Q:

How can I update the font size and background color of my Tooltip and make its icon smaller? 

I have tried for hours to get this to work trying multiple solved stack solutions and none have worked. 
I am assuming I am either missing something or I have my divs/spans incorrect.  Any help would be greatly appreciated.  Currently my text is very small and bg color is grey and slightly transparent. 
Also if it is possible to make the "info" icon smaller that would be great as well.  Thank you! 

  .mainContainer {
    width: 40%;
    float: left;
  }
  
    .info {
    display: flex;
  }

  .info .span {
    align-self: center;
  }

  .info-button{
    padding: 0;
    border: none;
    background: none;
    outline: none;
    background-color: transparent;
    top:-10px;
  }
      <div class="mainContainer">
        <h3>Main Title</h3>
        <div class="info">
          <h5>Title</h5><span class="info-span">
            <button mat-icon-button class="info-button" #tooltip="matTooltip" matTooltip="Tooltip Text">
            <mat-icon color="basic" >info</mat-icon>
            </button>
            </span>
        </div>
      </div>

A:

Try this
::ng-deep .mat-tooltip {
    background-color: transparent;
    font-size: 12px;
}

You can set ! important  if not working. 
Hope useful

