorsp

joint with **Frederik Kunstner**, **Nick Harvey**, and **Mark Schmidt**

**Theory Student Seminar @ University of Toronto**

Training/Fitting a ML model is often cast a **(uncontrained) optimization problem**

Usually in ML, models tend to be BIG

\displaystyle \min_{x \in \mathbb{R}^{d}}~f(x)

**\(d\) is BIG**

Running time and space **\(O(d)\) **is usually **the most we can afford**

First-order (i.e., gradient based) methods fit the bill

(stochastic even more so)

Usually \(O(d)\) time and space per iteration

\displaystyle f(y) \geq f(x) + \langle \nabla f(y), x - y\rangle + \frac{L}{2}\lVert x - y \rVert_2^2

**\(f\) is convex**

Not the case with Neural Networks

Still quite useful in theory and practice

\displaystyle \Bigg\{

More conditions on \(f\) for rates of convergence

**\(L\)-smooth**

\displaystyle \min_{x \in \mathbb{R}^{d}}~f(x)

**\(\mu\)-strongly convex**

\displaystyle f(y) \leq f(x) + \langle \nabla f(y), x - y\rangle + \frac{\mu}{2}\lVert x - y \rVert_2^2

\displaystyle x_{t+1} = x_t - \alpha \nabla f(x_t)

Which step-size \(\alpha\) should we pick?

\displaystyle \implies

\displaystyle f(x_t) - f(x_*) \leq \left( 1 - \frac{\mu}{L} \right)^t (f(x_0) - f(x_*))

Condition number

\displaystyle \alpha = \frac{1}{L}

\displaystyle \kappa = \frac{L}{\mu}

\(\kappa\) Big \(\implies\) hard function

If we know \(L\), picking \(1/L\) always works

**and is worst-case optimal**

What if we do not know \(L\)?

Locally flat \(\implies\) we can pick bigger step-sizes

\displaystyle x_{t+1} = x_t - \tfrac{1}{L} \nabla f(x_t)

If \(f\) is \(L\) smooth, we have

\displaystyle f(x_{t+1}) \leq f(x_t) - \tfrac{1}{L} \tfrac{1}{2} \lVert \nabla f(x_t) \rVert_2^2

**"Descent Lemma"**

**Idea:** Pick \(\eta\) big and see if the "descent condition" holds

(Locally \(1/\eta\)-smooth)

**Backtracking Line-Search**

Start with \(\alpha_{\max} > 2 L\)

\(\alpha \gets \alpha_{\max}/2\)

**If**

\displaystyle f(x_{t+1}) \leq f(x_t) - \alpha \tfrac{1}{2} \lVert \nabla f(x_t) \rVert_2^2

\(t \gets t+1\)

**Else**

**While** \(t \leq T\)

\displaystyle \alpha_{\max} \gets \alpha_{\max}/2

Halve candidate space

**Guarantee**: step-size will be at least \(\tfrac{1}{2} \cdot \tfrac{1}{L}\)

**Armijo Condition**

\displaystyle f(x) = x^T A x

\displaystyle A =
\begin{pmatrix}
1000 & 0 \\
0 & 0.001
\end{pmatrix}

\displaystyle \kappa = 10^{-6}

\displaystyle x_{t+1} = x_t -
\begin{pmatrix}
0.001 & 0 \\
0 & 1000
\end{pmatrix}
\nabla f(x_t)

**Converges in 1 step**

\(P\)

\(O(d)\) space and time \(\implies\) \(P\) diagonal (or sparse)

Can we find a good \(P\) automatically?

**"Adapt to \(f\)"**

**Preconditioer \(P\)**

\displaystyle x_{t+1} = x_t - P_t \cdot \nabla f(x_t)

Preconditioner at round \(t\)

**AdaGrad from Online Learning**

\displaystyle P_t = \Big( \sum_{i \leq t} \nabla f(x_i) \nabla f(x_i)^T \Big)^{1/2}

\displaystyle \mathrm{Diag}\Big( \sum_{i \leq t} \nabla f(x_i) \nabla f(x_i)^T \Big)^{1/2}

or

Better guarantees if **functions are easy**

while preserving optimal worst-case guarantees in Online Learning

Attains** linear rate in classical convex opt** (proved later)

But... Online Learning is** too adversarial**, AdaGrad is **"conservative"**

In OL, functions change every iteration **adversarially**

But... Online Learning is** too adversarial**, AdaGrad is **"conservative"**

"**Fixes**": Adam, RMSProp, and other workarounds

"AdaGrad inspired anincredible number of clones, most of them withsimilar, worse, or no regret guarantees.(...) Nowadays, [adaptive] seems to denoteany kind of coordinate-wise learning rates that does not guarantee anything in particular."

**Francesco Orabona** in "A Modern Introduction to Online Learning", Sec. 4.3

**Idea: **look at step-size/preconditioner choice as an optimization problem

Gradient descent on the hyperparameters

How to pick the step-size of this? Well...

Little/ No theory

Unpredictable

... and popular?!

P_t = \nabla^2 f(x_t)

Newton's method

is usually a great preconditioner

**Superlinear** convergence

...when \(\lVert x_t - x_*\rVert\) small

**Newton **may diverge otherwise

Using step-size with Newton and QN method ensures convergence away from \(x_*\)

**Worse than GD**

\displaystyle f(x_t) - f(x_*) \leq \left( 1 - \frac{1}{\kappa^2} \right)^t (f(x_0) - f(x_*))

\displaystyle \phantom{\kappa}^2

\(\nabla^2 f(x)\) is usually expensive to compute

P_t \approx \nabla^2 f(x_t)

should also help

Quasi-Newton Methods, e.g. BFGS

**(Quasi-)Newton**: needs Hessian, can be slower than GD

**Hypergradient methods**: purely heuristic, unstable

**Online Learning Algorithms**: Good but pessimistic theory

at least for smooth optimization it seems pessimistic...

**Online Learning**

**Smooth Optimization**

**1 step-size**

**\(d\) step-sizes**

(diagonal preconditioner )

Backtracking Line-search

Diagonal AdaGrad

Coordinate-wise

Coin Betting

(non-smooth opt?)

**Multidimensional Backtracking**

Scalar AdaGrad

Coin-Betting

**What does it mean for a method to be adaptive?**

\displaystyle \mu I \preceq \nabla^2 f(x) \preceq L I

\displaystyle \frac{1}{\kappa} P^{-1} \preceq \nabla^2 f(x) \preceq P^{-1}

**Optimal step-size**: biggest that guarantees progress

**Optimal preconditioner**: **biggest (??)** that guarantees progress

\displaystyle P = \tfrac{1}{L} I

\displaystyle \kappa = \tfrac{L}{\mu}

\displaystyle f(y) \geq f(x) + \langle \nabla f(y), x - y\rangle + \frac{L}{2}\lVert x - y \rVert_2^2

**\(L\)-smooth**

**\(\mu\)-strongly convex**

\displaystyle f(y) \leq f(x) + \langle \nabla f(y), x - y\rangle + \frac{\mu}{2}\lVert x - y \rVert_2^2

\displaystyle P_*

minimizes \(\kappa_*\) such that

\displaystyle \frac{1}{\kappa_*} P^{-1} \preceq \nabla^2 f(x) \preceq P^{-1}

**Over diagonal matrices**

**Line-search**

step-size is at least \(1/2\) the optimum \(1/L\)

# backtracks \(\leq\)

\displaystyle \log\Big(\alpha_{0} L \Big)

\displaystyle f(x_t) - f(x_*) \leq\Big(1 - \frac{1}{2 \cdot \kappa}\Big)^t (f(x_0) - f(x_*))

**Multidimensional Backtracking**

Condition number is at least \(1/\sqrt{2d}\) the optimum

# backtracks \(\lesssim\)

\displaystyle d \cdot \log\Big( \alpha_0 \cdot L\Big)

\displaystyle f(x_t) - f(x_*) \leq\Big(1 - \frac{1}{\sqrt{2d} \cdot \kappa_*}\Big)^t (f(x_0) - f(x_*))

\displaystyle \Bigg\{

\displaystyle \Bigg\{

**Worth it if \(\sqrt{2d} \kappa_* \ll 2 \kappa\)**

**Line-search**: test if step-size \(\alpha_{\max}/2\) makes enough progress:

\displaystyle f(x_{t+1}) \leq f(x_t) - \alpha_{\max} \tfrac{1}{2} \lVert \nabla f(x_t) \rVert_2^2

**Armijo condition**

If this fails, **cut out** everything bigger than \(\alpha_{\max}/2\)

**Preconditioner search:**

0

\alpha_{0}

\tfrac{\alpha_{0}}{2}

\tfrac{1}{L}

\tfrac{\alpha_{0}}{4}

Test if preconditioner \(P\) makes enough progress:

Candidate preconditioners \(\mathcal{S}\): diagonals in a box

\displaystyle f(x_{t+1}) \leq f(x_t)

- \tfrac{1}{2} \lVert \nabla f(x_t) \rVert_P^2

If this fails, **cut out** everything bigger than \(P\)

\langle \nabla f(x_t), P \nabla f(x_t) \rangle

**Preconditioner search:**

Test if preconditioner \(P\) makes enough progress:

Candidate preconditioners \(\mathcal{S}\): diagonals in a box

\displaystyle f(x_{t+1}) \leq f(x_t)

\langle \nabla f(x_t), P \nabla f(x_t) \rangle

- \tfrac{1}{2} \lVert \nabla f(x_t) \rVert_P^2

If this fails, **cut out** everything bigger than \(P\)

\(P\) does not yield **sufficient progress**

Which preconditioners can be thrown out?

All \(Q\) such that \(P \preceq Q\) works, but it is **too weak**

\displaystyle
h(P) \coloneqq f(x - P^{-1} \nabla f(x)) - f(x) + \tfrac{1}{2} \lVert \nabla f(x) \rVert_P^2

\(P \) does not yield sufficient progress \(\iff\) \(h(P) > 0\)

\displaystyle
h(Q) \geq h(P) + \langle \nabla h(P), Q - P \rangle

Convexity \(\implies\)

\displaystyle
h(P) + \langle \nabla h(P), Q - P \rangle > 0

\(\implies\) \(Q\) **is invalid**

A separating hyperplane!

\(P\) in this half-space

\displaystyle \Bigg\{

**Hypergradient**

Contraction of \(1/\sqrt{2d}\) from boundary

\displaystyle
\implies

Constant volume contraction

Theoretically principled adaptive optimization method for strongly convex smooth optimization

A theoretically-informed use of "hypergradients"

ML Optimization meets Cutting Plane methods

Stochastic case?

Heuristics for non-convex case?

Other cutting-plane methods?